fH3 ANALYTICAL mUSmaSKtlOB OP 
 SLB3TRIC BAIL7&Y SPEED - TIME ESATIONS 
 
 By 
 
 Lloyd Hash Robinson 
 
 B.B. (Union College) 1911 
 M.S. (University of California) 1917 
 
 THSSIS '- *:' 
 Submitted in partial satisfaction of the requirements for the degree of 
 
 DOCTOR OF PHILOSOPHY 
 
 in the 
 GRADUATE DIVISION 
 
 of the 
 
 UNIVERSE? Y OP CALIFORNIA 
 June, 1919 
 
 Approved by the 
 Sub-CoEBiit t ee-in-Charge , 
 
 Deposited in the 
 University Library, 
 
 19 JQ 
 
 Chairman. 
 

IOH Page 1 
 
 I FACTORS HI ELECTRIC TRAIN PROPULSION 
 
 A - Energy, Power and Tractive Bffort ....... 3 
 
 B - Cons>onont8 of Tractive Effort ........ 7 
 
 Train Resistance, Curve Reaistance, 
 Braking Effort of Friction Brakes, 
 Grade Reaistance, Accelerating Effort. 
 
 C - Train Acceleration .....* 11 
 
 II RAILY/AY MOTOR CHARACTERISTICS 13 
 
 III SPEED - TITJE FORMULAE 
 
 A - Train Cycle .................. 20 
 
 B - Starting ................... 21 
 
 - Normal Terminal Voltage Applied to JTotora. ... 24 
 Balancing Speed, Acceleration. 
 
 D - Coasting ..* 30 
 
 E - Braking 36 
 
 F - Sxwmary of Principal Formulae ......... 39 
 
 IV EXAMPLES 41 
 

ft I/-''" 
 
 IOTHODUGTION 
 
 ?ho iloai,-;;n or a Modoi-n olectric rail./v noceasarily 
 an acourato predet oral ration of the performance of the completed system 
 whan the oonat motion period shall hare passed and the traffic begin* 
 to move* Consequently, the speed of trains predominate* among the fac- 
 tors to be treated* The speed of a train, lllce that of any moving body, 
 varies as a function of time in accordance with the fundamental lairs of 
 mechanics. From these principles, it is possible and practicable to 
 jaBsdeterraine the speed of a given electrically propelled train at any 
 instant in the course of a run over a Known or asaroaed track* ?or high 
 speed roads with frequent stops* such aa subway systems, the converse 
 problem of determining the time required to attain a particular speed 
 may be of greater importance and of correspondingly more frequent 
 ooourr ence 
 
 The predeterminations of spoed-tlme relations necessarily 
 depend on the type of motive equipment and therefore must bo based 
 upon the characteristics of the motors in the case of electrically 
 propelled trains* The methods, in current use for carrying out these 
 determinations are step-by-step, graphical processes* The most used 
 
 -1- 
 
. 
 
 
 
 .; 
 
 " 
 -'. 
 
 f oc ^Bfct; e 
 ^J MBO orW 
 
 . 
 
 - 
 
method is that proposed by Mr* C. 0. Liailloux In his "notes on the 
 Plotting of Speed-571n Curves" In the Transact lona of the /jaorican 
 Institute of Sleotrioal Sngineers, Volume XIX (1902), page 984. 
 
 The present purpose is to dorivo formulae, by means of 
 the spoed-tl* relations may be determined directly by arithmetic sub- 
 stitutions and operations without recourse to graphs other then the 
 characteristic motor curves* Since the continuous ouri-ent aeries motor 
 Is used almost universally in American electric railway practice, only 
 the formulae applicable to this type of motor are presented although 
 the method of derivation should indicate hoar analogous formulae, suited 
 to other types of motors, may bo derived* 
 
 As a foundation for the subsequent derivations, a segregation 
 of the factors involved in electric propulsion of trains is essential* 
 
 - 2 - 
 
c , ; vc JkMoqp*!- 
 
 10 Illlt Triir>lT tft r.l ^MTWO fOJ - 
 
 ^RHI^'- - 
 
 IQ tOMe: <' ,o*xaptol 0tHaA ot ti io<t l Wt to 
 ojt& O'Slft Awctettlftb t>J ^srt a . 
 
 
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I 
 
 PAOTORS DT HLSflfBIC T&XV SROPULSlOsT 
 
 A - gnSBPT. POV.151 AITD TRACT 171 BWOBg 
 
 The movement of a oar or train along a track necessitates an 
 expenditure of energy. In the general case, the mechanical energy, 
 supplied to a train, simultaneously servos three well defined purposes, 
 and consequently must be treated aa the sura of three distinct oangponents* 
 These ttxspasjfsfta aret first, that dissipated tat to friction; second, 
 that stored or liberated as potential energy due to change in the eleva- 
 tion of the train; and third, that stored or liberated as kinetic mvegj 
 due to change in the speed of the train* 
 
 By the application of vihat is icoown as dynamic braking, that 
 is, operating the motors as generators hile descending grades or while 
 Bstidng stops, mechanical energy may be extracted froa * moving train, 
 converted to electric energy and returned to the electric distribution 
 system* Ifiysloally, the extraction of energy is the reverse of the 
 process of supplying: soargy. Thus, in calculations, energy extracted 
 must be treated as negative energy supplied* Hence, the energy supplied 
 in a chosen period of time will be positive or negative according to 
 operating conditions* 
 
 The dissipation of enorcy due to friction continues as long 
 aa a train is in motion* The dissipated energy can not be recovered as 
 useful mechanical energy in subsequent operations since it passes off 
 aa heat or possibly in other, less easily discernible forms. Its nosja 
 defines it as lost* The dissipated enercy In turn may conveniently 
 

 
 
 
 I 
 
 
 
 . 
 
 - 
 
 
 . 
 
 . 
 
be rosolvefi into throo c exponents, nts aret first, 
 
 due to friction ir. trsv l tr.r<r :*, taoli*Uy re- 
 
 ferred to as tee- to trrln roriot'nce: noconfl, tJtat du<) to an incrtcicnt 
 of friction tetrodlnced by trriSh crcrvrtnre, usually spfldosn of as duo 
 to curve resistance; and third, thet due to the application of friction 
 i, Known as tae~%t> tanking effc\ . . 
 
 Potential energy it stored wbila a train ia ascending a grade* 
 a subsequent descent, potential energy is liberated and either 
 extracted, dissipated or transfonasd to kinetic energy* kinetic energy 
 is stored v/nile the speed of a train is increasing* As the speed later 
 decreases, kinetic imrflj is liberated sad either extracted, dissipated 
 or trcnsformou to potential energy, depending on whether the decrease in 
 is due to dynamic braking, friction or to ascent of a grade* 
 Oxe potential energy of a body A is the sans ^>en it is at tto 
 
 , , - , > rt -<rVT^' 
 
 elevation* Also the kinotio energy of a body at^ steadstill is zero. 
 TlMirnifri n all the energy* that in supplied to a train during its run 
 between two stations ,7hioh are at the sons elevation, is ultimately 
 dissipated before the train stops unless some of it is recovered by means 
 of dynamic braiding* 
 
 Although the foregoing discrimination of the forms | in which . 
 
 . T 
 
 energy is expended, is an essential step in tho analysis of train pro- 
 
 f> 
 
 pulsion, the term, speed-time relation, accur&toly lilies instantaneous 
 
 values of speed and time* Consequently the determination of these re- 
 lations is iiiniouiatoly concerned with insJtentTjqoous rates of energy 
 expand iture, conawqption and absorption^ 'Jhat is to say, it depends 
 H>on the input and distribution of mechanical power, for power is the 
 instantaneous rate of energy expenditure, conversion, consumption or 
 
 -4 - 
 

 IS t< 
 
 . 
 
 
 
 
 . 
 
absorption. Tho mechanical energy input ha* bean divided into three 
 main OQBPponenta. Similarly, the mechanical power input is the sum of 
 the corresponding instentanooua ratea of energy dissipation and 
 storage* 
 
 The measure of mechanical power input to a moving body is 
 tho algo ornie produot of the applied force and the instantaneous speed 
 at which the point of application of the force io moving* In an elec- 
 trically propelled train, the mechanical power input is transferred 
 from the motors throi&i sears to tho axles at the surface of the latter* 
 Hence the measure of the mechanical poerar input to an axle is tho pro- 
 duot of the tangential force at the surface of the axle and the peri- 
 pheral speed of that surface* However, in railway calculations, it is 
 convenient to use the speed of the train aa a basis of reference so far 
 aa possible* 
 
 flu peripheral speed of a point on the tire of & oar wheel is 
 flat MM as the speed of the oar provided the wheel does not slip* HJhe 
 peripheral speed of a point on the tire of a wheel is to tho speed of 
 a point on the surface of its axle in tho ratio of the diameters of the 
 wheel and axle* Therefore the ratio of the train speed to the peripheral 
 speed of tho cyolindrioal surface of the axle is equal to the ratio of 
 the diameters of wheel and axle* 
 
 By the principle of moments, a tangential force at the surface 
 of an axle may be replaced by an equivalent force applied at t ho surface 
 of a concentric oyclindor of greater radius* Zhe tangential force, which 
 must be applied at a radius equal to the wheel radius, in order to pro- 
 duce the Maw moment aa the actual tangential force applied at the surface 
 
 - 5 - 
 

 
 
 9* Of 
 
 
of tho axle, must be to the latter In the Inverse ratio of tfw diameters 
 of the wheel cad axle* This equivalent force, hypothetically noting on 
 the axle at a radius equal to the serai-die T notQr of e. driving wheel, ia 
 ailed the tractive effort* 
 
 She above relations can be expressed concisely in algebraic 
 form: thus 
 
 ( Mechanical power } { Tangential force applied ) ( peripheral speed ) 
 ( input to train | ( at aurfaoe of a driving ) X ( of cyclindrical j 
 ( per motor ( axle } ( surface of axle ) 
 
 ( Tangent ir,l force applied ) ( peripheral speed ) 
 
 ( at surface of a driving ) ( of axle surface ) 
 
 ( axle X axle diameter [ * I I hel diameter ) 
 
 ( i vjhoel diameter ) ( i axle diameter ) 
 
 Motor tractive effort X train speed 
 
 Tli& stnndard /onerioan unit of measure of train speed is miles 
 per hour, and that of tractive effort Is pounds, avoirdupois* 25io total 
 tractive effort applied to a train is tfce sm of the tractive efforts 
 applied at the several driving axles* Hovever, for purposes of com- 
 parison and for simplicity in computations, the tractive effort is 
 usually referred to the welgit of the train* That is. tho tractive 
 effort is spoken of as so many pounds per ton of gross train weigit* The 
 phrase , pounds pex* ton, unfortunately contains a latent ambiguity because 
 tho number of pounds in a ton Is preranably constant, but there should be 
 no confusion \rhen the expression is confined to railroad parlance* 
 
 Because of tho proportionality of mechanical power aad tractive 
 effort, the applied tractive effort is obviously divisible into several 
 components corresponding to tho rates of energy dissipation nd storage* 
 

 
 
 
 
 
In geoex 1 ?-!. the trrctlvo effort applied to a train jaay be 
 
 j -~~V 
 
 rosolvoii Into the foil rains consonants t 
 
 1 - Oorroaponiline to the rate of enercy diasipatlon, 
 
 (a) - Train resistance, 
 
 (b) - Curve resistance, 
 
 (o) - Braiding effort of friction brakes; 
 
 2 - Corresponding to the rate of storage of potential energy, 
 
 (a) - Srado resistance! 
 
 3 - Corresponding to the rate of storage of Kinetic energy, 
 
 (a) - Accelerating effort* 
 
 :* 
 
 rescB 
 
 Vnen a tr&iu ic moving on Icvol tangent trf.c';, thoro 10 energy 
 dlaslpatc'.l In rail raid journal friction, eir resistance, etc* She son 
 of the forces, corresponding to tho rates of energy dissipation due to 
 these causes, is c? lloa the train resistance* 2h agaltado of the 
 train resistance at any inattuit dopends tepon the train weight, speed, 
 cross-sectional area, and the mc&er of oars* The relations of these 
 factors have been determined by oxtencivs eaqportooiite with diff ercmt 
 Classen of oqidptnent in operations under wide ranges of conditions* Prom 
 ftM oaqperlmental data, empirical fonanlae for trr-in resistance hiive 
 boon deduced* One of these, flfeidx 10 quite comonly uaoii, is tfu:t 
 developed by Mr* A* H* Armstrong, 
 
 In whidh 
 
 R Train resistance In poonfla per ton of gross train wel&it, 
 
 T * (Jroan wel^it of train la tons (3000 lbs), 
 
 7 3peod of train in miles per hour, 
 
 X Projeotec! croas-sectional area of train In sqwve feet, 
 
 H ITamber of cars In train, 
 
 > 3.3 
 
 VaT 
 
 - 7 - 
 
V 
 
 
 
 ' l' 
 
 . 
 
 - 
 
 
 
Quire 
 
 torixoat 
 
 Vhen a trr.in is proceeding along a horiaontal curve in tho 
 trade, there is introduced an additional resistance due to the side 
 
 jsurs of tho ufaecl flanges on the rails, tho unequal distribution 
 of weight of t&c trela on the two rails, etc* This additional reals- 
 is called curve resistance* 
 
 DM curvs real seance varies from = 0*5 D to 1*5 D 
 
 ., .. 
 pounds per ton of gross train weight, depending upon the condition 
 
 of tho tracfc and tihcela and vg>on ttu degree of a^eroleration of the 
 outside rail* D is the degree of tho curve j that ia, ,ho ntcd>or of 
 degrees of central angle subtended by a one hundred-foot chord of the 
 circular arc described by the center line of the track* It la clear 
 that, for average conditions, may be talon as mmsrioally equal to 
 the degree of the curve. 
 Braking Hffort of Friction 
 
 friction brakes are applied to the tfeeeola of a moving 
 train, energy is dissipated in accordance with the lava of aliuing 
 friction of natal on metal tdder pressure- However, since the pressure 
 of the braksa shoes is subject to the iraaadi&to control of the operator, 
 tho magnitude of tho braking effort does not bear any fixed relation 
 to the train speed* In applications of tho braioss on long domgrades, 
 the braking effort is iwpt practically oonstoitj and, when service 
 
 stops are being made, tho braking period is of such short duration 
 that no serious error in con^uations is introduced by treating the 
 braking effort aa constant during this period* Considerations of tho 
 aafety of tho equipment and of the comfort of passengers dictate the 
 
 - a - 
 

 
 
allowable braking effort* In subueouent formulae, the bralcing effort 
 
 of friction brakes is represented by B and, liice the other component* 
 
 of tractive effort, it is measured in pounds por ton of gross train 
 
 weight. 
 
 Grado Hesistanoe 
 
 In mounting a grade, a train absorbs energy which is return- 
 able when the train later descends to a lower elevation* Ihe component 
 of tractive effort, corresponding to the rate of storage, or liberation, 
 of potential energy, it 
 
 a - 2000 sin j* , 
 
 where 2000 is the number of pounds in a ton, and $ is tho vertical angle 
 measured upward from the horizontal to tho grade line of tho track* 
 
 Since the per cent grade, 100 tan /J, can readily be obtained 
 from the surveyors* data, it is convenient to express tho grade resis- 
 tance as 
 
 - 2000 sin ( tsfi 1 ( ?a* * grade 1 ) 4 
 ( ( 100 ) ) 
 
 Tor light grades, such as are met in steam railroad practice, there is 
 no appreciable error introduced by assuming that 
 
 in * tan 
 and using the approximation, 
 
 a * 0' - 2000 ( ^' *g* ***** ) - 20 X per cent grade. 
 But ranch heavier grades are common in urban and suburban electric sys- 
 tens and serious error may be introduced in calculations by applying 
 the approximation* 
 
 - 9 - 
 

 
 
 
 
 
 
 
While train roaiatanoo, OUFTO resistance and the explication 
 of friction braicoB all tend to retard tho motion of a train, It la evi- 
 dent ISiat grade resistance will tend to retard or to aocolerato depend- 
 ing on whothor the train la proceeding uphill or down. OJhia la taicon 
 Into aooount In the formulae by assigning to the value of G the positive 
 or negative algoUraio algtx ae diotatod by the above oonaiderations. 
 Chat Is to say, the vine of G la positive or negative acoordlng aa 
 that of ain fl la poaitivo or negative* 
 Accelerating Effort 
 
 If the magnitude of the propelling force applied to & body la 
 greater than that, undor tho action of which the exlatlng apoed of the 
 body will bo jnalntainod, the speed of the body will increase; and, con- 
 versely, the apoed will flecroaae if the propelling force la Insufficient 
 to maintain the ocfruit apeed of the body* Consequently, If the tractive 
 effort, applied to the driving wheola of a train, la more than sufficient 
 to ovorooiae the train reaiatance, curve roa is fence, braking effort and 
 grade raaiatance, tlio balance will operate to accelerate the apoed of 
 tho train* 2hia conpouont of the tractive effort nay be termed the 
 r ocol orating effort* 
 
 The accelerating effort, neoeaaary to produce an acceleration 
 of A miles per hour per second, la usually taJoan aa 100 A pounda per 
 ton of groaa train weight* The value of the accelerating effort la 
 poaitive or negative depending on whether that of A la poaltive or nega- 
 tive; thE-t is, according as the speed of the train is increasing or de- 
 er easing. 
 
 - 10 - 
 
". >Y: 
 
 . 
 . 
 ' 
 
 .-"(I 
 
 i ; 
 
 
 
 . 
 
 
of Oom^oaBnts 
 
 The algebraic sm of the above five components is the tractive 
 effort applied to the driving axles through the functioning of the motors 
 and gears* That is to say, 
 
 P XOO A+B+C+G+R , 
 in which P 
 
 
 
 
 B 
 
 I 
 
 Tractive effort applied at driving axles, 
 Acceleration of train speed in miles per hour per second, 
 Braking effort of friction brakes. 
 Curve resistance, 
 Grade resistance, 
 resistance* 
 
 The unit of neasure of all, except the acceleration A, is pounds (avoir 
 dupois) per ton (2000 Ibs.) of gross train weight* 
 
 - TRADT 
 
 Transposing 7 and A in the above tractive effort equation, 
 
 A - 0*01 (F-3-C-a-n) 
 
 This relation expresses the fact that the acceleration is determined by 
 the values of the applied tractive effort, braiding effort, ant curve, 
 grade and irain resistance* 
 
 ^f o_- \ M ** .** 
 
 Die acceleration is the instogfrtaeom W*+ rate of Change of 
 
 A^-ttv*! 'fcifc*' 
 
 the train speedy tiiat is, 
 
 dV 
 
 JL t n 
 
 dt 
 
 Henco - . o.Ol (P-S-C-a-a) 
 
 dt 
 
 Substituting for B its equivalent, given on page 7 above, 
 
 ,1) 
 
 - 11 - 
 
jjj frfrlt . 
 
 , 
 
 
 J 
 
This relation (1) la the fRr.damentr.1 fltff erratic! cjnation 
 of train spa*. Its solxttionB reader formolae for the direct |fl?e- 
 aotermlurtion of tho ej>0cv*-t.l"w relations for kBOm or agmmed w- 
 vloe conditions* As ha0 loen provtously stated, 3, C and a are inde- 
 pendent of tho train speed Y However, the applied tractive effort 
 F y vary as A function of the train speed as will "be seen from a 
 cons iflorat ion of tha <r>.ractoriatic curves of railway motor and the 
 natal methods of operatix the** 
 
 ' -. W.V. - 
 
 - 12 - 
 

 
XI 
 
 2AXL1AT MOTOR CKARACTJ2RI37IC3 
 
 Che ear lea motor derives its name from the fact that its 
 field circuit is connected in soriaa with the armature circuit aa is 
 shown in Pig* 1* It la clear that tSio motor current, armature cur- 
 rant and field current are identical* 
 
 The performance of a motor under operating conditions is 
 moat conveniently expressed in vfeat are teaown aa characteristic 
 curvoa. Tioae ourvoa for railway motora are determined from their 
 performance in teata which are usually made at the factory* Fig* E 
 shows tbe oharaot eristic curves of a continuous current aeries rail- 
 way motor* For normal voltage applied to the motor teminfus, these 
 curvoa display the relations between the motor current and each of 
 the following fact or at 
 
 1* Speed of oar or train in milea per hour* 
 
 2* Tractive effort of motor in pdnqfls, 
 
 3* Efficiency of motor and ita gears in per cent, 
 
 4* Power output of motor with ita goora in kilowatt a* 
 
 It ia clear that the relations, expreaaed by the curves, are 
 all affected more or leaa by the diameter of the driving wheels and by 
 the reduction ratio of the gears between motor and occlo* Also tho re- 
 lation of apeed to current la largely dependent upon tho motor tormina! 
 voltage* For those reaaona, the gear ratio, nheel diameter and terminal 
 voltr-o, for which the ourvoa apply, aro stated on the curve aheet* 
 
 Tho apd-current curve shows that tho current decreases aa 
 the apeed increases. 2hla is due to the direct proportion connecting 
 
 - 13 - 
 

 
 fin 
 
 
 
 
 dtoa M i .fe9$6 -{Xf 
 
 -TO tf ac feNto^i M* ,v;Xi-;' Mrrwb M* *JA 
 4s0- os* J4itm> wft (NcO WTJOA ?TJO va*iro-Jkie^ 
 
 lit- MV f trft :?1 Oi/T 
 
To controller 
 and supply 
 
 FIG. I 
 
 SCHEMATIC WIRING DIAGRAM 
 
 OF 
 CONTINUOUS CURRENT SERIES MOTOR 
 
 CHARACTERISTIC CURVES OF A ' .', : 
 CONTINUOUS CURRENT SERIES RAILWAY 
 
the apeed of rotation of th> armture with the tartueei! counter-electro- 
 motive force of a motor. Wxst ia to aay, an inoreaae of the afwature 
 apeefl produoea an increase of the electromotive force induced in tho 
 armature and this oppoaen the impreaeed electromotive force, tht re- 
 ducing tho current in the motor circuit* Tho curvea ahow alao that 
 the tractive effort decreaaea aa tho current decreaaea* ffl6 nat reault 
 is that the tractive effort input to the driving axlea deoreaaea a* the 
 peed of the trr.in inoroaaea* In other wor*s, tho tractive effort bear a 
 a fixed relation to the apeod aa long aa constant voltage la applied to 
 the terminala of the motor* Thla relation mat bo determined ao that an 
 expreaaion for tractive effort in terms of apoed can be aubatituted for 
 F in the differential equation (1) before the latter can be aolvod. 
 
 Although tho apeed-ourrent and tractive effort-current curvet 
 indicate the existence of perfectly definite, continuous relations bo- 
 ttreen speed and tractive effort, it ia fcq?oaaible to expreas these re- 
 lationa in a formal;: rationally derived from fundamental principles. 
 The alternative ia to uae an approximation that ia aufficlently exact 
 for engineering purposes* 
 
 The tractive effort-current and power output-current curvea 
 in Pig. 2 and these ourvoa are typical in this reapect although they 
 apply to a particular motor may be oloaely approximated by the 
 atraigat llnea AA and BB over the operating range of tho motor* It haa 
 already been ahoron that the power input to the train, which la equal to 
 the power outputs of all tho motors combined, ia the algebraic product 
 of the tractive effort and train apeed. 3y making uae of those relations 
 and the above straight lino approximationa, an algebraic expression of 
 the relation between tractive effort and apeed aay be obtained* 
 
 - 14 - 
 

 
 . 
 
 
 Alt* ftttEl 
 
 >-.j2 
 
 
 
equation of the line AA in Pig* 2 la 
 
 P - h} * hjl (2) 
 
 in vhitih 
 
 P* B Power output per motor in kilowatts, 
 
 I Current per jnotor la as$}3ros t 
 
 h* and b* are constants determined by the co-ordLrv.tou of any 
 ttro poJnta on tiio lino AA* 
 
 To roduoo the payer to kilowatts per ton of groae train weiflfrt, ib- 
 
 atUute 
 
 -1-p . pt (3) 
 
 In v^iioh 
 
 P mecfhanloal power input to driring axles In Kilowatts 
 
 per ton, 
 
 T - Oroaa weigit of train in tons, 
 H Vonb*r of rootore in the tr- 1 in. 
 
 P - h| + h|l (4) 
 
 or P--|~th| +h^I ) (8) 
 
 Since ? is e2preaed in pounds per ton and V in mile* per 
 hour, and ainoo it Is doaired to ezpreea tho power in term of ? and V t 
 the potroiv * kUowatta per ton.mnat be retnoea to mile pounds per hour 
 per ton* One idlowstt la oqulTe,lont to 
 
 503 9 80 * 36 W- mile pounda per hour. (6) 
 0*746 X 5280 
 
 Thus the poorer input to the driving axles la 
 
 y V " 503 P mile pounds per hour per ton* (7) 
 
 : J (8) 
 
 V-.B03Mh{ (9) 
 
 h.* 
 
 - 15 - 
 
1 
 
 I 
 
 1 
 
 
 . 
 ' 
 
 ' 
 
 . 
 
 t. .; ftf a?3*K;Sri 
 1 aar.T. oil: . 
 
 - ..iw jf.v:- 
 
 . "O* 1^[ rr 
 
The equation of the atrai^it line BB in ?ig 2 ia 
 
 P 1 - h 3 + h 4 I (10) 
 
 in which 
 
 P* Tract ire effort output per motor in pounda, 
 I = Our rent per motor in ao^eroa, 
 
 h, and h 4 are constants determined by the eo-ordinatea of 
 any two points on the line 3B 
 
 To reduce the tractive effort to pounds per ton of gross train woi&xt, 
 substitute -2- F - 7* (11) 
 
 rfMft >r M 
 
 In which 
 
 P Tract ire effort input to driTiug axles in pounda per ton, 
 TV} I Gross weigpit of train in tons, 
 H Umber of motors in train. 
 
 Then 
 
 or 
 
 JL p - hg + h 4 l 
 
 T ( h 4 T - 503 hj) 
 
 503 M 
 
 T "( V- 503h|/h 4 ) 
 
 P - It h, 
 
 - - tui 
 
 Combining equations (9) and (14), 
 
 t T Y - 503 II hj T F - II h g 
 
 503 Mh! Mh. * 
 
 2 ? 
 
 Then * P V a 4 - 603 M hjhj^ - SOSTPhg-SOSH hjig , (16) 
 
 * ( H 4 V - 503 hg ) P - 503 M ( hjh 4 - n|hg) , (17) 
 
 503 M ( ^A " h ?*3V. , (18) 
 
 - 16 - 
 

 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 '' 
 
 ( rf oa - T,.- 
 
 
 
 

 
 Letting hj - ( h| - V^ ) ( 
 
 t 4 
 
 h 2 
 and hg - 503 . (21) 
 
 F - -- . (> 
 
 aquation (22) 1* the approximate formula for tractive effort 
 F, in pounds per ton of gross train weiffct, in terras of the train speed V 
 in miles per hour, vfrum rated voltage is applied to the terminals of the 
 motors* 
 
 ?or reasons, that will appear presently, the tormlnal voltage 
 of tfro motors is roauoed below the rated voltage during certain periods 
 in the operation of a train* In fact, there are three conditions of 
 terminal voltage to be oonaideretl , namely t 
 
 1* Operation at rated voltage) 
 
 2* Operation with power shot off, that is, 
 with aero voltage applied; 
 
 
 3* Operation with applied voltage 
 rated voltage and not aero* 
 
 AM has bean pointed out, equation (13) applies in the first 
 of those throe oases* 
 
 While a train is moving with tbo por/or shut off, 13io terminal 
 , applied to the motors, is zero; henoo the motor current is sero* 
 output of tho raotoai however, is slightly negative* That is, a small 
 amount of energy is extracted from the train and dissipated due to fric- 
 tion in the motors and gears* However the rate of extraction of thia 
 
 is so snail that, for practical purposes, it is neglected so that 
 
 F (23) 
 
 -17 - 
 

 
 
 , 
 
 
 
 . 
 
Bceept rthilo starti^, trains ere seldom operated for any 
 appreciable length of time with any terminal voltage loss than normal 
 applied to the motors* In spools! oases, a oonatant partial voltage 
 nay be applied* But such oases are rare and, vfaon they most be treated, 
 an erprooalon for ? in the form of ecuatlon (22) can readily be derived 
 by making suitable modifications of the motor characteristic onrves to 
 accord with the partial volt^o* 
 
 Before a train starts, tho speed of the motors is sbro, so 
 the electromotive force, induced in the armature, la ero. Hence, if 
 voltage is applied to the motor terminals with tho train at stand- 
 still, the current is liTnitce only by the resistance of the field sad 
 armature circuits combine!!* This resistance is neoosearily so small 
 in railway motors that, if the rated voltage were applied to the motor 
 terminals while the train was at standstill, the motors would either 
 be d-juagod seriously or develop an excessive torcfue causing the wheels 
 to slip or the train to start trith a severe jerlc* (Therefore the motor 
 terminal voltage is varied so as to keep the currant within safe limits 
 during the starting period* 
 
 Ihe voltage, applied to a train+ls usually constant so. in 
 controlling continuous current series railway motors, the adjustment 
 of notor terminal voltage is procured liy connect*^ external resistance 
 in series -wittx the raotora. From several points of vlow t the ideal con- 
 trol would maintain tho motor current constant at its muxlsRxa permissible 
 value throughout the starting period* For practical reasons, however, 
 Hie ecatom is to vary the extcra^l resistance In a fev steps, keeping 
 tiae current within certain well defined limits v/hilo starting* Although. 
 

 
 
 
 
 
 
 
 . 
 
 
 
 
 
 MkitM 
 
 , 
 
it Is not maintained by thin raothod, it is practicable 
 to treat the our rent as though it were constant at an equivalent aver- 
 age value. It haa been sh.ovm that the motor tractive effort dopende 
 pon the notor current and is independent of tho motor terminal vol- 
 tcge except in ao far as tho letter affbcte tho magnitude of the 
 current* Hence, ?m equivalent constant starting current produces a 
 corresponding constant tractive effort* In other words, tho tractive 
 effort, ? in equation (1), is treated as constant from the instant 
 nhen tho train, beting to move until the train spoei attains the value 
 indicated in the motor characteristic curves as corresponding to the 
 averse stsurting current* 
 
 Tilth tho permissible average startiag current fixed, the 
 corresponding tractive effort is determined from the characteristic 
 curves. This constant tractive effort, expressed in pounds per ton, 
 i tho value of P in equation (1) during the starting period. 
 
 
 - 19 - 
 
. 
 
 ' 
 
Ill 
 
 MB -TIM POBMUUK 
 
 The ti'.-.lui. of a railway syaton operate in Mtoat are called 
 trips between b&iviL-uile, vjhich may be at the ondo of s lino of tra0fc 
 or Intoroediato as at the anfia of lUvloioua. In the course of a trip, 
 a train usually si&lcoa aovaral stops* The period of time, that olnpaM 
 botwoon r/^ion a train lonvtMi ona atopplag place and vhon It Icavoa the 
 nooct t is oalleo. a train cyolo* 
 
 The rirst Gvoiib la a train cycle la the at; rt , igiiich is f ol- 
 lotrod by xiocoasiTo perioils of opera* ior. oach imolTiaer nor* or less 
 different com"! it ions, ud finally cones tha stop* /ll tha normal 
 conditions of operation, thnt r.ro not in a train cycle, way be olaeei- 
 fiod tciilor a few typioal phaoat tvhloh ore ooaed largely in accordance 
 with the effects tha* they prodtuso tgjon the train apcot!* They aro 
 
 X. start log, 
 
 2. Accoloratine, 
 
 S* I5tc'uing at constant spood, 
 
 4. Coasting, 
 
 5 Braking, 
 
 6* Stop* 
 
 BMh of these phases of operation entails a unique interpret at ion of 
 certain terms, particularly the input tractive effort F, in the funda- 
 mental differential equation (!) And upon these interpretations rest 
 the solutions of this equation and thereby the formulae for the speed- 
 relations. 
 
 - 20 - 
 
For obvious reasons, the braiooa are not ordinarily applied 
 while tho motora are functioning and consequently, during tho greater 
 part of any train cycle, 
 
 I - . <f* 
 
 Also curves and grades are of incidental occurrence ao moat train 
 cycles inclttf o periods when 
 
 0-0, G = or C G " 
 .'.hen B, C or G la not aero, it nay be constant | but, if it is not 
 constant, 16 ie usually nocossary and sufficiently accurate for pruo- 
 tiot-1 purposes to treat it aa constant by taking average values over 
 definite intervals. Xhe terma, B, and G, will be carried aa constanta 
 
 in the present dor ivat ions in ordor that the resulting formulae nay be 
 
 . 
 univer aally applicable. 
 
 Having determined tho tractive effort for the three poaaible 
 conditions of applied motor voltage, and having divided the train cycle 
 into its component phases of operation, it remains to eolva the above 
 equation (1) for enoh of those several phasea* 
 
 It has been pointed out above that, during the starting 
 period, the voltage applied to the motor terminals is varied ao aa to 
 maintain practically constant tractive effort. This permits tho 
 solution of equation (1) on the assumption that F is constant. 
 
 
 -21- 
 
f .' 
 
 
 to 
 
 
 ta- 
 
 
 . 
 
 
 : 
 
 V I 
 
 
 
 
In order to facilitate manipulation, group the constants 
 and, for the starting period, let 
 
 /O, = O.OOO3 , 
 
 0.00002 X /, . N-l ) 
 7, r V /o J 
 
 edition (1) becou8 
 
 dt 
 dt = 
 
 1,dt = 
 
 . 
 
 ctV 
 
 Slnoe --is constant. 
 
 Integration of this equation renders 
 
 I 
 
 (26) 
 
 (27) 
 (23) 
 (B9) 
 (30) 
 
 (33) 
 
 (34) 
 
 where C' is a constant of integration. Dividing both members of 
 (34) by tf , 
 
 
 _c: 
 
 Transforain; to comon loearithma in order to facilitate confutation, 
 
 / . ^.00 7^^, 1 3d-' '' -' w -"* I y._o_ /3S) 
 
 / . ^=% ' S'0 I / . . ., 3-5 _ _ _, . , \ * \.-~ >f3 ) 
 
 V. 
 
 - 2S - 
 

 
 A= 
 
 
 
 
 
 
 
 
 
 (St) 
 
 (ee) 
 
 *v : 
 
 
 
 
 ^ ^x \ 
 
 UN. 
 
 
 1.1 
 \0 
 
 
 
 (> 
 
 
It the train speefi la 7. at the instant t ; tho tine t, In 
 
 starting period, at tfilch the apeefl will reach saay other vslue V, 
 it given "by 
 
 2.30 ^ . ^ y<M,t +/3?--t-/3,+z7, V. ^ ^ < 
 * " z ' 
 
 tl-.i.l: IB, 
 
 ^aaraiLl I 
 
 ^^ / /- / 0,-^^ J J 
 
 or 
 
 f = 
 
 log, (V4<x,<t-r#* +/3,+ay, V) - 
 
 w 
 
 t and t are aspraaaetl In secoMa and Y and V in miles per hour 
 Combining equations (36) and (36), and letting 
 
 C, 
 
 - 
 
 It la Men that 
 
 or 
 
 equation (39) reduces to 
 
 , t 
 
 If a train is started from atrndatlll and ti o la aaanred 
 from 1h Inatant of atarting, ao tfc&t Y Q " and t Q " | qwttion (38) 
 that tiie ntniber of seconds, reqtdrod for the train to attain the 
 
 V miles per hot*, la 
 
 . 23 - 
 

 
 IJK 
 
 9S' 
 
 
 
 v t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
4 - 
 
 2.30 
 
 j 
 
 * " ~ 
 
 , -A V- 
 
 , \ 
 
 (46) 
 
 mT.nAiia ivotiTn, 
 
 At the conclusion of the nbove starting poriod, \*.an foil 
 rated voltaffe la appliofl to the motor terminals, the motor prooooda 
 to function In accordance with its dharooteristlo curves. Ordinarily, 
 there will T>e a poriod of acceleration between th rd of the start- 
 Ing period and the time when fall speed is attained. Also, in subse- 
 q-uent periods of a train cycle, there may be acceleration, positive 
 or negative, as grades, curves, etc* arc encountered* As long as 
 rated terminal voltage is applied to the motors, the relation of 
 tractive effort and speed is approximately 
 
 Substituting thla In equation (1) glvos 
 
 dV 
 -dT = ' 
 
 -0.03V- 
 
 /O 
 
 o.o/ 
 
 V-h, 
 
 (22) 
 
 - 24 - 
 

 
 
 -j^V' 
 
 
 
 i \ \ v ^.iwv c\ 
 
 &- 
 
 
 vs*;}- 
 
 
Let 
 
 dt 
 
 Let 
 
 c = o.o/ F 
 
 o.ooax , /y-/ 
 
 7- (' + /O 
 
 A 
 
 O. 
 
 Q.QI 
 <G 
 
 o.oah* - 
 
 1 - 
 
 In railway parlnnce, a train operating at oozutaat 
 fated voltage applied to the motor terminals la apoloen of & 
 naming at balnnolng speed* The term arises from the fact that 
 
 / 
 
 iJhQn a train i operating at oonatant speed, the power li^ut is Jtiat 
 balancoi: by the power dissipated plus the rate of storage of potential 
 energy* In other words, thoro is no kinetic energy being stored in, 
 or extracted from, the tmin* 
 
 Since the speed, V in the foregoing equations* becomes con- 
 stant at the balancing spoed, the latter can be designated as a parti- 
 cular value of 7, say 7 
 
 Also, when the speed is constant, the rate of change of speed 
 
 is ero; that ia, when 7 - 7 2 . JH_ . t (56) 
 
 dt 
 

 
 
 
 .*> 
 
 .o 
 
 
 .. > 
 
 
 
 
 
 
 
 ' 
 
Than, from equation (55), 
 
 Dividing this Toy 
 
 f.5-7; 
 
 In order to determino tho raltw of V 
 
 let v; = tv 
 
 lk 
 
 Let 
 
 _ 
 
 3 
 
 + r 
 
 VY 
 
 is, let 
 
 
 is 
 
 How 
 
 or 
 
 - 26 - 
 
 t 
 
 (,1) 
 ( 62 ) 
 
 (63) 
 
 (68) 
 

 
 o = 
 
 
 
 
 * l^x * tj * ** 
 
 K 
 
 V -< 
 
 
 
 I 
 
 
 - 
 
 
 
 
 - 
 
 
 ItJ 
 
 
 
 Vf , 
 
In general, '; will have throe distinct values, et least one 
 
 of which Is real* That la, 
 
 WV q.W + r = O [63) 
 
 has tJureo roots, and at least one root is real* 
 
 Let m be a real value of 
 
 
 and lot n be a real value of 
 
 
 n * n will be a real root of 1. 
 
 Let 
 
 The^the roots of 
 are 
 
 And, ainoo 
 
 ths values of T g In 
 
 are 
 
 (a) - --3- 
 
 -A (a>n, 
 
 v z = 
 
 xj, = 
 
 / 
 
 --** 
 
 
 
 
 It 10 Men, from equation (73) and tho roots (77), that, 
 
 
 
 and the values of V7 arc 2?n, -^^ -m , all real. 
 
 Ihat is, all the values of W, and conae'iuontly of 7 2 wo ^ a 1)e reftl 
 
 (76) 
 
 (63) 
 (77) 
 
 (75) 
 
 '-- 
 
 - 27 - 
 

 
 "ic owl**? Jtan A erf a fol &n 
 
 V * * wf IlJbv K ' at /not 
 
 =-- = u> 
 C> >s -\ -v Hi ^ -v- "^VM ^ fooi tf9j9lB 
 
 9tttfwtt fcffd 
 . 
 
 (,) 
 
 
 * 
 
 ^ JL- = ,cx 
 
 c* 
 
 -V - - m p 
 
 . j<; . -v $*> -V JE -- 
 
 
 
 
ifelch migjvfc loot! to umbic'tiity. Howevor, in the problom at hand, an 
 investigation of the r-'mgoa of values of 
 
 show* that. In ordinary oases. 
 
 In other words, the solution yields only one real value of 7 g sad two 
 imaginary values* The real, or principal, value of Vg is the actual 
 balancing speed* 
 
 2 - Acceleration 
 
 Tho next step is to derive a formula by which the speed-time 
 relations can be detorminefl for periods when the speed of tho train is 
 increasing or decreasing with rated terminal voltage applied to the 
 motors* 
 
 In the preceding section, the fundamental aquation (1) was 
 adapted to the condition for full voltage applied to the notor terminals, 
 sod reduced to simplest terms, namely t 
 
 Then it was shown that, when 
 
 either \4 = />/ = -^-+7n-fn. , (ya) 
 
 t i *f / \ 
 
 V 2 = jOj.. =.-**+ win + & n , (73} 
 
 * m. / * \ ' / 
 
 i - /3 - ' + <*>7n + t-m (30) 
 
 were substituted for T in 
 
 then V t 3 + liVS+tfiVscti^ O . (sa) 
 
 - 28 - 
 

 
 ' 
 
 ' 
 
 
 . 
 
 
 
 al 
 
 ftUMI 
 
 
 t s- f 
 
 
 
 
 -v 
 
 -v 5_ = 
 
 -v 
 
 w 
 
 
 
 
 
Therefore 
 
 Alao a 2 = -/0,/ct,^ , (36) 
 
 , (ar) 
 
 (as) 
 
 Introducing tho relation (85) into equation (55) renders 
 
 That is, f(V> = 
 
 g(v) (i/ 
 
 aay be resolved into partial fractions, thus 
 
 vbaro 
 
 _ 
 
 ' 
 
 from the equations (90), (94) and (95), it U aoen that 
 
 Integration then reader a 
 
 where C 2 la the oonatant of integr&tion* 
 
 - 29 - 
 
 transposed is _ (V-hJdr _ , } 
 
 >il " ^-/ 3 -)^-/^^-/J 
 
 Let /ri/; = v-h* (91) 
 
 and y (V) = 
 
 (93) 
 
lS) < 
 
 V"^ , ^C-N, 
 
 (86) 
 
 (ae) 
 
 (ea) 
 
 (<) - fZ^ "^^ 
 
 M 
 
 (ee) . .,e^N,scs VNe * 
 
 &. 
 
 |^ 
 
 JS. !*1 5 1, 
 
 M 
 
 -^^ 
 
 
 
 
 
t = l**fc-/* 
 
 (/ot) 
 
 It (7V , t-tj) IB a point on the curve; that is, if 
 
 t **t, , 
 
 the time t at tfiich the speed will attain some other value 7 is given 
 
 by tflie equation, 
 
 t-t. = - 
 
 In oonnon lo^arithna and expanded for convenience in computation, 
 
 t = t, + ^^\l*lofr(V-/,) + r 
 
 - h log,, (y,-/' - 
 eqwtions (101) and (103} and letting 
 
 
 ,= -- = 4 
 
 , from equations (105) and (108), 
 
 t = C* + 
 
 JogsrCv-p,) + 7njog,.(l/-/> f ) + 
 
 (109} 
 
 p - ooiariiiG 
 
 stations, descending grades, s lowing down for 
 crossings, etc*, it is generally advantageous from the standpoint of 
 energy econoray to shut off the eloctric power supply and "llow the train 
 to coast for a period before the brakas are applied* Under this con- 
 dition, the motors 4o not supply any power to tho train; in fact, they 
 
 30 ~ 
 
. 1} ^, ^_Ni) 
 
 ^ 
 
 
 
 c 
 
 ** 
 
 
 --l * 
 
 ( 
 
 - 
 A 
 
 
 
 s - 
 
 
 r ^.-^V 
 
 
draw a oertr.ln aran.ll amount from the kinetic energy of the train in 
 ordor to ovoroosie tho friction and windage losses in the motors and 
 gear*. However, this draught is usually negligible, *o it my be 
 assumed that the tractive effort input to the train is zero during 
 coasting. Alto, by tho abovo definition, the br.:kinn effort is zoro 
 during coasting. Hence, with 
 
 1-0 (110) 
 
 and B - , (111) 
 
 the fundamental equation (1) reduces to 
 
 = 0.01 [-C-G-^- 0.03 V - ^^ (,+ ^L) V*} (,12) 
 
 55io Talue of tt will bo positive or negative according as the 
 train is going up-hill or clovm-hill* fhe magnitude and algebraic sign 
 
 of the vnltuj of 5 will determine ^Aether the acceleration, 3L , is 
 
 dt 
 
 positive, negative or zero* pressed algebraically t 
 if G > 
 
 if G =- 
 
 if G <- 
 
 In quation (112) lot 
 
 ct 3 = 0.01 (C + G + ^=r) * (lie) 
 
 /&, = 0.0003 , (in) 
 
 - dt 
 
 V*) , (119) 
 
 - SI - 
 

 
 
 
 
 
 
 
 
 
 - < a 
 
 o 
 
 - 
 
 -o 
 
 - 
 
 
 
 
 
 
A* 
 
 eat ttw law** 
 
 
 Honco 
 
 ('**) 
 
 ('**) 
 
 There aro throo possible solutions of equation (125) Ihe 
 algebraic sign of ^J^Jdet ermines which Is the proper solution. 
 The vnlttes of /3 3 and i 9 are positive r.s long aa the train is moving* 
 Hotv3T9r, the value of or, is positive or negative according as G > -(t 
 
 or 
 
 lTovortholosf;,ct, ,/3 3 and ? 3 are Constanta, fixed by 
 
 track and operating conditions so it can readily be determined Whether 
 
 et, 
 
 Oase I 
 
 If 
 
 the solution of 
 
 ia 
 
 
 
 sin 
 
 ,' , (>27) 
 
 
 or 
 
 V+ 
 
 OK '( 
 
 _ 
 
 J 
 * 
 
 -2 . -/I" ^7 r 3 v-i-/^* 1 _ c^ / . _\ 
 
 v<^3/3-// S/A ' ^/,r^^*^>*^*jj " * ' ( 
 
 ~ 
 
 - 
 
 , 
 
 

 . 
 
 ( 1v - 4 
 
 V& tn* 
 
 
 . 
 
 
 
 
 
 
 t O < -* 
 
 ' 
 
 ~ 
 
 lx) 4- 
 
 e \- SX: 
 
 
 FV 
 
 
 v 
 
where time la expressed in seconds and the Inverse sine (sin ) in 
 radian* 
 
 Since, in trigonometric tables, sines aro conr.iled as 
 fc&iotiona of angles expressed in degrees, it is more convenient to 
 modify the formula (130) ao that the inverse sine con be derived 
 directly from the tables in degrees and deoimal fractlona of a 
 degree* This ia accomplished by introducing the factor, 57*296, 
 tho number of degrees in a radian. Equation (130) thonbecc 
 
 t = w r ***** c; 
 
 -I - : -n' f *f*V+* 1 .. Ci ( /32 ) 
 
 .6V*<* 3 t 3 -tt*' \y+/ 3 (w+0,v+<*,)] 1 3 ^ ' 
 
 z, 
 
 where time ia expressed in seconds and the inverse sine in degrees* 
 
 If (t-t , W 3 ) be a point on the speed-time cwve during 
 the coasting period, that is, if 
 
 V - 7 a when t - 3 | (183) 
 
 tho time t, at which the train will attain any other speed T by 
 coasting, is given by 
 
 t-t,= 
 
 .-/ 3 * , - , 
 
 " ? \/**r+*v+ ' ^ 
 Combining: qn&tions (132) and (135), and letting 
 
 s 
 
 c 3 = -^ , 
 
 
' 
 
 
 -* 
 
 
 
 
 I 
 
 
 9*1 
 
 BPOCB Ml ellf 
 
 
 
 
 JV N f UWMJ *c -vj 
 
 vM 1\_ M y* 
 
 ^ j^ja * 
 
 ^Mrft^itakf 9^ttr -- * 
 
 - / W 
 
 
 
 
 r 
 
 
 
 M 
 
 
If 
 
 (125) becomes 
 
 The soltxtion of this Is , / 
 
 1*t = 
 
 -f 2 --^- = O , (139) 
 
 (141) 
 
 
 or 
 
 If V - V 3 tghen t - tp , (133) 
 
 tho tine t, at which tho train will roach auy other speed T by coasting, 
 is given by a 
 
 or 
 
 Combioing eqtjy.tion8 (142) and (144), and letting 
 
 Than 
 
 t = 
 
 
 f/7) 
 
 III 
 
 If 
 
 -^--^L < 
 
 / 3 ^/a 1 r 
 
 
 (>+*) 
 
 ihere tho parallel vortiwal lines signify that the absolute, or 
 niariaal, value or the quantity, that they enclose. Is t&Ken with 
 
 - 34 - 
 

 
 
 
 
 
 . 
 
 
 
 
 
 * = 
 
 
 
 . 
 
 ~e\ 
 
 
 
 . 
 
the positive algebraic sign* 
 
 Substituting tho relation (149) into' equation (125) f gives 
 
 -Ut = 
 
 The solution of tills Is 
 
 lot 
 
 9 v 
 
 *,f3-&\ 
 In cornaon loganlthaa, in order to facilitate confutation. 
 
 {, 
 
 2. 
 
 If 
 
 t - tg 
 
 (133) 
 
 the tisio t , ftt vthlcii the vraln will attain any other speed T by 
 coasting, is given by 
 
 f f 
 
 3 " 
 
 so that 
 
 t = t 
 
 " 
 
 equations (154.) and (156) , and letting 
 
 c 
 
 C 3 = 4 - 
 
 t = c 3 + 
 
 flf0 \ 
 
 MI 
 
 2. 
 
 - 35 - 
 

 
 
 
 
 
 
 
 
 .' arf 
 
 - ^ t IV 
 
 . 
 
 . 
 
 oe.s 
 
 
 
 
 r . 
 
 . 
 ~. 
 
 . 
 
 . 
 
 
 RV 
 
 - v- te^- <iV j 
 
 rJt&'-.Y.t^vi 
 
 .. 
 
 oe.s 
 
 
When tho brakes are applied to a train, the power surply la 
 uawlly 8t>-.i J G off 00 in coasting, o *k Input tractive effort, P In 
 equation (1) la zero during the braking period, nonce, aquation (1) 
 
 - 0.03V- 0+V*. 060) 
 
 Bratoea are most comnonly applied for tho purpose of retard- 
 ing the motion of a train. However, occaeionplly, as during tho descent 
 of grrdes, the braloas are partially applied and the speed allowed to 
 increase thoucft not to such an extent as It would if tho brakss wore 
 not applied, dtoeroforo, in order to be general, the formulae for sptA- 
 tine relations during braking not provide for positive, zero and naga- 
 tive acceleration. &qpresaed algebraically t 
 
 $<o * G> -[ B + C ^+ ^+^0+^'. 
 $-0 if G =-[B + C+^+O.OSV+!^(,+ ^ 
 
 >0 if G<- fo + C+ ^+ 0.03V+ ? (I +>)V*} - M 
 
 dt 
 
 She acceleration is 
 A - ^p = O.O/\-B-C-G-^~ 0.03V- 
 
 ot d 4 = O.OI(B + Q.+ 
 
 #, = 0.0003 , 
 
 0. 00002 X fl N-l^ 
 'U = -7=- -(^jQ-f ' 
 
 dV . 
 
 - 36 - 
 
-j- &-*-*-- 
 
 
 
 - 
 
 
 K t {l-VV ^.^XSOO.O .,___ 
 
 " L ^~oT ^~ -Meo.o - 
 
 - - f- 
 
 v ft 1 \O.O = JD 
 
 , eooo.o = >s 
 M r C^-^M5^ m + 
 
 (\) - ^M^-N^^-^y- 
 
 
A comparison of equations (116) to (119) and (165) to (16S) 
 Shows that fcho spood-ti'ne relations during the coasting period are 
 different from those in the bracing period only in tihat B f in 
 the latter; that la 
 
 = o.oi 
 
 while 
 
 a., = O.Oi '(C ' + G + 
 
 Hence, ainco and are both constants, the solutions for the two 
 periods v/lll be similar in form and tho reuniting formulae for the 
 braking period onn be written directly* 
 
 speed-time curve 
 
 Thus, if (t-t , V7 ) ^ a P oJjl * 
 
 in the brruring period, that is, if 
 
 V - 7. 
 
 t t 
 
 . . 
 
 4 4 
 
 the ti: t, at -nhioh tho speed xrill attain somo other value 7 under 
 constant -"--.ilication of tho brakes, is given by tho following 
 forrrnlaas 
 
 (169) 
 
 |f 
 
 - > , 
 
 
 ;,;' 
 
 Q = 
 
 In these formulae, tirao is expressed in seconds, speed in miles per 
 hour, and Inverse sines in degrees* 
 
 - 37 - 
 
* t 
 
 
 -vo 
 
 
 r 
 
 
 
 Iflsia 60 lllw sLc lie 
 
 
 v. 
 
 -gi| Jni6 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 e-, 
 
Case II 
 
 If 
 
 O 
 
 t = t + 
 
 Letting 
 
 ("*) 
 Q77) 
 
 Case III 
 
 If 
 
 <o , 
 
 so that 
 
 2.30 
 
 , 
 
 -* , 
 
 -- - ~ " 
 
 Letting 
 
 c. = 4 - 
 
 2.30 
 
 (ire) 
 
 0?o) 
 
 ?.3Q 
 
 ^rrm* *" 
 
 + /Q*.+2l4.V\ 
 
 Special Case, V " 
 
 In stopping a train by the application of friction brakes, 
 if the speed is VA at the instant t^ in the braking period, the tiir 
 t, at which the train will atop, is given by letting V " In 
 equation (171)* Then. 
 
 / 
 
 za. 
 
 sin 
 
 . -/ r &4 
 
 - sin \-j==~ 
 
 [V 4K+1, 
 
 (,33) 
 

 
 ^^rNJS* 
 
 
 .A^Nt^S 
 
 
 , 
 
 
 
 Pv) 
 
 
 
 
 
 \- 
 
 -V*s 
 
 ^* 
 
 5i^M 
 
 ']* 
 
 O&.S 
 
 - 
 
 1C * * - J 
 
 - 
 
 - ^ = J> 
 
 
 
 (68V, 
 
 
 
 
 
p - goaauiT o? PRINCIPAL 
 
 In conclusion, the principal formulae are grouped together 
 in order to facilitate reference to them* 
 
 The acceleration formula f or fundamental differential 
 
 qoatlon of train speed is 
 
 ff> t 
 - 0.03V- - 
 
 '^r)y'\ . (f) 
 
 
 9w formla for the tractive effort input to the train, 
 rated voltage la oppUed to the motor *;0rainals t la 
 
 For the starting period, that is, with constant tractive 
 effort innut to tho train, 
 
 **- 1 2 - 3 ^ 
 
 -A-zt V) 
 
 (39) 
 
 If V - and t " 
 
 4 = 
 
 During acceleration with rated voltage applied to the motors, 
 
 , / T y-fi 
 
 * ~ + ~J~ *^e TTTT" + " 
 
 (47) 
 
 T, 
 
 
 During conating. 
 
 
 t- t,+ j 
 
 * za.B-V^K^-ft 
 
 proridefl 
 
 
 - 39 - 
 
. 
 
 CfeOQ* J0 
 
 
 
 - N &<XQ - 
 
 \0.0 
 
 
 
 
 
 ^ ( 
 
 ^ 
 
 . < _ 
 
 ' * 
 
 ' ft to-- -ni 
 
 * 
 
 . . i. . 
 
 
 
 IX 
 
 
 

 *-4 
 
 if 
 
 ^L-^L <0 
 
 i* *& N 
 
 Durlnc 
 
 = t + 
 
 provided 
 
 sin 
 
 sm 
 
 If 
 
 t = t + - 
 
 (/S6) 
 0+8) 
 
 (170) 
 
 0*4 
 
 (/SO) 
 
 if 
 
 If the speed steadily deoreaaoa to ? = taider the application of the 
 the train will atop 
 
 v- / 
 
 ^.6 V^^^rTOT 
 
 -40 - 
 
**> 
 
 
 
 
 
 v 
 
 X 
 
 *^ 
 
 x ^ 
 
 
 Wi) 
 
 r * > - 
 
 o = 
 
 
 ^NJ5 
 
 O x ~ 
 
 ; 
 
 mt 
 
 m*-. 
 
IV 
 
 EXAMPLES 
 
 These examples are given for the purpose of illuatratine the 
 methods of applying the foregoing formulae* For simplicity, they are 
 all computed for the same train. 
 Train Data 
 
 Htnriber of cars, H " 6 
 Motor cars, 
 
 Trailers , 5 
 
 Motors per motor car, 2 
 
 Number of motors, II 6 
 
 Weight of three motor cars, 3 Z 28 84 tons 
 
 Weight of three trailers, 3 X 22 66 tons 
 
 Total weight of train, T 150 tons 
 
 Gross train weight per motor 150 f 6 25 tons 
 
 Area of projected cross-section X 110 sq*ft< 
 
 Average starting current per motor 350 amperes* 
 
 Motor Characteristics Fig* 2 
 
 . . . /.-.. 
 
 \, 
 
 -41 - 
 

 i 
 
 : XW tT "0* 
 
 nt <MBB crt 101 IT 
 
 . 
 
 t B*U3O 10 ' 
 , 1JU 
 
 , 
 
 ,-iciS v.ai-oir 
 
 ,W*0J ' .-JU 
 
 - ' - : Offl Wtri* ^O Jl^l 
 
 38 X 5 <n e^dt 1e - iv 
 
 r . ., -jjio-,* j 
 
 ;S 1 C3I ^0 %! : 
 
 ' X 
 
 
 
 * 
 
. 
 
 How much tl-ne will be consumed in the starting period on 
 level tangent track, that la, to accelerate the train from standstill 
 to 17 2 miles per hour, the speed, on the mot or a 1 normal speed curve, 
 corresponding to the average starting current of 350 anperoe ? 
 
 Prom the character i at ic curves, Fig 2, the tractive effort, 
 corresponding to 350 amperos, ia 5000 ll>s. per motor. Hence, the 
 tractive effort per ton of cross train weight is 
 
 P - 5000 / 23-200 Ibs. po- ton. 
 
 Jinoe the track la lovel and straight, and the tandOM are not applied 
 during starting, B-C-G-0 
 
 in equations (24), (25) and (26), 
 
 a, = 
 
 = 1.96 , 
 
 -3 
 
 /5, = O.O003 O.3x/0 , 
 
 O.OOOOZ 
 
 /SO 
 
 Then formula (47) Gives 
 
 -0.0003x17.2 + /7.Z 
 
 = a.Q 
 
 
 -42 - 
 

 
 
 
 
 -<A* eo.o ^- 
 
 ~" V 
 
2 - Balancing Speed 
 
 VShat will be the balancing speed on a one per cent up-grade ? 
 On the straight line AA, which approximates tho motor power 
 output current curve, Fig. 2, 
 
 P* - 160 hen I - 325 
 
 and P 1 - 49.4 when I - 100. 
 
 Substituting these values in equation (2) gives 
 
 160 - h^ + 325 h 
 
 and 49.4 - h| + 100 h^ 
 
 Solving these as simultaneous equations, 
 
 110.6 - 225 h' , 
 
 hj - 0.4915 
 and h| - 0.250 
 
 On the straight liao BB, which approximates the tractive 
 effort-current curve, Fig. 2, 
 
 P 1 4980 when I 350 
 
 and P 1 880 when I " 100 . 
 
 Substituting these values in equation (10) gives 
 
 4980 - h, +350 h. 
 3 * 
 
 and 880 hg + 100 h^ 
 
 Then 4100 250 h f 
 
 and h_ - - 760 . 
 
 Substituting these values of hj , hg , hg and h^ in equations 
 (20) and (21) gives 
 
 - 43 - 
 

 
 ' 
 
 JL 
 
 . 
 
 < 
 
 ' 
 
 
 
 *%i 
 
 :&r 068 - ' 
 * _ll * Uce* 
 
 r if! - OI^^H 
 
 
 ni fi IUBJ 
 
 |OS) 
 
iff * * . 760 X 0*4916 
 * ' 25 * 
 
 - 453.4 
 
 - 18.0 
 Ihua aquation (22) glrot 
 
 F . 463.4 9 
 T - 15.0 
 
 lor oae por cent 
 
 5 - 8000 aln( 
 
 - 20.0 . 
 (50), (52), (53) and (54) give 
 
 = -ZZxIO 6 , 
 O.OI 
 
 - - 374.8 x/0 3 , 
 
 A = - 
 
 = /0.74-x/O 3 , 
 
 = - 7.364- . 
 
 (61) ana (62) give 
 
 A 
 
 li aft \ 
 
 a * 10740 - 
 
 10.74 X 10 3 
 
 and r - -374800 
 
 I 
 10740 X 
 
 3 
 -369.9 X 10 3 . 
 
 HelPtlons (74) and (75) give 
 
 - 44 - 
 

 
 
 . 
 
 t, \ 
 ^-l- 
 
 
 
 
 
[369900 / (369900)* t (/074-oJ* 
 
 _^_ ^ ^ i __ ^ -^- i 
 2V* 27 
 
 = 74. / 7 , 
 
 f36S9OO / (369900)* (/O74O) 3 
 
 11 = V~~ '* ~^T~ 
 
 Honc, t>7 foRnoloe (78), tho balr.no ing 
 
 _ 
 
 ' 
 
 74J7-4-e.il 
 
 = 29,^52. niloa per hcrur* 
 
 3 - Aocelerati 
 
 If the trj-.in is aooolcwtted fjpom the oloao of the atarting 
 period on a one pr oent grade, Tiow ionpr after starting from atand- 
 atill will the speed bocomo 28*0 nilea per hour ? 
 
 From the preceding eaoanplee and equr.tiona (76), (78), (79) 
 
 and (80), 
 
 t, = 8.8 , 
 
 V, = /7.a , 
 V = 28.0 , 
 
 = - 13.38 +J24.3O -, 
 
 -y 
 
 = 13.58 -j 24.30 . 
 
 From <ntiona (96), (97) and (98), 
 
 13.58 -t-jZ.4-.aO- 1 5.O 
 
 3(-/3.5S+j Z+.30)* + Z 
 
 = - O.O033B+j'O.OOiaZ 
 - 45 - 
 
c^v/ .MTT\ 
 
 
 
 ^ <x 
 / 
 
 '. 
 
 - --^ - - s ^- * 
 
 l- 
 
 
 
 . 
 
 ftawi 
 
 c 
 
 } tarn 
 
 8.8 ^^ v i 
 
 , S.\\ 
 
 0.8S = \I 
 
_ 
 
 3(ZB.SZ)*-I- Z(-l.364Xza.SZ) + 107+0 
 O.OO/03 , 
 
 -/3. Sa -JZ+. 30 - /$. O 
 
 ~\+t074O 
 
 O.OO338 -jO.OOieZ . 
 
 thase value* la aqmtlon (104) 
 
 = 8.8 + 
 
 -22 
 
 +(-o. 00336 -t-jo.ooiaz) ?oge y^ 
 
 +(-O.OO33B-j 
 
 (I-J.Z 
 
 (za.oo 
 
 t = 8.8- 
 
 !. 
 
 22 
 
 0. 00103 
 
 0.^5Zx 
 
 11.32* 
 
 + (-O.0033&+jO.OO\&Z)logif i (4I.SQ-jZ4.30) 
 (-O.OO33B +/O.OO/SZ) 10g e (30.78 -J24.3O) 
 + (-0.00338 -JO. 00/82) lofcfrl.SB +J24.3O) 
 - (-O.O0338 -JO.OOI82) loge(3O-7B +J2.4.3O) 
 
 POP logwittane of the connloa: qur.ntltioa, in 
 1qge(x+jy) = Jog>\X+jy\ +jtan% + zjmTT , m 
 
 infinite 
 
 
 liowov r, oinoe m lo any intocer, thU fowrala 
 
 / . r^<J^ 
 
 imifttr of v.luoo of lofrlX+jy) . nat unly tto pgia^tpul trluos of 
 
 ^ "^^^ot_ 
 Jtflnitt li^mnli ar oonoernetl In apeecS-tlMe detormlwtiona, so 
 
 the last trm t a J * , wy b tMglootedi that is let - 
 Iha rMOltir*? nolublon UMB it 
 
 t 8*8 * 230*7 239*5 aeoor^bi 9 
 4*00 
 
 
 - . - 
 

 
 -oc 
 
 
 
 ( 
 
 (oe . vs\- e%>\v) ^>\ (ssvoo .o'i 
 
 \ 4 
 
 - -v 
 
 - 8.8 = 
 
 Ttmys -v 
 
 
 V 
 
 \s-v 
 
 ^ 
 
 
 
 
 
 &-, 
 
 
If thu pcwtir is shut off whan the speed re: dies 2C.O 
 per hour, iaul tho train is allowed to coast tip ths cao per ont grate, 
 
 long i-ftci- the train a tar toil v/ill tiie apoed bocoiae 10*0 railea 
 per hour ? 
 
 (11C), (117) and (118) ffivo 
 
 f S~O 
 
 or, = 0.01 ' ' 
 
 O.Z403 , 
 
 >0, = O.OO03 , 
 
 O.OOOOZ 
 
 ,50 "" 70 
 
 O.os x/cT 
 
 /O.9x/0 3 >O . 
 fflxerefore, fomsula. (135) applies and 
 
 5//1 
 
 _-6 
 
 I 
 
 lSxl6 6 (zZxl6"\ 2&.o ojxnt '28.o-o.Z4V6, 
 2 X 22 X /O^X/o +O-OO03 
 
 -sin 
 
 -\J+X 22.* A*(22 X If -h O.OOO3X fO.O + O.2.4-O8) 
 
 = 233.5 +70.6 . 
 = 3/O 5econ</s 9 
 = *5.I7 minutes 
 

 
 . 
 
 
 e* 
 
 , eooo.o = 
 
 SOOrK>JD 
 -\ 
 
 sa = 
 
 
 o\xs&x-s 
 
 ."^?V 
 
 GoOO-O^- 
 
 ; '. -t. I 
 
 ~^7d^ * t - ees 
 
 me- 
 
Prom the time the speed reuohoa iOG miles por hour until 
 the ti'.'ln stopa, an average broking effort of 200 potmts per ton of 
 &roas train w*igab la applied by friction brakes, the grade continuing 
 at + 1*00 per cent, now nruoh time will hare been oozunmd In making 
 the run froa start to stop ? 
 
 ?rom oeuationa (165), (166) tuoal (167), 
 
 = z.Z4oa , 
 
 = O.OOO3 , 
 
 o.ooooa 
 /so 
 
 - 22x70"* 
 2.Z40Q 
 
 110(1+ 
 
 0.09 x/0* 
 
 ZZxlti* 
 = lOZxIO 3 > O 
 
 Therefore fornjula (1R3) applies 
 
 t = 3/0 + 
 
 SLH 
 
 -sin 
 
 = 3/O+ 4.8 
 = 31-5 
 
 .o +0.0003 
 
 O.OOO3 
 

 
 
 
 +O.OS -v 
 
 \ 
 
 , eooo.o 
 
 v> 
 
 ,^s^v ^ 
 
 , O 
 
 
 
 
 : 
 
 , I 
 
 - -V 0\( 
 
 *o\ ^&. 
 
 s.v 
 .*-.-. 'i."^ 
 
14 DAY USE 
 
 RJBTURN TO DESK FROM WHICH BORROWED 
 
 LOAN DEPT. 
 
 This book is due on the last date stamped below, or 
 
 on the date to which renewed. 
 Renewed books are subject to immediate recall. 
 
 07 1Q66 T .' 
 
 JAM 367-UM 
 
 ' O 
 
 General Library 
 University of California 
 
YF Oil!