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Las diagrammas suivants illustrant la mAthoda. 1 2 3 1 2 3 4 5 6 ii^^-\ .U.V MICROCOPY RISOIUTION TfST CHART (ANSI and ISO TEST CHART No 2) |£ 12.8 ISO ^•B ■ 2^ 1^ |12 l£ ^lii. g US lyg 1.8 A /APPLIED IIVMGE I SFm 'S^-5 East Main Street "-S Rochestpr, New York i*609 USA ^S (^16) *fi2 - 0300 - Phone ^S ('16) 28« - 5989 - Fq» THE THEORY OF MACHINES This book is produced in full compliance with the government's regulations for con- serving paper and other essential materials. THE THEORY OF MvVCHlNES PART I THE PRINCIPLES OF MECHANISM PART 11 ELEMENTARY MECHANICS OF MACHINES BY ROBERT W. ANGUS, B.A.Sc, MEMBER or THE AMERICAN BOCIETT Or MECHANICAL BNaiNEEBa, rnorBsauR or mechanical enoineerino, cnivemitt or TORONTO, TORONTO, CANADA Carleton College Library - Ottawa Heconu Edition Ninth Imprkssion McGRAW-HILL BOOK COMPANY, Inc. NEW YORK AND LONDON 1917 ('OI'YHKIHT, lltl7, BT THK Mc(1kaw-Hill Hook Company, Inc phinteu in tbb rNiTro htatkh i>r amkhica PREFACE The present treatise deulinK with the I'rinciples of Mechaninm iiiul MethanirH (,f Maihinery is the result of a iiuiuIht of years' <"X|M«rienee in leaohinK the suhjeets and in prattisinK engineerinK and endeavors to deal witii problems of fairly oonunon occur- rence. It 18 intended to cover the needs of the }>eKinner in the Htudy of the science of machinery, and also to take up a numljer of the advanced problems in mechanics. As the engineer uses the drafting Uard very freely in the solution of his problems, the author has devised graphical solu- tu.ns throughout, and only in a very few instances has he used formula involving anything more than elementary trigonometry and algebra. The two or three cases involving the calculus may be omitted without detracting much from the usefulness of the book. The reader must remember that the book does not deal with machine design, and as the drawings have bet-n made for the special purpose of illustrating the principles under discussion the mechanical details have frequently l)een omitted, and in cer- tain cases the proportions somewhat modified so as to make the constructions employed clearer. The phorograph or motion diagram has U'en introduced in Chapter IV, and appeared in the first edition f(.r the first time in prmt. It has been very freely used throughout, so that most ot the solutions are new, and experience has shown that results are more easily obtained in this way than by the usual methods. As the second part of the book is much more difficult than the first, ,t w recommended that in teaching the subject most of the first part be given to students in the sophomore year, all of the second part and possibly some of the first part being assignetl in the junior year. The thanks of the author are due to Mr. J. H. Parkin for his careful work on governor problems, some of which are incorpor- ated, and for assistance in proofreading; also to the various firms and others who furnished cuts and information, most of which la acknowledged in the body of the book. yfi PRKFA OK The preaent edition ha« been rntiroly rewritten and enlarged and all of tin* proviniit exniiipleM ciiri'fully checked and corrected when- neccHNiiry. Tlie cu»h have hn-n re-ok. with tlie meanings usually attached to them. w = weight in pounds. g = acceleration <.f gravity = .Tilti f,. ,vr .s^.eon.l per second. _ "' m = mass = g V = velocity in feet jmt second. n = revolutions per minute. 2irn 60 X = 3.141(). a = angular acceleration in radians per .secon.l per secon.l. - crank angle from inner deatl center. / = moment of inertia about the center of gravity. k = radius of gyration in feet = y/l/m J = reduced inertia referred to primarv link. T = torque in foot-pounds. P, P', P" represent the point P and its imag.«>aring for the crank- .reutv eailcd the frame, and which is always fixed in tH,sition () The piston, piston rod and crosshead, whi.-h are Lis !s geiher as in large steam and g,.s engines, or of a single casting 3 4 THE THEORY OF MACHINES as in automobile engines, where the piston rod is entirely omitted and the crosshead is combined with the piston. It will bo "on- veniont to refer to this part as the piston, and it is to be notice wristpin, moves in a straight line, which latter motion is governed by the piston. All points on the rod mQve in parallel planes, however, and it is said to have plane motion, as has al.«<. the piston. The purpose of the rod is to transmit the motion of the piston, jn a modified form, to the remaining part of the machine, and for this purpose one end of it is bored out to fit the wristpin while the other end is bored out to fit a pin on the crank, which two pins are thus kept a fixed distance apart and their axes are always kept parallel to one another, (d) The fourth and last essential part is the crank and crankshaft or as It may be briefly called, the crank. This part also pairs with two of the other parts already named, the frame and the connect- ing rod, the crankshaft fitting into the bearing arranged for it on the frame and the crankpin, which travels in a circle about the crankshaft, fitting into the bored hole in the connecting rod available for it. The stroke of the piston depends upon the radius of the crank or the diameter of the crankpin circle, and is equal to the latter diameter in all cases where the direction of motion of the piston passes through the center of the crankshaft. The flywheel forms part of the crank and crankshaft. In many engines there are additional parts to those mentioned steam engines having a valve and valve gear, as also do many internal-combustion engines, and yet a number of engines have no more than the four parts mentioned, so that those appear to be the only essential ones. 3. Lathe.— Another well-known machine may be mentioned namely, the iHtho. All latlu-s conUvin h fixed {xut ur frame ...'■ ' By a motion of translation is i.ieaiit that all pointe on the part considered move u. parallel straight lines in the same direction atid sense and through the same distance. 'm^ THE NATVRK OF THK MACHINE 5 |.<-.l which hold.s the nx.Hl or tail oontcr, and which also contai„s K.nHl bean,.«8 for the live center and gearshafts. Then there L h(,hvo renter which rotates in the bearings in the frameand which Ir c^ the work, hcng itnelf generally operated by n.eans of a I.Ht from a countershaft. I„ ad.lition to thase parts there is he carriage which holds the tool ,K,st and h.s a slidfng motil alon^ the rame the gears, the lead screw, belts and other parts, all wh^Ih'l,""/ ""t 'r"" '""'^^""■^ ^" P^"-^--' ^ho details o which noed not be dwelt upon. 4. Parts of the Machine.-These two machines aro typical of .i very large number and from them the definition of the machine may be developed. Each of these machines contains more tiian that ft?' T "' ;^'"^'"S "f ^">' «ther machine it will be seen that It contains at least two parts: thus a crowbar is not a machine to conceive of anything whic-h was not a ma.-hine. The so-called sunple machines," the lever, the wheel and axle, and the wedg cause confusion along this line because the complete 1'^"; called a lever, it serves such a purpose onlv when along with it s a fulcrum; the wheel and axle acts as a machine only when ^^(dge. lhu.s a machine consists of a combination of parts. 6. Again, these parts must offer some resistance to change of «hape to be of any value in this connection. Usually the part used, and i is well known that these serve their proper purpose only when they are in tension, because only when they are used n this way do they produce motion since they offer resist^ce place ^^here it is in compression. Springs are often used as in a ve gc.u-s and governors, but they offer resistance wherever "s,d. Thus the parts of a machine must Ik, resistant 6. Relative Motion.-Xow under the preceding limitations a slip or building or any other structure couM readHy be inchid ■TL c n M ""f ""' ' '^'' ''■'""' ^ ^**^^""^^>'' ^» the other ■ . a,o capable of moving, and when the machine is serving pedals, etc., all move relatively to one another, and in all machine; THE TIlEOIiY OF MjU IIISKS the partH iiuiHt Imvo relative motion. It is to Im« home in niiiid that till the piirtH tlo not iiccc'ssurily move, iind iis a nuittor of fact there" arc very f«'\v inachincs in which ont> part, which is rcforrod to briefly as the frame, is not statiotiary, hut all parts must move relatively to one another. If on(! stood on the fraino of an enRine the motion of the eonneeting rod would lie (juite evident if slow tuioiinh; and if, on the other hand, one clunR to the conneeting rod of a very slow-moving eiiRine the frame would apiM-ar to move, th.at is, the frame has a motion relative to the (••)nnectinj? rod, and vice verm. 7. In a bicycle all parts move when it is going along a road, but still the different parts have relative motion, some parts moving faster than others, and in this and in many other similar cases, the frame is the part on which the rider is and which hjw no motion relative to him. In ciise of a car skidding down a hill, all parts have exiictly th(^ same tnotion, none of th(> parts having r(>Iativf motion, the whohf acting iis a solid body. 8. Constrained Motion. — Now considering the nature of tho motion, this also distinguishes the machine. When a body moves in space its direction, sense and velocity depend (>ntirely upon the forces acting on it for the time being, the path of a riflo ball depends upon the force of the wind, the attraction of gravity, etc., and it is impossible to make two of them travel over exactly the same path, because the forcfcs acting contiimally vary; a thrown ball may gt> in an approximately straight line until struck l)y the batter when its course suddenly changes, so also with a ship, that is, in general, the path of a free body varies with the external forces acting ui)on it. In the case of the machino, how- ever, tho matter is entirely different, for the path of each part is predeterminod by the designer, and he arranges the whole machine so that each part shall act in conjunctif)n with tho others to I)ro(luce in each a perfectly defined path. Thus, in a steam engine the piston moves in a straight line back and forth without turning at all, the crankpin describes a true circle, each \m\ni on it remaining in a fixed plane, normal to the axis of the crankshaft during the rotation, while also the motion of the connecting rod, although not so simple is perfectly delmite. In judging the (luality of the workmanship in an engine one watches to see how exact each of these motions is and how nearly it approaches to what was intended; for example, if a point on the crank does not describe a true circle in a fi.xed M^!^Si^^^'if^sw^^7.^:^^^^:s^^L:'^T. plane, or the cro««hoad does not move in h ,3«.rfoctly straiKht lino the onKitK. is not regurded as a goon H unchanged, and if .sufficient pre.ssun> is p od 'h hoZ, Jl T"" "■" '""n ""'^' "'""« ^''^^' ^^«""' whether tne tool which it carries is idle or subjected to consideml.I.. force due to the cutting of metal; should he carrhj- ,"" pu aside so that it wouhl not slide on the fran.... the laU e wo .1 ' djustcd. hese Illustrations might be n.ultiplie,i indefinitely »". the reader will think out ma.- *he,^ f„r himself. ^' n,ot ol"'r n"' ''I '"'"' ^""'"'■' "■ -' """^'''"^•' t'"'t the relative notions of all parts are completely h..d an.l do not depend i, anv ^ ths of h n\^ , '" u""'\ ^"''^'^ ^' ^P^"«^' the relative pains 01 tne parts are unalU'red n-'u^To'th!' T" f^**=^^«T'''here renuuns one other matter nutm to he machine, and that is its purpose. Machines ar always .lesigne.l for the special purpo.se of 'doing wo k , a .steam c-ngin," energy is supplied to the cyliruler by tho steam' f-m the bcile., the object of the engine is o converMhten^v pu nping u.iter. Power is delivered to the spindle of a lathe wough a belt and the lathe in turn u.ses thi.s energy in dng ^^OIk on a l.ir by cutting a thn>ad. Energy is supplL to -.|k on a windlass, and this energy, in tuit is takj^^ Uy tC (i( Mgne^l for the express purpo.se of doing work. resistant n«rlc Juu " ^^'*"»»on: A machine consists of to eaTotJlf ' H '^ ''*'^' * "'^'''^^ ^°^ ^^^on relative uergy may be made to do a desired form of work. 8 TUH TIIHiiliY (H- MACHISFS 11. Imperfect Machineii -Mrtny mnchinorftH'ti{| ()f \,\w hUmuii ••iiKiii" and \\w lufhc, wIuto all parts aro carefully inado and tin) motions aro all iw clow^ to thom; do«ir«Hl jus one could niako them. Hut tlwro arc many others, which althouKJi <()mmonly ami rorr«H-tly chwjk'd as nuichinctt, do not conin Htri a««umerfections are not uncommon in machines; the endlong motion of a rotor of an electrical machine, the "flapping" of a loose Ix'lt or chain, etc., are familiar to all p«;rsons who have stn^n machinery running; and even the unskilhwl ol)server knows that conditions of this kind are not good and are to \ki avoidi-d where possible, and the more these incorrect motions are avoidinl. the more m-rfect is the machine and the more nearly does it coriifily with the conditions for which it was dt^igned. DIVISIONS OF THE SUBJECT 12. Divisions of the Subject.— It is convenient to divide the study of the machine into four parts: 1. A study of the motions occurring in the inacliine without regard to the forces acting externally; this study deals with the kinematics of machinery. 2. A study of the external forces and their (>fTects on the parts of the machine assuming them all to be moving at uniform veliK-ity or to be in ei^uilibrium ; the balancing forces may then be found by the ordinary methods of statics and thii proi)lenis an- t hose of static eeuilibrium. 3. The study of m>,chanics of machinery takes into account the mass and acceleration of each of the i)arts as well as the external forcts. 1. I he ilufcrnunutioii of ihe |)ropcr sizes and shape's to be given the various parts so that they may l)e enabled to carry the loads and transmit the forces imposetl u[)on them from without, as well as from their own mass. This is machine design^ ■k'CtA' rnt: s.\n /,./., nt rnh .i/.ir///v/r 9 KINDS OF MOTION ol-rvod that i„ ..Hd. „.ovi,.« part ,1,. ,,,,1. of .. v ,',;,;,, - .n one piano, for .xan.plo. tho pat 1. .If a ,.,int .u. T 'ra ' n , I.OH on H plan. n„nnal t., the .-rankshaf,. L .!.„. ,, , ,;', "^'"7 ^r;" ••" • >'"«-ti..« ro,|. an.l also .1,;,,,, .f' rKunt on tlu, ..ro.s.slH.a,|. Sin.-o ,his is ,|.,. .-as., ,h..' J ,f , H,«.no nun.t.on...! ar. sai.I u. hav pl.„e motion 1. • J i L' m..nt .H Hirnply n.-ant ,hat .h. path of any poi t o„ ^ ." goNornor ba.ls, u, a lathe all ),arLs nsnallv hav piano n.otion tKoHame^tru..of an ohn-tri.. n.otor an.l. in fa.-t. ,h. ' v ^^ „ ! 1 14. Spheric Motion.-Thor.. ar,., h.,w,.v,.r. .■as.-s ^vluMV .liff.Mvnt motu^ns occur, for oxan.pl., th.-ro ar.. ,,art. .>f n.aohinos X or where the n.ot..,n is s,„-h that any ,H,int will uhvavs li.. on „„ urfaoe of a sphere ..f whi.-h th.- fix.-.l ,K.int is th.- .-on r ■ s sphenc motion ami .« not nearly .so ..o,.,,,,.., as tl... plan.. n..>tio„ 16 Screw Motion.-A thinl class ..f ,„otio„s o.lurs wlu^ro a body ha. a motion of rotation alu.ut an axis and a!s., a n . .,n of translation along the axis at the san.e tinu>. the nu, i. „ translation bearing a fix..! ratio to the nH>ti.„. of n.tatiT Tl ! In the onhnary n.nnkey wrench the movable jaw has a r,l.u,o notnn, relative to the part h.>ld in the han.l. it pl^l ,, ", -... on., o; tr.nslatH,n .r sli.Iin,. all points ..ntl^";;,.', plane uu,Unn relat.ve to the part hel.l, the nu.tio,. I,ei,„, ,„., „f >otat,.,n about the axis of the scr.nv, ar.,1 the screw has , hi.- motion relative to the movable jaw. an.l .ice versa wj-VA^^^sfi^^^a^mi^^^M^^s^msmm 10 THE THEOHY OF MACHINES PLANE CONSTRAINED MOTION It has been noticed already that plane motion is frequently constrained by causing a l)ody to rotate about a given axis or by causing the body to move along a straight line in a motion of translation, the first form of motion may be called turning motion, the latter form sliding motion. 16. Turning Motion. — This may be constrained in many ways and Fig. 1 shows several methods, where a shaft runs in a fixed bearing, this shaft carrying a pulley as shown in the upper left & (a) ;3 s Truck Fi(i. 1. Forms of turning pairs. figure, while the lowi-r left figure shows a thru.«t bearing for the propeller shaft of a boat. In the figure (a), tli(>ie is a pulley P keyed to a straight shaft S which passes through a bearing B, and if the construction were left in this form it would permit plane turning motion in the pulley and shaft, but would not constrain it, as the shaft might move axially through B. If, however, two collars C are .secured to :he shaft by screws as shown, then these collars effectually prevent the axial motion and make only pure turning po.ssible. On the propeller shaft at {h) the collars C are forged on the shaft, a considerable number being THE NATVItE OF THE MACniNE Ji and the „„ri„g ,„.. o.hc.'and.t:!' C aT t^TZr l-l .lep<.,>.lu,g upon oirc„ms,„,,.J Thr2^';''f'f^''' 1.1 oullino th,> ,„etho,l used i„ railroid ..^^ .kT * ' "" in .»,„...,, „i,h ,he shaft l^<7\ 2:^^77 r"-'' presses it info « r., fK- ? ' vertical shaft and i<. Chain and Force Closurp t.. +i,« / x . U. f,l (ft. . > ■ , *«»r»i' m gciicTal than the <,th(-r» ;^u i g^V'and'oZ "T '"'r "■"'■" '"* -«d '" -'" ° * W oo'nsSroftta' .t': I?r """'' """" '^ "'"" *^"'«' f» ont. *n,e,.t n,ay ia°i,„::t\h:rt*r::;r.*:: * 12 TH'l THEOHY OF ^^^r^TXh'S upon it. The construction of the forma (e) and (/) is evident The reader will see very many forma of this pair in machines and should study them carefully. Fl(i. 2. — Forms of sliding pairs. In the automobile engine and in all the smaller n.is and gasoline engines, the sliding pair is circular, because the crosshead is omitted and the connecting rod is directly attached to the piston, the latter being circular and not constraining sliding motion. nt nd is ci. THE NATUltE OF THE MACHINE 13 In this ca^e the sliding motion is constrained through the con necting rod, which on arrnimf «f *\.^ • ■ """»" ^"<' con- will not peU .he plton t„ J to ^7*,"!,!'^ '"■°.""* rifi. 3. — Sliding jxijrs. quonUyJjowevc.., foreo closuro is us.d a. in the c..e (^; shown at ,.,.,.1 i' r •' ^ '^ ^-^ ""^ ''^e^^i" u«<" uniil the t-il,l.. iv ' r „t ;;r ;; '|-' "■« ^-^ - -c. t,..t t,.!'!' " > n.i\(. piano motion, u con dt ion only nossiblo if n, .able ros,» ,„ ,„„ ^„„.„ .„ ,,„ ,„,„^, ,^ ^,,> l^^l^^ 14 THE THEORY OF MACHINES table is constrained by chain closure and this tail sliding piece of the piston rod in Fig. 3 (c) by force closure as is evident. 19. Lower and Higher Pairs.— The two principal forni.s of plane constrained motion are thus turning and sliding, (lios(^ motions being controlled by turning and sliding pairs resi^ct- ively, and each pair consisting of two elements. Where contact between the two elements of a pair is over a surface the' pair is called a lower pair, and where the contact is only along a line or at a point, the pair is called a higher pair. To illustrate this the ordinary bearing may be taken as a very common example of lower pairing, whereas a roller bearing has line contact and a ball bearing point contact and are examiiles of higher pairing, these illustrations are so familiar as to require no drawings. The contact between spur gear teeth is along a line and therefore an example of higher pairing. In general, the lower pairs last longer than the higher, because of the greater surface -exposed for wear, but the conditions of the problem settle the type of pairing. . Thus, lower pairing is used on the main shafts of large engines and turbines, but for automobiles and bicycles the roller and ball bearings are conunon. BIACHINES, MECHANISiJiS, ETC. 20. Formation of Machines. — Returning now to the steam engine, Fig. 4, its formation may he further studied. The valve gear and governor will be ornittt-vi at present and the remaining parts discussed, these consist of the crank, crankshaft and flywheel, the connecting rod, the i)iston, piston rod and Tin: NA TURK OF TIIK MA Cf/rXf, , , rrmshnul. and finally the frame and cylinder Talc.n. \ nienee for if ;< . erank eons sts esseiiti-illv ,.f < . • I'luutH. j luis the .-onneeted ''^•^^^''^'^''^ ^^^ <^vo turning elements ,,roperlv It; THE TIIKORY OF MM'IUNES eiiginc, the guides are omitted and the sliding clement is entirely in the cylinder. Of cours(>, the shape of the element dejH'nds upon the purjxwe to which it is put; thus in the ease last referred to it is round. Then, there is the crosshead r, with the turning element pairing with the connecting rod and the sliding element pairing with the sliding elemcMit on the frame. The sliding element is usually in two parts to suit those of the frame, hut it may he only in one if so desired antl conditions permit of it (see Fig. 2). Thus, the steam engini; (!onsisis of four parts, each part con- taining two elements of a i)air, in some cases the elements being for sliding, and in others for turning. Again, on examining the small gxsoline engine illustrated in Fig. 5, it will be seen that the same method is adopted here as in the steam engine, but the c-osshead, piston and piston rod are all combined in the single piston c. Further, in the Scotch yoke, Fig. 6, a schenie in use for jjuinps of small sizes as well as on fire engines of some makes and for other purposes, there is the crank a with two turning elements, the piston and crosshead c with two sliding elements, and the block h, and the frame d, each with one turning element and one sHding element. 21. Links and Chains. — The same will be found true in all machines having plane motion; each part contains at least two elements, each of which is paired with corresponding (>lements on the adjacent parts. For convenience each of these parts of the machine is called a link, and the series of links so con- nected as to give a complete machine is called a kinematic chain, or simply a chain. It must be very carefully borne in mind that if a kinematic chain is to form part of a machine or a whole machine, then all the Hnks nmst be so connected as to have definite relative motions, this being an essential condition of the machine. In Fig. 7 three cases are shown in which each link has two turning elements. Case (a) could not form part of a machine be- cause the three links could hi-.ve no relative motion whatever, as is evident by inspection, while at (b) it would be quite impossible to move any link without the others having corresponding changes of position, nd for a given change in the relative posi- tions of two of the linLs a definite change is produced in the others. Looking next at case (c), it is observed at once that both DC and OD could be secured to the ground and yet AB, BC, and OA moved, that is a definite change in .1 B produces no necessary I THK s.vrvin: ar tiik .maciiink 17 Hfi arrangomei.t could not for.n o-irt of -i 7.. . .*^'"°"- ^^"«'' otluT Stationary «,|v "','■""""' '»,«'"-' '" ■' f"m. nicchanisin is shown in Fig. 9 with the crank a, connecting rod b and piston c, the latter having one liding and one turning element anu representing the reciprocating masses, i.e., piston, p' 'on rod and crosshead. The frame d is represented by a straight line and although it is common, yet the line of motion of c does nnt always pass through O; however, a.s shown at (1), it represents the usu-d construction for the ordinary engine. If now, instead of fixing n from 6 by a pin c working in a slot in d. The arm .1 is attached to a tool holder at £. ¥ nth- \ AT in I'! OF Til/-: machine 21 The CJnouie nu.tur u^ed on aeroplanes is also an example of h w^artr'V-lK-'^ '^ ^'"^^" "' ^'«- ^2 and the cylinde as hi wiu J"';: "r"' "*" ''''^"" ^""" •'•« ^"••- "-han«,m IheHhafUnir '^"•^'*-™^"'-" '»«*•«". " '•«"'« the link between lt:;rhl":^^^^^^^^^^ Btud.vthe.echanisn, Fio. 13.— Simper iiu'olmiiism. SO that the line of motion of O Fiir on^ ,. .. i ^ ■ . f K^ „„L , . V, i ig. !> U), pawsses above O, fdviixsr engint X""' '" """'"'"' '"^ '""^'^'^'^ '" ^""'*'' ^""- "^ «-«'^- shoVn"rFL'\f ","' ""^-^-" °f ^he slider-crank chain is worth ,„. f ; T "'''''''"'^^- On comparing it with the Whit- north mo .on shown at Fig. U, and the engine shown at Fig JO It s ^en th.Mhc n.echanism of Fig. may be somewhat Xred tratid at"Fil irr?r '^ ''^ ""'^- ^he mechanism i lus- ^rated at Fig. 13 should be clear without further explanation D .s the driving pmion wo.Wng in with the large gear b, the ^i ■if: >?-x SSSK*S- 23 THE TIIHOIiV OF M AC If! \ US is Attnc-h«'il to H whirl) in tirivcn from c liy Ihc link .1. It if* roiulily HiHMi tliiit H iiiov»'8 fiiMtcr in mw iliri-ction tliuii the other. KurlhiT, itn iimiiiKfmciit in iiimli> for varying tht- Mlrokc «tf H ut ph'jwun- hy moving the ci'ntor of <• closer to, or fiirthor from, thiit of 6. 28. Double Slider-crank Chain.— A further ilhistriition of n ••hiiin whieh k>.,• CHAPTER II MOTION IN MACHINES 27. Plane Motion.— It is now desirable to siudy briefly certain of the characteristics of plane motion, a term which may be defined by stating that a body has plane motion when it moves in .such a way that any given point in it always remains in one and the same plane, and further, that the planes of motion of all points in the body are parallel. Thus, if any body has plane motion relative to the paper, tl- n any point in the body must remain in a plane parallel to the plane of the paper during the motion of the body. A little consideration will show that in the case of plane motion the location of a body is known when the location of any line in the body is known, provided this line lies in a plane parallel to the plane of motion or else in the plane of motion itself. The explanation is, that since all points in the body have plane motion, then the projection of the body on the plane is always the same for all positions and hence the line in it simply locates the body. For example, if a chair wore pushed about upon the floor and had points marked li and L tipon fhe bottoms of two of the legs, then the location of i\w. (;hair is always known if the positions of It and /., that is, of the (imaginary) lino HL is known. If, however, tho chair won- free to go up and down from the floor it, would 1k> nocos,sary to kno"- t'lo position of the projection of HL on tho floor and ;dso tho height of tho line above the floor iit .any iiistjmt. Further, if it were possible for the chair to Im; tilled l.iurkward alxmt the (imaginary) line RL, the positicm of tho latter would tell very little about the {.osition of the chair, jis tho tips of its legs might be kept stationary while tilting the chair back and forth, the position of RL being the same for v?.rious angular positions of the chair. If the case where a body not plane motion is considered, then the line will tell very h ,.; about the position of the body. In the case of an au-.ship, t r> .>lo, the ship may stand at various angles about a given lir-, say the axis of a pair of tho 24 1aaij'3!i'-'- .mrroN m machine,^ wheels, the ship dipping (l.mnwan) ,„• ming ut th v.ll r .. operator. ""nig m in„ vill of tJie 28. Motion Determined by that of a LinP q; *i , ill I \c' I'Ki. 10. P tJir. i.X^ • 1 '^Jiown wnen the location of ;--' of rl.o paper ot'THv/i T'"''"'' ^'" ''"-^J^^*'''" «» the the paper, whic-h is use .1 / ?"'"'^ *^ ''^" "' ^''^" P^'"'« «f the paths AA' and ^«' l^ ' *^"'.'"^"'t«'l and /^ move over "- i« at .i;/.. an^ he^r f7:::';^""^'r^ r-'-" *^« thi« line, and this locates tte posi- "'"" '" '^'""" ^^'^^^^ <'on of the body corresponding to the location A,B, of the line J 7i It will therefore follow that the •notion of a body is completely known piovided only that the motion of any line in the body is known. This proposition is of '■'^ RS'Si:^:^^ ^rr '^ -« -^ U''' ^^'^ forward, and a person on tCound ir'^ T " *''' ^'^'- ^"^""^ would say he w.is station-trv whil ., "^ ^"'' '''' ^"« »'^"'J "e was moving at ^:!:::;x^^r^i 'T r'" ^^•^■ and saw a ship go out one would say that til n, 1"'^ '" '^'''"' and yet a person on the ship wouS say thtt t ""' """'"^' These conflicting statements, which a e ho ''" "'"''• mon, would lead to endless r-nnV. ?' ^°''^^'«'-' very corn- endless confusion unless the essential differ- 'f 26 THK TfiKOL'Y nr u.u7//,vf;.s wn !! ences in tlic viirious cOvSes were gnisped, and it will be seen that, the real difference of view results from the fact that different, persons have entirely different standards of comparison. Stand- ing on th(! groimd the standard of rest is the earth, and anything that moves relative to it is said to be moving. The man on the tlat car would l)e described as stationary because he does not move with rejiard to the chosen standard— the earth, but th.e eiij!;iue driver would be thiiikinf!; of the train, and he would say the man moved bectause he moved relative to his standard- (he train. It is easy to multiply these illustrations indefinitely, but they would always lead to the same result, that whether a body moves or remains at rest depends altogether upon the standard of comparison, and it is usual to say that a body is at rest when it has the same motion as the body on which the observer stands, and that it is in motion when its motion is (litTerent to lliat t)f the body on which the observer stands. On a railroad train one speaks of the poles flying past, whereas a :nan on the ground says they are fixed. 30. Absolute and Relative Motion.- When the standard which is used is the earth it is usual to speak of the motions of other bodies s absolute (although this is iiu-orrect, for the eartli itself moves) and when any standard which moves on the earth is used, the iuotions of the other bodies are said to be relative. Thus the absolute motion of a body is its motion with regard to the earth, and the relative motion is the motion as compared with another body which is itself moving on the earth. Uidess these ideas are fully appreciated the reader will undoid)tedly meet with nuich difficulty with what follows, for the notion of relative motion is troublesome. In this coimection it should he pointed out that a body seemed to the earth may have motion relative to another body which is not so secured. Thus when a ship is coming into port the dock appears to move toward tlu* passengers, but to the jwrson on shore tlu! ship appears to come toward the shore, .hus the motion of the ship relative to the; dock is equal and opiJOsite to the motion of the dock relative to the ship. 31. Propositions Regarding Relative Motion.— ( ertain proposi- tions will now be self-evident, the first being that if two bodies have no relative motion they have the same motion relative to every other body. Thus, two passengers sitting in a train have no relative nu)tion, or do not change their positions relative to >« ^MmM:;^ Ww^^ ♦ • MOTION IN MACHINES 27 one another, and hence they have the same motion or cnange of position relative to the earth, or to another train or to any other body: the converse of this proposition is also t»-ue, or two bodies which have the same change of position -elative to other bodies have no relative motion. 32. Another very important proposition may be stated as follows: The relative motions of two bodies are not affected by any motion whi(^h they have in common. Thus the motion of the connecting rod of an engine relative to the frame is the same whether the engine is a stationary one, or is on a steamljoat or locomotive, simply Iwcause iri tho latter cases the motion of the locomotive or ship is co- imoii to the connecting rod and frame and does not affect their r(>l:itive motions. The latter proposition leads to flu; statement that if it be desired to study the relative motions in any machine it will not produce any change upon them to add the same motion to all parts. For example, if a bicycle were moving along a road it would be found almost impossible to study tlie relative motions of the various parts, but it is known that if to all parts a motion be added sufficient to bring tlie frame to rest it will not in any way affect the relative motions of the parts of the bicycle. Or if it be d(?sired to study the motions in a locomotive engine, then to all parts a common motion is added which will bring one part, usually the frame, to rest relatively to the observer, or to the observer and to all parts of the nuu^hine such a motion is added as to bring him to rest relative to them, in fact, he stands ui)on the engine, having added to hi'uself the motion which all parts of the engine have in common. So that, whenever it is found necessary to study the motions of machines all parts of which are moving, it will always be found convenient to ad(. to the observer thq common motion of all the links, which will bring one of them to rest, relative to him. To give a further illustration, let two gear wheels a and b run together and turn in opposite sense about fixed axes. Let a run at + oO revolutions per minute, and h at — 80 revolutions per minute; it is required to study the motion of b relative to a. To do this add to each such a motion as to bring a to rest, that is, — 50 revolutions per minute, the result being that a turns + oO - oO = 0, while b turns - 80 - 50 = - 130 revolutions per minute or b turns relative to a at a speed of 130 revolutions per minute and in opposi' 3 sense to a. Here there has simply 28 THE THEORY OF \f WHINES ii been added to each wheel the sain .lolioii, which does not affect their relative motiuns but has the effect of bringing one of the wheels to rest. To find the motion of a relative to h, bring h to rest by adding + 80 revolutions per minute, so that a goes + 50 + 80 = 130 revolutions per minute, or the motion of h relative to a is equal and opposite to that of a relative to h. 33. The Instantaneous or Virtual Center. — It has already been jiointod out in Sec. 27 that when a body has plane motion, the motion f)f the body is .omijletcly known provided the motion of any lino in th(> body in the plane of motion is known, that is, provided the motions or paths of any two points in the body are known. Now jet c, I'ig. 17, represent any body moving in the plane of the paper at any instant, the line AB being also in the plane of the paper, and let FA and BE repre- sent short lengths of the i)aths of .4 and B respectively at this instant. The direction of motion of A is tangent to the path FA &i A, and that of B is tangent to the path BE at B, the paths of A and B giving at once the direction of the motions of these points at the instant. Through A draw a normal AO to the direction of motion of A, then, if a pin is stuck through any point on the line AO into the plane of reference and c is turned very slightly about the pin it will give to A the direction of motion it actually has at the instant The same argument applies to BO a normal to the path at B, and hence to the point where AO and BO intersect, that is, if a pin is put through the [)oint in the body c and into the plane of reference, where the ))ody is in the position shown, the actual motion of the body is th«! same as if it rotated for an instaot about this pin. O is called the instantaneous or virtual center, because it is the point in the body c about which the latter is virtually turning, with regard to the paper, at the instant. In going over this discussion it will apiwar that may be found provided only the directions of motion of A and B are known at the instant. The only pur[)ose for which the paths of these points have been used was to get the directions in which A and B were moving at the instant, and the actual path is un..nportant in so far as the finding of is concerned. It will further appear that the point O will, in general, change for each new position ITIIIIIII I llllttiiil < ~ «. I- urther, there is a virtual center of 6 ^ c, that is be und also a center of c ~ «, which is ca, thus for the three bodies here are hree virtual centers. Now it will be assumed tha enough m onnation has been given about the motions of a b and c to deternune ah and ac only, and it is required to find 'be bince be ,s a point common to both bodies h and ., let it be ^supposed to he aiP', then P' is a point in both the bo.lies 6 ans, ^ce, however, this is not the point, it shows .It once that the point be is located somewhere along the hne cib-cn or say at P, because it is only such a point as> which has the same n.otion with regard to a whether considered as a W^Unb or m c; thus the ..enter be nmst lie on the same straight hm .u, the centers ah an.l ac. It is not possible to find the ex- ao os,t,on of be, h.>w,>v.M-, with.,ut further information, all that is known is the line on which it lies. This proposition may be thus stated: If in any mechanism these three centers must aU lie on one straight line Iwo of the centers may be permanent but not the third; in The sign ~ means "with regard to." Fi 3. When a person in an autonioliil •, which i.i naininn on a «tri'ct nir, Uwk.i at the latter without looking at the grountl, the i-ar appears I.. I..- coniinK toward him. Why? 4. Kxplain the difference between relative and iilisolute motion and Mt.ilc the propositions referring to tlu-se motions, 6. Tlie speeo8 of two pulleys are UK) ri'v.ihitions per minute in the .li.rk- wise sense and 125 rc/ohitions per minute in the npposil.' seiiii', rr.sp.-ctivilv what is the relative speed of the former to the hitter? 6. In a geared pump the pinion nuikes 90 revolutions per minute ,ind the pump crankshaft 30 revolutions in opposite sense; what is the motion uf ilic pinion relative to the shaft? 7. Distinguish between the instantaneous uiul complete motion of a body. What information gives the former completely? What is the virtuiil center? 8. W^hat is the virtual center of a wagon wh«H.'l (a) with regard to the earth, (6) with regard to the wagon? 9. A vehicle with JJG-in, wheels is moving at 10 miles an hour; what are the velocities in space ami the directions of motion of a point at tlie top of the tire and also of points at the ends of a horizontal diameter? Is the motion ttie same for the latter two points? If not, find their relative motion. 10. A wheel turns at 1.50 revolutions per minute; what is its angular velocity in radians per second? Also, if it is 20 in. diameter, what is the linear velocity of the rim? 11. Give the information necessary to locate the virtual center Ix'tween two bodies. 12. What is the difference between the virtual, permanent and fixed center? State and prove the theorem of the three centers. 13. Find the virtual centers for the stone crusher or any other somewhat complicated machine. CHAFrKR III VELOCITY DIAGRAMS 42. Applications of Virtual Center.~So,„o of tho main appli- cations of (1.0 v.rtuul ccMtor cli.sn..s.s,.d in tho last chaptor a o tn the (loternunat.on of tho volocitios of ,ho various points and links m n.ochan.sms, an.l also of the forces acting throughout tho mochanisni duo to external fon-os. The latter question will Ik- (hscusscd in a subs,.c,uent ehar.tor and the present chapter will »>o confined to the dotennination of velocities and to the repre- sentation of thesT velocities. 43 Linear and Angular Velocities.-Thorc are two kinds of velocities which are required in machines, the lin(>ar velocities of the various points and the angular velocities of the various links and It will be best to begin with the detcrmi.mtion of lin(.ar velocities. 44. Linear Velocities of Points in Mechanisms.-'l he linetr velocities of the various points may be required in one of two forms, either the absolute velocities may be required or else it may be only desired to compare the velocities of two points that 18, to determine their relative velocities. The latter problem may be always solved without knowing the velocity of anv point in the machine, the only thing necessary being the shape of the mechanism and which link is fixed, while for the determination of the absohite velocity of a i>oint in a mechanism that of some point or link must be known. Again the two points to be compared may be in one link, or in d.fTerent inks, and tho .solution will be made for each caM> and an A. « ""'''"' *" ''''*'''" solutions which are (,uite general 46 Points in the Same Link.-The first ca.se will bo that of the <.ur-hnk mechanism, frequently referred to, containing four urmng pairs and shown in Fig. 21, and the letter d will be iKsod to indicate the fixed link. As a first problem, lot the vHocity of any pomt A, ma be given and that, of another point .1.. in the same link be required. The six virtual centers have bee,; fouml mid marked on tlje drawing, and the link a has been selected for the hi.t example because it hjus a permanent center which is ad 36 3ti Till': TIIFAUiY OF MA( IllSKS Now the velocity of .li whieh is ausuiiictl given, iH the alwolute velo<;ity, that is the velocity with regard to the earth. From .1 , lay oiT AiK in any direction to represent the known velocity of A i and join lul-K and produce tluH line outward to meet the line AiF, parallel to AxK, in the jxjint F. Then A2F will reprcMiit the linear velocity of .l, on the same scale that AiK representH the velocity of .li. (It is juwumed in this construction that nd, Ai an, Fig. 22, the velocity of Bx in the same link being given an^l let d be the fixed link as ]>efore. Now since it is the alj.solute velocity of lii that is given, the first point is to find the center of bd about which b is turning with regard to the earth. The velocity of B\ then bears the same relation to that of /^j as the respective dis- tances of these points from bd, or Linear velocity of li hd - H, Linear velocity of B^ ~ bd - B2 It is then only neces.sary to get a simple gra|)hical method of obtaining this ratio and the figure indicates one way. First, mmsLjMm'z^%.Mmis^>:'rr'^v';m^&L^riKm*i!mMt^r Vt..AH ITY IHAdHAMS 37 u^th .■,.n..r W draw ar.-s of o.rcl.. through /?. and fl, cutting W - nh .„ //'and //',. Thon if /y'.r; ,,. drawn in anv direct on / ..// panillcl to «'/;, w.ll n.„r...scnt the li„oar velocity of fl, or '|.<- ra.H, of /y,/ to /yV/ in the ratio of the velocftieH of B, The only difTeren.-e l.etwe.-n this and the last e,i«e i, that in the us','''.'L!""\ /■;'"'"■■■/•'' """' ''"" P<-"Manent, whereas in th.n (Use 111,. c.Miter W used is a virtual center 46. Points in Different Links.-If it were required to compare .■ ...ear ve oc.ty of the poi..t .1. i„ . .,th that of Ji, i„ fIZ ""••f.od woul.l he as uulic-atci ,n Fig. 22. The two links con- eeitu'd are « and /> and r/ is the fixe(| link and these links have the three centers a,l, nh, b,l, fill on one line, also nh is a point coniinon to a and h, •'«'ing a j)oint on each link.' 'I'if.'King it as a i)oint in n, pioceed as in the first ex- ample to find its velocity. Thus set olT A J-: in any eet mIf in Vu^..'^:' .-.presents tlv. velocity of B. in the link b on the 1^ 1," fh^t AiE represents the velocitv of A, Notice that in dealing with the various links in finding relative 38 THE THEORY OF MACHINES i i velocities it is necessary to use the centers of thti links under con- si(i(M ttion with regard to the fixed link; thus the centers ad and bd and the comnaon center ab are used. The reason ad and bd are employed, is because the velocities under consideration are all absolute. To compare the velocity of any point ^i in a with that of d in c, Fig. 23, it would be necessary to use the centers ad, ac and cd. Proceeding as in the former case the velocity of ac is found by drawing the arc AyL with center ad and drawing LN in any direction to represent the velocity of vli on a chosen scale, than the line ac~ M parallel to LN meeting ad-N produced in M will represent the velocity of ac. Join cd - M, and draw the arc CiC'i with center cd, then ("lA' parallel tnuc-M, will represent the linear velocity of Ci. l''i(j. 23. A general proposition may be stated as follows: The velocity of any point A in link a being given to find the velocity of F in /, the fixed Unk being d. Find the centers ad, fd and af, and using ad and the velocity of .1 , find the velocity of af, and then treating af as a point in / and using the center fd, find the velocity ofF. 47. Relative Angular Velocities.- Similar inetliods to the procediiiK may be employed for finding uiiKular velocities in mechanisms. I>rt any body having plane motion turn through an angle (9 about any axis, either on or off the body, in time t, then the angular velocity of th<> body is defined by the relation to = -As all links in a mechanism move except the fixed Imk, there are in general as many different angular velocities as there are moving links. The angular velocities of the various links a, b, c, etc., VELOCITY DJACIRA U.S 39 will be designated by «„. o^, a,„, etc., respectively, the unit being the radian per second. As in the case of linear velocities, angular velocities may bo expressed either as a ratio, in which case the result is a pure tnitnber, or as a number of radians per second, the method de- I>endmK on th(> ki.ul of information sought and also upon the .lata Kivcn. Unless the .lata includes the absolute angular ve ocity of one link it is quite im,K)ssible to obtain the absolute velocity of any other link and it is only the ratio between these velocities which may be fountl. 48. Methods of Expressing Velocities.-In finding the lela- tive angular velocities between two bodies it is most usual to Wa express the result as a ratio, thus ^ , which result is, of course, a pure number, such a method is very commonly employed in connection with gears, pulleys and other devices. If a belt con- nects t^vo pulleys of 30 in. and 20 in. diameter their velocity ratio will be /30 = h, that is, when standing on the ground and count- ing the revolutions with a speed counter, one of the wheels will he found to make two-thirds the number of revolutions the other one does, and this ratio is always the same irrespective of the absolute speed of either {)ulley. It happens, however, that it may. be necessary to know the relativ-e angular vel<,cities in a different form, that is, it may be desired to know how fast one of the wheels goes considering the other as a standard; the result would then be expressed in radians per second. Suppose a gear a turns at 20 revolutions per minute ": - ^•^^f*^'^"^ !>«'• «*'''"'«1. and meshes with a gear h running at 30 revolutions per minute, a,, = 3.14 radians per second the .wo waeels turning in opposite sense, then the velocity of n with regard to 6 is co,. - «, = 2.09 - (-3.14) = +5.23 radians per second, that is, if one stood on gear h and looked at gear a the latter woul.l appear to turn in the opposite sense to b and at a rate of 5 23 radians per second. If, on the other hand, one stood on o, then since .. - .„ = -3.14 - 2.09 = -5.23, ft would appear to turn backward at a rate of 5.23 radians ,>er second, the relative motion of a to 6 being equal and opposite to that of 6 with regard to n. The first method of reckoning these velocities will alone be employed in this discussion and the construction will now be explained. 40 THE THEORY OF MACHINES 49. Relatiye Velocities of Links.— Given the angular velocity of a link a to find that of any other link b. Find the three centers ad, M and ab, then as a point in a, ab has the linear velocity (ad - ab) ua and as a point in b, ab has the velocity (bd - ab) «». But as ab must have the same velocity whether considered as a point in a or in b, then (ad - ab) «„ = (bd - a/>) wt, or "* = COa ad — ab ^ _ . • i he construction is shown in FiR. 24 and will require very little explanation. Draw a circle with center ab and radius (A - ad, which cuts ab - bd in G, lay off bd - F in any direc- Fia. 24. tion to represent «„ on chosen scale, then draw GE parallel to bd -F to meet ab - F in E, and GE will represent the angular velocity of 6 or a* on the same scale. Similar processes may be employed for the other links b and c, and no further discassion of the point will be given here. The general constructions are very similar to those for finding linear velocities. As in the case of the linear velocities the following general method may be conveniently stated: The angular velocity of any link a being given to find the angular velocity of a link /, d being the fixed link. Find the centers ad, fd and af, then the angular velocity u>„ is to w/ in the same ratio that/(i - af is to ad - af. 60. Discussion on the Method.— Although the determination of the linear and angular velocities by means of the virtual center •^SiW VELOCITY DIAGRAMS 41 i« simple enough in the cases just considered, yet when it is em- ployed m practice there is frequently much difficulty in getting convenient constructions. Many of the lines locaLg virtual centers are nearly parallel and do not intersect within the limits of he drafting board, and hence special and often troublesome methods must be employed to bring the constructions Sn ordmary bounds. Further, although it is common to have given the motion of one link such as a, and often only the mo ion of one other point or link say /, elsewhere in the mech- anism IS desired, requiring the finding of only three virtual centers, K a/ and d/, yet frequently in practice these cannot be obtmned without locating almost all the other virtual centers n the mechanism first. This may involve an immense amount of labor and patience, and in some cases makes the method unworkable. 61. AppUcation to a M Aamsm.-A practical example of a more comphcated mechanism in common use will be worked out here to illustrate the method, only two more centers af and 6/ being found than those necessary for the solution of the problem Fig. 25 shows the Joy valve gear as frequently used on locomotives and other reversing engines, more especially in England: a rep- resents the engine crank, h the connecting rod, and c the piston etc as in the ordinary engine, the frame being d. One end of a mk 6 IS connected to the rod h and the other end to a link / the atter link bemg also connected to the engine frame, while (o Hie link e a rod g is jointed, which rod is also jointed to a sliding block h, and at its extreme upper end to the slide valve stem V I he part in on which h slides is controlled in direction by the engineer who moves it into the position shown or else into the ho .lotted position, according to the sense of rotation desired in the cranLshaft, but once this piece m is set, it is left stationary and virtually becomes fixed for the time. A very useful problem in such a case is to find the velocity of L'f;' ^r '^uV" ^' ^^' ^ ^^"" P°^**'°" ^"^ «P^^ «f the crank- T\ i , ^ "" concerns three links, a, d and g, the upper the three centers arf, ag, and dg are required. First write on all ihe ccners which it is possible to find by inspection, »uch as ad, otters bvthe'^h'-^' '*"' rt '''" P^""^^ *° «"^ '""^ '^^^-^ cen e 1 ! H ^"^'""^ "V^'"'" '"'"'^'^ '^^^^ '" ^ec. 39. The centers ag and dg cannot be found at once and it will simplify 42 THE THKORY OF MACHINES the work to set down roughly in a circle (not necessarily accur- ately) anywhere on the sheet i)oints which are approxiniately equidistant, there being one point for each link, in this case eight. Now letter these points a, b, c, d, e, f, y, h, 'm correspond with the links. As a center such as ab is found join the points a and b in the lower diagram and it is possible to join a fairly large number l''i • ,ir ..,.„ I . ., * hnows that the centers ab, ac ad th ''/, are known, while tie centers /lA hn .1. > "f-. «a, "e, ,.ath»/„ - .^ a„j f„ _ „^ ,,,„^„ .^^_„ ^ welltrhe pa i , - e». Ihe center gc coulil not, however !», f„„„.l u t J .here „„uh. then he only one'path Z- ^'ZL^tj^^^^ ;;;.iM ;o .,e .,ut.n „, .he ...L prohiiX^'^rrrtU 63. Graphical Representation of Velocities — Tf ;. r 'lesirable to have a diairi , t velocities.— It is frequently 44 THE THEORY OF MACHINES methods are in fairly common use (1) by means of a polar dia- gram, (2) by means of a diagram on a straight base. To illustrate these a very simple case, the slider-crank mechanism, Fig. 26, will be selected, and the Unear velocities of the piston will be determined, a problem which may be very conveniently solved by the method of virtual centers. Let the speed of the engine be known, and calculate the linear velocity of the crankpin ab; for example, let a be 5 in. long, and let the Fio. 26. — Piston velocities. speed l)e 300 revolutions per minute, then the velocity of the 300 5 crankpm = 2t X g^ >^ 12 " ^'"^-^ ^^- P^*" second. Now be i.s a point on both the piston c and on the rod b and clearly the velocity of be is the same as that of c, the latter link having only a motion of translation, and further, the velocity of the crankpin ab is known, which is also the same as that of the forward end of the connecting rod. The problem then is: given the velocity of a iwint 06 in 6 to find the velocity of be in the same link, and from what has already been said (Sec. 35), the relation may be written: velocity of pist on _ velocity of be bd — 6c velocity of crankpin "" velocity of a6 ~ bd — ab :^^m;^ ik of ry et ty le VELOCITY DIAGRAMS But by similar triangles 45 so that hd — ah ac — ad ad — ab .jelocity of piston _ ad - ac velocity of crankpin ~ ad- ah evi^tfhaT t '"* ^"' all positions of the nmchine, it is evident that ad - ac represents the velocity of the piston on the same scale as the length of a represents the linear vSocity of he size then ad - ab = 5 in, and the scale will be 5 in. = 13 1 ft persecond or 1 in. = 2.62 ft. per second. nf^/°^^'^^"^'~^''''' '^ '' convenient to plot this velocity the result for the different crank positions, or vertically above the Strenf >" '' ^VV^ '"'^^^ *° "-^P^-"* ^^e velocity fo ctmZe r /'." °^ ^'^^ P"'°"- ^^ ^'^'^ determination for the complete revolution is made, there are obtained the two diagram! rind ^^^^.t ^''^^^'•^•" ^«"«'«ts of two closed curves passing through and both curves are similar; in fact the lower one can be ot r^n .'?'•"" ''' ""^'^•"« ^ *"•-•"« -' the latter and tur- K roHhe mo"f "'" l' " ''■ ^^'^ ^""^^ ^he connec - le^tkal throuTh n"'"J V *'^ ^"'•^^ symmetrical about the vertical through O, and for an infinitely long rod the curves are circles, tangent to the horizontal line at O la^ne^offT" ? v ^*^^* Base.-The diagram found by fnd nf.K ' "" ^^"'^ 'h^'^'"' '« egg-shaped with the small he 1 :tf IZT^T 'f *'\"'°'^ ^"^^^ ^'--*^^-' ^""" ine line ot travel of the piston. Increasing the length of the rod S?o:5l: -"iriii^r^ "'"■*"• -' ""■ '"^ '"«""" arf^f f ' ^r"*'''" ''^ "'"*'''" ^^ *h« Pi«*°" does not pass through mot on f^.' '"'"' ^''^ ^« "«^ symmetrical about^elbe of the pTst: .I'X^^^^ '"* *^'^^*•^« ^«- «hown at Fig. 27. whe^e clear frl' theS'h T"" ''"\' '^' " ^^"'«« ''^ ^^'^^ it is Piston ,r//^ '^'^^^"' *h** ^^^ '"ean velocity of the Sra;:hl"frt ^ yreateMhan on the out stroke, a.^ lay, therefore, be used a.s a quick-return motion in a shaper \* ^^^C'm- ■^■liMiita 46 THE THEORY OF MACHINES or other similar machine. Engines are sometimes made in this way, but with the cyUnders only slightly offset, and not as much as shown in the figure. 66. Pump Discharge.— One very useful application of such diagrams as those just described may be found in the case of pumping engines. Lot A be the area of the pump cylinder in square feet, and let the velocity of the plunger or piston in a given position be v ft. per second, as found by the preceding method, let Q cu. ft. per second be the rate at which the pumj) IS di.'-charging water at any instant, then evidently Q = Jy and as 4 is the same for all piston positions, Q is simply proportional Fic. 27.— Off-sot oylindor. to V, or the height of the iiiston velocnty diagruiii represents the rate of delivery of the pump for the corresiwriding piston position. If a pipe were connected directly to the cylinder, the water in it would vary in velocity in the way shown in the velocity dia- gram (a), Fig. 28, the heights on this diagram representing piston velocities and hence velocities in the pipe, while horizontal dis- tances show the distances traversed by the piston. The effect, of both ends of a double-acting pump is shown; this variation in velocity would produce so much shock on the pipe as to injure it and hence a large air chamber would be put on to equalize tlu> velocity. Curve (6) shows two pumps delivering into the same pipe, their cranks being 90° apart, the heavy line showing th.at the variation of velocity in the pipe Hne is less than before and re- quires a nuich smaller air chamber. At (c) is shown a diagiaui corresponding to three cranks at 120° or a three-throw pump, W^J^- a^l^- i VELOCITY h J AURA MS 47 in which case the variation in velocity in the nirv. !,„« „^ u i much «.^,e. .in, thi« ve.oc.t.v U^in/'^tn^ Tthe^h^g^ up to the heavy hne. Ail the curves are drawn for the cLe of a very long connecting rod, or of a purnp hke Fig 6 Thu8 the velocity diagram enal.les the study of such a problem purpos(,s to which It may he put, and which will appear in the course of the engineer's ex,K.rience. Angular veloS nmv of course, he plotted the «v.ne way as linear veloJtts '^' (a) single Cylinder Pump BeiuUant Vtlocily PiitoB Fstiiieai (b) Two Cylinder*. Cranks at 90 i lU.uIian, VclMU)' P(YYYYTfVyYYYYY\ (e) Three Cylinder.. Cranki at 120° Fio. 28.— Rate of discharge from pumps. rt^f^ r.t T '^''^'*'''' ''"'^ * ^'^^^' «"«g««tions are made ^is to further uses of these velocities in F)racti,-e. QUESTIONS ON CHAPTER III IsVn V^J" "'^hanism of Fig. 21, a, b, c and ,i are respectively 3 15 10 and -J. tmd also the annular velocities <,f the links in thB -<.,>. 48 THK THEORY OF MAC/IINhS 4. Find the linear velocity of the left rnd of the jaw in Fin. M, knowinn the angular velocity of the camshaft. 5. Plot a diagram on a straight baaw for every 1.5" of crunk angle for the piston and center of the rod in question 3; also plot a polnr dingnini in., h - -i', in. mikI /- l;i in., the line of motion of the table puNNJng llnout;)) a,i ? 8. Find the velocity of the tool in one of the rivi-ters Ki\ t n in ( Imiif cr I .\ , asHuming the velotrity of the piston at the instant to b.- know n. 9. Draw the iH)lar diagraiii for ihi- angular veloeitvof the valv.- stem in the meehaniHm of Fig 40 for tl of • of i.k- l;t Carleton College Library - Ottawa f'HAPTKR IV THE MOTION DIAGRAM ease of the flZho^rn ' ''"''' '* ""'^"^"^ «P^^' *« i" the 40 fiO rilH TIlEdHY (tF MMIllSKS (i(*t(>rininution of th«> rt'lulivc inoliuiiH ni° the viirioiis iMirU; the inothod dt'HcrilK'd !i«'rr iiiny Ih- iimciI in cithrr vvkuv. The construction ahout to In; explained hiw U'en culled by its dituK)verer the phorugraph method, and, uh the name 8U»'Ke»*tH, is a method for graphically representing the inotionH It iH a vector mcthoi! of a kind flimilur to that uHed in determining the Htr«>fM«e« in bridgcH and roofH with the important differences that the vector u*»eer alTected, while the vector repr»wnting velooity is in many CHsett normal to the direction of the link concerned and further that the diagram is drawn on an arbitrarily selected link of reference which ia itself moving. 69. The Phorograph. — The phorograph is a construction by which the motions of all points on :i machine may be represented in a convenient graphical manner. As dis- cussetl here the only applica- tion made is to plane motion although the construction nmy ? 'adily be modified so as to make it apply to non-plane motion, but in most cases of the latter kind any graphical construction iM'comes compli(;ated. The method is biised on very few important ijrinciples and these will first be explained. 60. Relative Motion of Points in Bodies. First Principle. — Th(? first principle is that any one point in a rigid Inxly can move relatively to any other point in the sainc Iwdy only in a direc- tion at right angles to the straight line joining them; that is to say, if the whole body moves from one position to another, then the only motion which the one point has that the other has not is ir a direction normal to the line joining them. To illustrate this take the connecting rod of an engine, a part of whit li is .shown at a in J'ig. 29 and let the two points B and C, and hence the line BC, lie in the plane of the paper. Let the rod move from o to a', the line taking up the corresponding new position B'C During the motion above described C has moved to C and B Fui. 29. TIIH MOTKtN I) I A' {HAM gi to B'. Now draw BB,, parallel to CC and CB,, paralle' to BC then an inspection of the figtjre »hows at once that if the rod had only moved to B,C' the points li arul (' w,.„I.I hav.. ha.l exmtly the same ni^t motion, that is, one of tnin«hition through VL" » BBx in the same dircetion and sense, and hence B and C would have had no relative motion. But whe: the rod has moved to a B has had a further motion which C has not had, namely Thus during the motion of «, H h,« had only one motion not shared by t, ox B h.is n.ov.Hl relatively to C through the are B,B , and at each stage of the motion the direction of this are was evidently at right angles to the radius from C, or at right angles to the line joining B and C. Thus when « body has pUme motion any point in the body can move reUtively to any other point in the body only at right angles to the line joining the two points. It follows from this that If the hne joining fho two p«i„ts should he normal to th,- nIotL '"°*'°"' **"*'" *''"' ^^''' '"""*' '"""''' ''"'''' ""^ ""^'"^'^'^ 61. Second Principle. -I,et Fig. ;«) i<.p,,.s..,.t a mechanism having four links, a, h, c and ,/, join...! together l,y four turning Fic. 30 Pmrs O, P Q and Ji as indicated. This mechanism is selected l>ecause of its common application ami the reader will find't used in many complicated mechanisms. For example ttl hah the meehamsm used in the beam engine, when he links are somewhat differently proportioned, a being he crank 6 h7cr Meeting rod and c on.^half " the walking beam l" ^ e .hah" :hi:;s^;" ''- "--^ --'- '--^^^ ^- - -rr ;r; The second principle uix.n which the phorograph depends mav now be explained by illustrating with the above mectnlsm " "^ 52 THE THEORY OF MACHINES II In this mechanism the fixed link is d which will be briefly referred to as the frame. Thus and R are fixed bearings or permanent centers, while P and Q move in arcs of circles about O and R respectively. Choose one of the links a^ the link of reference, usually o or c will be most suit;*ole as they both have a permanent center while h has' not; the one actually selected is a. Imagine that to a there is attached an immense sheet of cardboard extending indefinitely in all directions from O, and for brevity the whole sheet will be referred to as a. A consideration of the matter will show that on the cardboard on the link a there are points having all conceivable motions and velocities in magnitude, direction and sense. Thus, if a circle be drawn on a with center at O, all points on the circle will have velocities of the same magnitude, but the direction and sense will be different; or if a vertical line be drawn through O, all points on this line will move in the same direction, that is, horizontal, those above O moving in opposite sense to those below and all I)oints having different velocities. If any point on a be selected, its velocity will depend on its distance from 0, the direction of its motion will be normal to the radial Une joining it to 0, and its sense will depend upon the relative positions of the \vA\\i and O on the radial line. The above statements are true whether a has constant angular velocity or not, and are »lso true even though O moves. From the foregoing it follows that it will be possible to find a point on a having the same motion as that of any point Q in any link or part of the machine, which motion it is desired to study; and thus to collect on a a set of points each representing the motion of a given point on the machine at the given instant. Since the [K)int8 above described are all on the link a, their relative motions will be easily determined, and this thorefon; affords a very direct method of comparing the velocities of tiie various points and links at a given instant. If the motion of a is known, as is frequently the case, then the motions of all points on a are known, and hence the motion of anj* point in the mechan- ism to which the determined point on a corresponds; whereas, if the motion of a is unknown, only the relative motions of the different points at the instant are known. A collection of points on a certain link, arbitrarily chosen as the link of reference, which points have the same motions as points on the mechanism to which they correspond, and about SE^ii- 1 i THE MOTION DIAGRAM 53 which infonnatioii is desired, is caUed the phorograph of the mecbanism, because it represente graphicaUy (vectorially) the relative motions of the diflferent points in the mechanism. 62. Third Principle.— The third point upon which this graph- ical method depends is that tho very construction of the mechan- ism supphes the information necessary for finding in a simple way the representative point on the reference link corresponding to a given point on the mechanism; this representative point may be conveniently called the unage of the actual point. Looking at the mechamsm of Fig. 30, and remembering the first principle as enunciated in Sec. 60. it is clear that if it is desired to study the motion and velocity of such a point as Q, then the mechanism gives the following information at once: 1. The motion of Q relative to P is normal to QP since P and Q are both in the link b, and as P is ulso a point in link a, G'^B C-^Jf Fkj. 31. the motion of Q~/> in a is normal to QP, or the motion of Q in b with regard to a point P in a is known. nu ^i"?^.'' ^^^ ^ P°'"^ '" ' *h« '»"^'«" ofQ^Ris normal to QK. But i2 ,s a point in the fixed hnk d and hence R is stationary as O IS, so that the motion oi Q ^ R i.s the same as the motion of. W ~ O. Hence the motion of Q with regard to a second point in a IS known. ^ As will now be shown these facts are sufficient to determine on a point Q in a having tho same motion ae n an I n *u 1 r Hnin« tK.o ,. 11 u J "lounn as V anU tlic method of doing this will now be demonstrated in a general way 63. Images of Pomts.-Let there be a body A', Fig. 31, con- taming two points E and F, and let K have plane motion of any r)4 THE THEORY OF MACIITXES I S nature whatsoever, the exact nature of its motion being at pres- ent unknown. On some other body there is a point G also movmg m the same plane as K; the location of G is unknown and the only mformation given about it is that its instantaneous motion relative to E is in the direction G - 1 and its motion relative to F is in the direction G - 2. It is required to find a point C on A- which has the same motion as G; the point G' is called the image of G. Referring to the first principle it is seen that the motion of uiiy point in A' - E is per,)eiidicular to the line joining this iK)int to A , for example the motion olF^E is perpendicular to FE. Now a iwint G' is to be found in A' having the same motion as G, and as the direction of motion of (/ - E is given, this gives at once the position of the line joining /;: t<. the required point; it must be xwt- P«'ndicular to G - 1 and pass through E. The point could not he ut H for instance, because then the direction of motion woul.l be perpendicular to HE, which is different to the specifiwl dirc,- tu.n G- 1. Thus (," li.-s on a line EG' perpendicular to G - 1 through E. Similarly it may be sl,own that G' must he on a line through t perpendicular to (/ - 2. and hence it must lie at the intersec- tion of the lines through E and F or at G' as shown in Fig. 31 Ihen G is a pomt on A having the same motion as G in some external body. 64. Possible Data.-A little consideration will show that it is not possible to assume .t random the sense or magnitude of the motions, but only the Iwo directions. The point G' could, how- ever, be found by a.ssuming the data in another form; for example If the angular velocity of K were known and also the magnitude direction and sense of motion uf G - E, G' could be locatetl, and then the motion of (;~/.' could be determined, the reader will readily see how this is done. In general the data is given in the form stated first, as will appear later. Now as to the information given, the ft. per second, where G'H is in feet Further, the motion otG^Emir. the sense G^ - 1 and the veloc- ity of G ~ £ IS A'G'.a, ft. per second. 66. AppUcatlon to Mechamsms.-The application of the alwvo pnncples to the solution of problems in machinery will illustrate' the method very well, and in doing this the principles upon which he construction depends should be carefully studied, and atten- tion paid to the fact that if too much is assumed the different Items may not be consistent. The sin.ple mechanism with four links and four turning pairs Will be again .selected as the first example, and is shown in Fig 32, the letters a, b, c, ./, O P Q and It having the same significance ^vs in former figures and a is chosen as the link of reference or more conveniently, the primary link, a rough outline being shown to md.cate its wide extent. In future this outline will be omitted "f />, of T m c and of Q, also the angular velocities of b and c com- sT:::;m:h:t.:e.^ '-'-'-- ^^ -«- ^^-«^ ^^^ -s, Points will first be found on « having the same motions as Q cIh K ^r"*t '''"^ ''^ ^^^^ "f e well again to remind the reader that the point Q' is a point on o but .aat Its motion is identical with that of Q at the junction of the links 6 and c. If the angular velocity of o is « radians per second then the hnear velocity of Q' on a is Q'O.o, ft. per second and its direction m space is perpendicular to Q'O, and from the sense of rotation shown on Fig. 32 it moves to the left. Since the motion of Q IS the same as that of Q then Q also moves to the left in the direction normal to Q'O and with the velocity Q'O.o) ft per second. Fi(}. 32. n ^^\^T^f ^ °^ Links.-Since P' and Q' are the images of P and . "i' ' f ? ."'""^ "" regarded as the image of h, and will in futuro l>e denoted by V; similarly R'Q'{OQ') will be denoted by c'. Bv a similar process of reasoning it may be shown that since S bisects ?V''' ^3 J i)^?* ^'^' ^"^ ^^^ ^' "^•^y be found from the relation R'T : T'Q' = RT : TQ. Since the latter point is of importance and of frequent occur- rence, It may be weU to prove the method of locating S' The direction of motion of .5 ~ P is clearly the same as that of Q ~ /> that 18, perpendicular to PQ or b, but the linear velocity of Q ~ P IS twice that of 5~P, both being on the same link and S biseoung PQ. But the motion of P' is the same as P and of C is he same a« Q; hence the motion of Q' .^ P' is exactly the same as that of y ~ P. go that the velocity of S' ~ P' is one-half that THE MOTION DIAGRAM 57 of Q' ~ P', or S' will lie on P'Q' and in the center of the latter line. 68. Angular Velocitie8.-The diagram may be put to further use in determining the angular velocities of h and c when that of a IS known or the relation between them when that of a is not known. Let a% and co„ respectively, denote the angular velocities of 6 and cm space, ti.e angular veLcity of the primary link a being a. radians per second. Now Q and P are on one link 6 and the motion of or b' Similarly b' X w Wc = w = c X • c ^ c 69. Image of Link Represents Its Angular Velocity.— The above discussion shows that if the angular velocity of a is con- stant then the lengths of the images b' and c' represent the angular velocities of the links 6 and c to the scale '^ and ^ respectivety, 8ince b and c are the same for ail positions of the inechanism. On the other hand, even though « is variable, at any instant '^'' = ^'. TewlV/T* '^"f '' ^, ^''■''* '""'*^°^ °f ««"i"8 the rdaticm between the angular velocities in such cases 70 Sense of Rotation of Links.-The diagram further shows he sense in which the various links are turning, and by the formulas for the angular velocities these are readily inferred. Thus (-» = ^0,, and starting at the point P, P'Q' = b' is drawn to the left and PQ = b to the right, hence the ratio ^ is negative, or the link 6 is at the instant turning in opposite sense to a or m a clockwise sense. In the case of the link c the lines R'Q' and 58 THE TUKOHY OF MACHIXKH 9 V. RQ are drawn upward from R and R', that is ^ is positive and hence a and c are turning in the same sense. 71. Phorograph a Vector Diagram —The figure OFQ'R' is evidently a vector diagram for the mechanism, the distance of any pomt on this diagram from the pole being a measure of the velocity of tho corresponding point in the mechanism. The direction of the motion in space is normal to the line joining the image of the point to (), and the sense of the motion is known from the sense of rotation of the primary link. Further, the lengths of the sides of this vector diagram, b'(P'Q'), c'('r'Q') and d'iOR') are measures of the angular velocities of these links the sense of motion being determined as explained. As d is at rest, OR' has no length. In Fig. 33 other positions and proportions of a similar mechan- ism are shown, in which the solution is given and the results will be as follows: Fio. 33. b' At (1) the ratio ^ is positive as is also ^^ or all links are turning in the same sense; at (2) the link a is parallel to c and hence Q' and P' lie at P, so that 6' = or «» = *'« = o, that i.s, at this in- stant the link 6 has no angular velocity and is either at rest or has a motion of translation. Evidently it is not at rest since the velocity of all points on it are not zero but are OP.tj = au ft per second. As shown at (3) the links a and 6 are in one straight hne^and in the phorograph Q' and R' both lie at 0,so that Q'R' = c' = 0, and hence «. = 0, in which case the link c is for the instant at rest, since both Q' and R' are at O, the only point at rest in the figure. At (4) both the ratios |' and '^' are negative so that b and c both turn in opposite sense to a and therefore in the same sense as one another. At (.5) the parallelogram used commonly on the side rods of locomotives, is shown and the THE MOTION DIAGRAM 59 phorograph shows that Q' and P' coincide with P so that 6' = Th!!' "'^h 'f'^ i**^ *n"l°''°" °^ translation, as is well known. There is the further well-known conclusion that since c' = c the links a and c turn with the same angular velocity. It IS to be noted that if the image of any link reduces to a Single pomt two explanations are possible : (a) if this ,x)int falls at U the link is stationary for the instant as for d in the former not at O, the inference is that all points in the link move in the same way or the link has a motion of translation at the instant, as for the hnk h in (2). ' JJ'%'"!!u°'^ J'" """^ ^^ employed in a few typical cises. 72. Further Eiounple.-The mechanism shown in Fig. 34 is a little more complicated than the previous ones. Here /" Q' Fiu. 34. andii!'arefoumlaslK,fore,andsincethemotionsof6'~OandS~P are perpendicular respectively to SQ and SP, therefore S'P' and ^ Q are drawn parallel respectively to .S'P and -SO th.„ KT.TQi^c^t). Nextsincethemotionsoft'~.Sand6^~raro given, draw ^'f/' parallel to SU and U'T parallel loVT, fhc ntersection locating U'. Assuming a to turn at angular veb ny « m the sense shown, then the angular velocity of /r Is Slj « in the same sense as a and that of (/r is ^ co in opiK,8it.. sense^to a (Sec. 70). The linear velocity of U is OU'.. ft. pe. 73. Image is Exact Copy of Link.-There is an important point which should be emphasized here and it is musSS in t'on of i to P Q IS the same as that of S to PQ, or the image 'r:^SIiiy(i-h: ^^iSm^Smi^SS^^S^ 60 THE THEORY OF MACHINES I of the link 18 an exact copy of the Unk itaelf. and although it may be inverted and is usually of different size, al! Unes on the image are parallel with the corresponding lines on the mechanism. Whenever the image of the link is inverted it simply means that the link, of which this is the image, is turning at the given in- slant in the opposite sense to the link of reference; if the image is the same size as the original link, then the link has the same angular velocity as the link of rofonMurc. 74. Valve Gear.— The mechanism shown in Fig. 35 is very commonly used by some engine builders for operating the slide valve, OP being the eccentric, RS the rocker arm pivoted to the frame at R, ST the valve rod and T the end of the valve stem which has a motion of sliding. OP has been selected as the link of reference and P', Q', R' and S' are found as before Fio. 35.— Valve gear. t^elZ' r'""°". ^r""^ ^ ^ "^°^" horizontally in space, and tZ %l T^^- ^ir * """ **''"°"«** ^ perpendicular ti the mo ion of r, that is r is on a vertical line through O, and further J'n °" * ^"'' *^''°"8*^ ^' P"a"el to ST, which fixes T Ja Tlu\ ?u^ "'^I^^tions given regarding former cases, it is ev dent that the velocity of the valve is OT'.o, ft. per second, « being the angular velocity of OP. While the other velocitie^ are not of much importance yet the figure gives the angular velocity of ST as -^j,-« in the opposite sense to a and the linear velocity of S is greater than that of Tin the ratio OS'OT Vil\aT^ Engine.-'l he steam engine mechanism is shown in iig. 6b, (a) where the piston direction passes through O and (b) where it passes above with tiic cylinder offset, but the same I THE MOTION DIAGRAM 61 letters and description will apply to both. Evidently Q' lies on P'Q" through P', parallel to PQ, that is, on QP produced, and also since the motion of Q in space is horizontal, Q' will lie on the vertical through 0. The velocity of the piston is OQ'.u in both cases and the angular velocity of the connecting rod is ^ « in the opposite sense to that of the crank, since P'Q' lies to the left of P' while Pk, lies to the right of P, and it is interesting to note that in both cases when the crank is to the left of the vertical line through 0, the crank and rod turn in the same sense; further that the rod is not turn- '^^•w l' $ ^^■^eing similarlv treated. The scale will then Xm OC = W\co ft. per second or ^u : 1. The points < ', /.', F', //', /;' and ./' are readil v located. Further choose M on the curved link A(W directly below A' on the rocker arm U)K and d,aw lines F'A', F'li', H'G' and D'K', of unknown lengths but parallel respectively to EA, FB, HG and DK It has already been seen (Sec. 73) that the image of each link is snnilar to and similarly divided to the link itself (it is, in fact a phot,.graphic image of it); hence the link AGB must have Ln image sinnlar to it, that is, the (imaginary) straight lines ^'6" and G'B' must l)e parallel to ir/ and GB and the triangle A'G'B' must be similar to AGB. But the lines oi whic^h A', G' and B' lie are known; h.'nce the problem is ..impl> the geometrical one of drawmg a triangle A'G'B' similar an.i parallel to AGB with its vertices on three known lines. The reader may easily invent a geometrical method of doing this with very little efTort, the pro- cess Wm^ as simple as the one shown in Fig. ;i7, but the con- struction is not shown F^ecause the figure is already complicated Having now located the points A', G' and B' the curved link .1 G B may l)e made by copying AGB on an enlarged scale and on -^^E^^m ^a.^nsf': 64 II ft I. THE TIIKOHY OF MACHINES It the point M ,n.y » located m.niUrly to M in the actual link. For the purpo-e of illustrating the problem the image of the curved Imk ha. been drawn in on the flgure although this is not then A M ,a .Irawn normal to the curved link at M' which L/D'*D'K^ '^' '" '""'"^ '"""'" ''**" '*''*^''"* ^''^ '■ ^^ " The construction shows that the curved link is turning in the tamo sense as the crank with angtdar velocity }>i ■ ^j^-' „ ginco the scale is such that OE' == 20E. again. :he valve is moving to »— ch B«4 d'h'j' Fni. 39.~8tephen!)on link niutiun the right at the velocity represented by QU, and further the Sl-'V both"V!H '^ ''"'V" ^'^ ^"'"^^ '"'^ •« -1— 't-' oy A M , both of these on the same scale that OC = 2 Of represents the linear velocity of the crankpin. The n.ethod gives a very d.reet means of studying the whole link n.otion. Reevefr''" I' ^'"f-The Corliss valve gear used on the Keeves engine is shown d.agrammatically in Fig 40, there being necerry-O Tail's "^T'T ''"^' ""^*^ ^^^^""^^-" '^ necessary O, H and 6 are fixed in space, S being the end of the rocking valve stem; hence R' and S' are at 0, and the link OP I driven direct from the crankshaft through the e^-centric conne " tnsMesj ^^na^j^p^fsa^Mmm^smg^vi^^^t^f^m^^assB 77/ A" MOTIUX DIMIHAM 06 pp ltc«»iftc I 0«BU«CliOB Fi(i. 40. — R»»evP8 valve K^ar. Fia. 41. fid TIIK rilKOnv OF MACHINES ^on. The point Q on the shding block e is directly over a mova Wo pomt T on the lever / which le^er is keyed to the valveTem T .8 located by drawing Q'T perpendicular to S'T or to // ll this fK>sition the angular velocity of the valve is ^J tin,es that ^f^^^Cnr ' '''' ''"' ''' '-- -^-'y «^ ^'^ valve Of.!!; ^°^ ^"f^^f ^^"•-''^his chapter will be concluded bv one other exa„.ple here, although the n.ethod has been used a Jod deal throughout the book and other examples appear tter" , The example chosen is the Joy valve gear, shown in Fig 4? this gear havmg been largely used for locomotives and rete'r^n" rod Tth. TJ '° *'"' '«""' ° •« ''^' --'J^' b the connec ng link but may be inserted rehdve to it The toaJe m., be *! -B icu» ai me unk. If this statement is kept in i I I THK MOTION DIAGRAM 67 mind, it will great?, aid in the solution of prohlenis and the iinderstandinR of the method. .If ^ # 1 QUESTIONS ON CHAPTER IV L Prove that any point in a body can only move relatively t„ anv other point, m a direction perpendicular to the line joininf? them. 2. Define the phorograph and state the principles involved 3. A body a rotates about a fixed center O; show that all points in it have (1 lerent velocities, either in magnitude, sense or direction 4. Show that the phorograph is a vector diagram. What quantities may be determined directly by vectors from it? 5. If the image of a link is equal in length to the link and in the same sense what is the conclusion? What would it be if the image was a point? 6. In the mechanism (3) of Fig. 9, lot d turn nt constant speed, as in the Gnome motor; find the phorograph and the angi-lar velocitv of the rod 6 7. In an engme of 30 in. stroke the connecting ro'« pressed'togethe^ Itels cl led " "- V'*'""^ ^" '^"^^«' «'•' ^g^'"' toothed ^hoe s called gear wheels may te used on the two shafts, as in street cars and m inost automobiles. Any of th..se n.othods is possible in a few cases, but usuallv h n.ethods the n.ore preferable. If the shafts are far apart a be and pulleys may be used, but as the drive is not positive the belt may shp, and thus the relative s,>eeds may change, the speed oters of the pulleys would indicate. Where the shafts are fairiy ose together a belt does not work with satisfaction and then a Cham and sprockets are sou.etimes used whic-h cannot slip, an hence the speed ratio r,-quire-. -- of the most in.porta bemg the distar.ce apart of the shafts concerned in it, another bemg the question as to wh,>ther the velocity is t.) be ai-curat .fy Z ;: IrSmuT ^^ ""^'"'''"''' ^"" ="'"^^- ••-« *'^^^-- 82. Spur Gearing.-The di.scussion here deals only with drives between shafts which must turn with an exact velocity ratio which 68 TOOTHED GEARING eg must be known at any instant, and they are generally used when the shafts are fairly close together. It will be convenient to deal firet with parallel shafts, which turn in opposite senses, the gear wheels connecting which are called spur wheels, the larger one being commonly known as the gear, and the smaller one as the pimon. Kmenmtically, spur gears are the exact equivalent of a pair of smooth round wheels of the same mean diameter, and which are pressed together so as to drive one another by friction Ihus If two shafts 15 in. apart are to rotate at !()() revolutions per minute and 200 rcv(,lutions per minute, respectively, thev n.ay be connected by two smooth wheels 10 in and 20 in. in diameter one on each shaft, which are pressed together so that they will not shp, or by a pair of spur wheels of the same mean diameter, both methods producing the desired results. But if the power to be transmitted is groat the friction whe-ls are inadmissible on ac- count of the great pressure between them nece.ssarv to prevent 8hpp,ng. If slipping occurs the velocity ratio is variable, and such an arrangement would be of no value in such a driv,- as is used on a street car, for instance, on account of the jerkv motion It would produce in the latter. 83 Sizes of Gears.- In order to begin t!,.. problem in the simplest possible way con..ider first the very common .as." of a pair of spur pears connecting two shafts which are to have a con- stant velocity ratio. This is, the ratio between the spee.ls ,,. and n, is t^ be constant at every instant that the shafts are re- volving ,.et / be the distance from center to center of the shafts, hen If friction wheels were used, the velocities at their rims diame ers o the wheels in inches, and it will be clear that the ^locty of the run of each will be the same since there is to be no supping. riVi = rjwj where r, and r, are the radii. '■i + ro = I Tlierefore since and But Hence n. '•2 + r2 = / or and ri = '12 "1 + "a m. 70 THK THKORY OF MACHINES &i--' I t W whatever actual sha{K^ is give., of these wheels, the motion oi the shafts n.ust be the same as if two smooth wheels, of sizes as determined above, rolled together with.,ut slippine I„ other words, whatever shape the wheels a.^tuallv have the re- sultiiiR motion must be equivalent to that obtain.>d bv the roll- ing together of two cylinders centered on the shafts." In gear wheels thes.- cylinders an- calhul pitch cyUnders, and their pro- jections on a plane normal to their axes, pitch circles, and the circles evidently, touch on a line joining their- cente,-s, which point IS called the pitch point. 84. Proper OutUnes of Bodies in Contact. -I.(>t a small part ot the actual outline of each wheel be as shown in the hatched lin(>s of iMg. 42, the projections on the wheels being i-e(,uin'd to I'l.i. 4_'. prevent slippinK of the pit.-h lines. It is requi.-ed t,. tin.l th.- necessary shape which thes.. pn.jections must hav<'. f-et the actual outlines „f il,.- -.v,, wheels touch at /' and let i.e joined to th<. pit.-h point C; it has been aheadv expl:uned that there inust be no .slipping «.f th.- circK-s at C. Now /> is a point in both whcvls, and as a pomt in the gear /, it ,noves with regard to r on the pinjoti a at right angh-s to PC, whil<> as -i point on tl„. ,,.„ion a it ,„ov<"s with regard t.. the gear /. also at right angles to /T. Whether, therefore, /' is cth.T the direetioi. of sliding m,^* ai > . s l>e along the eoninu.n tungent to their .su,-f:u....s .t th. ^-.t ^- .ontact, that IS, the direction of sliding must in this ..is, '„ , /'■ But /', IS the point „f c.i.iuet and is therefore ., ixm- .v :u-h wheel' and the moli.,,, „f tl„. two wheels must be the ^nth pit(h cireh's rolled tog(>ther, having contact at the case, if the two projections shown are pla* . f I .,r the direction of motion at their ])()int of conM.^ IKTpendicular to 1\C. whereas here it is i,erpendi.-u!ar I his would cause slipping at C, and would gu ■!„ pr^^^-r shape for pitch ciivles of radii AC and Ii( \ which w^.iid .■,„■. spond to a different velocity ratio. Thus r' shouhi he at ' nui / 1/ should be normal to P,r. Another method of dealing with this matter is !.v meu.s of the virtual center. Calling the frame which supports the Ix^ar- uigs of a and b, the link ,/, then .1 is the center a.. 4«.— Cycloi.Ijil tt-eth. ing. the o.hor h,U.. ... ,1.,. ...oU, on the wheel .(', .nav Ik, pricked hrough w,th u n..,.,II,.. The sun.e n.e.hod n.av be enfpbved for the ic-th on wheel /rA. "'piox.u The method of .Irawingthese curveson the drafting board is not .liffin.lt, and .nay he de^erihed. Le, r . . . ., in Fi^. 44 repre- ...^ one c^ the pMeh ,...,es and the snudi.-r ein-les th 'describing rd . Chonding chords. Dra.. radial hne« from ^ as shown, and locate points G',/V,A-,L at TOOTH Kb OKAHlmi 75 distance from the pitch cirHe equal to the radius of the describ- u.g oarcle. and frouj th,.. ,Kun,H ,,ruw in a number of dr e« I>omt8 C M N li, S. which are points .n tiie desired epicycloidal "a7be dZn ' ' ' ''^'^^^^'-d'^' «-ve below the pitch Xt . f fh.^^' 0' Describing Circle-Nothing has so far been said of the sizes ,.f the .lescribing oircles, a.ui, indeed, it is evident hat any size of d..cnhi,.« ..,>,.,e, ., long .« it is somewhat small" than the pitch cnc-le. may be ,ise,|, and will produce a curve ful' Fio. 45. Hlliiig the desired co.ulitions, but it may be shown that when the .l^'scnbinK .urcle i« one-half the diameter of the corre pond „' ucle, and for reasons to be explained later this is undesirable Ihe n.aximu,n sue of the describing circle is for tl^r e!son Cs:t :?"'"' ^'^' "' '''' corresponding pitch ;;:c.ca:d h.n a set of g,.us are to run together, the describing circle for .og;,,;::" '"•'''' "'•^- ^^^^ ^^"-^'^^ -^ *»- -- ^^ -rk prop^nly ci^ie^;;^ Jh'' '1^^ ''^'T^"'"'^ '^ '' ^^^^''^' '- 'f the describing iho 1 ' ^ V'''''' "'•^■'" '"'^' ^''^^' ^J- <»"«'-ribing circle P JnJ 1 he arc Bt is equal to the arc PC by construction, and hence u ii 76 57/ A' TIIEOHY OF MACUISKS tho angle PEC u( thr ccriU'r A' „f DPC in twice the angle BDC ^(•ause the ra.lius i„ the latter case is twice that in the former! But the angles HOC and PEC are lK)th in the smaller circle the one at the circumference .md the other at the center, and mnee the latter is double the former they must stand on the same un- / ( In other words BP is a radial line in the larger circle smi e DP and DB must coincide. 89. Teeth of Wheels.— In the actual gear the tooth profiles are not very long, l.ut are limited between two ciroU-s concentric with the pitch circle in e:uh gear, and called the addendum and root circles resp<.ctively, for the tops and bottoms of the teeth, the distances between these circles and the pitch circle tmng quite arbitrarily chosen b> the manufacturers, although cer- tain proportions, as given later, have Ix-en generally adopte.l. Ihese circles are shown on Fig. 46 and they limit the path of oontatt to the reverse,! curve PCP, and the amount of slipping of each pair of teeth to PR - PD + P,E - P^F = PI{ + P,E - {PD + 1\F), the distances being measured along the profiles of ' ho teeth in all eas(>s. Fun her, since the common normal to the ^(■eth always i)asses through C, then the direction of pressure between a given pair of teeth is always along the line joining their point of contact to C, friction l,eing neglected, the limiting direc- tions of this line of pressure thus being PC and P,C. TOOTH EU GEARING 77 The arc PC is called the arc of approach, being the locus of the point, of contact down to the pitch point C, while the arc CPj ui the arc of receaa, P, being the last point of contact. Sim- ilarly, the angles DAC and CAE arc called the angle of approach and angle of recess, roflptHtivciy, for the left-hand gear. The reversed curve PCP, in the arc of contact and its length depends to some extent on the size of the doscrihinK circles among other things, being longer hm the r(>lative size of the describing circle increases. If this arc of contact is shorter than the distance be- tween the centers of two adjaeent teeth on the one gear, then only one pair of teeth can Ik> in eontaet at once and the running is uneven and unsatisfaotory, while if this arc is just equal to the dis ance Jn-tween the centers of a given pair of teeth on one gear or he circular pitch, as it is called (see Fig. o2), one pair of teeth will just be going out of contact as the second pair is coming in, which will also cause jarring. It is usual to make PCP, at least 1 .5 times the pitch of the teeth. This will, of course, increase the amount of slipping of the teeth. With the usual proportions it is fo.u.d that when the number of teeth in a wheel is less than 12 the teeth arc not well shaped for strength or wear, and hen.e, although thev will fulfil the kinematic conditions, they are not to be recommende,! in pra<.tiee I»»T°»»t« Teeth.-Th.. second a„,l perhaps the most com- nH,n method of forming the curves for gear teeth is by nwans of ntvolu e curves. I^t .4 and B, Fig. 47, represent the axes of the gears, the pitch circles of which tou.-h at C, and through C draw a secant Z)f K at any angle 6 to the normal to ^fl, and with centers A and H draw circles to touch the secant in D and E Now (Sec. K.'i Wi liC AC HI) - ^^., .so that the new circles have the «a.„e ,s,K.<..l ratios as the original pitch circles. It then a string 18 run from one dotte,! circle to the other and us.hI as a belt lHoint P on the l>clt DE and attach at this pomt a iK^nc.l, and as the wheels revolve it will evidently mark nn tlu- ongmal wheels having centc-rs at A and li, two curves I a and Pb respectively, a being reached when the pencil gets down to E, and b btung the starting point just as the pencil leaves MICROCOPr RESmUTION TEST CHAUT (ANSI and ISO TEST CHART No. 2) 1^ 1^ A APPLIED IM/1GE '653 EosI Main StrMl Rochester. New York 14609 USA (716) «82 - 0300- Ptone (716) 288- 5989 -Fox 78 It i i 1 ^ THh THEORY OF MACHINES A and since the point P traces the curves simultaneously thev ^ncUa^P .' ^'p '^ '""''^' ^'•""^^'•"^ ^«— rd with (.e pencl at P. Since P can only have a motion with regard to the Fi«. 47. — Involuto teoUi. Wheel aE normal to the string PE, and i»s motion with re,.ard to the wheel Db is at right angles to PD, it will bo ^ at once evident Fig. 48.— Involute teeth, that these two curves have a common nonnal at the point where (Sec 85 ""■'' ""'' '' "^'^ ^ ^'^« P-fi'- «f g«ar teeth TOOTHED GEARIXa 79 The method of describing these curves on the drafting board IS as follows: Draw the base circle 6-5 with center b Fig 48 and lay off the short arcs fc - 1, 1 - 2, 2 - 3, 3 - 4, etc., all of equal length and so short that the arc may be regarded as equal in length t9 the chord. Draw the rad-al lines Db,B-l B-2 etc and the tangents bD, 1 -£, 2-F, S-G, 4~H any 'length, and lay off 4 - i/ = arc 5-4,3- (/ = urc 3 - 5 which equals twice arc 5-4, 2-F equal three times arc 5-4, l-E equal four times arc 0-4, etc.; then 1), K, F, G, H and 5 are all points on the desired curve and the latter may now be drawn in and extended, if desired, by choosing more points below h. 91. Involute Curves.— The curves Pa and Pb, Fig. 47, are called mvolute curves, and when they are used as the profiles of gear teeth the latter are involute teeth. The angle (? is the angle of obUquity, and evidently gives the direction of pressure between the teeth, so that the smaller this angle becomes the less will be the pressure between the teeth for a given amount of power trans- nutted. If, on the other hand, this angle is unduly small, the base circles approach so nearly to the pitch circles in size that the curves Pa and Pb have very short lengths below the pitch circles Many firms adopt for the angle 14^°, in which case the diameter of the base circle is 0.968 (about, m^) that of the pitch circle. If the teeth are to be extended inside the base circles, as is usual the inner part is made radial. With teeth of this form the dis-' tance between the centers A and B may be somewhat increased without affecting in any way the regularity of the motion Recently some makers of gears for automobile work have in- creased the angle of obliquity to 20°, in this way making the teeth much broader and stronger. Stub teeth to be discussed later are frequently made in this way, largely for use on auto- ""•biles. A discussion of the forms of teeth appears in a lat(.r section. 92. Sets of Wheels with Involute Teeth.— Gears with involute teeth are now in very common use, and if a set of these is to be made, any two of which are capable of working together, then all must have the same angle of obliquity. The arc of contact IS usually about twice the ci. jular pitch and the number of teeth in a pinion should not be less than 12 as the teeth are liable to bo weak at the root unless the angle of obliquity is increased. A more complete drawing of a pair of gears having involute teeth IS shown in Fig. 49. Taking the upper gear as the driver. 80 i I I I s - If • I 1 THE THEORY OF MACHINES the line of contact will be along DPCE, but the addendum circles usuaUy Umit the length of this contact to some extent, contac takmg place only on the part of the obliquity line DE inside the addendum hne. The larger the addendum circles the longer the hnes of contact will be and the proportions are such inTg 49 that contact occurs along the entire lino DE. No contacts can fxjssibly occur inside the base circlos. 7Btarr>raDC* Lisa Upper Oeu -iDterforeoca Line Lomr 0«*r Fi(!. 49.— Involute toetli. 93. Racks.-When the radius of one of the gears becomes mfimely large the pitch line of it becomes a straight line tangl" The teeth ot the rack m the cycloidal system are made in exactW the same manner as those of an ordinary gear, but both the de^ smbmg circles roll along a straight pitch line, generating cycloiil curves, havmg the same properties as those on the ordinal For the involute system the teeth on the rack simply have . teeth formmg the angle e with the radius line AC drawn from the center of the pinion to the pitch point 94^ Annular or Internal Gears.-In all cases already discusesd fnZT ^""'' ^°f ^"« *"«^*^^^ ^^^« been assumed to turnin EentlT'' '•^^"^!;"«/" *he use of spur gears, but it not in- frequently happens that it is desired to have the two turn in the teeth ms de the nm and is called an annular or internal gear evide'tltT ""t "'*' ' ^P"^ P*"^""' ^"^ '^ -'" be self: th^Jhe pfnion '"" " '''' ""^* ^'^^^^ '^ ^^^^^at larger A small part of annular gears both on the cycloidal and in- volute systems is shown at Fig. 50 and the odd appearance o^ the i! ^t TOOTHED GEARING 81 involute internal gear teeth is evident; such gears are frequently avoided by the use of an extra spur gear. ^.^^^ Creloidal Teeth Involute Teeth Fio. 50. — Internal gears. 96. Interference.— The previous discussion deals with the cor- rect theoretical form of tee^h required to give a uniform velocity Fig. 51. — Interference. Ik*'°'^^"*7'**' *^^ "^"''' proportions adopted in practice for the addendum, pitch and root circles, it is found that in certain I'i I 82 I 1 I I i t i Is I S i •:: I I' i i i THE THEORY OF MACHINES cases parts of teeth on one of the gears would cut into the teeth on the other gear, causing Interference. This is most common .> 1! nf";r ' ''''*"" ^"^ °'""™ "™°«* ^^^'^ the difference wor t wh ^'^",/" ''^"*^'* •' ^''^^^^t' **>"« interference is worst where a small pinion and a rack work together, but it •nay occur, to some extent with all sizes of gears An example will make this more clear. The drawing in Fig 51 represents one of the smaller pinions geared with a rack in the nvdute system and it is readily seen that the point of the rack order that the pair may work together it will be necessary either to cut away the bottom of the pinion tooth or the top of the (Se^c^'tm on * '^?f"?' ^"^/^.'"^'"bering the former discussion t£ linP if Vr^^ ', '"'**'' '' '' ''''' '^^' «^"*««t ^i" be along nroJuced h '"'*'^ ^'"'" '^ '^ ^ '^"^ ^''^^ ^^'^ °" t^is line cl produced have no meamng in this rogard. so that if BC denote the pitch line of the rack, the teeth of the rack can only be use- lT!l I, * u .' *^^ P*"'* ^'^ ^'^^ '•^''^ *<^«*b between ^Z) and f G s"m swt ;.? " T^"*' °" *'^ ''^'' -"-* be made the same shape a the involute would require but must be modified n order t^ clear the teeth of the pinion. The usual practice s to modify the teeth on the rack, leaving the lower parts of th" te LeTh of'tr'^r^'^"'''' '^"^ *^^ ^'^'^^ «^-« dotted how Interference will occur where the point E, Fig. 51 (a) falls below the addendum lino KG, the one tooth cutting int' the ot "er a^^ onthemeofoblic,.ity. Where a pinion mosL with a g a wh l^ •s not too large, then the curvature of the addendum fine Jtho Koar may be sufficient to prevent contact at the point ^ in w« ase mterforenoe will not ocvur. As has already been explained n^er^renoe occurs most when a pinion, meshes with a get ^^h,ohls very much larger, or with a rack. Where a 1™ gear meshes with a rack as in the diagram at Fig 51 (6) X interference hne DE is above the addendum line FG and hence no modification is necessary. add"ndum'r^'' '"^7^"^";^ ""^ f"'- the lower gear is inside the adduuhim hne and hence the points of these teeth must be cut TOOTHED GEARING S3 away, but the points of the teeth on the upper gea. would be correct as the interference line for it coincides with its adden- dum line. 96. Methods of Making Gears.— Gear wheels are made in various ways, such as casting from a solid pattern, or from a pattern on a moulding machine containing only a few teeth, neither of which processes give the most accurate form of tooth I he only method which has been devised of making the teeth with great accuracy is by cutting them from the solid casting and the present discussion deals only with cut teeth. In order to produce these, a casting or forging is first accurately turned o the outside diameter of the teeth, that is to the diameter of the addendum line, and the metal forming the spaces between the teeth IS then carefully cut out by machine, leaving accurately formed teeth if the work is well done. Space does not permit the discussion of the machinery for doing this class of work for various principles are used in them and a number of makeL of the machines will produce theoretically correct tooth outlines The reader will be able to secure information from the builders of these machines himself. Bggy Fia. .52. eith!; ^ Teeth.-The various terms applied to gear teeth, i • TK ^'^/^^^^^t^ ^' ^y^'Joidal form, will appear from Fig. 5^. The addendum line is the circle whose diameter is that of the outsme of the gear, the dedendum line is a circle indicating he depth to which the tooth on the other gear e.xtends; usually the addendum and dedendum lines are equidistant from the wV K r ?^ '""'^ "'"^"y '*'■« «"t ^^^-ay to the root circle whKih ,s shghly inside the dedendum circle to allow for some Clearance, so that the total depth of the teeth somewhat exceeds 84 TIfK TIIKOliY OF MACIIIXES the workiiiK d<'|)th or distance iHJtween the addendum and de- denduni circles. The dimension or length of the tooth purailel to the shaft is the width of face of the gear, or often only the face of the gear, while the face of the tooth is the surface of the latter above the pitch line and the flank of the tooth is the surface of the tooth below the pitch line. The solid part of the tooth outside the pitch line is the point and the solid part below the pitch line is the root Two systems of designating cut teeth arc now in use, the one most commonly used l)eing by Brown and Sharpe and it will first be described. Let d be the pitch diameter of a gear having t teeth, A, the depth of the tooth between pitch and addendum circles, and h, the depth below the pitch circle, so that the whole depth of the tooth is h - hr + hi, while the working depth is 2hx. The distance measured along the circumference of the pitch circle from center to center of teeth is called the circumferential or circuUr pitch which is denoted by p and it is evident that pt = rd. In the case of cut teeth the width W of the tooth and also of the space along the pitch circle are equal, that is, the width of the tooth measured around the circumferenc(> of the pitch circle is equal to one-half the circular pitch. The statements in the present paragraph are true for all systems. 98. System of Teeth Used by Brown and Sharpe.— Brown and Sharpe have used very largely the term diametral pitch which is defined as the number of teeth divided by the diameter in inches of the pitch circle, and the diametral pitches have been largely confined to whole numbers though some fractional numbers have been introduced. Thus a gear of 5 diametral pitch means one in which the number of teeth is five times the pitch diameter in inches, that is such a gear having a pitch diameter of 4 in. would have 20 teeth. Denoting the diametral pitch by q then g = - and from this it follows that pg = t or the product of the diame- tral and circular pitches is 3.1416 always. The circular pitch IS a number of inches, the diametral pitch is not. The standard angle of obliquity used by Brown and Sharpe is U}4° and further />,= in.,hi V 20 + 9n in» clearance = { t 20 in., and the width W of the tooth is ^ so that there is no side clearance or back lash between the sides of the teeth. TOOTH Kn (iKAUIXd 85 99. Stub Tooth System.— Rec^ently M.o v.-ry great use of goiira for automobiles and the severe sorvicp to which thest' gears have Ijcen put has caused manufacturers to introduce what is often called the "Stub Tooth" system in which the teeth are not proportioned as adopted by Brown and Shari)e. Stub teeth are made on the involute system with an obliquity of 20° usually, and are not cut as deep as the teeth already dps(!ribed. The di- mensions of the teeth are designated by a fraction, the numerator of which indicates the diametral pitch used, wliile tlic denomina- tor shows the depth of tooth above the pitch line. \-^>j goar is one of ,5 diametral pitch and having a tootli of deptli A, = H ill. above the pitch line (in the Brown and S'larpe system hi would be ^-^ in. for the same gear). The usual pitches with stub tooth gears arc 4< 5' ej 7,^ 8 ' /il, Via and 1^14. Some httle difference of opinion Hj)|,ears to exist with regard to the clearance between the tops and roots of the teeth, the Fellows fJear Shaper Co. making the cl(>ar- ance equal to one-quarter of the depth //,. Thus, a ')j gear would have the same A, as is used in the Brown and Sharp<' system for a 7 diametral pitch gear, that is h^, = 0.142«> in and a clearance equal to 0.25 X if = 0.0357 in., which is much greater than the 0.0224 in. which would be used in the Brown and Sharpe system on a 7 pitch gear. 100. The Module.— In addition to the methods already ex- plamed of indicating the size of gear teeth, by means of the cir- cular and diametral pitch, the module has also to some extent been adopted, more especially where the metric system of meas- urement 18 in use. The module is the number of inches of diameter per tooth, and thus corresponds with the circular pitch, or number of inches of circumferences per tooth, and is clearly the reciprocal of the diametral pitch. Using the symbol m for the module the three numbers indicating the pitch are related as follows: Hi VI : also 7 = ' TO q ' m p The module is rarely expressed in other units than millimeters. 101. Examples.— A few illustrations will make the use of the formulas clear, and before working these it is necessary to re- member that any pair of gears working together must have the same pitch ..nd a set of wheels constructed so that any two may m M 11 f I I 8 t I I 77//-; ruhjoifv OF .yr.u'iifxhs n and ., ro.s,K.c.t,vol.v ,uui lot (hoso .„• p|....,, „„ si." i Zt -1 n,rn.nK at, ». ,u,.l „, rovoluti..,. .k. uU. T j Zn Nv. 8J, vvJuTi' spur RearH only aro used. iin, ,, ,.., ,„.. .^tsule diamc^c^^ the gears are 12^ in. and 01, i„.. ,he tooth 20 = 0.0393 in. The module w.Mdd be ., in. clearances = — _ ^JiV ~ 24"T^ith' If the gears have stub teeth of four-fifths size t h«n * i, of teeth will be 48 an r1 9.1 .. » t f °', ^^^^' <"^en the numbers th«t th I , , ^'' ''''^°'"^' •'"* ^1 ^"" be !-< = 2 in so hat he outside diameters will be 12.4 in. and 6.4 in resnec ivdv on th;B:or:S'Sh::;^:;s:::: ^"^^ ^-^^^^ ^"- ^- ^^^ ^-^ of a3ttrthe"n-f !;' r""" ""^' ^'^ "" *^^ -*«'de diameter verj aesirable ihat the reader become so familiar with fhp proportions as to be able to know instantlv ih. ITl u ! T(M)Tl/KI) ({KAHINd 87 102. Diicufdon of th« Gear System8.-The involute form of tooth M now more generally used than the cycloidal form In the firHt place the profile is a single curve instead of the double one re luired with the cycloidul shape. Again, Ix^cause of its construe tion, It IS possible to separate the centers of involute gears without causing any uiu^venness of r.inning, that Is, if the gears are de- signed for shafts at certain distance apart this distance may be Hh;«hMy incr.'us..rofore the loss tho frictional loss. But this is also accompanied bv a docroaso in f ho arc of contact and hence the number of teeth in contact at nnv one time will, for a given pitch, be decreased, which mav caus'e unevenness in the motion. If, however, the whole width of tho fTu u T^"'^^ "'^''^ "P °^ * '°* °f th'» ^iscs, and if, after the eeth had been cut across all the discs at once, they were then slightly twisted relatively to one another, then the whole width Of the gear would be made up of a series of steps and if these 88 THE THKOKY OF MACHINES Ht,.,« wrro made small enough the teeth would run aorc«, .he « ftelical gear. Ihe advantage „f «uch gears mil appear very of the face of a tooth at one inntant, the tooth will only gradual y .on.e .nto contact the action In^ginning at one end and woriing graclua ly over to the other, and in this way very groat eve^sj The T'Tir'' "?*' «'"^* ^'^^^ of consTd e'rahle S The profile of the teeth of such gea« is n.ade the san e, on a Fi t" '•»""<'• S' "Stll I;!,''.!*"'' /!•""'**'" "' "'" *"«''• "' "'^"""'^y "" involute teeth'.' 11 ^» "« the relative merits of cyckmlal and involute teeth? 14. Obtain all the dimensions of the following gears: (a) Two smir wheels, velocity ratio 2, pitch 2, shafts 9 in. apart I) Outside diL.et; o gear 4 in.. diamHral pitch 8. (c) Gear of 50^eeth. 4 pitch/ (d) Wh e s Llt'6?n "Tt/'^T"' f '"''''• ^'> P'"' ''f «'■'-' -ti" 3 to t smaller 8 in. pitch diameter with 30 teeth a-ltant^gt'l^s itf' '^''' """' ""'^ ''"' ^•"' "'*"'*' P--^'"'"-' ^^hat 'i«";h^i;!::;,t:::n thr ""'' " ''-"'' ^"•' ^'^- -^ -^ -•" -•• centers 4,^ m., 6 pitch; draw the correct teeth on the 20= stub system 18. Explain the construction of tho helical gear an.i state its advantages :*' CHAPTER VI f I 11 t, t BEVEL AND SPIRAL GEARING 104. Gears for Shafts not ParaUel.— Froqiientl:- in practice tho shafts on which gears are placed are not parallel, in which case the spur gears already described in the former chapter cannot be used and some other form is required. The type of gearing used depends, in the first place, on whether the axes of shafts intersect or not, the most common case being that of intersecting axes, such as occurs in the transmission of autf mobiles, the con- nection between the shaft of a vertical water turbine and the main horizontal shaft, in governors, and in very many other well- known cases. On the other hand it not infrequently happens that the shafts do not intersect, as is true of the crank- and camshafts of many gas engines, and of the worm-gear transmissions in some motor trucks. In many of theje cases the shafts are at right angles, as m the examples quoted, but the cases where they are not are by no means infrequent and the treatment of the present chapter has been made general. 106. Types of Gearing.— Where the axes of the shafts inter- sect the gears connecting them are called bevel gears. Where the axes of the shafts do not intersect the class of gearing depends upon the conditions to be fulfilled by it. If the work to be done by the gearing is of such a nature that point contact between the teeth is sufficient, then screw or spiral gearing is used, a form of transmission very largely used where the shafts are at rightangles although It may also be used for shafts at other angles. One peciharity of this class is that the diameters of the gears are not determined by the velocity ratio required, and in fact it would be quite possible to keep the velocity ratio between a given pair of shafts constant and yet to vary within wide limits the relative diameteid of the two gears used. Where it is desired to maintain line contact be< - een the teeth of the gears on the two shafts, then the sizes of the gears are exactly determined, as for spur gears, by the velocity ratio and 90 BEVEL AND SPIRAL GEARING 91 also the angle and distance between the shafts. Such gears are called hyperboloidal or skew bevel gears and are not nearly so common as the spiral gears, but are quite often used. The different forma of this Rearing will now be discussed, and although a general method of dealing with the question might be given at onco, it would soem better for various reasons to defer the general case for a while and to deal in a special way with the simpler and more common case afterward giving the general treatment. BEVEL GEARING 106. Bevel Gearing.— The first case is where the axes of the shafts intersect, involving the use of bevel gearing. The inter- secting angle may have any value from nearly zero to nearly 180°, and it is usual to measure this angle on the side of the point of intersection on which the bevel gears are placed. A very common angle of intersection is 90" and if in such a case both shafts turn at the same speed the two wheels would be identical and are then called mitre gears. The type of bevel gearing corresponding to annular spur gearing is very unusual on account of the difficulty of construction, and because such gears are usually easily avoided, how- ever they are occasionally used. Let A and B, Fig. 54, rep- resent the axes of two shafts intersecting at the point C at angle 6, the speeds of the shafts being, respectively, m and n, revolutions per minute; it is required to find the sizes of the gears necessary to drive between them. Let E be a point of contact of the pitch lines of the desired gears and let its dis- tances from .1 and B bo r, and n, these being the respective radii. Join EC Now from Sec. 83 it will be seen that run = /•,«, since the i If ill n I" 92 ft I THE THEORY OF MACHINES pitch circles must have the same velocity, there being no slipping Mween them, and hence l"^^ = ^^ is constant, that is at any ,K,i„t where the pitch surfaces of the gears touch, ^ must bo constant, a condition which will be fulfilled by any pJint on the Une EC in the case of bevel gearing, therefore, the pitch cylinders re- ferred to in Sec. 83 will 1m3 replaced by pitch cones with apex at t , which cones are generated by the revolution of the line CE about the axes A and B. Two pairs of frustra are shaded in on Fig. o4 and both of these would fulfil the desired conditions for the pitch surfaces of the gear wheels, so that in the case of bevel gearing the diameteis of the gears are not fixed as with spur wheels, but the ratio between these diameters alone is fixed by the velocity ratio desired. The angles at the apexes of the two pitch cones are 20, and 26^ as indicated. If r ''''' '''" ""' ^PP^^'- ^ P-^"^' «t-«' t orme „,ane OP " "'"""" P^^P^^^icular, while on fh. and t. xstence 0'P'1 fv. ^/ f ',^"^^' ^'*^"""" ^^e shafts and 0'P'~TZv ^u '*''*^"'' ^"*^"^^" them and when ^ given Th"e need fT ^'/""'* P""*^""^ °^ ^^« ^^afts are ripU ,, P '^ °^ the shafts n, and n, must also be known .us well as the sense in which they are to turn ' In stating the angle between the shafts it is'always intended r^tl^ ::"'%" "''^' ^'^ ''"^ ^^ ^-^-* must^ieXs n tac; m!^ ?r'' ""^ '"*"''"" ^'""•^ ^"^"^^t^ that the line of con- T£^tT rr'''^' ^" ''''' ^"^'^ ^^'^^ -"d not in aZ L shaft ;oTut Ih^""^ '^ ""' ^"^*^^^ °^ ^'^^- Should Shaft AO turn m the opposite to that shown, then the line BEVEL AND .SPIRAL GEAKIAV U5 of contact would be in the angle, AiOB, such iis r,Q (since an- nular gears are not used for this type) and then the angle A lOB would be called 0. 110. Data Asstuned. — It is assumed in the problem that th(^ angle 6, the distance h, and the speeds r?, and »i2 or the ratio ni/nj, are all given and it is required to design a pair of gears for the shafts, such that the contact between the teeth shall be along a straight line, the gears complying with the above data. 111. Determination of Pitch Surfaces.— Let the line of contact of the pitch surfaces be CQ and let it be assumed that this line passes through and is normal to OP, so that on the right-hand projections A'O', C'Q' and B'P' are all parallel. The problem then is to locate CQ and the pitch surfaces to which it corre- sponds, the first part of the problem being therefore to deter- mine hi, hi, Bi and B-i, Fig. 57, and this will no'v be done. Select any point R on CQ, Fig. 57, R being thus one point of contact between the required gears, and from R drop perpendicu- lars RT and RV on OA and BP respectively. These perpendicu- lars, which are radii of the desired gears, have the resolved parts .ST and VV, respectively, parallel to OP, and the resolved parts RS and RU perpendicular both to OP and to the respective shafts. These resolved parts are clearly shown in the figure, and a most elementary knowledge of descriptive geometry will enable the reader to understand their locations. Further, it is clear the RT^ = RS^ + S'T"' and RV^ = RU' + U'V. At the point of contact R, the correct velocity ratio must be maintained between the shafts, and as /? is a point of contact it is u point common to both gears From the discussion in Sec. 84 it will be clear that, as a point on the gear located on OA the motion of /e in a plane normal to the line of contact CQ must be identical with the motion of the same point R considered as a point on the wheel on BP, that is, in the plane normal to the line of contact CQ, the two wheels must have the same motion at the point of contact R. Sliding along CQ is not objectionable, however, except from the point of view of the wear on the teeth and cau.ses no unevenness of motion any more than the axial motion of spur gears would do, it being evident that the endlong motion of 5=pur gears nill in no way affect the velocity ratio or the steadiness of the motion. In designing thi« class of gearing, therefore, no effort is made to prevent slipping along the line of contact CQ. Imagine now that the motion of R in each wheel in the plane 96 THE THEORY OF MACHINES ■ I f normal to CQ is dividcl into two part«, namely, those normal to P at normll „ r^ ''' I""""' *" ''^'^ ^'•«* P'""«^ ""^ "> ♦ho piano normal to CQ, its motion as a iwint in the wheel nn DA But i?s = 0/e sin i and if the distance apart of the shafts 4 in. and h, = 16 in., and the angle e, is tan di or „^ =K = 0.5, Oi 26° 34' and (Ja = 90 - 0, = g3° 26' so that the line of contact is located Case 3.-Parallel shafts at distance h apart. This dveK fh. orchnary case of the spur gear. Here \ = L^ Sf ^i - 02, hence, sm 0, = Q = «„ ,, and cos 0. = i = cos fl„ BEVEL AND SPIRAL OKA RING 99 BO that there are only two conditions to satisfy: hi + hi =- h and hiTii =- htn^. Solving these gives hi = hi nt and substituting in A, -f A, = A gives and "' I h, «l + "3 I. "' 1 fit = ,-- h, ni + 712 formulas which will be found to agree exactly with those of Sec. 83 for spur gears. Case 4.— Intersecting shafts. Here /i = 0, therefore A, = and hi = 0. Referring to Fig. 60, draw OM = m and ON = n, Fig. 60. then MN is in the direction of the line of contact OC. There are only two equations here to satisfy: fli + ^2 = ^ and «i sin di = ni sin 0i and these are satisfied by MN. Then dmw OC parallel to MN (compare this with the case of the bevel gear taken up at the beginning of the chapter). Case 5.— Intersecting shafts at right angles. Here 6 = 90°. Further let n^ = n, then H = 45°, thus the wheels would be equal and are mitre wheels. I ,| % I- E- I I:. lUO THE THEORY OF MACHINES whioJ' To In ^"'^•'^•"•-^^••^"rning to the general problem in wh oh the location of the line of contact VQ is found bv th^ method de«cri.K.d for finding A.. A, ,. and J Now "It as L he c^e of the spur and l.vol gears, a nhort part oT'the ine of con act IS selected to use for the pitch surfaces of the gears ae cording to the width of face which is decided upon, the width o ace largeb. deiK-nding up<,n the power to be ZsmUtTd am to j^\vr? ?kT'" ^"''"""^'y ^^^^ '^ *h« J'"« CQ were secured lilv ^'''''' '■'^■"'^'^•' '^"^ ^«'-'"'''- •'"« would describe a surface known as an hyperboloid of revolution and a second hyp^r' \ / / >4 P^ \ ^" / 1^ *.,' f — — d, 1^ '-'. / I'lo. 61. boloid would be described bv securing the line CO fn «P .k curved lines in the drawin.r F.V „/'*^ '"®.""*' ^V *« "P> the hyperboloids bv nl ^' ^ ' '^"""''"^ '*''*^°"« ^^ ^^ese A, fhT ^\P''"»'s P'-i^sed through the axes ^0 and BP cnl Lrr^ of developing the hyperboloid is somewhat dfffi: cult and lonj,. the reader is referred to books on descriptive .uch problems as the present one. however, it is quite unneces- 7^ on its a;is .s ^r:;:^a ^ ^:'::^ ^t^^^^nditig upcn the size of the teeth desired, the power to be transmitted, the velocity ratio etc., m which case true curved surfaces will have to be used more especially if th(! gears are to have a wide face. If the face is not wide, it may Ik; possible to substitute frustra of approximafelv similar conical surfaces. If the distance h is great enough, and other conditions i>ermit of It, It IS customary to use the gorges of the hyperboloids as shown at F and G and where the width of gear face is not great cylindrical surfaces may often be substituted for the true curved' surfaces. For the wheels F and G the angles 0, and 5, give the mclmation of the teeth and the angles of the teeth for D and E may he computed from Bi and di. 116. Example. — To explain more fully, take Case (2), Sec. 114, for which = !K)°, «, = 2«2, and /(, = 2() in., then the formulas give Ai = 4 in. and hi = 16 in. I^t it be assumed that the drive is such as to allow the use of the gorge wheels corres- ponding to F and G, then the wheel on OA will have a diameter d, = 2Ai = 8 in. and that on BP will be dj = 32 in. diame- ter. Further the angles have been determined to be Oi = 26° 34', «2 = 63° 26'. As the numbers of teeth will depend on the power transmitted, etc., it will here be assumed that gear F has , ,„„ ^, „ /,^,„^^^ _ 1.25(Joo8 26»34' - 1.123 in ^.r . u.K.;ar(/.h..„u,nlH.roft,H,th(, . 40Hince„. - 2nVancl ^V;^:!';^'"^""''^'"- ^ "^^^toh or the gear';t H7. Form of Teeth.-Murh .liHrussion has ariw-n over th,. "••-.■'lo t.. „..ko H ..K.U. which will 1... theoroti a ; 0^^^^^^^^^^^ ir . ' '" ^''^^ '''"'"' """»"' t» tJie line of contaot ^:zzz;z:t:::;:^ ^^^ -^ «»^-'^ ^ave the e^;!; w.. hut s..h «oann«, if wH. n.a.le wii, run vor^^:! h^ quMli Although ,t IS difficult to construct there are c^s Hhcre the „o«mon« of the shafts n.ake its uho in.porative SPIRAL OR SCREW GEARING Thk Tkkth of which havk Point C'ontact 118. Screw Gearing.-In speaking of gears for shafts whi.h lit with In screw geanng there is no necessary relation -U^cen the chan.eters of the wheels and the velocity ratb UK camshaft of a gas engnu- runs at half the speed of the crank- d tt '' TVTT '"'" '"■•"'""'•■« ^ho .Iri^' are o/thc^Zt the san.e wor^ llldd W a!^t':\';:y'^^'^' ^"' ^^"^«^^™ ^- is Iho'Z^Z ^"*'^«-'^h> '"-t. fan.ihar form of this gearing Fig 03 In"d it""; T"" ""' """" "^^^'' ^^-h'^h is sketched in •^0 m 'ni.. , '"';" ^''«*'"^"'«hed by the name of the dtt^c ' m t . , T";';' "'1 '^ ^'^'^ ^'*^^ corresponds to the distance nom tluead to thread along the worm parallel to ita HHVHf, AM) SPItAL (IF.MUMI lo;' ftxiH, thfi threml in tingle pitch. If the (listuixo from ono throu.! to the next is one-half of the iixiiil pitch tho thrriid if. double pitch, an^ Fii,. 64. — Double and triple pitoh worms. 120. Ratio of Gearing.— Let p, he the axial pitch of the worm and D be the pitch diameter of the wheel measured on a piano through the axis of the worm and normal to the axis of the wheel. '^S^^ '^nu 104 THE THEORY OF MACHINES ,1 s i i I i Then the circumference of the wheel is rrD, and since, by defini- tion of the pitch, one revolution of the worm will move the gear forward pi in., hence there will be ^ revolutions of the worm for one revolution of the wheel, or this is the ratio of the gears Let t be the number of teeth in the gear, then if the worm is single pitch < = — or the ratio of the gears is simply the number of teoth in the wheel. If the worm is doable pitch, then pi the distance from center to center to teeth measared as before is given by p, = 2/>', where p' is the axial distance from the center of one thread to the center of the next one, and t = -^^ and as ttZ) '^' the ratio of he gears is ^ - , in the double pitch worm this is equal to ^' and for triple pitch it is ^ etc. 121 Construction of the Worm.— A brief study of the matter will show that as the velocity ratio of the gearing is fixed by the pitch of the worm and the diameter of the wheel, hence no matter how large the worm may be made it is possible still to retain the same pitch, and hence the same velocity ratio, for the same worm wheel. The only change produced by changing the diameter of the worm is that the angle of inclination of the spiral thread is altered, being decreased as the diameter increases, and vice versa. The p. .gle made by the teeth across the face of the wheel muse be the same as that made by the spiral on the worm, and if the pitch of the worm l)e denoted by p, and the mean diameter of the thread on the latter by d, then the inclination of the thread is given by tan 6 = ^J, and this should also properly be the in- clination of the wheel teeth. From the very nature of the case there will be a great deal of slipping between the two wheels, for while the wheel moves forward only a single tooth there wil'l be slipping of amount ird, and hence considerable frictioual loss, so that the diameter of the worm is usually made as small as possi- ble consistent with reasonable values of $. The worm is often immersed in oil and the friction loss is frequently below 5 per cent but in poorly made worms it may be much highei. When both the worm and wheel are made parts of cyhnders Fig. 6.5, thei: there will only be a very small wearing surface on the wheel, but as this is unsatisfactory for power transmission BEVEL AND SPIRAL GEARING 105 the worm and wheel are usually niade as shown in section in the left-hand diagram in Fig. 65 where the construction increases the wearing surftice. The usual method of construction is to turn the worm up in the lathe, cutting the threads as accurately as may be desired, then to turn the wheel to the proper outside finished dimensions. The cutting of the teeth in the wheel rim may then be done in various ways of which only one will be described, that by the use of a hob. 122. Worm and Worm-wheel Teeth. — A hob is constructed of steel and is an exact copy of the worm with which the wheel is to work, and grooves are cut longitudinally across the threads so as to make it after the fashion of a milling cutter; the hob is then hardened and ground and is ready for service. The teeth on the wheel may now be roughly milled out by a cutter, after which the hob and gear are brought into contact and run at proper relative speeds, the hob milling out the teeth and gradually being forced down on the wheel till it occupies the same relative position that the worm will even- AA/irvA Fk;. OG. — Proportions of worm. t daily take. In this way the best form of worm teeth are cut and the worm and wheel will work well together. The shape of teeth on the worm wheel is determined by the .vorm, as explained above. The A.G.M.A. standard for :-ingle- aiid double-threaded worms is a 14 '2' angle, but angles of 20° or more are us(h1 on triple threads, etc., as this gives greater clearance for machining and grinding the worms. For auto- mobile and high-speed work nearly 30" may be used. A sec- : :'.■?■■:■ i:^'-.--Ji'- f(r-.-. ■ 'V>:>^nJ::i 106 THE THEORY OF MACHINES M I If I I j-t i tion of the worm thrciid is shown at Fig. 66 in which the propor- tions used by Brown and Bharpc are the same as in a rack. 123. Large Ratio in this Gearing.— Although the frictional Josses in screw gearing are large, even when the worm works immersed in oil, yet there are great advantages in being able to obtain high velocity ratios without excessively large wheels. Thus if a worm wheel has 40 teeth, and is geared with a single- threaded worm, the velocity ratio will be ^^, while with a double- threaded worm it will be ^ = ~^^, so that it is very convenient for large ratios. It also finds favor because ordinarily it cannot be reversed, !bat is, the worm must always be used as the driver and cannot be driven by the wheel unless the angle 6 is large. In cream separators, the wheel is made to drive the worm. 124. Screw Gearing.— Consider now the case of the worm and wheel shown in Fig. 65, in which both are cylinders, and suppose that with a worm of given size a change is made from a single to double thread, at the same time keeping the threads of the same size. The result will be that there will be an increase in the angle and hence the threads will run around the worm and the teeth will run across the wheel at greater angle than before. If the pitch be further increased there is a further increase in B and this may be made as great as 45°, or even greater, and if at the same time the axial length of the worm be somewhat de- creased, the threads will not run around the worm completely, but will run off the ends just in the same way us the teeth of wheels do. By the method just dosciil>ed tho diameter of the worm is un- altered, and yot the velocity ratio is gradually approaching unity, since the pitch is increasing, so that keeping to a given diameter of worm and wheel, the velocity ratio may be varied in any way whatever, or ihe velocity ratio is independent of the diameters of the worm and wheel. When the pitch of the worm is increased and Its length made (juite short it changes its appearance from what It originally had and takes the form of a gear wheel with teeth running in helices across the face. A photograph of a pair of these wheels used for driving the camshaft of a gas engine is shown m Fig. 67, ami in tiiis t-asr the wheels give a velocity ratio of 2 to 1 between two shafts which do not intersect, but have an angle of 90° between plain's passing through their axes. This BEVEL AND SPIRAL (IKAHINC 107 form of gear is very extensively used for such jjurposes as afore- said, giving quiet steady running, hut, of course, (lie frictional loss is quite high. Some of the points mentioned may be mad(! clearer Ijy an illus- tration. Let it be required to design a pair of gears of this type to drive the camshaft of a gas engine from the crankshaft, the velocity ratio in this case being 1 to 2, and let both gears be of the same diameter, the distance between centers being 12 in. From the data given the pitch diameter of each \VheeI will be Fh;. G7. — Scr gears 12 in. and since for one revolution of the canisluift the crankshaft must turn twice, the pitch of the thn^ad on the worm must In; '•i X ttX 12 = 18.8.-) in. For the gear on the crankshaft (cor- responding to the worm) the "teeth" will run across its face at 18 8") an angle given by tan ^ = ^ ^ J2 " "•"'' '"" ^ = -'*" •^•*'' '^'''^ "''■^ angle is to be measured between the thread or tootii and the plane normal to the axis of rotation of the worm (see Fig. 04). The angle of the teeth of the gear on the cam.-,luift (corresponding to ZI^:z:£:iL'>i'L k^t :d^ii; IJ 1 i i I 108 T//A' THEORY OF MACHINES the worm wheel) will be 90 - 2G° 34' = 63" 26' measured in the same way as before (compare this with the gear in Sec. 116) It will be found that the number of teeth in one gear is double that m the other, also the normal pitch of both gears must be the same. The distance between adjacent teeth is made to suit the conditions of loading and will not be discUvSsed. Spiral gearing may be used for shafts at any angle to one another, although they are most common in practice where the angle is 90°. A more detailed discussion of the matter will not be attempted here and the reader is referred to other complete works on the subject. 126. General Remarks on Gearing.— In concluding this chap- ter It IS well to point out the differences in tiie two types of gear- ing here discussed. In appearance in many cases it is rather diffi- cult to tell the gears apart, but a close examinatic will show the decided difference that in hyperboloidal gearing contact be- tween the gears is along a straight line, while in spiral gearing contact is at a point only. A study of gears which have been in operation shows this clearly, the ordinary spiral gear as used in a gas engine wearing only over a very small surface at the centers of the teeth. It is also to be noted that the teeth of hyper- boloidal gears are straight and run jxTfoctly straight across the face of the gear, while the teeth of spiral gears run across the fac' in helices. Again in both classics of gears where the spiral gears have the form shown at Fig. 67, the ratio between the numbers of teeth on the gear and pinion is the velocity ratio tranmiitted, but in the case of the spiral gears the relative diameters may be selected as desired, while in the hyperboloidal gears the diameters are fixedilv about the hxed gear as in the Weston triplex pullev block or the (hfferential on an automobile when one wheel stops and the other spms m. the mud, the arrangemoit is called an epicycUc tram huch a tram may be used as a simple train of only two Wheels, but IS much more commonly compounded and reverted so that the axis of the last whe., coinci.les with that of the first i^ or the ordinary train of gearing the velocity ratio is the num- t'cr u{ tUHLs uf the laM wiu-el divided by the number of turn-. of the first wheel in the same Mme, whereas in the epicvclic tram the velocity ratio is the mmiber of turns of the last wheel in the 110 m )IL ■ i""TifirfrTT"''frTr" 3^^ms^si^sasat,''wrviKz*fSi ■ THAINS OF (iKMilSd 111 train divided i)y the numher of turns in tlio same time of the frame carrying the moving wheels. 128. Ordinary Trains' of Gearing.— It will be well to begin this discussion with the most common class of gearing trains, that is those in which all the gears in the train revolve, and the frame carrying their axles remains fixed in space. The outline of such a train is shown in Fig. 68, where the frame carrying the axles is shown by a straight line while only the pitch circles of the gears are drawn in; then" are no annular gears in the train shown, although these may b<> treated similarly to spur g«;ars. Let n-t be the number of revolutions per minute made by the Vu.. (is. last gear and «i the corresponding number of revolutions [kt minute made by the lin;t gear, then, from the definition already given, the rat-o of the traui is ,. ^ n2 _ the number of revolutions per miimte of the last gear ni the number of revolutions per minute of the first gear. The figure shows a train consisting of six spur gears marked 1, a, 6, c, e and 2, and let 1 be considered the first gear and 2 the last gear. The following notation Avill be employed: n^, tia = J/*, Vc = v, and na will represent the revolutions per minute, ri, r„, n, r^ r, and rz the radii in inches, and ti, L, tb, tc, t, and U the numbers of teeth for the several gears used. The gears a and b and also the gears c and e are assumed to be fastened together so that the train is compounded, a statement true of any train of over two gears. Any pair of gears such as b and c, which mesh with one another, must have the same type and iiitch of teeth, l)ut both the type and pitch m.ay bf» different for any oihrr pair which mesh together, such as e and 2; the only requirement is that each gear 'In wliat follows in this chiiptor refprcrife is iiiailf' to spur and bevel gears only. ..)« wmm .i-?*.^' >■ k>TAVt^-^.»- r J-.-. \7,- i I i 112 THF TIIFoh'Y OF M.\( IHSFS alsMi "' '■' 'a n,, r t. ri < I'li.Tflni-f. til,, r.-ifio .If th.- rntiii /t- uinl tilt rt'l'ore \ ".. a,, tit 'l-a 'Ir n-i . '•■ >^ r., X r, ': X '•.: X r,. ■'; '< f/. V C, t amn« the ni-.r wheel u, .a.^li pair ..... i, H an.i «, the dnver Ittr^ K r^ ***f *^ ^" ^* P"''^"^ °^ ^« "^ "--^'=^ "• ■ =-^— ■--«....: - ,^ -"• -•' " •'•^' "— 'nu oucne. .e i, -* ..., ^ for .he nan. ' Uiu t, > anu ,u,o • uu, : "e.pe.nveiv. Then -. . is, ■„ = .J(. /f '» \ i < .') V^s n •i '■ : '^ ■rum Uie rmiii 1*1 :*iifa»i«»!f^ K^%t'^ii-i^SESaBS}»^«l««-Jt^SS«fcI'. 77.'. WAN Oh- <;KAhI\n 113 »r H IK X IN X K(> M) X ;<2 X IS 1 frmu the iiiinilMMs nf tc^-th. Kiirtli.T. il vvlir.-l I turn at h simm-.I of »,, ^ .-,() rcvf.hitK.ns [«-r ininulo tlu' himt.Is of tl.r other ncars will Ik- n, = r<^ = S() n.-volu- tioris, »j„ - „, - rj(» revolutions and n, = 2()0 revolutions ]mt minute. If the (list.ance I.etween the axe.s of pears 1 and 2 \ver«^ fixH liy some exi.-rnal conditions at the distance apart corresj^.n-ijiiir to the above train, then tlu- whole train could !«■ nplac«'d hv a pair of pears having radii of l>>.-,;{ in. and l.S,S in., and thes<. would give the sain«' velocity ratio as tlu> train, hut would oft^n i-- objectionaMe on aiu-ount of the larpc' : Ize of the larper gear. The stMise of rotation of the various pears may now U- ex- amined. Lookinp apain at Fip. (IS, i; is ohst^rved that for two spur wheels (which h.ave one contact ) the sense is re\ers«'etwe*.n 1 and 2 the sense is revenwd and the rule for determininp the Mi-xi\' sense of rotacion of the first and Ia.st wheels may 1^ statH thui^: In any spur-wheel train, if the number of contacts between the first and last gears are even then both turn in the same sense. and if the number of contacts is odd, then the first and last wheels turn in the opposite sense. Sh<.uld the train vnnxmu annular pears, the same rule will apply if it is rem.-mUred tha: any contact with an annular pear has the same effect us tw. contacts }>etween spur pear. The same rule a!s<. apj.H.- ;r. .•:^., Wts are used, an ojien belt correspondinp to an annular p.ur i.uc a cro.s,sed one to spur pears. The rules botii for ratio and s. -se of rotaticjn are th. h^;-:;* -..• U'vel gears as for spur pears. 130. Idlers.— It not infrequently ImpiKus that in a coi: fx-unc; train the two gears on an intermediate axle are made of tht sor.i? size and comlnned into one; thus /„ may tn- made (-qual v.. *, or 'a = tb. This single intermediate wheel, then, lia.s uu effec: ur the velocity ratio U, as an insix-ction of the formula for K wi] show, .and is therefore callt-d an idler. 'i"h such wheels is either to ch c so; purj^.twr ill aSi increase the distance bet mcreasing their diameters anpc the sen.s<' of rotat at Kin <.>r H;tt een the centers of other wLefl;;^ witb.> v::%:f^'?^^c;l£€i..%7>sxiKa^i>see«^^ 114 I f THE THKOUY OF MACHINES 131. Ewmples.— The application of tJie forimila nmy Ik^ \^Ht explained by some examples which will now Im- Kiven • 1. A wheel of 144 teeth drives one of 12 teeth on u shaft whi.l, makes one revolution in 12 sec, while a second one .Iriven l.y i( turns once m 5 sec. On the latter shaft is a 4()-in. pulley con- nected by a crossed belt to a 12-in. pulley; this latter pulley turns rr U /]! °"",'^,T'-"' r'^ '* *""- three times. Show that the ratio IS 144 and that the first and lust wheels turn in the san.e sense. 2. It is required to arrange a train of Kearing haying a ratio of 250 13 2^ M^^ 1^ ! T **"' '"■"'''"'" '••^' "•'''"« ''''' «'^"'-« having wt Sl"f •'""'V^'P'""''''''''^'- '"'^ '" K^'"^--'^' tho larger wheel wdl be too hg and it will be well to make up a train of four or six gears. Break the ratio up into factors, thus: R = ^^ = 50 60 ^3 13 ^ 12 ''"^ -eferring to the formula for the latio of a train it ieeth '^n V'?.* Z^'t' ''! "^"'''^ ''^ "^ ^""'- ^'^'^ having 50 teeth 13 teeti). 60 teeth and 12 teeth, and these would be ar- ranged wuh the first wheel on the first axle, the gears of m teeth and 13 teeth would be keyed together and turn on the inter- mediate axle and the 12-tooth gear would be on the last axle and whed" "'""''^ ^"^ ^^"^ ^" ^" '*"' ''^ ''"'' **''' ^^ *" '^^ '^-t*'""' Evidently, the data g^•en allow of a >j,,.at n.anv solutions for this problem, another with six wheels being, 40 h = 250 5 >. 13 ~ iX 4 ^ 13 60 15 40 12 ^ 12 ^ IS 1 his would gu-e a tram sinular to Fig. 68 in which the gears are ' ~ SL^' ': =: ^^' '* = 1^ '^ = 12, K = 40 an,! /, = 13 teeth. 3. ro design a tram of wheels suitabl.. for connecting the sec- ond hand o a watch to the hour hand. Here the ratio is senle andT T' '^^ /^"' "''^^'^ '""''* ^"^" ^^ ^he same sense, and as annular wheels are not used for this purpose the number of contacts must be even. The following two sohlHo'^ Would be saMslactory for eight wheels: " "" R = 720 4X4X5X1) 1" •56 48 50 108 14 "^ 12 ^ 10 ^ 12 "Wr^^rimi^^T ■mmwi'^mm TRAINS (fF (iKMilSt; II.' or H 720 = 1-><5 _ 72 ^ 60 ^^ 52 _ 60 12' 12 ^ 10 ^ 13 ^ Attention i« called to the Htatenicnt in Keo. m that it is un- usual to have wheelH of Icsh than 12 teeth. 4. Required the train of gearH suitable for connectinR the min- ute and hour hands of a clock. Here /i = 12 and the; train must be reverted; further, since both hands turn m th(. same sense there must be an even numln-r of contacts, and four wheels will Im' selected. In addition to obtainuiK the correct velocity ratio it i.^ necessary that r, + r^ = H -f- /•.. and if all the wheels have the same pitch <, + t„ = t„ + 1^ The followiuK train will evidently produce the curreci result- ft = 12 = 4 X .3 48 ' 45 1 12 >^ 15 or U - 48, t„ = 12, I, = 4.-) and t, = 15 teeth, the hour hand carrymK the 48 teeth and the minute hand the 15 teeth 132. AutomobUe Gear Box.-\'ery many applications of trams of gearing have been made to automobiles and a drawing of a variable-speed transmission is shown in Fig. 69. The draw- ing shows an arrangement for three forward si)eeds (one without using the gears) and one reverse. The shaft K is the crankshaft and to It IS secur(.d a gear A having also u part of a jaw clutch B on Its right-hand si'^^s^m ]U) Tin: TiiKouY OF \r.\ciij\h:s s iff i in I tH.ntact wilh //, the ratio of neurs then beinji A to C, and // to D- A- reinaiiiH ,in hIk.wh. For low,.«t 8i)oo(l I) jh pl„,.,.,| as „how„ i,! the fiKuro and A' slid into contact witli J; tho shaft /' and tho car are reverHed by nioving F \o tho right until it nienhes with L the gear ratio l>einK '1 to G and A" to L to F. Builders of autonu.l.iloH m dcHign the op0. — Autoiri()l)ilo gear 1k)x. 133. The Screw-cutting Lathe.— Most lathes are arranged for cutting threads on a piece of work. uikI as this forms a very interesting application of the principles already described, it wiil be used as an illustration. The general arrangement of the headstock of a lathe is shown in I'lg. 70, and m this case in or-h-r to make the present discussion as simple as possible, it is assumed that the back gear is not in use. The cone C is connected by belt to the countershaft whidi supplies the power, the four pulleys permitting the operation of the lathe at four different speeds. This cone is secured to the spindle 5. which carries the chuck K to which the work is at- Tit.\r\s or cF.Miisa ii; t , Uched and by which it ia (irivt-n at the same rate as the c«iie C. On the other en^i -jf 5 is a gear e, which drivca the K«*ar h through one idler n or two idlers / and g. The ^haft which cairies h also has a gear 1 which is keye„s,., an.l ii" (he leudin. •screw has a nght-hand thread, as is usual, then the thread , ut on the work will also he right-hand. The idl.s f an.l , n provided to facilitate this matter, and if a right-hand th-Ac so t^at , alone connects e ar.d h, while, if a left-han.l thread is in ni. ''" ^^"'•^..'^'l^^^^ tlH^ setting for a right-hand throa.i. An Illustration will show the method of setting the gears to do eut'lo/rr "' :"■'• • '"'^"""^ '''''' ^ ^^'^he has a leading screw veb 20, 40, 4o, oO, f,.,, 00, Go, 70, 7o, 80 and 115 teeth. Assume 1. It is required to out a right-hand screw with 20 tluva.ls per inch. Then - = '' X '- wherr> / - 4 , i • . , s /a /2 ^^'"'^' ' - ■* '"HI s IS to he 20. Thus '' y '^ = ■* ta h 20 ;ihis ratio may be satisfied by using the following gears t, = 20 used f!' '* ^.f '"'/ 'r ?"• ^'"'-^' *^"' "^^" ""«^ '^ ^vouhl be used to give the right-hand thread. 2 To cut a standard thread on a 2-in. gas pipe in the lathe. I he oroper number of threads here would be llj.^ per inch an.l hence I = 4, s = 11 1^ and '' X - = "^ - - * . n.,. , , U"^ i. 111,- - 23- Jhis..ould as one' U^ " "' '' = '' "'^ '= == '^''' ^"'^ '* - ^« '-*h acting as one id'er. " 3. If it were required to cut 100 * tds per inch then I = 4, mti v be divided into two s = 100 and ^' -'" 4 1 ia ^ /j 100 2.V whi parts, thus 25 = 4 X 6^' '° *^^* ™* '-"« ^^ = 20, «, = 80, ittJEC^^!:!HB^.^isi7i!ysaeeAi%'Vi "^■P'-^^'-'^tM fk ^Hhiqg!^Sahi^;&i A%^fiaSieS£iUgSii^.' v^iTV;,ift 7'/^.l INS OF UEA RIMi j , 9 <2 = 75, would require an extra gear of 12 teeth to take th.' place of 6, as h = \2. The axle h ,' 1, and a detailed drawing in Fig. 72 in the latter of which IS silown a belt .one with four pulleys, P, running frc-lv on the live spmdle S. Keyed to the same spindle is the gear shown at (A and secured to the cone P is the pinion T, and Q and T mesh when required, with corresponding gears on the back gearshaft K. When the cone is driving the spindle directlv the pin IF shown in the gear Q, is left in the position shown in the drawing' thus forcing P to drive the .spindle through Q, Imt when the back Rear IS to be used, the pin W is drawn back out of contact with the cone pulley, the shaft R is n>volved by means of the handle ^ so as to throw the gears on it into mesh %vith T and Q and rti 120 THE THEORY OF MACHINES i < 1-^ 1 -■ i.S « I ' tL. ;&. iriVHK>iiSfi::^Uo^r<£?^HK;''cv If': 122 ill M i THE THEORY OF V u ffjyj^s L, K, H, etc. The numbers of teetl .lowri in the various gears correspond to those used in the 16-in. and 18-in. hithes. The handles shown control tlie gear ratios; thus U controls the positions of the gears L, K, and // and the figure shows the three possible positions provided by the maker and corresponding to the three holes 1, 2 and 3 in Fig. 72. The gear B is provided with a feather running in a long key seat cut in the shaft shown and the handle I is arranged so as to control the horizontal position of the gear B and its tumbler gear; that is the handle \ enables the operator to bring B into gear with any of the 12 gears on the lead screw. The lead screw has 6 threads per inch U ith the handle V in No. 3 hole and the handle V in the fourth hole as shown in the right-hand diagram of Fig. 73 the ratio is / I{ = «i = '^ X '"^ y '"■■ V '' V '^' ^ ''< ^ X ^^ X ^'^ X ^'^ V ''^ ^ -^O 12 7 ■ ■ *" "^ 12 X<' = 12 X «) = 34 or the lathe would i)e set to cut 3^ threads per inch 134. Cutting Special Threads, Etc.-When odd numbers of threads are to be cut, various artifices are resorted to to get the required gearing, sometimes approximations onlv being employed Ihus If It were required to cut threads on a 2-in. gas pipe, which has properly 11>^' tlireads per inch, and the lathe had not gears for the purpose, it might be possible to cut 11^ threads per inch or 11^4 threads per inch, either of which would serve such a purpose quite well. There are cases, how(,>ver, where (>xact threads of odd pitches must be cut and an example will show one method of getting at the proper gears. m!^Q?,n= '""T'^i^^ «"<' ^ s^'-e^^- with an exact pitch of 1 mm. (0.0393708 in.) with a lathe having 8 threads per inch on the leading screw, and assume t, = th. A convenient means of working out this probl.>m is the m(>tho,i of continued fractions. The exact value of the ratio I is rt 1 R H 0.0393708 0.125 0.0393708* '^^^^j.'-t.^^mM t,jti^. TRAINS OF GEARING 123 The first approximation is 1 j^ = 3, the real vah.e being ^J = 3 + 3^^!^^- The second is 3 + g, the real value 'oinK 3 + ^^ ^^g ^ "^ 68,876 and in this way the third, fourth, fifth, etc., approximations are readily found. The sixth is 1 R = 3 + 1 5 + 1 1 + 2 + 3 40 " 40 ■ 1 + 1 Thus with a gear of 40 teeth at 1 or /i= 40 and U = 127 on the leading screw, and an idler in place of a and b the thread could he cut. 125 127 (Note tiiat „ noo'jync = 3.17494 while .„ = 3.175, so that the arrangement of gears would give the result with great accuracy.) Problems of this nature frequently lend themselves to this method of solution, but other methods are sometimes more con- venient and the ingenuity of the designer will lead him to devise other means. 135. Hunting Tooth Gears. — These are not much used now but were formerly employed a good deal by millwrights who thought that greater evenness of wear on the teeth would result when a given pair of teeth in two gears came into contact the least number of times. To illustrate this, suppose a pair of gears had 80 teeth each, the velocity ratio between them thus being unity; then a given tooth of one gear would come into contact with a giveii tooth of the other gear at each revolution yf each gear, but if the number of teeth in one gear were increased to 81 then the velocity ratio is nearly the same as before and yet a given pair of teeth would come into contact only after 80 revolu- tions of one of the gears and 81 revolutions of the other. The odd tooth is called a hunting tooth. Compare the case where the gears have 40 teeth and 41 teeth with the case cited. - If 124 THE THEORY OF MACHINES EPICYCLIC GEARING. ALSO CALLED PLANETARY GEARING 136 EpicycUc Gearing.~An epicyclic train has been defined at the beginning of the chapter as one in which one of the gears in the train is held stationary or is prevented from turning while all the other gears revolve relative to it. The frame carrying the revolving gear or gears must also revolve The train IS called epicyclic because a point on the revolving gear describes epicychc curves on the fixed one, and the term planetary gearing appears to be due to the use of such a train by Watt in his "sun and planet" motion between the crankshaft and connecting rod of his early engines. An epicyclic train of gears is made up in exactly the same way as an ordinary trair. already examined, the only difference •t Y\(\. 74. — Kj>icyclic ir.'iins. between the two is i„ (he part of the cond.ination (hat is fixed- in the ordinary train the axles on which the gears revolve are fixed in space, that is, the frame is fixed and all the gears revolve whereas in the epicyclic train one of the gears is prevented from' turning and all of the other gears and the frame revolve. This IS another example of the inversion of the chain explained in y hapter I. The general purpose of the t^ain is to obtain a very low veocity ratio without the use of a large number of gears; thus a ratio of >i 0^000 may very simply be obtained with four gears, the largestof whichcontains lOlteeth. It also has other apph nations. Any number of wheels may be used, although it is unusual to employ over four. In discussing tlie train, the term "first wheel " will correspond w,th wheel and last wheel " with wheel 2 in the train sho^vn in l-ig. b8, and It will always be the first wheel which is prevented TRAINS OF CEARTNO 125 from turning. The ratio of the train is the number of turns of the hiHt wheel for each rcvoUition of the frame. In Fig. 74 two forms of the train, each containing two wheels, are shown. In the left-hand figure, wheel 1 is fixed in space and the frame F and wheel 2 revolve, whereas in the right-hand figure the wheel 1 is fixed only in direction, being conne'jted to links in such a way that the arrow shown on it always remains vertical (a construction easily effected in practice), that is wheel 1 does not revolve on its axis, and the frame F and wheel 2 lM)th revolve about the center B. The following discussion applies to either case. 137. Ratio of Epicyclic Gearing. — ^I^et the gears 1 and 2 con- tain ti and ti teeth respectively; then as a simple train the ratio is R = '^ and is negative, since the first and last wheels turn in the opposite sense. The method of obtaining the velocity ratio of the corresponding epicyclic train "lay now be explained. Assume first that the frame and both wheels are fastened together as one body and the whole given one revolution in space; then frame F turns one revolution, and also the gears 1 and 2 each turn one; revolution on their axes (not axles). But in the ei)icyclic train the gear 1 must not turn at all, hence it must be turned back one revolution to bring it back to its original state, and this will cause the wheel 2 to make R revolutions in the same sense as iMifore, since the ratio li is negative. During the whole operation gear 1 has not moved, the frame F has made 1 revolution and the last wheel 1 + R revolutijns in the same sense, hence the ratio of the train is I + R Revolu tions ma de by the last^ wheel E = Revolutions made by the frame. A study of the problem will show that if R were positive then E - I — R and in fact the correct algebraic formula is E = I - R and in substituting in this formula care must be taken to attach to R the correct sign which belongs to it in connection with an ordinary train. Owing to the difficulty presented by this m.atter the following method of arriving at the result may be helpful, and in this case a train will be considered where R is positive, i.e., there are an even number of contacts. I I 126 1 lei. r III! i. f THE THhOHY OF MACHINES First wheel makes + 1 revolutions. Last wheel makes + R revolutions. But the epicyclic train is one in which thp fircf «,!,» i a revoke, .„d th ,„, ^ ,,„„ ., ...^'''JtfS : ' ^t turned one revolution in o,„K,»ite .cnse to the former motTon tf rnrsxiirr, t— "- --- Frame has made - 1 revolutions. Fu-st wheel has nmde + 1-1 revolutions. wm"^ i! I ^^l'""'^' + ^'' - 1 revolutions, which has brought the wheel 1 to rest; hence — 1 ~ '^ ''•"* '«'fore. 138. Ewmples.-The followinR exantples will illustrate thp njean^ of the forn,ula and the applLtion J^'ZX 1. Let wheel 1 have 60 teeth and wheel 2 have .5<) teeth- fl"'n t, = GO, <2 = 59 and therefore H ^ -\ 60 60 Hence, E = 1 ~ R = i or the 59 (- ^^) = 1 + ^0 = ll''> 2. Suppose m,w that an idler is inserted between 1 and 2, keeping R = ,^ still, but making it positive. Then J^' = 1 - i^ = 1 _ /+ 60\ 60 _ _ i rp, \ 09/ 59 en' 3. If in e.an,ple (2) wheels , ,.nd 2 are i,„ereha„„ed, then - 60 ^"^ '•'* positive, so that 60 ^ 60 rir:aT,t';::ri.r' '"^ •■•• ™- -- - «>« 4. To design a train having a rx)sitive ratio of ,, ^ that is 10,000' ^^ '^' li TRAIXS OF liKARJNd 127 one in which the last wheel tur.js in the same senHe as the frame and at iQQQn ♦''« si>ee(l. Here A' = 1 - ft = ^^\^ or li = \ - 1 io,bo() \ 1(X)/ \ ^ 100/ 100 ^ 101 100* The train thus consists of four gears \, a, b and 2 and the nuinl)ers of teeth are t^ = !)!), /„ = 1(K), h = 101, h = 100. In practice such a train <'ouI(l easily he revertini, although the numbers of teeth are not exactly suited to it, and would work quite smoothly. The train is fre, the ciankslmft mad,, two revohiti.ms f„r ,m.I. two strokes of the piston. 139. Machines Using Epicyclic Gearing.- There are a greiit many illustrations of this interesting arrang(>,nent and space permits the introduction of only a very few of these (a) The Weston Triplex Pulley Block.-A form of this block which contams an epicyclic train of gearing, is shown in Fig. 7(5 Ihe frame D contains bearings which carry the hoisting sprocket A, and on the casting carrying the hoisting sprocket arc axl(>s eacli carrying a pair of compound gears B(\ the smaller one C gearing with an annular gear made in the frame /;, while the otlu-r and larger gear B of the pair meshes with a pinion A on Ih.. end of the shaft ,S to which the hand sprocket wheel // is attached When a workman pulls on the hand sprocket chain he revolves // and with It the pinion A on the other end of the shaft, which in turn «ets the compound gears BC in motion. As one of these gears meshes with the fixed annular gear on the frame D the only motion possible is for the axles carrying the compound gears to revolve and thus carry with them the hoisting sprocket F In the one ton Weston block the annular wheel has 49 teeth he gear fi has 31 teeth, C has 12 teeth and the pinion A has 13 teeth For the train, then, evidently R is negative since one wheel is annular and li = Hence 49 31 12 ^ 13 = - 9.73. ii' = 1 - a; = 1 - (- 9.73) = 10.73. ■ > ll ii II I I mil I iMi miiiii I ■mill i i <^;ETXK^H:vunBKsn«£x^aaL^f^l3cS^ TRM\S OF f!RMlI\r; 129 So that there must be 10.73 tuniH of the hjiiu! wl.eel to cause one turn of the hoist uik wheel F. As these wheels are respec- tively O-'j in. and .'i',s i'l. (iiuincter, the hand chain nuist Imj moved .., X 10.71^ = ',V.\.2 ft. to eavise the hoisting chain to Fi, which latter revolves at the same speed as li and operates the vulve f«»r the motors. The gear (' is keyed to a shaft which has another gear E also secured to it, the latter meshing with pinions /'' which in turn mesh with the internal near (! secured to the frame ,1. The dears F run freelj' on shafts ./ which are in turn secured in a flange on the socket .S, which carries the drill. .\s the motors operati* on the crankshaft causing it to n-volve, the pinion li turns with it and also th(> gear T and with it the gear E. As E revolves it sets the gears F in motioy and as these mesh with the fixed gear (1 the only thing i)ossil)le is for the spindles J carrying F to revolve in a circle ahout the center of E and as these n-volve they carry with them the drill socket S. In one of these drills the motor runs at l,27o revolutions jier minute and the munhers of teeth in the gears are /« = 14, t, ^ "(), /,, = 15, tp — 1,'> and tf this cul. (ft -3 «: I ! i 132 THE THEORY OF MACHINES and It IS boyond the present purpose to discuss the action of these m detail, but it may be explained that these control band brakes, one about the drum /, another about the drum E and a third about the drum C, and in addition the pedals and lever control the disc clutch between C and M. In this mechanism the gears have the following numbers of eeth: /, = 27 teeth, fo = 33 teeth, t, = 24 teeth, t, ^ 27 teeth. tr = 21 teeth and tj = 30 teeth. Fig. 78.— Old rnocJol IV>nl tran.>iiiii.ssic)n. Should the driver wish the car to travel at maximum speed he throws the disc clutch into action which connects .1/ and C and thus causes the power shaft /' to run at the same speed as that of the engine crankshaft. If ho wishes to run at slow speed he operates the pedal which applies the band brake to the drum E causing the latter to come to rest. The gears F, G, H and 8 then form an epicyclic train and for this w 21 ^^ 27 33 ^ 27 =^ "-^^'^^ '"'^' A' = 1 - ft = o.m. So that the power shaft P will turn forward, making 36 revolu- tions for each 100 made by the crank. TRAINS OF GEARING 133 If the ciir is to be reversed, drum / is brought to rest and the train consists of gears J, K, H and B. 1 Then „ 30 27 5 , „ , 24 27 ^ 4 ^ E = I R= ~ or the power shaft P will turn in opposite sense to the crank and at one-fourth its speed. The brake about C is for applying the brakes to the car. (d) Automobile Differential Gear.— The final illustration is the differential used on the rear axle of Packard cars. This is 1: Vutoinobile differential gear. sh«)wn in Fig. 79 which is from a Packard pamphlet. The power shaft /' attached to the Ixjvel pinion A drives the bevel gear B wliicii has its axis at the rear axle but is not directly connected thereto. The wheel B carries in its web bevel pinions C, the axlea of which are mounted radially in B, and the pinions C may rotate freely on these axles. The rear axle S is divided where it passes B and on one part of the axle there is a bev( 1 gear D and on the other one the bevel gear K of the same size jus I). When the car is running on a straight smooth road the two wheels and therefore the two parts ;f|| 11 ':^pimHKmi^i»fw>.si^M 134 THE THEORY OF MACHINES S of the reiir axle run at the same speed and then the power is transmitted from P through A and B just as if the gears C, D and B formed one solid body. In turning a corner, however, the rear wheel on the outer part of the curve runs faster than the inner one, that is D and E run at different speeds and gear C rotates slowly on its axle. WTien the one wheel spins in the mud, and the other one remains stationary, as not infrequently happens when a car becomes stalled, the arrangement acts as an epicyclic train purely. QUESTIONS ON CHAPTER VII 1. Find the velocity ratio for a train of gears as follows: A gear of 30 teeth drives one of 24 teeth, which is on the same shaft with one of 48 teeth • this last wheel gears with a pinion of 16 teeth. 2. The handle of a winch carries two pinions, one of 24 teeth, the other of 15 teeth The former may mesh with a 60-tooth gear on the rope drum or, If desired, the 15-tooth gear may mesh with one of 56 teeth on the same shaft With one of 14 teeth, this latter gear also meshing with the gear of 60 teeth on the drum. Find the ratio in each case. 3. Design a reverted train for a ratio 4 to 1, the largest gear to be not over 9 in. diameter, 6 pitch. 4. A gear a of 40 teeth is driven from a pinion c of 15 teeth, through an Idler h of 90 teeth. Retaining c as before, also the positions of the centers of a and c, it is required to drive a 60 per cent, faster, how may it be done' 6. A car is to he driven at 15 miles per hour by a motor running at 1,200 revolutions per minute. The car wheels are 12 in. diameter and the motor pinion has 20 teeth, driving through a compound train to the axle; design the tram. ^ 6. In a simple geared lathe the lead screw has 5 threads per inch, gear « = 21 teeth, h = 42 teeth, 1 = 60 teeth and 2 = 72 teeth; find the thread cut on the work. 7. It is desired to cut a worm of 0.194 in. pitch with a lathe as shown at lig. 70, using these change gears; find the gears necessarj'. ^ 8. Make out a table of the threads that can be cut with the lathe in Vit ( with different gears. 9. Make a similar table to the above for the Hendev-Xorton lathe illustrated. 10. Design an auto change-gear box of the selective type, with throe speeds and reverse, ratios l.S and .3.2 with % pitch stub gears, shaft centers not over 10 in. 11. A motor car i.s to have a speed of 45 miles per hour maximum with an engine .speed of 1,400 revolutions per minute. What reduction will be required at the n'.^r nx!,. beve! geai's, 36-in. tires? At the same engine speed find the road speed at reductions of 4 and 2 respectively. 12. Design the gear box for the above car with % stub-tooth gears, shafts 9 m. centers. TRAINS OF GE A KIM! 135 18. Prove that the velocity ratio of an epicyclic train is E = \ - R. 14. Design a reverted epicyclic train for a ratio of 1 to 2,500. 15. In a train of gears a has 24 teeth, and meshes with ii 12-tooth pinion 6 which revolves bodily about o, and 6 also meshes with an internal gear c of 48 teeth. Find the ratio with o fixed and also with c fixed. ii m \i\ . v: ■ itl _rK.'.nlc= ,*■><; mKs.vjf." !5-T?!?^f?^^vfi: CHAPTER VIII CAMS 140. Purpose of Cams. — In many classes of machinery certain parts have to move in a non-uniform and more or less irregular way. For example, the belt shifter of a planer moves in an irregular way, during the greater part of the motion of the planer table it remains at rest, the open and crossed belts drivmg their respective pulleys, but at the end of the stroke of the table the belts must be shifted and then the shifter must operate quickly, moving the belts, after which the shifter comes again to rest and remains thus until the planer table has completed its next stroke, when the shifter operates again. The valves of a gas engine afford anotner illustration, for these must be quickly opened at the proper time, held open and then again quickly closed. The operation of the needle bar of a sewing machine is well known and the irregular way in which it moves is familiar to everyone. In the machines just described, and indeed in almost all machines in which this class of motion occurs, the part which moves irregularly must derive its motion from some other part of the machine which moves regularly and uniformly. Thus, in the planer all the motions of the machine are derived from the belts which always run at steady velocity; further, the shaft operating the valves of a gas engine runs at speed proportional to the crankshaft while the needle bar of a sewing machine is operated from a shaft turning uniformly. The problem which presents itself then is to obtain a non- uniform motion in one part of a machine from another part which has a uniform motion, and it is evident that at least one of the links connecting these two parts must be imsymmetrical in shape, and the whole irregularity is usually confined to one part which is called a cam. Thus a cam may be defined as a link of a machine, which has generally an irregular form and by means of which the uniform motion of one part of the machine may be made to impart a desired kind of non-uniform motion to another part. 136 .i^mi^,^^:-:- CAMS 137 Cams are of many different forms and designs depending upon the conditions to be fulfilled. Thus in the sewing machine the cam is usually a slot in a flat plate attached to the needle bar, in the gas engine the cam is generally a non-circular disc secured to a shaft, whereas in screw-cutting machines it often takes the form of a slot running across the face of a cylinder, and many other cases might be cited, the variations in its form being very great. Some forms of cams are shown in Fig. 80. Several problems connected with the use of cams will explain their application and method of design. Ul Stamp-miU Cam.— The first illustration will Ix; that of the stamp mill used in mining districts for crushing oras, and a general view of such a mill is shown in Fig. 81. Such a mill con- sists essentially of a number of stamps A, which are merely Fra. 80. — P'orma of cams. heavy pieces of metal, and during the operation of the mill these stamps are lifted by a cam to a desired height, and then suddenly allowed to drop so as to crush the ore below them. The power to lift the stamps is supplied through a shaft B which is driven at constant speed by a belt, and as no work is done by the stamps as they are raised, the problem is to design a cam which will lift them with the least power at shaft B, and after they have been lifted the cam passes out of gear and the weights drop by gravity alone. Now, it may be readily shown t.dt the force required to move the st.amp at any time will depend upon its acceleration, being least when the acceleration is zero, because then the only force necessary is that which must overcome the weight of the stamp alone, no force being required to accelerate it. Thus, for the (iS 138 Ml r'^ if- 11 II THE THEORY OF MACHINES minimum expenditure of energy, the stamp must be Ufted at a uniform velocity, and the problem, therefore, resolves itself into that of designing a cam which will lift the stamp A at uniform velocity. The general disposition of the parts involved, is shown in Fig. 82, where B represents the end of the shaft B shown in Fig. 81^ Fio. 81.— Stamp mill. and IT represents the center line of the stamp A, which does not pass through the center of B. Let the vertical shank of the stamp have a collar C attached to it, which collar comes into direct contact with the cam on B- then the part C is usually called the follower, being the part of the machine directly actuated by the cam. It will be further assumed that the stamp is to be raised twice CAMS 139 for each revolution of the shaft B, and as some time will be taken by he stamp m falhng. the latter must be raised its full distance while the shaft B turns through less than 180°. Let the total lift occur while B turns through 102°. Further, let the total lift of the cam be h ft., that is. let the The construction of the cam may now be begun. Draw BF ^rpendicular to K/r and lay off the angle FBE equal to^of Next divide the distance - 6 = A, and also the angle FBE. into Fia. 82.— Stamp mill cam. IZiTT ""'"^''' °^ '^"'^^ P^'''' *^« «^'"« ""'"ber being used m each case; six parts have been used in the drawing «nH 7 \l t f nsideration will show that since the stamp A and also_the shaft 5 are to move at uniform speed, the distaic^ ^,1 ^,^-6, etc. and also the angles FBG, GBH HB J do r Ltrv^l T;' *'""«V^ *^^ ^^™^ interv'a/of Jif a';d'ali these intervals of time must be equal. With center B and radius ^^:trtr'?f-;L^*^."r*°*^«"^^«-6-ddra;^^^^ HN etc., tangent to this circle at G, H, etc. Now while the tZ:^r' " nl'^r r ' *^^ ^^^^^ ^ ^ --1-d through ^e angle FBG and then the line GM will be vertical and must be long enough to reach from F to 1 or GM should equal FL The construction is completed by making HN = F - 2 JP 'I t i .1 III s ]■ I- ;11 u] ill If lie 140 THE THEORY OF XfACHINEf^ [ 5?^ Si"? II I! '. -=/!'- 3, etc., and '- this way locating the points 0,M, N, P,Q, R and S and a smooth curve through these points gives the face of the cam. As a guide in drawing the curve it is to be remem- bered that MG, NH, etc., are normals to it. A hub of suitable size is now drawn on the shaft, the dimen- sions of the hub being determined from the principles of niachinn design, and curves drawn from S and down to the hub coinplcte the design; the curve from S must be so drawn Ihat Iho follower will not strike the cam while falling. The curve OMN . . . S is clearly an involute having a base circle of radius BF, or the curve of the cam is that which would be described by a pencil attached to a cord on a drum of radius Fia. 83. — Uniform velocity cam. BF, the cord being unwound and kept taut. The dotted line shows the other half of the cam. In this case there is line contact between the cam and it.s follower, that is, it is a case of higher pairing, as is frequently, though not always, the case with cams. 142. Uniform Velocity Cam.— As a second illustration, take a problem similar to the latter, except that the follower is to have a uniform velocity on the up and down stroke and its line of motion is to pass through the shaft B. It will be further assumed that a complete revolution of the shaft will be necessary for the up and down motion of the follower. CAMS 141 Let - 8, Fig. 83 (a) represent the travel of the follower, the latter being on a vertical shaft, with a roller where it comes into contact with the cam. Divide 0-8 into, say, eight equal parts as shown, further, divide the angle OBK (= 180°) into the same number of equal parts, giving the angles OBV, l'B2', etc. Now since the shaft B turns at uniform speed the center of the follower is at 1 when Bl' is vertical and at 2 when B2' is vertical, etc., hence it is only necessary to revolve the lengths Bl, B2, etc., about B till they coincide with the lines BV B2' etc., respectively. The points 1', 2', 3', will be obtained on the radial lines BV, B2', etc., as the distances from B which the center of the follower must have when the corresponding lino 13 vertical. With centers 1', 2', 3', etc., draw circles to represent the roller and the heavy line shown tangent to these will be the proper outline for one-half of the cam, the other half being exactly the same as this al)out the vertical center line. Here again there is higher pairing and some external force ia supposed to keep the follower always in contact with the cam. A double cam corresponding to the one above described is shown at Fig. 83 (b), this double cam making the follower i)erf.)rni two double strokes at uniform speed for each revolution of the camshaft. 143. Cam for a Shear.— The problem may appear in many different forms and the ca.se now under consideration a-ssumas somewhat different data from the former two, and the shear shown in Fig. 84 may serve as a good illustration. Suppose it is required to design a cam for this shear; it would usually be desir- able to have the shear remain wide open during alwut one-half the time of rotation of the cam, after which the jaw should begin to move uniformly down in cutting the plate or bar, and then again drop quickly back to the wide-open position. With the shear wide open, let the arm l)e in the position .4,5, where it is to remain during nearly one-half the revolution of the cam; then let It be required to move uniformly fnun X,B, jo A.B, while the cam turns through 120°, after which it must drop back again very quickly to ^,jBi. An enlarged drawing of the right-hand end of the machine is shown at Fig. 85, the same letters l)eing used as in Fig. 84, the lines AiBi and A^B^ representing the extreme positions of the arm AB. Draw the vertical line QB^B^ and lay off the angle BiQB'2 equal to 120°; this then is the angle through which the i ,1 m 1^ UI 142 THE THEORY OF MACHINES ^1 camshaft must turn while the arm is moving over its range froni AiBi to AiBf Now (Hvide the angle BiOB,, Fig. 84, into any numlwr of equal parts, say four, by the lines OC, OD, and OE; these lines are shown on Fig. 85. Next, divide the angle Fio. 84. BxOBi into the same number of parts as BiOH^, that is four, by the lines QC, QD' and QE'. Now, when the line QBi is vertical as shown, the cam must he tangent to .4 :Bi. Next, when the cam turns so that QC, becomes ill I i. V . \ Fia. 85. — Cam for shear. vertical, the arm must rise to C, and hence in this position the line OC must be tangent to the cam and the corresponding out- line of the cam may thus be found. Draw the arc CC with center Q, and through C" draw a line making the same angle CA MS 143 a, with QC' that OC does with QC. The lino through C is a tangent to the cftm. Similarly, tangents to the cam through D', E' and B't may be drawn and a smooth curve drawn in tangent to these lines, as shown in Fig, 85. The details of design for the part B't G may be worked out if proper data are given, and evidently the part GFB is circular and corresponds with the wide-open {Kjsition of the shear, 144. Gas-engine Cam. — It not infrequently happens that the follower has not a straight-line motion hut is pivoted at some jKjint and moves in the arc of a circle. This is the case with some gas engines and an outline of the exhaust ram, camshaft Flu. 86. — Gas-engine cams. lever and exhaust valve lor such an engine is shown at Fig. 86 (o), where A is the camshaft and B is the pin about which the fol- lower swings. This presents no difficulties not already discussed but in executing such a design care must be taken to allow for the deviation of the follower from a radial line, and if this is not done the cam will not do the work for which it was intended. As this problem occurs commonly in practice, it may be as well to work out the proper form of cam. The real difficulty is not in making the design of the cam, but in choosing the correct data and in determining the conditions which it is desired to have the cam fulfil. A very great deal of discussion has taken 144 THE THKOHY OF SfACH1SF.fi 1% r ?■ i p R 1 ■ i' t ■■•■ «■ .- - 'v i ■ ': ■' £. ■ Ul place on this point, and as the matter depends primarily on the conditions set in the engine, it is out of place here to enter into it at any length. Such a cam should optm and close the valve at the right instants and should push it opoti far enough, but in addition to these requirements it is neccHsary thai the valve should come back to its seat quietly, and that in moving it should alwa.vs rtMnuin in contact with the cam-actuated oi^rating lever. Further, there hIiuuM Im- no imdue strain at any part of the motion, or the pressure of the valve on the lever should b<^ as nearly uniform and us low u.s possible, during its entire motion. Tne total force required to move the valve at any instant is that necessary to overcome the gas pressure on top of it, plus that necessary to overcome the spring, plus that necessary to lift and accelerate the valve if it has not uniform velocity. The gas pressure is great just at the moment the valve is opened (the exhaust valve is here s(K)ken of) and ininu'u.itely falls almost to that of the atmosphere, while the spring force is least when the valve is closed and most when the valve is wide open. The weight of the valve is constant and its acceleration is entirely under the control of thi designer of the cam. Under the above circumstances it would seem that the acceleration should be low at the moments the valve is ooened and closed, and that it might be increased as the valve is raised, although the increasing spring pressure would prevent undue increase in acceleration. Again, the velocity of the valve at the moment it returns to its seat must be low or there will be a good deal of noise, and the cam should be so designed that the valve can fall rapidly enough to keep the follower in contact with the cam, or the noise will be objectionable. The general conditions should then be that the follower should start with a small acceleration which may be increased as the valve opens more, and that it must finish its stroke at comparatively low velocity. In lieu of more complete data, let it be assumed that the valve is to remain open for 120° of rotation of the ca!n, and is to close at low velocity. The travel of the valve is also given and it is to remain nearly wide open during 20° of rotation. It will first be assumed that the follower moves on a radial line as at Fig. 86 (6) and correction made later for the deviation due to the arc. From the data a.ssumed the valve is to move upward for 50° of rotation of the cam and downward during \W same interval, CAMS 145 and M the catnnhaft turns at coimtant speed each degree of rota- tion represents the same interval of time. Let the acoelerat on be as shown on the diagram Fig. 87 (a) on a base representing degrees of camshaft rotation, which is also a time base; then the assumed form of acceleration curve w. mean that at first the acceleration is zero but that this rapidly increases during the first 5" of rotation to its maximum va.ie at which it remains for H^ 10" H" 1?0 I>*«r— II I /alao Hvcoad* alao Saoonda alw Second! Fill. 87. the next 1.")°. It then drops rapiilly to the great (>st ncKiitivc value where it also remains constant for a short interval and then rapidly returns to zero at which it remains for 20°, after which the process is repeated, ^uch a curve means a rapidly increasing velocity of the valve to its maximum value, followed by a rapid decrease to zero velocity corresponding to the full opening of the valve and in which position the valve remains at rest for 20". The valve then drops rapidly, reaching its seat at the end of 120' at zero velocity. to 146 THE THEORY OF MACHINES By integrating the acceleration curve the velocity curve ia found as shown at (6) Fig. 87, and making a second integration gives the space curve shown at (c), the maximum height of the space curve representing the assumed lift of the cam. These curves show that the valve starts from rest, rises and finally comes to rest at maximum opening; it then comes down with rapid acceleration near the middle of its stroke and comes back on its seat again with zero velocity and acceleration and therefore without noise. 'I ■I ■t ¥ Fio. 88. — Gas-engine cam. Having now obtained the space curve the design of the cam is made as follows; In Fig. 88, let .1 represent the camshaft and the circle G represent the end of the hub of the cam, the diameter of which is determinetl by considerations of strength. There is always a slight clearance left between the hub and follower so that the valve may be sure to seat properly and this clearance circle is indicated in light lines by C. Lay off radii (say) 10° ap.art a.s shown; then AD and AE, 120° apart, represent the angle of action of the cam. Draw a circle F with center A and at distance from C equal to the radius of the roller; then this circle F is the base CAMS 147 circle from which the displacements shown in (c), Fig. 87, are to be laid oflf, and this is now done, one case being shown to indicate the exact method. The result is the curve shown in dotted lines which begins and ends on the circle F. A pair of compasses are now set with radius equal to the radius of the roller of the follower and a series of arcs drawn, as shown, all having centers on the dotted curve. The solid curve drawn tangent to these arcs is the outline of the cam which would fulfil the desired conditions provided the follower moved in and out along the radial line from the center A as shown at Fig. 86(6). Should the follower move in the arc of a circle as is the case in Fig. 86 (a), where the follower moves in the arc of a circle described about B, then a slight modification must be made in laying out the cam although the curves shown at (a), (h) and (c). Fig. 87, would aot be altered. The method of laying out * - cam from Fig. 87 (c) may be explained as follows: From c^ter A draw a circle H (not shown on the drawings) of radius AB equal to +he distance from the center of camshaft to the center of the fulcrum for the lever. Then set a pair of compas.ses with a radius equal to the distance from B to the center of the follower, and with centers on H draw arcs of circles outward from the points where the radial lines AD, etc., intersect the circle F; one of these is shown in Fig. 88. All distances such as a are then laid of! radiaUy from F but so that their termini will be on the arcs just described; thus the point A' will be moved over to L, and so with other points. The rest of the procedure IS as m the former case. For ordinary proportions the two cams will be nearly alike. Should the follower have a flat end without a roller, as is often the case, then the circle F is not used at all and all distances such as a are laid off on radial lines from the circle C and on each radius a line is drawn at right angles to such radius and of length to represent the face of the follower. The outline of the cam is then made tangent to these latter lines. Lack of space prevents further discussion of this very interest- ing machine part, which enters so commonly and in such a great variety of forms into modern machinery. No discussion has heen given of cams having reciprocating motion, nor of those used very commonly in screw machines, in which bars of various shapes are secured to the face of a drum and form a cam which may be easily altered to suit the work to be done by simply •h M \n Hi if 148 THE THEORY OF MACHINES t '■k remo>ring one bar and putting another of different shape in its place. After a careful study of the cases worked out, however, there should be no great difficulty in designing a cam to suit almost any required set of conditions. The real difficulty, in most cases, is in selecting the conditions which the oam should fulfil, but once these are selected the solution may be made as explained. QUESTIONS ON CHAPTER ViU 1. Design a disk cam fur a stamp mill, for a flat-fa<;iHl follower, the line of the stamp being 4 in. from the camshaft. The stamp is to he lifted 9 in. at a uniform rate. 2. Design a disk cam with roller follower to give a uniform rate of rise and fall of 3 in. per revolution to a spindle the axis of which passes through the renter line of the shaft. 3. Taking the proportions of the parts from Fig. 84, design a suitable cam for the shear. i. A cam is required for a 1 in. shaft to give motion to a roller follower ?i in. diameter, and placed on an arm pivoted 6 in. to the l(;ft and 2 in. above the camshaft. The roller (center) is to remain 2 in. above the cam- shaft center for 200° of camshaft rotation, to rise ^i in. at uniform rate during 65°, to remain stationary during the next 30°, and then to fall uni formly to its original position during the next 65°. Design the cam. S. Design a cam similar to Fig. 88 to give a lift of 0.375 in. during 45°, a full open period of valve of 25° and a closing period of 45°. Base radius of cam to be 0.625 in. and roller 1 in. diameter. m CHAPTER IX FORCES ACTING IN MACHINES 146. External Forces. — When a machine is performing any useful work, or even when it is at rest there are certain forces acting on it from without, such as the steam pressure on an engine piston, the belt pull on the driving pulley, the force of gravity due to the weight of the part, the pressure of the water on a pump plunger, the pressure produced by the stone which is being crushed in a stone crusher, etc. These forces are called external because they are not due to the motion of the machine, but to outside influence, and these external forces are trans- mitted from link to link, producing pressures at the bearings and stresses in the Hnks themselves. In problems in machine design it is necessary to know the effect of the external forces in producing stresses in the links, and further what the stresses are, and what forces or pressures are produced at the bearings, for the dimensions of the bearings and sliding blocks depend to a very large extent upon the pressures they have to bear, and the shape and dimensions of the links are determined by these stresses. The matter of determining the sizes of the l)earings or links does not belong to this treatise, but it is in place here to deter- mine the stresses produced and leave to the machine designer the w^k of making the links, etc., of proper strength. In most machines one part usually travels with nearly uniform motion, such as an engine crankshaft, or the belt wheel of a shaper or planer, many of the other parts moving at variable rates from moment to moment. If the links move with variable speed then they must have acceleration and a force must be exerted upon the link to produce this. This is a very important matter, as the forces required to accelerate the parts of a machine are often very great, but the consideration of this question is left to a later chapter, and for the present the acceleration of the parts will be neglected and a mechanism consisting of light, strong parts, which require no force to accelerate Iheni; will be assumed in place of the actual one. 'I 1*1 IS 15() THE TIIKORY OF MACHINES \ r i f- 146. Machine is Assumed to be in Equilibrium. — It will be further assumed that at any instant under consideration, the machine is in equilibrium, that is, no matter what the forces acting are, that they are balanced among themselves, or the whole machine is not being accelerated. Thus, in case of a shaper certain of the parts are undergoing acceleration at various times during the motion, but as tlie belt wheel makes a constant number of revolutions per miruito there must be a balance l)e- twcen the resistance due to the cutting and friction on the one hand and the power brought in by the belt on the other. In the case of a train which is just starting up, the speed is steadily increasing and the train is being accelerated, which simply means that more energy is being supplied through the •'team than is being used up by the train, the balance of the power being free to produce the acceleration, and the forces acting are not balanced. When, however, the train is up to speed and running at a uniform rate the input and output must l)e equal, or the locomotive is in equilibrium, the forces acting upon it being balanced. 147. Nature of Problems Presented.— The most general form of problem of this kind which comes up in practice is such as this: Given the force required to crush a piece of rock, what belt pull in a crusher will be required for the purpose? or: What turning moment will be required on the driving pulley of a punch to punch a given hole in a given thickness of plate? or: Given an indicator diagram for a steam engine, what is the result- ing turning moment produced on a crankshaft?, etc. Such prob- lems may be solved in two ways: (a) by the use of the virtual center; (6) by the use of the phorograph, and as both iij.ethods are instructive each will be discussed briefly. 148. Solutions by Use of Virtual Centers.— This method de pends upon the fundamental pnnciples of statics and the general knowledge of the virtual center discussed in Chapter II. The essential principles may be summed up in the following three statements : If a set of forces act on any link of a nuKjhine then there will be equilibrium, provided: 1. That thf- r-^sultant of the forces is zero. 2. That if the resultant is a single force it passes through a point on the Imk which is at the ineiant at rest. Such a point FORCES ACTING IN MACHINES 151 may, of course, be jxirmanently fixed or at rest, or only tempora- rily so. 3. That if the resultant is a couple the link has, at the instant, a motion of translation. The first statement expresses a well-known fact and requires no explanation. The second statement is rather less known but it simply means that the forces will be in equilibrium if their resultant passes through a point which is at rest relative to the fixed frame of the machine. No force acting on the frame of the machine can disturb its equiHbrium, for the reason that the frame is assumed fixed and if the frame should move in any case where it was supposed to remain fixed, it would simply mean that the machine had been damaged. Further, a force passing through a point at rest is incapable of producing motion. The third statement is a necessary consequence of the second and corresponds to it. If the resultant is a couple, or two parallel forces, then both forces must pass through a point at rest, which IS only possible if the point is at an infinite distance, or the link has a motion about a point infinitely distantly attached to the frame, that is the link has a motion of translation. Let a set of forces act on any link b of a mechanism in which the fixed link is d; then the only point on b even temporarily at rest is the virtual center bd, which may possibly be a permanent center. Then the forces acting can be in equilibrium only if their resultant passes through bd, and if the resultant is a couple l)oth forces must pass through bd, which must therefore be at an infinite distance, or b must at the instant, have a motion of translation. These points may be Ijest explained by some examples. 149. Examples.— 1. Three forces /^, P^ and 1\ Fig. 89, act on the link b; under what conditions will there be equilibrium? In the first place the three forces must all pa.s.s through the same point A on the link, and treating Pi as the force balancing P, and Pi, then in addition to Pa passing through A it must also pass through a point on the link b which is at rest, that is the point bd. This fixes the direction of Pj, by fixing two points on It, and thus the directions of the three forces Pi, Pj and P, are fixed and their magnitudes may be found from the vector triangle to the right of the figure. 2. To find the force Pj acting at the orankpin, in the direction 152 THE THEORY OF MACHINES IS of the connecting rod, Fig. 90, which will balance the pressure P, on the piston. In this case P, and P, may both be regarded as forces acting on the two ends of the connecting rod and the problem is thus similar to the last one. Pi and P, intersect at 6c; hence their resultant P must pass through 6c and also through the only point on b at rest, that is bd, which fixes the position and Fia. 89. direction of P and hence the relation between the forces may be determined from the vector triangle. This enables P, to be found as in the upper right-hand figure. The moment of P, on the crankshaft is P, X OD, which may readily be shown by geometry to be equal to P, X - cc Pi OD smce p^ = Q-:_— > that is, the turning eflFect on the crankshaft Fio. 90. due to the piston pressure P, is the same as if P, was transferred to the point ac on the crankshaft. Let Ps, acting normal to the crank a through the crankpin, be the force which just balances P,; it is required to find P,. Now P, and P, intersect at H, and their balancing force P' must pass through H and through 6d which gives the direction FORCES ACTING IN MACHINES 153 and poaition of P' and the vector triangle EFG gives P» corre- sponding to a known value of P,. The force P, is called the crank effort and may be defined as the force, passing through the crankpin and normal to the crank, which would produce the same turning moment on the crank that the piston pressure does. More will be said about this in the next chapter. 3. Forces Pi and P, act on a pair of gear wheels, the pitch circles of which are shown in Fig. 91; it is required to find the relation between them, friction of the teeth being neglected. Since friction is not considered, the direction of pressure between Fia. 91. the teeth must be normal to them at their point of contact, and is shown at Pj in the figure, this force always passing through the point of contact of the teeth and always through the pitch point or point of tangeney of the pitch circles.' For the involute system of teeth P, is fixed in direction and coincides with the line of, obliquity, but with the cycloidal system Pj becomes more and more nearly vertical as the point of contact approaches the pitch point. Knowing the direction of Pt from these considera- tions, let it intersect Pi and Pj at A and B respectively. On the wheel a there are the forces Pi, P, and P, the latter acting through A and ad, and their values are obtained from the vector triangle; and on b the forces Pj, P, and P', the latter acting through fl ' For a complete diwuasion on these points see Chapter V. f In m m 01 154 THE THEORY OF hfACHTNES and bd, and representing the bearing pressure, are found in the same way, the vector polygon on the left giving the values of the several forces concerned and hence P, if P, is known. 4. The last example taken here is the beam engine illustrated in outline in Fig. 92, and the problem is to find the turning moment produced on the crankshaft due to a given pressure P, acting on the walking beam from the piston. Two convenient methods of solution are available, the first l)eing to take moments about cd and in this way to find the force Pj acting through />.• which is the equivalent of the force P, at C, the remainder of the problem there being solved as in Example 2. Fio. 92. It is more general, however, and usually simpler to determine the equivalent foice on the crankshaft directly. Select any point D on Pi and resolve Pi into two components, one P passing through the only point on the beam c at rest, that is cd, the other. Pa, passing through the common point ac of a and c. The positions of P and P3 and their directions are known, since both pass through D and also through cd and ac respectively; hence the vector triangle on the right gives the forces P3 and P. But Ps acts through ac on a, and ii ad — E = h, he drawn from ad normal to Pi, the moment of P, about the crankshaft is Pih, which therefore balances the moment produced on the crankshaft by the pressure Pi on the walking beam. The magnitude of this moment is, of course, independent of the position of the point D. FORCh'S ACTING IN MACHINES 155 160. General FonnuU.— The general formula for the solu- tion of all such problems by use of virtual centers is as follows : A force Pi acts through any point B on a link 6; it is required to find the magnitude of a force P,, of known direction and posi- tion, acting on a link e which, will exactly balance Pi, d being the fixed link. Find the centers bd, be and ed. Join B to be and bd and resolve Pi into P» in the direction B -be and P* in the direction B — bd; then the moment of Pj about de must be the same as the moment of P2 about the same point and thus Pj is known. 161. Solution of Such Problems by the U^e of the Phoro- graph. — In solving such problems as are now under considera- tion by the use of the phorograph the matter is approached from a somewhat different standpoint, and as there is frequent occasion to use the method it will be explained in some detail. It has already been pointed out that the present investigation deals only with the case where the machine is in equilibrium, or where it is not, on the whole, being accelerated. This is always the case where the energy put into the machine per 8Pcon / '> the lef*-hand end of which is a roller resting OA. The crusher jaw is pivoted on the frame ' '»pp the jaw and the arm HH a fixed ' rns, the eccentric imparts a motion •auses the jaw to have a pendulum -ressure P on a stone to be crushed. II"' ill. rela- .'Uli \ id the f-»— - *i • ii.i. of ref- t" i)tKHi>graph :•. i'.g. 6d, mak- As the device two chains attached t^ on the e at 7 an •• distan" ;) toth' ■ mc .. ' It ti( press Select (I I (ts ; erence am! . il.; of double scH ing OA' = 2 OA. simply employs similar to Fig. 32, viz., OAHCH and JDCH, the images of all the points may readily be found and these are shown on the figure. Then P is tran-'Tred from G to 6" and Q from £ to E' and then P !ind Q both act on the one link and hence their moments muft 1)0 equal, or Q X OE' = mo ment of P about 0, from which P is readily found for a give value of Q. 4. The application to a governor' is shown in Fig. 96 which represents one-half of a Proell governor, and it is required to find the speed of the vertical spindle which will hold the parts m equilibrium in the position shown. In the sketch the arm OA is pivoted to the spindle at and lo the arm BA &t ^ the latter arm carrjdng the ball C on an extension of it and being attached to the central weight W at B. The weight of each revolving ball at C is g lb. and of the central weight is W lb. ' A complete diacussion of governors is niven in Chapter XII. -A, ►] Fig. 96. — Proell governor. 160 THE THEORY OF MACHINES i !i II t. Treating OA as the link of referenc- and G as the center of it, W find the images of /I' at /I and also B' and C, then transfer y w (one-half the central weight acts or each side) to B' and ^ to C", and if it ia desired to allow for the weights Wa and u% of the arms OA and AB the centers of gravity G and H of the latter are found and also their images G' and H', then u^t is transferred to H', but as G' is at G, Wa is not moved. If the balls revolve with linear velocity v ft. per second in a circle of radius r ft., then the centrifugal force acting on each ball will be P = x- X - pds. in the horizontal direction, and this force P is trans- ferred to C Let the shortest distances from the vertical line through to B', C, G' and H' be hi, hj h^ and A* respectively, and let the vertical distance from C to OB' be ^5, then for equili- brium of the parts (neglecting friction), taking moments about O. W w w v" ^•^1 + g/'i + Waha + Whhi= ^ X X ^6 which enables the velocity v necessary to hold the governor in equilibrium in any given position to be found, and from this the speed of the spindle may be computed. 5. The chapter will be concluded by showing two very interest- ing applications to riveters of toggle-joint construction. The first one is shown in Fig. 97, the drawing on the left f,howing the construction of the machine, while on the right is shown the mechanism involved and the solution fo»- finding the pressure P at the piston necessary to e :ert desired rivet pressure R. The frame d carries the cylinder g, with piston /, which is con- nected to the rod e by the pin C. At the other end of e is a pin A which connects e with two links a and 6, the former of which is pivoted to the frame at O. The link b is pivoted at B to the slide c which produces the pressure on the rivet. Select a as the primary link because it is the only one having a fixed point; then A' is at A, and since ii has vertical motion, therefore B' will lie on a horizontal line through O and also on a line through A' in the direction of h, that is, on b produced so that B' is found. C" lies on a line through O normal to the direc- tion of motion of C, that is to the axis of the cylinder (/, and since FORCES ACTING IN MACHINES 161 it also lies on a line through A' parallel to e, therefore C is found. By transferring P to C and R to B' as shown dotted, the rela- tions between the forces P and R are easily found, since their moments about O must be equal, that is, P X OC = R X OB'. By comparing the first and later positions in this and the following figures the rapid increase in the mechanical advantage of the mechanism, as the piston advances, will be quite evident. 6. Another form of riveter is shown at Fig. 98 and the solution for finding the rivet pressure R corresponding to a given piston pressure P is shown along with the mechanism on the right in •1 L«t«r PoiitioQ Fio. 97.— Riveter. two iMsitions. The proportions in the niefliani.sm have been altered to make the Illustratioa more clear. The loose link 6 contains four pivots, C, B, A, F; C being jointed to the frame at D by the link e; B having a connection to the link c, which link is also connected at E to the sliding block e acting directly on the rivet. A is connected to the frame at O by moans of the link o, and F is connected to the piston g at G by means of the link /. Either links o or e may be used as the link of reference, as each has a fixed center, the link a having l)een chosen. The images are found in the following order: C" is on yl'Cand on D'C parallel to DC; B' is next found by proportion, as is also F' and thus the image of the whole link h. Next E' is on B'E', parallel to BE and on O'E' drawn perpendicular to the motion of the slide e, while G' is on a line through perpendicular to the motion of the piston g and is also on the line F'G' parallel to FG. 11 -A 162 THE THEORY OF MACHINES Transfer the force P from G to G', and the force R from E to E', and then the moment about oi R through E' must equal the moment of P through G', that is, R X OE' = P x OG' from which the relation between R and P is computed and this may be done for all the different positions of the piston g. W^lt,D L«ter Poiitlon R Fia. 9.S.— Riveter. QUESTIONS ON CHAPTER IX A\'hat effects do they produce in 1. Why are external forces so named ? the machine? >%lve the /allowing by virlual centers: 2. Determine the crank effort and torq.ie when the crank alible is 45° m an 8 m. by 10-in. engine with rod 20 in. long, the steam pressure l)einK 40 pds. per square inch. 3. In a pair of gears 15 in. and 12 in. diameter respectively the direction of pressure between the teeth is at 75>r to the hne of centers, which is hori- zonta . On the large gear there is a pressure of 200 pds. sloping upward at « and Its hne of action is 3 in. from the gear center. On the smaller gear is a force / actmg downward at 10° and to the left, its line of action being 4 n from the gear center. Find P. 4. Inamechanisn>likeFig.37o = 15 in., 6 = 24in.,d = 4 in. and e = 60 in. and the link a is .Iriven by a belt on a 10-iu. pulley sloping upward at 60" I-ind the relation between the net belt pull and the pressure on/ when a is 6 The connecting rod of a 10 in. by 12-in. engine is 30 in. long and weighs M lb.. Its center of gravity being 12 in. from the crankpin. VVhut tumia^ ellect liotrs trie rol, or what corresponds to one, which will produce a steadying eflFect and the size of the wheel derwuids on tl»> type of engine very largely. Thus, a singhvcylinder engine would have a heavy wheel, a tandem compound engine would also h.ive a heavy one, while for a cross-corn iK)uiKi engine for the same purpose the flywheel could he mucli smaller and lighter. Again, a single-cylinder, four-cycle, single-acting gas engine " aia iia," a iiiuc!! i.ifgcr Wueei iniiii any lonu of steam ert?int;, 164 CHA NK-KFFOHT A SD Tl H SING MOMENT 1G6 and the flywheel si^p would Ammwki «^ the number of cylinders increased, or as the engine was m,»^. ci^.u bio-acting or made to run on the two-cycle principle, mt^Ay bf,-^Hu«. the input to the pistons l^ecomes more constant from irwtan< -o instant, and the energy (iclivcivd, by the fluid becomes m*^. .s*. .Ay. In order that the engineer may undoMa*Ki tr^ auses of these (Ufferenccs.and may know how the n^iaohin.- i^ ^^ *-• designed (he matter will here be dealt with in d.-tail .nA -.m^ first ca«i examined will lie the steam engine. 154. Torque.— An outline of a steam engine i.^ x>x,w„ in Fig IM), and at the instant that the machine is in tin. ^mfmn let Vui. 9!). the steam produce a prc>ssure P on the piston as indir-ated (the method ot arriving at F will l,e explained later), the., it i« re- qunr.i to find the turning moment produced by this pressure on the rr,..nkshaft. It is assumed that the force P acts through the renter of the wrist pin li. Construct the phorograph of the machine and find the image B of li by the prinriples laid down in ("hapier IV. Now m Chapter IX it is showi» that, for (h,- purposes of determining the equihbnur>. of ,M machine, any acting fon-o mimv be transferred from Its actua' point of application to the imago of its [wnnt of "PplK-ation. Honce, th<' force /' acting through H will pro.iuce I he sam(> eflr(>,.t as if this for.v were transferred t<, H' on the OB' is measured in feet. 156. Crank Eflfort.— Xow l(.t the torque T be divided bv the length « of the crank in feet, then since a is constant for all crank , ■ ■ '""■=- •—■-•rrt- -.vhien Ki proportional to the torcjue T produced by tiie steam on the ciankshaft. This 166 THE THEORY OF MACHINES ^5 force id usually termed the crank effort and may be defined as tho force which if acting through the crankpin at right angles to the crank would produce the same turning effect that the actual steam pressure does (see Sec. 149 (2)). Let E denote the crank effort; then Exa = T = PxOB' ft.-pds, or E P- ^-pds. It is evident that the turning moment produced on the crank- shaft by the steam may be represented by either the torque T ft.-pda. or by the crank effort E pds., since these two always bear a constant relation to one another. For this reason, crank efforts and torques are very frequently confused, but it must be re- membered that they are different and measured in different units, and the one always bears a definite relation to the other. The graphical solution for finding the effort E corresponding to the pressure P is shown in Fig. 99. It is only necessary to lay off OH along a to represent P on any convenient scale, and to draw HK parallel to A'B', and then the length OK will represent E on the same scale that OH represents P. The proof is simple. Since the triangles OB' A' and OKH are similar, it is evident that: OK OB' OB' E . ^, „ ^., OH^OA ^ a = P'^'"«« hXa^Px OB'. 166. Crank Effort and Torque Diagrams. — Having now shown how to obtain the crank effort and torque, it will be well to plot a diagram showing the value of these for each position of the crank during its revolution. Such a diagram is called a crank- effort diagram or a torque diagram. In drawing these diagrams tho usual method is to u.sc a straight base for crank positions, the length of the bsise being equal to that of the circumfcretioi; of tho crankpin circle. 167. Example.— Steam Engine.— The nietiiod of plotting such curve from the indicator diagrams of a steam engine is given in detail .so that it may be quite clear. Let the indicator diagrams be drawn as shown in Fig. 100, an outline of the engine being sliovvn in the same figure, and the crank effort* and torques will be plotte u l> Fia. 100. 1 in. in height on the diagram represents s pds. per square inch pressure on the engine piston; thus if s = tiO then each inch in height on the diagram represents a pressure of 60 pds. per square inch on the piston. The lenKths 01 the head- and crank-end diagrams are assumed as li and l^ in. (usually li = It) and these lengths rarely exceed 4 in. irrespective of the size of the engine. Now place the diagrams above the cylinder as in Fig. 100 with the atmospheric lines parallel to the line of motion of the piston. The two diagrams have been separated here for the .sake of clearness, although often they arc superimposed with the atmospheric lines coinciding. Further, the indicator dia- gram lengths have been adjusted to suit tJie length representing the travel of the piston. Whilt this ir- not liec^-s-ary, it wiU fre- quently be found convenient, but all that is really required is to 1 f- 168 THE THEORY OF MACHINES *fr~= atmospheric prewure on ,rale , ^""' '° ""^ of the i.„.,:^ df, ItiorlXt'^Z'^^tT' ""' -. .. ^ .. ,, 1, j,^^ , I'm. lOl.-Crank effort and torque diagram. hence .he piston i, ,„„,,„, ,„™a J Ldef J'plh'ive Lulrof '' = *, X»Xyl,-A, x»XA, pds. .. -.. aacthou already explained and the process repeated CRANK-EFFORT AND Tl'RNING-MOAfENT 109 for each of the 24 crank positions, obtaining in thi« way 24 values of E. These will Ik? found to vary within fairly wide Umits. Then, using the axis of Fig. 101, having a base OX equal the circumfcironce of the crankpin circle, plot the values of E thus found at each of the 24 positions marked and in this way the crank-effort diagram OAfNRSX is found, vertical heights on the diagram representing crank efforts for the corresponding crankpin iwtiitions, and these heights may also be taken to repre- sent the torques on a proper scale determined from the crank- eflfort scale. 168. Rektiun between Crank-effort and Indicator Diagrams. — From its construction, horizontal distances on .ho Cjank-efTort diagram represent space in feet travelled by the pin, while verti- cal distances represent forces in pounds, in the direction of motion of the crankpin, and therefore the area under this curve represents the work done on the crankshaft in foot-pounds. Since the areas of the indicator diagrams represent foot-pounds of work delivered to the piston, and from it to the crank, there- fore the work represented by the indicator diagrams must be exactly equal to that represented by the crank-effort diagram. The stroke of the piston has been taken as L ft. and hence the length of the base OX will represent r X L ft., while the length of each indicator diagram will represent L ft. Calling p„ the mean pressure corresponding to the two diagrams and Em the mean crank effort, then 2 L X p^= w X L X Em ft.-pds. or 2 A-'v = -/>« |)ds., that is, the mean heightofthecrank-effortdiagram . • 2 . in pounds is tunes the mean mdic-ated pressure as shown by the indicator diagrams. In this way the mean crank-effort lin(> LU may be located, and this location may be checked by finding tht* area under the crank-(>ffort diagram in foot-pounds, l)y planimeter and then dividing this by nX will give Em. The crank-effort diagram may also l)e taken io represent torques. Thus, if the diagram is drawn on a vertical scale of K pds. equal 1 in., and if the crank radius is a ft. then torques may be scaled from the diagram using a stale of E X a ft.-pds. equal to I in. The investigation above takes no account of the effect of inertia «>f iiu- jnuis as (his matter is treated extensively in Chapter XV iindfr aeceleratioiis in machinery. 170 THE THEORY OF MACHINES Fi. m^^ ?^"u°' ^'^^ Engine..-An examination of Hk. 10 «how8 that the t..r„inK moment on the crankshaf" h .he 0MK.ne d«cu«sed. i« v. ry variable indeed and thiswou d ^•a.iHo .ortam vanations i„ the operation of the engine which W.I1 .>e d«cu««ed later. In the meantime it may be ^Jted Z . cHignen, try to arrange the machinery aa far as powible to prt duce uniform effort and torque. H^wioie lo pro- ■'■ullut Ttr^a* . Fia 1()2.-T..rquc diagrams for rross-com pound or.iTne. ritean, . ngines arc frequently designed with more than one cy nuler. sometames as compound engines and ^metimes as tmn arrungenaents, as in the locomotive and in many rolling-m^ engines. ( ompound engines may have two or three and Le- times four expansions, requiring at least two, three or four cvlin- either with the cylinders tandem and having both pistona Fig. 103.— Torque diagrams for tandem engine w th h^ V T T" r'"''^'"^' "^ ""^ cross-compound engines with the cylinders placed side by side and each connected through ts own crosshead and connecting rod to the one crankshaft he cranks being usually of the same radius and being set at 90» to one another In Fig. 102 are shown torque diagrams for twhi eng,nes as used u. the locomotive or for a cro.<,-compound engine with cranks at 90°, the curve A showing the f.rque c^respond ig iZ.*.. r-r^!E«.JH!!T7.' fEK^'-"*'^!^ CRANK-EFFORT AND TVRNINO.MOMENT 171 to the high-pressure cylinder with leading crank and B that for the low-presstire cylinder, while the curve C in plain lined gives the resultant torque on the crankshaft, and the horiaontal dottetl line D shows the corresponding mean torque. The very great improvement in the torque diagram resulting from this arrange- ment of the engine is evident, for the torque diagram C varies very little from the mean line I) and is never negative as it was with the single-cylinder engine. On the other hand, the tandem engine shows no improvement in this respect over the single-cylinrler machine as is shown by the torque diagram corresponding to it shown in Fig. 10.3, the dotted curves corresponding to the separate cylinders and the plain curve Iwing the resultant torque on the shaft. ^ !!•»« Torquf /' K y'^ ''^ /^- /'\ X. Fia. 104.— Torque diagram for tripie-txpanaion engine. Increasing the number of cylinders and cranks usually smooths out the torque curve and Fig. 104 gives the results obtained from u triple-expansion engine with cranks set at 120°, in which it is seen that the curve of mean torque diflfers very little from the actual torque produced by the cylinders. 160. Internal-combustion Engines.— It will be well in con- nection with this question to examine its bearing on internal- combustion engines, now so largely used on self-propelled vehicles of all kinds. Internal-combustion engines are of two general classes, two-cycle and four-cycle, and almost all machines of this class are single-acting, and only such machines are discussed here as the treatment of the double-acting engine offers no difficulties not encountered in the present case. In the case of four-cycle engines the first outward stroke of the piston draws in the explosive mixture which is compressed in the return stn.ke. At the end nf this stroke the charge is ignited and the presiure rises sufficiently to drive the piston forward on .'Avr^i'i^ :...id°: MMOa MKROCOPV tfSCMUTION TEST CHART (ANSI ond ISO TEST CHART No. 2) A /APPLIED IIVHGE 1653 Cast Mom Street Rochester, New York U609 uSA (716) 482 - 0300 - Phone (716) 288 - S989 - Tq» 172 THE THEORY OF MACHINES the third or power stroke, on the completion of which the exhaust valve opens and the burnt products of combustion are driven out l)y the next instroke of the piston. Thus, there is only one power stroke (the third) /or each four strokes of the piston or for each two revolutions. An indicator diagram for this t> p^ of engine is shown in (a) Fig. 105, and the first and fourth strokes are represented by straight lines a little below and a little 'ibove the atmospheric line respectively. The indicator diagram from a two-cycle engine is also shown in (6) Fig. 105 and differs very little from the four-cycle card Fig. 105. — Gas-engine diagrams. except that the first and fourth strokes are omitteil. The action of this type may be readily explained. Imagine the piston at its outer end and the cyli:ider containing an explosive mixture then as the piston moves in the charge is compressed, ignited near the inner dead point, and this forces the piston out on the next or power stroke. Near the end of this stroke the exhaust is opened and the burnt gases are displaced and driven out by a fresh charge of combustible gas which is forced in under slight pressure; this charge is then compressed on the next instroke. In this cycle there is one power stroke to each two strokes of the piston or to each revolution, and thus the machine gets the same number of power strokes as a single-acting steam engine. The torque diagram for a four-cycle engine is shown in Fig 106 and Its appearance is very striking as compared with thos.- for the steam engine, for evidently the torque is negative for th^e out of the four strokes, that is to sav, there has to be sufficient energy in the machine parts to move the piston during these strokes, and all the energy is supplied by the gas through the one power or expansion stroke. The torcpie has evidently very large variations and the total resultan* nunn torque is very small indeed. For the two-cycle engine the torque diagram will he similar to the part of the curve shown in Fig. 106 and included in the com- pression and expansion strokes, the suction and exhaust strokes CRANK-EFFORT AND TURNING-MOMENT 173 being omitted. Evidently also the mean torque line will be much higher than for the four-cycle curve. Returning now to the four-cycle cngiuc; it is soon that the turn- ing moment is very irregular and if such an engine were used with a small flywheel in driving a motorcycle or dynamo, the motion would be very unsteady indeed, and would give so much trouble that some special means must be used to control it. Various methods are taken of doing this, one of the most common Fic. 1(H). — Torque diagram for four-cycle f,aa engine. in automol)ilos, etc., being to increase the number of cylinders. Torque diagrams from two of the more common arrangements are shown in P'ig. 107. The diagram marked (n) gives the results for a two-cylinder engine where those are either opposed or are placed side by side and the cranks are at 180". Diagram (ft) gives the results from a four-cylinder engine and corresponds Fig. 107. — Torque diagrams for multicylindcr gasoline engines. to the use of two opposed engines on the same shaft or of four cylinders side by side, each crank being 180° from the one next it. All of the arrangements shown clearly raise the line of mean torque and thus make the irregularities in the turning moment very much less, and by sufficiently increasing the number of cylinders this moment may be made very regular. Some auto- mobiles now have twolvo-cylinder engines resulting in very uniform turning moment and much steadiness, and the arrange- 174 THE THEORY OF MACHINES II ments made of cylinders in aeroplanes are particularly satis- factory. Space does not permit the discussion of the matter in any further detail as the subject is one which might profitably form a subject for a special treatise. 161. General Discussion on Torque Diagrams.— The unsteadi- ness resulting from the variable nature of the torque has been referred to already and may now be discussed more in detail, although a more complete treatment of the subject will be found in Chapter XIII under the heading of "Speed Fluctuations in Machinery." For the purpose of the discussion it is necessary to assume the kind of load which the engine is driving, and this affects what is to be said. Air compressors and reciprocating pumps produce variable resisting torques, the diagram representing the torque required to run them being somewhat similar to that of the engine, as shown in Fig. 101. It will be assumed here, however, that the engine is driving a dynamo or generator, or turbine pump, or automobile or some machine of this nature which requires a constant torque to keep it moving; then the torque required for the load will be that represented by the mean torque line in the various figures, and this mean torque is therefore also what might be called the load curve for the engine. Consider Fig. 101 ; it is clear that at the beginning of the revolu- tion the load is greater than the torque available, whereas be- tween M and N the torque produced by the engine is in excess of the load and the same thing is true from R to S, while NR and SU on the othei hand represent times when the load is in excess. Further, the area between the torque curve and MN plus the corresponding area above RS represents the total work which the engine is able to do, during these periods, in excess of the load, and must be equal to the sum of the areas between LM, NR and SU and the torque curve. Now during MN the excess energy must be used up in some way and evidently the only way is to store up energy in the parts of the machine during this period, which energy will be restored by the parts during the period NR and so on. The net result IS that the engine is always varying in speed, reaching a m ;ximum at N .nnd S and minimum values at M and R, and the amount of these speed variations will depend upon the mass of the moving parts. It is always the purpose of the designer to limit these variations to the least practical amounts and the torque curves CRANK-EFFORT AND TURNING-MOMENT 17:» show one means of doing this. Thus, the tandein compound engine has a very decided disadvantage relative to the cross- compound engine in this respect. Internal-combustion engines with one cylinder are also very deficient because Fig. 106 shows that the mean torque is only a small fraction of the maxinmm and further that enough energy must ba stored up in the moving parts during the expansion stroke to carry the engine over the next three strokes. When such engines are used with a single "ylinder they are always constructed with very heavy flywheels in order that the parts may be able to store up a large amount of energy without too great variation in speed. In automobiles large heavy wheels are not possible and so the makers of these machines always use a number of cylinders, and in this way stamp out very largely the cause of the difficulty. This is very well shown in the figures representing the torques from multicylinder engines, and in such engines it is well known that the action is very smooth and even and yet all parts of the machine including the flywheel are quite light. It has not yet been possible, however, to leave off the flywheel from these machines. QUESTIONS ON CHAPTER X 1. Using the data given in the engine of Chapter XIII and the indicator diagrams there given, plot the crank effort and torque diagrams. For what crank angle are these a maximum? 2. What would be the torque curve for two engines similar to that in question 1, with cranks coupled at 90°? ,\t what crank angle would these give the maximum torque? 3. Compare the last results with two cranks at ISO" and tlireo cranks at 120°. 4. Using the diagram Fig. 161 and the data connected therewith, plot the crank-effort curve. 6. In an automobile motor 3}f in. bore and 5 in. stroke, the rod is 12 in. long; assuming that the diagram is similar to Fig. 161, but the pressures are only two-thinls as great, draw the Ciank-effort curve and torque diagram. Draw the resulting curve for two cylinders, cranks at 180°; four cylinders, cranks at 90°; six cylinders, cranks at 60° and at 120° respectively. Try the effect of different sequence of firing. t CHAPTER XI THE EFFICIENCY OF MACHINES 162. Input and Output.— The accurate determination of the efficiency of machines and the loss by friction is extremely com- plicated and difficult, and it is doubtful whether it is possible to deal with the matter except through fairly close approxima- tions. All machines are constructed for the purpose of doing some specific form of work, the machine receiving energy in one form and delivering this energy, or so much of it as is not wasted, in some other form; thus, the water turbine receives energy from the water and transforms the energy thus received into electrical energy by means of a dynamo, or a motor receives energy from the electric circuit, and changes this energy into that necessary to drive an automobile, and so for any machine. For convenience, the energy received by the machine will be referred to as the input and the energy delivered by the machine as the output Now a machine cannot create energy of itself, but is only used to change the form of the available energy into some other which is desired, so that for a complete cycle of the machine {e.g., one revolution of a steam engine, or two revolutions of a four-cycle gas engine or the forward and return stroke of a shaper) there must be some relation between the input and the output. If no energy were lost during the transformation, the input and output would be equal and the machine would be perfect, as it would change the form of the energy and lose none. However, if the input per cycle were twice the output then the machine would be imj)erfect, for there would lie a loss of one-half of the energy available during the transformation. The output can, of course, novrr exceed the input. It is then the province of the designer to make a machine so that the output will be as nearly equal to the input as possible and the more nearly these are to being equal the more perfect will the machine be. 176 THE KVFICIENCY OF MACHINES 177 163. Efficiency. — In dealiiiR with machinery it is customary to use the term mechanical efficiency or efficiency to denote the ratio of the output per cycle to the input, or the efficiency »; = r - , — P*^_ y . The maximum value of the efficiency is mput per cycle unity, which corresponds to the perfect machine, and the mini- mum value is zero which means that the machine is of no value in transmitting energy; the efficiency of the ordinary machine lies between these two limits, electric motors having an efficiency of 0.92 or over, turbine pumps usually not over 0.80, large steam pumping engines over 0.90, etc., while in the case where the clutch is disconnected in an automobile engine the efficiency of the latter is zero, all the input being used up in friction. The quantity 1 — »j represents the proportion of the input which is lost in the bearings of the machine and in various other ways; thus in the turbine pump above mentioned, r; = 0.80 and 1 — t; = 0.20, or 20 per cent, of the energy is wasted in this case in the bearings and the friction of the water in the pump. The amount of energy lost in the machine, and which helps to heat up the beanngs, etc., will dejiend on such items as the nature of lubricant used, the nature of the metals at the bear- ings and other considerations to be discussed later. Suppose now that on a given machine there is at any instant a force P acting at a certain point on one of the links which point is moving at velocity v\ in the direction and sense of P; then the energy put into the machine will be at the rate of Pvi ft.-pds. per second. At the same instant let there be a resisting force Q acting on some part of the machine and let the point of application of Q have a velocity with resolved part vz in the direc- tion of Q so that the energy output is at the rate of Qvt ft.-pds. per second. The force P may for example be the pressure acting on an engine piston or the difference between the tensions on the tight and slack sides of a belt driving a lathe, while Q may repre- sent the resistance offered by the main belt on an engine or by the metal being cut off in a lathe. Now from what has been P.Vi already stated the efficiency at the instant is »? = .- mput and if no losses occurred this ratio would be unity, but is always less than unity in the actual case. Now, as in practice Qi'j is always less than Pvi, choose a force Po acting in the direction of, and through the point of application of P svh that Povi = 13 178 THE THEORY OF MACHINES Qvt, then clearly Po is the force which, if applied to a friction- less machine of the given type, would just balance the resist- ance Q, and V = Pv, Povi PVi Po p Po so that evidently the efficiency will be ^/ at the instant, and Po will always bo less than P. The efficiency may also be expressed in a different form. Thus, let Qo be the force which could be overcome by the force P if there were no friction in the machine; then Pvi = QoVt and therefore Qf2 Pi'i Qi2 Q Q V ^ ^'^^ ^" ^^ always greater than Q. 164. Friction. — Whenever two bodies touch each other there is always some resistance to their relative motion, this resistance being called friction. Suppose a pulley to be suitably mounted in a frame attached to a beam and that a rope is over this pulley, each end of the rope holding up a weight w lb. Now, since each of these weights is the same they will be in equiUbrium and it would be expected that if the slightest amount were added to either weight the latter would descend. Such is, however, not the case, and it is found by experiment that one weight may be considerably increased without disturbing the conditions of rest. It will also be found that the amount it is possible to add to one weight without producing motion will depend upon such quantities as the amount of the original weight w, being greater as w increases, the kind and amount of lubricant used in the l)earing of the pulley, the stiffness of the rope, the materials used in the bearing and the nature of the mechanical work done on it, and upon very many other considerations which the reader will readily think of for himself. One more illustration might be given of this point. Suppose a block of iron weighing 10 lb. is placed upon a horizontal table and that there is a wire attached to this block of iron so that a force may be produced on it parallel to the table. If now a tension is put on the wire and there is no loss the block of iron should move even with the slightest tension, because no change is being made in the potential energy of the block by moving it from place to place on the table, as no alteration is taking place in its height. THE KFFICfHNCY OF MMlIINRfi 179 It will be found, however, that the block will not begin to move until considerable force is produced in the wire, the force possibly running as high as 1.5 pds. The maRnitude of the forci' necessary will, as before, depend upon the material of the table, the nature of the surface of the table, the area of the face of the block of iron touching the table, etc. These two examples serve to illustrate a very important matter (ronnected with machinery. Taking the case of the pulley, it is found that a very small additional weight will not cause motion, and since there must alwayis be equilibrium, there must be some resisting force coming into play which is exactly equal to t'lat produced by the additional weight. As the additional weight increases, the resisting force must increase by the same amount, but as the additional weight is increased more and more the resist- ing force finally reaches a maximum amount, after which it is no longer able to counteract the additional weight and then motion of the weights begins. There is a peculiarity about this resisting force then, it begins at zero where the weights are equal and increa.ses with the inequality of the weights but finally reaches a maximum value for a certain difference l)etween them, and if the difference is increased beyond this amount the weights move with acceleration. In the case of the block of iron on the table something of the same nature occurs. At first there is no tension in the wire and therefore no resisting force is necessary, but as the tension in- creases the resisting force must also increase, finally reaching a maximum value, after which it is no longer able to resist the tension produced in the wire and the block moves, and the motion of the block will be accelerated if the tension is still further increased. This resisting force must be in the direction of the force in the wire but opposite in sense, so that it must act parallel to the table, that is, to the relative direction of sliding, and increases from zero to a limiting value. The resisting force referred to above always acts in a way to oppose motion of the parts and also always acts tangent to the surfaces in contact, and to this resisting force the name of friction has been applied. Much discussion has taken place as to the nature of the force, or whether it is a force at all, but for the present discussion th's idea will be adopted and this method of treatment will give a satisfactory solution of all problems con- nected with machinery. ISO THE THKOIiY (W MACHINVfi IP Wherever motion exists friction is always acting in a sense opposed to the motion, although in many cases its very presence is essential to motion taking place. Thus it would be quite impossible to walk were it not for the friction between one's feet and the earth, a train could not run were there no friction between the wheels and rails, and a Mi would l)e of no use in transmitting power if tl.. "c were no friction l)etween the belt and pulley. Friction, therefore, acts as a resistance to motion and yet without it many motions would be imiK)Ssible. 165. Laws of Friction. — A great many exjM'riments have Jn-en made for the purpose of finding the relation between the friction and other forces acting between two surfaces in contact. Morin stated that the frictional resistance to the sliding of one body upon another depended upon the normal pressure between the surfaces and not uixin the areas in contact nor upon the velocity of slipping, and further that if F is the frictional resistance to slipping and N the pressure between the surfaces, then F = nN where n is the coefficient of friction and depends upon the nature of the surfaces in contact as well as the materials composing these surfaces. A discussion of this subject would be too lengthy to place here and the student is referred to the numerous experiments and discussions in the current engineering periodicals and in books on mechanics, such as Kennedy's "Mechanics of Machinery," and Unwin's "Maciiine Design." It may only be stated that Morin's statements arc known to be quite untrue in the case of machines where the pressures are great, the velocities of sliding high and the methods of lubrication verj' variable, and special laws must be formulated in such cases. In machinery the nature of the rubbing surfaces, the intensity of the pressures, the velocity of slipping, methods of lubrication, etc., vary within very wide limits and it has been found quite impo.ssible to devise j',ny formula that would include all of the cases occurring, or even any great number of them, when condi'ions are so variable. The only practical method seems to be to draw up formulas for each particular class of machinery and method of lubrication. Thus, before it is possible to tell what friction there will be in the main bearing of a steam engine, it is necessary to know by experiment what laws exist for the friction in case of a similar engine having similar materials in the shaft and bearing and oiled in the same way, and if the machine is a horizontal Corliss THE EFFICIENCY OF MACHINES 181 engine the laws would not Ihj the same a» with a vertical high- Hjieed engine; again the laws will depend ui>on -vhether the lubri- cation is forced or gravity and on a. great many other things. For each type of bearing and lubrication there ,vill be a law for determining the frictional loss and these laws niUHt in each case l)e determined by careful expiirinient. 166. Friction Factor. — Following the method of Keimedy and other writers, the formula uh

F = /A' for tietermining the fricti(jnal force /' corn'S|K)nding to a normal pres- sure A' between the rubbing surfaces, where / is called the fric- tion factor and differs from the (HKffiicient of friction of Morin in that it depends upon a greater numlier of elements, and the law for / must be known for each class of surfaces, method of lubrication, etc.. from a series of ex|x?rinjent8 ix^formed on similarly constructed and operated surfaces. In dealing with machines it luus l)een shown that they are made up of parts united a.sually by sliding or turning pairs, so that it will he well at first to study the friction in thcwe pairs separately. FRICTION IN SLIDING PAIRS N d Fig. 108. 167. Friction in Sliding Pairs. — Consider a pair of sliding elements as shown in Fig. 108 and let the normal comi)onent of the pressure between these two elements \xi N, and let K be the resultant external force acting ujjon the upjKjr element which is moving, the lower one, for the present being considered stationary. Let the force li act pnrallel to the surfaces in the sense shown, the tendency for the body is then to move to the right. Now, from the previous discussion, there is a certain resistance to the motion of a the amount of which is fN, where / is the friction factor, and this force must in the very nature of the case act tangent to the surfaces in contcict (Sec. 164); thus, from the way in which R is chosen, the friction force F = fN and R are parallel. Now if R is small, there is no motion, as is well known, for the maximum value of F due to the normal pressure A"" is greater than R; this corresponds to a sleigh stalled on a level road, the horses being unable to move it. If, however, R be increased steadily it reaches a point where it is equal to the maximum value of F and then the body will begin to move, and so long as m IS2 THE TllFJiliY OF MM'UINEfi ^m R and F are equal, will continue to move at uniform speed bocauBe the force R is just balanced by the reHiMtuncc to motion; thi>« corresponds to the case where the sleigh is drawn along a level road at uniform 8|)eed by a team of horses. Should R be still further increased, then since the frictional resistance F will Ix; less than R, the body will move with increasing speed, the acceler- ation it has de|)ending upon the excess of R over /'; this corre- sponds to horses drawing a sleigh on a level road at an increasing speed, and just here it may In; pointed out that the friction factor must depecau»e the horses soon reach a speed lieyond which they cannot go. These results may be sunmiarized a.i> 'ollows: 1. If R is less than F, that is RfN, there will be accelerated motion, relatively, between the bodi»'s. R is the resultant external force acting on the body and "s parallel to the surfaces in contact. Consider next the case shown in Fig. 109, where the resultant external force R acts at an angle to the normal to the surfaces in contact, and let it be assumed that the motion of a relative to d is to the right as shown by the arrow. The bodies are taken J^^ to be in equilibrium, that is, the velocity Fig. 109 "^ slipping is uniform and without accelera- tion. Resolve R into two components AB and BC, parallel and normal resrwctively to the surfaces of contact, ^lien since BC = A' is the normal pressure between the surfaces, the frictional resistance to slipping will be F = fN, from Sec. 1G6, where /is the friction factor, and since there is equilibrium, the velocity being uniforn., the value of F nmst be exactly equal and op{)osite to AB, these two forces being in the 'ame direction. Should AB exceed F = fX there would be ac- celeration, and should it be less than/iV there would be no motion. Now from Fig. 109, AB = R sin and also AB = BC tan = N tan . Hence, since AB = fN, there results the relation fN = N tan ov f = tan 0; this is lo suy, in order that two bodies may have relative motion at uniform velocity, the re- sultant force must act at an angle to the normal to the rubbing surfaces, and on such a side of the normal as to have a resolved THE KFFiriKNCY OF MACHINES 18.-) part in the direction of motion. The angle ^ it fixed by the fact that its tangent ir the friction factor /. 168. Angle of Friction. — The all-in tnay he con vctiiontly callod the angle of friction iiiul wliorcviT thi- syinliol <(> occurs in the rest of this chapter it Htuiids for the alible of friction and is such that its tanRcnt is the friction factor/. The angle is, of course, the limiting inclination of the resultant to the normal and if the re- sultant act :it any other angle less than to the normal, motion will not occur; whereas if it should act at an angle greater than 4> there will he accelerated motion, for the simple reason that in the latter case, the resolved part of the resultant parallel to che sur- faces would exceed the frictional resistance, and there would then be an unbalanced force to cause acceleration. 169. Ezamples.~A few examples should make the |)rinciples clear, and in those first given all friction is neglecteu excejjt that Fi(i. 111). — C'r()s.sli('ii to the normal to the surfaces in contact, (Sec. 167); thus R has the direction shown. Note that the side of the normal on which K lies must be so chosen that R ha^ a component in the direction of motion. Now draw AB = P the steam pressure, and draw AC and BC parallel respectively to R and Q, then BC = Q the thrust of the rod and AC = R the resultant pressure on the crosshead shoe. 184 THE THEORY OF MACHINES If there were no friction in the sliding pair R would be normal to the surface and in the triangle ABD the angle BAD would be 90°; BD is the force in the connecting rod and AD is the pressure on the shoe. The efficiency in this position will thus be _ BC _ Q "' BD~ Qo necessary to overcome Q if there were no friction by drawing CE P BF normal to AB then Po = BE and v = ~p = 'nj' 2. A cotter is to l)e designed to connect two rods, Fig. Ill; Or it is just as direct to find Po the force .1' f li Fui. 111.— Cotter pin. it is required to find the limiting taper of the cotter to prevent it slipping out whon the rod is in tension. It will be assumed that both parts of the joint have the same friction factor /, and hence the same friction angle <^, and that the cotter tapers only on one side with an angle 9. The sides of the cotter on which the pressure comes are marked in heavy lines and on the right-hand side the total pressure Ri is divided into two parts by the shape of the outer piece of the (connection. Both the forces i?i and ftj act at angle to the normal to their surfaces and, from what has already been said, it will be understood that when the cotter just begins to slide out they act on the side of the normal shown, so that by drawing the vector triangle on the left of height AB = P and having CB and BD respectively parallel to Ri and R2, the force Q necessary to force the cotter out is given by the side CD. THE EFFICIENCY OF MACHINES IS,") In the figure the angle ABC = and ABD = - e. Therefore Q = P [tan + tan (<^ - e)\ The cotter will slip out of itself when Q = 0, that ^s tan + tan {<(> — $)= O, or 8 = 2 This angle d is evidently independent of P except in so far as is affected by the tension P in the rod. l''i + tan (tf + <^)] and Q' increases with 6. Small values of d make the cotter easy to drive in and harder to drive out. 3. An- interesting example of the friction in sliding connections is given in Fig. 112, which shows a jack commonly used in lifting automobiles, etc.; the outlines of the jack only are shown, and i! 186 THE THEORY OF MACHINES li no details shown of the arrangement for lowering the load. In the figure the force P applied to lifting the load Q on the jack is as- sumed to act in the direction of the pawl on the end of the handle, and this would represent its direction closely although the direc- tion of P will vary with each position of the handle. The load Q is assumed applied to the toe of the lifting piece, and when the load is being raised the heel of the moving part presses against the body of the jack with a force Ri in the direction shown and the top pressure between the parts is R-2, both Ri and Ri being inclined to the normals at angle . At the base of the jack are the forces Q and Ri, the resultant of which must pass through A, while at the top are the forces Ri and P, the resultant of which must pass through B; and if there is equilibrium the resulti.nt Fi of Q and Ri must balance the resultant Fi of Ri and P, which can only be the case if Fi passes through A and B; thus the direction of Fi is known. Now draw the vector triangle ECG with sides parallel to Fi, Ri and P, and for a given value of P, so that Fi = EC and Ri = CG. Next through E draw ED parallel to Ri and through C draw CD parallel to Q from which Q = CD is fouad. If there were no friction the reactions between the jack and the frame would be normal to the surfaces at the points of contact, thus A would move up to A o and B to Bo u.nd the vector diagram would take the form EDoCjG where EG = P aa before and DqCq = Qo so that Qo is found. Q The efficiency of the device in this position is evidently r; = q^ It is evident that with the load on the toe, the efficiency is a maximum when the jack is at its lowest position because AB is then most nearly vertical, while for the ve;y highest positions the efficiency will be low. 4. One more example of this kind will suffice to illustrate the pnnciples. Fig. 113 shows in a very elementary form u (luick- return motion used on shapcrs and machine tools, and illustratetl at Fig. 12. Let Q be the resistance offered to the cutting tool which is moving to the right and let P be the net force applied by the belt to the circumference of the belt pulley. For the present problem only the friction losses in the sliding elements will be considered leavuig the other parts till later. Here the tool holder g presses on the upper guide and the pressure on this guide is fti, the force in the rod e is denoted by /''i. Further the THE EFFICIENCY OF MACHINES 187 pressure of 6 on c is to the right and as the former is moving downward for this position of the machine, the direction of pressure between the two is Rt through the center of the pin. Now on the driving link a the forces acting are P and R2, the resultant F2 of which must pass through and A. In the vector diagram draw BC equal and parallel to P, then CD and BD parallel respectively to F2 and R2 will represent these two forces '/w77ZV7; ^-^-^— / -^ IH. l''u;. 113. — Quick-return mot'on. SO that Ri is dotcrniined. Again on c the forces acting are R% and Fi, and their resultant passes through Ox and also through E, the intersection of Fi and Rt, so that drawing BG and DG in the vector diagram parallel respectivel3' to Fz and Fx gives the force Fx = IXi in the rod e. Acting on the tool holder q are the forces Fx, Q and Rx and the directions of them are known and also the magnitude of Fx, hence complete the triangle GHD with sides parallel to the forces concerned and then GH = Q and HD = Ri which gives at once the resistance Q which can be overcome at the tool by a given net force P applied by the belt. If there were no friction in these sliding ps'rs then the forces Rx and ^2 would act normal to the sUding surfaces instead of at angles <^i and j to the normals so that A moves to Ao and H 1 ; 188 THE THEORY OF MACHINES E to Eo and the construction is shown by the dotted linew, frf>ni which the value of Qo is obtained. The efficiency for this posi- tion of the machine is 17 = v; • The value of »j should be found for a number of other positions of the machine, and, if dtjirable, a curve may be plotted so that the effect of friction may be properly studied. Before passing on to the case of turning pairs the attention of the reader is called to the fact that the greater part of the problem is the determination of the condition of static equilibrium as described in Chapter IX, the method of solution being by means of the virtual center, in these cases the permanent center being used. The only difficulty here is in the determination of the direction of the pressures I{ ii between the sliding surfaces, and the following suggestions may be found helpful in this regard. Let a crosshead o. Fig. 1 14, slide between the two guides di and dt, first find out, by inspection generally, from the forces acting whether the pressure is on the guide di or da. Thus if the cou- in compression the pressure 111. necting is on di are rod and piston rod if both are in tension it is on di, etc., suppose for this case that both are in compression, the heavy line showing the surface bearing th r sure. Next find the reiu.. . to the normal where = tan"'/, / being the friction factor, so that the resultant must be either in the direction of Ri or R'l. Now Ki the pressure of a upon do .acts downward, and in order that it may have a resolved part in the direction of motion, then Ri and not Ri' is the correct direction. If Ri is treated as the pressure of dj uiK>n a then Ri acts upward, but the sense of THE EFFICIENCY OF MACHINES 189 motion of dj relative to a is the opposite of that of a relative to di, and hence from this point of view also Ri is correct. It is easy to find the direction of Ri by the following simple rule: Imagine either of the sliding pieces to be an ordinary carpenter's wood plane, the other sliding piece being the wood to 1)6 dressed, then the force will have the same direction as the tongue of the plane when the latter is being pushed in the given direction on the cutting stroke, the angle to the normal to the surfaces being . 170. Turning Pairs. — In dealing with turning pairs the same principles are adopted as are used with the sliding pairs and should not t'HUse any difficulty. Let a, Fig. 115, represent the outer Pio. 115. element of a turning pair, such as a loose pulley turning in the sense shown upon the fixed shaft d of radius r, and let the forces P and Q act upon the outer element. It must be explained that the arrow shows the sense in which the pulley turns relatively to the shaft and this is to be understood as the meaning of the arrow in the rest of the present discussion. It may Ije that both elements are turning in a given case, and the two elements may also turn in the same or in opposite sense, but tho arrow indicates the relative sense of motion and the forces P a ad Q are assumed to act upon the link on which they are drawn, that is upon a in Fig. 115. 190 TJIE TIIKORY OF MAC II INKS I I ^ I li i ^ ' F if \ - ;- 1 r If there were no friction then the resultant of P and Q would pass through the intersection A of these forces and also through the center of the bearing, so that under these circumstances it would be possible to find Q for a given value of F by drawing the vector triangle. There is, however, frictional resistance offered to motion at the surface of contact, hence if the resultant R of P and Q acted through 0, there could be no motion. In order that motion may exist it is necessary that the resultant produce a turning moment about the center of the bearing equal an«i opposite to the resist- ance offered by the friction between the surfaces. It is known already that the frictional resistance is of such a na- ture as to oppf>se motion, and hence the resultant force must act in such a way as to produce a turning mo- ment in the sense of mo- tion equal to the moment offered by friction in the opposite sense. Thus in the case shown in the figure the resultant nuist pa.ss through .1 and lie to the left of O. In Fig. 116, which shows an enlarged view of the bearing let V be the iKTpendi. .ila. distance from to R, so that the moment of /,• about O is R X p. Ihe point C may be conveniently called the center of prersure, being the point of intersection of R and the surfaces under pressure. Join CO. Now resolve R into two components, the first, F, tangent to the surfaces at C and the second, X, normal to the surfaces at the same point ' Fol- lowmg the method employed with sliding pairs. A' is the normal r)ressure between the surfaces and the frictional re.'^istance to motion will be/.V, where /is the friction factor (Sec. 160), and since the parts are assumed to In- in equilibrium, there must be no unbalanced force, so that the irsolved part F of the resultant R must be equal in magnitude to the frictional resistance, or t - /A. But / = tan , anil hence the resultant R must make an angle with the radius r at the center of pressure C. 171. Friction Circle. — With center O draw a circle tangent to R as shown dotted; then this circle is the one to which the result- ant R must be tangent to maintain uniform relative motion, and the circle may be designated as the friction circle. Tho radius p of the friction circle is p = r sin , where r is the radius of the journal, and this circle is concentric with the journal and much smaller than the latter, since is also small, and hence tan (j) = sin nearlv, so that approximately p = rsin <^ = r tan <^ = rf. ' Four difforcnt iirrangeineiits of the forces on a turning pair !ire shown at i'ig. 117, similar letters being used to Fig. 11.'). At (rt) I' and Q act on the outer element but their resultant R acts in oi)posite sense to the ^ormer cjuse and hence on opposite side of the friction circle, since the relative .sense of rotation is the same. In case (c) P and Q act on the inner element and the relative sense of rotation is reversed from (a), hencf! R passes on the right of the friction circle; at (b) conditions are the same as (a) except for the relative sen.se of motion which also changes the position of R; at (d) the forces act on the outer element and the sense of rotation and position of R are both as indicated. 172. Examples. — The construction already shown will be applied in a few practical cases. 193 THE THEORY OF MACHINES 1. The first case considered will Ije an ordinar>' bell-crank lover, Fig. 1 18, on which the force P is assumed to act horizontally and Q vertically on the links a and c respectively, the whole lover b turning in the clockwise sense. An examination of the figurt^ shows that the sense of motion of a relative to b is counter- clockwise as is also the motion of c relative to b, therefore /' will lie tangent to the lower side of the friction circle at l>earing I, and Q will be tangent to the left-hand side of the friction circle I i i\ ^ : Ft... 11>J. :it UvHring :i, and tho rt>^ultiint ot P and Q must pass through .1 atui must Ih> tangent to the upf>er side of the friction circle on the pair 2, so that the direction of R becomes fixed. Now draw Dt: in the direction of P to reprt^st^nt this force and then draw KF and DF parallel respectively to Q and R and intersecting at F. then FF = Q and DF = ft. In case there was no trictioti and assuming the direc^ons of /' and Q to ronuiin uiichange:ii,'r>." is ar once ooiaiuttl tue lorce i^o = ct'. Then the etticieiicy ot the lever in this position is rf = ^- and for any other ^Kisttion tnay In? sinularly found. THK KFFK'IKNCY OF AfAflllNES 193 The friction circles are not drawn to scale but are made larger than they should l>e in order to make the drawing clear. 2. I^t it lie required to find tlie liiu; of action of the force in the connecting rod of a steam engine taking into account friction at the crank- and wristpins. To avoid confusion the details of the rod are omitted and it is represented by a line, the friction cirelos being to a very mii<-h cxuggorated scale. Let Fig. 119(o) represent the rod in the position under consideration, the direc- Fni. 119. — Stj'am-fngine rnefhanism. tion of the crank is also shown and the piston rod is assumed to be in compression, this being the usual condition for this position of the crank. Inspection of the figure shows that the angle a is increasing and the angle /3 is decreasing, so that the line of action of the force in the connecting rod must be tangent to the top of the friction circle at 2 and also to the Ixjttom of the fric- tion circle at 1, hence it takes the position shown in the light line and crosses the line of the rod. This position of the Hnf of action of the force is seen on examination to be correct, because in Ijoth cases the force acts on such a side of the center of the bear- ing as to produce a turning moment in the direction of relative motion. 104 THE THEORY OF MACHISES u r i Two other positions of the ongttio are shown in Fig. 119 at (fcl and (f), the direction of revolution U'ing the same as l^eforo and the line of action of the force in the ro?ht Unes. In the case {b), the rod is assumed in compression and evidently l>oth th»> angles a and (i are decreasinjj so that the line of action of the force lies below the axis of the rod; while in the position shown in u'l, the coiuurtiiig hkI is assumed in tension, a is decrea.-*- ing, and 3 is increasing so that the line of the forct> int(Ts«M,'ts the rod. In all cases the determining factor is that the force must lie on such a side of the center of the phi as to prtxluee a turnini: moment in the direction of relative motion. 173. Governor — Turning Pairs Only.— A complete device in which turning paii-s alone occur is shown at Fig. 120, which J- G" a '■".' V R \ \ \ N H ■""^ ,c "/ / 1 • '■F, Fn;. rJO. — (.I'lverriur. represents one oi the novertiors discassed fully in the followimi chapter, except tor the clTiMr ot iVtctinn. 'I'he guviTiior herewith IS siiowu also at Vn. 12."' and .)ii!\ ()ii.--halt of it ha^i been drawn in, the total weiiiht ui the two rotatuiij bulls is w lb. while that k.n the central weight including the pull of the valve gear is taken as H lb. lu *.'haptcr XII no account has been taken of friction ur pin pressures while thtse are essential to the [iresent purpose. There will be no iVietional rt-sistaace between the central weight and the spiiidle and the fricfi..>u circles at A, B and C are drawn exaggerated in order tu make the construction m«.>re clear. It is assumed that; the bails are moving slowly outward anti that when pa^ising through the position illustrated the spindle rotates at n revolutions per rmiiute or at a; raiiiaris per second; It IS requirt^-d to find n and also the speed n' oi the spindle as the THE EFFICIENCY OF MACHINES i!»r. balls pass throiiKh this sniiin position whrn trllV(■llin^ inwiinl. The (lifTcrnicc ln'twrcii tlicsc two simmhIb indicates l<» sonir cxtciil the f|uality of llic governor, as it sliows what chango must Ix' made Itcfon- thr balls will rcvcrso their motion. On one l>all there is a eentrifuKal force ,, pds., where ,, = n ru>', C r heinu 'he raiiins of rotation of the halls in feet, also „ acts V horizontally while the weight of the hall ,, ll>. acts vertically, and their resultant is a force I' inclined as shown in the left-hand figure. The arms AH and liC are both in t«'nsion evidently, and as the halls are moving outward, a is increasing and /3 is decreasing (see Fig. 120); henci? the direction of the; force in the arm BC crosses the axis of the latter as shown, Fi representing the force. Now the direction of the force /* is unknown and it -innot C lie determined without first finding „ which, however. ■nds upon n, the quantity .sought. An approximation to the sIojm; of /' may he found by neglecting friction and with this approxi- mate value the first trial may he made. With the assumed direction of P the point //, where P intersects F\, is determitied and then the resultant R of Fi and P must pass through // and also l>e tangent to the friction circle a.i A. (If there were no friction, R would pass through the center of .1, .Sec. 170.) Turn- W mg now to the vector diagram on the right make I)K = „ anil If ^ EL = ., ; then draw /Xr' horizontally to meet K(r, whicih is paral- lel to Fi in G. The length EG represents the force Fi in the arm liC, while IX; represents the tension on the weight W which is balanced by the other half of the governor. Next draw GJ and /.'./ parallel resi)ectively to R and P, whence these forces are found. If the slope of P has l)een properly a.ssumed, the point ./ will be on the horizontal line through L, and if J docs not lie on this line a second trial slope of P must bn made and the process continued until ./ does fall on the horizon- tal through /-. The length /../ then represents 9 - .^ '"'*'"' from wiiich w is readily computed, and from it the sp*"<*d n in revolutions per minute. I- i- ■ I THE THEORY OF MACHINES 'I Ih« lUittcd liiicB nhow the case where the mechanism patwca tlirough tho Willie ixwition hut with the balU moving inward and from the length /^' the value of w' and of n' may be found. If oidy the relation U'tween n and n' m required, then ~, - yjj^i' The meaning of this is that if the balls were moving outward due to a decreased load on the prime mover to which the governor was connected then they would pass through the position shown when the spindle turned at n revolutions per minute, but if the load were again increased causing the balls to move inward the speed of the spindle would have to fall to n' before tho balls would pass through the position shown. Evidently the best governor is one in which n and n' most nearly agree, and the device would be of little value where they differed much. In rei\ding this problem reference should^lso l)e made to the chapter on governors. 174. Machine with Turning and Sliding Pairs.— This chapter may be very well concluded by giving an example where both turning and sliding pairs are used, althou ,h there should he no difficulty in combining the principles already laid down in any machine. The machine considered is the steam engine, the barest outlines of which are shown in Fig. 121. The piston is assumed combined wiih the crosshead and only the latter is shown, and jjj t>,p problem it has been assumed that the engine is lifting a weight from a pit by means of a vertical rope on a drum, the resistance of the weight being Q lb. Friction of the rope is not considered. The indicator diagram gives the information necessary for finding the pressure P acting through the piston THE EfFlCIKNCY OF MACUINKS 107 on the croB«ht!ud, iiiid the problem i« to fiiul Q and tho officieticy. Froru the principles already laid down, the direction of //» the |)re8t*uro on the crosshead is known, also the line of ai^tion of F\ and of Rt. For equilibrium the forces Fi, R, and /* must interw'ct at one point which is evidently A, as f, the force due to th(! steam pressure, is taken to act along the center of the piston nxl. On the crankshaft there is the force Fx from the connectinK rod, and the force Q due to the weight lifted, and if there were no friction, their resultant would pass through their point of intersention li and also through O the center of the crank- shaft. To allow for friction, however, Rt must Ik? tangent to the friction circle at the crankshaft and nmst touch the top of the latter, hence the \r ii»ion of /?i is fixed. Thus the locations of the five forces, P, h fii, Rt and Q are known. Now draw the vector diagram, laying off CD = P and drawing CE and DK puralU'l respectively to Ri and Fi, which gives these two forces, next draw EF parallel to Rt and DF parallel to Q which thus determines the magnitude of Q. If there were no friction, Fi would be along the axis of the rod, and Ri normal to the guides, both forces passing through .4o the center of the wristpin. Further, Rt would pass through Po tho intersection of Fi and Q, it would also pass through as shown dotted, so that the lines of action of all of the forces are known and the vector diagram CEoFoD may be drawn obtaining the resistance Qo = DFo, which could be overcome by the pressure P on the piston if there were no friction. The efficiency of the machine in this position is then r; = !:-, and may be found in a similar way for Vo other positions. If desired, the value of the efficiency for a numlwr of positions oi the machine may be found and a curve plotted similar to a velocity diagram. Chapter III, from which the efficiency pei cycles is obtained. In all illustrations the factor / is much exaggerated to make the constructions clear and in mr.ny actual cases the eflSciencj will be much higher than the cuts show. Where the efficiency is very close to unity, the method is not as reliable as for low efficiencies, but many of the machines have such high efficiency that such a construction as described herein is not necessary, nor is any substitute for it needed in such ccses. i\ V til f 1 m n 198 THE THEORY OF MACHINES QUESTIONS ON CHAPTER XI 1. In the engine crosshcad, Fig. 110, if the friction farttir is 0.0.5, what .size is the friction angle? If the piston pressure is 5,000 pd.^., and the connect- ing rod is at 12° to the horizontal, what is the pressure in the rod and the efficiency of the crossheai, neglecting friction at the wristpin? 2. Of two 12-in. journals one has a friction factor 0.002 and the other 0.(MW. What are the sizes of the friction circles? 3. What would be the efficiency of the crank in Fig. 118 if the seal*' •>( the drawing ia one-quarter and the pins are l^i in. diaine'ier? 4. Determine the direction of the force in the side rod of a locomolive in various positions. ii. A thrust bearing like Fig. l(^) has five collars, tlie mean bearing diain- eter of which is 10 in. If th" shaft runs at 120 revolutions per iniiniteaiid has a bearing pressure of 50 lo. per square inch of area, find the power lost if the friction factor is 0.05. 6. In the engine of Fig. 121, taking the scale of tlie drawing as oi.c-six- teenth and the friction factor as 0.06, find the value of Q when P = 2,.")00 pds. the diameters of the crank and wristpins being 3'2 and 3 in. respec- tively. 7. In a Scotch yoke, Fig. 6, the crank is 6 in. long and the pin 2 in. diam- eter, the slot being 3 in. wide. With a piston pressure of 500 pds., f;nd the efficiency for each 45° crank angle, taking/ = 0.1. PART II MECHANICS OF MACHINERY I, CHAPTER XII GOVERNORS 176. Methods of Governing. — In all prime movers, which will be briefly called engines, there must be a continual balance be- twr- 1 the energy supplied to the engine by the working fluid and the energy delivered by the machine to some other which it is driving, e.g., a dynamo, lathe, etc., allowance being made for the friction of the prime mover. Thus, if the energy delivered by the working fluid (steam, water or gas) in a given time exceeds the sum of the energies delivered to the dynamo and the friftion of the engine, then there will be some energy left to accr' ite the latter, and it will go on increasing in speed, the friction also in- creasing till a balance is reached or the machine is destroyed. The opposite result happens if the energy coming in is insufficient, the result being that the" machine will decrease in speed and may eventually stop. In all cases in actual practice, the output of an engine is con- tinually varying, because if a dynamo is being driven bj' it for lighting purposes the number of lights in use varies from time to ti ne; the same is true if the engine drives a lathe or drill, the demands of these continually changing. The output thus varying very frequently, the energy put in l)y the working fluid must be varied in the same way if the desired balance is to be maintained, and hence if the prime mover is to run at constant speed some means of controlling the energy ad- mitted to it during a given time must be provided. Various methods arr emploj-ed, such as adjusting the weight of fluid admitted, adjusting the energy admitted per pound of fluid, or doing both of these at one time, and this adjustment may be made by hand as in the locomotive or automobile, or it may I)e automatic as in the case of the stationary engine or the water turbine where the adjustment is made by a contrivance called a governor. A governor may thus be defined as a device used in connection with prime movers for so adjusting the energy admitted with the 201 202 THE THEORY OF MACHINES working fluid that the speed of the prime mover will be constant under all conditions. The complete governor contains essentially two parts, the first part consisting of certain masses which rotate at a speed proportional to that of the prime mover, and the second part is a valve or similar device controlled by the part already described and operating directly on the working fluid. It is not the mtention in the present chapter to discuss the valve or its mechanism, because the form of this is so varied as to demand a complete work on it alone, and further because its design depends to some extent on the principles of thermo- dynamics and hydraulics with which this book does not deal. This valve always works in such a way as to control the amount of energy entering the engine in a given time and this is usuall; done in one of the following wa3's: (a) By shutting off a part of the working fluid so as to admit a smaller weight of it per second. This method is used in many water wheels and gas engines and is the method adopted in the steam engine where the length of cutoff is varied as in high-speed engines. (6) By not only altering the weight of fluid admitted, but by changing at the same time the amount of energy contained in each pound. This method is used in throttling engines of various kinds. (c) By employing combinations of the above methods in various waj's, sometimes making the method (a) the most im- portant, sometimes the method (6) The combined methods are frequently used in gas engines and water turbines. The other part of the governor, that is he one containing the revolving masses driven at a speed proportional to that of the prime mover, will be dealt with in detail because of the nature of the problems it involves, and it will in future be })riefly referred to as the governor. 176. T3rpes of Governors. — Governors are of two general classes depending on the method of attaching them to the prime mover and also upon the disposition of the revolving masses, and the speed at which these masses revolve. The first type of governors, which is also the original type used by Watt on his engines, has been named the rotating-pendulum governor because the revolving masses are secured to the end of arms pivoted to the rotating axis somewhat similar to the method of construction of a clock pendulum, except that the clock pendulum swings in one ({(JVHHNUHS 203 plane, while the governor masses revolve. In this type there are three subdivisions: (a) gravity weighted, in which the centrifugal force due to the revolving masses or balls is largely balanced by gravity; (b) spring weighted, in which the same force is largely balanced by springs; and ('•) combination governors in which both methods are used, (iovernors of this general class are usually mounted on a separate frame and driven by belt or geai's from the engine, but they are, at times, made on a part of the main shaft. The second tyipe is the inertia governor which is usually made on the engine power shaft, although it is occasionally mounted separately. The name is now principally used to designate a class of governor with its re- volving masses differently dis- tributed to the former class; its equilibrium depends on centrifugal force but during th(! changes in position the inertia of the masses plays a prominent part in producing rapid adjustment. The name shaft-governor is also much used for this type. 177. Revolving-pendulum Governor. — Beginning with the revolving-pendulum type, an illustration of which is shown at Fig. 122 connected up to a steam (>ngine, it is seen that it consists essentially of a spindle A, caused to revolve by means of two bevel gears B and C, the latter being driven in turn through a pulley D which is connected by a belt to the crankshaft of the engine; thus the spindle A will revolve at a speed proportional to that of the crankshaft of the engine. To this spindle at F two balls G are attached through the ball arms K, and these arms are connected by links J to the sleeve //, fastened to tho rod R, which rod is free to move up and down inside the spindle A as directed by the movement of the balls and links. The sleeve H with its rod R is connected in some manner with the valve V, in this illustration a very direct connection being indicated, so that a movement of the sleeve will open or close the valve V. Pio. 122. -Sirii|)lo governor f HI 204 THE THEORY OF MACHINES The method of operation is almost self-evident; as the engine increases in speed the spindle A also increases proportionately and therefore there is an increased centrifugal force acting on the balls G causing them to move outward. As the balls move outward the sleeve H falls and closes the valve V so as to prevent as mu( !i steam from getting in and thus jau.sing the speed of the engine to decrease, upon which the reverse s(>ries of opera- tions takes place and the valve ojxjns again. It is, of course, the purpose of the device to find such a position for the valve V that it will just keep the engme running at uniform speed, by admit- ting just the right quantity of steam for this purpose. 178. Theory of Governor. — Several different forms of the governor are shown later in the present chapter and will be dis- Fid. 123. cussed subsequently, but it may be well to begin with the simplest jform shown in Fig. 123, where the connection of the sleeve to the valve is not so direct as in Fig. 122 but must be made through suitable linkage. The left-hand figure shows a governor with the arms pivoted on the spindle, while the right-hand figure shows the pivots away from the spindle, and the jame letters are used on both. Let the total weight of the two balls be xv lb., each ball w therefore weighing ^ lb., and let these be rotated in a circle of radius . »t., the spindle turning at n revolutions per minute corre- 2xn sponding to w = -^ radians per second. For the present, friction will be neglected. GOVERNORS au5 Three forces act upon each ball and determine its |K)8ition of equilibrium. These are: (a) The attraction of gravity, which will act vertically downward and will therefore be parallel with the spindle in a governor where the spiiidle ia vertical as in the illus- w tration shown. The magnitude of this force is „ pda. (6) The second force is due to the centrifugal effect and acts radially and at right angles to the spindle, its amount being ' • r. w' pds. (c) The third force is due to the pull of the ball arm, and will be in the direction of the line joining the center of gravity of the ball to the pivot on the spindle, which direction may be briefly called the direction of the ball arm. These three forces must be in equilil)riuni so that the vector triangle ABC may be drawn where AB = „, BC = -„ ru^ and w must be such that AC is parallel to the arm. Now let D be the point at which the ball arm intersects the spindle and draw AE perpendicular to the spindle DE; then AE = r, the radius of rotation of the balls and the distance DE = h is called the height of the governor. The triangles DAE and ACB are similar and therefore: or DE AB EA ~ BC w h 2 r w . which gives 2g ro)' h = g Thus, the height of the governor depends on the speed alone a..d not on the weight of the balls. The investigation assumes that the resistance offered at the sleeve is negligible us indeed is the case with many governors and gears, but allowance will be made for this in problems discussed later. 179. Defects of this Governor. — Such a governor possesses several serious defects. In the tirst place, the sleeve must move in order that the valve may be operated, and this movement of the slecvo will evidently correspond with a change in the height u-i 20«> TIIH THKORY OF MACHIXKS US: jind honcc with a change in 8{)ocd w. Thus, cucli jiosition of Ihe l)ulla, corresponding to u a;i^■<•Il vulve position, means a different speed of the gov<'rnor and th. refore of the engine; this is what the governor tries to prevent, for its purpose is to kiH?p the speed of the engine constant, although the valve may have to ht opened various amounts corresponding to the load which the engine carries. This d(>fect may he briefly expressed by saying that the governor is not isochronous, the meaning of isochronism being that the speed of the governor will not vary during the entire range of travel of the sleeve, or in other words the valve may be moved into any position to suit the load, and yet the engine and therefore the governor, will always run at the same speed. The second defect is that for any reasonable speed h is extremely small. To show this let the governor run at 120 revolutions per minute so that u> 27rn ,„._,. , = 12.,>< Hidians per second; 2.44 in., a dimension which is so small, that if the balls were of any rea.sonable size, it would make the practical con- struction almt>st impossible. 180. Crossed-arm Governor. — Now it is the desire of all builders to make their governors as nearlj isochronous as is consistent with other desirable characteristics, which means that the height h must be constant, and to serve this end the crossed-arm governor shown in Fig. 124, has been built somewhat extensively. The propor- tions which will produce isochronism may be found mathematic- ally thus: Inspection of th(> figure shows i,'.iat /( ; I cos B — (I (!ot, B. For isochronism /( is to remain constant for changes in the angle or (Ih t;. = = —I sin B -\- a cosec^ B. 60 then h = Fi 12 4 . — C" r <) .s s (; <1 - a r m governor. From whit^li ,IB a - I sin' B h ^ I cos' nOVERXORS 207 and thort'fore a = /sin' fl = /i turr'^ ., tun* 6; which formuIuH Kiv(> t\w rol.'itioiis l)otwooii a, I and 0, and it will he noticed that the weight w doea not enter intt) the calculation any more than it does into the time of swing of the pendulum. As an exam[)le let the 8i)eed he w = 10 radians per second (corresponding to 07 revolutions per minute) and let $ = 30°. Then the formulus give a = 0.0(118 ft. or 0.74 in., I = 0.495 ft. or 5.94 in. and the value of h corresponding to 9 = 30° is 0.322 ft. With these propoitions tlu^ value of h when d l)ecomes 35° will be 0.317 ft., a decrease of l.oti per cent., corresponding to a change of speed of ahout 0.8 ])er cent. With a governor as shown at Fig. 123 and w = 10 as before, a change from 30° to 35° produces a change in speed of ahout 3 per cent. It is possible to design a governor of this type which will maintain absolutely constant speed for all positions of the balls, and the reader may prove that for this it is only necessary to do away with the ball arms, and place the balls on a curved track of parabolic form, .so that they will always remain on the surface of a paraboloid of revolution of which the spindle is the axis. In such a case, h and therefore w will remain constant. A perfectly isochronous governor, however, has the serious defect that it is unstable or has no definite position for a given speed, and thus the slightest disturbing force will cause the balls to move to one end or other of their extreme range and the gov- ernor will hunt for a position where it will finally come to rest. Such a condition of instability is not admissible in practice and designers always must sacrifice isochronism to some extent to the very necessary feature of stability, because the hunting of the halls in and out for their final position means that the valve is being opened and closed too much and hence the engine is changing its speed continually, or is racing. In the simple governor quoted in Hec. 178 it is evident that while it is not isochronous it is stable, for each position of the l)alls corresponds to a different but definite speed belonging to the corresponding value of the height h. 181. Weighted or Porter Governor. — In order to obviate these difficulties Charles T. Porter conceived the idea of plac- ing on the sleeve a heavy central weight, free to move up and down on the spindle and having its center of gravity on the 208 THE THEORY OF MACHINES axis of rotation. This modified governor is shown in Fig. 125, with the arms pivoted on the spindle, although sometimes the arms are crossed and when not crossed they pre fro(juently huh- ScaIo of Incbel 1 : 3 t S • 7 • Fi(i. 12.'). — Porter governor. penyr;ii-fi ,v.-"--L *-i- ^■r"-^': ^: wsis:r:-^\^ (iOVKUSOHS an h' -h v ■ iw The relation ^ m evidently the sensitivenesH of the governor' and the smaller the ratio the more sennitive is the governor. For an isochronous governor 6w = 0. To compare the wtighted and unweighted governors in regard to sensitiveness tak«' the angular velocity w - 1(" radians per second and let W --= 00 Ih. and W ^ % lb. I^t the ch'uige in height necessary t(t move the sleeve through its entire lift he } 2 in. (rt) Unweighted Governor. For the data given h = a.8»» in., and, therefore, = « hh 0.5 h 3.86 •'"'• Hence, 2- = 0.129 or " = O.OtW or {\A per cent., so that the variatioii in sjK'ed will he 0.4 per cent. ih) Weighted Governor. — For this governor and or , 2ir + w ^ ,^^. 2 X 00 + H bh h 0.5 01.70 . = 0.008 _ 5w „ Su 2 = 0.008 giving = 0.(H)4 or 0.4 i>er cent. the variation in speed heing only 0.4 per cent. Such a gover- nor would therefore he very nearly isochronous. A third property of this weighted governor is that it is power- » This may he simply shown by the calcuhi.s thus: and differentiating, 21F + w g W a-' _. '2\V 4- 11' 26c, «n = - g- ■, w u^ •■•T = -2- Of SI, I, where Su and Sh represent the small changes taking place in o> and h. » The negative sign appearing before Sh on the preceding formula mere' • means that an increase in speed corresponds to a decrease in ••"=->;t E^^n^^^^^^mmm alve gear or to overcome friciion. Powerfulness is a very desirable feature, for it is well known in practice that the force required to operate the valve gear is not constant and therefore produces a variable effect on the governor niechanisui, which, unless the governor is iwwerful, is sufficient to move the weights, causing hunting. The Porter governor thus enables the designer to make a very sensitive governor, of practical proportions and one which may be made as powerful as desired, so tiiat it will not easily be disturbed by outside forces. THE CHARACTERISTIC CURVE 183. A number of the results and properties of governors may l)e giapiiically represented by means of characteristic curves, and it will be convenient at this stage to explain these curves in connection with the Porter governor. lA^t Fig. 127(m) represent the right-hand part of a Porter governor, the letters having the same significance as before. Choose a pair of axes, 0(' in the direction of the spindle and OA at right angles to the spindle, and let ths centrifugal force on the ball be plotted vertically along OC, as igainst radii of rotation of the balls, which are plotted along OA, ri and ra representing respectively the iiuier and outer limiting radii, the resulting figure will usually be a curved line somewhat similar to CiCd in Fig. 127(a). Let the angular velocities corresponding to the -acUi -i ami rj be wi and W2 radians per second respectively, and let w = H (wi + W2) represent the mean angular velocity to which the corresponding ra.lius of rotation is r ft. Then w C\ = - ri CO,* Ca = -; rj coj* and (' r u' where the forces C, Ci and Ci are the total centrifugal forces acting on the two balls. The properties of this curve, whi^^h may be briefly called the C curve, may now be discussed. OOVERXOR.^ 213 1. Condition for Isochronism. — If the governor is to be iso- chronous then the anrv \i velocity for all positions of the balls must be the same, i id i,s uj - c>- - toj and hence the centrifugal force depends only u *he radius ot rotation (see formulas above) or - = "^ -" 7 = a constant, a condition which is fulfilled by a C curve forming part of a straight line passing through 0. Thus any part of OC would satisfy this condition and the part ED corresponds to the radii ri and rj in the governor selected. ,c c, / y /'' / / '•l. r*«t (6) l''ni. 127. — Chiinu'ttrristic curve. 2. Condition for Stability — Although the curve ED will give an isochronous governor, it produces instability. The curve CS^d indicates that the speeds are not the same for the various positions of the balls, and a little consideration will show that Ci corresponds to a lower speed and Cj to a higher speed than C. This is evident on examining the conditions at radius ri, for the point E corresponds to the same speed as C, but since E and Ci are both taken at the same radius, and since the centrifugal force FE is greater than FCx it is evident that the angular velocity «, corresponding to Ci is less than the angular velocity w correspond- i -'"-I 214 THE TIIKORY OF MACIITXES ing to E. Thus a curve such as C1CC2, which is steeper than the isochronous curve where they cross, indicates that the speed .)f the governor will increase when the balls move out, and it may similarly be shown that such a curve as Ci'Cd', which is flatter than the isochronous curve, shows that the speed of the governor decreases as the balls move out. Now an examination of these curves shows that the one CiCCt belongs to a governor that is stable, for the reason that when the ball is at radius ri it has a definite speed and in order to make it move further out the centrifugal force must increase. But on account of the nature of the curve the centrifugal force must increase faster than the radius or the speed must increase as the ball moves out, and thus to each radius there is a corresponding speed. On the other hand, the curve CiCC\' shows an entirely different state of affairs, for at the radius r^ the centrifugal force is greater thaii '-'it,' or the ball has a higher speed than w and thus lus the ball moves out the sj^ed will decrease. Any force that would disturb the governor would cause the ball to fly outwanl under the action of a resulta force Ci'K, and if it were at radius r-i any disturbance would cau.se the ball to move inward. Another way of treating this in that f'^r the curve CiCd the energy of the ball tlue to the centrifugal force is increasing due both to the increase in r and in the speed, and as the Weights W and (/' are being lifted, the forces balance one another and there is equilil)rium; whereas with the curve C/CTo' there is a decrease in speed ami also in the energy of the balls while the weights are being lifted and the forces are therefore unbalanced and the governor is unstable. Thus, for stability the C curve must be steeper than the line joining any point on it to the origin O. Sometimes governors have curves such as those shown at Fig. 127(6) and curve (\C('i indicates a stable governor, (\'CCi an unstable governor, CiCCi partly stable and partly unstable and finally d'CCt ,^artly unstable and partly stable. 3. Sensitiveness. — The shape of the curve is a measure of the sensitiveness of the governor. If N indicates th'h .sensitive- ness, then by definition i> = - — Now wi — wi _ {uii — Wi)(a)ii + oil) w w(a)» + Wi) C02 % _ 2w» «i' GOVERNORS 215 since wj + wi = 2w nearly. 'J'herefore S = 1 0)2 8 _ Wl' »Ut and and C, = ?(^ Tj 0)2* M' „ II' C = w rw' Hence, l)y substituting in the formula for ,S', the result is w S = g r2 w c r Tj c; 1 ri ri ~ 2 c r Hefcrring now to Fig. 127 it is seen that :» nd or = tan $2 = C r OA ' = tan d ^ - II Irtan $2 — tan ^,1 1 tan e ■hll = tan di = DA OA (\A - HA DA HA OA 1 CtB 2 da' Thus the (■ curve is also valuable in showing the sensitiveness of the governor. For an isochronous governor d, Ji and D coincide and .S = O. Evidently the njore stable the governor is the less sensitive it is, and in a general way an unstai)Ie governor IS more sensitive than a stable one. At C,, Fig. 127(«), the stable governor is most nearly isochronous, and evidently a fair degree of stability and sensitiveness could both be obtained in a governor havmg a reverse curve with point of inflexion near C, the part CCt being concave to OA, the part dC conve to 0^. 4. P;.werfulness.— The C curve also shows the powerfulnoss of the governor, since in this curve vertical distances represent 21B rHK THEORY OF MACHINES '%k M the centrif'.igal forces acting on the balls, while horizontal distances represent the number of feet the balls move horizontally in the direction of the forces. Thus, an elementary area represents the product C.^r ft.-pds. and the whole area between the C curve and the axis OA gives the work done by the balls in moving over their entire range, and is therefore the work available to move the valve gear and raise the weights. The higher the curve is above OA the greater is the available work, and this clearly cor- responds to increased speed in a given governor. 6. Friction. — The effect of friction has been discussed in the previous chapter and need not be considered here. Some writers treat friction as the equivalent of an alteration to the central weight, and if this is done the effect is very well shown in Fig. 128 where the C curve for the frictionless governor is shown at CiCCi. As the weight W is lifted the effect of friction when treated in this way is to increase W by the friction/ with the result that the C curve is raised to 3-4, whereas when the weight W is falling the friction has the effect of decreasing the weight W and to lower the C curve to 5-6. The effects of these changes are evident without discussion, 184. Relative Effects of the Weights of the Balls and the Central Weight — For the purpose of further understanding the governor and also for the purpose of design, it is necessary to analyze the effects of the weights separately. Referring to Fig. 120 and finding the phorograph by the principles of Chapter IV the image of D is at D' and taking moments about A, remember- iiig that ^ may be transferred from D to its image D', H^Vb + Viwe - 1 2 Ch = 0. (Sec. 151, Chapter IX). Now let Cw be the part of C neces- sary to supjwrt W and r„. the corresponding quantity for w, so that Feet Flli. 128. Cw + (\ = C where C = w r«« But <- «' = C. That is Ct^ aoVERXORS = W , and Cu, = wt- 217 The graphical construction is shown in Fig. 129. Draw JH and IJj horizontally at distances helow .4 to represent w and W respectively, then join AK, the line D'E being » vertical through D'. Then it may Ix; easily shown that (\ =- AF and C„. = .tA'. ''lFt .•O .92 Fi.i. 130. is stable. From this curve it ap{x?ars that the sensitiveness is I2 X .,.p.^ = 0.0850 or 8..")0 ikt con*., whicli checks very well with the speeds as shown in the last column, and which indi- cates a sensitiveness of 8.G5 per cent. » If it is desired to find the position of the bills for a speed of 150 revolutions per minute, then u> = 15.7 radians per second and the force (' = " ro;- = 0.933 X r X 240.5 = 230 r. Then draw the line for which the tangent is - = 230 and where it cuts the C curve is the radius of the balls corresponding to this speed. Assuming the mean height of the V curve to be 207 pds. the work done in the entire travel of the balls is 207 (0.98 — 0.83) = 31 ft.-pds. (;OVK UXORS 219 186. Design of a Porter Governor. — These curves may be eon- . "iiiently used in the design of a Porter governor to satisfy given conditions. Let it be recinired to design a governor of this typ«' to run at a mean siM-ed of 200 r(>vohitions per minute with an overall varii'tion of k-ss than 10 \)vr cent, for the extreme range. The sleeve is to have a travfl of 2 in. and the governor is to have a powerfulness n^presented by 20 ft.-pds. From general experience select the dimensions o,, oj, /,, /, and BM in Fig. 129. Thus take /i = /, = 10 in., and BM = 3 in.; also make oi = aj = 1 in. Draw the governor in the central position of the sleeve with the arms at 90°, as this angle gives greater uniformity than other angles, and measure the extreme radii and also that for the central oridm position of the sleeve. The C ° curve may now be constructed and at Fig. 131 the three radii are marked, which are r^ = 9..") in., r = 10.22 in. and rj = 10.82 in. Now the power of the governor is 20 ft.-pds., and divid- ing this by rj - ri = 0.11 ft. gives the mean height of the C curve as 182 pds. Plot this at radius r making HG = 182 pds. and join to 0; it cuts rj at D. Now the sensitiveness is to be 10 per cent., so that T and U are found such that DT = DC = 0.10 X AD = 19.30 pds. Join T and U to 0, thus locating 1' and the resulting C curve will be VGT shown dotted. Next, since the centrifugal force Gil = 182 pds. corresponds to a radius r = 10.22 in. and a speed of 200 revolutions, the weight w may be found from the formula C = — rw* and eivef w = 15.75 lb. Hy the use of such a diagram as Fig. 129 the three values of C,„ are measured for the three radii and the C„ curve is drawn in Fig. 131, and then the valuea of CV are found. Thus, 13.6 R vx hl"0 60 . 20 r,- 9.5" r - 10.22" >-5-10j«" l''ii;. 131. — Governor ik-sign. 220 THE THEORY OF MACHINES RF scales off as 1 53 pds. and hence CV, = 153 - 13.6= 139.4 pds., and similarly the other values of Cr are found, and from them Wi = 104.6 pds., W = 107.6 pds. and Wt = 110.6 pds. are obtained, as shown at Fig. 129. As a trial assume the mean of these values W = 107.6 pds., as the value of the central weight and proceeding as in Sec. 185 find the three new values of Cv and also of C = Cw + C„ and lay these off at the various radii giving the plain curve in Fig. 131. This will be found to correspond to a range in speeds from 192 II Fi(i. 132. — I'roell governor. to 207.5 rcvolution.s per minute, and as this gives less than a 5 per cent, variation either way from the mean si)eed of 200 revolutions it would usually be satisfactory. If it is desired to have the exact value of 5 per cent., then it will be better to start with a little larger variation of say 6 per cent, and proceed as above. 187. ProeU Governor.— The method already descrilwd may be applied to more complicated forms of governor with the same ease a.s is used in the Porter governor, the phorograph making nOVRRXOItS 221 theae cases quite simple. As an illustration, the Proell governor is shown in Fig. 132 and is similarly lettered to Fig. 129, the difference between these governors being that in the Proell the hall is fastened to an extension of the lower arm DB instead of the upfKir arm AH as in the Porter governor. As before, Ali is chosen as the link of reference and the images found on it of the jwints 1) and M by the phorograph, Chapter IV. The force '2 IT is then transferred to D' and J^C and i-a"' ^^ ^^' from { 'haptcr IX, but in computing C the radius is to be measured from the spindle to M and not to M', since the former is the radius of rotation of the ball. The meanings of the letters will appear from the figure and by taking moments about A the same relation is found as in Sec. 184. The results for the complete travel of the balls is shown on the lower part of Fig. 132. SPRING GOVERNORS 188. Si)ring governors have been made in order to eliminate the central weight and to make po-ssible the use of a nearly isochronous and yet sensitive and powerful governor. These governors always run at high speed and are sometimes mountetl on tlie main engine shaft, but more frequently on a separate spindle. 189. Analysis of Hartnell Governor. — One form of this gover- nor, frequently , ribed to Hartnell of England, is shown in Fig. 133 and the action of the governor may now be analyzed. Let the total weight of the two balls be w lb., as l)ef(jre, and let W denote the force on the ball arms at BB, due to the weight of the central spring and any additional weight of valve gear, etc. In this case 11' will remain constant as in the loaded governor. Now let F be the pressure produced at the points B by the spring, F clearly increasing as the spring is compressed due to the outward motion of the balls. In dealing with governors of this class it is best, to use the mo- menta of the forces about the pin A in preference to the forces tltomseh OS, and hence in place of a C curve for this governor a moment or M curve will be plotted in its place, the radius of rotation of the balls being used as the horizontal axis. The symbols M, Mp, Mw and Mw indicate, respectively, the mo- ments about A of the centrifugal force C, the spring force F, the weight of the balls w and the dead weight W along the spindle. Then M = My + Mw + M„ i 222 TlfR TlIKlHiY OF MAf'IITXES or Ca cos = Fb cm 9 + Wb cos 6 — wa sin $. The inomont rurvoH may he drawn and take the general shajXJH Hhown in Fig. I'-i'-i and Miniihir statcmentH may be mada about these eurves as about those for the 'Sorter governor. If it is desired, the corresi)onding C curves may readily be drawn from the formula Ca cos $ - Fb cos 6 + Wb cos d — wa sin 6 or C = F + W - w tan 6 a a = fV 4- Cw + ('u, *-T *V Our,, Fio. ISS."— Hartnell governor. and a graphical method for finding these values is easily devised. The curve for W is evidently a horizontal line since W, b and a are all constant, while that for w> is a sloping line cutting the axis of r under the pin A and the Cf curve may be found by differences. 190. Design of Spring. — The data for the design of the spring mav be worked out from the Cr curve found as above. Evidently Cf = F a or F = Cf X f = Cf X & constant, and thus from the curve for Cy it is possible to read forces F to a suitable scale. wm (iOVEHXOHS 22:1 These forces /' may now' l)c plot tod an ut Fig. I'M which gives the values of F for the difTereiit radii of rotation. As the line tJGL thus found is slightly curved, no .spring could exactly fulfil the n quiremcnta, but by joining K an compressed through //./ in., for the inner position of the balls. In ordinary problems it is safe to a.s8ume for preliminary cal- culations that the effect of the weights W and w can be neglected and the spring may be designed to balance the centrifugal force alone. In completing the final computations the results may be modified to allow for these. In the diagrams here shown their effects have been very much ex- aggerated for clearness in the cuts. 191. Governors with Hori- zontal Spindle. — Spring gover- nors are powerful, as the complete computations in the next case will show, and are therefore well adapted to cases where the move- ment of the valve gear is difficult and unsteady. When such a governor is placed with horizontal spindle such as Fig. 135 the effects of the weights are balanced and the spring alone balances the centrifugal force. 192. BeUiss and Morcom Governor. — One other governor of this general type may be discussed in concluding this section. It is a form of governor now much in use and the one shown in the illustration. Figs. 136 and 137, is used by BeUiss and Morcom of Birmingham, England, in connection with their high-speed engine. The governor is attached to the crankshaft , and therefore the weights revolve in a vertical plane, so that their gravity effect is zero. There are two revolving weights W with their centers of gravity at G and these are pivoted to the spindle by pins A. I-1 ! m^mp^^^-^^M^w^^^'^^s^ 224 THE THHOHY OF MACHINES i I t'lci. I'.ii't. — (idvprnor with gravity effeot neutnilizcl. Fiu. 13t3. — Belliss anil Mort'oni governor. {♦i aOVKRXORS 22.^ r"? — M\t\N^\ Betwwn the weights there arc two springs S fastened to the f(,-iner by mentis of pins at li. The balls operate the collar C, wliich slides along the spindle, thus operating the boll-erank lever UFV, which is pivoted to the engint! frame at F and connected at V', by means of a v«'rtical rod, to the throttle valvo of the engine. There is an additional com|M>nsating spring Sc with its right-hand end attache*! to the frame and its left-hand end con- nected to the l»ell-crank lever DFV at //, there being a hand wheel at this connection so that the tension in the spring may 1x5 changed within certain limits and thus the engine spe«'d may l>e varied to some extent. This spring will easily allow the operator to run the engine at n» arly 5 per cent, above or below normal. The diagrammatic sketch of the governor, shown in Fig. 137, enables the different parts to be distinctly seen as well a.s the eolations of the various points. It will be noticed that this governor differs from all tlii> others already described in that part of the centrifugal force is directly taken up by the springs S, while the forces acting on the sleeve are due to tile dead weight of the valve V and its rod, and the slightly unl)alanced steam pressure (for the valve is nearly balanced against steam pres- sure) on the valve, and in addition to tiicse forces there is the pressure d-ie to the spring S^. Tiie governor is very efficiently oiled and it is found by actual experiment thai the frictional effect may be practically neglected. In this case it will be advisable to draw the moment curve for the governor a.s well as the C curve and from the latter the usual information may be obtained. As this moment crve presents no difficulties it seems unnecessary to put the investi- gation in a mathematical form as the formulas become lengthy on account of the disjxisition of the parts. An actual case has 15 Fi( 137. — Btlliss and MorcDtn governor. 226 THK THEORY OF MACHINES been worked out and the results are given herewith and show the effects of the various parts of the governor. 193. Numerical Example. — The governor here selected, is attached to the crankshaft of an engine which has a normal mean speed of 525 revolutions per minute although the actual speed depends upon the load and the adjustment of the spring iS,;. The governor spindle also rotates at the same speed as the engine. The two springs >b' together require a total force of 112 pds. for each inch of extension, while the spring *S„ requires 220 pds. {mm- inch of extension, the springs having been found on calibration to be extremely uniform. Each of the revolving masses has an effective weight of 10.516 lb. and the radius of rotation of the center of grav'ty varies from 4.20 in. to 4.83 in. The other dimensions are: e = 4 in. radius, b = 3.2 in., c = 3.5 in., a = 3.56 in., d = 10.31 in.,/ = 4.67 i i., g = 5.31 in. The weight of the valve spindle, valve and parts together with the luibalanced steam pressure under full-load normal conditions is 20 lb. The following table gives the results for the governor for four different radii of the center of gravity G, all the moments being expressed in inch-pound units, when reduced to the equivalent moment about the pivots A of the ball-. Beli.iss and Morcom Governor III ' ,1 Speed, revolutit>nH per minute 508 526 529 532 RudiuB of rota- tion off/, inchett Centrifu- gal force, pounds Moment about A, inch- pounds (1) Moment of main springa about A, inch- pounds 4.47 345.5 1,220 1,083 4.73 392.0 1,364 i 1,196 4.78 400.0 '. 1,388 1,223 4. S3 410.0 1,418 1)2.50 (2) Moment of spring St about A, inch- pounds 115 143 145 148 (3) Monient due to valve wcigiit about A, inch- pounds 16 16 16 16 Sum of (1), (2) and (3), inch- pounds 1,214 1,355 1,384 1,414 In examining the table it will be observed that the am of columns (1), (2) and (3) is always a little less than the moment due to the ueiitrifugal force. As the results are all computed from measurements made on the engine during operation, there is possibly a slight error in the dimensions, and further the effect of centrifugal force on the springs S will make some difference. GOVERNORS 227 The results agree very well, however, and show that the calcula- tions agree with actual conditions. The results are plotted in Fig. 138, the left-hand part of the curves being dotted. The reason of this is that the observations below 508 revolutions per minute were taken when the engine was being controlled partly by the throttle valve, and do not therefore show the action of the governor fairly; the points are, however, useful in showing the tendency of the curves and represent actual positions of equilibrium of the governor. The effect of the weight of the valve and unbalanced steam pressure are almost negligible, so that the power of the governor Inch a/l Poondt 1000 wo /C 600 -»0 200 ~ y»lT« Welihl F=— I— i fiht I Moment 3.TO 3JS 3.9 U t.1 4.2 U PlO. 138. U 4^ 13 4.7 4.8 Badlul r Inchei ,«•, does not need to be large, but the spring Sc produces an appre- ciable effect amounting to about 11 per cent, of the total at the highest speed. If the compensating spring .S^ were removed, the governor would run at a lower speed. Joining any point on the moment curve to the origin 0, as has been done on the figure, shows that the governor is stable. The sensitiveness and powerfulness may be found from the (' curve .shown at Fig. 139. At the radius 4.47 in. the cciitri- fiigui force is 345.5 pds., and if the origin be joined to this point and the line produced it will cut the radius 4.83 in. at 373.5 pds., whereas the actual C is 410 pds. The sensitiveness then is 228 THE THEORY OF MACHINES tIttt^-t-IS^ = 0.0465 or 4.65 per cent. From the speeds 3^(410 + 373.5) the corresponding result would be 1/(530 + 508) "^ 0.0442 or 4.42 per cent., which agrees quite closely with the former value. The moment curves cannot be used directly for the determina- tion of the power of the governor because areas on the diagram do not represent work done. If the power is required, then the base must be altered either so as to represent equal angles passed through by the ball arm, or more simply by use cf the C curve plotted on Fig. 139. It will be seen that the C curve differs Clk r Poandt 400 -300 , Otl«» . 200 100 _l_ 3." 3.^ 3.9 1.0 4.1 1.2 U 4.1 4w 4.6 4.7 i3 InchM Pio. 139. very little in character from the moment curve. The power of the governor is only about 11.6 ft.-pds. The computations on this one governor will give a good general idea of the relative effects of the different parts in this style of governor, and also show that spring governors of this class possess some advatitages. THE INERTIA GOVERNOR, FREQUENTLY CALLED THE SHAFT GOVERNOR 194, Reasons for Using this Tjrpe. — The shaft governor was proi^ably originally so named because it is usually secured fo the crankshaft of .an etigine and runs therefore at the engine s|x;ed. In recent practice, however, certain spring governors, such as the Belliss and Morcom governor, are attached to the crankshaft t GOVERNORS 229 and yet these scarcely come under the name of shaft governors. The term is more usually restricted to a governor in which the controlling forces differ to some extent from those already dis- cussed. This tj'pe of governor is not nearly so old as the others and was introduced into America mainly as an adjunct to the high-speed engine. On this continent builders of high (rotative)-speed engines have almost entirely governed them by the method first men- tioned at the beginning of this chapter, that is by varying the point of cut-off of the steam, and in order to do this they have usually changed the angle of advance and also the throw of the eccentric by means of a governor which caused the center of the eccentric to vary in position relative to the crank according to the load, the result will be a change in all *he events of the stroke. The eccentric's position is usually directly controlled by the governor, and hence it is necessary to have a powerful gover- nor or else the force required to move the valve may cause very serious disturbances of the governor and render it useless. Again as the governor works directly on the eccentric, it i» convenient to have it on the crankshaft. Governors of this class also possess another peculiarity In- those already described the pins about which the balls swung were in all cases perpendicular to the axis of rotation, so that the balls moved out and in a plane passing through this axis. In the shaft governor, on the other hand, the axis of the pins is parallel with the axis of rotation and the weights move out and in in the plane in which they rotate. While this may at first appear to be a small matter, it is really the point which makes this class of governor distinct from the others and which brings into play inertia forces during adjustment that are absent in the other types. Such governors may be made to adjust themselves to their new positions very rapidly and are thus very valuable on machinery subjected to sudden and frequent changes of load. 196. Description. — One make of shaft governor is shown at Fig. 140, being made by the Robb Engineering Co., Amherst, Nova Scotia, and is similar to the Sweet governor. In this make there is only one rotating weight W, the centrifugal effect of whirh is partly counteracted by the flat leaf spring S, to which the ball is directly attached. The eccentric E is pivoted by the pin P to the flywheel, and an extension of the eccen- tric is attached by the link 6 to the ball W. The wheel •4- r 23U THE THEORY OF MACHINES j< ' I I t i' rotating in the sense shown, causes the ball to try to move out on a radial line, which movement is resisted by the spring S. As the ball moves out, due to increased speed, the eccentric sheave swings about P, and thus the center of the eccen- tric will take up a position depending upon the speed. Two stops are provided to limit the extreme movement of the eccentric and ball. Other forms of governor are shown later at Fig. 143 and at Fig. 147, those having somewhat different dispositions of the parts. I'lii. 140. — Hohh inertia governor. Powerfulncss in such governors is obtained by the use of heavy weights moving at high speed, for example in one governor (he revolving weight is 80 lb. and it revolves in a circle of over 29 in. radius at 200 revolutions jxrr minute, dimensions which should be compared with those in the governors already discussed. 196. Conditions to be Fulfilled. — The conditions to be ful- filled are quite similar to il-ose in other spring governors so that only a brief discussion will be necessary, which may be illustrated in the following example. Let A, Yig. 141, represent a disc rotating alwut a center at n revolutions ikt minute, ami let this disc have a weight U) GOVERNORS 231 mounted on it so that it may move in and out along a radial line as indicated, and further let the motion radially be resisted by a spring - rium (Sa = C or whether r is cfiual to, greater than or less than a, ajid these will result as follows: S = 0.000341 X 25 X J X 200» = 341 pds. 1 1. /• = a = lft., 2. r = lft.,a = 0.57ft.,(S' = 0.000341X25X 0.57 X 200* = 600 pds. 1 M9 X 2()0= = 288 3. r = 1 ft.,a = 1.19 ft., 5 = 0.000341 X 25 pds. So that, as the formula shows, the spring strength depends upon the relation of r to a. The resulting conditions when the ball is 10 in., 12 in. and 14 in. respectively from the center of rotation with the three springs, are set down in the fol'owing table and in Fig, 142. "t'i m 232 THE THEORY OF MACHINES ■1 Radius r, ii CentrUugkl Sprins pull oheB {oroe kt 200 , '•»• 1 S - 288 1 1 1 5-341 1 1 S - 600 ii 10 1 12 : 14 284 293 341 341 398 389 284 341 j 398 241 341 441 For the spring S = 288 pda. per foot of extension it is seen that at the smaller radius the spring pull is higher than the centrifugal force or the disc must run at a higher speed than 200 revolutions for equilibrium, while at the outer radius the spring pull is too lo\\ and the speed must Ihj below 200 revolutions for equilibrium, Fio. 142. that is the speed should decrease as the ball moves out. With spring S = 600 exactly the reverse is true, or the speed must increase as the ball moves out, while for spring the moments about P due to the centrifugal force C and to the npring pull .S must be equal if, f«»r the present, the effect of gravity and of the forces required to move the valve are neglectetl. That is: • Cdi = Sdt in.-pds. or — rw'di = Sdi in.-pds. g In such an arrangement as shown the effect of the forces re- quired to move the valves is frequently quite appreciable and is generally also variable, as is also the effect of friction and gravity, although usually gravity is relatively so small that it may be neglected. If it is desired to take these into account then rw*di = Sdt + moment due friction, valve motion and gravity. Denote the distance AP by a and the shortest distance from G to AP by x; thus a is constant but x depends on the position of the balls. From similar triangles it is evident that rdi = ox and therefore g 9 Thus the moment due to the centrifugal force is, for a given speed, variable only with x and hence the characteristics of the governor are very well shown on a curve' in which the base repre- sents values of x and vertical distances the centrifugal moments — w'ax. Such a curve is shown below, Fig. 144, and the shape of the curve here represents a stable governor since it is steeper at all points than the line joining it to the origin 0. From this curve information may be had as to stability and sensitiveness, but the power of the governor cannot be determined without either placing the curve on a base which represents the angular swing of the balls about P or else by obtaining a C curve on an r base jis in former cases. If u is constant, or the governor is isochronous, 3/ varies directly with x or the moment curve is u struiglit line j):is.sing through the origin 0. Having obtained the M curve in this way the moment curves ' For more complete discussion of this method see Tolle, " Die Kegelung der Kraftmaschinen.'' ■hi a- 7--r 336 THE THEORY OF MACHINES i m I t 111 about P oorresponding to gravity, friction of the valves and parts and also thoae necessary to operate the valves are next found, these three curves also being plotted on the x base, and the difference between the sum of these three moments and the total centrifugal moment will give the moment which must be pro- vided by the spring which is Sd, in.-pds. From the curve giving Sdt the force S may be computed by dividing by dt and thoso values of S are most conveniently plotted on a base of spring lengths, from which all information for the design of the spring may readily be obtained (see Sec. 190). In order that the relative values of the different quantities may be understood. Fig. 145 shows these curves for a Buckeye lack V»1t« 0«ar Beactiou ^ V«l»« friction "^ Values at « iDcbei Bcootftc rtlcllaa Fia. 145. governor, in which the gravity effect is balanced by usin^; two revolving weights symmetrically located. Friction of the vtilvi' and eccentric and the moment required to move the valve iin- all shown and the curves show how closely the spring-moment and centrifugal-moment curves lie together. The curves an- drawn from the table given by Trinks and Housom, in who.st'' treatise all the details of computing the results is shown so as to be clearly understood. The governor has a powerfulness of nearly 600 ft,-pd8. RAPIDITY OF ADJUSTMENT 198. The inertia or shaft governor is particularly well adapted to rapid adjustment to new conditions and it is often made so « Thinks and Housom, "Shaft Goveroore." aOVERNOHS 387 that it will move through its entire range in one revolution, which often means unly u small fraction of a second. The rapidity of this adjustnipiit depends almost entirely upon the distribution of the revolving weight and not nearlj' so much upon its magnitude. For a given position of the parts the only force acting is centrif- ugal force already discus8(>d but during change of position the parts are being accelerated and forces due to this also come into play. Fig. 146 represents seven different arrangements of the weights; in five of those the weight is concentrated into a ball with center of gravity at G and hence with very small moment of inertia about G, so that the torque required to revolve such a weight at any moderate acceleration will not be great; the opposite is true of the two remaining cases, however, the weight Fui. 146. being much eioiiguted and having a large moment of inertia about G. A.ssuming a sudden increase of speed in all cases, then at (a) this onlj' increases the pressure on the pin B because BG i% normal to the radius AG, at (6) an increase in speed will produce a relatively large turning moment about the pin which is shewn at A. Comparing (c) and (d) with (o) and (6) it is seen that the torque in the former cases is inc»eased at (c) because in addition to ihti acceleration of G there is also an angular acceleration about G, whereas at (d) G is stationary and yet there is a decided torque due to its angular acceleration. At (e), (/) and (g) the sense of rotation is important and if an increase in speed occur in the first and last cases the acceh-rating forces assist in moving the ^1 338 THK THEORY OF MACHINES u weJghtH out rapidly to their new positions, whereas at (/) the acceleratinK forces opijose the rnoveineiit. Space preventH further diHcusHion of this inatt<»r here, but it will appear that the acceleratinK fi.rees nuiy Ixi adjusted in any dcHired way to produce rapid changes of |H>sition, the weights Iwing first determined from principles already stated and the distribution of these defM'nding on the inertia effects desired. Chapter XV will assist the reader in understanding these forces more definitely. ill f I'lci. 147.— Rites governor. A form of governor made by Rites, in which the inertia forces play a prominent part during adjustment is shown at Fig. 147. The revolving weights are heavy and arc set far apart, but their (ienter of gravity (/ is fairly close to A so that the centrifugal moment is relatively small. In a governor for a 10 by 10-in. engine the weight W was over 120 lb. and the two weight centers were 32 in. apart. QUESTIONS ON CHAPTER XII 1. I>'liiii- a n<)v«Tin»r. What is the difference between the fiinetione of a governor and a flywheel? aOVERNORS 230 S. What ia th« height of a Miinple K<>vfrnor ruiiiiiiiK at 9A n'vohitiotiH per niinut«? S. What ia meant by an iaochroiuxiM Kovcriinr'' N xm-h h Kovcrnor dt-Htr- able or not? Why? 4. Explain fully the ternia stability ami |><>werfuliuHH. 5. Prove that in a govenior where the biilU move in it piirilniloiil nf revo- lution, h is constant "nd the govt-rniir Ih iiwH-lironouN. 6. What are the aclvnntageH of the I'orter Kovernor? 7. Umnn the data, rii - 100, n, = 110, provi- that , -= 2 '*• 8. CoiiipHre the m-iiMitivenoiw of ii Himple anil ii I'orter K»verii 120 lb. and to - IT) lb. 9. Analyze the following Kovcrnor for w iiMitivcnoMH iinil powt-r '>•• •> Fit;. 129): n - 130, W - 110, w = 12, I = 12' a, HM = ;i'i, /, = 10' 'I I 0. «>« ~ ^Hi sleeve travel 2Jj in. 10. Design a Porter governor for a sp«?e per cent, each way, travel 2'2 in, power :i.") ft.-jHiN 11. In a governor of the type of Fig. i:{3, o =■ 2. 1 in.; b = 0,7"> in., dis- tance between pivots 2?4 in. inner raditm of bull 1 (i in., weight per ball 1*^ lb., travel J-^ in. and speed 2.')0. I3esign the .spring for ."» per cent, viiriiition. IS. What are the advantage-i of tlio shaft governor? Sliow iiow the iIIh- tribution of the weight affects the rapidity of adjustment. ;k leaasBT^^^- CHAPITER XIII ■t SPEED FLUCTUATIONS IN MACHINERY 199. Nature of the Problem.— The preceding chapter deals with governors which are used to prevent undue variations in speed of various classes of machinery, the governor usually con- trolling the supply of energy to the machine in a way to suit the work to be done and so as to keep the mean speed of the machine constant. The present chapter does not deal with this kind of a problem at all, but in the discussion herein, it is assumed that the mean speed of the machine is constant and that it is so controlled by a governor or other device as to remain so. In addition to the variations in the mean speed there are variations taking place during the cycle of the machine and which may cause just as much trouble as the other. Everyone is famil- iar with the small direct-acting pump, and knows that although such a pump may make 80 strokes per minute, for example, and keep this up with considerable regularity, yet the piston moves very much faster at certain times than others, and in fact this variation is so great that larger pumps are not constructed in such a simple way. With the larger pumps, on which a crank and flywheel are used, an observer frequently notices that, although the mean speed is perfectly constant, yet the flywheel speed during the revolution is very variable. Where a steam engine drives an air compressor, these variations are usually visible, at certain parts of the revolution the crankshaft almost coming to rest at times. These illustrations need not bo multi- plied, but those quoted will suffice. The speed variations w hich occur in this way during the cycle are dealt with in this chapter. 200. Cause of Speed Fluctuations.— The flywheel of an engine or punch or other similar machine is used to store energy and to restore it to the machine according to the demands. Consider, for example, the steam engine; there the energy supplied l)y the steam at different parts of the stroke is not constant, but varies from time to time; at the dead centers the piston is stationary and hence no energy is delivered l>y the working fluid, whereas when 240 M SPEED FLUCTUA TIONS IN MACHINERY 241 the piston has covered about one-third of its stroke, energy is being delivered by the steam to the piston at about its maximum rate, since the piston is moving at nearly its maximum speed and the steam pressure is also high, as cutoff has not usually taken place. Toward the end of the stroke the rate of delivery of the energy by the steam is small because the steam pressure is low on account of expansion and the piston is moving at slow speed. During the return stroke the piston must supply energy to the steam in order to drive the latter out of the cylinder. Now the engine above referred to may be used to drive a pump or an air compressor or a generator or any other desired machine, but in order to illustrate the present matter it will be assumed to \ie connected to a turbine pump, 3ince, in such a case, the pump offers a constant resisting torque on the crankshaft of the engine. The rate of deli ing energy by the working fluid is variable, as has already been explained; at the beginning of the revolution it 7or4u« Baqotnd is much less than that required to drive the pump, a little further on it is much greater than that required, while further on again the steam has a deficiency of energy, and so on. At this point it will l)c well to refer again to Fig. 101 which has been reproduced in a modified form at Fig. 148 and shows in a very direct and clear way these important features. During the first part of the outstroke it is evident that the crank effort due to the steam pressure is less than that necessary to drive the load; this being the case until M is reached, at which point the effort due to the steam pressure is just equal to that necessary to drive the load; thus during the part Oil/' of the revolution the input to the engine being less than the output the energy of the links themselves must be drawn upon and must supply the work represented by OML. Jiut the energy which may be obtained from the links will depend upon the mass and velocity of them, the energy l)eing greater the larger the mass and the greater the velocity, the result is that if the energv of the links is decreased ■16 , - 1 •i ^2 TIII'J rilKORY OF MACHINES m ■ -i V'l f by drawing from them for any purpose, then since the mass of the Unks is fixed by construction, the only other thing which may happen is that the speed of the Hnks must decrease. In engines the greater part of the weight in the moving parts is in the flywheel and hence, from what has been already said if energy is drawn from the links then the velocity of the flywheel will decrease and it will continue to decrease so long as energy is drawn from it. Thus during OM' the speed of the flywheel will fall continually but at a decreasing rate as M' is approached, and at this point the wheel will have reached its minimum speed. Having passed M' the energy supplied by the steam is greater than that necessary to do the external work, and hence there is a balance left for the purpose of adding energy to the parts and speeding up the flywheel and other links, the energy available for this purpose in any position being that due to the height of the torque curve above the load line. In this way the speed of the parts will increase between M' and N' reaching a maximum for this period at .V. From .V to R' the speed will again decrease, first rapidly then more slowlj-, reaching a minimum again at R' and from R to S', there is increasing speed with a maximum at S'. The flywheel and other parts will, under these conditions, be continually changing their speeds from minimum to maximum and vice versa, producing much unsteadiness in the motion during the revolu- tion. The magnitude of the unsteadiness will evidently depend upon the fluctuation in the crank-effort curve, if the latter curve has large variations then the unsteadiness will be increased; it will also depend on the weights of the parts. In the case of the punch the conditions are the reverse of the engine, for the rate of energy supplied by the belt is nearly con- stant but that given out is variable. While the punch runs light, no energy is given out (neglecting friction), but when a hole is Ijeing punched the energy supplied by the belt is not sufficient, and the flywheel is drawn uix>n, with a corresponding decrease in its speed, to supply the extra energy, and then after the hole is punched, the belt gradually sjweds the wheel up to normal again, after which another hole may be punched. To store up energy for such a purjwse the flywheel has a large heavy rim running at high speed. It will thus be noticed that a flywheel, or other part servi.g the same purpose, is required if the supply of energy to the ma- 4: 1 SPEED FLVCTUATIONS IN MACHINERY 243 chine, or the delivery of energy by the machine, i.e., the load, is variable; thus a flywheel is required on an engine driving a dyna- mo or a reciprocating pump, or a compressor, or a turbine pump; also a flywheel is necessary on a punch or on a sheet metal press. It is not, however, in general necessary to have a flywheel on a steam turbo-generator, or on a motor-driven turbine pumping set, or on a water turbine-driven generator set working with constant load, because in such cases the energy supplied is always equal to that given out. The present investigation is for the purpose of determining the variations or fluctuations in speed that may occur in a given machine, when the methods of supplying the energy and also of loading are known. Thus, in an engine-driven compressor, hoth the steam- and air-indicator diagrams are assumed known, as well as the dimensions and weights of the moving parts. THE KINETIC ENERGY OF MACHINES 201. Kinetic Energy of Bodies.— In order to determine the speed fluctuations in a machine it is necessary, first of all, to find the kinetic energy of the machine itself in any given posi- tion and this will now be determined. If any body has plane motion at any instant, this motion may be divided into two parts: (a) A motion of translation of the body. (6) A motion of rotation of the body about its center of grav- ity. Let the weight of the body be w lb., then its mass will be m = -, where g is the attraction due to gravity and is equal to 32.16 in pound, foot and second units, and let the body be moving in a plane, the velocity of its center of gravity at the instant being v ft. per second. Further, let the body be turning at the same instant at the rate of w radians per second, and assume that the moment of inertia of the body about its center of gravity is /, the corresponding ratii«is of gyration being k ft., so that / = mkK Then it is shown in books on mechanics that the kinetic energy of the body is £" = y^mv^ + >2 /^^ = y^mr'^ + l^wd-^'w^ ft.-pds!, and, hence, in order to find the kinetic energy of the body it is neces- sary to know its weight and the distribution of the latter because of its effect on k, and in addition the velocity of the center of Hi I 1 244 THE THEORY OF MACTJIXEFi giftvity of the weight and also the angular velocity of the body. 202. Application to Machines.— Let Fig. 149 represent a mechanism with four links connected by four turning pairs, the links being a, h, c and d, of which the latter is fixed, and let I a, h and I, represent the moments of inertia of a, b and c respective- ly about theii centers of gravity, the masses of the links being ma, nn, and Wc. Assuming that in this position the angular Fig. 149. IJ velocity w of the link a is knowr., it is required to find the corre- siwnding kinetic energy of the machine. Find the images of F, Q, a, b, c, d and of G, H and N, the centers of gravity of a, b anJ c respectively, by mouiis of the phorograph ♦liscussed in Chapter IV. Now if Vo, v„ and Vy be used to rep- resent the linear velocities of G, H and ;V and nlso if «6 and &>« be used to denote the angular velocities of the links b and c, it is at once known, from the phorograph (Sees. 66 and 68), that: Vo = OG'.u; vu = OH'm and Vi^ = ON'.u ft. per second, and wi = -r-w and uc = — « radians jier second, so that all the neces- c sury linear and angular velocties are found from the drcwing. 203. Reduced Inertia of the Mschme.— The investigation will be confined to the determination of the kinetic energy of the link b. which will be designated by Et, and h ^ing found this quantity the energy of the otl^ier links may l^e found by a similar W^- *i. SPEED FLUCTUATIONS /.V MACIITNERY 24r) process. Since for any body the kinetic energy at any instant is given by the formula: E = H »'"* + H /«' ft.-pds. Therefore, Ei, = }4m,,.v^„ + ^ I^^^ ft.-pds. /b' \' /h' \ * Now, liuii* = 7M6A-»«c*» = OTtA-i^ (^"j = mt ( jA;J«*. Following the notation already adopted, it will be convenient b' h' to write k\ for ^A:6, since the length ^h is the length of the image of ^6 on the phorograph. The magnitude of k'b is found by draw- ing a line HT in any direction from // to represent h and find- ing T' by drawing H'T' parallel to HT to meet TP produced in T' as indicated in Fig. 149; then H'T' is the corresponding value of A;' Hence ^»= H v„.v„' + HIi>^i- = j .^ Wfc [Oir + k/\ a,« ft.-pds. Let the quantity in the square bracket be denoted by Kb*; then evidently Ki, may be considered as the radius of gj'ration of a body, which if Kocured to the link n and having a mass mt would have the same kinetic energy as the link 6 has at thia instant. It is evidently a very simple matter to find Kb graphi- cally since it is the hypotenuse of the right-angled triangle of which one side is O//' and the other At,'; this construction is shown in Fig. 149. Thus Eb = }4 vibKb'ui^ ft.-pds. Similarly E^ - >2'"oA'„2„= ft..piis. and K, = >^m,A',2a,2 ft.-pds.; constructions for A'„ and K^ are shown in the figure. For the whole machine the kinetic enerKv is E = K^ + Eb + A\ = >^maA'^a,' + yi'nbK\i^'- + yinirKKJ* = M[w«A*a + mbK\ + w.A'Ma)> = K[/'. + /'6 + /'cl «» = M«/w' ft.-pds. i m *3 ' 2U» THE TllKOnV OF M.\CIII};EFi A study of these formulas jind a comparison with the work just covered, shows that /'„ is the moment of inertia of the maas with center of gravity at and rotating with angular velocity « which will have the same kinetic energy as the link o actually has; in other words, /'„ may be looked upon as the reduced mo- ment of inertia of the link a, while similar meanings may be attached to I\ and /'.. Note that /'„ and 7„ differ because tlie former is the inertia of the corresponding mass with center of gravity at 0, whereas /„ is the moment of inertia about the center of gravity G of the actual link. The quantity J is, on the same basis, the reduced inertia of the entire machine, by which is meant that the kinetic energy of the machine is the same as if it were replaced by a single mass with center of gravity at 0, and having a moment of inertia / about 0, this mass rotat- ing at the angular speed of the primary link. It will be readily understood that J differs for each position of the machine and is also a function of the form and weight of the links. The foregoing method of reduction is of the greatest importance in solving the problems under consideration, because it makes it possible to reduce any machine, no matter how complex, down %'- in f " to a single mass, rotating with known speed, about a fixed center, HO that the kinetic i-nergy of the machine is readily found from the drawing. 204. Application to Reciprocating Engine. — The method may be further illustrated in the common case of the reciprocating engine, which in addition to the turning pairs contains also a sliding pair. The mechanism is shown in Fig. 150 and the saaie notation is employed as was used in the previous case, and the only peculiarity about the mechanism is the treatment of the link c. The link c has a motion of translation only and therefore w, = and /,w,* = so that the kinetic energy of the link is 4 SPEED FLVCTVATIONS IN MACHINERY 247 E, = }4 mc- r.« = lyi m. OQ''o>' or /.' = m„OQ'' since the point Q has the same linear velocity as all points in the link. The remainder of the machine is treated as before. Lack of sF)aoe prevents further niultiplieation of these illus- trations, hut it will be found that th»' method is easily applied to any machine and that the time required to work out the values of J for a complete cycle is not very great. SPEED FLUCTUATIONS 206. Conditions Affecting Speed Variations.— One of the most useful applications of the foregoing theory is to the determination of the proper weight of flywheel to suit given running conditions and to prevent undue fluctuations in speed of the main shaft of a prime mover. Usually the allowable speed variations arc set by the machine which the engine or turbine or other motor is driving an«l these variations must be kept within very narrow limits in order to make the engine of value. When a dynamo is being driven, for example, fluctuations in speed afifect the lights, causing them to flicker and to become so annoying in certain cases that they are useless. The writer has seen a particularly bad case of this kind in a gas engine driven generator. If alternators are to be run in parallel the speed fluctuation must be very small to make the arrangement practicable. In m;uiy rolling mills motors are being used to drive the rolls and in such cases the rolls run light until a bar of metal is put in, and then the maximum work has to b • done in rolling the bar! Thus, in such a case the load rises suddenly from zero to a maxi- mum and then falls ofT again suddenly to zero. Without some storage of energy this would probably cause damage to the motor and hence it is usual to attach a heavy flywheel somewhere l)etweon the motor and the rolls, this flywheel storing up energy as it is being accelerated after a bar has passed through the rolls, and again giving out part of its stored-up energy as the bar enters and passes through the rolls. The electrical conditions determine the allowable variation in speed, but when this is known, and also the work required to roll the bar and the torque which the motor is capable of exerting under given conditions, then it is necessary to be able to determine the proper weight of flywheel to keep the speed variation within the set limits. In the case of a punch already mentioned, the machine runs ill M wym 248 THE THEORY OF MACHINES n l! !'* !■! ft"' light for some time until a plate is pushed in suddenly and the full load is thrown on the punch. If power is being supplied by a belt a flywheel is also placed on the machine, usually on the shaft holding the lielt pulley, this flywheel storing up energy while the machine is light and lussisting the belt to drive the punch through the plate when a hole is lieing punched. The allowable percentage of slip of the belt is usually known and the wheel must be heavy enough to prevent this amount of slip being exceeded. The present discussion is devoted to the determination of the speed fluctuations with a given machine, and the investiga- tion will enable the designer to devise a machine that will keep these fluctuations within any desired limits, although the next chapter deals more particularly with this phase. 206. Determination of Speed Variation in Given Machine. — Let El and Ei be the kinetic energies, determined as already explained, of any machine at the beginning and end of a certain interval of time corresponding with a definite change of posi- tion of the parts. Then the gain in energy, L'j - Ei, during the interval under consideration represents the difference between the energy supplied with the working fluid and the sum of the fric- tion of the machine and of the work done at the main shaft on some other machine or object during the same interval, because the kinetic energy of the machine can only change from instant to instant if the work do le by the machine differs from the work done on it by the working fluid. In order to simplify the problem friction will be neglected, or assumed included in the output. Consideration will show that A'j - Ei will be alternately posi- tive and negative, that is, during 'le cycle of the machine its kinetic energy will increase to a maximum and then fall again to a minimum and so on. As long as the kinetic energy is in- creasing the speed of the machine must also increase in general, so that the speed will be a maximum just where the kinetic energy begins to decicase, and conversely the speed will be a minimum just where the kinetic energy begins to increase again. But the kinetic energy of the machine will increase just so long as the energy put into the machine is greater than the work done by it in the same lime; hence the maximum speed occurs at the end of any period in which the input to the machine exceeds its output and vice versa. The method of computing this speed fluctuation will now be considered. y SPEED FLUCTUATIONS IN MACHINERY 249 It has already been shown that the kinetic eiuTKy of the madiine ia given by E = iy«» ft.-pds. from which there is obtained by differentiation' or hE = }i \2J. w.&u + u,K6J] ft..pr second in the interval of time in which the gain of energy of the machine ia 5E and that in J is 5J. Of course, any of these changes may be positive or negative and they arc not usually all of the same sign. The values of J and u used in the formula may, without sensible error, be taken as those at the beginning or end of the inter- val or as the average throughout the interval, the latter being preferable. 207. Approximate Value of Speed Variation.— The calculation is frequently simplified by making an approximation on the assumption that the variation in y mry be neglected, i.e., that SJ = 0. The writer has not found that there is enough saving in time in the work involved to make this approximation worth while, but since it is often assumed, it is placed here for con- sideration and a slightly tlifferent method of deducing the resulting formula is given. Let Ex, Et, ui and wa have the meanings already assigned, at the beginning and end of the interval of time and let the reduced moment J be considered constant. If » To those not familiar with the oalculus the foHowirig iiicthud may hoof value. Let E, J and u be the values of the quantities at the beginning of the interval of time and E + 6E. J + SJ and u -f ««, tlie corresponding values at the end of the same interval. Then E = }^Jo,' E + SE =}4(J + SJ) (a. + «a.)» — /'2W***' "i~ 2J.W0W -f* uj'oJ) where in the multiplication such terms as (Su)*, and SJ.Su are neglected aa being of the second order of small quantities. By subtraction, then, E -\- SE — E = SE ■= Vo \'2JuiSiji> + a)*4./| as above. if ill 250 Then or THE THEORY OF MACHINES K, - E\ = HJiW - «i') Ei — E\ 1 -/ « l^ — J- = ,'2(w» - «i ) = V'jCw" + wi)(wj — Wl) = w(a>i — «i) W > ~f" Wl where CO hiis 1hh»ii writt(Mi for " ,, ,!i siilwtitution which causes little inucciinicy in practice, Therefore Ei — El 0»« — 0)1 = - -,— or 6w = 5/i' This is the same result as would have Ih>«'u obtained from the former formula by making jj = 0. 208. Practical Application to the Engine. — The meanings of the different quantities can best \te explained by an example p'? Fio. 151. which will now be worked out. The steam enguie has been selected, because all the i)rinciples are involved and the method of selecting the data in this case may be rather more readily understood. The computations have all bei-n made by the exact formula, which takes account of variations in J. Consider the duublc-aeting engine, wliicli is shown with the indicator diagrams in Fig. 151 ; it is required to find the change of speed of the crank while passing from A to B. Friction will lie neglected. For simplicity, it will be assumed that the engine is driving a u SPEED FLUCTVATlOSfi IN MACHINERY 251 turbine pump which offers a uniform resisting turning moment and hpnc«' the work done by tho engine during any interval is proportioniil to \\\v crank angle pas««>d through in the given intorval. If the work done per rovolution aH computed from the diagram is U' ft.-ptls., then he work done by the engine during the interval from A to B will W. -'„.,,, 'H' ft. AM) jmIs. T«) make the Oi = 18"; then the case as definite as poHHible .sup|)OHe that flj work done by the engine will Ihj }in\V ft.-|)d8 209. Output and Input Work.—Again, let A , and /I, represent the areas in square inciies of the head end and the crank end of the cylinder resfjectively, U and U being the lengths t)f the cor- responding indicator diagr.ims in inches. The stroke of the pis- ton is taken as L feet and the indicator diagrams are assumed drawn to scale s pds. per square inch = 1 in. in heiglit. With these symbols the work represented by each square inch on the tliagram is nAi / ft.-pds. forth«; head-<^nd and nAi, ft.-p«ls. for the «i fj crank-end diagram. Now suppose that during the crank's motion from A io B Oie area of the head-end diagram reckoned above the zero line is «! sq. in., see ¥\g. 151, and the corresponding area for the crank- end diagram a^ sq. in. Then the energy delivered to the engine li}' the steam during the interval is aisAi, — atsAi, ft.-ixls. h 12 while the work done by the engine is W 20 ^*"P^^- Note that the work W is the total area of the two diafirams i . square inches multiplied by their corresponding constants to bring the quantities to foot-pounds. Then the input work exceeds the output by UisAi y — QisAif — 2Q ft.-pds. which amount of energy must be stored up in the moving parts during the interval. That is, the gain in energy dunng the period is El — Ei = (iisAi f — UisAi -J — s?i ft.-pds. 80 that the gain in energy is thus known. 262 TIIK THEORY OF MACHINES 1 >■■ ¥ Agniii, (he method descrilH'd CHrlior in the chaj>»er unablen thn values (»f Ji and .'j to be found and hence the value of Jt — J\. SubstitutinK these in the formula, iE - \WiJ the gain in angular velocity is readily found. The values are Et- El ~ 6E, Ji - Jx " hJ, HiJt + Ji) " J and for « no error will result in practice by using tue mean speed of rotation of the crank. During a complete revolution the values of Su will sometimes be positive and sometimes negative, and in order that the engine may maintain a constant mean speed the algebraic sum of thti^e must be zero. Should the algebraic sum for a revolution Ix; positive, the conclusion would be that there is a gain in the mean speed during the revolution, that is the engine would l>e steadily gaining in speed, whereas it has Ixjcn assumed that the governor prevents this. 210. Numerical Example on Single-cylinder Engine. — A nu- merical example taken from an .actual engine will now Imj given. Fio. 152. 'Ihc engine used in this computation hail u cylinder 12H6 "»• diameter with a piston rod 1 J^ in. diameter and a stroke of 30 in. The connecting rod was 90 in. long, center to center, weighed 175 lb. and had a radius of gyration about its center of gravity of 31.2 in. The piston, crosshead and other reciprocating parts weighed 250 lb., while the flywheel weighed 5,820 lb. and had a moment of inertia about the shaft of 2,400, using pound and foot unit',. The mean speed of rotation was 86 revolutions per minute. Using the notation employed in the earlier discussion, the data may be set down as follows: a = 1.25 ft., b = 7..') ft., h = 2.t)0 ft., /, = 2,400 pd.(ft.)', iv- The specil w III,, -= 2 jr« 00 IHi, 111,, = ">.44, ///c = 7.7S. ^ ;,;r— = racuans iH>r second. tiO SPUED FLIK'TVATIONS IN MACIIINKRY 2r.a IJaitig the atjove data thn following quantitioH were iii«>ti«un> <7 a« 5.442 r>4 I 7 -JCK) 5,488 .'>7S I . i'li ft -4 . HI +r. » Zero Line Fl«i. lo.-]. a i =.o-C Sij. In. Thus, during the IS" under eonsidi'raiion there is u gain in the reduced inertia of 5.9, although as the complete tahle given later on shows, there is a loss in other parts of \\' revolution. The indicator diagrams for the engine are shown in Fig. 153 and the areas corresjwnding to the crank motion considered are S« if 254 THE THEORY OF MACHINES shown hatched and marked Ui and 02. These areas were meas- ured on the original diagrams which were drawn to 60-ptl. scale, although these have been somewhat reduced in reproduction. Data for computations from the indicator diagrams are as follows: Cylinder areas: Head end, Ai = 114.28 sq. in. Crank end, A J = 111.52 sq. in. Diagram lengths: Head-end diagram, U = 3.55 in. Crank-end diagram, I2 = 3.58 in. Stroke of piston, L = 2.5 ft. Hence, each square inch on the diagrams represents L 2 "S sAi^ = 60 X 114.28 X 3 gg = 4,829 ft.-pds. for the head end, and sA-jT = 60 X 111.52 X 3^5g = 4,673 ft.-pds. for the crank end. The original full-sized diagrams give ai = 0.550 sq. in. and Oj = 0.035 sq. in., from which the corresponding work done will l>e: 0..550 X 4,82!) = 2,656 ft.-pds. for the head end, and 0.035 X 4,673 = 163 ft.-pds. for the crank end. It is a.s?u.ned that the engine is driving a turbine pump or electric generator which offers a constant resisting torque, .so 18 1 that the corresponding work output is ^^ = ^ of the total work represented by the two diagrams, and is 1,079 ft.-pds. The quantities are set down in the table below. Diagram arpaa Work doiii' on piston Work done .^ ""'■' I""-, ,,,„ j" " , " bvrrank, 'l»nnK chang.. of H..a.l.i,, Crank a,, Mead, ' Crank, Total. ft.-pda. kiru-tic rm-rgy. »■!. in. «ic|. in. , ft.-pilji. , ft.-pds. ft.-pds. 1 ii.-pus. ^^ J 0.5.50 I 0.0;J5 I 2,t>;j6 16;{ 2,4'.);{ 1,070 1,414 'Ihe total combined ;ircas of (he two diagrams represent 21,584 ft.-pds., and since the speed was 8() revolutioTis imt minute the indicated horsepower was yg'jww. X 86 = 56.2 hp. The quantity in (he last colunm is the difference between 2,493 and 1 ,079 and would evidently cau.se the machme to sjM'ed up. SPEED ILVCTVATIONS IN VACHINKRY 255 ill 5co = therefore Then the work available for increasing the energy is 1,414 ft.-pds. and this must represont the gain in kinetic energy of the machine, or ht: = +1,414 ft.-pds. The gain in anguhir volocitj- may now he computed. The average value of J is J = }2 [2,410.9 + 2,416.8] = 2,413.8 hence J.oi = 2,413.8 X 9 = 21,724.6 and li w^SJ = 3^X 9' X 5.9 =-■ 238.9 dE - li w^J Ju 1,414 - 238.9 21,724.6 = 0.054 1 radians per second which is the gain in veiocitj- during the period considered. Sim- ilarly the results may be obtained for othci periods, and thus for the whole revolution. These results are set down in the table given on page 257. 211. Speed-variation Diagram. — The values of du thus ob- tained are then plotted on a straight-line base. Fig. 154, which has been dividfvl into 20 ecjual parts to represent each 18° of crank angle. If it is assumed that the speed variation is small, as it alwaj's must be in engines, then no serious error will be made by assuming that these crank angles are passed through in equal times, and hence that the base of the diagram on whi«'h the values of 5w r'.re plotted is also a time base, ecpial distances along which represent ecpial intervals of tim(>. If desin'd, the equal angle base m.-iy be corrected for the varia- tions in the v<>loeity, using the valu(>s of 5u! already found, so as to make the base exactly represent time intervals, but the author does not think it worth the labor and has made no correction of this kind on the diagram shown. Attention should here be drawn to the fact that the height of the original base used for plotting the speed-variation curve has to l>e chosen at random, but after the curve has been plotted, it is necessary to find a line on this diagram representing the mean speed of rotation, w = 9. This may be readilj- done bj- finding the area under the curve by a planimeter, or otherwise, and then locating the line w = 9 so that the positive and negative areas :%\ i i:il 256 THE rUEORY OE MACHINES m •3»S i»• a « T. H ■J 3 oS.'.SS c b t» * « O rt .-5 V o s "' t> e S » C4 ^? ^ n 5 o o o o o d o o o d d ^ X *r ;c If §0 — M T o o o o O C S O 3 o o * 3 ai fft'l'E, «■? , .i. q „ M JO L >r ? £ i, > , - * y. u- ^ Ji i .? T o o f '"^ 31 t* '.O h. -T f^ + ++-(- r^ 3j » ^ ^ ^ jC 3> JO C ?■*«'* X C4 C^ -^ 9) ix ^ -N -f I-- O ' U5 I I I I I I 4 to « ri » 00 ^ + + + I o) o 1" T o a> M ri — ^ I I I 1 + + •f ri — — JO 4 I « 9 I I 4 + 1~> CO ■o » I I r- — M Si I I _ 'jO r^ Q X c X -• CO o» c r*. 00 -- " ^ S i. T ti X o o •- M t» n M o t- OS 'O -. — -o e> - ■^< « -r ;0 I . I + f 4 f I fiiO'T iv«:ir I -I- 7 + JU\J.>)II1 ({■> I I I ■i -o CO ae "O X X re — -rcoor-ox — ren X •* -i* r^ ?i JO o -f M f i" *j ,-j « '* 'O ?i — h- -r -1* re /■- o 1- -r X cc O r* '* "« « « '^ — — /) 51 -* -»• "» !•* ^ -f* '-2 '^» '"^ ^ "* ^ — OS X 'C •'; ri re r *5 — t^ p I- X X -t- — — '*. !■• 51 X '"S o cj -^^ o r^ o ^: S x_ *c ci *" tj Ci * ro m" fi -^ « ^ ^" N -J — — ?i It 31 r^ ri M X X »0 c^ :s 'T X X ^ tM t-- Ci o r» ^ T) 'O o re o rj o ^ ri ^ — ■ n f 'CC!OC'2re oo — — ^»?^ — — o ocoo — — -l*» — — — — .« ^ ^ -f Tt- t "f "•■ "^ *»■, f '••''. t T "t T "^ "". "t **. Tt :' rt ?i M* ri *i n t» ?t ?i :t ti ?i ri ri ?i m ri .-f ?t 5*5?SS5§§§SSSS8iiSii5 « ^ ^ ^ ^ •¥• — -^ — -*••* -r ■^, 1^ ^ —.-? -r ''■.'*_ •*. ri ri ri r* ri ri ri "i ri ?"' ri ri ri rl « ri ri rj ri ri ri c ^ r; X 't '■^ r: X ci 'r 5 >~. n X T 'r '^ :r 3; x = 5 .-, :^ ,0 « -^ 31 u- — X 5 -^ — ■« 5j — -^ '-e ■* ".-i 5 5 -. o " ri r* ri o :^ ~ c o "e -^ 31 ri ri j-- »r — c .-; — ^ ;2 i, — X -) — 1^ I'; t- ■- ?i X — r- o ■»• :; '" - X f "i re -r -r — I- o -^ «r I' — ■" •" rr r< ^ X — • 'O r-. X X t- ■J -r O «e T '.-^ X ■- re — 'i> — r" J X '-r •- .-r C :^ =5 t- X O — ri ri — D 3i -^ f- / r- O — -» "I — C X r- r X o ■* r* C ' - -^ ri o X --i) -• r» o >: r ••■ r4 — -e »i-5 i" 3: o r* f o X -r- — ce o r~ r o r» f - — —. — -. — — rj MM rir*rerere7i IT 258 THE THFOHY OF MACHINES * between this new line and the velocity-variation curve are equal. It is to be remembered that th(! computation gives the gain in v( locity in each interval, and the result is plotted from the end of the curve, and not from the base line in each case. 212. Angular-space Variation.— Now sinc(^ the s[)ace traversed is the product of the correspondiiiR velocity and time, the angular- space variation, be in radians, is found by multiplying the value of 5a) by the time / in seconds required to turn the crank through the corresponding IS°. that is bd - t.bu radians. But t.6u is evidently an area on the curve of angular-velocity variation, so that the angular space variation in radians up t^o any given crank angle, say 54°, is simply the area under the angular-velocity variation curve up to this point, the area l)eing taken with the mean angular velocity as a base and not with the original base line. In this case the area between the mean speed line, 0) =' i), and the speed-variation curve, from to 54°, when reduced to proper units, represents 0.275 radian as plotted in the lower curve of Fig. 154. The upper curve shows that the minimum angular velocity was 8.922 radians per second; while the maximum was O.Ofti radians jx-r second, a variation of 0.141 radian per .second, or 1.57 per cent. The lower curve shows the angular swing of the flywheel about its mean jmsition, and shows tiiat the total swing between the two extremes was 0.58°, although the swing from the mean posi- tion would be only ai)()ut one-half of this. The complete computations which have been given here in full for an engine, will, it is hoped, clearly illustrate the method of })roce(lure to be followed in any case. The method is not as lengthy as would appear at first, and the results for an engine; mav be (piickly obtained by the use of a slide rule and drafting board. i(. the case of engines, all moving pails have relatively high veios, such lor exanii)le as a belt-driven punch, ail parts are very slow-moving with the -xeeption of the shaft carrying the belt pulleys and fly- wheel. ;ind in such a case it is onlv necessaiy to take account of tlie uiertia of the high-speed parts. When>ver the parts are of SPEED FU'CTdATIONS LY MACHISEHY 250 large sizi- or \v(>ight or run iit high speed, account must .j*- -ikt' of their effect on tlie machine. '2, owaiani 2 o ® puoo.'»g ja iicccs.sary, alternators which ;u> to ivork. in parallel. - ; ;.. 1 1 .. i •11 ir iftjuii;-;. t;ul as in tlie case of 260 THE THEORY OF MACHINES 213. Factors Affecting the Speed Fluctuations.— A general dig- oiission has lx;en given earlier in this chapter of the factors that aflF(!c-t the nmgnitude of the speed fluctuations in machinery and as an illustration here Fig. 155 has l)een drawn. This figure shows three s{)eed-fluctuation curves for the engine just referred to, and for the same indicator diagrams as are shown in Fig. 153, but in each case the engine is used for a different purpose. The curve in the plain line is an exact copy of the upper curve in Fig. 154 and represents the fluctuations which occur when th«« engine is direct-coupled to an electric generator, the total fluc;- tuation l)eing 0.14 radian per second or about 1.57 per cent. The dotted curve corresponds to a water pump connected in tandem with the engine, a conmion enough arrangement, al- tb 'igh the piston speed is rather too high for this class of work. 1 • speed fluctuation here would be less than before, amounting < > 123 radian or about 1.37 per cent., this being due to the •t hat the unbalanced work is not so great in this class of re- iiice as in the generator. he broken line corresponds to an air-compressor cylinder in I iem with the steam cylinder and the resulting variation is <' ^^1 radi a per second or 3.38 per cent., which is over twice ; ach ;. the first case. QUESTIONS ON CHAPTER XIH '-■ '^ ound Cii3t-iron disk 2 in. thick has a linear velocity of 88 ft. I> iid its kinetic energy. What would be its kinetic energy if it a!-.i i, f\d at 100 revolution/; per minute? 2. A light ateel rwl 2 ft. long, IVg in. diameter, rotates about an axis normal to its f »nter line, and 6 in. from its end, at .50 revolutions per minute. What IS its kinetic energy? 3. Find the kinetic energy of a wheel 12 in. diameter, density 2 14, at .500 revolutions per minute. 4. What is the kinetif energy of a cast-iron wheel .'{ ft. diameter, l' ft. long and of uniform .section 2' 2 l)y ,5 in., with drivers (iO in. diameter, and a .stroke of 2\ in., find the kin(!tic energy of the rod in the upper and lower jjositions. 6. Show how to find the kinetic energy of the tool sliding block of the Whitworth riuick-retiim motion. 7. .Suppose the whct'l in (piestion .'i is a grinder us«"il to sharpen a tool and that its speed is decrciscd in the iircicss to 4.'il) revolutions iti 1 sec ; what is till change in kinetic energy ? 8. I'lot the'specii ,'thd Miigular Velocity-variation curves for two engines lik-; iiiai di.siussed in the text with cranks at (Ml', only one llywiieel being us«'d. 9. Ilepeat the above with cranks at ISO". CHAPTER XIV THE PROPER WEIGHT OF FLYWHEELS 214. Purpose of Flywheels.— In the preceding chapter a oom- plete discussion has been given as to the causes of speed fluc- tuations in machinery and the method of determining the amount of such fluctuation. In many cases a certain machine is on hand and it is the province of the designer to find out whether it will satisfy certain conditions which are laid down. This being the case the problem is to be solved in the manner already discussed, that is, the speed fluctuation corresponding to the machine and its methods of loading are to be determined. Frequently, however, the converse problem is given, that is, it is required to design a machine which will conform to certain definite conditions; thus a steam engine may be required for driving a certain machine at a given mean speed but it is also stipulated that the variation in speed during a revolution must not exceed a certain amount. Or a motor may be required for driving the rolls in a rolling mill, the load in such a case var>'ing so enormously, that, if not compensated for would cause great fluctuations in speed in the motor, which fluctuations might be so l)ad as to prevent the use of the motor for the purpose. In a punch or shear undue fluctuation in speed causes rapid destruc- tion of the belt In all the above and similar cases these varia- tions must be kept within certain limits depending upon the machine. In all machines certain dimensions are fixed by the work to be done and the conditions of loading, and are very little affected by the .speed variations. Thus, the diameter of the piston of an engine depends upon the power, pressure, mean speed, etc., and haviiiK , because it c(mtains both turning and sliding elements and gives a fairly general treatment. In almost all machines there are certain parts which turn at uniform speed about a fixed center and which have a constant moment of in- ertia, such :is the crank and flywheel in an engine, while other parts, such as the (ionneoting rod, piston, etc., have a variable motion about moving centers and a correspondingly variable reduced nionieiit of inertia the tal)le in the jireceding chapter illustratfs this. It will be . Mivenieiit to use the symbol Ja to represent the nioiiieiit of iiertia of the former parts, while Jt represents thjit ni the hn •, and thus J,, is constant for all |X)sitions oi iiie in;i< iiinc, a, 111 Jb is Variable. The total reduced inertia of the machme is J J,, + -h- Both of these quantities THE PROPER WEIGHT OF FLYWHEELS 263 Ja and ^6 urc independent of the speed of rotation and depend only upon the mass and shape of the links, that is upon the rela- tive distribution of the masses about their centers of gravity. Suppose now that for any machine the values of J are plotted on a diagram along the x-axis, the ordinates of which diagram repri'srnt the corrosporuling value of the energy E; this will give a diagram of the general shajx? shown at Fig. 156. where the curve ff 1 Fio. 1.56. reprt'sonts J for the corresponding value of E shown on the vertical line.' Looking now at the figure KFGHK, it is evident from construc- tion that its width depends on the values of J at the instant and is thus indepcnik'iit of the speed. Also, the height of this figure depends on the diflference between the work put into the machine and the work delivered l)y the machine during given intervals, that is, it will depend on sueh matters as the shai)es of the indi- cator and load curves. The shape of the input work diagrams within certain limits defK'nds on whether the machine is run by gas or st<>ain, and on whether it is simple or compound, etc., but for a given engine this is also, generally speaking, independent of the speed: tlie load curve will, of course, depend on what is being driven, whether it i.s dynamo, compressor, etc. Thus the height of the figure is ahso independent of the sneed. ' This form of dianraiii appears U) be due to Wittk.vbaukr; se« "Zeit- schrift des Vereinea deutscher Iiigenieure" for 1905. t 2M TIIK TllKiHiY OF MACHINKfi 4 It will further l»o noted that the shape of the figure «ioe« not iie|H>iitJ on Ja, which is ronKtmit f<»r ii given nmchine, but only on the values of the viiriuble /*; hence the mIiuim; of this figure will be independent of the weight of the flywheel and speed, in so far as the input and load curves are indei)ondent of the spi'ed, de|M>nding solely on the reciprocating niasMca, th«! connecting nwl, th<' input-work diagrams and the load curves. Now draw from (J the two tangents, OF and OH, to KFdll, touching it at F and // resi)ectiv<'ly, then for OH tin- energy A', = HH', an «i. and since at is the least value such an angle can have it is evident that wi is the minimum sjx^ed of the engine. Similarly, A'j = FF' F'> and Ji = OF', and '^wj' = -'' = tan aj and hence, wa would be the maximum si)ce«l of the engine, since 02 is the maxin\um value of a. 216. Dimensions of the Flywheel. — Suppose now that it is required to find the dimensions of a flywheel necessary for a given engine which is to be used on a certain class of service, the mean speed of rotation being known. The class of service will fix the variations allowable and the mean speed; in engines driving alter- nators for parallel operation the variation must be small, while in the driving of air compressors and plunger pumps very much larger variations are allowable. Thus, the class of service fixes the speed variation wj — wi radians jw^r second, and the mean speed 0) = — _ - is fixed by the requirements of the output. Experience enables the indicator diagrams to be assumed with considerable accuracy and the load curve will again depend on what class of work is being done. The only part of the machine to be designed here is the fly- wheel, and as the other parts are known, and the indicator and load curves are assumed, the values of E and Ji are found as explained in Chapter XIII and the E — Ji, curve is drawn in. In plotting this curve the actual value of E is not of imix)rtance, but any point may arb trarily In? selected as a starting point ami then the values of 5E, or the change in E, and Jb will alone give the desired curve. Thus, in Fig. 156 the diagram KFGH has been so drawn and it is to i>e observed tliat the exact position of this figure with regard to the origin is unknown until J a is THE PHOFKU WEIGHT OF FLYWHEELS 265 known, but it is J, that is sought. A little considcrtttion will show, however, that an axis E'O^ niuy Im- selected and used as the axis for plotting J^, values of whieh amy Ik; laid off to the right. rurth.T, any horizontal axis O' - ./^ may Ik- selected, and for any value of J^ a point may b«! arbitrarily selected to represent the eorrcsiwnding value of IC and the meaning of this point may be later d(!termined. Having selected the first jxjint, the remain- ing points are definitely fixed, since the change in E corresponding to each change in J» is known. Thus, the curve nmy be found m any case without knowing /„ or the 8|)eed, but the origin O has Its position entirely dep<>ndent uinni both, and cannot he deter- nuned without knowing them. Thus the correct position of the axes of E and J are as yet unknown, although their directions are fixed. Having settled on «, and wj, two lines may be drawn tangent to the figure at // and F and making the angles a, and a, respec- tively, with the direction O' - J,,, where tan oi = l^w,» and tan aj = ' .^wj*. The intei-section of these two lines gives O and hence the axis of E, so that the required moment of inertia of the wheel may be scaled from the figure, since /„ = OOi. It shoiild, however, be pointed out that if the position of the axis of E is known, and also the mean speed w, it is not possible to choose w, and wj at will, for the selection of either E or the speeds will determine the position of 0. In making a design it is usual to select « and -^-^', which give «, and «,, and from the • hosen values to determine the position of O and hence the axes of /:.■ and ./. The mean speed w corresponds with the angle o. Draw a line NMLK iK-rpcndicvilar to OJ, close to the E - J r,,, LR '" OR ~ **'• diagram but in any convenient position. tan o, so that on some scale which may SR ^ , MR OR = **"' "^ '""' OH l)e found, LR represents cor, or the square of the speed /;, in revolutions jx-r minute, SR represents w^' and MR represents the square of the mean speed n all on the same scale. As in engines the difference between «i and /*, is never large it is fairly safe to a.«sume 2/i= = /i,' + n,- or that M is midway J>etween iV and L. 217. Coefficient of Speed Fluctuation.— Using now 5 to denote MICIOCOPV RBOIUTION TEST CHART (ANSI and ISO TEST CHART No. 2) u (37 116 i« 112.0 1.8 ^ /APPLIED INA/^GE I ^K 165J Eos! Main Stre« g^a Rochester. New York '4609 USA jS ("5) ♦''2 - 0300 - Phon- es ("6) 288 - 5989 - Fo< 266 THE THEORY OF MACHINES the coefficient of speed flucttiation, then 8 is defined by the relation tii — til S = n Now Therefore or nj — Ui Ui— 111 _ o "^'^ — "■' 6 = = ,-; V" \ — ^ 6 =2 (2n)' 25 = 2n« n2^ — ni* It But it has already been shown that El , . 2irni l^ui* = -^ = tan ai and since wi = ^cq"' therefore 4ir2ni' = tan ai, or 2 X60« „ J = ^^ V tan a, = 182.3 tan a,. Similarly, nj* = 182.3 tan ai) thus the speeds depend on a only. Since in Fig. 156 the base OR is common to the three triangles with vertices at A'^, M and L, it follows that RL = OR tan ai ^^ OR X ^§23 = ^"711^ Off and flA^ = Cnj^ where (' = j^^ in both cases. Further gen- erally, RM = Cn^ Then, referring to the fornmla for 25, which is 25 = 2 - m' — » Hi' — Hi' this may be put into the following form: ^ RN RL C _J& RN - RL NL RM " RM RM C 25 = 'I'hus NL = 25 X RM. 'I'hese arc marked in Fig. 156. ' It is instnictive to coiiipure this invest igatiin with the correspondinn one for governors given in Sec. 183 and Fig. 127a. THE PROPER WEIGHT OF FLYWHEELS 267 In general, a-i — ai is a small angle in practice, in which case M may be assumed midway between N and L without serious error, and on this assumption NM = 3/Z. = n^ X 5. The foregoing investigation shows that the shape of the E — J diagram has a very important effect on the best speed for a given flywheel and the best weight of flywheel at the given speed. Thus, Fig. 158 shows one form of this curve for an engine to be discussed later, while Fig. 160 shows two other forms of such curves for the same engine but different conditions of loading. With such a curve as that on the right of Fig. 160, the best speed condition will be obtained where the origin is located along the line through the long axis of the figure. In order to make this more clear, this figure is reproduced again on a re- duced scale at Fig. 157 and several positions of the origin O are drawn in. This matter will now be discussed. 218. Effects of Speed and Flywheel Weight.— Two variables enter into the problem, namely the best speed and the most economical weight of flywheel. Now, the formula connecting the speed with the angle a is ^^w^ = tan a. Sec. 215, so that the speed depends upon the angle a alone, and for any origin along such a line as OF there is the same mean speed since a is constant for this line. To get the maximum and minimum speeds corre- sponding to this mean speed, tangents are drawn from to the figure giving the angles ai and pe2 and hence toi and w«. A glance at Fig. 157 shows that the best speed corresponds to the line OF and that for any other origin such as Oi, which represents a lower mean speed, since for it a and hence tan a is smaller, there will be a greater difference between wi and 0)2 in relation to w than there is for the origin at 0. A few cases have been drawn in, and it is seen that even for the case O4 which represents a higher mean speed than O the value of 5 will be increased; thus the best speed corresponds to the line OF and its value is found from i^^w' = tan a. But the speed variations also depend on the weight of the flywheel and hence upon the value of J or the horizontai distance of the origin from the axis O'E'. If the origin waa at O4, there would be no flywheel at all but the ppeed variation taken from a scaled drawing, would be prohibitive as it is excessively large. For the position the inertia of the flywheel is represented by ll I i to i 2G8 THE THEORY OF MAClIIXEFi O - Oi and the speed variations would be comparatively small, but if the origin is moved up along OF to O.-,, the sjieed being the same as at 0, the variations will be increased vt^y slightly, but the flywheel weight also shows a greater corresped variations if the speed is not the best one, and increa.s- ing the speed may produce the s:une result, but at the speed represented by OF, the heavier the wheel the smaller will bo the variation, althougli the gain in steadiness is not nearly balanced by the extra weight of the wheel beyond a certain point. Frequently the operating conditions prevent the best THE PROPER WEIGHT OF FLYWHEELS 269 speed being selected, and if this is so it is clear that the weight of the wheel must l)e ncitiier too large nor too small. These results may be slated as follows: For a given machine and method of loading there is a certain readily obtained s^^eed which corresponds to minimum speed variations, and for this best value the variations will decrease slowly as the weight of flywheel is increased. For a certain flywheel weight the speed variations will increase as the speed changes either way from the best speed, and an increase in the weight of the flywheel does not mean smaller fluctuation in speed unless the mean speed is suitable to this condition. 219. Minimum Mean Speed. — The alx)ve results are not quite so evident nor so marked in a curve like Fig. 158 but the same Plain Line is for Outward Stroke £ I Dotted Line is for Return Strulce .♦J I i' i I Fig. 158. — Steam engine with generator or turhinc ])unip loud. conditions hold in this (ase also. The best speed is much more deiinitely hxeil for an elongated E-J curve and becomes less marked as the l)oundary of the curve comes most nearly to the form of a circle. The foregoing investigation further shows that no point of the E - J curve can fall below the axis -J, because if it should cut this axis, the machine would stop. The minimum mean speed at which th(^ machine will run with a given flywheel will be found l)y making the axis O -J touch the bottom of the curve, and finding the corrcs} juding mean speed; the minimum t 1 ■li •I :: ■J » ij » 'i ■: ii = 270 THE THEORY OF MACHINES speed will, of course, be zero, since 02 = 0. The miniinuin speed of operation may be readily computed for Figs. 158 and 160 and it is at once se^n that the right-hand diagram of Fig. 160 corrc- i)onds to a larger minimum speed than any of the others, that is, when driving the air compressor the engine will stop at a higher mean speed than when driving the generator. 220. Numerical Example of a Steam Engine.— The principles already explained may be very well illustrated in the case of tlie steam engine used in the last chapter, which had a cylinder 12.' iV, in. diameter and 30 in. stroke antl a mean speed of 87 revolutions per minute for which w = 9 radians per second. The form of indicator diagrams and loading are assumed as before and the engine drives a turbine pump which is assumed to offer constant resisting torque. The weight of the flywheel is required. Near the end of Chapter XIII is a table containing the values of J and &E for equal parts of the whole revolution and for con- venience these results are set down in the table given herewith. Tablb of Valuk-s ok J and E for 12>{6 "^ 30-iN. Enoink t, degrees J, total 2,4a3.2 18 2,405.4 36 2,410.9 54 2,416.8 72 2,420.5 90 2,420.7 108 2,418.5 126 2,412.7 144 2,407.9 162 2,404.4 180 2,403.2 198 2,404.4 216 2,407.9 234 2,412.7 252 2,418.5 270 2,420.7 288 i 2,420.5 306 I 2,416.8 324 2,410.9 342 2,405.4 360 2,403.2 2,400 3.2 5.4 10.9 16.8 20.5 20.7 18.5 12.7 7.9 4.4 3 2 4.4 7.9 12.7 18.5 20.7 20.5 16.8 10.9 5.4 3 2 m, foot-pounds - 233 + 1,387 + 1,414 + 699 + 186 - 191 - 461 - 646 - 858 -1,077 - 520 + 678 + 1,360 + 738 + 294 - 33 - 332 - 546 - 797 -1,0,58 THE PROPER WEIGH r OF FLYWHEELS 271 Selecting the axesO' — E' andO' — J't, Fig. 158, the corresponding E J curve is readily plotted as follows: The table shows that when e = 0°, Jb = 3.2 and when d = 18°, Jt = 5.4, the gain in energy which is negative, during this part of the revolution being &E = - 233 ft.-pds. Starting with Jt = 3.2 and arbitrarily calling E at this ix)int 1,000 ft.-pds. gives the first point on the diagram; the second point is found by remembering that when Jb has reached the value 5.4 the energy has decreased by 233 ft.-pds., so that the point is located on the line Jb = 5.4 and 233 ft.-pds. below the first point. The third point is at ^6 = 10.9 and 1,387 ft.-pds. al)ove the second point and so on. Now draw on the diagram the line QM to represent the mean vpeed « == 9, its inclination to the axis of Ji, being o where tan a = l^w* = }i X 9* = 40.5. The actual slope on the paper is readily I'aund by noticing that the scales are so chosen that the same length on the vertical scale stands for 1,000 as is used on the hori- 1 000 zontal scale to represent 5, the ratio being -^ — = 200; then the 5 40 5 actual slope of QM on the paper is — ^ = 0.2025, which enables the line to bo drawn. This line may be placed quite accurately by making the j)eriK>ntlicular distance to it from the extreme lower point on the figure equal the perpendicular to it from F (see Figs. 15G and 158). Thus the position and direction of the mean speed line QM are known. Now suppose the conditions of operation require that the max- imum speed shall be 1.6 \ier cent, above the minimum speed, or that the coefficient of speed fluctuation shall be 1.6 per cent. Then, from Sec. 217, 5 = 0.016, that is 6 = Wo — Wl CO 0.016 and the problem also states that the mean speed shall be w =9 0)2 "f- Wl = o' • ^^" comparing these two results it is found that W-, = 9.072 and wi = 8.928. On substituting these two values in the equations for the angles, the results are tan ai = j-jui^ = Ji X 79.709 = 39.854 and tana2 = H«2* = H X 82.301 = 41.150 which enables the two lines HL and AF to be drawn tangent to the figure at H and F and at angles ai and aj respectively to the axis of Jb (on the paper the tangents of the slopes of these lines will be, for ^^■^^^ 0.2058 and for HL = -^f^- = 0.1993). These AF = 200 200 272 THE THEORY //•' MACHINES linert are so nearly parallel that their distance apart vertically can Iw measured aiiywhe-e on the figure, and it hius actually txjen measured along NML, the distance NL representing 3,180 ft.-pds. Referring again to P'ig. 156 it is seen that NR = OR tan «•• and LR = OR tan ai and by combining these it maybe shown that 011= ^^\ .. ■ Substituting the results for this im)blem tan tan a give OR = 2,45:1 = ./.. + J>. = ./.. + 25. 3,180 H.15() - 3!>.85l Hence, the moment of inertia of the flywheel should be 2,430 approximately, which gives the desired solution of the problem. 221. Method of Finding Speed Fluctuation from E-J Dia- gram.— The converse problem, that of finding the speed varia- tion corresponding to an assumed vah:3 of J„, has been solved in the previous chapter but the diagram may l>e used for this pur- pose also. Thus, let w = 9, the same mean speed as before, and Ja = 2,000. Then, since }>W = 7 = tan a the value of E at M is (2,000 + 25) X }4 X 9' = 82,012. The points N and L will be practically unchanged and hence at N the value of Ei is 82,012 + >^(3,180) = 83,002 ft.-pds. and the value of 83 602 ui^ may be computed from the relation 3 2^2^ = 2,025 *"^^ '" ^ similar way co, may be found ami the corresponding sikhhI varia- 0)2 — a)i tion 5 = ■ A somewhat simpler method may be used, however, by refer- ring to Fig. 156, from which it appears that NL = 2n^. Thus, 2n^5 is represented by 3,180 ft.-pds. and «« by 82,012 ft.-pds., from which the value of 5 is found to be 0.0194 which corresponds to a speed variation of 1.94 per cent. In order to show the effect of making various changes, let the speed of the engine be nmch increased to say 136 revolutions per minute for which w = 14.1, and let the speed variation be still limited to 1.6 per cent. The line QM will then take the position Q'M' for which the tangent on the paper is }i and the distance corresponding to LN measures 2,400 ft.-pds. On completing the computations the moment of inertia of the flywheel is found to be Ja = 740, that is to say that if the wheel remains of the same Uw- 1 THE I'lfOPKH WKICHT OF FLY WHEELS 21^ diameter it need Ije lesH than one-third of the weiglit icquire«i for the speed of 87 revolutions. The diagram Fig. 158 hafs been placed on flic correct axis and is shown in Fig. l.W which gives an idea of the iwsition of the origin for the value J„ = 2,400 and w = !). Ficj. 159. 222. Effect of Form of Load Curve on Weight and Speed.—To show the effects of the form of load curve on this diagram and on the speed and weight of the flywheel, the curves shown in Fig. 100 have been drawn. The two diagrams shown there were made for the same engine and indicator diagrams as were used in Fig. 158, the sole difference is in the load applied fo the engine. The left-hand diagram corresponds to a plunger j)ump connected in tandem with the steam cylinder, while the right-hand diagram is from an air compressor connected in tandem with the steam cylinder. The effect of the forni of loading alone on the E-J diagi-am is most marked and the air compr(>ssor especially pro- duces a most peculiar result, the best speed here being definitely fixed and being much higher than for either of the other cases, and if the machine is run at this speed it is dear that the weight of the flywheel is not very important so long as it is not extremely small. It is needless to say that the form of indicator diagram also produces a markt ! effect and both the input and output diagrams are necessary for the. determination of the flywheel weight and the speed of the machine. The curves mentioned are sufficient 18 1 = i! f; ; 274 r///; THEORY OF MACHINES to show that the weight of wheel and the best speed of operation depend on the kind of engine and also on the purpose for which it is used. It is frequently impossible, practically, to run an engine at the speed which givos greatest steadiness of motion and then the weight of wheel must be selected with care as out- lined in Sec. 218. Fifj 160 — U>ft-han(l figure is for a plunder pump in tandem with sjonm engine; right-h^nd figure ia for an air compressor m tandem with steam engine. 223. Numerical Example on Four-cycle Gas Engine.— An illus- trntion of the application to a gas engine of the four-cycle tyiwi is shown at Fig. 162, this bei.ig taken from an actual case of an engine diroct-connectcd to an electric generator. The engim^ had a cylinder U}4 in. diameter and 22 in. stroke and was single- acting; the indicator diagram for it is shown at Fig. 161. Ihc piston and other reciprocating parts weighed 360 lb., while the weight of the connecting rod was 332 lb., and its radius of gyra- tion about its center of gravity 1.97 ft., the latter point being 24.3 in. from the center of the crankpin, and the length of the rod between centers was 55 in. nn ik There were two flywheels of a combined weight of 7,000 lb. and the combined moment of inertia of these and of the rotor of the generator was 1,600 (foot-pound units). TIIK PHOPKU ]yKiailT OF FLYWHEELS 27r, The form of the E-J diagram for this case is given in Fig. 162 and differs materially in apiieiirance from any of those yet shown, and the best speed is much more difficult to determine because of the shape of the diagram. The actaal speed of the engine mr FOUR-CYCLE QAS ENGINE DIAQRAM FlO. 161. '2 4 6 8 10 12 14 16 Fig. 162. — Oiia engine driving dynamo. 18 jX wa.s 172 ri'Vulutiuii.s per minute and fur this viiluc the .sloping lines on the diagram have been drawn. The mean-speed line would have an inclination to the ax's of Jb given by tan a = }io)^ = 162 and its slope on the paper would be ; -^^ of this, 1 f J- \ 270 THH rniioitv of m.uiiixks i n !1 :l i ' ) 1 4 Bince til.* vortical scaUf is ^^OO tiiiu'H thi' hori/niilal; thus llip tangont of the actual «1(»!m' is j .^*j - O.IOS ami I lie iiiu-s aiv drawn with this inclination. The total height of the diaRram is 31,2()0 ft.-ixls. and using tho value Ja = 1,«)0(), the mean value of A' is 1(12 X 1,9,2(H) ft.-pds. so that the sjMH'd variation is The engine here deserilnnl was ii'stalled to |)roduce eUn-tric light and it is jx-rfw^tly evident that .t was entirely unsuit»'d to ita purpose aa such a large 8|)eed variation is quite inadmissible. Owing to the peculiar shape of this diagram and the fact that the tangent points touch it on the left-hand side, it appears that the distance between them will not be materially changed by any reasonable change of sIoik; of the lines, so that if the speed re- mains constant at 172 re olutions per minute the value of J„ or the flywheel weight is inversely proixjrtional tf) the speed varia- tion and flywheel? of double th<( weight would reduce the fluc- tuation to about 3 per cent. A change in speed would bring an improvement in conilitions and the results may readily be worked out. QUESTIONS ON CHAPTER XIV 1. Show the effect of the followiiiR: (rate it at a^ ImkIi a siMt'cl us poHHJ- ble in order to increase its output. Where tiie niuchinen con- tain parts that are not moving at a uniform hjmhhI, such an th<' connecting rod of an ciiKinc or the swinging jaw of a rock crusher, the variable nature of the motion requires alternate? acceleration and retardation of these parts, to produce which forces are re- quired. These alternate accelerations and retardations cause vibrations in the machine and disturb its equilibrium; almost everyone is familiar with the vibrations in a motor l)oat with a single-cylinder engine, and many law-suits have resultetl from the vibrations in buildings caused by machiixTy in shops and factories nearby. These vibrations are very largely due to the irregular motions of the parts and to the accelerating forces due to this, and the forces increase much more rapidly than the speed, so that witi high-speed machinery the determination of these forces becomes of prime importance, and they are, indeed, also to be reckoned with in slow-spted machinery, as there are not iv few cases of slow-running machines where the accelerating forces havj caused such disturbances as to prevexit the owners opt-rating them. Again, in prime movers such as reciprocating engines of all classes, the effective turning moment on the crankshaft is nnich modified by the forces necessary to accelerate the parts; in some cases these forces rre so great that the fluid pressure in the cylin- der will not overcome them and the flywheel has to be drawn upon for assistance. The troubles are particularly aggravated in engines of high rotative sjjeed and appear in a most marked way in the high-speed steam ciigine and in the gasoline engines used in automobiles. The forces required to accelerate the valves of automobile engines may also be so great that the valve will not always re- main in contact with its cam but will alternately leave it and re- turn again, thus causing very noisy and unsatisfactory operation. ( '• 278 THE THEORY OF MACHINES LP ti. Specific problems involving the consiciprations outlined alwve will be dealt with later but before such problems can be solve«l follow it with .45 = % = V. the negative sign having been taken into account by the sense in which these are drawn. The polygon from 5 to may now be completed by adding the vectors Qt and Rt, and as the directions of these are known, the process is evidently to draw from O the line OC in the direction Rt, that is normal to c, and from B the line BC normal to b, which is in the direction of Qt, these lines inter- secting at the point C. Then it is evident that BC represents Qr on the scale — w^ to 1, and that OC represents Rt on the same scale, so that in the diagram OPP"ABCQ"0 it follows that OP = Ps, PP" = Pt, P"A = Qy, AB = R^, BC = Qt and CO = Rt, all on the scale — w" to 1. Complete the parallelogram CBAQ"; then OP" = P^ + Pt, P"Q" = Qn + Qt and Q"0 = Rs + Rt, and therefore, the vector triangle OP"Q"R" gives the vector acceleration diagram of all the hnks on the machine. 229. Ac-^leration of Points. — The linear acceleration of any point s. ;. G on b is readily shown to be represented by OG" and to be tqual to G"0.co=, where the point G" divides P"Q" in the same way that G divides PQ, the direction and sense of the acceleration of G is G"0. Similarly, the acceleration of // in c is n"0.(ji^ in magnitude, direction and souse where //" divides 0Q"{R"Q") in the same way as H divides RQ. In this way the linear acceleration of any point on a machine may be directly determined. Angular Accelerations of the Links. — The angular accelera- tions of the links may be found as follows. Since Qr ^ -AQ" X — w" w* = bub, then ba^ = - AQ".u^ or - ak = AQ" X -^ so that the length AQ" represents oa,, the angular acceleration of the link ACCELERATIONS IN MACHINERY 283 b, and similarly CO represents the angular acceleration Oc of 2 c or ae = — CO X—- The sense of these angular accelerations c may be found by noticing the way one turns to them in going from the corresponding normal acceleration line; thus, in going from Pn to Pj. one turns to the right, in going from Qf/{P"A) to Qt{AQ") th(j turn is to the loft and ht-not; a^ is in o{)posite sense to a, and by a similar process of reasoning Uc is in the same sense as a. Thus, in the position shown in the diagram, Fig. 165, the angular velocities are increasing for the links a and c, and that of the link b is also increasing since both 06 and wj are in opposite sense to a and u. It wir be found that the method described may be applied to any machine no matter how complicated, and with comparative ease. The construction resembles the phorograph of Chapter I\', which it emplovs, and hence this latter chapter must be care- fully read. Simple graphical methods for finding --i etc., may be made up, f)ne of which is shown in the applications given hereafter. THE FORCES DUE TO ACCELERATIONS OF THE MACHINE PARTS 230. Th(! real object of determining the accelerations of points and links in a machine is for the purpose of finding the forces which must be applied on the machine parts in order to produce these accelerations and also to learn the disturbing effects pro- duced in the machine if the accelerations of the parts are not balanced in some way. The investigation of these disturbing effects will now be undertaken, the first matter dealt with being the forces which nmst 1)0 applied to the links to produce the changing velocities. It is shown in books on dynamics, that if a "oody having plane motion, has a wcidit «• lb. or mass in = and an acceleration of g its center of gravity of /ft. per second per second, then the force necessary to produce this acceleration is 7«/ pds., and this force must act through the center of gravity and in the direction of the acceleration/. In many cases the body also rotates with variable angular velocity, or with angular acceleration, in which case a torque nuist act on the body in any position to produce this variable rotary motion, and if the body has a moment of inertia 284 THE THEORY OF MACHINES I about its center of gravity and angular acceleration a radians per second per second this torque must have a magnitude of I X a ft.-pds. Let the mass of the link be so distributed that its radius of gyration alxiut the center of gravity is k; then / = vik^ and the torque is nik'^a. For proof of this the reader is referred to books en dynainics. To take a specific case lot a machine with four links Ix; selected, as illustrated in Fig. 166, and lot the vector acceleration diagram U > Fk 160. — Disturbing forces due to mass of rod. OP"Q"0, as well as tho phorograph OP'Q'O be found, as already explained; it is required to find the force which must be exerted on any link such as h to produce the motion which it has in the given position. Lot G be the center of gravity of the link and lot its weight be wt, lb. and its moment of inertia about G be represented by h in feet and pound units; then h = mijcb^ where nib = — and kb is the radius of gyration about the point G. From if the vector diagram it is assumed that the angular acceleration offc has been found; also the acceleration of G, which is G"0 X w*. To produce the acceleration of G a, force must act through it of amount F = tn X G"0 X w- in the direction and sense G"0, while to produce tho angular acceleration a torque T must act on the link of aniouiit T = hab = fUbkb'ab- The torque T may ACCELERATIONS IX MACHINERY 285 be produced by a couple consisting of two parallel forces acting in opposite sense and at proper distance apart, and these forces may have any desired magnitude so long as their distance apart is adjusted to suit. For convenience let each of the forces be selected equal to F; then the distance x ft. between them will be found from the relation T = Fx. Now, as this couple may act in any position on the link b let it be so placed that one of the forces passes through G and the two forces have the same direction as the acceleration of G. Further, let the force passing through G be the one which acts in opposite sense to the accelerating force F; this is shown on Fig. 166. Now the accelerating force F and one of the forces F composing the couple act through G and balance one another and thus the accelerating force and the couple producing the torque reduce to a single force F whose magnitude is Tnb.G"0.u*, whose direction and sense are the same as the acceleration of the center of gravity G of b, and which acts at a distance x from G, determined by the relation T = Fx, and on that side of G which make? the torque act in the same sense as the angular accelera- tion Ob. The distance x of the force F from G may be found as follows: Since Q^ = &«<. = Q"A X w^Fig. 165, then a* = Q"A X j, because the line AQ" represents Qr on a scale — w'' : 1. Also and T = hat, tribki, ^ X Q"A F = m6.G"0.«2, mjib^ therefore x = „ = — mb.G"OM^ b X «= Q"A G"0 where IS a constant, so that x = const. Q"A X ^,,rk which ratio can readily be found for any position of the mechanism. This gives the line of action of the single force F and, having found the position of the force, let M be its point of intersection with the axis of link b. Now find M' the image o* M and move the force from M to its image M'; then the turn'ng moment necessary on the link a to ac- celerate the link b is FA, where h is the shortest distance from to the direction of F, Fig. 166. 28G THK THEORY OF MACHINES li 4 m This completes the problem, giving the force acting on the link and also the turning monjcnt at the link a necessary to produce this force. The same construction n>ay be ap{)liod to each of the other links such as a and c and thus the turning moment on a necessary to accelerate the links may be found as well as the necessary force on each link itself. DETERMINATION OF THE STRESSES IN THE PARTS DUE TO THEIR INERTIA 231. The results just obtained may be used to find the bending moment produced in any link at any instant duo to its inertia. -■' Fio. 167.^Bonrling forces on rod due to its inertia. Any part such as the connecting rod of an engine is subject to stresses due to the transmission of the pressure from the piston to the crankpin, but in addition to this the rod is continually being accelerated and retarded, these changes of velocity pro- ducing bending stresses in the rod, and these latter stresses may now be determined. To make the case as general as possible, let OPQR, Fig. 1G7, represent a machine for which the vector acceleration diagram is OP"Q"0, it is required to find the bending moment in the rod b due to its inertia. Lay off at each point on 6 the acceleration of that point; thus make PAi, GCi, QB\, etc., equal and parallel re- spectively to OP", OG", i)Q", etc., obtaining in this way the curve A\C\Bi. Now resolve the accelerations at each point in h into two parts, ACCELERATIONS IN MACHINERY 71X7 one normal to h and the other imriillel to the link. Thus PA is the acceleration of P nornial to 6, and GC and Qli are the corre- sponding accelerations for the jwints G and Q resiM'ctivcly. In this way a second i-urve ACH may be drawn, and the perpen- dicular to 6 drawn from any jxiint in it to the line ACH repn-sents the acceleration at the given point in h in the direction normal to the axis of the latter, the scale in all cases being — w' : 1. Thus the acceleration of P normal to h is AP.u'^, and so for other points. Now let the rod be placed as shown on the right-hand side t)f Fig. 167 with the acceleration curve ACH above it to scale. Imagine the rod divided up into equal short lengths one of which is shown at D, having a weight 5w lb. and mass hm = , and let the normal acceleration at this point be represented by DE. Should the rod be of uniform section throughout its length all the small masses like 5m will be equal since all will be of the same weight hw, but if the rod is larger at the left-hand end than at the right-hand end, then the values of bm will decrease in going along from P to Q. Now the force due to the acceleration of the small mass is equal to hm multiplied by the acceleration corresponding to DE and this force may be set off along DE above D. Proceeding in this way for the entire length of the rod gives the dotted curve as shown which may be looked upon as the load curve for the rod due to its acceleration. From this load curve the bending moments and stresses in the rod may be (ietermined by the well-known methods used in statics. For a rod of uniform cross-section throughout the acc('lerati<,m curve ACB will also be a load curve to a properly selected scales but with the ordinary rods of varying section the work is rather longer. In carrying it out, the designer usually soon finds out by experience the position of the inechanisni which corresponds to the highest position of the acceleration (;urve ACB, and the accelerations being the maximum for this i)osi1ion the roil is designed to suit them. A very few trials enable this position to be quickly found for any mechanism with which one is not familiar. The proce.ss must, of course, be carried out on tiie drafting board. 232. To Find the Accelerations of the Various Parts of a Rock Crusher.— In order to get a clearer gra.sp of the principles in- a 3 288 THK THEORY OF MACHINES ik voived, a few applications will be made, the first cane being that of the rock crusher shown in Fig. 168, The mechanism of th«f crusher is shown on the left and has not been drawn closely to scale as the construction is more clear lor the proportions shown. A crank OP is driven at uniform speed by a belt pulley on th«i shaft and to this crank is attached the long connecting rod PQ. The swinging jaw of the crusher is pivoted to the frame at T and connected to PQ by the rod SQ, while another rod QR is pivoted to the frame at R. As OP revolves Q swings in an arc of a circle about R, giving the jaw a swinging motion about T and crushing between the jaw and the frame any rocks falling! into the space. In large crashers the jaw is very heavy and its variable velocity, or acceleration, sometimes sets up very serious vibrations in buildings in which it is placed.' The acceleration diagram is shown on the right and there is also drawn the upper end of the rod b and the whole of the crank a. It is to be noted that the actual mechanism may be drawn to as small a scale as desired and the diagram to the right to as large a scale as is necessary, because in the phorograph and the acceleration diagram only the directions of the links are required and these may be easily obtained from the small scale drawing • See article by Prof. O. P. Hood in American Machinist, Nov. 26, 1908. ACCELERATIONS IN MACHINERY 289 nhown. The phorograph of the raechaniam and acceleration dia- gram sliould give no difficulty because the mechanism is simply a combination of two four link mechanisms, OPQR and RQiiT exactly similar to that shown in Fig. 165 and already dealt with. The crank OP has l)een chosen as the primary link. The crank OP is assumed to turn at uniform speed of w radians per second. For the phorograph, OQ' parallel to RQ meeting b producetl gives Q' and P'Q' = b' and OQ' = c'\ further OS' par- allel to .ST meeting Q'S' parallel to QS gives S' and Q'S' » e' while S'O gives/'. The points R' and T' lie at 0. For the acceleration diagram P" lies at P since a is assumed to run at uniform speed; then, following the method already de- H(Til)ed in Sec. 228, lay off 1"'A = , and AH parallel to c and of length AB = —, and finish the diagram by making BC per- c pendicular to 6 and OC perpendicular to c, these rsecting at C. Complete the parallelogram ABCQ" and joii. 'Q" and Q"0; then in the acceleration diagram OP" = a", P"Q" = b" and Q"0 = c" which gives the vector acceleration diagram for the part OPQR. Then starting at Q", which gives the accelera- tion of Q on the vector diagram, draw Q"I) = - and parallel 7"»S'* to e\ this is followed by DE puraih'l to TS and . anil the vector diagram is closed t)y drawing A7' perpendicular to e to meet OF peri)endicular to / in F. On completing the par- allelogram DEFS", the point S" is found and then S" is joined to O and to Q". The line S"Q" represents e on the acceleration diagram while OS" = /" represents / on the same figure. The length OS" represents the acceleration of S on a scale of — w* : 1 and the acceleration of any other point on / is found by locating on OS" or R"S" a point similarly situated to the desired point on ST. If the angular acceleration of the jaw is required, it may be found as described at Sec. 229 and evidently is a/ = -FOX J. Calling G the center of gravity of the jaw / and locating G" in the same way with regard to .S'T" that G is located with regard 19 290 run TIIKOR) OF MA. niXFS to ST, thp Hccolorution of G i» G"0 X w» iind tho forcp roqiiircH to cauw thi« ju'(H'l«Tution and therefore Mhakinn the machine Ih ...... . weiRht of jaw , parallel to 0"0 and is ecjual to O'V X w' X 32 16 '' Or the torque required for tho pur{K)W! is // X a/ where // Ih the moment of inertia of/ with reRard to (!. 233. Application to the Engine. — Thiu conHtruction and the determination of the aeiH'leratioiw and forces has a very uw'ful application in the caw^ of the reciprocating engine and this ma- chine will now l)e taken up. Fig. Ui9 represents an engine in Fio. 169. which is the crank.shaft, P the crankpin and Q the wrist pin, the block c representing the crosshead, pi.ston and piston rod. Let the crank turn with angular velocity w radians per second and have an acceleration a in the sensi hown, and let G be the center of gravity of the connecting rod b. To get the vector accelera- tion diagram find P" exactly as in the former construction, OP representing the acceleration PO.u- and PP" the acceleration aa, both on the scale —u^ to 1. Now the motion of Q is one of .sliding .and thus Q has only tangential acceleration, or acceleration in the direction of sliding, in this case QS, the sense being determined later. Hence, the total acceleration of Q must he represented by a line through in the direction, QS therefore Q" lies on a line through the center of the crankshaft, and the diagram is reduced to a simpler form ACCKLKRATinXS IN .dACtllSKRY 2t)l than in the inoro general case. Having found P", draw P"A parallel to 6, of length . » to represent Qh, and also draw AQ", normal to P"A, to meet the line Q"0, which is parallel to QS, ill Q". Then will AQ" represent the value of the angular acoel- eration of the rod 6. Since hai, » Q"i4.w' or a» - Q"A.-^, and since AQ" lies on the same side of P"A that PP" does of OP, therefore o^, is in the same sense as a; thus since tan is opposite to w, the angular velocity of the rod is decreasing, or the rod is being retarded. The acceleration of the center of gravity of h is represented by or?" and is ecjual toG"0.«', and similarly the acceleration of the end Q of the rod is represented by OQ" and is equal to Q"0.w', this being also ii acceleration of the piston. It will Im? obsti ved that all of these accelerations increase as the square of the number oi revolutions per nunutc of the crank- shaft, so that while in slownspeed engines the inertia forces may not produce any very serious troubles, yet in high-sjwed engines they are very important and in the case of such engine" as are used on automobiles, which run at as high speeds as 1,.')00 revo- lutions per minute, thes<> accelerations are very la^ge and the forces necessarj' to produce them cause considerable disturb- ances. Take the piston for example, the force required to move it will depend on the product of its weight and its acceleration, so that if an engine ran normally at 750 revolutions per minute and then it was afterward decided to speed it up to 1,500 revo- lutions per minute, the force required to move the piston in any position in the latter case would be four times as great as in the former case. 234. Approximate Construction. — In the actual case of the engine, the calculations may l»e very much simplified owing to certain limitations which arc imposed on all designs of engines driving other machinery, these limitations being briefly that the variations in velocity of the flywheel must l>c comparatively small, that is, the angular acceleration of the fljTvheel must not be great, and in fact, on engines the flywheels are made so heavy that rt cannot be large. To get a definite idea on this subject a case was worked out for a 10 by 10-in. steam engine, running at 310 revolutions per min- ute, and the maximum angular acceleration of the crank was 292 THE THEORY OF MACHINES found to be slightly less than 7 radians per second per second. For this case the normal acceleration of P is rw" = K2 X 1,100 = 458 ft. per second per second, while the tangential acceleration is ra = Yo X 7 = 5.8 ft. per second per second, which is very 12 small compared with 458 ft. per second per second, so that on any ordinary drawing the point P' would be very close to P. Thus without serious error ra may be neglected coni pared with ru^ and hence P" is at P. With the foregoing modification for the engine, the complete acceleration diagram is shown at Fig. 170, the length PA repre- 0' X' a ^h i Piston acceleration. senting and AQ" is normal to b, thus P'Q" is the acceleration diagram for the connecting rod and OQ" represents the accelera- tion of the piston on the scale -u^ to 1. Two cases are shown: (o) for the ordinary construction; and (6) for the offset cylinder. The acceleration of any such point as G is found by finding G", making the line GG" parallel to QQ", the acceleration then is It should be noticed that the greater the ratio of b to a, that is the longer the connecting rod for a given crank radius, the more nearly will the point A approach to P because the distance b' PA represents the ratio , and this steadily decreases as h m- creases, and at tlie same time AQ' becomes more nearly vertical. In the extreme case of an infinitely long rod, carried out practi- cally as shown at Fig. 6, the point A coincides with P and AQ" ACCELERATIONS IN MACHINERY 293 is vertical and then the acceleration of the piston which is OQ" is simply tho projection of o on the line of the piston travel or the acceleration Q"0 X w* = a . cos X u^ where B is the crank angle POQ". 235. Piston Acceleration at Certain Points. — Taking the more common form of the mechanism shown at Fig. 170(a) the num- erical values of the acceleration of the piston may be found in a few s[)e(ti:il cases. When the crank is vertical, b' is zero and there- fore .1 is at /* verticallj' above 0, so that when AQ" is drawn, Q" lies to the left of showing that the piston has negative ac- celeration or is bf'ing retarded. For this position a circle of diam- eter QQ" will pass through P and therefore Q"0 X OQ = OP' or OP^ o* — and the acceleration of the piston is 2 — n2 ^ Q"0 w" X QO a' ■y/b^ -a' Vb'^ - a' ft. per second per secoiid. ^ = -^. so At both the dead centers b' = a hence P"A that for the head end, Q"0 = a + , and the piston has its maxi- mimi acceleration at this point, which is ia + l)<*'' toward 0, while for the crunk end, Q'O = a - -y and the acceleration is (a — I ) w* toward 0, or the piston is being retarded. Example. — Let an engine with 7-in. stroke and a connecting rod 18 in. long run at 525 revolutions per minute. Then a = *.^ = 0.29 ft., ^ = 19 = 1-^ ft., and w = 55 radians per second. At the head end th(> acceleration of the piston would be: {,1 + "^)w" = (o.20 + j'g ) X 552 = 931 ft. per second per second. At the (Tank end the acceleration would be: (a — , ) oj- = (0.29 — " ^-j X 55^^ = 623 ft. per second per second. At the tiiiio wiiLii t\K- crank is vertical tiit- result is: second. 0.292 ^ \/l.5'2 - 0.29^ X 55^ = 173 ft. per second per 294 THE THEORY OF MACHINES The angular acceleration of the connecting rod, being deter- mined by the length AQ", is zero at each of the dead points but when the crank is vertical it has nearly its maximum value; the formula for it is Q"A X %-• When the crank is vertical a dia- gram will show that Q"A "^ \/b^ - a' an, which, if properly placed, and if of proper weight, will have the same inertia and weight as the it'i original rod. Let these masses be m\ and vit where Wi = -— -' in which i/i and m'j are the weights of the masses in Q and m-i = pounds. Further, let muss Wi be concentrati'd at Q, it is required Fkj. 174. 11 to fiiul the weights ifi and w'o and the position of the weight u'l. Let r-i be the distance from the center of gravity of the rod to mas.s m-i. These masses are determined by the following three conditions: L The sum of the weights of the two masses nuist be equal to the weight of the rod, that is, u\ + w'2 = «"(., or vh + '"2 = rrih. 2. The two ma.sses Wi and rtii, must have their combined center of gravity in the same place as before; therefore, Wiri = 3. The two masses must have the same moment of inertia about tb.eir combined center of gra\ 'y G as the original rod has about the same point; hence For convenience these are assembled here: nil + WI2 = w* (1) viiTi - miTi (2) '"I'l^ + W2r2* = mi^h^ (3) ACCELERATIONS IN MACHINERY 299 Solving these gives: ra wii = m* X ,' and wij = m* X — ; — ri + Ti ri + rj and rifj = kb* or rj ri" Thus, for the purposes of this problem the whole rod may be replaced by the two masses w'l and ma placed as shown in Fig. 174. 'rh(! one ma.ss OTi merely has the same (effect as an increase in the weight of the piston and the method of finding the force required to accelerate it has already been described. Turning then to the mass ?«2, which is at a fixed distance ri from G; the center of grav- ity of m-i is K and the acceleration of K is evidently K"0 X u)', K"K being parallel to G"G. The direction of the force acting on wtj is the same as that of the acceleration of its center of gravity and is therefore parallel to K"0, and the magnitude of this force is iTii X K"0 X w^. The force acts through K, its line of action being KL parallel to K"0. The whole rod may now be replaced by the two masses vii and Tn-i. The force acting on the former is vii X Q"0 X w* through Q parallel to (?"0, that is, this force is in the direction of motion of Q and passes through L on Q"0. The force on the mass ?«2 's m-i X K"0 X w", which also passes through L, so that the resultant force/*' acting on the rod must also pass through L. Thus the construction just described gives a convenient graphical method for locating one point L on the line of action of the resultant force F acting on the (connecting roj. Having found the point L the direction of the force F has been already shown to be parallel to G"0 and its magnitude is wj(, X G"0 X «". Let F intersect the axis cf the rod at //, find the image //' of H, and transfer F to //'. The moment required to produce the acceleration of the rod is then Fh. A number of trials on different forms and proportions of en- gines have shown that the point L remains in the same position for all crank angles, and hence if this is determined once for a given engine it will be only necessary to determine G"0 for the different crank positions; as this enables the magnitude and direction of F to be found and its position is fixed by the point L. 239. Net Turning Moment on. Crankshaft. — For the position uf the machine shown in Fig. 174, let P be the total pressure on 300 THE THEORY OF MACHINES the piston due to the gas or steam pressure; then the net turning moment acting on the crankshaft is PXOQ' -[mcX Q"0 X «» X OQ' + m* X G"0 X «' X A) after allowance has been made for the inertia of the piston and connecting rod. This turning moment will produce an ac- celeration or retardation of the flywheel according as it exceeds or is leas than the torque necessary to deliver the output. All of these quantities have been determined for the complete revolution of a steam engine and the results are given and dis- cussed at the end of the present chapter. 240. The Forces Actix^ at the Bearings.— The methods de- scribed enable the pressures acting on the bearings due to the inertia forces to be easily determined, and this problem is left for the reader to solve for himself. In high-speed machinery the pret jures on the bearings due to the inertia of the parts may become very great indeed and all care is taken by designers to decrease them. Thus, in automobile engines, some of which attain as high a speed as 3,000 revolutions per minute, or over, during test conditions, the rods are made as light as possible and the pistons are made of aluminum alloy in order to decrease their weight. In one of the recent automobile engines of 3-in. bore and 5-in. stroke the piston weigh.s 17 oz. and the force necessary to accelerate the piston at the end of the stroke and at a speed of 3,000 revolutions per minute is over 800 pds., corresponding to an average pressure of over 110 pds. per square inch on the piston and the effect of the connecting rod would increase this approximately 50 per cent; thus during the suction stroke the tension in the rod is over 1,200 pds. at the head- end dead center and the compressive stress in the rod is much less than that corresponding to the gas pressure. At the crank- end dead center the accelerating force is also high, ^ lough less than at the head end, and here also the rod is in compression due to the inertia iv^rces. If the gas pressure alone were considered, the rod would be in compression in all but the suction stroke. 241. Computation on an Actual Steam Engine. — In order that the methods may be clearly understood an example is worked out here of an engine running at 525 revolutions per minute, and of the vertical, cross-compound type with cranks at 180°, and developing 125 hp. at full load. Both cylinders are 7 in. stroke and 1 1 in. and 15^ in. diameter for the high- and low- ACCELERATIONS IN MACHINERY 301 pressjjre sides respect ivoly. The weight of each ';et of recip- roeufiiiK parts iiiehidinK jHstorv, piston rod and crossheiid is 101 II)., while the coniieetinK rod weighs 47 lb. has a length between centers of 18 in. and its radius of gyration al)oiit its cen- -'■»'^36 Fio. 175. ter of gravity is 7.56 in., the latter point being located 13.3 in. from the center of the wrist pin. From the above data u = 55 radians per second, 1 ft 1 A7 T K.£i. m. = 32 16 "^ ^' "^ "^ 32 16 ^ ^'^^ *"^^ ** ^ 12 " *'^^ '*' Alsor,= ^^2^ = 1.11 ft., r, = y^y = <*'^'» ^^- »"^^ wi = i.40 X 0.36 1.11 +0.36 = 0.35 while w, = 1.11. Fig. 176. —Effect of connecting rod ami piston. The construction for the crank angle 36° is shown in Fig. 175 with all dimensions marked on, and the complete results for the entire revolution for one side of the engine are set down on the Hccouipanying table, ail of the quantities being tabulated. The point L for this engine is located 0.44 in. from O and on the cylinder side of it. The table shows that at the hea,202 It), would be rerjuired to accelerate each piston which corresponds to a mean pressure for the high-pressure 302 THE THEORY OF MACHINES 'f-. u ^ J' side of 55 lb. per square inch., in other words if the net steam pressure fell below 55 lb. at this point the high-pressure rod would be in tension instead of compression. The disturbing effect of the connecting rod is much less nmrked as the table shows, but in accurate calculations cannot be neg- lected. The coml)in(Ml effect of the two as shown in the last column is quite decided. In order that the re.si'lts may be more clearly understood they have been plotted in Fig. 176, which shows the turning moment at the crank.shaft recjuired to move the piston and the crosshead separately, and also the combined effort required f'"- both. The turning effort required for the rod is not quite one- twelfth that recpiired for the piston. Tabi.k SiiowiNd TiiK Ekfkct DiTK T<) TUB Inkktia uv thk I'arts or AN 11 UY 7-iN'. Stkam E.vcink HiiiNMNu AT 52'i Revolutions I'KU Mini'tk roniirctiiiK rml Total turn- • i • Pist (■ r<»M«hra»l, ftc. 'ing moment reciuircd ll -fee ! < *^ f 1.053 4 . r 5.262 II % B J at crank to * ?!, niovt all „J, parU, Crank Q'O. ft 34(1 -. 1 *■, '■ 1 1,0,56 1„542 a c B |ig; + ds. +0 S4H 1 H .■(2.5 ttH3 4.1)16 107 J i ,526 297 898 1.311 0.0,33 + 43 569 M . 253 765 3,827 198 7.58 272 823 1,201 0.052 62 820 54 0.1.W 463 2,314 0.264 611 "242 732 1,069 0.050 53 664 72 +0.047 + 142 + 711 0.295 +210 0.217 050 958 0.023 + 22 232 90 -0 060 1 -181 -908 292 ; 1 -265 218 650 950 i0.023 -22 287 108 0.137 414 1 1 2,072 0.260 1 539 '0.228 690 1.007 !o.045 45 584 !26! (1.187 566 2,829 0.208 588 So. 248 750 1,095 {0.048 48 636 144 217 656 3,282 0.1431 469 i 0.267 808 1,179 0.035 41 510 162 -0 232 -702 -3„509 073 -2.56 0.273 824 1,205 0.017 -20 276 ISO 235 711 3,555 O' 1 , 0.235 711 1,038 : 1 108 +0 2.32 + 702 1 1 + 3,.')09 0.073 +2.56 0.273 824 1,205 'o.017 + 20 276 J16^ 0.217 656 3,282 0.143 469 0.267 808 1,179 0.035 41 510 234' 0.187 566 2,829 0.208 ,588 0.248 750 1,095 0.048 48 636 2.52 0.137 414 2,072 0.260 .539 0.228 690 1007 0.045 45 584 270 +0.060 ! + 181 + 908 2P2 +265 0.215 1 650 950 0.023 + 22 287 288 -0.047 -142 -711 0.295 -210 jo. 217 656 958 |0.023 -22 232 306' 153 463 2,314 0.264 611 'o 242 732 1,069 0.0.50 53 664 324 253 765 3,827 0.198 7.58 0.272 823 1,201 0.52 02 820 342 0.325 9S3 4,910 0.107 - 526 297 898 1.311 0.033 - 43 569 360 -0.348 i - 1 .0.53 ' - 5,262 i 1 i 0.349 1 ,0.56 1,542 <» i I 1 '.-''^/'iX-a ACCFLFRATIONS IX MACmXERY 303 I The relative effects of these turnii>K tnomentH is shown more clearly at Fig. 177 in which separufi! curves are drawn for the high- and low-preasure sides. The dotted curves in both cases show the torque due to the steam pressure found as in Chapter X, while the broken lines show the toniue required to accelerate the parts and the curves in solid lines indicate the net resultant torque acting on the crankshaft. The n-ader will lie at once S H «iu sou I'JOU . SOO C 400 800 1200 High Prntara Cylindar I TOrnoll \ /f Torqo« Bi-quirvd to \ \ 7 —' : ikccelcmtu Ftrtt V / — — . - . ^.^^* Low Preuure Cylindar N- /f _^orqu« Hcqulrcd to i » AccelerAtu Pkrtt vy- Fio. 177. — Torque diuKruius allowing for inertia of parts. struck with the modification produced by the inertia of the parts, but it must always be kei)t in mind that this only modifies the result but produces no net change, a.s the energy used up in accelerating the masses for one part of the revolution is returned when the nias.ses are retarded later on in the cycle of the ma- chine. That these forces nmst be reckoned with, especially in high-speed machinery, is very evident. 242. The Gnome Motor. — One further illustration of the ;iinciples stated here may be give., in the Gnome motor, which has had much application in aeroplane work. The general form of the motor ha.s already been shown in Fig. 12 in the early ..iiiL., i..,....^ 304 THE THKOHY OF MA('HI\KS part of this book and it ha« been explained that the mechaniBm is exactly the name as in the ordinary reciprocating engine except that the crank iH fixed and the connecting rod, cyUnder and other parts make complete n^volutions. The mechiiniHm in shown in Fig. 178 in which a is the cylinder and parts secured to it, b is the connecting nxl und c is the piston, fti •' power is delivered from the rotating link a, which is iiHwuni to turn at (sonstant speetiiiK (hi'Hi-alc ff P'Q' = CR' and make Q'A and AR' eqtial respectively to P'B and flC. Then OP' = /*a, /"Q' = Qr, (?'.! = «a, AR' = /^ and (^*/i* represents the rod b vectorially on the acceleration diagram, G' corresponding to its center of gravity (!. The accel- eration of the center of gravity G o{ b inG'O X w* and the an- gular acceleration of the rod is R'A X-,- as given in Sec. 229. 6 The pull on b due to the centrifugal effect of the piston is Q'O weight of position X w« X in tl •': -lirection of a. 32.1ti The resultant force F on the rod b may be foui.d as in Sec. 230 and is in the direction G'O, that is, aloiig a. Its position is shown on the figure and the pressiire between the piston and cylinder due to this force is readilj- found knowing the value and position of F. QUESTIONS ON CHAPTER XV 1. A woiuhl of 10 11). is iittacln'fl l>.v h rod 1") in. lonn to a shaft, rotating Ht 100 rcvoliitiotis per riiiiiiitc; find the acceloration of the weight and the t.pn- sidii in the nxl. 2. If the shaft in question 1 increases in speed to 120 revolutions per minute in 40 sec., find the tangential acceleration of the weight anil iiNo its total acceleration. 3. A niilroad train weighiiifr -ttX) tons is hrought to r- '., from •'^' . .iles per hour in 1 mile. Find the average rate of retardation and the inei, ' ' .distance used. 4. At each end of the stroke the vekx-ity of s piston is zero; how is its acceleration a nuixinuini ? 5. Weigh and measure the parts of an automobile engine und compute the maximum acceleration of the parts and the piston pressure necessary to produce it. 20 30'> THE THEORY OF MACHINES 6. Find the bending stresses in the connecting rod of tlie same engine, dw to inertia, when the crank and rod are at right angles. 7. Divide the rod in question 6 up into its equivalent nuisurs, locating ouv at the wristpin. 8. Make a complete deterniination for an automobile engine ul tlic result- ing torque diagram due to the indicator diagram and inertia of parts. ( HaPTER XVI BAlAI'.riNG OF MACHINERY 243. General Discussion on Balancing.— In all machines the parts have relative motion, as discussed in Chapter I. Some of the parts move at a uniform rate of spwd, such as a crankshaft or belt-wheel or flywheel, while other parts, such as the piston, or shear blade or connecting rod. have varial)le motion. The motion of any of these parts may cause the machine to vibrate and to unduly shake its foiuuhition or the l)uilding or vehicle in which it is used. It is also true that the annoyance caused by this vihratio.. may be out of all proportion to the vibration itself, the results being .so marked in some cases as to disturb buildings many blocks away from the i)lace where the machine is. This disturbance is frcHjuently of a very .serious nature, sometimes forcing the abandonment of the faulty machine alto- gether; therefore the cause of vibration in machinery is worthy of careful examination. It is not possible in the present treat is(» to discuss the general question of vibratiu-s, as the matter is too exten.sive, but it may he stated that one of the most conmion causes is lack of balance in different parts of the machine and the present chapter is devoted entirely to the jM-oblem of balancing. Where any of the links in a machine' undergo acceleration forces are set up in the machine tending to shake it, and unless these forces are balanced, vibrations of a more or less serious nature will occur, l)ut balancing need only be applied where accelerations of the parts occur. It must be borne in mind, however, that the accelerations are not confined solely to such parts as the piston or the connecting rod which have a variable motion, but the particles compos- ing any mass which is rotating with uniform velocity about a fixed center also have acceleration' and may throw the macihine out of balance, because, as explained in Sec. 226, ' In connection with this (lie first part of Chapter XV should be read over again. 307 J^ a ;l ai U " 'N' 30S THE THEORY OF MAC TIT YES a mass has acceleration along its path when its velocity is changing, and also acceleration normal to its curved path even when its velocity is constant. In discussing the sub- ject it is most convenient to divide the problem up into two parts, dealing first with links which rotate about a fixed center and second with those which have a different mot.on, m all cases plane motion lieing assumed. THE BALANCING OF ROTATING MASSES 244. Balancing a Single Mass —Let a weight of iv lb. which has a mass m = ''' rotate about a shaft with a fixed center, at a fixed radius r ft., and let the radius have a uniform angular velocity of 0, radians per second. Then, referring to Sec. 22fi this mass will have no acceleration along its path since w is assumed con- stant, but it will have an acceleration toward the axis of rotatioii of ru^ ft. per second i-er second, and hence a radial force of amount - rw^ = mru^ pds. must 1)0 applied to it to maintain it at the give', radius r. This force must l)e applied by the shaft to which the weight is attached, and as the weight revolves there will be a pull on the shaft, always in the radial direction of the weight, and this pull will thus produce an unbalanced force on the shaft, which must be balanced if vibration is to be avoided. Let Fig. 179 represent the weight under consideration in one of its positions; then if vibration is to be prevented another weight w, must be attached to the same shaft so that its acce ora- tion will be always in the same direction but in opposite sense to that of w, and this is possible only if uh is placed at some radius r, and diametrically opposite to «-. CMearly, the relation be- Fio. 179. 1/' tween the two weights and radii is given by ^ roi- - — riw or no = nwr, since "' is common to both sid-s, from which the product r,uu is found, and having arbitrarily selected one of these quantities such as r„ the value of u'l is easily determined. BALANCIXG OF MACHINERY 300 If the two weiglits arc placed as explained there will be no resul- tant pull on the shaft during rotation, ami hence no vibration; in other words the shaft with its weiglits is balanced. It sometimes happens that the eonstruetion prevents the phuung of the balancing mass directly opposite to the weight «', as for example in the case of the crankpin of an engine, and thei\ the balancing weights must be divided ])etween two planes which are usually on opposite sides of the disturbing mass, although they may be on the same side of it if desired. * Let Fig. 180 represent the crankshaft of an engine, r.- d let the crankpin correspond to an unbalanced weight w lb. at radius r. The planes A and B are those in which it is possible to plac(? counterl)alancc weights and the magnitude and positiv < of the weights are Fio. 180. — Crank-shaft balanoing. required. Let the weights be u'l and u-o lb. and their radii of rotation be r^ and rj respectively; then clearly the vector sum iwiTi -\- WiTi — wr) — = 0, or iriri + icjrs = irr. Let all the masses be in the plane coiitaining the axis of the shaft and the radius r. Now it is not sufficient to have the r^-'ation between the masses and radii determined by the formula u'lri + Wifi = ur alone, because this condition only means that the shaft will be in static equilibrium, or will be balanced if the shaft is sup- ported at rest on hor'^ontal knife edges. When the sliuft re- volves, however, there may be a tendency for it to "tilt" in the plane containing its axis and the radii of the three weights, and this can only be avoided by making the sum of the moments of 310 THE THEORY OF MACHi ..ii the qiiantitiea r X w X — about an axi3 through the shaft normal to the last-mentioned plane, equal to zero. For convenience, select the axis in the plane in which W\ re- volves, and let a and a-i be the respective distances of the planes of rotation of w and Wj from the axis; then the moment equation gives (hto — WiT-iai ) — = or xvra = Wirtat. * y Combining this relation »vith the former one i/'iri + wir-y = ur gives WiTi = ht(1 — ) and tViVi = wr — Oj so that u'lVi and w-zTi are readily determined. As an example let lo = 10 lb., r = 2 in., a = 4 in. and 02 = 10 H2 in.; then u'2''2 = ?{2) ^^^ if '"2 be taken as 4 in. H'2 = -^'^ = 8, Ti n = 1, from which if «"i = 2 lb., since the radii are to be in feet. Further, the value of /10_ 4' l'^"--<^\ in V 2/12 12 \ 12 be arbitrarily chosen as 4 lb., it will have to revolve at a radius of 1 4 ft. or 3 in. from the shaft center. In this way the two weights are found in the selected planes which will balance the crankpin. 246. Balancing Any Number of Rotating Masses Located on Different Planes Normal to a Shaft Revolving at Uniform Speed. — ^Ijct th(>re be any number of masses, say four, of weights u'l, W2, w's and V4, rotating at respective radii ri, rj, r-^ and Vi on a shaft with fixed axis and which is turning at w radians per second, the whole being as shown at Fig. 181. It is required to balance the arrangement. As before, this may be done by the use of two additional weights revolving with the shaft and located in two planes c'' revolution which may be arbitrarily selected; these are shown in the figure, the one containing the point O, and the other at A, and the quantities Oi, «2, oj, a^ and a^ represent the distances of the several planes of revolution from 0. BALANCING OF MACHINERY 311 It is convenient to use the left-hand plane, or that through 0, !is the plane of reference and, in fact, the reference plane must always contain one of the unknown masses, and it will be evident that if the niass(>s are balanced the vector sum - X r X w- nuist be zero. Further, the vector s .m of the tilting mr .ents )/• X r X a X of the various masses in planes containing the masses and the shaft must also be zero;. otherwise, although the system may be in equilibrium when at rest, it will not be so while it is in motion. Now, since u"^ and g are the same for all the I'i(i. 181. — Balancing revolving masses. masses, therefore, the above ecpiations may be reduced to the form: (l)vectorsum of the products w X rnmst be zero; and (2) vector sum of the products wra must be zero. Sinc^ the first of these is the condition to be observed if the shaft is stationary, it may be called the static condition, while the second is the dynamic condition noming into play only when the shaft is revolving. Now the tilting moment w X r X a has a tendency to tilt the .shaft in the plane containing r and the shaft, and it will be most convenient to represent it by a vector parallel to ♦he trace of 312 THE THEORY OF MACHINES »f ' this plane on the plane of revolution, or what is the same thing, by a vector parallel with the radius r itself, and a similar method will be used with other tilting momenta. Two balancing weights will be required, ir at an arbitrarily selected radius r in the normal plane through 0, and w» at a selected radius r^ in the normal plane through A. Now from the static condition the vector sum nr + w'lri + wir^ + t/Jjfj + Wtr^ + w^ti — where w and w'5 are unknown, and these cannot yet be found l)ccause the directions of the radii r and rs are not known. Again, since the reference plane passes through 0, tilting moments about must balance, or and here the only unknown is iv^rtai which may therefore be determined. The vector polygon for finding this quantity is shown at (o) in Fig. 181 and on dividing by at the value of w^& is given. The force polygon shown at (6) may now be completed, and the only other unknown w X r found, and thus the magnitude and position.s of the balancing weights w and u'b may be found. The construction gives the value of the prod- ucts wr and w'srs so that either w or r may be selected as desired and the remaining factor is easily computed. By a method similar to the above, therefore, any number of rotating masses in any positions may be balanced by two weights in arbitrarily selected planes. Many examples of this kind occur in practice, one of the mosi common being in locomotives (see Sec. 253), where the balancing weights must be placed in the driving wheels and yet the disturbing masses are in other planes. 246. Numerical Example on Balancing Revolving Masses. — Let there be any four masses of w^eighis Wi = 10 lb., w% = ^ lb. Wi = 8 lb. and W4 = 12 lb., rotating at radii ri = 6 in., rj = 8 in., rj = 9 in. and r^ = 4 in. in planes located as shown on Fig. 12.8 It is required to balance the system by two weights in the pknse through the points and A respectively. The data of the problem give Wiri = 10 X 77i = 5, Wsr? = 6 X 8^ 12 9 4, wiTt = 8 X j2 = ^ *"d WiPi = 12 12 X 12 4, and BALANCING OF MACHINERY 313 it further u),riai = 5 X Tg == 2.5, m ..roaj = 4 X ^^ = ^•'*^' "'»''»"» 15 . . 18 „ = 6 X JO = 7.5 and w^ria* = 4 X jg = *^- T*^o first thing is to draw the tiltinR-couple vector polygon as shown on the left of Fig. 182 and the only unknown here is WiTiOi which may thus be found and scales off as 4.65. Divitling by aj 12 12 1 ft. gives WiXi = 4.65 and the direcrtion of r^ is also given as parallel to the vector w-^r-jH. Next draw the vector diagram for the products u\r as shown on the right of Fig, 182, the only unknown being the product ut «'»r,o,-4.g;" Fi(!. 182. for the plane through 0. From the i)olygon this scales off as 2.9 and the direction of r is parallel to the vector in the diagram. In this way the products wr and w^^ are known in magnitude and direction, and then, on a.ssuming the radii, the weights are easily found. This has boen done in the diagram. It is advis- able to check the work by choosing a reference plane somewhere between O and .4. and making the calculations again. BALANCING OF NON-ROTATING MASSES 247. The Balancing of Reciprocating and Swinging Masses. — The discussion in the preiseding sections shows that it is always 314 THE THEORY OF MACHISES ', r a possible to balance any number of rotating; masses by i cans of two properly pluijed weights in any two desired planes of revolution, and (he uiethotl of deterniiniiiK tliose weights has l)cen fully explained. The present and following se(!tions deal with a much more difficult problem, that of balaiuting musses which do not revolve in a circle, but have either a motion of translation at varial)le sjx'ed, such as the piston of an engine or else a swinging motion such as that of a connecting rod or of the jaw of a rock crusher or other similar part. Such problems not only present nuich difficulty, but their exact solution is usually imjwssible and all that can generally be done is to i)arti- ally l)alance the parts an- tained in this way, that is, a single revolving mass cannot be made to balance a reciprocating mass. There is only one way in which such a mass can be completely balanced and that is by duplication of the machine. Thus, if it were possible to use Pi as a crank and place a second piston, as shown in Fig. 184, the mai?ses would V^e completely balanced. If the second machine cannot be placed in the same plane nor- mal to the shaft as the first, then balance could l)e obtained by dividing it into two parts each having reciprocating weights rW-\l- aid THH TUFOHY Oh' }rA('f{lXKS '^1 ^ :iiii|iii(iist)itit from the i)liinc of tho firHt, inaoluiu*. Wlicn the rwiprociitiiiK mass moves in such ii wuy that its position may be represented by such a relation as a cos d it is said to have simple harmoiiie motion and its acceleration may always W represent»'d i)y the formula a cos 6 X w-. lialanciuK jjroblems connected with this kind of motion are problems in primary balancing and are applicable to cases where the connect- iuK rod is very lonj?, giving approximate results in such cases, anil, exact results in cases where the rod is infinitely long, and in tlic case shown in Fig. 183, just discujssed. One method in which revolving weights mav be used to produce exact balance in the case of a part having simple harmonic Oeun. ~^- Ocanxl to CruDk Shaft Itatlo 1:1 (^of Kngtu Tiii. IS"). — KriKine hahtiiring — primary Imlaiioe. motion is shown in Fig, 18o, where the two weights J^w are equal and revoIv6 at tho sjxjed of the crank and in opposite sense to one another, their combined weight being equal to the weight «• of the reciprocating parts. Evidently here the vertical components of the two weights balance one another, leaving their horizontal components free to balance the reciprocating parts. Taking the combined effective weights as equal to that of the reciprocating parts, then they must rotate at a radius equal to that of the crank, and must be 1S0° from the latter when it is on the dead center. 249. Reciprocating Parts Operated by Short Connecting Rod. — The general construction adopted in practice for moving reciprocating parts differs from Fig. 183 in that the rod imparting BALA X( ISa OF MA ( UINKK Y 3i; the inutiuii is not w) loan that iho parts move with Himph* hur- inonic motion, and in the usual proportions adopted in euKines the variation is (juite mariced, for liie rods are never lonner than six times the erank radius and are often as small as four and one- half times this radius. The method to he adopteei in sueh eases is to determine the a<'eeleration of the reeiproeatinn parts an^ + 711) = 882, which is the mean value for the true curve heights at 0" and at 180° crank angles. The difference between these two curves has been plotted in the broken line curve C and will be found on examination to be almost a true sine curve, in fact, it diffeiu so little from a sine curve that it would be impossible to distin- guish between them on the scale of this drawing.' It will be observed that the ctirve C is also in ])hase at thb ' See Appendix A fur mathematical proof of thept- statements. 318 Tllh' THEORY OF MACHINES inner dnul ccnlcr with lli<» curve .1 but Iuih twice the froquenoy and ninxiniuin height on the drnwiuK «»f 171 ft. jht second p<'r Hecond. It will also 1m' found that 171 is '/ X HH'J or ' .^ X 882 = 171. The reciprocfttinR parts of this euKine could also Ix- balanced in the manner shown in Fig. 185, but it would require that in- stead of one pair of weijjlits, two pairs sh -uld 1)C used; one pair rotating at the siKvd of the crank and 180° from it at the dead centers, and another pair in phase at the inner dead center with the first but rotating at double the speed of the crank. The w(>ight rotating at the s|M'ed of the crankshaft should l)o the same as that of the pi.ston, namely 1(»1 lb. {m = 5), if placed at at 3*2 in. radius, while the weight making twice the speed (w = 110) of the crank might also be placed at a radius of 3' 2 ''i-i 5 X 171 X 32.2 in whicli case it would weigh 3.5 12 = 7.7«) lb. In X (110)- order that these weights could rotate without interference they might have to be divid d and separated axially, in which case the two halves of the same weight would have to be placed equi- distant from the plane of motion of the connecting rod. It is needless to say that the arrangement sketched above is too complicated to be used to any extent except in the most urgent cases, where some serious disturbance results. C'ounter- weights attached directly to the crankshaft are sometimes used, i)ut at best these can only balance the forces corresponding to the curve H and always produce a lifting effect on the engine. The reader mu.st note that the above method takes no account of the weight of the connecting rod, which will be considered later. If the method already described cannot be used, then the only other method is by dupUcation of the parts and this will be dt;- .>s RivinR the resolved parts of the acceleration in two planes and balance each sejiarately, i>ut usually an approximate result is all that is desired, and, as the shakinn forces are mainly in one plane, the resolved part in this plane alone is all that is usually balanced. In engines, the method of balancing the connecting rod is somewhat different to that outlined above. The usual plan is to divide the rod up into two equivalent mass«'S in the manner described in Sec. 238, one of the masses v)i being assumed as located at the wrist pin and the location of the other nuiss w» is found as described in the section referred to. In this way the one nuiss ?«i may be repardeil simjjly as an addition to the weight Hi will l)e only 0.36 in. away from the crankpin; for long-stroke engines, however, the mass W'j may be some distance from the crankpin and in .such cases the method described \k'\o\\ will not give good results. In automobile engines the usual practice is to make the crank end of the rod very much heavier and the crankpin larger than the same quantities at the piston end and hence the first statement of this p.nragranh is not true. In one rod examined the length between centers was 12 in., and the center of gravity 3.03 in. from the crankpin center; the weight of the roJ was 2.28 lb. and selecting the mass J«i at the wristpin its weight would be 0.43 lb. and the remaining weight would l>e 1.85 lb. concentrated 0.92 in. 320 THE THEORY OF MACHINES \cU from the crankpin and on the wristpin side of it, so that consider- able error might result by assuming the latter mass at the crank- pin center. The fact that the mass mj does not fall exactly at the crankpin has been already explained in Sec. 241, and in the engine there discussed the resultant force on the rod passes through L, slightly to the right of the crankshaft, instead of passing through this center, as it would do if the mass wis fell at the crankpin. If an approximation is to be used, and it appears to be the only thing to do under existing conditions, rrit may be assumed to lie at the crankpin, and thus the rod is divided into two masses; one, wii concentrated at the wristpin and balanced along with the recip- rocating masses, and the other, mj, concentrated at the crank- pin and balanced along with the rotating masses. It should be pointed out in passing, that the method of divid- ing the rod /iccording to the first of two equations of Sec. 238, that is, so that their combined center of gravity lies at the true center of gravity of the rod, to the neglect of the third equation, leads to errors in some rods. Much more reliable results are obtained by findijig tw, and Wj according to the three equations in Sec. 238 and the examples of Sec. 241, except that Wj is assumed to be at the crankpin center. Dividing the ma&s m so that its components Wj and wj have their center of gravity coinciding with that of the actual rod will usually give fairly good results, if the diameters of the crankpin and wristpin do not differ unduly. 261. Balancing Reciprocating Masses by Duplication of Parts. —Owing to the complex construction involved when the recipro- cating masises are balanced by rotating weights, such a plan is rarely used, the more common method being to balance the recip- rocating masses by other reciprocating masses. The method may l>est l>e illustrated in its application to engines, and indeed this is where it finds most common use, automobile engines being a notable example. For a single-cylinder engine the disturbing forces due to the reciprocating masses are proportional to the ordinates to such a curve as A, Fig. 186, or what is the same thine to the sum of the ordinates to the curves B and C. Suppose i.^w a second en- gine, an exact duplirute of the first, was attached to the same shaft as the former engine and let the cranks be set 180° apart lis at Fig. 187^»). Mion it is nt onoo evident that there will be a tilting moment, normal to the shaft in the plane passing through BALANCING OF MACHINERY 321 the axis of the shaft and containing the reciprocating masses, and further, a study of Fig. 186 will show that while the ordinates to the two curves B belonging to these machines neutralize, still the two curves C are additive and there is unbalancing due to the forces corresponding to curves C. In Fig. 187 arc shown at (6) and (c) two other arrangements of two engines, both of which eliminate the tilting moments; in the arrangement {hi) the cranks are at 180° and the unbalanced forces are completely eliminated, producing perfect balance, whereas at (c) the sum of the crank angles for the two opposing engines is 180° and the forces cor- responding to curve B are balanced, while those corresponding to C are again unbalanced and additive, so that there is still an imbalanced force.' Since the disturbing forces are in the direc- (o) ^«a- -^a- (c) Fig. 187. — Different arrangements of engines. tion of motion of the pistons, nothing would be gained in this respect by making the cylinder directions different in the two cases. If a three-cylinder engine is made, with the cylinders side by side and cranks set at 120° as in Fig. 188, an examination by the aid of Fig. 186 will show that the arrangement gives approxi- mately complete balance, since the sum of ordinates to the curves B and C for the three will always be zero, but there is still a tilting moment normal to the axis of the crankshaft which is unavoidable. Four cylinders side by side on the same shaft, with the two outside cranks set together and the two inner ones also together and set 180° from the other, does away with the tilting moment of Fig. 187(a) but still leaves unbalanced forces proportional to the ordinates to the curves C. A six-cylinder arrangement with* cylinders set side by side and made of two ' In order to get a clear grasp of these ideas the reader is advised to make several separate tracings of the curves B and C, Fig. 186, and to shift these along relatively to one another so as to see for himself that the statements made are correct. 21 322 THE THEORY OF MACHINES parts exactly like Fig. 188, but with the two center cranks parallel gives complete balance with the approximations used here. In what has been stated above the reader must be careful to remember that the rod has been divided into two equivalent masses and the discussion deals only with the balancing of the reciprocating part of the rod and the other reciprocating masses. The part acting with the rotating masses must also be balanced, usually by the use of a balancing weight or weights on the crank- shaft according to the method already described in Sec. 245. It is to be further understood that certain approximations have been introduced with regard to the division of the connecting rod, and also with regard to the breaking up of the actual acceleration curve for the reciprocating masses into two simple harmonic curves, one having twice the frequency of the other. Such a division is a fairly close approximation, but is not exact. Fig. 188. The shape of the acceleration curves A, Fig. 186, and its com- ponents B and C, depend only upon the ratio of the crank radius to the connecting-rod length, and also upon the angular velocity w. For the same value of v the curves will have the same shape for all engines, and the acceleration scale can always be readily determined by remembering that at crank angle zero the accelera- tion is la + "r) w^ ft. per second per second. These curves also represent the tilting moment to a certain scale since the mo- ment is the accelerating force multiplied by the constant distance from the reference plane. The chapter will be concluded by working out a few practical examples. 262. Determination of Crank Angles for Balancing a Four- Cylinder Engine. — An engine with four cylhidcrs side by side and of equal stroke, is to have the reciprocating parts balanced by setting the crank angles and adjusting the weights of one of the pistons. It is required to find the proper setting and weight, motion of the piston being assumed simple harmonic. The BALANCING OF MACHINERY 323 dimensions of the engine and all the reciprocating parts but one set are given. Let Fig. 189 represent the crankshaft and let Wt, Wj, Wi repre- sent the known weights of three of the pistons, etc., together with the part of the connecting rod taken to act with each of them as found in Sec. 250. It is required to find the remain- ing weight Wi and the crank angles. Choose the reference plane through Wi, and all values of r are the same; also the weights may be transferred to the respective crankpins, Sec. 248, as harmonic motion is assumed. Draw the xvra triangle with sides of lengths W2ra2, Wtrat and Wtrot which gives the directions of the three cranks 2, 3, and 4. Next draw the ur polygon, from which Wir is found, and thus Wi, and the rr, =2so\bt. 700 Momenti wra Force* wr Fig. 189. — Balancing a four crank engine. corresponding crank angle. The part of the rods acting at the respective crankpins, as well as the weight of the latter, must be balanced by weights determined as in See. 245. The four recip- rocating weights operated by cranks set at the angles found will be balanced, however, if harmonic motion is assumed. Example.— Let Wi = 250 lb., w-j = 220 lb. and u't = 200 lb., r = 6 in. and the distance between cylinders as shown. Then u'ir2 = 125, WiTj = 110, wiTt = 100, WirsPi = 250, w^ga- — 6.50 and w^^ai = 700. The solution is shown on Fig. 189 which gives the crank angles and weight wi = 216 lb. The rotating weights would have to be independently balanced. 263. Balancing of Locomctives. — In two-cylinder locomotives the cranks are at 90°, and the balance weights must be in the 324 THE THEORY OF MACmNER driving whrcls. In onler to avoid undue vertical forces it is usual to balance only a part of tiie reciprocating masses, usually ;il)out two-thirds, by means of weights in the driving wheels, and these balancing weights are also so placed as to compensate for the weights of the cranks. Treating the motion of the piston as simple harmonic, this problem gives no difliculty. Example. — L<'t a locomotive be proportioned as shown on Fig. IIM). The piston stroke is 2 ft. and the weight of the revolving masses is etjuivalent to 620 lb. attached to the crankpin. The reciprocating masses are assumed to have harmonic motion and to weigh 550 lb. and only 60 per cent, of these latter masses are to be balanced, so that weight at the crankpin corresponding to both of these will be 620 + 0.60 X 550 = 9501b. for each side. O i wi l -f tliit RaJ Fui. 190. — Locomotive halimcing. r!t The reference plane for the tilting moments nmst always pass through one of the unknown masses, and the plane is here taken through the wheel 2. Note that the crank 1 being on opposite side of the reference plane to wheel 3 and crank 4, the sense of the moment vector must be opposite to what it would be if it were in the jwsition 4. The crank 1 is thus drawn from the shaft in opposite sense to the vector WiTiai. A diagrammatic plan of the locomotive in Fig. 190 gives the moment arms of the masses and the values of the corresponding moments are plotted on the right. Thus u'lria, = 950 X 1 X 1.12 = 1,069 and w^rAQA = 950 X 1 X 6.04 = 5,740 and from the vpftnr diagram tcsfjOj scales off as 5,840. Since o^ = 1.92 ft., "'»'"3 = 1,187. The force polygon may now be drawn with sides 950, 950 and 1,187 parallel to the moment vectors and then u'srj is scaled off as 1,187. Selecting suitable radii r2 and rs give the weights Wi and Wt and the end view of the wheels and BALANCING OF MACHINERY 325 axle on the right shows how the weights would l)e placed in accordance with the results. The al>ove treatment deals only with primary disturbing forces, only part of which are balanced, and further, it is to Ikj noticed that there will be considerable variation in rail pressure, which might, with some designs, Hft the wheel slightly from the tracks at each revolution, a very bad condition where it occurs. 254. Engines Used for Motor Cycles and Other Work. — In recent years engines have been conwtructe*! having more than one cylinder, with the axes of all the cylinders in one plane nor- mal to the crankshaft. Frequently, in such engines, all the connecting rods are attached to a single crankpin, and any number of cylinders may be used, although with more than five, or seven cylinders at the outside, there is generally difficulty in making the actual construction. The example, whown in Fig. Fi(i. 191. — Motor cyi It en(riiie. 191, represents a twfx'vlinder engine with line>^ 90° apart and is a construction often used in iiiotetw(-en tliem is 120°. These constructions intnxluce a nuin}j«?r of difficult problems in balancing, which can only l>e touch or —X = a cos 6 + b cos — (6 + n). Nowco8<^= \l — sin^<> =-Jl — tj sin* tf ; since sin (^ = .sin 0. a} Further, since j-„ sin* 6 is generally small compared with unity the value of V^-ft^' sin* Sis equal 1 — ^^ sin* fl approximately. (It is in making this assumption that the approximation is intro- duced and for most cases the error is not serious.) Thus -X = o cos ff -f 6 1 - ^\ sin* tf - (6 4- o) 329 330 THK THEORY OF MACHINES or — X = « cos uil — sill' ut — a 2b therefore — and dx dl ao> sin ti)t — (ji sin ut cos ut = 00) win «< — w sin 2ci)< rf'x a* J - = aw' cos w< — , w' cos 2(o< ht -1 r i = aw- cos d — , w* cos 28. Therefore the acceleration is — / = aw' cos — -,- w' cos 2$ = aw' r ]■ cos — , cos 2tf Since x is negative and the acceleration is also negative, the latter is toward the crankshaft, in the &;<.me sense as x. The above expression will be found to be exactly correct at the two dead centers and nearly correct at other points,. It shows that the acceleration curve for the piston is composed of two simple harmonic curves starting in phase, the latter of which has twice the frequency of the other, and an amplitude of . times the former's value. This has been found to be the case in the curves plotted from the table at the end of Chapter XV and shown on Fig. 186, and the error due to the factor neglected is found very small in this case. In the fiase shown in Fig. 186, w is 55 radians per second and 3>^ n = .„ = 0.292 ft., so that the value of aw' cos 6 at crank angle = is aw' = 0.292 X (55)' = 882 ft. per second per second, and this is the maximum height of the first curve. At the same angle B = the value of aw' X ? cos 2fl = 882 X ^ = 171 ft. per second per second, which gives the ma.ximum height of the curve of double frequency. These values are the same as those scaled from Fig. 186. APPENDIX B Experimental Method of Finding the Monient of Imrtia of Any Body For the convonionoe of those using this hook the experimental method of finding the moment of inertia and radius of gyration of a body about its center of gravity is given herewith. Suppose it is desired to find these quantities for the connecting rod shown in Fig. 193. Take the plane of the paper as the plane of motion of the rod. Balance the rod carefully across a knife edge placed parallel with the plane of motion of the rod, and the center of gravity G will be directly above the knife edge. Fia. 193.— Inertia of rod. Next secure a knife edge in a wall or other support so that its edge is exactly horizontal and hang the rod on it with the knife edge through one of the pin holes and let it swing freely like a pendulum. By means of a stop watch find exactly the time required to swing from one extreme position to the other; this can be most accurately found by taking the time required to do this say, 100 times. Let t sec. be the time for the complete swing. Next measure the distance h feet from the knife edge to the center of gravity, and also weigh the rod and get its exact weight w lb. Then it is shown in books on mechanics that / = - X wh - ,^2 X h^ in foot and pound units, gives the moment of inertia of the rod about its center of gravity. 331 332 THE THEORY OF MACHINES As an example, an experiment was made on an autoniohilo HMJ of 12-in. centt?™ and weighing 2 lb. 4,4 ot. or 2.281 lb. The crank and wristpins were respectively 2 in. and i)-i« in. diamcti r, and when placed sideways on a knife edge it was found t'' bal- ance at a point 3.03 in. from the center of the crankpin. The rod was first hung on a knife edge projecting through the crank- pin end, so that h = 4.03 in. or 0.336 ft., and it was found that it took 94^ sec. to uiake 200 swings; (htis 94.6 200 t = = 0.473 w'c. (0.473)' • .. 2.281 Then / - '^\^^i^y X 2.2H1 X 0.336 - .^^_^ = 0.00037 in foot and ikuuuI unitw. X (0.336) « Wh«'n .suH|M>nded from the wristpin end, it is evident that h = 9.315 in. or 0.776 ft., while t was found to be QJd'A^ ec. giving the value (0.539)' / = (3.1416)' 0.00940. X 2.281 X 0.776 - 0.0709 X (0.776)' The average of these is 0.0094 which may Iw taken as the moment of in«'rtiiv al)Out the center of gravity. The square of the radius of gyration about the same point is 0.0094^X 32.16 2.281 w 0.132 or k =- 0.36 ft. J* w INDKX AliiMiluto iiiutiun, 2ti AccelerutiDri in iiiuiliiiiery, Chuii. XV, 277 uiigular, 282 l)eii(linK "lonient due to. 2S7 ronnecting rod, 292, 2s, 320, etc. primary, 316 recipr teeth of. 92 bicydc^ 6 Brown and Sharpe gear gysteic, M Buckeye governor, 233 curves for, 23ti 833 334 INDEX \h Cams, Chap. VIII, 136 gaa engine, 143 general solution of problem, 144 kinds of, 137 purpose of, 136 shear, 141 stamp mill, 137 uniform velocity, 140 Center, fixed, 30 instantaneous, 28 permanent, 30 pressure, 190 theorem of three, 30 virtual, 28 Chain closure, 11 compound, 17 double slider crank, 22 inversion of, 17 kinematic, 16 simple, 17 slider crank, 18 Characteristic curve for governor, 212 Chuck, elliptic, 22 Circle, base, 77 describing, 72 friction, 191 pitch, 70 Circular or circumferential pitch, 77, 84 Clearance of gear teeth, 83 Cleveland drill, 129 Clock train of gears, 114 Closure, chain, 11 force, 11 ('oefficient of friction, 180 speed fluctuation, 265 Collars, 10 Compound chain, 16 engine, 170 gear train, 110 Cone, back, distance, 93 pitch, 92 Connecting rod, 4 acceleration of, 292 balanring of, 320 friction of, 193 Connecting rod, velocity of, 61 Constrained motion, 6, 10 Contact, arc of, in gears, 77 line of, skew bevel gears, 90 path of, in gears, 77 Continued fractions, application, 122 Cotter design, 184 Coupling, Oldham's, 22 Crank, 4 Crank angles for balancing, 322 Crank effort, 162, 165 diagrams. Chap. X, 166 effect of acceleration on, 295, 299 of connecting rod, 297 of piston, 295 Crankpin, 4 Crossed arm governor, 206 Crosshead, friction in, 183 Crusher, rock, 158 Cut teeth in gears, 83 Cycloidal curves, 73, 80 teeth, 72 how drawn, 74 Cylinders, pitch, 70 '>edendum line, 83 Describing circle, 72, 75 Diagrams of crank effort, 166 E-J (energy-inertia), 262 indicator, 167, etc. moticm. Chap. IV, 49 polar, 45 straight base, 45 * torque, 166, 169 vector, phorograph, 68 velocity, uses of, 49 Chap. Ill, 35 Diametral pitch, 84 Differential gear, automobile, 133 Direction of motion, 29, 30, 32 Discharge, pump, 46 Divisions of machine study, 8 Drill with planetary gear, 129 Drives, forms of, 68 INDEX 33.-) E Efficiency, engine, 196 governor, 194 machine. Chap. XI, 177 mechanical, 177 shaper, 186 Effort, crank, 152, 165 E-J (energy-inertia) diagram, 262 for steam engine, 270 Elements, 11 Elliptic chuck, 22 Energy available, 164 kinetic of bodies, 243 of engine, 246 of machine, 244 producing speed variations, 251 Engirr -celeration in, 290, 291, 300 be .1.1, 154 compound, 170 crank effort in .sti am, 166, etc. diagram of speed variation, 250, 252, 2.55 efficiency, 196 energy, kinetic, 246 internal combustion, 171 multi-cylinder, 321, etc. oscillating, 25 proper flywheel for steam, 270, 274 for gas, 274 Epicyclic gear train, 110, 124, etc. ratio, 110, 125 Epieycloidal curve, 73 Equilibrium of machines, 150 static, 8 External forces, 149 Face of gear, width of, 84 tooth, 84 Factor, friction, 181 Feather, 11 Fixed center, 30 Fluv tuations of speed in machinery, Chap. XIII, 240 apprt>x!rnat« determination of, 249 Fluctuations of speed, cause of, 242 conditions affecting, 247, 260 diagram of, 255 energy, effect on, 240, 247, 251 in any machine, 248 in engine, 250, 252 nature of, 240 Flywheels, weight of, Chap. XIV, 261 best speed, 267 effect of power on, 261 of load on, 261 gas engine, 274 general discussion, 262 given engine, 270 minimum mean speed, 269 purpose, 261 speed, effect on, 264, 267 Follower for cam, 138 Forces in machines. Chap. IX, 149 accelerating, 283, 300 effect on bearings, 300 general effects, 283 causing vibrations, 314 closure, 17 external, 149 in machine, 151 in shear, 152 Ford transmission, 131 Forms of drives, 68 Frame, 3, 6 Friction, 178 angle of, 188 circle, 191 coefficient of, 180 crosshead, 183 factor, 181 in connecting rod, 193 in cotter, 184 in governor, 194, 216 laws of, 180 sliding pairs, 181 turning pairs, 189, 192 Gas engine cam, 143 crank effort, 172 flywheel, 274 336 INDEX Gears and gearing, Chap. V and VI, 68,00 Gears, annular, 80 bevel, Chap. VI, 90, 94 Brown and Sharpe system, 84 conditions to be fulfilled in, 71 when used, 68 diameter of, 69 examples, 85 face, 84 hunting tooth, 123 hyperboloidal, 90, 94 interference of teeth, 81 internal, 80 methods of making, 83 path of contact of teeth, 73 proportions of teeth, 84 sets of, 79 sizes of, 69 spiral. Chap. VI, 90 spur, 68 systems, discussion on, 87 toothed, Chap. V, 68 types of, 90 Gears for nonparallel shafts, 91 Gears, mitre, 91 screw, 90, 102 skew bevel, 90, 93 spiral, 90, 102 Gears, worm, 90, etc. construction, 104 ratio of, 103 screw, 106, 107 Gearing, trains of, Chap. VII, 110 compound, 110 definition of, 110 epicyclic, 110, 124, etc. kinds of, 110 planetary, see Epicyclic. reverted, 110 Gleason spiral bevel gears, 93 Gnome motor, 20 - acceleration in, 303 Governors, Chap. XII, 201 Belliss and Morcom, 223 curves for, 227 Buckeye, 233 characterintic curves, 212 crossed arm, 206 Governors, definition, 201 design, 210 efficiency of, 194 friction in, 194, 216 Hartnell, 221 design of spring for, 222 height, 205 horizontal spindle, 223 inertia, 228, 229 properties of, 229 iaochronism, 206, 213, 230 McEwen, 233 pendulum, theory of, 203, 204, etc. Porter, 207 powerfulness, 211, 212, 215 Proell, 159, 220 Rites, 238 Robb, 230 sensitiveness, 210, 214 spring, 221 design of, 222 stability, 207, 213 types of, 201, 202 weighted, see Porter governor. Governing, methods of, 201 Graphical representation, see Mailer desired. II Hartnell governor, 221 Height of governor, 205 Helical motion, 9 teeth, 87 uses of, 88 Hendey-Norton lathe, 119 Higher pair, 14 Hunting in governors, 207 tooth gears, 123 Hyperboloidal gears, 91, 94 pitch surfaces of, 95, 101, 108 teeth of, 102 Hypocycloidal curve. 73 I Idler. 113 Image, 53, 55 INDEX 337 Image, angular velocity from, 57 copy of link, 59 how found, 55 of point, 53 Inertia-energy (E-J) diagram, 262 Inertia governor, 203, 228, etc. analysis of, 233 distribution of weight hi, 237 isochronism in, 234 moment curves for, 234 properties of, 229 stability, 234, 235 work done by, 236 Inertia of body, 331 parts. Chap. XVI, 3()7 re«luced, for machine, 244 Input work, 176, 251 Instantaneous center, 28, 32, 35 Interference of gear teeth, 81 Internal gears, 80 Inversion of chain, 18 Involute curves, method of drawing, 78, 79 teeth, 78 Isochronism in governors, 206, 213, 230 Jack, lifting, 185 Joy valve gear, 41 velocity of valve, 41, 66 K Kinematic chain, 16 Kinematics of machinery, 8 Kinetic energy of bodies, 243 of engine, 246 of machine, 2 14 Lathe, 4; 116, etc. Hendey- Norton, 119 screw cutting, 1 16 .^e of contact in gears, 90 Linear velocities, 29, 35 Link, 16 Link, acceleration of, 282 angular velocity of, 68 kinetic energy of, 243 motion, Stephenson, 63 primary, 66 reference, 49 velocity of, 3(5, 37, 40 general i)ri>pnH, 3.S Ixjad, effect on flywheel weight, 261, 273 Locomotive )>altincing, 323 Lower pair, 14 .\I Machine, definition, 7 design, 8 efficiency of, Chap. XI, 176 equilibrium of, 150 forces in, Chnp. IX, 150 general discussion, 3 imperfect, 8 kinetic enorpy of, 244 nature of, 3 parts of, 5 purpose of, 7 reduced inertia of, 244 simple, 17 Machinery, fluctuations of speed in Chap. XIII, 240 cause of speed variations, 240 effect of load, etc., 240, 247 kinematics of, 8 McEwen governor, 233 Mechanical efficiency, 177 Mechanism, !7, 65 Mitre gears, 91 Module, 85 Motion, absolute, 26 constrained, 6 diagram. Chap. IV, 49 direction of, 29, 30 helical, 9 in machines. Chap. II, 24 plane, 4, 9, 24 quick-return, 20, 62 relative, 5, 25, 26 propositions on, 26, 27 screw, 9 338 INDEX Motion, sliding, 11 spheric, 9 standard of coinparisoii, 25 translation, 3 turning, 10 Motor cycle ba' noing, 325 Multi-cylinder engines, 320, 322, 325, etc. N Normal acceleration, 278 Numerical examples; see Special subject. O Obliquity, angle of, 79 Oldham's coupling, 22 Oscillating engine, 19 Ouvput work, 176, 251 Pair, 3 friction in, 167, 170 higher, 14 lower, 14 sliding, 12, 33 friction in, 181 turning, 11 friction in, 189 Permanent center, 30 Phorograph, "^0, 52, 58, Chap. IV for mechanism, 55 forces in machines by, 155 principles of, 50-53 property of, 66 vector velocity diagram, 58 Pinion. 69 Piston, 3 acceleration of, 291-296, 329 velocity, 44, 61 Pitch, circular or circumferential 77, 84 circle, 70 cones, 92 cylinder, 70 diametral, 84 Pitch, normal, 101 point, 84 surfaces, 95, 100 Plane motion, 4, 9, 24 Planetary gear train, see Epieyclie train. examples, 126 purpose, 124 ratio, 125 Point, acceleration of, 282 image of, 53 of gear tooth, 84 Polar diagram, 45 Porter governor, see also Weighted governor. advantages, 209 description, 207 design of, 219 height, 209 lift, 210 sensitiveness, 210 Power, effect on flywheel weight, 273 Powerfulness in governors, 211, 215 Pressure, center of, 190 Primary balancing, 316 Primary link, 56 Proell governor, 159, 220 Pump discharge, 46 Quick-return motion, 20, 62, 186 Whitworth, 20, 62 R Racii.g in governors, 207 Rack, 80 Ilapidity of adjustment in governors, 236 Recess, angle of, 77 arc of, 77 Reciprocating masses, bp lancing of, 314, 320 balancing by duplication, 320 Reduced inertia, 244 Reeves val . e gear, 64 Referr 1 ■> link. 49, see Primaru link. Relative motion, 5, 25, 26 INDEX 339 Relative velocities, 38, 40 Resistant parts of machines, 5 Reverted gear train, 110 Rigid parts of machines, 5 Rites governor, 238 Riveters, toggle joint, 161 Robb governor, 230 Rock crusher, 158 acceleration in, 287 Rocker arm valve gear, 60 Root circle, 76, 83 of teeth, 84 Rotating masses, balancing of, 308 pendulum governor, 202, etc. defects of, 205 theor>- of, 204 Rotation, sense of, in gears, 1 13 sense of, for links, 57 Screw gears, 90, 102 motion, 9 Secondary balancing, 319 Sensitiveness in governors, 210, 214 Sets of gears, 79 Shaft governor, 203, 229; see also Inertia governor. properties of, 228 Shaper, efficiency of, 187 Shear, cam, 141 forces in, 157 Simple chain, 17 Skew bevel gears, 90, 94 pitch surfaces of, 96 Slider-crank chain, 18 double chain, 22 Sliding motion, 11 friction in, 181 pair, 12, 33 Slipping of gear teeth, 76 Space variation, angular, 258 Speed fluctuations in machines. Chap. XIII, 240 approximate determination, 249 cause of, 240 coefficient of, 265 conditions affecting, 247, 260 diagram of, 255 Speed fluctuationtt, effect of load on, 240, 247 energj' causing, 251 in engine, 250, 252 in any machine, 248 minimum, mean, 269 nature of, 240 Speed of fly-wheel, best, 267 Spheric motion, 9 Spiral bevel teeth, 90, 93 gears, Chapter VI, 90 Spring governor, 221 Spur gears, 68 Stal)ility in governors, 196, 207, 213 Stamp-mill cam, 137 Static equilibrium, 8, 150 Stephenson link motion, 03 Stresses due to acceleration, 286 Stroke, 4 Stub teeth, 79, 85 Sun and planet motion, 128 Swinging masses, balancing of, 313 T Tangential acceleration, 278 Teeth, cut, 83 cycloidal, 72, 74 drawing of, 74, 78 face, 84 flank, 84 helical, 87 hyperboloidai, 117 interference, 81 involute, 77, 79 of gear wheels, 76 parts of, 84 path of contact, 73, 78 point, 84 profiles of, 73 root, 83 slipping, amount of, 76 spiral bevel, 93 stub, 79, 85 worm and worm-wheel, 103, 104 Theorem of three centers, 30 Threads, cutting in lathe, 122 Three-throw pump, 47 Toggle-joint riveters, 161 340 ISDEX Toothed gearing, Chap. V, 68 Torque on cranksha/t, 166, etc. effect of acceleration on, 209, 301 Total acceleration of point, 280 Trains of gearing. Chap. VI, 90 automobile gear box, 116, 133 change gears, 117 clock, 114 definition of, 110 epicyclic, 124 examples, 114 formula for ratio, 112 lathe, 116 planetary, 124 ratio of, 112 rotation, sense of, 1 13 Translation, motion of, 3 Transmission, Ford automobile, 131 Triplex block, Weston, 128 Turning motion, 10 moment. Chap. X, 164 pair, 11 frict'on in, 180 U Uniform velocity cam, 140 Unstable governor, 207 V Valve gears, Joy, 41 60 Reeves, 64 rocker arm, 60 Velocity, diagram, Chap. Ill, Hf> graphical representation, 43 linear, 20, 35 of pointa, 36 relative, 38, 40 piston, 44 Velocities, angular, 35, 37, 39 how expressed, 39 Vibration due to acceleration, 277 Virtual center, 28, 32, 35 W Watt sun an