GIFT OF 
 
 V;illiam a Muart Smith 
 
 ENGINEERING LIBRARY 
 

 
THE 
 
 RELATIVE PROPORTIONS 
 
 OF THE 
 
 STEAM-ENGINE: 
 
 BEING 
 
 A RATIONAL AND PRACTICAL DISCUSSION OF THE DIMEN- 
 SIONS OF EVERY DETAIL OF THE STEAM-ENGINE. 
 
 BY 
 
 WILLIAM DENNIS MARKS, 
 
 \i 
 
 WHITNEY PROFESSOR OF DYNAMICAL ENGINEERING IN THK 
 UNIVERSITY OF PENNSYLVANIA. 
 
 WITH NUMEROUS DIAGRAMS. 
 
 SECOND EDITION. REVISED AND ENLARGED. 
 
 J. B. LIPPINCOTT & CO. 
 
 1884 
 
Entered according to Act of Congress, in the year 1878, by 
 
 J. B. LIPPINCOTT & CO., 
 In the Office of the Librarian of Congress, at Washington. 
 
 COPYRIGHT, 1883, BY J. B. LIPPINCOTT & Co. 
 
 GIFT OF 
 
 -v. 
 
 ENGINEERING LIBRARY 
 
 ' PMuduphiu. 
 
PREFACE. 
 
 IT is a source of regret to the author of these Lectures 
 that none of the distinguished writers upon Mechanics or 
 the Steam-engine have undertaken to give, in a simple and 
 practical form, rules and formulae for the determination of 
 the relative proportions of the component parts of the 
 steam-engine. 
 
 The authors of the few works as yet published in the 
 English language either entirely ignore the proportions of 
 the steam-engine or content themselves with scanty and 
 general rules Rankine excepted, who in the attempt to be 
 brief is sometimes obscure, leaving many gaps in the im- 
 mense field which he has attempted to cover. This defi- 
 ciency in the literature of the steam-engine is remarkable, 
 because the problem which the mechanical engineer is most 
 frequently called upon to solve is the determination of the 
 dimensions of its various parts. 
 
 From time to time hand-books of the steam-engine have 
 been published giving practical (?) rules, the result of obser- 
 vation of successful construction ; and with these rules the 
 practising engineer, who has little time for original investi- 
 gation, has had to content himself. It is of course reasonable 
 to limit the correctness of these rules to cases in which all 
 
 869476 
 
4 PREFACE. 
 
 the conditions are the same, as in the case or cases from 
 which these rules have been derived, thus placing a serious 
 obstruction in the way of improvement or alteration of de- 
 sign, and rendering the rules worse than useless even dan- 
 gerous in many cases. 
 
 " The usual resource of the merely practical man is pre- 
 cedent, but the true way of benefiting by the experience of 
 others is not by blindly following their practice, but fry 
 avoiding their errors, as well as extending and improving 
 what time and experience have proved successful. If one 
 were asked, What is the difference between an engineer and 
 a mere craftsman ? he would well reply that the one merely 
 executes mechanically the designs of others, or copies some- 
 thing which has been done before, without introducing any 
 new application of scientific principles, while the other 
 moulds matter into new forms suited for the special object 
 to be attained, and lets his experience be guided and aided 
 by theoretic knowledge, so as to arrange and proportion 
 the various parts to the exact duty they are intended to 
 
 fulfil. 
 
 "'For this is art's true indication, 
 
 When skill is minister to thought, 
 When types that are the mind's creation 
 The hand to perfect form hath wrought.' " 
 
 STONEY'S Theory of Strains. 
 
 Zeuner, in his elegant Treatise on Valve Gears, transla- 
 ted by M. Miiller, has laid the foundation for the treatment 
 of slide-valve motions for all time, and in his Mechanisehe 
 Wdrmetheorie has carried the application of the mechan- 
 
PREFACE. 5 
 
 ical theory of heat to the steam-engine as far as the present 
 state of the science of Thermo-dynamics will permit. 
 
 Poncelet, in his Mecanique appliquee aux Machines, has 
 most thoroughly treated some members of the steam-engine, 
 neglecting others of as great practical importance. 
 
 Him, in his Theorie mecanique de la Chaleur, gives 
 us, besides a very able treatise on the science of Thermo- 
 dynamics, a valuable series of experiments upon the steam- 
 engine itself, confirming Joule's results. 
 
 A translation of Der Constructeur, by F. Reuleaux, 
 would, if made, add much to our knowledge of the proper 
 proportions of the steam-engine, as well as of other ma- 
 chines. 
 
 A rational and practical method of determining the 
 proper relative proportions of the steam-engine seems as 
 yet to be a desideratum in the English literature of the 
 steam-engine; and these Lectures have been written with 
 that feeling, purposely omitting the consideration of such 
 topics as have already in many cases been over-written, and 
 considering only those which have not received the atten- 
 tion which their importance demands. 
 
 In the choice of a factor of safety a matter wherein 
 opinions widely differ the author, guided by considerations 
 set forth in Weyrauch's Structures of Iron and Steel, trans- 
 lated by DuBois, has fixed upon 10 as being the most cor- 
 rect value. If any of our readers should prefer a different 
 factor, the formulae deduced will be correct if the actual 
 steam-pressure per square inch is divided by 10 and multi- 
 i* 
 
6 PREFACE. 
 
 plied by the preferred factor of safety, and the result used 
 in the place of the actual steam-pressure. 
 
 In reducing all the required dimensions of parts of the 
 steam-engine to functions of the boiler-pressure or mean 
 steam-pressure in the cylinder per square inch, of the diam- 
 eter of the steam-cylinder, length of stroke, number of 
 strokes per minute, and horse-power, he trusts that he has 
 put the formulae in the simplest possible form for immediate 
 use. 
 
 It is indeed in this transformation of the formulae for the 
 strength of materials that the usefulness of the book lies ; 
 for the practitioner, once satisfied of their correctness, has 
 but to insert quantities fixed at the commencement of his 
 design, and derive from the formulae the required dimen- 
 sions, being relieved of many formulse and details connected 
 with the applications of statics to the strength and elasticity 
 of materials. 
 
 The constant references to the fourth section of Weis- 
 bach's Mechanics of Engineering are necessary, as it is 
 no part of the author's plan to discuss the strength and 
 elasticity of materials any further than it is necessary to do 
 so in their application to the steam-engine. Those unac- 
 quainted with this branch of mechanical engineering will 
 nowhere find it treated with greater simplicity and thor- 
 oughness. Other references have been made for the pur- 
 pose of directing the reader to such sources as have been 
 drawn upon in the consideration of topics discussed in this 
 work. 
 
PREFACE. 1 
 
 "We who write at this late day are all too much in- 
 debted to our predecessors, whether we know it or not, to 
 complain of those who borrow from us ;" and each of us is 
 only able to make his relay, taking up his work where 
 others have left it. 
 
 The lack of accurate experimental data has, in many 
 cases, forced the writer to make, perhaps, bold assumptions 
 which may not prove entirely correct ; however, as in all 
 cases the method of reasoning is given, the reader, where 
 he is in possession of more accurate data, can modify by 
 substitution. 
 
 The accidental loss of all of the original manuscript and 
 drawings of these Lectures, and the necessity of rapidly re- 
 writing them for use in daily instruction, have caused the 
 work to be more abbreviated than was originally intended. 
 
 Deeply sensible of the many unavoidable deficiencies of 
 this little work, even in the limited field covered, its author 
 still hopes that it will aid in the diffusion and advancement 
 of real knowledge, upon whose progress the prosperity of 
 our civilization depends. 
 
 W. D. M 
 
 UNIVERSITY OF PENNSYLVANIA, 
 Philadelphia, 1878. - 
 
PREFACE TO THE SECOND EDITION. 
 
 THE kind reception of the First Edition of this work 
 has afforded the author an unexpected pleasure, for which 
 he is very grateful. 
 
 The pleasure is the greater because it has come from an 
 unlooked-for direction. Those actively engaged in design- 
 ing engines rarely turn to books for instruction, and are 
 quite frequently heard to complain that they find in them 
 a good deal that they do not care to know, and nothing 
 that they can utilize. Among the added lectures the 
 author wishes particularly to call the attention of readers 
 to The Cheapest Point of Gut -Off and to The Errors of the 
 Zeuner Diagram. The former will prove of interest to all, 
 whether designers or users. The latter is not written with 
 a desire to detract from the honor due to Prof. Zeuner, but 
 rather to perfect the great discovery of the law of valve- 
 motions due to his genius. 
 
 W. D. M. 
 
 UNIVERSITY OP PENNSYLVANIA, 1883. 
 
CONTENTS. 
 
 ART. CHAPTEE I. PACK 
 
 1. INTRODUCTORY 13 
 
 2. The Steam-Cylinder 13 
 
 3. Indicated Horse-Power 15 
 
 4. Thickness of the Steam-Cylinder 17 
 
 5. Thickness of the Cylinder-Heads 19 
 
 6. Cylinder-Head Bolts 20 
 
 CHAPTEK II. 
 
 7. Standard Screw-Threads for Bolts 22 
 
 8. The Steam-Chest 25 
 
 9. The Steam-Ports..... 28 
 
 10. The Piston-Head 29 
 
 11. The Piston-Eod 29 
 
 12. Wrought-Iron Piston-Kod 30 
 
 13. Steel Piston-Kod 34 
 
 CHAPTER III. 
 
 14. Comparison and Discussion of Wrought-Iron and Steel 
 
 Piston-Rods 37 
 
 15. Keys and Gibs 37 
 
 16. Wrought-Iron Keys 40 
 
 17. Steel Keys 43 
 
 18. The Cross-Head..... 44 
 
 19. Area of the Slides..., . 46 
 
10 CONTENTS. 
 
 ART CHAPTER IV. 
 
 20. Stress on and Dimensions of the Guides 49 
 
 21. Distance between Guides 51 
 
 22. The Connecting-Rod 53 
 
 23. Wrought-Iron Connecting-Rod 55 
 
 CHAPTER V. 
 
 24. Steel Connecting-Rod 59 
 
 25. General Remarks concerning Connecting-Rods 62 
 
 26. Connecting-Rod Straps 63 
 
 27.. Wrought-Iron Straps 63 
 
 28. Steel Strap , 65 
 
 CHAPTER VI. 
 
 29. The Crank-Pin and Boxes 66 
 
 30. The Length of Crank-Pins 70 
 
 31. Locomotive Crank-Pins, Length and Diameter of 73 
 
 CHAPTER VII. 
 
 32. Diameter of a Wrought-Iron Crank-Pin for a Single Crank. 74 
 
 33. Steel Crank-Pins 75 
 
 34. Diameters of Crank-Pins from a Consideration of the Pres- 
 
 sure upon them 76 
 
 35. Of the Action of the Weight and Velocity of the Recipro- 
 
 cating Parts 76 
 
 CHAPTER VEIL 
 
 36. The Single Crank 85 
 
 37. Wrought-Iron Single Crank 90 
 
 CHAPTER IX. 
 
 38. Steel Single Crank 93 
 
 39. Cast-Iron Cranks 95 
 
 40. Keys for Shafts 95 
 
 41. Wrought-Iron Keys for Shafts 98 
 
CONTENTS. 11 
 
 ART. PAGB 
 
 42. Steel Keys for Shafts 99 
 
 43. The Crank or Main Shaft 99 
 
 44. Shaft Subjected to Flexure only 100 
 
 45. Wronght-Iron Shaft, Flexure only 101 
 
 CHAPTEK X. 
 
 46. Steel Shaft, Flexure only 103 
 
 47. Shaft Subjected to Torsion only 103 
 
 48. Wrought-Iron Shaft, Torsion only 104 
 
 49. Steel Shaft, Torsion only 104 
 
 50. Shaft Submitted to Combined Torsion and Flexure 105 
 
 51. Flexure and Twisting of Shafts 108 
 
 52. Comparison of Wrought-Iron and Steel Crank-Shafts Ill 
 
 53. Journal-Bearings of the Crank-Shaft Ill 
 
 CHAPTEK XI. 
 
 54. Double Cranks 115 
 
 55. Triple Cranks 118 
 
 56. The Fly- Wheel 120 
 
 57. Fly-Wheel, Single Crank 123 
 
 58. Fly- Wheel, Double Crank, Angle 90 127 
 
 CHAPTEE XII. 
 
 59. Fly- Wheel, Triple Cranks, Angles 120 129 
 
 60. Of the Influence of the Point of Cut-Off and the Length of 
 
 the Connecting-Hod upon the Fly- Wheel 131 
 
 CHAPTEE XIII. 
 
 61. The Weight of the Eim of Fly- Wheels 137 
 
 62. Value of the Coefficient of Steadiness 138 
 
 3. Area of the Cross-Section of the Eim of a Fly- Wheel 139 
 
 64. Balancing the Fly- Wheel 140 
 
 65. Speed of the Eim of the Fly-Wheel 142 
 
12 CONTENTS. 
 
 ART. CHAPTER XIV. PAGK 
 
 66. Centrifugal Stress on the Arms of a Fly- Wheel 143 
 
 67. Tangential Stress on the Arms of a Fly- Wheel for a Single 
 
 Crank 147 
 
 68. Work Stored in the Arms of a Fly- Wheel 150 
 
 69. The Working-Beam 151 
 
 70. General Considerations 151 
 
 71. Note on the Taper of Connecting-Rods 154 
 
 CHAPTER XV. 
 
 72. The Limitations of the Steam-Engine 161 
 
 CHAPTER XVI. 
 
 73. The Cheapest Point of Cut-Off. 176 
 
 CHAPTER XVH. 
 
 74. The Errors of the Zeuner Valve-Diagram 187 
 
 TABLES. 
 
 75. TABLE V. The Elasticity and Strength of Extension and 
 
 Compression 212 
 
 76. TABLE VI. The Elasticity and Strength of Flezure or 
 
 Bending 213 
 
 77. TABLE VII. The Elasticity and Strength of Torsion 214 
 
 78. TABLE VIII. The Proof-Strength of Long Columns 215 
 
 NOTE. These Tables will be found to contain almost all the formula 
 referred to in these Lectures. 
 
.-, ' -' * ,> 
 THE J 
 
 RELATIVE PBOPOETIONS 
 
 OF THE 
 
 STEAM-ENGINE. 
 
 CHAPTER I. 
 
 (1.) Introductory. In the present work, unless spe- 
 cially stated, the single-cylinder, double-acting steam-engine 
 only will be the subject of discussion. 
 
 In making use of the following rules and formulae, if a 
 non-condensing engine is under consideration, the pressures 
 per square inch above the atmosphere, or as registered by 
 an ordinary steam-gauge, must be used. If a condensing 
 engine be considered, fifteen pounds per square inch must be 
 added to the pressures above the atmosphere. 
 
 While it is impossible to take up every form of steam- 
 engine which has been invented, the formulae are sufficiently 
 general to admit of adaptation to any form of engine which 
 the engineer may wish to devise. 
 
 (2.) The Steam-Cylinder. In many cases occurring 
 in practical Mechanics other considerations than economy 
 of steam determine either the stroke or the diameter of the 
 steam-cylinder. When, however, these dimensions are not 
 fixed by other considerations, that of economy of steam 
 should have the precedence, as being a constant source of 
 gain ; and it being demonstrable by the differential calculus 
 that the surface of any cylinder closed at the ends and en- 
 closing a given volume is a minimum when the diameter 
 2 13 
 
14 
 
 THE RELATIVE PROPORTIONS 
 
 of that cylinder/ "is equfeLto jits; length,* it follows that any 
 given volume *oi" 'steam *ha a' inittimum of surface of con- 
 densation /ftnji'jJon^QcJuSnS^j loses; Jess ;by condensation than 
 it would in any 'other 'cylmcfef of 'equal volume and differ- 
 ent relative dimensions ; and therefore that the best relative 
 proportions of the stroke and diameter of the steam-cylinder 
 are attained when they are equal. 
 
 The importance of the action of the walls of the steam- 
 cylinder in condensing steam, and the inability of a steam- 
 jacket to do more than keep the cylinder warm, without 
 actually communicating any appreciable amount of heat to 
 the enclosed steam, are becoming more clearly recognized 
 among engineers, and are forcing them to adopt higher pis- 
 ton-speeds, "f to take means of reducing the surface of con- 
 
 * Demonstration. Let the surface of the cylinder = S nyx + -~ . (1) 
 
 " " volume " " =F=-^. (2) 
 
 47 
 From eq. (2) we have 7rx = r or ^xy = 4.Vy~ l . 
 
 y 
 
 Fio. I- Substituting in eq. (1), we have 
 
 Differentiating, we have 
 
 xL-4Fjr 2 + 7ry = i 
 
 giving 
 
 4V 
 
 Substituting this value of y in eq. (2), we have also z= \/-^r-> show- 
 ing, since the second diff. coef. is positive, a minimum. 
 
 f For a full discussion of the consumption of steam, see A Treatise 
 on Steam, chap, iv., Graham ; " Experiments with a Steam-Engine," 
 translated from Bulletin de la Societe Industriel de Mulhouse, in Journal 
 of the Association of Entfg Societies, May, J.883, Hirn-Smith; also 
 Chapters XV. and XVI. 
 
OF THE STEAM-ENGINE. 15 
 
 densation in the cylinder, and to superheat the steam to a 
 limited extent before its introduction into the engine. 
 
 If the cylinder and heads be of a uniform thickness, the 
 quantity of metal required to form a cylinder of the de- 
 sired volume is nearly a minimum : we say "nearly," because 
 sufficient length must be added to the cylinder to provide 
 for the thickness of the piston-head and the required clear- 
 ance at the ends. 
 
 The use of shorter cylinders than has hitherto been cus- 
 tomary has the advantage of reducing the piston-speed, and 
 consequently the wear upon the piston-packing, for any 
 given number of revolutions, although the wear upon the 
 interior of the steam-cylinder is constant for any constant 
 number of strokes per minute. 
 
 (3.) Indicated Horse-Power. In the present work 
 the indicated horse-power only will be referred to or made 
 use of. 
 
 Let (HP) = the indicated horse-power. 
 
 P =the mean pressure of steam on the piston- 
 
 head in pounds per square inch. 
 L = the length of stroke in feet. 
 A = the area of the piston-head in square inches. 
 JV = the number of strokes ( = twice the number 
 
 of revolutions of the crank) per minute. 
 We have the well-known formula, 
 
 If in formula (1) we make the length of stroke in inches 
 equal to the diameter of the steam-cylinder, it becomes, 
 letting d = the diameter of cylinder in inches, 
 
 12x33000 ' 
 
16 THE RELATIVE PROPORTIONS 
 
 and reducing, we have for the common diameter and stroke 
 of a steam-cylinder of any assumed horse-power, 
 
 d- 79.59 -J^p. (2) 
 
 This formula gives, for any assumed horse-power, mean 
 pressure, and number of strokes per minute, the common 
 diameter and stroke of the steam-cylinder in inches. 
 
 The reduction in size, and the consequent economy in 
 using steam, resulting from assuming the pressure per 
 square inch, P, and the number of strokes per minute, N, 
 as large as circumstances will permit, in designing an engine 
 of any desired horse-power (-HP), will at once be perceived 
 upon inspection of formula (2). The advantages resulting 
 from high pressures, early cut-off, and rapid piston-speed 
 will be more thoroughly discussed in Art. 35 of this work 
 
 Example. In a given cylinder, 
 Let L = 4 ft. = 48 inches. 
 " d = 32 inches. 
 " P = 40 Ibs. per square inch. 
 " N= 40 per minute = 20 revolutions of the crank. 
 Using formula (1), we have 
 
 PLAN 40x4x804.25x40 
 
 =156 > approx - 
 
 3000 
 
 If now we assume the horse-power (-HP) = 156, and 
 Let, as before, N= 40 per minute, 
 
 P=40 Ibs. per square inch, 
 we have, using formula (2), 
 
 d = 79.59 V^P - 79.59 \l-^- = 36.73 inches, 
 * PN v 40x40 
 
 for the required common stroke and diameter of a cylinder 
 of equal power with the first. 
 
OF THE STEAM-ENGINE. 17 
 
 (4.) Thickness of the Steam-Cylinder. Steam-cylin- 
 ders are usually made of cast iron ; and in order that the 
 engine may be durable, this casting should be made of as 
 hard iron as will admit of working in the shop. Steel lining- 
 cylinders for ordinary cast-iron cylinders have sometimes 
 been used, and have well repaid in durability their greater 
 cost. 
 
 Large steam-cylinders should always be bored in either a 
 horizontal or vertical position, similar to that in which they 
 are to be placed when in use. 
 
 Weisbach, in Art. 443, Vol. ii., of the Mechanics of En- 
 gineering, gives the following formula for the thickness of 
 steam-cylinders : 
 
 Let t = the thickness in inches of cast-iron cylinder- walls. 
 " P b = the boiler-pressure in pounds per square inch. 
 " d = the diameter of the cylinder in inches. 
 Then 
 
 I = 0.00033P 6 d + 0.8 inch, (3) 
 
 which makes 0.8 inch the least possible thickness of a steam- 
 cylinder. 
 
 Van Buren, in Strength of Iron Parts of Steam-Machinery, 
 page 58, establishes the following formula by means of a 
 discussion of a 72-inch English steam- cylinder which had 
 been found to work well : 
 
 V/2D 
 
 
 
 Reuleaux, in Der Constructeur, page 561, gives the fol- 
 lowing empirical formula for the completed thickness of 
 steam-cylinders : 
 
 < = 0.8 inch + ~^. (5) 
 
18 THE RELATIVE PROPORTIONS 
 
 Inspection of these differing formulae, all founded upon 
 successful practice, would lead to the conclusion that it is 
 best first to calculate the thickness necessary to withstand 
 the pressure of the steam, and then to make an addendum 
 sufficient to provide for boring* and re-boring, and also to 
 give the cylinder perfect rigidity in position and form. 
 
 Good cast iron has an average tensile strength of 18,000 
 pounds per square inch cross- section, and with a factor of 
 safety of 10 gives 1800 pounds per square inch as a safe 
 strain. That this factor is not too large will be conceded 
 when we consider that with some forms of valve-motion the 
 admission of steam to the cylinder partakes of the nature of 
 a veritable explosion. 
 
 From Weisbach's Mechanics of Engineering, Vol. i., Sec. 
 vi., Art. 363, we have, if we take safe strain = 1800 pounds, 
 
 0.00028^. (6) 
 
 Example. For a locomotive cylinder, 
 
 Let P b = 150 pounds per square inch. 
 " d = 20 inches. 
 
 From formula (3) we have t = 1.8 inches. 
 " (4) " " 2 = 1.34 " 
 
 (5) " " 4-1.00 ' l 
 
 (6) " " t= .83 " 
 
 Any of the thicknesses given would probably serve success- 
 fully, and about U inches is the best practice. It is not 
 
 * The best means of securing an approximately true cylinder is to 
 finish to size with a shallow broad cut, giving a rapid feed to the lathe 
 or boring-mill tool. If the cylinder is bored with a fine feed and deep 
 cutting-tool, the gradual heating and subsequent cooling are apt to 
 make the interior tapering in form, as well as to require the running 
 of the shop out of hours in order to avoid stopping, and thereby caus- 
 ing a jog in the cylinder. 
 
OF THE STEAM-ENGINE. 19 
 
 advisable to make a steam-cylinder of less than 0.75 inch 
 thickness under any circumstances. 
 
 In deciding upon the thickness to be given to any cylin- 
 der, the method of fastening it, as well as the distorting 
 forces that are likely to occur, should be' carefully con- 
 sidered. 
 
 (5.) Thickness of the Cylinder -Heads. It is demon- 
 strated in Weisbach's Mechanics of Engineering, Vol. i., Sec. 
 vi., Art. 363, that if the cylinder-heads were made of a 
 hemispherical shape, they would need to be of only half 
 the thickness of the cylinder-walls ; and in designing, the 
 attempt is sometimes made to attain greater strength by 
 giving to the cylinder-heads the form of a segment of a 
 sphere. 
 
 In considering rectangular plane surfaces subjected to 
 fluid pressure in Weisbach's Mechanics of Engin eering, Vol. 
 ii., Sec. ii., Art. 412, the following formula is deduced for 
 square plane surfaces, which will of course be, with greater 
 safety, true for a circular inscribed surface : 
 
 Let ti = the thickness in inches of the cylinder-heads. 
 " P b = the boiler-pressure in pounds per square inch. 
 " d = the diameter of the steam-cylinder in inches. 
 
 <! = 0.003d \Sp t > (7) 
 
 In large cylinders the heads are stiffened by casting radial 
 ribs upon them. 
 
 Example. 
 
 Let d = 20 inches. 
 " P b = 150 pounds per square inch. 
 
 We have, from formula (7), 
 
 *! = 0.003dl/^ = 0.003 x 201/150 = 0.73 inches. 
 Comparing this with the result derived from formula (6), we 
 
20 THE RELATIVE PROPORTIONS 
 
 observe it to be less ; but we also find, comparing formulae 
 (7) and (6), that 
 
 .003^ P* 10 
 
 Prom formula we see that i v = t when P b = 100 pounds ; that 
 $! is greater than t when P b < 100 pounds ; and that ^ is less 
 than t when P b > 100 pounds. 
 
 A good practical rule for engines in which the pressure 
 does not exceed 100 pounds per square inch is to make the 
 thickness of the cylinder-heads one and one-fourth that of 
 the steam-cylinder walls. 
 
 (6.) Cylinder-Head Bolts. Having assumed a conve- 
 nient width of flange upon the steam-cylinder, the diameter 
 
 of the boll should be assumed at one-half that width, and 
 thoroughfare bolts used preferentially to stud bolts, as a 
 stud bolt is likely to rust and stick in place, and be broken 
 off in the attempt to remove it. 
 
 The bolts fastening the cylinder-head to the cylinder 
 should not be placed too far apart, as that would have a 
 tendency to cause leaks. 
 
 Taking 5000 pounds per square inch of the nominal area 
 of a bolt as the safe strain,* in order to cover fully the 
 strain upon the bolt due to screwing its nut firmly home, as 
 well as the strain due to the steam-pressure, and dividing 
 the total pressure of the steam upon the cylinder-head by it, 
 we will obtain the area of all the bolts required ; and divid- 
 
 * The Baldwin Locomotive Works use eleven |-inch stud bolts to 
 secure the head of an 18-inch steam-cylinder. If we assume the 
 greatest steam-pressure to be 150 pounds per square inch, we have 
 for the stress per square inch of nominal area of the bolts about 5800 
 pounds, and we are therefore well within limits which have been 
 found thoroughly practical by a tentative process. 
 
OF THE STEAM-ENGINE. 21 
 
 ing this latter area by the area of one bolt of the assumed 
 diameter, we have the number of bolts required. 
 
 Let P 6 = the boiler-pressure in pounds per square inch. 
 " d = the diameter of the steam-cylinder in inches. 
 " c = the area of a single bolt of the assumed diameter 
 
 in square inches. 
 
 " b =the number of bolts required. 
 We have 
 
 
 5000c 
 
 Example. In a given steam-cylinder, 
 Let d = 32 inches. 
 
 " P b = 81 pounds per square inch. 
 
 " c = 0.442 square inches (diameter of bolt J inch). 
 
 We have, from formula (9), 
 
 5 = 0.000^^30, 
 
 showing about 30 three-quarter inch bolts to be required. 
 With so wide a margin as is given by the assumption of 
 5000 pounds per square inch as a safe strain, considerable 
 variation may be made from this number. Mr. Robert 
 Briggs, in a paper in the Journal of the Franklin Institute 
 for February, 1865, page 118, says: "Ordinary wrought 
 iron, such as is generally used in bolts, can be stated to be 
 reliable for a maximum load under 20,000 pounds per 
 square inch, and the absolute (ultimate ?) tensile strength of 
 any bolt may be safely estimated on that basis." 
 
22 
 
 THE RELATIVE PROPORTIONS 
 
 CHAPTER II. 
 
 (7.) Standard Screw-Threads for Bolts. The stand- 
 ard American pitch and dimensions of head and nut of bolts 
 as now used in all the mechanical workshops of the United 
 States was first proposed by Mr. Wm. Sellers. (See Report 
 of Proceedings, Journal of the Franklin Institute for May, 
 1864 ; Report of Committee, Journal of the Franklin In- 
 titute, January, 1865. For a full critique and comparison 
 with other systems, see Journal of the Franklin Institute, 
 February, 1865; On a Uniform System of Screw-Threads, 
 Robert Briggs.) 
 
 The advantages of uniformity of dimensions in an element 
 of a machine so frequently occurring do not need discussion. 
 
 The " Committee on a Uniform System of Screw-Threads " 
 reported as follows : 
 
 "Resolved, That screw-threads shall be formed with 
 straight sides at an angle to each other of 60, having a 
 flat surface at the top and bottom equal to one-eighth of the 
 pitch. The pitches shall be as follows, viz. : 
 
 Diameter of bolt.. 
 No. threads pr. in. 
 
 Diameter of bolt.. 
 No. threads pr. in. 
 
 A 
 
 18 
 
 16 
 
 14 
 
 13 
 
 3* 
 
 12 
 
 31- 
 
 11 
 
 3' 
 
 10 
 
 4* 
 
 21 
 
 H 
 
 4f 
 
 1 
 
 21 
 
 Ifll! 
 
 5| 
 
 2| 
 
 " The distance between the parallel sides of a bolt head 
 and nut, for a rough bolt, shall be equal to one and a half 
 diameters of the bolt plus one-eighth of an inch. The thick- 
 ness of the heads for a rough bolt shall be equal to one-half 
 the distance between its parallel sides. The thickness of the 
 nut shall be equal to the diameter of the bolt. The thick- 
 ness of the head for a finished bolt shall be equal to the 
 
OF THE STEAM-ENGINE. 
 
 23 
 
 thickness of the nut. The distance between the parallel 
 sides of a bolt head and nut and the thickness of the nut 
 shall be one-sixteenth of an inch less for finished work than 
 for rough." 
 
 FIG. 2. 
 
 The required dimensions of bolts and nuts can be best 
 expressed in a general way by means of formulae and 
 Fig. 2. 
 
24 THE RELATIVE PROPORTIONS 
 
 Let D the nominal diameter of any bolt in inches. 
 Then p = the pitch = 0.24 V /Z) + 0.625 - 0.175. 
 
 n = the number of threads per inch = -. 
 
 P 
 d = the effective diameter of the bolt i. e., at root 
 
 of thread = D-1.3p. 
 =the depth of thread = 0.65p. 
 H= the depth of nut = D. 
 d n = the short diameter of hexagonal or square nut 
 
 = -Z>+0".125. 
 
 A = the depth of the head of bolt = |D+ T V inch. 
 
 g 
 d h = the short diameter of the head of bolt = -D + 0.125. 
 
 The value of the pitch p, in terms of D, was derived from 
 a graphical comparison of the then existing threads as used 
 in the most prominent workshops in the United States. The 
 depth of the thread S is deduced as follows : 
 
 Since the angles of a complete V thread are each = 60, 
 its sides and the pitch would form an equilateral triangle, 
 and we would have for its depth p sin 60 = 0.866p ; but in 
 the actual thread is taken off at the top and bottom ; leav- 
 ing only J of the depth of a complete V thread. 
 
 Therefore, S - -QMGp - 0.65p. 
 
 4 
 
 Were it possible or always convenient to have uniformly 
 close work, such an accident as the stripping of the thread 
 from a bolt with the dimensions stated would be impossible. 
 
 The proportions established are the result and an average 
 of the practical requirements of machine-shop practice, and 
 are therefore to be preferred to proportions which might be 
 established from a theoretical consideration of the strength 
 
OF THE STEAM-ENGINE. 25 
 
 of materials. The appended table of dimensions (pp. 26, 
 27) is furnished by Messrs. "Wm. Sellers & Co. 
 
 (8.) The Steam-Chest. In deciding upon the dimen- 
 sions of the steam-chest, it must be borne in mind that it 
 ought to be as small as the dimensions and travel of the 
 valve will permit, in order to avoid loss by condensation 
 of steam. 
 
 The chest is subject to many modifications of form, but 
 usually consists of the ends and two sides of a cast-iron box, 
 resting upon a rim surrounding the valve-face, upon which 
 a flat cover is placed, and the whole firmly secured to the 
 steam-cylinder by means of stud-bolts passing through the 
 cover outside of the sides of the box, in order to avoid 
 rusting of the bolts as well as to diminish the contents of 
 the box. 
 
 The number of bolts required can be determined, as shown 
 in Art. (6), from a consideration of the steam-pressure upon 
 the steam-chest cover. 
 
 It is customary to make the sides and cover of the steam- 
 chest of the same material and thickness as the cylinder- 
 walls, sometimes strengthening the cover by casting ribs 
 upon it. 
 
 To deduce the theoretical thickness, we have, from Weis- 
 bach's Mechanics of Engineering, Vol. ii., Sec. ii., Art. 412, 
 the following formula : 
 
 Let I = the longest inside measurement of chest in inches. 
 " b = the breadth of chest in inches. 
 " P b - the boiler-pressure in pounds per square inch. 
 " T - the safe tensile strain upon cast iron per square 
 
 inch - 1800 pounds. 
 " <2 = the required thickness of the steam-chest cover in 
 
 inches. 
 
THE RELATIVE PROPORTIONS 
 
 o 
 
 PL. 
 O 
 
 PH 
 
 55 
 
 paqeiaij 
 
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 }.ioqs 
 
 paqeiutj 
 
 -jjoqg 
 
 UO UIBI^S 3JBg 
 
 jo q?P!M. 
 
 pr?aiqi jo 
 
 -raBip imtuojn 
 
 < M o-.H< |tocoH< lMH" H* N 
 
 i -- - "":- ' ' ^M^w^txu-ste 1 ^ 
 <M <M C< C<J CO CO CO CO -^ 
 
 <M C<J C<J C<1 C< CO CO 
 
 I <M C<J <N (M (M 
 
 N <M (^ <M (M (M 
 
 TH r-i T-( r-l TH N 
 
 rlC<l<M<M<MC^COCO 
 
 <M <M C<J C<l N 
 
 r-l rH (N (M <N (N (M (M 
 
 23 2SS22S2- 
 
 
 COOOO OGOOOOOCOt^O 
 (OJCO ^t^t^-OOiOi 
 
 oooooooo 
 
OF THE STEAM-ENGINE. 
 
 27 
 
 paqsjtnj 
 
 (M CO CO CO CO ^f 
 
 rH rH rH (M 
 
 co'co co"co 
 
 OO OS OS O O i-i rH 7<l 
 
 a'jauiBipSao'j 
 
 to < 
 
 |r-(rH|MI 
 
 O iO CD CD l>- t>- t- GO 
 
 a** 
 
 paqsmij 
 
 u.auiBip ?aoqg 
 
 CO CO CO -^ 
 
 CD CO CO t^ t> GO 00 GO OS 
 
 paqstuij 
 
 >oto to toi-ito 
 
 ioto 
 
 r-JM 
 
 10 
 
 aaBtibg 
 
 Saoq; 
 
 paqsiuij 
 J 4 auiBip l-ioqg 
 
 j,anmp vioqg 
 
 rH|rt|N 
 
 CO CO 
 
 uo UIBJ^S ajBg 
 
 ^ O 
 tO CO 
 
 oooo 
 oo *o ^> ^ 
 
 O t~- OS rH 
 rH rH rH C<l 
 
 w 
 
 t^t-(M(M t^t^-^CO 
 <M (M CO CO CO CO CO ^ 
 
 pBajq^ jo 
 
 JOOJ ^B J,aiUBIQ[ 
 
 rHCOl>.C<l C<JI>-O'-H 
 l>-OSi-H^ COOOi-HCO 
 
 r-i I-H <M' c<i c<i c<i co co 
 
 COCOCOCO W(M!M<M C<l(N C<J <M 
 
 ?loq 
 
 tUBlp [BOIU 
 
 IO 10 10 10 CO 
 
28 THE RELATIVE PROPORTIONS 
 
 
 
 Example. In a given steam-chest, 
 
 Let f = 40". 
 " 5 = 20". 
 " P b = 81 pounds. 
 
 Then, substituting in formula (10), we have 
 
 . 40 / 1600x400 C1 Q . , 
 
 /2 = -v / 81 = 3 inches. 
 
 60 \ 2560000 + 160000 
 
 (9.) The Steam-Ports.* D. K. Clark, in Railway 
 Machinery, page 108, states that with a piston-speed of 600 
 feet per minute a port area equal to one-tenth of the piston 
 area is found sufficient to admit steam to the cylinder with 
 sufficient facility for all practical purposes, and with but a 
 slight reduction of pressure. The size of the port may be 
 increased or diminished proportionally to the piston-speed 
 from these data with good success, bearing in mind that all 
 tortuousness of direction that can be should be avoided in 
 ports. 
 
 Example. Let the piston-speed = 500 feet per minute ; 
 
 * For a thorough treatise on link- and valve-motions, as well as in- 
 dependent cut-offs, see Zeuner's Treatise on Valve-Gears. Auchin- 
 closs (Link- and Valve-Motions) treats the same subject in a simpler 
 manner, avoiding the use of formulae as much as possible. King 
 (Steam, Steam-Engine Propellers, etc.) gives good descriptions of some 
 purely American forms of valves. Burgh (Link- Motions) gives a 
 large number of examples of English link-motions which have been 
 constructed. The author refrains from entering upon the wide field 
 of valve-motions, as one which has been thoroughly covered by the 
 splendid work of Zeuner mentioned above. 
 
OF THE STEAM-ENGINE. 29 
 
 then the port area should be = f x ^ piston area = T V piston 
 area. 
 
 (10.) The Piston-Head. It is very doubtful if any 
 formula can be derived which will give a correct value for 
 the thickness of the piston-head under all circumstances. 
 If the steam-cylinder be laid horizontally, other things being 
 equal, the piston-head should be broader than if the cylinder 
 is vertical, and an extra breadth of piston should be given 
 in all cases of rough usage or very rapid piston-speed. 
 
 The writer offers as a guide only the following formula : 
 
 Let -L = the length of stroke in inches. 
 " d = the diameter of cylinder in inches. 
 " 3 = the thickness of the piston-head in inches. 
 
 Then (.-v'Zar, (11) 
 
 or if L = d, (,-j/J. (12) 
 
 Example. Let L = 24 inches. 
 " d = 20 inches. 
 
 Then, ^ = y 24 x 20 = 4J- inches, approx. ]/ 
 
 (11.) The Piston-Rod. The piston-rods of the best 
 engines are made of steel, although in many instances ham- 
 mered wrought iron is still used. The rod, being fastened at 
 one end to the piston-head and at the other keyed to the 
 cross-head, sustains an alternate thrust and pull while pass- 
 ing accurately through the stuffing-box and gland at one 
 end of the cylinder. 
 
 It is obvious that the rod must, in addition to being strong 
 enough to sustain these stresses, be possessed of sufficient 
 rigidity not to bend in the slightest degree while in action. 
 3 * 
 
30 THE RELATIVE PROPORTIONS 
 
 As the piston-head is aided in holding its position acros3 
 the cylinder by means of the rod, and the cross-head is 
 always liable to have a slight amount of lateral play, due 
 to imperfections of workmanship or wearing of the guides, 
 we must regard it as a solid column fixed at one end and 
 loaded at the other, which is free to move sideways. 
 
 (12.) Wrought-Iron Piston-Rod, For the purpose of 
 determining the proper diameter of the piston-rod, either of 
 the two following formulae may be used, both taken from 
 Sec. iv. of Weisbach's Mechanics of Engineering : 
 
 Let $=the stress upon the whole piston-head due to the 
 
 pressure of the steam. 
 " di = the diameter of the rod in inches. 
 " .F=the cross-section of the piston-rod in square inches. 
 " -BT=the ultimate crushing strength per square inch of 
 
 wrought iron = 31000 pounds (Weisbach). 
 " W= the measure of the moment of flexure of a round 
 
 " J = the modulus of elasticity of wrought iron = 
 
 28000000 pounds per square inch. 
 " I = the length of the rod in inches = L = the stroke. 
 
 We have, for columns breaking by direct crushing, 
 
 S=FK, (13) 
 
 and for long columns tending to break by bending i. e., 
 buckling 
 
 (14) 
 
 Formula (13) must be used when the resulting diameter 
 of the rod is greater than one-twelfth of its length. If the 
 
OF THE STEAM-ENGINE. 31 
 
 use of formula (13) results in a diameter less than one-twelfth 
 the length of the rod, formula (14) must be used.* 
 
 If we assume 5 as the least allowable factor of safety for 
 materials subjected to a constant strain in one direction only, 
 then will 10 be the proper factor for members subject to 
 alternate and equal stresses in opposite directions i. e. t ten- 
 sion and compression. (See Weyrauch, Structures of Iron 
 and Steel, translated by Du Bois, chapters iv. and xiii.f) 
 
 As before, 
 
 Let P b = the steam-pressure in boiler in Ibs. per square inch. 
 " d = the diameter of the steam-cylinder in inches. 
 
 The greatest stress which the piston-rod has to sustain is 
 that due to the initial pressure of the steam in the cylinder, 
 which can be taken as equal to the boiler-pressure. 
 
 Introducing a factor of safety of 10 and substituting in 
 formula (13), we have, 
 
 * To determine the ratio of the length to the diameter of a wrought- 
 iron rod at which its tendency to crush is equal to its tendency to 
 break by buckling, we place formula (13) = formula (14). Let d = 
 
 ftd'* I TT \ 2 Trd* 
 the diameter of rod in inches, ^-K=( j - E. Therefore, 
 
 = A/ , and substituting the values of .E'and K, we have = 12. 
 
 f The ever-varying chemical constituents of the various brands 
 of wrought iron ; the fact now fully ascertained that, however care- 
 fully made and worked, wrought iron may be heterogeneous in its 
 composition ; that the reduction of the area from pile to bar has a 
 very great influence upon its strength (the greater the reduction, 
 the greater its strength) ; that a high tensile and compressive 
 strength is often obtained by loss, to a great extent, of ductility 
 and welding power all being considerations of a purely practical 
 nature, as well as the theoretical one mentioned above, point to a 
 higher factor of safety than 6 or 8, which is generally recommended. 
 
32 THE RELATIVE PROPORTIONS 
 
 lOP^L ^31000. 
 4 4 
 
 Therefore, 4 = 0.0179(^2* 
 
 Or supposing the pressure P b uniform throughout the stroke, 
 we have from formula (1) 
 
 *f 33000(gP) 
 ~ ~ 
 
 if we take the stroke in inches. Therefore, from formula (15) 
 (HP) 33000x12x4 
 
 ~~~ 
 
 Therefore, d, = 12.753 \ (16) 
 
 * LN 
 
 Note that the stroke is taken in inches. 
 . Substituting in formula (14) the values as stated above, 
 we have, using a factor of safety of 10, 
 
 64 
 
 Therefore, 4 = J r/WI7P b = 
 
 \ TT'X 28000000 v 
 
 0.03901 {/WI7F b inches. (17) 
 
 If we take L = d, formula (17) becomes 
 
 4 = 0.03901d i/P b inches. (18) 
 
 If we suppose the pressure of the steam to be uniform, 
 formula (17) becomes, in terms of the horse-power (JJP), 
 
 ! - 1.039^ (gP) inches. (19) 
 
OF THE STEAM-ENGINE. 33 
 
 From the consideration of the preceding formulae we 
 deduce the following rule for wrought-iron piston-rods: 
 
 Deduce the diameter of the piston-rod by means of formula 
 (15) or (16), as may be most convenient. Should this diameter 
 be less than one-twelfth of the stroke, then use the most conve- 
 nient of formulas (17), (18), and (19). 
 
 Example. What should be the diameter of a piston-rod 
 for a steam-cylinder ? Data as follows : 
 
 L = 48 inches. 
 d = 32 inches. 
 
 P = 40 pounds per square inch. 
 N = 40 per minute. 
 (.HP) = 156, approx. 
 
 Using formula (15), we have 
 
 * - 0.0179 x 32 i/30 - 3.6 inches. 
 Using formula (16), we have 
 
 d l - 12.753 J 156 = 3.63 inches. 
 * 48 x 40 
 
 As the diameter, 3.6 inches, is less than one-twelfth of the 
 stroke, 48 inches, we must use formula (17) or (19). 
 Substituting in formula (17), we have 
 
 d l = 0.03901 1 / / 40x32 2 x48 2 = 3.84 inches. 
 Substituting in formula (19), we have 
 
 d, = 1.039 ^ = 3.84 inches. 
 
 NOTE. It must be borne in mind that the piston-rod will 
 almost always be proportioned from the initial or boiler- 
 
34 THE RELATIVE PROPORTIONS 
 
 pressure of the steam, P b , and not from the mean pressure, 
 P, as in all the better forms of engine the mean pressure is 
 much less than the initial pressure. 
 
 (13.) Steel Piston-Rod. When steel is the material 
 used for piston-rods, its greater modulus of ultimate strength 
 and differing coefficient of elasticity will alter the numerical 
 values in the formulae given above. 
 
 Hodgkinson, in his experiments on long columns of 
 wrought iron and steel, found that a steel column of the 
 same dimensions as a wrought-iron column would bear one 
 and one-half times as great a load. 
 
 A consideration of formula (14) shows that these experi- 
 ments would require that E, the modulus of elasticity for 
 steel (which is 28,000,000 pounds for wrought iron), should 
 become 42,000,000 pounds. 
 
 This value agrees very nearly with that given by Reu- 
 leaux in Der Constructeur, page 4, for cast steel. 
 
 Kupffer, Styffe and Fairbairn place the value of E at 
 between 30 and 31 million pounds per square inch for steel. 
 
 As, however, we shall use the value E= 42,000,000 Ibs. 
 under exactly the same conditions as the experiments from 
 which it is derived, it is probably the most correct value for 
 our purposes. 
 
 The crushing strength of steel is so much greater than its 
 tensile strength, where the material is not permitted to de- 
 flect, that it need not be taken into consideration in formula 
 (13). 
 
 As a mean of eleven experiments made by Kirkaldy on 
 steel taken at random from merchants' stores, in all cases 
 the extension being upward of 10 per cent., we have for the 
 tensile breaking strain (equal to K) about 90,000 pounds 
 per square inch. 
 
 The elongation of 10 per cent, or upward would indi- 
 cate steel of sufficient toughness for machinery purposes. 
 
OF THE STEAM-ENGINE. 35 
 
 In the absence of reliable experiments determining the 
 ultimate crushing strength of soft steel, we are safe in as- 
 suming that it will bear 100,000 pounds per square inch 
 without deformation, which assumption makes the crushing 
 strength of steel equal to that of cast iron, and which it 
 probably much exceeds. 
 
 Substituting in formula (13) and using the same notation 
 as in Art. (12), we have 
 
 4 4 
 
 Therefore, ^ = (M)105dv<F 6 . (20) 
 
 Or if we suppose the pressure uniform throughout the 
 whole stroke, we have, in terms of the horse-power, 
 
 (21) 
 
 LN 
 
 Substituting in formula (14), we have 
 
 =42000000,. 
 
 4 4L* 64 
 
 and reducing, we have 
 
 d 1 -0.03525|^3 I Z^; (22) 
 
 or if in (22) we suppose L = d, 
 
 (23) 
 
 If we suppose the initial pressure to be uniform through- 
 out the whole stroke, we have 
 
 (24) ' 
 
36 THE RELATIVE PROPORTIONS 
 
 Formulae (20) and (21) should be used when the result- 
 ing diameter of the piston-rod is greater than J- of the length 
 of the stroke ; if it is less, then formula (22), (23) or (24) 
 must be used.* 
 
 Example. What should be the diameter of a steel piston- 
 rod for steam-cylinder ? Data as before. 
 
 L = 48 inches. 
 d = 32 inches. 
 
 P = 40 pounds per square inch. 
 N= 40 per minute. 
 = 156approx. 
 
 Using formula (20), we have ^ = 0.0105x32/40 = 2.12 
 
 I -t en 
 
 inches. Using formula (21), we have di = 7.48^/ 
 
 4o x 4U 
 
 2.13 inches. 
 
 We observe that the diameter of rod resulting from form- 
 ulae (20) and (21) is less than -|- of the stroke, and we must 
 therefore use formula (22) or (24). 
 
 Substituting in formula (22), we have 
 
 <?! = 0.03525|/32 2 x 48 2 x 40 = 3.47 inches. 
 Substituting in formula (24), we have 
 
 d, = 0.9394^/ 156x48 - 3.47 inches. 
 * Referring to note to Art. (12), we find the following general 
 
 / TT n? 
 
 formula: ^\'-=. Substituting in this the assumed values of E 
 a o > K 
 
 i . E= 42000000 , I 
 and Kfor steel, , we have - = 8. 
 
OF THE STEAM-ENGINE. 37 
 
 CHAPTER III. 
 
 14. Comparison and Discussion of Wrought-Iron 
 and Steel Piston-Rods. If, considering the formulse for 
 rupture by crushing or tearing only for wrought iron and 
 steel, we divide (21) by (16), we have 
 
 For steel c^ 7.- 
 
 = 1_^L = .58. (25) 
 
 I7HF) 
 For wrought iron d l 12.75^ 
 
 That is, the steel rod would have but 0.58 the diameter 
 of the wrought-iron rod, and 0.34 the area. 
 
 If we treat the formulse (24) and (19) for rupture by 
 buckling in a similar manner, we have 
 
 For steel di 
 
 \ i\ 
 
 0.90. (26) 
 
 For wrought iron d+ 
 
 That is, the steel rod has 0.9 the diameter and 0.81 
 the area of a wrought-iron rod working under the same 
 conditions. 
 
 In either case the use of steel is productive of economy 
 in weight, varying from 34 to 81 per cent. 
 
 Letting V= the volume of the piston-rod in cubic inches, 
 " y = the weight of a cubic inch of wrought iron 
 or steel = 0.27 lb., approx., 
 
 we have, for purposes of comparison, 
 
38 THE RELATIVE PROPORTIONS 
 
 F-^Z. (27)* 
 
 Letting C represent the numerical constant, we have, by 
 substituting for d? its value, derived equation (16) or (21) 
 for crushing, 
 
 and we see from (28) that the volume, and consequently 
 the weight, of a piston-rod are inversely as the number of 
 strokes per minute for any assigned horse-power. 
 
 Substituting formulae (19) and (24) for rupture by buck- 
 ling in (27), we have 
 
 If in this we assume L = d,vre have from equation (2), 
 letting Ci represent the constant in that equation, 
 
 giving an expression similar to (28), but affected by the 
 steam-pressure, and showing an economy in weight to be 
 derivable from high pressure as well as piston-speed for this 
 particular case that is, when the stroke and diameter of 
 the steam-cylinder are equal, and the diameter of the rod 
 does not exceed for wrought iron ^ and for steel of its 
 length. 
 
 If we multiply the results of formula (28), (29) or (30) 
 by y = 0.27, and by the factor representing the ratio of the 
 
 * This statement is not accurately true, as the piston-rod must be 
 somewhat longer than the stroke. Multiplying (27) by a factor of 
 from | to 2 would give an approximate result. 
 
OF THE STEAM-ENGINE. 39 
 
 total length of the piston-rod to the stroke, we obtain its 
 weight in pounds for a uniform pressure. 
 
 Referring to formula (15) or (20) for crushing or tearing, 
 and to formula (17) or (22) for rupture by buckling, we see 
 that the diameter of the piston-rod increases with the square 
 root of the pressure, or its area increases directly as the 
 pressure of the steam in the first case i. e., for crushing or 
 tearing and that the diameter of the piston-rod increases 
 with the fourth root of the steam-pressure, or its area with 
 the square root of the pressure, in the second case. 
 
 All of these formulae alike show that a rapid increase in 
 the boiler-pressure does not cause a correspondingly rapid 
 increase in the diameter of the rod, and explain the success 
 of empirical rules giving a constant ratio between the diam- 
 eters of piston-rods and steam-cylinders regardless of the 
 steam-pressure. 
 
 For the purpose of showing how little variation in the 
 diameters of piston-rods results from great changes in the 
 steam-pressure, the 'following short table of second and 
 fourth roots of the usual steam-pressures is given : 
 
 TABLE II. 
 
 Number. 
 
 Square root. 
 
 Fourth root. 
 
 25 
 50 
 75 
 100 
 125 
 
 5.00 
 7.07 
 8.66 
 10.00 
 11.18 
 
 2.236 
 2.659 
 2.949 
 3.163 
 3.344 
 
 Thus we see that for a pressure increased 5 times formulae 
 (15) and (20) increase the diameter of the rod but a little 
 over 2 times, and formulae (17) and (22) increase the diam- 
 eter of the rod but a little less than one-half. 
 
 Generally it will be found that the formula for crushing 
 (15) will give the greatest diameter for a wrought-iron 
 
40 THE RELATIVE PROPORTIONS 
 
 piston-rod, and that formula (22) will give the greatest 
 diameter of a steel piston-rod. 
 
 (15.) Keys and Gibs. Reserving keys for shafts for 
 discussion in Article (40), we will consider that form of 
 key used in making the connection between the piston-rod, 
 piston-head and cross-head, and also for the connecting-rod. 
 
 If we wish to avoid weakening the piston-rod at the point 
 where the key passes through it a precaution not found to 
 be practically necessary in most cases we must increase the 
 diameter of the rod at that point. 
 
 It is customary to thicken the rod where it enters the 
 piston-head, but for convenience not to do so where the 
 piston-rod enters the cross-head. If it is desired to thicken 
 the rod at both ends, it is best to make it of a uniformly 
 enlarged diameter from end to end. 
 
 !Let d' = the diameter of the enlarged end of rod in inches. 
 " di = the diameter of the piston-rod, as derived in Art. 
 (12), (13) or (14), in inches. 
 
 The average dimensions of keys are assumed as follows : 
 
 h = the breadth of the key = d f . 
 
 t = the thickness of the key = . 
 
 4 
 
 d' 9 
 Therefore, 2ht = , which nearly equals the cross-section 
 
 2i 
 
 of the enlarged end - - = 0.5354<r. 
 
 Bearing in mind that there are two shearing-sections for 
 every through-key, and assuming the shearing strength of 
 wrought iron equal to its tensile or compressive strength, as 
 is customary in practice. 
 
OF THE STEAM-ENGINE. 
 
 41 
 
 We can place, if the enlarged end of the rod weakened 
 by the key-way and the key are to be of equal strength, 
 
 444 
 
 Reducing, 2.141 6<T = 3.1416^. 
 Therefore, <T= 1.2114. 
 
 (31) 
 
 These various dimensions and measurements are shown in 
 Fig. 3. 
 
 FIG. 3. 
 
 If one or two gibs are used in connection with the wedge- 
 form of key, their mean area must be used in the calcula- 
 tions as for a single key. Various methods are used to 
 prevent keys from falling out, as split pins, set screws, and 
 bolts attached to one end of the key and held by a nut on 
 the gib. 
 
 (16.) Wrought-Iron Keys. The method given in the 
 preceding paragraph, although the usual one in practice, is 
 liable to cause some error in results. 
 
 4* 
 
42 THE RELATIVE PROPORTIONS 
 
 Let jP=the mean area of a key in square inches. 
 " K= the shearing strength (ultimate) of wrought iron 
 
 per square inch = 50000 pounds. 
 " P b = the boiler-pressure in pounds per square inch. 
 " d = the diameter of the steam-cylinder in inches. 
 " 10 - the factor of safety. 
 
 Kecollecting that every through-key has two shearing sur- 
 faces, we have 
 
 4 
 Therefore, F= 0.000078e* 2 P 6 . (32) 
 
 Or, if we assume the steam-pressure to be uniform through- 
 out the stroke, and take the length of stroke in inches, we 
 have 
 
 / i"T"rv\. 
 
 (33) 
 
 , 
 
 and placing 2F= --- , 
 
 we have ^-1.9 (34) 
 
 Example. Let P 6 =the boiler-pressure = the pressure 
 
 throughout the stroke = 40 
 pounds per square inch. 
 " d = 32 inches. 
 
 " L= 48 inches. 
 jy = 40 per minute. 
 " (J3P) = 156, approx. 
 From formula (32) we have 
 
 F= 0.000078 x 1024 x 40 = 3.19 square inches. 
 From formula (33) we have 
 
OF THE STEAM-ENGINE. 43 
 
 "1 E\ 
 
 .F=39.6 - = 3.21 square inches. 
 48x40 
 
 Therefore, from formula (34,), we have 
 d' - 1.9 /33 = 3.40 inches. 
 
 Comparing d' = 3.40 inches with d l in the example at the 
 end of Art. 12, we see that we are able to key the rod with- 
 out thickening it at either end, if we suppose the safe shear- 
 ing and tensile strength equal, and 5000 pounds per 
 square inch. 
 
 (17.) Steel Keys. Adopting the same notation as in 
 the preceding paragraph, we have only to change the value 
 of K. The shearing strength of steel is equal to three- 
 fourths of its tensile strength ; therefore, K= f 90000 = 
 67500 pounds per square inch area. 
 
 We have 2 x 6750 x JP= -dlP 6 . 
 
 4 
 
 Therefore", F= 0.000058d 2 P 6 . (35) 
 
 Or if we suppose the steam-pressure to be uniform through- 
 out the length of the stroke, and take the stroke in inches, 
 
 (36) 
 
 Comparing formulae (33) and (36), we have 
 
 For steel, F __ 29.33 . 
 For wrought iron, F~ 39.6 " 
 
 and therefore that the mean area of a steel key is 74 per 
 cent, of the mean area of a wrought-iron key. 
 
 Bearing in mind that the shearing strength of steel is but 
 three-fourths of its tensile strength, we can place Art. (15), 
 
44 THE RELATIVE PROPORTIONS 
 
 Therefore, d' = 1.68 yT. (37) 
 
 Example. Let P t = 40 pounds per square inch = uni- 
 
 form pressure. 
 " d =32 inches. 
 " L =48 inches. 
 " N = 40 per minute. 
 " (J7P) = 156, approx. 
 
 We have, from formula (35), 
 
 F= 0.000058 x 1024 x 40 = 2.37 square inches. 
 We have also, from formula (36), 
 
 F= 29.33^^ -- 2.37 square inches. 
 48x40 
 
 From formula (37) we have 
 
 d' = 1.68 /237 - 1.73 inches. 
 
 Comparing this latter result with the value of d l in the 
 example Art. 13 = 3.47, we find that no enlargement of the 
 ends of the rod is needed. 
 
 (18.) The Cross-Head. The cross-head or motion- 
 block, as it is sometimes called to one end of which the 
 piston-rod is keyed, and extending laterally from which are 
 the slides or slide, which by pressure upon the guides pre- 
 serve the rectilinear motion of the end of the piston-rod after 
 leaving the cylinder, and to the other end of which the con- 
 necting-rod is attached by means of a pin, usually of the 
 same diameter as the crank-pin, though not necessarily so, 
 has many different forms. 
 
 The proportions of the cross-head or motion-block are 
 with a few exceptions rather a matter of experience and 
 good taste than of calculation. Reuleaux, in Der Con- 
 
OF THE STEAM-ENGINE. 
 
 45 
 
 strueteur, page 546 et seq., gives many very good examples 
 of forms of cross-head. A few of the more common forms 
 
 are shown in Fig. 4, a, b and c, although the determination 
 
46 THE RELATIVE PROPORTIONS 
 
 of dimensions, rather than the suggestion of ingenious 
 arrangements, is the purpose of this work. Arthur Rigg, 
 in A Practical Treatise on the Steam-Engine, suggests (Plate 
 41) one or two good forms of cross-head. 
 
 The material used in the construction of cross-heads is, 
 according to circumstances, cast or wrought iron or steel. 
 
 Example a is a form of cross-head used for direct-acting 
 pumping and blowing engines, and its dimensions can be 
 calculated from the formula for a beam fixed at one end 
 and loaded at the other, given in Table VI., the load being 
 one-half the stress due to the steam-pressure upon the 
 piston-head. 
 
 Example b is a common form for engines having but two 
 guides. 
 
 Example c, sometimes called the slipper form, having but 
 one guide, is well adapted to" engines running in one direc- 
 tion only. 
 
 Many other forms of cross-head exist, and in fact almost 
 every new construction demands special adaptation of the 
 form of cross-head. 
 
 (19.) Area Of the Slides. When a horizontal engine 
 " throws over " while the piston-head is moving toward the 
 main shaft, all of the pressure and wear comes upon the lower 
 slide. If the motion of the engine be reversed, all of the 
 strain and wear comes upon the upper slide. 
 
 Since lubricants spread and flow over the lower guide 
 more easily than the upper, and since it is easier to resist a 
 strain in compression than tension by means of fastenings to 
 the bed-plate, it is customary and proper to cause engines 
 having motion in one direction only to throw over rather 
 than under. The slipper-guide (see Fig. 4, c) can be used 
 in this case with great propriety. 
 
 The greatest pressure upon a slide occurs when it is near 
 the middle of its stroke (assumed at the middle for conve- 
 
OF THE STEAM-ENGINE. 47 
 
 nience), and can be deduced with little trouble from the 
 triangle of forces. 
 
 Let S - the total steam-pressure on the piston-head in 
 pounds = -d?P b . 
 
 " Si = the pressure upon the guide in pounds. 
 
 / = the length of the connecting-rod in inches. 
 
 r ?= the length of the crank = in inches. 
 
 ft 
 
 n = - = the rates of the length of the connecting- 
 rod to the crank. 
 
 P b = the boiler-pressure in pounds per square inch. 
 d = the diameter of the steam-cylinder in inches. 
 L = the length of stroke in inches, 
 " (HP) = the indicated horse power. 
 
 Referring to Fig. 5, 
 
 FIG. 5. 
 
 we have S : $ : : T/ nV - r 2 : r. 
 
 Therefore, fl - S . (38) 
 
 Vn*-\ 
 
 The value of n commonly varies between 4 and 8. 
 For n = 4, formula (38) becomes 
 
 " 0.2582& 
 1/16-1 
 
48 THE RELATIVE PROPORTIONS 
 
 For n = 8, it becomes 
 
 4--=L=#-o.i2aa 
 
 1/64-1 
 
 If in this we substitute the value S=-d 2 P b , we have 
 
 4 
 
 3,0.7854gR (39) 
 
 I/ n 2 -! 
 
 Or, supposing the pressure per square inch, P 6 , to be uni 
 form throughout the stroke, we have 
 
 3%000_(flP) 
 
 "We have, then, from either equation (39) or (40) the 
 maximum pressure upon the slides. 
 
 One hundred and twenty-five pounds per square inch is 
 as high a pressure per square inch as should be used, and 
 the most modern English locomotive practice takes forty 
 pounds per square inch as the proper pressure, the prac- 
 tical result being a great diminution in the wear upon both 
 guides and slides. 
 
 Let A = the area of a slide in square inches. 
 
 " b = the pressure per square inch allowed. 
 Then will we have for the area of a slide, 
 
 Example. Let P 6 =P=40 pounds per square inch. 
 
 " d = 32 inches. 
 
 " N = 40 per minute. 
 
 " L =48 inches. 
 " (HP) = 156 horse-power. 
 " b = 1 25 pounds per square inch. 
 
 " n =5. 
 
OF THE STEAM-ENGINE. 49 
 
 Substituting in formula (40), we have 
 
 c 39600Q 156 
 
 01 - - - = 6568 pounds. 
 
 48x40 
 
 Substituting in formula (41), we have 
 
 6568 KOK 
 A = - = oz.o square inches. 
 
 CHAPTER IV. 
 
 (20.) Stress on and Dimensions of the Guides.* 
 
 The first requisite of a guide is that it shall be perfectly 
 rigid under all circumstances. .In many cases the guides 
 are so attached to the bed-plate or frame- work of the engine 
 as to require no calculation of their rigidity. 
 
 Cast iron is used for guides where they are firmly fastened 
 throughout their length to a bed-plate or framing. 
 
 Under other circumstances wrought iron or steel is to be 
 preferred as having greater moduli of elasticity, and conse- 
 quent rigidity. 
 
 When it is considered necessary to calculate the dimen- 
 sions of a guide, the following method will give a result 
 which is safe: 
 
 * Parallel motions, which take the place of guides in some engines, 
 are discussed in Willis's Principles of Mechanism, pp. 350-363. How 
 to Draw a Straight Line, by A. B. Kempe, is a particularly interest- 
 ing little book, giving all the later discoveries in parallel motions 
 which have followed the invention of Peaucellier's perfect parallel 
 motion. 
 
 5 
 
50 THE RELATIVE PROPORTIONS 
 
 Let Si = the stress upon the guide [see formulae (39), (40) 
 
 Art. (19)] in Ibs. 
 
 " /' = the length of guide in inches. 
 " JF=the measure of the moment of flexure of the 
 
 guide. 
 " E = the modulus of f For wro't iron = 28000000 Ibs. 
 
 elasticity, 1 For steel =30000000" 
 " a = the deflection in inches. 
 
 We have (Weisbach's Mechanics of Engineering, Sec. iv., 
 Art. 217), for a beam supported at both ends and loaded in 
 the middle, 
 
 m^L'-SE^ (42) 
 
 48 WE 
 
 Since perfect rigidity is unattainable, let us concede a 
 deflection, a = ^ of an inch, and formula (42) becomes for 
 
 wrought iron, 
 
 -- T_ 
 
 13440000 
 Assuming a rectangular cross-section for the guide, 
 
 Let b = the breadth in inches. 
 " h = the depth in inches. 
 
 Then W=^-, 
 12 
 
 and formula (43) becomes 
 
 
 134400006 
 
 Substituting the value of $, derived from equation (39), 
 and extracting the cube root, (44) becomes 
 
 (45) 
 
OF THE STEAM-ENGINE. \ 
 
 and substituting the value of S l from equation (40), 
 
 k - Mvnr %ii=*' (46) 
 
 Example. Let t = 60 inches. 
 d =32 inches. 
 L =48 inches. 
 P b =P = 40 Ibs. per square inch. 
 
 6=4 inches. 
 
 " (HP) = 156 indicated horse-powers. 
 N = 40 per minute. 
 n = 5. 
 
 Using formula (45), we have 
 
 h = 0.00889 x 60 x p= = 6.82 inches. 
 Mv/25-JL 
 
 Using formula (46), we have 
 
 3 I I K> 
 
 h = 0.7071 x 60 X - - - 6.82 inches. 
 
 v 4x48x40^/25-1 
 
 (21.) Distance between Guides. It is important to 
 know at what angle of the crank the connecting-rod requires 
 the greatest distance between the guides, if the plane of 
 vibration of the connecting-rod intersects them. 
 
 We can then determine the least possible distance be- 
 tween the guides, or that position of the connecting-rod 
 which, clearing all parts of the engine, will make it impos- 
 sible for it to touch with the crank at a different angle. 
 
 Solution. (Approximate.) 
 
 Let r = radius of crank = CB. 
 " l = nr = length of connecting-rod = AB. 
 
 We have EFiBD:: [2r - <1 - cos a)] : j/F - r* sin 2 a. 
 
52 TEE RELATIVE PROPORTIONS 
 
 FIG. 6. 
 
 Let EF=x, we have BD = r sin a, 
 
 r 2 sin a (1 + cos a) 
 then x = - v ^, 
 
 rj/Ti 2 sin 2 a 
 and neglecting for the present sin" a in the denominator, 
 
 we have x = - sin a (1 + cos a). 
 
 n 
 
 But sin a = 2 sin - cos and (1 + cos a) = 2 cos 2 -, which 
 22 2 
 
 gives a; = sin a cos 3 -. Differentiating, we have 
 
 dx 4r r . a a a i 
 
 3 sin 2 - cos 2 - + cos 4 - , 
 da n L 22 2J 
 
 and placing this equal to and dividing by cos 2 -, 
 
 2 
 
 3 sin 2 ?- cos 2 J 
 or tan 2 = 
 
 tan -!/$-. 578, 
 
 ^ = 30 and a = 60 approximately. 
 z 
 
 Showing that when the crank forms an angle of 60 with 
 the centre line of the cylinder, we have the maximum dis- 
 
OF THE STEAM-ENGINE. 53 
 
 tance from that centre line required to make the centre line 
 of the connecting-rod clear the guides, 
 
 1.5x0.866 1.3r 
 
 ____ 
 
 -I/V-. 
 
 To get the whole distance between the guides we multi- 
 ply by 2, giving &ff=2.6-. 
 n 
 
 To this value must be added the thickness of the connect- 
 ing-rod. For any point, as K, on the connecting-rod, we 
 have, knowing the angle a = 60, the proportion, 
 
 AL :AD::KL: BD, 
 ^ T 
 
 to which we must add the half thickness of the connecting- 
 rod and multiply by 2 to get the whole distance between 
 the guides. 
 
 It must be borne in mind, when arranging the guides, 
 that room must be made for the introduction of the key 
 into the stub end of the connecting-rod, if that end cannot 
 be slid outside of the guides at one end of the stroke. 
 
 (22.) The Connecting-Rod. The connecting-rod of a 
 steam-engine is usually made from 4 to 8 times the length 
 of the crank that is, 2 to 4 times the length of the stroke 
 of the steam-cylinder. 
 
 FIG. 5. 
 
 Referring to Fig. 5, we see that when the crank is at 
 5* 
 
54 THE RELATIVE PROPORTIONS 
 
 right angles to the centre line of the piston-rod, the strain 
 upon the connecting-rod is a maximum. 
 
 Let $ = the total steam -pressure upon the piston-head 
 
 in pounds. 
 
 " $2 = the stress upon the connecting-rod in pounds. 
 " / = the length of the connecting-rod in inches. 
 " r = the radius of the crank. 
 
 " n = - = the ratio of the length of the connecting-rod 
 
 to the crank. 
 We have S : S 3 : : i/n 2 r* - r 2 : nr. 
 
 Therefore, Sz = S . (47) 
 
 If in (47) we let n = 4, we have 
 
 &-fl 4 =1.0328& 
 
 If we let n = 8, we have 
 
 S, = S - =1.0008& 
 t/64-1 
 
 These numerical results show the rapidity with which the 
 value of the maximum stress on the connecting-rod ap- 
 proaches the constant stress upon the piston-rod as the ratio 
 of the connecting-rod to the crank is increased, and further 
 by what a small percentage = .03 the stress upon the con- 
 necting-rod is greater in the extreme case for the value 
 of n = 4. 
 
 A relatively short connecting-rod is productive of econ- 
 omy of material, and the increased pressure upon the sides 
 can be provided for by increased area. (See Art. 19.) 
 
 The connecting-rod, being free to turn about its pins at 
 
OF THE STEAM-ENGINE. 55 
 
 either end, must be regarded as a solid column not fixed at 
 either end, but neither end free to move sideways. 
 
 To determine the point at which a column of this cha- 
 racter has an equal tendency to rupture by crushing or 
 buckling, we place the formulae for crushing and buckling 
 equal to each other (Weisbach's Mechanics of Engineering, 
 sec. iv., art. 266), and letting d 2 = the diameter of the con- 
 necting-rod, 
 
 or, substituting the values of jP and W, 
 
 I \~E 
 
 which gives = .7854-* / : . 
 j \ K 
 
 I /2800000(> 
 
 For wrought iron, - .7854^-^^- - 23J, (48) 
 
 and we see that a wrought-iron column will rupture by 
 crushing when its diameter is greater than Jj of its length, 
 and will rupture by buckling when its diameter is less than 
 % of its length, and that a steel column will rupture by 
 crushing when its diameter is greater than y 1 ^ of its length, 
 and will rupture by buckling when its diameter is less than 
 T V of its length. 
 
 (23.) Wrought-iron Connecting -Rod. Let ^=the 
 cross-section of the rod in square inches. 
 
 Let JfT=the ultimate crushing strength of wrought iron 
 per square inch. 
 
56 THE RELATIVE PROPORTIONS 
 
 Referring to formula (47), we have for crushing, 
 
 n (50) 
 
 Let P 6 = the boiler-pressure in pounds per square inch. 
 " d 2 = the diameter of the connecting-rod in inches. 
 " d = the diameter of the steam-cylinder in inches. 
 " the factor of safety be 10, as before. 
 
 Then formula (50) becomes 
 
 * 
 
 4 
 
 ^31000. 
 
 Therefore, ^ = 0.0179^^x1 -7^7- (51) 
 
 ^ rr 1 
 
 In this formula for n = 4 we have 
 
 and as the value of n increases this quantity becomes more 
 and more nearly equal to unity, and can therefore be neg- 
 lected in all cases in which n = 4 or a greater number. 
 
 Formula (51) then becomes 
 
 d t = 0.0179 rfv/F,. (52) 
 
 Let (-HP) = the indicated horse-power. 
 " L = the length of the stroke in inches. 
 " N = the number of strokes per minute. 
 
 Formula (52) becomes, in terms of the horse-power, 
 
 ^ = 12.753^^. (53) 
 
 The formula for rupture by buckling for long solid 
 
OF THE STEAM-ENGINE. 57 
 
 columns, not fixed at either end, is Weisbach's Mechanics of 
 Engineering, sec. iv., art. 266. 
 
 Letting W= the measure of the moment of flexure, 
 
 " E = the modulus of elasticity = 28000000 pounds 
 per square inch, 
 
 I = the length of the rod in inches = nr n , 
 
 (54) 
 
 Substituting in this the values of $ = n & W= 
 
 and .E, we have, with a factor of safety, 10, 
 
 r 2800000 - 
 z 64 
 
 n 
 Reducing and neglecting the term */ - - , we have 
 
 <Z 2 - 0.02758v / ^P 6 inches. (55) 
 
 If in this we substitute the value of I n , we have 
 
 d 2 - 0.01 95 v'nWET* inches. (56) 
 
 If L-d 9 (56) becomes 
 
 d z = 0.0195rf|/ / n 5 P 6 inches. (57) 
 
 If we suppose the pressure of the steam to be uniform 
 throughout the stroke, formula (56) becomes, in terms of the 
 horse-power, 
 
 (58) 
 
58 THE RELATIVE PROPORTIONS 
 
 The following rule may be given for the determination of 
 the diameters of round wrought-iron connecting-rods : 
 
 Deduce the diameter of the connecting-rod by the use of 
 formula (51) or (53), as may be most convenient. Should this 
 diameter be less than -% of the length of the rod, then use 
 formula (55), (56), (57) or (58), as may be most convenient. 
 
 Example. Let L = 48 inches. 
 " d =32 inches. 
 
 n =5. 
 
 " P b = P= 40 pounds per square inch. 
 " N = 40 per minute. 
 " (-HP) = 156 approximately. 
 
 Substituting in formula (52), we have 
 
 d 2 = 0.0179 x 32^40 - 3.62 inches, 
 or substituting in formula (53), we have 
 
 4 = 12.753 r~T5T~ 
 
 / = 3.63 inches. 
 
 \48x40 
 
 48 
 
 Since the length of the connecting-rod = x 5 = 120 
 
 2i 
 
 inches, we see that its diameter is less than -fa of its length, 
 and that we must use formula (56) or (58). 
 Substituting in (56), we have 
 
 d t = 0.0195]/-25 x 1024 x 2304 x 40 = 4.30 inches. 
 Substituting in formula (58), we have 
 
 40 
 
OF THE STEAM-ENGINE. 59 
 
 CHAPTER V. 
 
 (24.) Steel Connecting-Rod. The same course of rea- 
 soning as that followed in Art. (13) gives us, for the proper 
 diameter of a steel piston-rod to resist safely a strain in 
 tension, 
 
 d, = 0.0105(VT 6 inches, (59) 
 
 which is the same as formula (20), or 
 
 rf, - 7.481^|p inches, (60) 
 
 LN 
 
 which is the same as formula (21) for a constant pressure 
 of the steam throughout the length of the stroke. 
 
 Considering next the strength of a. connecting-rod to re- 
 sist rupture by buckling, and letting .#=42000000 pounds 
 per square inch for steel in formula (54), substituting in a 
 similar manner to that in Art. (23) and reducing, we have 
 
 ^42000000. 
 I- 64 
 
 8 I ~ 
 
 Therefore, neglecting the term 
 
 
 , 
 = 0.02492 v 4/ dWV (61) 
 
 Substituting l = nr = n , we have 
 2 
 
 c/ f - 0.0176 lT&Fi (62) 
 If in (62) we make 
 
 n = 4, or I = 2 A we have d, - 0.0352 ytflfP* ; (63) 
 
 n = 5, or I = 2J A " rf, = 0.0394 ^UP b ; (64) 
 
 n - 6, or J - 3 A " ^ 2 = 0.0432 ^"OTJ^ ; (65) 
 
 n = 7, or I = 3 J L, " ^ 2 - 0.0466 ^"OTTA ; (66) 
 
 w = 8, or I - 4A " ^ - 0.0496 y 7 ^!^ . (67) 
 
60 THE RELATIVE PROPORTIONS 
 
 81 -^3 
 
 It must be remembered that the term -i / ;=r is neglect- 
 
 V l/w 2 -l 
 
 ed in these formulae, and that it should be taken into consid- 
 eration when n is less than 4. 
 
 If in the above formulae we let L = d, the radical reduces 
 to the form d j/VP 6 . 
 
 If we consider the steam-pressure uniform throughout the 
 stroke, and substitute for P^Ld* in (62) its value in terms 
 of the horse-power, we have 
 
 = 0.469^^^. (68) 
 
 N 
 
 From their experiments upon steel, Kupffer and Styffe 
 have arrived at the conclusion that the percentage of carbon 
 has no effect upon the modulus of elasticity of steel, and 
 give as its value E= 30000000 per square inch. 
 
 In the present discussion we have assumed the value of 
 E= 42000000, as given by Reuleaux, and the ultimate 
 strength to resist crushing ^=100000 pounds per square 
 inch a value far less than would be considered safe from 
 the experiments of W. Fairbairn on the mechanical prop- 
 erties of steel. (Report of the British Association for 1867, 
 or Stoney's Theory of Strains, vol. ii., page 480.) 
 
 It is more than probable, from a consideration of these 
 facts, that the ratio of I to d* should be much less than 16, 
 and require the use in all cases of formulae (61) to (68), 
 inclusive, for the determination of the diameter of steel 
 connecting-rods. 
 
 Comparing formula (22) for rupture by buckling of steel 
 piston-rods with formulae (63) to (67), inclusive, for rupture 
 of steel connecting-rod, we find that when 
 
OF THE STEAM-ENGINE. 61 
 
 (69) n = 4, the diameter of the connecting-rod = 1.00 times 
 
 the diameter of the piston-rod ; 
 
 (70) n = 5, the diameter of the connecting-rod = 1.12 times 
 
 the diameter of the piston-rod ; 
 
 (71) n = 6, the diameter of the connecting-rod = 1.28 times 
 
 the diameter of the piston-rod ; 
 
 (72) n = 7, the diameter of the connecting-rod = 1.30 times 
 
 the diameter of the piston-rod ; 
 
 (73) n = 8, the diameter of the connecting-rod = 1.44 times 
 
 the diameter of the piston-rod. 
 
 If the length of the connecting-rod be taken = twice the 
 stroke, the diameters of piston- and connecting-rod are equal. 
 
 The following rule may be given for the determination 
 of the diameters of round steel connecting-rods : 
 
 Deduce the diameter of the connecting-rod from formula 
 (59*) or (60), as may be most convenient. If the resulting 
 diameter be less than one-sixteenth of the length of the rod, 
 then use some one of the formulas (61) to (73*), inclusive. 
 
 Example. Let L =48 inches. 
 
 " d =32 inches. 
 
 n = 5 inches. 
 
 P 6 = P=40 pounds per square inch. 
 . " N = 40 per minute. 
 " (.HP) = 156. 
 
 Substituting these values in formula (59), we have 
 
 d 2 = 0.0105 x 32 1/40 = 2.12 inches, 
 or, substituting in formula (6.0), we have 
 
 d, = 7.481 -- = 2.13 inches. 
 \48x40 
 
 jJg- of 24x5 = 120 = 7^ inches, and we see that we must 
 use one formula, (61) to (73). 
 
62 THE RELATIVE PROPORTIONS 
 
 Substituting in formula (64), we have for n = 5 
 
 d. 2 = 0.0394 y' 1024x2304^40 = 3.88 inches ; 
 or, substituting in formula (68), we have 
 
 c? 2 = 0.469 *! - = 3.89 inches. 
 
 Referring to example for Art. 13, we find the diameter of 
 a steel piston-rod to be d l = 3.47 inches, and substituting in 
 formula (70), we have 
 
 d, = 1.124 = 3.47 x 1.12 = 3.89 inches. 
 
 (25.) General Remarks concerning Connecting- 
 
 Rods. The discussion of Art. (14) will apply with little 
 modification to connecting-rods.* 
 
 If we take formula (27), as before, for the volume of a 
 connecting-rod, we have 
 
 "-(?) () 
 
 Substituting in this the value of d 2 from equation (68), 
 we have, letting C= constant, 
 
 Reducing V, = W^^-. (76) 
 
 Equation (76) shows that the volume of the connecting- 
 rod is affected by the varying conditions in a similar man- 
 ner to a piston-rod, excepting that the square of the ratio n 
 enters in and affects the volume of the rod. 
 
 * It is customary to make round connecting-rods with a taper of 
 about one-eighth of an inch per foot from the centre to the necks, 
 which should be of the calculated diameter. Experiment does not 
 show an increased strength from a tapering form. 
 
OF THE STEAM-ENGINE. 63 
 
 Thus, if we first take n = 4, and then n = 8, we see that 
 the volume in the first case is but J of the volume in the 
 latter case. 
 
 The increase of the area of the slides to provide for 
 shorter connecting-rods is quite slow (see Art. 19). It is 
 customary to make connecting and parallel rods of a rectan- 
 gular cross-section in locomotive practice. When this is 
 done it will be safe to make the smaller of the rectangular 
 dimensions equal to the diameter of a round rod suitable 
 to withstand the strains to which it will be subjected. The 
 resistance of a long rectangular column to rupture varies as 
 the cube of its smallest dimension of cross-section. (See 
 Weisbach's Mechanics of Engineering, sec. iv., art. 266.) 
 
 (26.) Connecting-Rod Straps. By means of gibs and 
 keys the straps draw the brasses solidly against the stub-end 
 of the connecting-rod. 
 
 If necessary the uniform cross-section of the straps can 
 be preserved by thickening them- at the point where they 
 are slotted to receive the gib and key. (See Art. 15.) 
 
 Fig. 7 a shows the usual form of strap for locomotives, 
 and Fig. 7 b a solid stub-end in which the keys are used 
 only to set the brasses without moving the strap. Fig. 7 c 
 represents the ordinary form of strap in which the brasses 
 and strap are held by means of the gib and key. 
 
 (27.) Wrought-Iron Strap. 
 
 Let FI = the area of one leg of the strap in square inches. 
 " the safe strain per square inch = 5000 pounds. 
 " d = the diameter of the steam-cylinder in inches. 
 " P b = the pressure per square inch of the steam. 
 
 We have 2 x 5000^ = 0.7854P 6 d 2 ; 
 
 therefore F l = 0.000078P 6 d 2 square inches. (77) 
 
64 THE RELATIVE PROPORTIONS 
 
 or if we assume the pressure to be uniform, and 
 
 Let L = the length of the stroke in inches, 
 .Af=the number of strokes per minute, 
 " (.HP) = the horse-power (indicated), 
 
 FI = 39.6^^ square inches. (78) 
 
 LN 
 
OF THE STEAM-ENGINE. 65 
 
 Example. Let d = 32 inches. 
 L = 48 inches. 
 N= 40 per minute. 
 P 6 = P = 40 pounds per square inch. 
 " (HP) =156 indicated horse-powers. 
 
 Substituting in formula (77), we have 
 
 F l - 0.000078 x 40 x 1024 = 3.19 square inches, 
 or substituting in formula (78), we have 
 
 F, = 39.6 156 = 3.21 square inches. 
 48 x 40 
 
 (28.) Steel Strap. Let 9000 pounds per square inch 
 equal the safe working-strain in tension. 
 
 Let F l = the area of one leg of the strap in sq. inches. 
 d = the diameter of the steam-cylinder in inches. 
 P b =- the steam-pressure per square inch. 
 L = the length of stroke in inches. 
 JV=the number of strokes per minute. 
 " (-HP) = the indicated horse-power. 
 
 We have, as in the preceding article, 
 
 2x9000^ = 0.7854^, 
 therefore P x = 0.0000437d 2 P 6 square inches, (79) 
 
 or assuming the stream-pressure to be uniform throughout 
 the stroke, and substituting for d?P b its value in terms of 
 the horse-power, 
 
 F, = 22.034^2 square inches. (80) 
 
 Dividing formula (80) by formula (78), we find that the 
 cross-section of a steel strap is but -J of the cross-section of 
 
 a wrought-iron strap of equal strength. 
 6* 
 
66 THE RELATIVE PROPORTIONS 
 
 Example. Let the data be the same as in the example 
 appended to the preceding article. 
 Substituting in formula (79), we have 
 
 Jl = 0.0000437 x 1024 x 40 = 1.77 square inches. 
 
 Substituting in formula (80), we have 
 "i r^^? 
 
 Fx - 22.034 -4- - 1.78 square inches. 
 48x40 
 
 CHAPTER VI. 
 
 (29.) The Crank-Pin and Boxes. The crank-pin has 
 ever been one of the most troublesome parts of the steam- 
 engine to the mechanical engineer. The mere determination 
 of its proportions, so that it will not break under the strain 
 put upon it by the pressure of the steam upon the piston- 
 head, does not suffice, and often results in trouble from 
 heating when the engine is at work. It therefore becomes 
 a first consideration to so proportion crank-pins as to pre- 
 vent heating, their strength being a matter of secondary 
 importance, to be afterward investigated if it is deemed 
 necessary to do it. 
 
 Before taking up the mathematical part of our consid- 
 eration, it will be of practical value to quote, from the 
 writings of General Morin, the following remarks : 
 
 " But it is proper to observe that from the form itself of 
 the rubbing body (cylindrical) the pressure is exerted upon 
 a less extent of surface according to the smallness of the 
 diameter of the journal, and that unguents are more easily 
 expelled with small than with large journals. This circum- 
 stance has a great influence upon the intensity of friction, 
 and upon the value of its ratio to the pressure. 
 
OF THE STEAM-ENGINE. 67 
 
 " The motion of rotation tends of itself to expel certain 
 unguents and to bring the surfaces to a simply unctuous 
 state. The old mode of greasing, still used in many cases, 
 consisted simply in turning on the oil or spreading the lard 
 or tallow upon the surface of the rubbing, and in renewing 
 the operation several times in a day. 
 
 " We may thus, with care, prevent the rapid wear of 
 journals and their boxes ; but with an imperfect renewal 
 of the unguent, the friction may attain .07, .08, or even .10, 
 of the pressure. 
 
 " If, on the other hand, we use contrivances which renew 
 the unguent, without cessation, in sufficient quantities, the 
 rubbing surfaces are maintained in a perfect and constant 
 state of lubrication, and the friction falls as low as .05 or 
 .03 of the pressure, and probably still lower. 
 
 " The polished surfaces operated in these favorable con- 
 ditions became more and more perfect, and it is not sur- 
 prising that th'e friction should fall far below the limits 
 above indicated." (Bennett's Morin, pp. 307, 308.) 
 
 If the unguents are expelled by extreme pressure, so that 
 the surfaces are simply unctuous, the friction increases rap- 
 idly, and the surfaces begin to heat and wear immediately. 
 
 These statements apply with equal force to cast iron and 
 cast iron ; cast iron and wrought iron ; cast iron and brass 
 or Babbitt's metal; or with steel or wrought iron in the 
 place of cast iron. 
 
 The supposed superiority of brass or Babbitt's metal lined 
 boxes over iron boxes in positions very liable to heating 
 lies in their greater softness and conductivity for heat. 
 Brass will conduct heat away from two to four times as 
 rapidly as iron. However, the film of unguent interposed 
 may render the conductivity of brass of less avail than is 
 generally supposed, and the advantage lies only in the fact 
 that, being a softer metal, in case of heating, the surface of 
 
68 THE RELATIVE PROPORTIONS 
 
 the softer metal receives the principal damage. Phosphor 
 bronze, which is a patented alloy, being nothing more than 
 brass or gun-metal in which the formation of oxide has been 
 prevented by the introduction of phosphorus, is coming into 
 general use for positions in which the wear is very great. 
 
 It is, perhaps, good practice to use brass or soft metal 
 wherever the pressure exceeds 125 pounds per square inch 
 of projected area. At lower pressures a good lubricating oil 
 may be relied upon to form a film and run without breaking 
 at ordinary speeds. (The continuity of the film of lubricant 
 is affected by so many different conditions that it is impos- 
 sible to fix any exact limit of pressure.) 
 
 With soft metal or brass bearings good results can be ob- 
 tained at pressures of 1000 pounds or more per square inch 
 of projected area. (See Hand-Book of the Steam-Engine, 
 Bourne, page 183, where 1400 pounds per square inch is 
 given as the greatest pressure per square inch of projected 
 area allowable on crank-pins. Arthur Kigg, A Practical 
 Treatise on the Steam-Engine, page 147.) If, however, the 
 film of unguent does break at these higher pressures, heat- 
 ing begins almost instantly ; and if the surfaces in contact 
 are both of hard metal, as iron and iron, injury to both at 
 once results, while, if the boxes are brass or some of the 
 softer metals, the continuity of the surface film may be re- 
 stored by increased lubrication or by stopping and cooling 
 as soon as heating is observed. 
 
 Several expedients are used to keep bearings which have 
 a tendency to heat cool until they have worn smooth. 
 
 The introduction of rotten stone or sulphur with oil is 
 perhaps the best. Quicksilver or lead-filings, introduced 
 with oil, coat the rubbing surfaces and diminish the heat- 
 ing where the rubbing surfaces are very much scored. (See 
 The Working Engineer's Practical Guide, pages 48 and 49, 
 Joseph Hopkinson.) 
 
OF THE STEAM-ENGINE. 
 
 69 
 
 A great increase of the velocity of the rubbing surfaces 
 renders bearings more liable to heat than a great increase 
 in pressure, although the total amount of work done by 
 friction is the same in both cases, and is probably accounted 
 for by the more rapid expulsion of the lubricant. 
 
 As the cause of the heating of bearings, when they are of 
 tolerably good workmanship, is the transformation of the 
 work of friction into heat, we see that it is necessary to 
 reduce the friction as much as possible by the perfect 
 smoothness of surfaces in contact, the interposition of lubri- 
 cants, and the reduction of the speed and pressure upon the 
 rubbing surfaces. 
 
 In all machines there is a limit below which we cannot 
 reduce the speed and pressure of the rubbing surfaces, and 
 we must, therefore, so proportion journal-bearings as to 
 cause no more work due to friction i. e., heat to be pro- 
 duced than can be conveyed away by the unguents, the 
 atmosphere and the conductivity of the metals without 
 raising the temperature of the bearing appreciably. 
 
 From the statistics of the working of the crank-pins of 
 four screw propellers in the United States Navy (Van 
 Buren, Strength of Iron Parts of Steam Machinery, page 24) 
 we take the following statement and table : 
 
 " The crank-pins of these vessels worked cool, giving but 
 little trouble, which is the ex- 
 ception rather than the rule for 
 screw-engines." 
 
 The projected area of crank- 
 pin journal, given in column 7, 
 is that rectangular area formed 
 by a central section of the crank-pin journal in the direc- 
 tion of its length. Shown cross-hatched in Fig. 8. 
 
 Columns 1, 2, 4, 5 and 6 are given. Columns 3, 7, 8, 9, 
 10 and 11 are calculated from them. 
 
 FIG. 8. 
 
70 
 
 THE RELATIVE PROPORTIONS 
 
 In calculating column 9 from columns 3 and 8 we have 
 assumed the coefficient of friction at .05, which is the high- 
 est value given by General Morin for constant lubrication, 
 and probably greater in the present cases.* 
 
 Column 11 is derived from columns 3 and 7. Column 
 10 is derived from columns 7 and 9. 
 
 TABLE HI. 
 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 
 | 
 
 2 
 
 11 
 
 1 
 
 
 d 
 
 Q, 
 
 s 
 
 t. 
 2.2 
 
 j. 
 
 Hi 
 
 . 
 
 
 a 
 
 a 
 
 % 
 
 1 
 
 P. 
 
 
 
 1 
 
 s S 
 
 C.-C S 
 
 If 
 
 
 1 
 
 t 
 
 
 I 
 
 
 
 a 
 
 ^ 
 
 u 
 
 S| 
 
 f? 
 
 
 NAME OP VESSEL. 
 
 Diameter of 
 inder. 
 
 Pressure per 
 inch. 
 
 h 
 111 
 
 Numbur of st 
 minute. 
 
 Length of era 
 
 1 
 
 IP rejected aret 
 pin. 
 
 11 
 
 Total work of 
 foot-pounds 
 
 Work of frict 
 inch of proj 
 in foot-pound 
 
 i 
 
 2, 
 
 
 in. 
 
 Ibs. 
 
 Ibs. 
 
 
 in. 
 
 in. 
 
 sq. in. 
 
 
 
 
 
 Swatara. 
 
 36 
 
 40 
 
 40716 
 
 160 
 
 12 
 
 8.5 
 
 102. 
 
 178. 
 
 362372 
 
 3552.6 
 
 399.5 
 
 Saco 
 
 30 
 
 40 
 
 28274 
 
 180 
 
 9 
 
 75 
 
 67.5 
 
 176.7 
 
 249801 
 
 3700.7 
 
 419. 
 
 Wampanoag... 
 
 100 
 
 40 
 
 314160 
 
 62 
 
 27 
 
 16. 
 
 432. 
 
 129.8 
 
 2038898 
 
 4719.7 
 
 727.2 
 
 Wabash 
 
 72 
 
 28 
 
 114002 
 
 100 
 
 16 
 
 15. 
 
 240. 
 
 196.3 
 
 1118910 
 
 4662.1 
 
 475. 
 
 Avera 
 
 
 4159. 
 
 505. 
 
 
 From column 10 of the table we find the average amount 
 of work per square inch of projected area of crank-pin 
 journal, which, in the cases cited, has been borne without 
 heating, to be 4159 foot-pounds per minute; and in making 
 use of this quantity in our subsequent calculations, we are 
 on the safe side if the coefficient of friction (assumed at .05) 
 has not been taken too small. 
 
 (30.) The Length of Crank-Pins. Let d = the diam- 
 eter of the piston-head in inches, P=the mean pressure in 
 the steam-cylinder in pounds per square inch. 
 
 * It is probable that the coefficient of friction, for crank-pins of 
 marine propellor-engines under ordinary conditions, is 9 or 10 times 
 greater than the assumed 0.05. 
 
OF THE STEAM-ENGINE. 71 
 
 Then 7854cPP = the mean pressure on the piston-head in 
 pounds. 
 
 Let / = the coefficient of friction. 
 " 4 = the length of the crank-pin journal in inches. 
 " d a = the diameter of the crank-pin journal in inches. 
 
 The mean force of friction at the rubbing surfaces of any 
 crank-pin journal per square inch of projected area is 
 
 Let jY=the number of strokes per minute (equal twice 
 
 the number of revolutions). 
 " w = the work of friction per minute. 
 The space passed over by the force due to friction in one 
 minute 
 
 1.5708 Nd 3 inches, 
 
 and we have for the work of friction i. e. } heat per minute, 
 
 w = 1.5708 x .7854/- inch-pounds. 
 4 
 
 From this formula the diameter of the crank-pin journal 
 has vanished. Why it has vanished will be understood 
 when we observe that the force per square inch of project- 
 ed area due to friction is inversely as the diameter of the 
 journal ; while the space passed over by this force is directly 
 as its diameter. 
 
 Replacing w in the last formula by the mean value de- 
 rived from column 10 of the table, equal 4159 foot-pounds 
 equal 49908 inch-pounds, we have 
 
 49908 = 1.2337/^^ 2 . 
 4 
 
 Therefore, 4 = .0000247 /P^cf = 12.454/^^. (81) 
 
72 THE RELATIVE PROPORTIONS 
 
 Considering formula (81), we see that the length of the 
 crank-pin increases and decreases with the coefficient of 
 friction, the mean steam-pressure per square inch the num- 
 ber of strokes per minute, and with the square of the diam- 
 eter of the steam-cylinder. 
 
 A consideration of the component formulae of (81) shows 
 that as the crank- pin journal decreases in size the pressure 
 per square inch becomes greater ; but if this reduction in 
 size is obtained by a diminution of the diameter (d s ) of the 
 crank-pin journal, the work per square inch of projected 
 area is not increased, for the velocity of the rubbing sur- 
 faces by this means is decreased in the same ratio as the 
 pressure is increased. 
 
 Within reasonable limits as to pressure and speed of rub- 
 bing surfaces, the general law may be enunciated : 
 
 The longer any bearing which has a given number of 
 revolutions and a given pressure to sustain is made, the 
 cooler it will work, and its diameter has no effect upon its 
 heating. 
 
 Example. Letd = 30", JV=180, and P=40 pounds per 
 square inch. We have, by substitution, in formula (81), 
 
 / 3 - .0000247 x/x 40 x 180 x 900 - 160. /. 
 If in this we take/=.03 to .05 for perfect lubrication, 
 we have 4 = 4.8" to 8". 
 
 If we take/=.08 to .10 for imperfect lubrication, 
 we have 4 = 12.8" to 16". 
 
 The results show the great advantages arising from con- 
 stant oiling of bearings and smoothness of surfaces. 
 
 NOTE. If 0.05 be taken as the coefficient of the force of friction, 
 we obtain the average length of the crank-pins quoted in Table III., 
 
OF THE STEAM-ENGINE. 73 
 
 Art. (29). About one-quarter of the length required for propellor 
 crank-pins will serve for the pins of side-wheel engines with good 
 results, and one-tenth for locomotive or stationary engines. 
 
 (31.) Locomotive Crank-Pins, Length and Diameter. 
 
 If for locomotive crank-pin journals we assume N, the 
 number of strokes per minute = 600. 
 
 P the pressure per square inch in pounds = 150, the form- 
 ula (81) being changed to 
 
 by removing the decimal point one place to the left, will 
 give the length of journal commonly assumed in successful 
 practice, if we assume the coefficient of friction at .06. 
 The above formula then becomes 
 
 4 = .013d 2 . (82) 
 
 This formula would prove the amount of heat per square 
 inch of projected area conveyed away from the crank-pins 
 of locomotives to be ten times greater than in the case of 
 marine engines, did not the variations of speed and frequent 
 stoppages of a locomotive prevent comparison. 
 
 Example. Let d = 18 inches. We have 
 It = .013x324 = 4.21 inches. 
 
 The diameters of locomotive crank-pins are usually taken 
 equal to their length. 
 7 
 
74 THE RELATIVE PROPORTIONS 
 
 CHAPTER VII. 
 
 (32.) Diameter of a Wrought-Iron Crank-Pin for a 
 Single Crank.* It is necessary, first, to determine its 
 length by formula (81), and with this length to determine 
 the proper diameter. 
 
 Let a = the deflection of pin under stress in inches. 
 " $ = the stress on pin in pounds. 
 " E = the modulus of elasticity of wrought iron = 28000000 
 
 pounds. 
 
 " JF= the measure of the moment of flexure of the pin. 
 " 1 3 = the length of journal in inches. 
 
 The deflection of a beam fixed at. one end and loaded at 
 the other (Weisbach's Mechanics of Engineering y sec. iv., 
 art. 217) is 
 
 SP 
 
 ZWE 
 
 If, again, the beam be supposed to be uniformly loaded 
 and fixed at one end, its deflection will be (Weisbach's 
 Mechanics of Engineering , sec. iv., art. 223) 
 
 _ 
 = 
 
 and if for the load at the end we concede a deflection of 
 j-J-g- of an inch, we have for the same load under the two 
 cases above mentioned 
 
 <Zi = .01 inch, 
 2 = .0038 inch. 
 
 * In Arts. 54 and 55 will be found a discussion of the stresses on 
 crank-pins for double and triple cranks. 
 
OF THE STEAM-ENGINE. 75 
 
 Then, taking the most unfavorable case i. e., the load at 
 the end we have, 
 
 TT d 2 
 letting S= P 6 , 
 
 letting P 6 = the greatest pressure of steam in cylinder equal 
 the boiler-pressure, 
 
 letting W-^' 
 
 o4 
 
 "nnr = "g^TTI 
 
 64 
 
 , (84) 
 for a constant steam-pressure. 
 
 Example. Let P 6 = 60 pounds per square inch. 
 
 " / 3 = 8 inches. 
 
 " d = 30 inches. 
 Substituting in formula (84), we have 
 
 d 3 = .066|/' 60x51 2x900 = 4.79 inches. 
 
 There is no need of an investigation of the strength of a 
 crank-pin, as the condition of rigidity gives a great excess 
 of strength. 
 
 (33.) Steel Crank-Pins. The length of a steel crank- 
 pin is just the same as that of a wrought-iron pin, and the 
 modulus of elasticity of steel is so nearly equal to that of 
 wrought iron as to make formula (84) serviceable for both 
 steel and wrought iron alike. 
 
76 THE RELATIVE PROPORTIONS 
 
 The advantages of steel crank-pins over wrought iron are 
 their greater strength, and the possibility of obtaining a 
 much smoother surface because of the homogeneous struc- 
 ture of steel. Their disadvantage is their liability to sud- 
 den fracture when not working truly for any reason, as inac- 
 curacy of workmanship or wrenching as in a marine-engine. 
 
 (34.) Diameters of Crank-Pins from a Considera- 
 tion of the Pressure upon them. Keferring to the table, 
 we find the average pressure per square inch of projected 
 area to be about 500 pounds. 
 
 If now we divide the whole pressure upon the crank-pin 
 by 500, we obtain the projected area required when this 
 limit is not to be exceeded. 
 
 The equation tt -- gives d 3 - .00157. (85) 
 4 x out) '3 
 
 Example. Let P= 40 pounds, 4 = 8", d = 30", 
 
 d, = .00157 9QO Q X4 - 7.06 inches. 
 8 
 
 This latter method is perhaps the most practical, and 
 has the advantage of limiting the pressure. It will almost 
 always give larger results than the preceding method. 
 
 If we assume the pressure to be uniform throughout, the 
 stroke (85) becomes, in terms of the horse-power, 
 
 (86) 
 
 (35.) Of the Action of the Weight and Velocity of 
 the Reciprocating Parts. By the reciprocating parts 
 we mean the piston-head, piston-rod, cross-head or motion- 
 block, and the connecting-rod ; also, in a vertical engine, the 
 working-beam if one is used. 
 
OF THE STEAM-ENGINE. 
 
 77 
 
 We are obliged to neglect the action of friction from 
 the impossibility of determining it, and we will also at first 
 neglect the influence of gravity and the angular position of 
 the connecting-rod i. e., suppose it to be of infinite length. 
 
 In order to clearly comprehend the motion of the recip- 
 rocating parts in a horizontal direction for a horizontal en- 
 gine, Fig. 9 (A), lay down a horizontal line, O X, and 
 divide it into 12 equal spaces, O, 1, 2, 3, etc., to 12; at these 
 points draw ordinates at right angles to O X. 
 
 With any centre upon the line O X, as 2, and any radius 
 as 2 C, describe the circle O C B 4 D O, and beginning at O 
 divide the circumference of this circle likewise into 12 equal 
 parts. 
 
 Let 2 C represent the position of the centre line of the 
 crank, let 2 be the centre of the crank-shaft, and let C be 
 the centre of the crank-pin. 
 
 Let the angle a = C 2 O be the variable angle formed by 
 the centre line of the crank with the horizontal O X. 
 
 r* 
 
78 THE RELATIVE PROPORTIONS 
 
 Let r = the radius of the crank 20. 
 " $=the space passed over by the piston-head (neces- 
 
 sarily in a horizontal direction OX) in the 
 
 time t. 
 " V=* the angular velocity of revolution of the crank (as- 
 
 sumed constant). 
 " T=the time of one revolution of the crank. 
 
 We have F= , 
 
 = r(l-cosa). (87) 
 
 Differentiating (87), we have 
 
 dS=rsmada. (88) 
 
 Letting v represent the velocity of the piston-head in a 
 horizontal direction, and dividing (88) by dt, we have 
 
 f -'-* 
 
 dS , da 2n 
 
 and since = v and y= , 
 
 (90) 
 
 Differentiating equation (90), we have 
 
 - 27TT x/v^s. 
 
 dv = COS a d a, (91) 
 
 and dividing by dt, as before, we have for the acceleration 
 
 dv^_2nr ^?./??V a (92^) 
 
 dt~ T "dt \ T i 
 
 If we assume the angular velocity = F= = 1, we can 
 graphically compare the curves of these equations, Fig. 9 (A). 
 
OF THE STEAM-ENGINE. 79 
 
 The ordinates to the curve O B E F 12 show the distance 
 of the piston-head from its starting-point during one revo- 
 lution, which can be calculated also from equation (87). 
 The ordinates to the curve O B 6 G 12 show the velocities 
 of the piston-head during one revolution, which can also be 
 calculated from equation (90). The ordinates to the curve 
 H 3 K 9 L show the accelerations of the velocity of the 
 piston-head during one revolution, which can also be calcu- 
 lated from equation (92). All of the reciprocating parts 
 are supposed to move in conjunction with the piston-head. 
 
 If now we wish to determine the acceleration of the pis- 
 ton-head for every position, Fig. 9 (B), on the line X, we 
 lay off from O toward X the ordinates to the curve of dis- 
 tances for six points, and at these points erect ordinates 
 taken from the same positions and equal to the ordinates to 
 the curve of accelerations. The extremities of these ordi- 
 nates can be joined by a straight line, A B. For, if we sub- 
 stitute in equation (87) the value of cos a derived from 
 
 equation (92) we have, letting y = = the acceleration, 
 
 at 
 
 Therefore S=r -> ( 93 ) 
 
 which is the equation of a straight line cutting O X at a 
 distance r from the origin. 
 
 Referring to equation (92), we observe that the accelera- 
 tion varies with the cosine of a, and therefore is a maxi- 
 mum for cos a = 1. This gives 
 
 jr-FV, (94) 
 
80 THE RELATIVE PROPORTIONS 
 
 which is the expression for the acceleration due to centrif- 
 ugal force. If, therefore, we wish to balance a horizontal 
 engine at its dead points, we must use a counter- weight so 
 placed that its statical moment is equal to the statical mo- 
 ment of the reciprocating parts supposed to be concentrated 
 at the centre of the crank-pin. The engine cannot be bal- 
 anced for any other than its dead points : and when the 
 crank is at right angles to the centre line of the cylinder, 
 nearly the full centrifugal force of the counter-weight is felt. 
 In engines driven at widely different speeds as, for in- 
 stance, a locomotive the use for counter-weights seems to 
 be the only practical method, and therefore the recipro- 
 cating parts should be made as light as is consistent with 
 sufficient strength to resist the stresses coming upon 
 them. 
 
 In the case of engines running at a constant speed, the 
 weight and velocity of the reciprocating parts affect the 
 stress upon the crank-pin in a manner which can be deter- 
 mined, and are therefore worthy of consideration. 
 
 Referring to Fig. 9 (B), we see that the piston resists, 
 leaving each end of the steam-cylinder with a force equal 
 to its centrifugal force (equation 94), 
 Fio.9(B). and further, that the intensity of 
 
 this force diminishes uniformly from 
 the end to the centre of stroke 3, 
 where it is zero. It will, therefore, 
 be at once recognized that an 
 amount of work represented by the 
 area of the triangle A3O is sub- 
 tracted from the work impressed 
 
 upon the piston by the steam during the first half stroke, 
 and that an equal amount, represented by the area of the 
 triangle 3 X B, is added to the work impressed upon the 
 piston during the last half of the stroke. 
 
OF THE STEAM-ENGINE. 
 
 81 
 
 In an ordinary indicator diagram, Fig. 10, we have the 
 means of measuring the force acting upon the piston-head 
 
 at every point of the stroke. (For a thorough discussion 
 and explanation of the steam-engine indicator refer to The 
 Richards Steam-engine Indicator, Porter, or The Engine- 
 Room, and who should be in it.) 
 
 Let the line O X represent the atmospheric line of an 
 indicator diagram taken from a non-condensing engine cut- 
 ting off at one-half stroke. 
 
 Let P b the initial pressure of the steam upon the piston- 
 head in pounds per square inch. 
 
 " P f = the final pressure of the steam upon the piston-head 
 in pounds per square inch. 
 
 " y = the resistance due to the inertia of the reciprocat- 
 ing parts in pounds per square inch. 
 
 If now we impose the condition that the initial and final 
 pressures upon the crank-pin be equal, we must have 
 
 Therefore 
 
 y 
 
 (95) 
 (96) 
 
 Let A = the area of the piston-head in square inches. 
 " G = the weight of the reciprocating parts. 
 
82 THE RELATIVE PROPORTIONS 
 
 We have, equating equations (94) and (96), 
 
 Transposing and substituting the values, 
 
 A = 0.7854cP square inches, 
 
 d = diameter of steam-cylinder in inches, 
 
 g = 32.2 feet per second, 
 JV=the number of strokes per minute, 
 
 F N feet per second, 
 60 
 
 r = the radius of the crank in feet, 
 we have 
 
 (98) 
 
 Considering formula (98), we see that if the initial and 
 final steam-pressure P b and P f become nearly or quite equal, 
 the weight of the reciprocating parts should be nothing. 
 Of course, the only method in such a case is to make the 
 reciprocating parts as light as possible. We further ob- 
 serve that the weight of the reciprocating parts increases as 
 the difference between P 6 and P f increases, and is inversely 
 as the square of the number of strokes. 
 
 A glance at the cross-hatched portion of Fig. 10 shows 
 that the pressure upon the crank-pin is by no means uni- 
 form, being greatest at the point of cut-off. 
 
 We further know that the angular position of the con- 
 necting-rod causes the straight line A B to become a curve. 
 The effect of the connecting-rod is discussed at considerable 
 length in The Richards Steam-engine Indicator. The curve 
 departs more widely from the straight line as n, the ratio 
 
OF THE STEAM-ENGINE. 83 
 
 of the connecting-rod to the crank, becomes less. We see 
 then that, for early cutting off of the steam, either the recip- 
 rocating parts should be made heavy or the number of revo- 
 lutions increased or decreased until the assumed weight G 
 of the reciprocating parts is obtained. 
 Transposing equation (98), we have 
 
 b / , (99) 
 
 GT 
 
 which enables us to determine approximately the number 
 of strokes per minute for any assumed pressure and weight 
 of reciprocating parts, which will give a nearly uniform 
 stress on the crank-pin. 
 
 P f may be determined from P b with sufficient approxima- 
 tion for practical purposes from Mariotte's or Boyle's law 
 for gases that the pressures are inversely as the volumes. 
 
 The initial and final pressures can be taken from an indi- 
 cator diagram, and should be determined after the erection 
 of the engine. 
 
 Example. To determine the proper weight of the recipro- 
 cating parts of a horizontal engine, data as follows (non- 
 condensing cut-off = stroke) : 
 
 P b = 125 pounds per square inch. 
 P f = 25 pounds per square inch. 
 d = 12 inches. 
 JV= 200 per minute. 
 
 Substituting in formula (98), we have the weight =- 
 4612.3x100x144 
 
 40000 
 
 = 3321 pounds. 
 
 Noting that this weight is very large, and assuming 800 
 pounds as the approximate weight of the 'reciprocating parts 
 
84 THE RELATIVE PROPORTIONS 
 
 of a steam-engine of one foot stroke and one foot diameter 
 of cylinder, we have, substituting in formula (99) the 
 proper number of strokes, 
 
 N= 67.914 x 12 J - = 407 strokes per minute. 
 ^ 800 x -g- 
 
 In an unbalanced vertical engine the weight of the recip- 
 rocating parts lessens the resistance with which the piston 
 leaves the upper end of the cylinder and adds to the resist- 
 ance with which it leaves the lower end of the cylinder, 
 thus shifting the line A B, Fig. 10, parallel to itself above 
 or below, but not introducing any greater irregularity of 
 pressure upon the crank-pin during one stroke. 
 
OF THE STEAM-ENGINE. 85 
 
 CHAPTER VIII. 
 
 (36.) The Single Crank. The crank is made of either 
 cast or wrought iron or steel ; the first-named metal is but 
 little used, and should be avoided in any but the very rough- 
 est machinery, because it is very liable to hidden defects, is 
 much weaker and has a lesser modulus of elasticity, thus re- 
 quiring to be heavier than wrought iron or steel. Further, 
 it will not admit of the crank-pin being " shrunk in " to the 
 eye without great danger of cracking. 
 
 This latter fact is illustrated very practically by an oc- 
 currence described by William Pole in his Lectures on Iron 
 as a Material of Construction, p. 123. The italics are ours : 
 
 "You have probably heard of the process of drawing 
 lead tube by forcing it in a semi-fluid (or sometimes in a 
 nearly solid) state through a small annular hole. The lead 
 is contained in a cylinder and pressed upon by a piston, and 
 the force required is enormous, amounting to 50 or 60 tons 
 per square inch. 
 
 " The practical difficulty of getting any cylinder to with- 
 stand the pressure was almost insurmountable. Cast iron 
 cylinders 12 inches thick were quite useless ; they began to 
 open in the inside, the fracture gradually extending to 'the out- 
 side, and increased thickness gave no increase of strength. 
 
 "Cylinder after cylinder thus failed, and the makers 
 (Messrs. Eaton & Amos) at length constructed a cylinder 
 of wrought iron 8 inches thick ; after using this cylinder 
 the first time, the internal diameter was so much increased 
 by the pressure that the piston no longer fitted with a suffi- 
 cient closeness. A new piston was made to suit the enlarged 
 cylinder ; and a further enlargement occurring again and 
 again with renewed use, the constant requirement of new 
 
86 THE RELATIVE PROPORTIONS 
 
 pistons became almost as formidable an obstacle as the 
 failure of the cast-iron cylinder. 
 
 " The wrought-iron cylinder was on the point of being 
 abandoned, when Mr. Amos, having carefully gauged the 
 cylinder both inside and out, found to his surprise that 
 although the internal diameter had increased considerably the 
 exterior retained precisely its original dimensions. He con- 
 sequently persevered in the construction of new pistons, and 
 found ultimately that the cylinders enlarged no more, and 
 so the last piston continued in use for many years. 
 
 " Here, therefore, the permanent set operated first in the 
 internal portions of the metal ; as they expanded it was then 
 gradually extended to the surrounding layers, and so at last 
 sufficient material was brought into play with perfect elas- 
 ticity, not only to withstand the strain, but to return back 
 to the normal state every time after its application, and thus, 
 by the spontaneous and unexpected operation of what was 
 then an unknown principle, an obstacle apparently insur- 
 mountable, and which threatened at one time to render 
 much valuable machinery useless, was entirely overcome." 
 
 It is this very property of wrought iron or soft steel 
 which renders it useful for cranks ; the metal when heated 
 possesses increased ductility or viscidity, and adapts itself to 
 the strain put upon it, while shrinking in a very perfect 
 manner. 
 
 The following expansions in length for one degree Fah- 
 renheit are given in Table XIV., p. 15, of Box's Practical 
 Treatise on Heat: 
 
 Cast iron 000006167 
 
 . Steel 000006441 
 
 Wrought iron 000006689 
 
 Neglecting cast iron as being unsuitable for cranks, let ua 
 take up the case of a wrought-iron crank. 
 
OF THE STEAM-ENGINE. 87 
 
 The value of E for wrought iron i. e., the hypothetical 
 weight which would stretch a bar one square inch in area 
 to twice its original length if it were perfectly elastic is 
 28000000 pounds. 
 
 If we take the rupturing strain in tension for wrought 
 iron at ordinary temperatures at 50000 pounds per square 
 inch area, 
 
 Letting x = the extension at the point of rupture, we have 
 1 : x : : 28000000 : 50000, and 
 
 x = = .0017857 of its length. 
 
 2800 
 
 If further we divide this amount by the lengthening of a 
 bar for one degree Fahr., we have 
 
 .0017857 = 267 Agrees Fahr., 
 
 .000006689 
 
 and we see that if it were not for the viscosity of wrought 
 iron, a bar heated 267 and fastened so as not to be able 
 to contract would rupture in cooling, and that further a 
 difference of 26.7 degrees Fahr. would in the contraction 
 resulting strain a bar 5000 pounds per square inch area. 
 
 Example. Let us assume the diameter of that part of the 
 crank-pin which is to be inserted into the eye of the crank at 
 5 inches, and further that the eye of the crank is* bored to 
 a diameter of 4.98 inches an accuracy which it is quite 
 practicable to obtain in any good machine-shop. 
 
 We then have for the required number of degrees through 
 which the eye of the crank must be raised 
 
 09 
 
 630 degrees Fahr. 
 
 .000006689x5 
 
 More accurately, 4.98 should have been used in the place 
 of 5 inches. 
 
88 THE RELATIVE PROPORTIONS 
 
 Knowing that at a temperature of 977 degrees Fahr. iron 
 is just visibly red, we may use the following formula to 
 determine the difference in the diameters of the eye of the 
 crank and of the inserted part of the crank-pin : 
 
 Let d c = diameter of inserted part of pin, 
 " d e = required diameter of eye ; 
 
 we have, with sufficient approximation, 
 d.-d. 
 
 .000006689^ 
 
 900, 
 
 Should this formula give a greater difference than is 
 necessary in practice, reduce it according to judgment by 
 using less heat, unless a particularly strong grip of the 
 eye upon the pin is desired. A high heat is apt to warp a 
 forging, and no more should be used than is necessary. This 
 formula cannot be regarded as accurate, and should be used 
 as a guide only ; it applies with sufficient approximation to 
 machinery steel. 
 
 If the entering part of the crank-pin be made very slightly 
 taper and larger than the eye of the crank, and the pin be 
 forced in by strong hydraulic pressure, we avoid the risk of 
 warping occasioned by heat, and secure almost as good a 
 result so far as strength of grip is concerned. 
 
 Crank-pins are sometimes fastened by a key " cutter " at 
 the back, and sometimes held by a nut. 
 
 Special care must be taken to have a sufficient thickness 
 of metal around the eye to hold the pin firmly. In prac- 
 tice the thickness of the ring around the eye is usually 
 taken equal to one-half the diameter of the eye. The depth 
 of the eye is usually from 1 to 2 times its diameter, thus 
 giving a cross-section of ring around the eye of from 
 
OF THE STEAM-ENGINE. 89 
 
 to 2 T 5 ^g- greater area than that of the inserted part of the 
 crank-pin at right angles to its axis. 
 
 In some cases the pin, crank and crank-shaft are forged 
 in one solid piece out of wrought iron or made of cast steel 
 in one solid mass ; this method is more expensive, but more 
 satisfactory. 
 
 The web of the crank is that part connecting its hub and 
 eye, and is made of many differing shapes, and always with 
 its cross-section increasing from the eye to the hub. Econ- 
 omy of material, as will presently be shown, dictates that 
 this cross-section be increased by an increase of the breadth 
 of the face of the crank i. e., in a direction at right angles 
 to the centre line of the crank-pin, rather than in the direc- 
 tion of the crank-pin. 
 
 If now we assume that the crank has a constant thick- 
 ness in the direction of the centre line of the crank-pin, 
 the longitudinal profile of the web of the crank should be 
 a parabola (Weisbach's Mechanics of Engineering, sec. iv., 
 art. 251-253), with its vertex at the centre of the pin, 
 and, providing we neglect the torsional strain on the web 
 due to the moment of the crank-pin, will be of uniform 
 strength. 
 
 The torsion of the web due to the force acting at the 
 crank-pin is much slighter, in fact, than might at first appear, 
 as the pin is fastened in the connecting-rod as well as the 
 crank ; and if the pin is a little loose, as it never should be, 
 its springing tends, by throwing the centre of pressure nearer 
 the crank, to reduce the moment. We can, therefore, neg- 
 lect the torsion of the web, and consider it as a beam re- 
 quired to be of uniform strength, fixed at one end and loaded 
 at the other. 
 
 To trace the parabolic outline of the crank would be 
 tedious and result in an ugly shape, but the property of the 
 tangent of a parabola permits us, with an increase of less 
 
90 THE EELATIVE PROPORTIONS 
 
 than six per cent, of material, to locate correctly the straight 
 sides of the web. 
 
 The interior diameter of the hub of the crank depends 
 upon the diameter of the main or crank-shaft ; the depth of 
 the hub is usually made equal to the diameter of the shaft, 
 or somewhat greater, and the thickness of metal around the 
 shaft equal to J of its diameter. 
 
 These proportions are varied very frequently, and are to 
 a great extent a matter of judgment with the designer. 
 
 The crank is frequently shrunk on the shaft, just as de- 
 scribed in the case of the crank-pin, and in such a case 
 similar precautions must be taken. 
 
 (37.) Wrought-Iron Single Crank. Referring to 
 Fig. 5, we see that in addition to a constant stress S in a 
 
 FIG. 5. 
 
 horizontal direction the crank is subjected to an alternate 
 
 o 
 
 maximum stress in tension and compression, Si" 
 
 which, for the values of n = 4 to 8, gives Si = 0.26 to 0.13 S. 
 See formula (38), Art. 19. 
 
 Let S = the stress in pounds upon the crank at extremity, 
 
 tending to cause flexure. 
 " Si = the stress in pounds upon crank, tending to cause 
 
 extension or compression. 
 " -F=the cross-section of the crank in inches. 
 " b = the assumed thickness of the crank in inches at 
 
 right angles to its face. 
 
OF THE STEAM-ENGINE. 91 
 
 Let v = the variable width of the face of the crank in inches. 
 " x = the variable length of the crank from the centre 
 
 of the eye. 
 " JF= the measure of the moment of flexure of the cross- 
 
 section of the crank at any point. 
 " 2"= the safe strain per square inch upon the crank - 
 
 5000 pounds. 
 
 " e = the half width of the face of the crank = . 
 
 " r = the radius of the crank in inches = . 
 
 We have, if we suppose, as is usually assumed, the con- 
 necting-rod to be of infinite length (i. e., n = oo ), Si = 0, but 
 /% is too large to be neglected in practice. 
 
 Placing ._" r -> + lP . (102) 
 
 See Weisbach's Mechanics of Engineering, sec. iv., art. 272. 
 Therefore, J=f +| f (103) 
 
 7p / 
 
 For a rectangular cross-section = . 
 
 W v 
 
 Therefore, F=-( - - - +-V (104) 
 
 irvyirri +) 
 
 Substituting for F its value bv, we have 
 
 , 81 I 6x\ 
 bv = 4 -- + I. 
 T\ V ^=l v) 
 
 Therefore, *- - ? - + J ^ + ggg. (105 ) 
 ^1 >46 2 T 2 (n 2 -l) bT 
 
 We have then the means of computing the value of V for 
 a series of assigned values of b and x. 
 
92 
 
 THE RELATIVE PROPORTIONS 
 
 We see further that the first two terms of the second 
 member of equation (105) will give small values which do 
 
 not greatly affect the value 
 
 FIG. 11. 
 
 of v. 
 
 sume 
 
 If in (105) we as- 
 
 L r ^ 
 
 x = = -, we have 
 
 the width of the face of the 
 crank at a point midway 
 between the centres of the 
 eye and hub, and equation 
 (105) becomes 
 
 8 
 
 SSL 
 
 (] 
 
 If we suppose n = oo , equa- 
 tion (106) becomes 
 
 [ML 
 
 (107) 
 
 In making use of equa- 
 tions (105) and (106) it is 
 most convenient to calcu- 
 late the value 
 
 separately. If in (107) we 
 substitute the values of S 
 and T % we have 
 
 0.0153d 
 
 (108) 
 
OF THE STEAM-ENGINE. 93 
 
 and for a uniform pressure P we have, in terms of the 
 horse-power, 
 
 Vl = 10.864 . (109) 
 
 * bN 
 
 If now we assume the value b = the thickness of the web 
 as uniform throughout its length, and recollecting that the 
 tangent to a parabola intersects the axis of abscissa at a dis- 
 tance from the origin (vertex) equal to the abscissa of the 
 point of tangency, we have a ready means of graphically 
 constructing the web of the crank, Fig. 11. 
 
 From equation (106) or (108) (the first is the more ex- 
 act) calculate the value of v lt Draw the centre line 
 ABC through the centres of the eye and hub. Bisect 
 D C in B and lay off D A = D B ; at the point B erect 
 the double ordinate v ly and through the extremities of this 
 ordinate and through the point A draw the lines A F G and 
 A H K to intersections with eye and hub. 
 
 CHAPTER IX. 
 
 (88.) Steel Single Crank. Formula? (105) and (106) 
 apply in the same manner, with the exception that T= 9000 
 pounds per square inch for steel. Formula (107) becomes 
 for this value of T, and the substitution of the value of S, 
 
 = 0.0114^ 
 
 and for an uniform pressure P we have, in terms of the 
 horse-power, 
 
 (Ill) 
 
94 THE RELATIVE PROPORTIONS 
 
 Examples. Let .L = 48 inches. 
 d = 32 inches. 
 " P b = P= 40 pounds per square inch. 
 
 
 " (HP) = 156 approximately. 
 " JV= 40 per minute. 
 " 6 = 7 inches. 
 
 We have # = 0.7854x1024x40 = 32170 pounds. 
 
 For a wrought-iron crank, ^=5000 pounds per square 
 inch. 
 
 Substituting in formula (1 06), we have 
 
 _ 32170 
 
 Vl ~ 2x7x5000x4.899 
 
 (32170) 2 3x32170x48 
 
 4 x 49 x 25000000 x 24 2 x 7 x 5000 
 
 = 0.094 + 1/0.0088 + 66.18 = 0.094 + 8.25 = 8.344 inches. 
 If we substitute in formula (109), we have 
 
 Vi = 10.864 X /-^L = 8.H inches. 
 
 \ 7x40 
 
 We see that the difference between the results of formulae 
 (106) and (109) is trifling, and this difference will be less for 
 all values of n > 5 ; it will be greater for all values of n< 5. 
 
 For a steel crank we should take b = 6" and T=9000 
 pounds per square inch. 
 
 Substituting in formula (111), we have 
 
 The use of these values of Vi in the graphical construc- 
 tion of the web of the crank, Fig. 11, Art. 37, is obvious. 
 
OF THE STEAM-ENGINE. 
 
 95 
 
 FIG. 12. 
 
 FIG.IS. 
 
 The use of formula (105) for the computation of a series 
 of values of v for assigned values of x and b is tedious, but 
 easy to understand. 
 
 (39.) Cast-Iron Cranks. Although subject to a greater 
 liability to accidental frac- 
 ture than wrought iron or 
 steel, cast-iron cranks are 
 frequently used in the 
 cheaper forms of engines. 
 
 Fig. 12 gives a rather 
 neat-looking unbalanced 
 crank, and Fig. 13 is an 
 example of the disc crank, 
 
 which allows of being balanced with great ease by means 
 of a counter-weight placed 
 on the opposite side from 
 the pin. 
 
 The proportions of cast- 
 iron cranks depend to a 
 great extent upon the cha- 
 racter of the iron used. 
 Formulae . can hardly be 
 applied to so uncertain a 
 material as cast iron. 
 
 (40.) Keys for Shafts. In every case a key is used 
 to prevent the crank from slipping on the shaft. The best 
 position for a key is in line with and between the centres 
 of the eye and hub of the crank. If more than one key is 
 used, their combined shearing strength should be equal to 
 that of a single key. Great care should be taken that the 
 sides of the key fit the keyway perfectly, and that these 
 sides be perfectly parallel, not taper. The very slight taper 
 given to the key should be at the top. 
 
 Considering the case of a single key, 
 
96 
 
 THE RELATIVE PROPORTIONS 
 
 Let bi its breadth in inches. 
 
 " I = its length in inches. 
 
 " t = its thickness in inches. 
 
 " K= the safe shearing strain per square inch area. 
 
 " d = the diameter of the steam -cylinder in inches. 
 
 " L = the length of stroke in inches. 
 
 " P b = the steam-pressure per square inch. 
 
 " di = the diameter of the shaft in inches. 
 Since the moment of the safe shearing strain of the key 
 should be equal to the maximum moment upon the crank, 
 
 we have 
 
 _._ 0.7854 
 
 2 
 
 Therefore, 
 
 (112) 
 
 Or, if we suppose the steam-pressure to be uniform, we 
 have, in terms of the horse-power (HP), and number of 
 strokes per minute N t 
 
 396000 
 
 (HP} 
 
 FIG. 14. 
 
 (113) 
 
 To determine the proper thickness of the key t we may 
 
 adopt the following method : 
 
 Let the key be sunk one- 
 half of its thickness in the 
 shaft, Fig. 14, and let the 
 thickness of the key be such 
 that the shearing surface of 
 the shaft on a line with the 
 inside surface of the key 
 N M equals the shearing sur- 
 face of the key ON. If t 
 be not determined graphi- 
 cally, it can be determined 
 
 approximately in an analytical way. 
 
OF THE STEAM-ENGINE. 97 
 
 We have t = 2KL = 2( CK- CL). (114) 
 
 If now we substitute these values of CK and CL in 
 formula (114), we have 
 
 and reducing we have, with sufficient approximation, 
 
 4& L 2 105^ 
 
 = -7- +-77-+ etc. (115) 
 
 di df 
 
 Example. Let e? 4 = 16 inches. 
 " ^ = 3 inches. 
 
 Substituting in formula (115), we have 
 
 4x(3) 2 10x(3) 4 810 
 
 t = + - ~~ 8J5* -- = 2.45 niches. 
 16 (16) s 4096 
 
 Where more than one key is used the sum of their 
 widths should equal the width of a single key fitted to 
 withstand the given stress. 
 
 The thickness of the keys will be much less, as, for in- 
 stance, two keys 1^- inches broad and -^ of an inch thick 
 will take the place of the single key given in the example 
 with equal safety. 
 
98 THE RELATIVE PROPORTIONS 
 
 It is a point worthy of particular notice that, neglecting the 
 last term of equation (115}, we see that the thickness of keys for 
 shafts varies directly as the square of their breadth, and there- 
 fore that the use of two or more keys is attended with great 
 economy of material in the keys, and less reduction of the 
 cross-section of the shaft, while the safety is equally as great. 
 
 In establishing the proper thickness of metal for the hub 
 of the crank the reduction due to the keyways must be 
 considered. 
 
 (41). Wrought-Iron Keys for Shafts. If in formula 
 (112) we place if =5000 pounds per square inch, we have 
 
 fti- 0.000157^-^. (116) 
 
 Idt 
 
 Letting K= 5000 pounds per square inch, we have, from 
 formula (113), 
 
 ^ = 79.21 (117) 
 
 Example. Let d = 32 inches. 
 
 " P b = P= 40 pounds per square inch. 
 
 L- 48 inches. 
 " 1 = 20 inches. 
 
 " d< = 16 inches. 
 
 J\T=40 per minute. 
 " (HP) - 156 approximately. 
 
 Substituting in formula (116), we have 
 
 6l i 0.000157 1024x4 x48 - 0.96 inches. 
 16x20 
 
 Substituting in formula (117), we have 
 
OF THE STEAM-ENGINE. 99 
 
 We have also for the thickness of the key, from formula 
 (115), 
 
 t = 4x0.92 
 
 16 
 
 (42.) Steel Keys for Shafts. Since the shearing 
 strength of machinery steel is found to be f of its tensile 
 strength, we have for a safe shearing stress f of 9000 
 pounds = 6750 pounds per square inch. Substituting this 
 value for JTin formulae (113) and (114), we have 
 
 (118) 
 i 
 
 or 
 
 ^ = 58.667^1 (119) 
 
 Comparing formulae (117) and (119), we find 
 
 b v for steel _ = 58.667 ^ Qy4 
 >! for wrought iron 79.2 
 
 Example. Data as before in Art. (41). We can calcu- 
 late as there shown, or abbreviate by taking 74 per cent, of 
 the breadth of a wrought-iron key for the breadth of a steel 
 key of equal strength, 
 
 h = 0.74 x 0.96 = 0.71 inches. 
 
 The thickness would be the same as before for a wrought- 
 iron shaft, and for a steel shaft, formula (115), we have 
 
 16 
 
 This latter value is too small for convenience, and the 
 key should be made of J inch thickness, or as would be 
 advisable greater in both cases. 
 
 (43.) The Crank or Main Shaft. The first requisite 
 of the crank-shaft is rigidity. Any flexure will be accom- 
 
100 THE RELATIVE PROPORTIONS 
 
 panied by injury and final destruction to the bearings of the 
 crank-pin and of the shaft itself. 
 
 The stresses to which the crank-shaft is subjected are as 
 follows : 
 
 (1.) At the beginning of each stroke it receives the thrust 
 due to the pressure of steam upon the piston-head, which 
 in some cases is a veritable blow, as will be noticed when 
 the journals are not properly fitted, and also with some 
 valve-motions without compression. 
 
 (2) At the middle of the stroke the crank-shaft is sub- 
 jected to a maximum torsional stress (whose moment is 
 measured by the length of crank multiplied by the steam- 
 pressure on the whole of the piston), in addition to the thrust 
 due to the whole steam-pressure on the piston. 
 
 (3.) When a fly-wheel propellor or paddle-wheels are 
 attached to the shaft it is deflected by their weight, as well 
 as strained by the torsional stress mentioned above. 
 
 These three cases will be discussed separately. 
 
 (M.) Shaft Subjected to Flexure only. Round 
 shafts only are used for steam-engine crank-shafts, and will 
 therefore be the only form of shaft considered. 
 
 Shafts are subject to stress in flexure only, either when at 
 rest or in some cases at the dead points that is, when no 
 fly-wheel is used. The parts subject to flexure are the over- 
 hanging ends of the shaft to which the crank propellor or 
 paddle-wheel is attached, or between those bearings which 
 support the fly-wheel propellor or paddle-wheel. 
 
 Let T=the safe stress per square inch upon the exterior 
 
 fibres of the shaft. 
 " JF=the measure of the moment of flexure for a 
 
 round-shaft = -. 
 64 
 
 * d 4/ = the required diameter of the shaft to resist flexure, 
 in inches. 
 
OF THE ,ST^AM-M r GIN^ >V /!; 101 
 
 Let I = the length of that part of the shaft under con- 
 sideration in inches. 
 " G = the load upon the shaft in pounds. 
 
 We have (Weisbach's Mechanics of Engineering, sec. iv., 
 art. 235), for a shaft fixed at one end and loaded at the 
 other, 
 
 Transposing, 
 
 32 I 
 32 Ol 
 
 = 7T T' 
 
 (120) 
 
 (121) 
 
 FIG. 15. 
 
 (45.) Wrought-Iron Shaft Flexure only. In the 
 
 case of a wrought-iron shaft 2 7 =5000 pounds per square 
 inch, and formula (120) becomes 
 
102 'j?&E RELATIVE PROPORTIONS 
 
 <V = - 002037 <& (122) 
 
 Formula (121) becomes 
 
 d 4/ = 0.1268 i/~GL~ (123) 
 
 In the case of an overhanging weight, as at E, Fig. 15, 
 these two formulae can be applied directly by substituting E 
 for G and 1 3 for I. 
 
 In the case of a load C between two supports B and Z), 
 we can determine the reactions at B and D as follows : 
 
 Let /i = the distance BC in inches. 
 " / 2 = the distance CD in inches. 
 " C= the weight in pounds. 
 " B the reaction at support B in pounds. 
 " D = the reaction at support D in pounds. 
 
 We have Cl 
 
 Therefore, D = --. (124) 
 
 ( li + 4) 
 
 We have then to substitute the derived value of D for G 
 and 4 for /, in equation (122) or (123), in order to determine 
 the diameter for flexure of the part CD of the shaft. Simi- 
 larly, we can treat the part BC of the shaft. 
 
 The overhanging part AB of the shaft should be estimated 
 from the centre of the crank-pin, and for G the maximum 
 pressure of the steam upon the piston-head = 0.7854P 6 cf 
 should be substituted. 
 
 Making these substitutions in formula (122), we have 
 
 (125) 
 (126) 
 
 = . 6 . 
 If we suppose the pressure P b uniform, we have 
 
 cV-806J^2*. 
 X/iV 
 
OF THE STEAM-ENGINE. 103 
 
 CHAPTEE X. 
 
 (46.) Steel Shaft Flexure only. For a steel shaft 
 T= 9000 pounds per square inch, and formula (120) becomes 
 
 d 4 / = 0.00113 Gl (127) 
 
 Formula (121) becomes 
 
 d = 0.1042 i/~GT (128) 
 
 The method of applying these two formulae is the same 
 as in Art. (45), equation (124), etc. 
 
 For the overhanging part of the shaft AB, Fig. 15, to 
 which a single crank is attached, we have by substitution in 
 
 formula (127) 
 
 dj - 0.000889 P b cP 1. (1 29) 
 
 If in this we suppose the steam-pressure P b to be uniform, 
 we have 
 
 dj = 448.18^^- 
 
 (47.) Shaft Subjected to Torsion only. Shafts are 
 subjected to a stress in torsion from the point where the 
 power is received to the point at which it is given off, and 
 not beyond that point. 
 
 The torsional stress in crank-shafts is zero at the dead 
 points, and a maximum when the crank stands at right 
 angles to the centre line of the steam-cylinder for single 
 cranks. (For double and treble cranks see Arts. 54 and 55.) 
 
 We need only to consider the maximum torsional stress 
 upon the shaft, leaving out of consideration the deflecting 
 stress occurring at the point of maximum torsional stress, 
 and due to the angular position of the connecting-rod. 
 (See Art. 37.) 
 
104 THE RELATIVE PROPORTIONS 
 
 Let r = the length of the crank in inches = . 
 
 2 
 " ^ = the required diameter of the shaft to resist torsion, 
 
 in inches. 
 " T = the safe stress per square inch upon the exterior 
 
 fibres of a round shaft. 
 " S = 0.7854P 6 d 2 = the stress at the extremity of the 
 
 crank i. e., upon the crank-pin. 
 We have (Weisbach's Mechanics of Engineering, sec. iv., 
 
 art. 264) 
 
 3== 16/Sr 
 
 Therefore, substituting for S and r the values given above, 
 ^ 8 = 2; ZVfZ, 
 
 /TT ' ' 
 
 and <4 = 1.26^T (m) 
 
 If in formula (131) we consider the pressure of the steam 
 to be uniform, we have, in terms of the horse-power," 
 
 e^ 3 = 1008660.^-2 (133) 
 
 (48.) Wrought-Iron Shaft Torsion only. For wrought 
 iron we have, with a factor of safety of 10, T*= 5000 pounds 
 per square inch, and formula (131) becomes 
 
 d 3 = 0.0004P 6 dlL, (134) 
 
 and formula (132) becomes 
 
 Formula (133) becomes 
 
 ( JTJ-*\ 
 
 7 3 OfM TO^"^ ' f"\ QJ\ 
 
 Ujt = ZU1.U . llOD) 
 
 N 
 
 (49.) Steel Shaft Torsion only. Since the stress upon 
 a steel shaft at the outer layer of fibres is of the nature of 
 
OF THE STEAM-ENGINE. 105 
 
 a shearing stress, we can take, with a factor of safety of 10, 
 T= 6750 pounds per square inch. 
 Formula (131) becomes 
 
 dj - 0.0002964PM. (137) 
 
 Formula (132) becomes 
 
 d - 0.06667y / P^Z: (138) 
 
 Formula (133) becomes 
 
 eV- 149.43^2. (139 
 
 N 
 
 (50.) Shaft Submitted to Combined Torsion and 
 Flexure. The most frequent case for crank-shafts is where 
 they are submitted at the same time to stresses of flexure 
 and torsion. 
 
 Let T=the safe stress per square inch. 
 " $ = the whole stress of the pressure of the steam upon 
 
 the piston-head, in pounds. 
 " G the load causing flexure, in pounds. 
 " / = the distance in inches of the point of application 
 
 of the load to the point of support. 
 
 " r = the length of the crank in inches . 
 
 2i 
 
 " d = the required diameter of the shaft, in inches, to with- 
 stand the combined stresses of flexure and torsion. 
 We have ( Weisbach's Mechanics of Engineering, sec. iv., 
 art. 277) the following approximate formulae: 
 
 (140) 
 
106 THE RELATIVE PROPORTIONS 
 
 mi 
 
 or 
 
 d t (141) 
 
 Considering formulse (140) and (141), we observe that 
 
 Q9 C 1 ? 
 c^. Kefer to Art. (47) ; and that T f- = rf v 8 . Ke- 
 
 Tt TTjT 
 
 fer to Art. (44). 
 
 Substituting these values, we have 
 
 
 *- r ,' -,. ( 143 > 
 
 M4 
 
 As in both formulae (142) and (143) c? 4 is greater than 
 either d 4/ or d 4t , we must, in order to obtain the first approx- 
 imation, substitute for d 4 in the second member the greater 
 resulting diameter from a consideration of the stresses in 
 flexure and torsion singly. 
 
 A single approximation will generally be sufficient for 
 practical purposes. 
 
 Therefore we have the following simple rules for calcu- 
 lating the diameter of a shaft submitted simultaneously to 
 torsion and flexure: 
 
 FIRST. Calculate the diameters for torsion and flexure 
 singly. 
 
OF THE STEAM-ENGINE. 1Q7 
 
 SECOND. If the diameter due to torsion be the greater, 
 divide it by the sixth root of the expression, Unity minus the 
 quotient of the cube of the diameter due to flexure divided by 
 the cube of the diameter due to torsion, and the result will be 
 the required diameter. 
 
 THIRD. If the diameter due to flexure be the greater, divide 
 it by the cube root of the expression, Unity minus the quotient 
 of the sixth power of the diameter due to torsion divided by 
 the sixth power of the diameter due to flexure, and the result 
 will be the required diameter. 
 
 Claudel, Formules a Vusage de VIngenieur, p. 288, gives 
 the following rule : " Calculate the diameter of the shaft 
 to resist each strain separately. Take the greatest of the 
 two values. If the largest diameter is given by the effort 
 of torsion, augment it by to -$" This rule gives too 
 small values. 
 
 Formulae (142) and (143) can be expanded into a series 
 giving an approximate value. Thus : 
 
 Therefore (142) becomes when torsion gives the greater 
 diameter 
 
 and in the same manner (143) becomes when flexure gives 
 the greater diameter 
 
108 THE RELATIVE PROPORTIONS 
 
 (51.) Flexure and Twisting of Shafts. The shaft is 
 deflected by the load placed upon it, and also twisted 
 through a greater or less angle by the action of the crank. 
 
 Let d = the diameter of the shaft in inches. 
 " (r = the load in pounds at the extremity of the part 
 
 considered. 
 " I = the length of that part of the shaft under consider- 
 
 ation in inches. 
 " E=the modulus of elasticity = 28000000 pounds per 
 
 square inch for wrought iron. 
 " a = the deflection in inches. 
 
 We have (Weisbach's Mechanics of Engineering, sec. iv., 
 art. 217), for a beam fixed at one end and loaded at the 
 other, 
 
 64 OT 
 
 giving the deflection from a straight line. 
 
 If the weight of the shaft be taken into consideration, we 
 must add or subtract it, as the case may require. 
 
 For the angle through which a wrought-iron shaft will be 
 twisted (Weisbach's Mechanics of Engineering, sec. iv., art. 
 263) we have, letting a = the number of degrees and the 
 notation be as before, 
 
 (147) 
 or supposing the pressure P b constant, 
 
 a = 12.807 . (148) 
 
 J\Cl/4 
 
 Example. Fig. 15, to determine the proper diameter of 
 
OF THE STEAM-ENGINE. 
 
 109 
 
 a wrought-iron shaft having a fly-wheel C, weighing 70000 
 pounds, supported between the plumber-blocks B and D. 
 
 Fro. 15. 
 
 Let 1 = 36 inches. 
 
 " /! = 48 inches. 
 
 " f a = 36 inches. 
 
 " P b = P 40 pounds per square inch. 
 " d = 32 inches. 
 
 w L = 48 inches. 
 
 " N= 40 per minute. 
 " (fiP) - 156 approximately. 
 
 The Overhanging Part AB. From formula (126) we 
 have for deflection 
 
 = 806.7 
 
 156x36 
 48x40 
 
 10 
 
110 THE RELATIVE PROPORTIONS 
 
 From formula (136) we have 
 
 Observing that the diameter due to flexure is the greater, 
 and extracting the cube root, we have 
 
 d tf = 13.31 inches, 
 
 and using formula (143), or, more conveniently, (145), we 
 have 
 
 giving the required diameter for the part AB of the shaft. 
 The Part EC of the Shaft. From Art. (45) we have the 
 
 reaction at 
 
 <% 70000x3 Qnn __ 
 B = - - - = 30000 pounds. 
 Ja + 4 7 
 
 We then have, formula (122), 
 
 dj - 0.002037 x 30000 x 48 = 2933.28, 
 
 dy= 14.31 inches. 
 Using formula (145), we have 
 
 giving the required diameter of the part BC of the shaft. 
 
 The Part CD of the Shaft. By substitution in formulae 
 (124) and (123) we find the required diameter of the part 
 CD to be = 14.31 for flexure alone, if we suppose the power 
 to be taken off the fly-wheel and neglect the stress from a 
 belt or gearing. 
 
 The Part DE.lf we suppose the weight at .=40000 
 
OF THE STEAM-ENGINE. Ill 
 
 pounds and 1 3 = 36 inches, we have, by substitution in form- 
 ula (123), dt = 14.31 inches for flexure alone, or, by substi- 
 tution in formula (145), if we suppose the shaft submitted 
 to torsion also, 
 
 c? 4 = 14.6 inches. 
 
 For torsion only d = {/7S6.75 = 9.23 inches. 
 
 (52.) Comparison of Wrought-iron and Steel Crank- 
 Shafts. Comparing formulae (128) and (123), we see 
 
 For steel d 4/ 0.1042 
 
 For wrought iron d 4f ~ 0.1268 ~ 
 
 Therefore, a steel shaft, to withstand the same stress in 
 flexure, requires to be but 0.82 of the diameter and 0.67 of 
 the weight of a wrought-iron shaft. 
 
 Comparing formulae (138) and (135), we have 
 
 For steel d, t 0.06667 
 
 For wrought iron d^ t 0.07368 
 
 Therefore, a steel shaft withstanding the same stress in 
 torsion requires to be 0.90 the diameter and 0.81 the weight 
 of a wrought-iron shaft. 
 
 Because of the near equality of the moduli of elasticity 
 of wrought iron and steel, the deflection and torsional angle 
 of wrought iron and steel under stress will be practically 
 the same in all cases. 
 
 (53.) Journal-Bearings of the Crank-Shaft. In the 
 
 various hand-books of mechanical engineering, giving em- 
 pirical rules for the length of the journals of shafts, we are 
 advised to make the journal-bearing from 1 J to 2 times the 
 diameter of the shaft. In Art. (30), formula (81), we have 
 the means of determining the least allowable length of jour- 
 nal of shaft under the most favorable circumstances that 
 is, when the shaft is submitted to torsion only, and does not 
 
112 
 
 THE RELATIVE PROPORTIONS 
 
 bear the additional weight of a fly-wheel, screw-propelloi 
 or paddle-wheels. 
 
 If to the stress due the steam-pressure on piston be added 
 the weight of a fly-wheel, etc., 
 
 Let Q = reaction at bearing due to weight. 
 " S = stress due steam-pressure on piston. 
 
 Referring to Fig. 16, we see that the force S always acts 
 in the direction of the centre line of the cylinder, and that 
 the force Q acts downward, and further that the small force 
 S, for ordinary lengths of connecting-rods, causes the re- 
 
 FlG. 16. 
 
 sultant R of the forces Q and S to vibrate between the posi- 
 tions and values R and R l when the crank moves in the 
 direction of the arrow i. e., throws over and tends to lift 
 the shaft from its bearings. If the crank throws under, the 
 small force Si tends to press the shaft down upon its bearings. 
 We have for the value of the resultant force, neglecting /Si, 
 
 .R-l/tf+e^etfcoBa. (149) 
 
 For the angle a = 90 degrees that is, for a horizontal 
 engine we have 
 
OF THE STEAM-ENGINE. 115 
 
 For a = degrees that is, for a vertical engine we have 
 
 jR-6+Sor "I 
 
 For a = 180 degrees, we have > . (151) 
 
 R=Q-S. ) 
 
 Let R = pressure on journal from above formulae. 
 " / = coefficient of friction. 
 " (4 = diameter of journal of shaft. 
 " / 4 = length of journal of shaft. 
 " W= work (or heating) allowed per square inch of pro- 
 
 jected area per minute = 49908 inch-pounds. 
 " N= number of strokes per minute. 
 
 Then, by a similar course of reasoning to that in Art. 30, 
 we have 
 
 (152) 
 
 Example. Let us take a horizontal cylinder. 
 Let S= 60000 pounds. 
 " = 30000 pounds. We have 
 
 R - V Q* + 2 - T/4500000000 - approx. 6700 Ibs. 
 
 Let N=4Q per minute and/=.08. 
 
 Then, by formula (152), we have 
 
 / 4 = .0000325 x .08 x 67000 x 40 = 6.9 inches, 
 which is the minimum length of shaft-journal allowable. 
 
 The importance of the influence of the number of turns 
 upon the length of the bearing has not hitherto been 
 noticed, and the use of empirical rules has resulted in bear- 
 ings much too long for slow-speeded engines and too short 
 for high-speeded engines. 
 
 To cover the defects of workmanship, neglect of oiling, 
 and the introduction of dust, it is probably best to take 
 /= .16, or possibly even greater if we make use of formula 152. 
 
 Five hundred pounds per square inch of projected area 
 10* 
 
114 THE RELATIVE PROPORTIONS 
 
 may be allowed for steel or wrought-iron shafts in brass 
 bearings with good results, if a less pressure is not attain- 
 able without inconvenience. 
 
 Babbit or soft-metal linings, which are moulded by pour- 
 ing in the metal around the shaft and allowing it to fit in 
 cooling, are used in some forms of engine with great economy 
 and good results. 
 
 For great pressures the shaft is sometimes cased in gun- 
 metal and the casing run in lignum-vitse bearings, which 
 are lubricated with water. This expedient is commonly 
 used for propellor-shafts, and an aperture communicating 
 with the hold of the vessel causes a constant stream of water 
 to flow through the bearing. 
 
 Hollow or " lantern " brasses, through the interior of 
 which a constant stream of water is kept flowing in order 
 to convey away the heat, are also used for great pressures 
 with good success. 
 
 It is best,* where great pressures are used, to have some 
 means of feeling of the shaft as it turns, and any spot which 
 feels rough (" ticklish ") should at once be lubricated by 
 means of a long-nosed oil- can, with a wick in the end of 
 the nose placed in contact with it. 
 
 By means of these expedients pressures of 1000 pounds per 
 square inch of projected area have been successfully used. 
 
 With very slow speed even greater pressures are some- 
 times used. 
 
 Arthur Rigg gives in A Practical Treatise on the Steam- 
 engine many forms of plumber-blocks and bearings from 
 English models. 
 
 Warren's Elements of Machine Construction and Drawing 
 gives some good forms of the French and American types. 
 Inspection of Fig. 16 shows at what points the bearings are 
 liable to the greatest wear, being the points at which the 
 resultant E intersects the circumference of the bearings. 
 
OF THE STEAM-ENGINE. 115 
 
 CHAPTER XI. 
 
 (54.) Double Cranks. For the purpose of obtaining 
 greater regularity of revolution of the crank-shaft, as well 
 as to enable the engine to start in any position, two cranks 
 at right angles are frequently used. 
 
 We shall neglect the obliquity of the connecting-rod, and 
 also consider the pressure upon the crank-pin to be uniform 
 throughout the stroke, as has been shown possible to render 
 it approximately in Art. (35). 
 
 r/ 
 
 Let a = the half angle between the two cranks I and II. 
 " = the variable angle B C O formed by the line C B, 
 
 which bisects the angle I C II. 
 " $ = the force acting upon each crank-pin. 
 " r = the radius of the crank. 
 
 Then, Fig. 17, we have for the combined moments of the 
 crank = y, when both cranks are on one side of the line O C, 
 
 y = Sr[sin (#-) + sin (# + )] (153) 
 = 2rsin cos . (154) 
 
 Differentiating, and equating with 0, we have 
 
 at 
 
116 THE RELATIVE PROPORTIONS 
 
 giving a maximum for 6 = 90 or 270. When the cranks 
 C I and C II are on opposite sides of the line O C, we have 
 
 y = Sr [sin (0 - a) + sin (0 + a - 180)] (155) 
 - 2Sr cos d sin a. (156) 
 
 Differentiating, and equating with 0, we have 
 
 dO 
 
 giving a maximum for = or 180. 
 
 Further, it will at once be seen, if we consider a as a vari- 
 able as well as 0, that the value of expressions (154) and 
 (156), which express all values of the combined moments, 
 will be a maximum, and therefore the possible minimum 
 value of the expressions differ least from the maximum 
 values if we make sin = cos and cos = sin a condition 
 which can only be fulfilled by letting = a = 45, giving 
 for the angle between the cranks 90, and for the position 
 of least moment 6 = 45 that is, when one of the cranks is 
 at its dead point. 
 
 We have then for a maximum value of equations (154) 
 and (156) 
 
 y = 2Sr* .7071 - 1.414ft-, (157) 
 
 and for a minimum value 
 
 y = '2Srx.5 = Sr. (158) 
 
 If, as in the case of a marine-engine, the power of the 
 first crank is communicated through a second bent crank, 
 we see that that crank and the shaft leading from it at no 
 time sustain twice the torsional stress exerted by one crank, 
 but at a maximum 1.414 times as much ; and in determin- 
 ing the dimensions of the after-crank and of the shaft,, we 
 should regard the stress as that upon the first multiplied 
 by 1.414. 
 
OF THE STEAM-ENGINE. 117 
 
 Multiplying the numeri- 
 cal coefficient of equation FIG. 18. 
 
 (132) by i/lA\A, we have, <* 
 
 with the same notation for 
 the diameter of the shaft 
 
 (159 ) 
 
 In the case of double en- 
 gines it is customary to make 
 
 the after-crank-pin of the same size as the shaft, for the 
 reason that it is subjected to many unforeseen stresses. 
 
 The length of the after-crank-pin should be the same as 
 that of the forward pin. See Art. (30). 
 
 The stress upon the after-pin due to its cylinder may be 
 regarded as constant, and in the direction of the centre line 
 of its cylinder. 
 
 The stress upon the after-pin from the forward crank-pin 
 is a maximum in a tangential direction to its circle of revo- 
 lution, and equal to S when the forward pin makes an angle 
 of 90 with its cylinder centre line. Fig. 18. 
 
 If we make the supposition, as may happen to be the case, 
 that both these stresses act simultaneously at the forward 
 end of the after-pin without support from the forward crank,* 
 we have for the maximum stress upon the pin 1/2$= 1.414$, 
 and for the diameter of the after-pin, from formula (84), 
 Art. (32), 
 
 4 = 0.072^4^; (160) 
 
 or, if we regard the steam-pressure as constant, 
 
 (161) 
 
 * This supposition does not give greater results than are often 
 found practically necessary by an expensive tentative process. 
 
118 THE RELATIVE PROPORTIONS 
 
 It should, however, be remembered that, although the 
 most unfavorable suppositions possible for the known stresses 
 have been made in the present case, unforeseen stresses are 
 liable to occur, which can be guarded against only by making 
 the after-pin the same size as the shaft if possible. 
 
 (55.) Triple Cranks. Triple cranks 120 degrees apart 
 are sometimes used to attain still greater regularity of 
 motion in the engine. (Fig. 19.) 
 
 Let = the angle made by 
 one of the cranks 
 with the line AB. 
 " S = the pressure on the 
 piston-head in Ibs. 
 " r = the radius of the 
 
 crank in inches. 
 We have for the tor- 
 sional moment of the three 
 cranks, = y, 
 
 y = Sr[sin a + sin(a + TT) + sin(a + f TT)], (162) 
 
 in which the sines are taken as positive because the cylinders 
 are double-acting. 
 
 Further, we see that the sum of the sines is not increased 
 
 7T 
 
 when we increase each of them by - = 60 degrees, and it is 
 
 3 
 
 then only necessary to consider the equation for the angle a 
 
 between the limits and -. 
 3 
 
 Reducing equation (162), we have 
 
 y = Sr [sin a + l/3~cos ]. (1 63) 
 
 Neglecting Sr, differentiating, and placing the first differ- 
 ential coefficent = 0, to find the maximum and minimum 
 values, we have 
 
OF THE STEAM-ENGINE. 119 
 
 = cos a - l/3sin a = 0, 
 da 
 
 ~- = - sin a - 1/3 cos a, 
 da* 
 
 giving for a maximum value 
 
 cos a = 1/3 sin a. 
 
 Therefore, tan a ; therefore, a = 30 = -. 
 
 1/3 6 
 
 The minimum values of equation (163) occur when a = 
 and = - = 60. 
 
 o 
 
 We see that that part of the shaft attached to the third 
 crank is subjected to a maximum torsional stress twice as 
 great as that due to one cylinder. Its diameter can be most 
 readily calculated by doubling the actual steam-pressure in 
 equation (132). 
 
 In the same manner, the proper proportions of the crank 
 can be calculated by Art. (37). 
 
 In the case of three cranks, in order to find the maximum 
 stress to which the second crank and shaft may be submitted, 
 we must find the maximum of the expression 
 
 / 2 \] 
 sin + sin j a+-7r ) (164) 
 
 Reducing, and neglecting &r, 
 
 y = - sin a + 1L_ cos a. 
 
 Differentiating, and placing = 0, we have 
 
 dy 3 j/3". 
 
 = - cos a - ^ sin a = 0. 
 da 2 2 
 
120 THE RELATIVE PROPORTIONS 
 
 Therefore tan a = T/37 therefore a = 60 = -, 
 
 o 
 
 gives the greatest maximum to which they are subjected, 
 and equation (164) becomes 
 
 2/ = r[0.866 + 0.866] = 1.732 /Sr, 
 
 which is also the minimum value of the torsional stress for 
 3 cranks together. 
 
 We can calculate the proper proportions of the crank, 
 si] aft and crank-pin by multiplying the steam-pressure by 
 1.732 in equation (132). 
 
 (56.) The Fly- Wheel. Before taking up the subject of 
 the fly-wheel mathematically, it will perhaps be best to give 
 a general idea of its function. 
 
 It is impossible to control the speed of any engine for any 
 considerable length of time by means of a fly-wheel, or to 
 render the motion of any engine exactly uniform for any 
 period of time. 
 
 It is, however, possible, by properly proportioning the 
 weight, diameter and speed of rim of a fly-wheel to the 
 work given out by the steam-cylinder, to confine the varia- 
 tion of the speed of the engine from any assigned mean 
 speed during the time of one stroke, within any assigned 
 limits. 
 
 A fly-wheel serves to store up work, or to give it out when 
 required, just as a mill-pond fed by a stream of variable dis- 
 charge serves to store up water for the mill-wheel ; the larger 
 the pond, the less the effect upon it of any sudden increase 
 or diminution of the water flowing into it, and so with the 
 fly-wheel : the larger it is, and the more rapid its motion, the 
 more steadily it will run, so that it would hardly be possible, 
 where a uniform motion in one direction only is desired, to 
 make a fly-wheel too large, were it not for the circumstances 
 that the loss of work due to increased friction and its greater 
 cost limit us in that direction. 
 
OF THE STEAM-ENGINE. 121 
 
 In considering the weight and speed of a fly-wheel, that 
 only of its rim will be considered, and it is also proper to 
 state here, to avoid leading our readers astray, that, unless 
 specially noted, the formulae to be subsequently established 
 do not take cognizance of the variation in work given out by 
 the steam-cylinder produced by the angular position of the 
 connecting-rod ; that the suppositions are also made that the 
 mean pressure of the steam is uniform throughout the stroke ; 
 and that the work given out by the fly-wheel is given out 
 uniformly. 
 
 The variation in the work given out by the steam-cylin- 
 der, produced by the angularity of the crank, is specially 
 considered. 
 
 When any body of a weight W is in motion with a ve- 
 
 TFv 2 
 locity v, it has stored up in it work = w = , and all of 
 
 this work must be given out before it can come to rest. If 
 this weight is not entirely brought to rest, but its speed re- 
 duced to !, it will, while being retarded, give out work 
 
 ( 1 65) 
 
 Or if the body be moving with a velocity v 1} and by the 
 action of a force its speed be increased to a velocity v, it will ' 
 store up work 
 
 Now, (v j Vi 2 ) =*(v+Vi*)(v v^), and if we take the mean 
 uniform speed of the body = u = -, when v and Vi are 
 
 supposed not to differ greatly, and denote by m the frac- 
 11 
 
122 THE RELATIVE PROPORTIONS 
 
 tional part of the mean velocity u, by which v and Vi are 
 allowed to differ, we have mu =*v v 1} and we have 
 
 (v 2 - Vj 2 ) = (v + Vi~) (v - vj 
 and formula (165) takes the following form : 
 
 w - m *. 9 (166) 
 
 9 
 
 in which ^ = 32.2, and which gives the amount of work 
 gained or lost by the body when its speed is increased or 
 decreased by (v - 1^) = mu. 
 
 For the present it will suffice to say that m is taken from 
 i to TOIT> according to the degree of regularity required, and 
 that u is the mean speed per second in feet. 
 
 We will next take up the amounts of work lost and 
 gained by the fly-wheel while receiving work from the 
 steam-cylinder in periodically varied quantities, and part- 
 ing with it, on the other hand, in uniform quantities to the 
 machinery driven. 
 
 If the work is not given out in uniform quantities, as is 
 sometimes the case, special provision should be made for it 
 by increasing the weight of the fly-wheel, so as to meet and 
 overcome this source of irregularity, or, better, by the use of 
 a fly-wheel at the point at which the variations occur, cal- 
 culated to meet and overcome the irregularities at that 
 point. 
 
 "With an early cut-off of the steam, the irregularity due 
 to the variable pressure on the piston is superadded to that 
 due to the angular position of the crank and the connecting- 
 rod, and will be specially noted hereafter in Table IV. for 
 such cases as may occur in which the weight of the piston 
 and appurtenances, is not or cannot be proportioned to the 
 speed of the reciprocating parts and the steam-pressure. 
 See Art. (35). 
 
OF THE STEAM-ENQINE. 123 
 
 To establish a clear understanding between the reader 
 and ourselves, we here define work to be force multiplied by 
 the space passed over by that force. 
 
 (57.) Fly- Wheel, Single Crank. Let JP=the steam- 
 pressure per square inch upon the piston-head. 
 
 Let d = the diameter of the steam-cylinder in inches. 
 " /S^the total pressure upon the piston-head. 
 
 The force pressing upon the piston-head is, as before 
 stated, for a single cylinder, 
 
 S = ^P. (167) 
 
 4 
 
 (Assuming the engine to have attained its average speed, 
 and neglecting the small variation produced by the connect- 
 ing-rod,) 
 
 Letting r = radius of crank, 
 
 " a = angle formed by centre line of crank with the 
 
 centre line of cylinder, 
 " s = space passed over by the piston, 
 
 the distance passed over by the piston is 
 
 s = r(l-cos). (168) 
 
 Therefore, we have for the work derived from the pis- 
 
 ton = Wi 
 
 cosa). (169) 
 
 To find the increments of work corresponding to each 
 increment of arc, we differentiate (169), giving 
 
 dw^Srsin ada. (170) 
 
 The total amounts of work gained and lost by the fly- 
 wheel during one stroke equal SL. \_L the length of the 
 stroke.] 
 
124 THE RELATIVE PROPORTIONS 
 
 As, however, the work is gained in varied increments, as 
 shown by (170), and on the other hand is lost in assumed 
 uniform quantities to the machinery driven, there are points 
 at which the increments of the gained and the lost work are 
 equal. 
 
 The lost work during the passage of the crank through 
 the angle a is 
 
 ra. (171) 
 
 8 
 
 Since = the lost work for the unit of arc, and differ- 
 entiating (170), we have 
 
 dw^-Srda. (172) 
 
 7T 
 
 Placing equations (170) and (172) equal to each other, 
 we have 
 
 2 
 Sr sin a = - Sr. 
 
 7T 
 
 Therefore, sin a - - - 0.636618, (173) 
 
 7T 
 
 which gives the following values, 
 
 Angle - 39 32' 25" 
 or = 140 27' 35" 
 or = 219 32' 25" 
 or = 320 27' 35" 
 
 We see from equation (170) that the increment of gained 
 work is also equal to for a = or 180, and a maximum 
 for a = 90 or 270, while the increment of lost work, equa- 
 tion (172), is a constant. 
 
 If now we substract the value of equation (169) for 
 
OF THE STEAM-ENGINE. 125 
 
 a - 39 32' 25" from its value for a - 140 27' 35", we have 
 
 Wi = $r[(l + cos a) - (1 - cos a)] ; 
 therefore, w, = 2Sr cos a - 1.54232r, (175) 
 
 and (175) is the total amount of work received from the 
 steam-cylinder by the fly-wheel from the point where the 
 increment of gained work is equal to the increment of lost 
 work until they are again equal. In order to find the sur- 
 plus of work absorbed by the fly-wheel, we must subtract the 
 total amount of lost work (lost uniformly) during the same 
 interval. 
 
 If now we subtract the value of equation (171) for 
 a = 39 32' 25" from its value for a - 140 27' 35", we have 
 
 100.91945 \ 
 
 IT: 
 
 180 / 
 
 Substracting (176) from (175), we have for the work 
 stored by the fly-wheel between the two values of a, above 
 
 stated, = i0 3 , 
 
 W3 - ( Wl _ w 2 ) = 0.42099r, 
 
 and substituting for r its value -, we have, for a single 
 
 2 
 
 cylinder, 
 
 fK,-.2104aSL, * (177) 
 
 or about .21 of the work done during the whole stroke. 
 
 If we substitute for S its value given in equation (167), 
 and take L in feet, we have, in terms of the pressure diam- 
 eter of steam-cylinder, letting c represent the numerical 
 coefficient derived from calculation or Table IV., 
 
 i0 3 = .7854P<f cL. (178) 
 
 To prove that the work lost by the fly-wheel, in passing 
 from crank-angle 140 27' 35" to 219 32' 25", is equal to 
 11* 
 
126 THE RELATIVE PROPORTIONS 
 
 that gained in passing from crank-angle 39 32' 25" to 
 140 27' 35". 
 
 Referring to equation (175), we see that the total work 
 gained between the limits a = 140 27' 35" and a = 219 32' 
 25", if we properly alter the signs, is 
 
 and, referring to equation (171), proceed as in equation (176), 
 we have for the total lost work of the fly-wheel 
 
 = 2^(0.4393). (180) 
 
 cos a = 0.77116, . . (1 - cos a) = .22884. 
 Subtracting equation (179) from (180), we have 
 (w t - = 2$r(0.4393 - .2288) - .421 Sr = w 3 the work lost, 
 
 and we see that the amount of work gained by the fly-wheel 
 in an arc of 100.92 degrees is equal to that lost in an arc 
 of 79.08 degrees. 
 
 Now, for the sake of a clear understanding of this topic, 
 let us follow the crank through one revolution. (See Fig. 20.) 
 From crank-angle 39 32' 25" to 140 27' 35" the an- 
 gular velocity of the crank and fly-wheel increases, attaining 
 FIG ^ a maximum at 140 + , the fly- 
 
 wheel storing up work. From 
 
 140 + to 219 + the an S ular ve ~ 
 locity of the crank and fly- 
 
 wheel decreases, reaching a min- 
 imum at 219 + , the fly-wheel 
 giving out work equal in amount 
 to that stored up in the first arc 
 mentioned ; from 219 + to 320 + 
 
 the fly-wheel again stores work, and from 320 + to 39 + 
 
 gives the same amount up. 
 
OF THE STEAM-ENGINE. 127 
 
 (58.) Fly-Wheel, Double Crank, Angle 90. The 
 
 most advantageous angle between the cranks being 90, as 
 shown in Art (54), the total work of the two cranks will be, 
 if we 
 
 Let a. = the angle between the centre line of the cylinder and 
 
 the first crank, 
 
 " S = the pressure in pounds upon each piston-head, 
 " r = the radius of the crank, 
 " M! = the work given out by the two cylinders, 
 
 since cos (90 + a) = sin a, 
 
 -l)]. (181) 
 
 Reducing and differentiating, we have the increment of 
 gained work for each increment of arc 
 
 dwi = /Sr(sin a + cos a)da. (182) 
 
 Multiplying equation (172) by 2, we have for the incre- 
 ment of the lost work for each increment of arc 
 
 dw^-Srda. (183) 
 
 7T 
 
 Equating equations (182) and (183), we have 
 
 4 
 Sin a + COS a = -. 
 
 7T 
 
 Squaring both members, 
 
 /4\ 
 sin 2 a + cos 2 a + 2 sin a cos = [-]. 
 
 w 
 
 Therefore, sin 2 - (* Y - 1 - 0.62114. (184) 
 
 Giving 2a - 38 24' 
 
 and = 19 12'. 
 
 Remembering that after passing through 90 of arc the 
 
128 THE RELATIVE PROPORTIONS 
 
 same variations are repeated, we have for the various values 
 of a, when the increments of lost and gained work are equal, 
 
 19 12 '.. ...51 36' 
 
 109 12' 51 o 36 , 
 
 160 48' ' 
 
 199 12' 
 
 250 48' 
 
 289 12T 
 
 340 48', etc. 
 
 From (182) we further see that the increment of gained 
 work is a maximum for a = 45 and a minimum for a = 
 or 90. 
 
 To determine the excess of work lost or gained, we have, 
 equation (181) reduced, 
 
 Wi = $r[l cos a -i- sin a]. 
 
 For a = 70 48' this equation .................... =Sr* 1.61 5509 
 
 For a = 19 12' this equation .................... =rx 0.384491 
 
 The total amount of work gained ........... = Sr* 1.231018 
 
 K"1 O/J 
 
 The total amount of work lost is 4Sr - = Sr x 1.146667 
 
 180 
 Giving for the gained or lost work ............ Sr x 0.084351 
 
 If for r we substitute its value , we have 
 
 w,-0.0422SL. (185) 
 
 As in the preceding article, we see that the angles deter- 
 mined correspond to points of minimum and maximum an- 
 gular velocity of the fly-wheel, and it is easy to prove that 
 the work gained by the fly-wheel in passing from say 19 12' 
 through 51 36' is equal to the work lost in passing from 
 70 48' through an arc of 38 24'. 
 
OF THE STEAM-ENGINE. 129 
 
 CHAPTER XII. 
 
 (59.) Fly-Wheel, Triple Crank, Angle 120. Let- 
 ting the notation be the same as in the preceding article, we 
 have, for the work given out by the three cranks, 
 
 Wi = >Sr[(l - COS ) + (!- COS (a + Jrr) - f ) 
 
 + l-cos(+f7r)-i)], (186) 
 
 s 
 
 which, after reduction, becomes 
 
 W= Sr[l - cos a + i/Fsur a], (187) 
 
 Differentiating, and neglecting the constants, we have 
 
 (188) 
 da 
 
 n 
 
 which can be placed equal to - = the work lost in each ele- 
 
 ment of arc, to find the points at which the increments of 
 the gained and the lost work are equal. 
 
 -|/3~cos a = sin a cos 60 + cos a sin 60 
 since - -j/Fand cos 60 = J. 
 
130 THE RELATIVE PROPORTIONS 
 
 We then have 
 
 sin (a + 60) = -. 
 
 7T 
 
 Therefore, ( + 60) = 72 44' and a = 12 44', 
 or (a + 60) = 107 16' and a = 47 16'. 
 
 Since the same phases are repeated after the cranks have 
 passed through an angle of 60 degrees, we need consider 
 this equation only between and 60 degrees. 
 
 Expression (188) becomes a minimum for = or = 60, 
 and is a maximum for = 30. 
 
 To determine the amount of work lost or gained by the 
 fly-wheel, we substitute in equation (187) the values 
 a = 47 16' and a = 12 44'. 
 
 For = 47 16' equation (187) becomes .......... rxl.5936 
 
 For a = 12 44' equation (187) becomes ......... . Sr* 0.4063 
 
 The total amount of work gained. is ............... $rx 1.1873 
 
 The total amount of work lost is for an arc | 
 
 of 34 32'=34.533 ......... : ............... }_ 
 
 Giving for the gained work absorbed by the "> 
 
 fly-wheel in an arc of 34.533 ............. } 
 
 To find the work lost by the fly-wheel as a check, we have 
 
 For the work lost in an arc of 25 28' ............ $rx 0.8488 
 
 For the work gained in an arc of 25 28' ......... Srx 0.8126 
 
 Giving for the lost work given out by the fly- ) 
 
 wheel in an arc of 25 28' .................. j ^rxO.0362 
 
 If for r we substitute its value , we have 
 
 2 
 
 (189) 
 
 "We can, in a similar manner to that already explained, 
 determine the six points of maximum and the six points of 
 minimum velocity of the fly-wheel. 
 
OF THE STEAM-ENGINE. 
 
 131 
 
 (60.) Of the Influence of the Point of Cut-off and the 
 Length of the Connecting-Rod upon the Fly-Wheel. 
 
 Let $ = the total pressure of the steam upoa the piston-head. 
 " r = radius of crank. 
 " l = nr = the length of the connecting-rod. 
 " a = the angle between the crank and the centre line O X. 
 " <p = the angle between the connecting-rod and the cen- 
 tre line OX. (See Fig. 21.) 
 FIG. 21. 
 
 r 
 
 Referring to equation (170), Art. (37), we see that the 
 increments of work given out with a varying velocity are 
 proportional to sin a, the force S being assumed constant ; 
 as the crank-pin may be assumed to move in a tangential 
 direction with a constant velocity, the condition of the 
 equality of the increments of work at the points O and B * 
 requires that the tangential or torsional force T shall vary 
 as the sin a. (This can be proved graphically also.) 
 
 CASE I. Let S be constant and 
 the length of the connecting-rod 
 be assumed infinite. 
 
 With a radius AB, Fig. 22, 
 representing the constant force S 
 acting on the piston to a con- 
 venient scale, describe the semi- 
 circle A 3 C. Divide this semi-perimeter into any number 
 
 * Theorem of virtual velocities. 
 
 FIG. 22. 
 
 1 
 
132 
 
 THE RELATIVE PROPORTIONS 
 
 FIG. 23. 
 
 of equal spaces, as 1, 2, 3, 4, 5, C. From these points drop 
 verticals to the line AC; these verticals will represent the 
 tangential forces T at these points. 
 
 If we lay off a horizontal line, as O X, Fig. 23, representing 
 the semi-perimeter of the crank-circle to any convenient 
 scale, and divide it into six equal parts, and upon these 
 divisions erect verticals of equal length to the verticals in 
 
 Fig. 22, we obtain the 
 curve of sines O D 6, and 
 the area of this figure, 
 O D 6 O, represents the 
 work done in one stroke. 
 
 u ' * J * * * The work lost by the en- 
 gine, being assumed lost 
 
 uniformly, can be represented by the rectangle O E G X O, 
 in which the vertical OE = G6 = the work done in one 
 stroke divided by the distance O 6, and the sum of the 
 areas of EHO + KG6 = area DK H. 
 
 CASE II. Let & be variable (steam cut-off) and the 
 length of the connecting-rod assumed infinite. 
 
 ' The different pressures caused by cutting off steam can 
 be calculated by Mariotte's law, or more correctly taken 
 from an indicator diagram, as shown in Fig. 24, by laying 
 off versines, say for each 30 degrees of arc on a diameter 
 equal to the length of the indicator diagram. 
 FIG. 24. 
 
 V 
 
 With the greatest force as a radius = O a, Fig. 24, describe 
 
OF THE STEAM-ENGINE. 
 
 133 
 
 the semi-circle A 3 C with 3 as a centre, Fig. 25 ; divide this 
 into six equal arcs and draw the radii 1, 2, 3, 4, 5 to 3; 
 upon these radii lay off the forces as measured from the cor- 
 responding ordinate of the indicator diagram, Fig. 24. Con- 
 necting these extremities we have the curve A 1, 2, 3, 4j, 5 1} 
 61, and the verticals 11, 22, 33, 4 1; 4, 5^ 5 ; from the intersec- 
 tions of this curve with the radii to the line A C give the 
 tangential forces acting upon the crank. 
 
 FIG. 26. 
 
 O / 
 
 If we divide the line O X, Fig. 26, representing the length 
 of half the crank-circle to any convenient scale, into six equal 
 parts, and at the points of division erect verticals equal in 
 length to the verticals in Fig. 25, we obtain the irregular 
 solid curve ODX. The area of this figure, ODXO, 
 equals the work done in one stroke. 
 
 CASE III. Let the force S be variable and the length 
 of the connecting-rod be taken into consideration. 
 
 FIG. 21. 
 
 Referring to Fig. 21, we see that the effect of the connect- 
 12 
 
134 THE RELATIVE PROPORTIONS 
 
 ing-rod of finite length is to cause the piston to move farther 
 than the crank-pin in a horizontal direction in the first and 
 fourth half strokes, and to move a less distance than the 
 crank in a horizontal direction in the second and third half 
 strokes. This can be shown in the following manner : 
 
 We have, for the work done in the cylinder with a vari- 
 able velocity and on the crank-pin with a constant velocity, 
 
 w = r[(l - cos ) +n(l -"cos ?)] J (190) 
 
 . 2 sin 2 a . 
 sm 2 <p = , since r sin a = nr sin <p ; 
 
 /Sri -cos + 
 
 sin 
 
 2n 
 
 Neglecting quantities containing greater than the second 
 
 power of - , and differentiating, we have 
 % n 
 
 o 
 
 - or tern a + 2 sin a cos a\ , 
 da. 2ra 
 
 and the tangential force which is proportional to the first 
 differential coefficient of the work, since the velocity of the 
 crank-pin is constant, is proportional to 
 
 sin2a /ini\ 
 
 sin a+ -- . (191) 
 
 2n 
 
 Attention must be paid to the signs of the circular func- 
 tions. 
 
 Laying down a straight line O X, Fig. 26, dividing it, 
 and erecting ordinates as before, we lay off the length of 
 
OF THE STEAM-ENGINE. 
 
 135 
 
 these ordinates, which in the first quadrant are greater and 
 
 in the second quadrant less by than for the solid 
 
 curve O D X. 
 
 FIG. 26. 
 
 In the third quadrant the ordinates will be less and in 
 
 the fourth quadrant greater by than in Case II., 
 
 11 
 
 giving the broken line curve O b c d e 6. 
 
 The work lost, being lost uniformly, can be represented by 
 the rectangle O E G X O, and the height of this rectangle 
 = O E = G X is equal to the work done in one stroke, repre- 
 sented by the figure O b D d X O, divided by half the 
 length of the crank-circle = O X. 
 
 The areas of these figures, as also the equal amounts of 
 work lost and gained by the fly-wheel, can be calculated by 
 means of Simpson's rule, or more conveniently by the use 
 of the polar planimeter. 
 
 Table IV. gives the work lost and gained by the fly-wheel 
 in terms of the fractional part of the whole work done by one 
 cylinder in one stroke, or, what is the same thing, the value 
 of the coefficient c in equation (178) for the common values of 
 
 n = -, and the usual points of cut-off when the influence of 
 
 the weight and velocity of the reciprocating parts is dis- 
 regarded. 
 
 If we wish to use this table when the indicated horse-power 
 of an engine is given, divide the horse-power of one cylinder by 
 
136 
 
 THE RELATIVE PROPORTIONS 
 
 the number of strokes per minute, and multiply it by 33000 
 foot-pounds, since the work done in one stroke 
 
 33000 
 
 and multiply the result by the coefficient, given in the table. 
 
 TABLE IV. 
 [From Des Ingenieurs Taschenbuch, page 379.] 
 
 Engine without 
 expansion. 
 
 Engine with expansion. 
 Steam cut off at- 
 
 I 
 
 Single 
 
 Two 
 
 Three 
 
 
 4L 
 
 * 
 
 1 Z 
 
 4Z, 
 
 j 
 
 *L 
 
 \L 
 
 * 
 
 n> = r 
 
 crank. 
 
 cranks. 
 
 cranks. 
 
 
 
 2 
 
 3-^ 
 
 * 
 
 7 
 
 
 4 
 
 0.2717 
 
 0.1672 
 
 0.0693 
 
 (f Single 
 1 crank 
 
 0.3741 
 
 0.4076 
 
 0.4372 
 
 0.4523 
 
 0.4625 
 
 0.4702 
 
 5 
 
 0.2577 
 
 0.1422 
 
 0.0594 
 
 
 
 
 
 
 
 
 6 
 7 
 
 0.2489 
 0.2489 
 
 0.1256 
 0.1136 
 
 0.0504 
 0.0453 
 
 (Two 
 (cranks 
 
 0.2044 
 
 0.2250 
 
 0.2412 
 
 0.2495 
 
 0.2552 
 
 0.2594 
 
 8 
 
 0.2384 
 
 0.1046 
 
 0.0414 
 
 /Single 
 M crank 
 
 0.3252 
 
 0.3594 
 
 0.3916 
 
 0.4088 
 
 0.4216 
 
 0.4300 
 
 Infinite 
 
 0.2105 
 
 0.0422 
 
 0.0181 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 \f Two 
 (cranks 
 
 0.0652 
 
 0.0720 
 
 0.0786 
 
 0.0820 
 
 0.0846 
 
 0.0862 
 
 It is the best plan to make use of the coefficient of single 
 cranks for double cranks, and of the coefficient for double 
 cranks for treble cranks, since it is sometimes necessary to 
 disconnect one cylinder for repairs, which we should be able 
 to do without creating such irregularities as will render the 
 engine unserviceable. 
 
 If we take the pressures upon the crank-pin as altered by 
 the weight and velocity of the reciprocating parts, as shown 
 in Fig. 10, Art. 35, and treat them as explained in this 
 article, we can obtain an exact representation of the work 
 lost and gained by the fly-wheel. For ordinary practice, 
 with the weight and velocity of the reciprocating parts 
 properly adjusted, we can use the coefficients for engines 
 without expansion in Table IV. 
 
OF THE STEAM-ENGINE. 137 
 
 CHAPTER XIII. 
 
 (61.) The Weight of the Rim of Fly-Wheels. 
 
 Referring to equation (166), we see that the amount of work 
 which any body can store up without having its velocity in- 
 creased, or give out without having its velocity decreased 
 more than the fraction m of the mean velocity u, is 
 
 w = Wu\ 
 9 
 
 If now we place this amount of work, equal to the amount 
 given by equation (178), as the excess or deficiency of work 
 to be stored or given out by the fly-wheel, we have 
 
 in .^ 
 
 9 ' 
 
 therefore, W,= 25.29 ^, (192) 
 
 m v? 
 
 in which L is the length of stroke in feet, and u the mean 
 velocity of the rim of the wheel in feet per second. 
 
 If for u we wish to substitute the mean diameter of the 
 fly-wheel rim D and the number of strokes per minute N, 
 
 -. 02618 DJV. 
 
 2x60 
 Squaring and substituting in formula (192), we have 
 
 (193) 
 
 The mean diameter of a fly-wheel D is usually taken at 
 from 3 to 10 times the length of the stroke. Inspection 
 12* 
 
138 THE RELATIVE PROPORTIONS 
 
 of formula (193) shows that, other things being equal, it 
 will diminish the weight and cost of a fly-wheel to increase 
 the number of strokes the diameter of the wheel, or to in- 
 crease the fractional coefficient m. 
 
 (62.) Value of the Coefficient of Steadiness m. 
 Rankine recommends taking m = -fa for ordinary ma- 
 chinery, and = -fa to fa for machinery requiring unusual 
 steadiness. 
 
 Watt's rule, given by Farey, and regarded by him as 
 giving sufficient regularity for the most delicate purposes, is, 
 " Make \ the vis viva i. e., the work stored in the fly-wheel 
 equal to the work done by the engine in 3f strokes," and 
 gives m = -fa. 
 
 Bourne's rule for all engines is, " Make the work stored 
 in the fly-wheel equal to that developed by the steam- 
 cylinder in six strokes," which would give m little greater 
 than -fa, a quantity much too small for ordinary purposes, 
 but nearer right than he is usually. Nystrom suggests 
 making m,- in practice, to vary between -fa and yjg-. 
 
 Morin gives, for engines requiring great regularity, the 
 value of m at from -fa to ^, which conforms with the best 
 practice, for engines of great steadiness of motion. 
 
 m = -fa is a good value for engines in which small tem- 
 porary fluctuations of speed are of little consequence. 
 
 The following values of m are taken from Des Ingenieurs 
 Taschenbuch, Hiitte, page 378 : 
 
 For machines which will permit a very uneven motion, as 
 for hammer-work, m = . 
 
 For machines which permit some irregularity, as pumps, 
 shearing-machines, etc., m = -fa to -fa. 
 
 For machines which require an approximation to a uni- 
 form speed, as flour-miDs, m = -fa to fa. 
 
OF THE STEAM-ENGINE. 139 
 
 For machines which demand a tolerably uniform speed, 
 as weaving-machines, paper-machines, etc., m = -$ to -fa. 
 
 For machines which demand a very uniform speed, as 
 cotton-spinning machinery, m =- -fa to ^j. 
 
 For spinning machinery for very high yarn numbers, 
 
 Example. Let n = - = 5 ; then from column first of the 
 r 
 
 table we have c = .2577 =- \ approx. 
 
 Let the uniform pressure be P=61.5 pounds per square 
 inch, rojfo, i-1 foot, d = 12 inches, D = 5 feet, JV-lOO 
 strokes per minute. 
 
 Substituting in formula (193), we have 
 
 ^ 36899x20x61.5x144 
 W f = - - --- = 6535 pounds. 
 4x25x10000 
 
 Upon reflection, we see that this would be comparatively 
 a very great weight, and, if possible, it will be best to in- 
 crease the diameter of the fly-wheel, so as to lessen its weight. 
 Letting D = 10 feet, we have 
 
 W f = 1634 pounds, 
 with an equal coefficient m of steadiness. 
 
 (63.) Area of the Cross-Section of the Rim of a Fly- 
 Wheel. Given the weight and mean diameter of ike rim of 
 a fly-wheel to determine the area of its cross-section. 
 
 Let D be the mean diameter in feet of a fly-wheel rim. 
 " W f be the weight in pounds. 
 
 " F be the cross-section of a fly-wheel rim in square inches. 
 We have for the mean perimeter in inches 
 
 127TD, 
 
140 THE RELATIVE PROPORTIONS 
 
 and for the volume of the rim in cubic inches 
 
 One cubic inch of iron weighs about .26 of a pound, and 
 
 we therefore have 
 
 TF 7 =. 26x12^1). 
 
 Therefore, ^=.10176^. (194) 
 
 Examples. Let W f = 6535 pounds. 
 " D = 5feet. 
 
 We have F= .10176 - - 133.05 square inches. 
 
 Again, Let W f = 1634 pounds. 
 " D = 10 feet. 
 
 We have F= .10176 - = 16.63 square inches. 
 
 (64.) Balancing the Fly-Wheel. When the fly-wheel 
 is erected, great care should be taken that its centre of gravity 
 coincides with the centre of the shaft upon which it is placed. 
 
 It is not necessary that it be perfectly circular, so long as 
 its centre of gravity coincides with its axis, as will presently 
 be shown. 
 
 With regard to a plane passing through its centre of 
 gravity at right angles to the axis, the wheel must be per- 
 fectly symmetrical. If it is not, the centrifugal force will 
 give rise to a couple tending to place it in that position, and 
 of course straining the shaft and the fly-wheel unnecessarily. 
 In both cases these disturbing forces will increase with the 
 square of the velocity. 
 
 If any two masses, as W and w, are in statical equilibrium 
 with regard to an axis C; if caused to revolve, they will have 
 no tendency to leave their common axis. 
 
OF THE STEAM-ENGINE. 141 
 
 FIG. 27. 
 
 C 
 
 
 Let W= weight of the larger mass. 
 " w = weight of the smaller mass. 
 " R distance of the larger mass W from C. 
 " r distance of the smaller mass w from C. 
 
 Since these bodies are assumed in statical equilibrium, we 
 have 
 
 wv* 
 
 The centrifugal force of the smaller body = - . 
 
 gr 
 
 The centrifugal force of the larger body = 
 
 gR 
 
 Letting N represent the number of turns of the connected 
 bodies about C per second, we have 
 
 V=2*RN, v 
 
 TFF 2 
 
 and - * - , and 
 
 gr gr gR gR 
 
 And dividing, we have 
 
 wp 
 
 gr wr 
 WV Z WR' 
 gR 
 
 But wr = WR. Hence, the centrifugal forces of the two 
 masses are equal and opposite, which was to be shown. 
 
 The fly-wheel is sometimes purposely erected out of bal- 
 ance, in order to force the crank over some particular point, 
 
142 THE RELATIVE PROPORTIONS 
 
 but this method of proceeding results in strains upon the 
 shaft and its bearings, which tend to injure them by causing 
 wear and vibration, and should not be used when avoidable. 
 Eefer to Art. (35). 
 
 (65.) Speed of the Rim of a Fly- Wheel. It has been 
 stated in article (61) that increasing the speed of the rim of 
 a fly-wheel, or, what is the same thing, increasing its diam- 
 eter and number of revolutions, one or both, is productive of 
 economy in its weight. 
 
 There is, however, a limit beyond which the speed of rim 
 cannot be driven without bursting the rim. 
 
 If we suppose the rim to be solid, and neglect the sup- 
 port that it receives from its arms, we first have the case in 
 which centrifugal force acts in a similar manner to the out- 
 ward bursting-pressure of water. (Weisbach's Mechanics 
 of Engineering, sec. vi., art. 363.) 
 
 Letting T=ihe tensile strength per foot of area, 
 " = mean radius of rim in feet, 
 
 " u = velocity of rim in feet per second, 
 
 G = weight per cubic foot, 
 
 " p = radial force of each cubic foot due to centrif- 
 ugal force, 
 
 we have for the centrifugal force of each cubic foot 
 
 = p = -- , and also, as shown in Weisbach's Mechanics of 
 
 gD 
 
 2T 
 Engineering, the resisting force is p = - . Equating these 
 
 values, we have T= - . 
 
 9 
 
 Therefore, "'' (195) 
 
OF THE STEAM-ENGINE. 143 
 
 Examples. For a cast-iron rim, assuming T= 2592000 
 pounds per square foot, and its weight G = 450 pounds per 
 cubic foot, we have for its bursting speed 
 
 32.2x2592000 , Qn -, . , 
 
 u = -*/ = 430.7 feet per second. 
 
 If we use a factor of safety of 10, we have 
 
 132.2x259200 10CO - , 
 
 u = -*/ = 136.2 feet per second 
 
 for a safe speed. 
 
 The speed of rim of fly-wheels is in some cases pushed to 
 about 80 feet per second, but is probably not often exceeded* 
 
 It is of interest to note that if 5000 pounds per square 
 inch be regarded as a safe strain for a railway tire when 
 subjected to shocks occurring while in motion, the greatest 
 speed of a locomotive with safety may be deduced. 
 
 We have _ . . 
 
 490 
 
 u = */ - = 217.|- feet per second, 
 
 4yu 
 
 or about 2.47 miles per minute. 
 
 CHAPTER XIV. 
 
 (66.) Centrifugal Stress on the Arms of a Fly- 
 Wheel. In the case just discussed the fly-wheel has been 
 supposed to rely entirely upon the strength of its solid rim, 
 regardless of any support it might have from the arms; 
 when, however, large wheels are constructed, the difficulties 
 attendant upon handling them, as well as the expense of 
 making large castings, make it necessary to cast the rim in 
 sections, which are put together in place, the arm sometimes 
 
144 
 
 THE RELATIVE PROPORTIONS 
 
 FIG. 28. 
 
 being cast with its section of rim, which is the best method 
 when practicable, and sometimes in a separate piece. These 
 segments on being put in place are 
 caused to abut firmly upon each other, 
 and held in position by means of a 
 prisoner and keys, as shown at B, or 
 by means of links, as shown at A, 
 Fig. 28. A double-headed bolt some- 
 times takes the place of the link. Both 
 link and bolt when used are heated 
 and then allowed to contract in place, 
 drawing the ends of the segments 
 solidly together. 
 
 For various practical mechanical 
 reasons, the strength of the bolts, 
 prisoners or links cannot be relied 
 upon, and they should only be considered as fastenings to 
 preserve the form of the fly-wheel. 
 
 The arm for each segment should be so proportioned as 
 to support with perfect safety its weight when at its lowest 
 position, and also the stress due to its centrifugal and tan- 
 gential force, a slight taper being given to the arm, increas- 
 ing from the rim to the hub, to allow for the weight and 
 centrifugal force of the arm itself. 
 
 The centrifugal force of any mass is exactly what would 
 result if its whole mass is supposed concentrated at its cen- 
 tre of gravity and the motion of this point alone considered. 
 We have (Weisbach's Mechanics of Engineering, sec. iii., 
 art. 115), for the distance from the centre of the wheel to 
 the centre of gravity of the segment, = x, 
 
OF THE STEAM-ENGINE. 
 
 145 
 
 in which e is the thickness of rim, a = the angle A C B, and 
 D is its mean diameter. Fig. 29. 
 
 FIG. 29. 
 
 Let W f the weight of the rim in pounds. 
 " M = the number of sections = number of arms into 
 
 which the rim is divided. 
 " (7= the centrifugal force of each segment. 
 " u = the velocity of the rim in feet per second. 
 " v - the velocity of the centre of gravity of each 
 
 segment. 
 
 We have 
 and since 
 we have 
 
 M gx 
 u:v::D:2x, 
 4zV 
 
 Substituting the value of x given above, we have 
 
 4 rr / " 
 Jf ^D 
 
 13 
 
146 THE RELATIVE PROPORTIONS 
 
 In this latter formula, if e is very small in comparison to 
 jD, we can neglect the small term ^ ( J , and the equation 
 
 i n A W f u? sin .j 
 
 becomes = 4- - . 
 
 M gDa 
 
 If to this expression for the intensity of the centrifugal 
 force of each segment we add its weight for its lowest posi- 
 tion, we have, calling ythe strain on any arm in the direc- 
 tion of the radius of the wheel, 
 
 (196) 
 
 and sioce w 2 - .0006853924 D*N\ we have 
 . 00274156 D 
 
 sinrl ( 
 
 ga 
 
 in which g = 32.2 feet per second. 
 
 If we divide the intensity of the force Y by the safe strain 
 in tension per square inch of the material of the arm, we 
 obtain the required cross-section to resist rupture from the 
 centrifugal force and weight of each segment. 
 
 Example. Let F^=1634 pounds. Let M=6 i.e., the 
 rim be divided into six parts, each having its arm. Let 
 D = 10 feet and JV- 100 strokes per minute. 
 
 We also have - - 30, and a - y = 1.0472. 
 
 1388 pounds approximately. 
 
OF THE STEAM-ENGINE. 147 
 
 (67.) Tangential Stress on the Arms of a Fly 
 Wheel for a Single Crank. In addition to the stress on 
 the arm due to its weight and centrifugal force, each arm 
 sustains a tangential strain at its extremity due to the inertia 
 of the rim, which, in case of sudden stoppages, is sometimes 
 of very great intensity. 
 
 For the case of any ordinary fly-wheel, whose only office is 
 to equalize the work given out in each element of time or arc, 
 
 Letting R = the mean radius of the rim in feet, 
 
 " JT=the sum of the tangential forces at the extrem- 
 ities of the arms in pounds, 
 
 we have, for the work given out or absorbed for each 
 element da of arc, 
 
 XRda. 
 
 We also have, equation (33), for the work received from 
 the steam-cylinder for each element da of arc, 
 Sr sin a da, 
 
 and, equation (35), for the work given out uniformly for each 
 
 2 
 element da of arc, Sr-da. 
 
 it 
 
 We see that the rim is called upon at certain points or 
 during certain arcs to assist or resist the work given out by 
 the steam-cylinder. 
 
 If we place 
 
 XRda = Sr sin a da - Sr-da, (198) 
 
 7T 
 
 v ^ 
 or X 
 
 we obtain the measure of the tangential force X. 
 39 32' 25" 
 
 320 27' 35" 
 
148 TEE RELATIVE PROPORTIONS 
 
 . For a = 90 = 270 we have 
 
 X= +.36338. 
 For a = = 180 we have 
 
 X= -.636618. (199) 
 
 This last value of X represents the maximum value of X, 
 being its value for the two dead points ; and if we divide 
 this by the number of arms, we obtain the force at the ex- 
 tremity of each arm which tends to bend or to break it at 
 its junction with the hub of the wheel, or the rim. We have 
 (Weisbach's Mechanics of Engineering, sec. iv., art. 272) the 
 following equation for a beam fixed at one end and loaded 
 at the other: 
 
 '-ZJ*\ * 
 F \M)' w 
 
 w 
 
 in which M = the number of arms ; F= cross-section of an arm 
 in square inches ; W= its measure of the moment of flexure ; 
 / = the length of arm in inches ; T= the proof (or safe) stress 
 per square inch ; Y= the radial stress on each arm ; X= the 
 tangential stress on each arm ; and e = the half diameter of 
 the arm in the plane of the fly-wheel; or inversely, 
 
 H{-(i)4 
 
 In this formula, for round and elliptical arms 
 
 Fe 4 
 
 ~W~e' 
 
 For rectangular arms, 
 
 ~W = e' 
 
 (Weisbach's Mechanics of Engineering, Art. 236, Sec. iv.) 
 In formula (200) the value of e remains to be determined 
 
OF THE STEAM-ENGINE. 149 
 
 approximately. This can be done by substituting in either 
 of the two following formulae, and taking the greater value, 
 
 :,': -*!?-. (201) 
 
 (Weisbach's Mechanics of Engineering, Art. 235, Sec. iv.) 
 
 F-Jr- ( 202 ) 
 
 Example. Let us assume the shape of the arm of the fly- 
 wheel already discussed to be elliptical. 
 
 X 
 
 Y= 282 pounds, = 141 pounds. 
 M 
 
 We have, equation (202), 
 
 Therefore, 
 
 in which b = the smaller -^-diameter of the arm assumed, 
 
 and, equation (201), since TF= for an ellipse, 
 
 4 
 
 (Weisbach's Mechanics of Engineering, Art. 231, Sec. iv.) 
 
 Let 6-1", T=1800, and J = 60 inches. 
 
 282x7 
 
 / 
 
 V 
 
 141x60x7 
 22x1800 
 
 13* 
 
150 THE RELATIVE PROPORTIONS 
 
 The second value of e = 2.44 inches must be substituted in 
 formula (200), and we have 
 
 If 4/X\ 1 4x141x60 
 
 = 7.862 square inches. 
 We have assumed 6 = 1 inch ; and since F= neb, we have 
 
 e = x 7.86 = 2.5 inches. 
 
 We thus see that each arm of a fly-wheel of the dimen- 
 sions indicated should be of an elliptical form, whose major 
 and minor axes are respectively 5 and 2 inches.* 
 
 It is customary to give the arms a slight taper from the 
 hub to the rim. 
 
 (68.) Work Stored in the Arms of the Fly- Wheel. 
 If we wish to take into account the weight of the arms in 
 estimating the work stored in the fly-wheel, we have, let- 
 ting u = velocity of rim, TP a = the total weight of the arms, 
 and w = work stored, 
 
 W 
 
 w = 0.325 -u 2 , approximately, 
 
 which can be added to the work stored in the rim. 
 
 For a more general and less practical analytical discus- 
 sion of fly-wheels, reference may be made to the works of 
 Morin, Dulos, Poncelet and Resal. 
 
 Dr. R. Proel, in his Versueh einer Graphischen Dynamik, 
 gives very clear and elegant graphical methods of represent- 
 
 * If the power of the engine is conveyed by means of a band or 
 geared fly-wheel, we must calculate the tangential stress upon the 
 arms by means of the theorem of moments, regarding the crank as 
 the short lever at whose extremity the whole steam-pressure acts. 
 
OF THE STEAM-ENGINE. 151 
 
 ing the work lost and gained by a fly-wheel under various 
 conditions. 
 
 It perhaps appears superfluous to some of our readers to 
 enter into detail to so great an extent as has here been done, 
 but the danger and loss resulting from the accidental breakage 
 of a fly-wheel demand the most painstaking care in estab- 
 lishing its dimensions. 
 
 (69.) The Working -Beam. The working-beam is be- 
 coming less used as the speed of the steam-engine is increased ; 
 it is preferably constructed of wrought iron or steel, or, if 
 made of cast iron, is in many instances bound around with 
 wrought iron. Its form, if solid, should be parabolic, with 
 the vertex at the point where the connecting-rod joins it, 
 and the load at that point is the total pressure of the 
 steam upon the piston-head. (Weisbach's Mechanics of En- 
 gineering, sec. iv., arts. 251-52-53.) 
 
 The working-beam is supposed to be fixed at its central 
 bearing, and thus becomes a beam fixed at one end and 
 loaded at the other. See Table VI. 
 
 Where web-bracing is used in working-beams, the graphi- 
 cal method will afford the simplest solution. (See Graphical 
 Statics, Du Bois.) 
 
 (70.) General Considerations. The recent improve- 
 ments in parallel motions will probably lead to their more 
 general use in the place of guides and slides. A most in- 
 teresting and instructive little work, How to Draw a Straight 
 Line, by A. B. Kempe, suggests to the mechanician many 
 forms which can be adapted to the steam-engine with little 
 trouble. 
 
 In the foundation and framework of engines every pre- 
 caution must be taken to obtain RIGIDITY and immova- 
 bility. 
 
 Too much stress cannot be laid upon this point ; an in- 
 
152 THE RELATIVE PROPORTIONS 
 
 secure foundation inevitably injures, and perhaps ruins, the 
 engine upon it. 
 
 The centrifugal governor has not been considered, because 
 it forms one of the principal topics in almost every work on 
 the steam-engine. 
 
 The practical defects of the centrifugal governor are in- 
 surmountable when a perfectly regular speed is desired of 
 the engine. They are as follows : 
 
 (1.) The engine must go fast in order to go slow, or the 
 reverse, since the balls cannot move without a change of 
 speed in the engine and themselves. 
 
 (2.) The opening of the steam-valve is dependent upon 
 the angle which the arms attached to the balls form with 
 the central spindle around which they revolve. 
 
 Thus, an engine having its full amount of work, and gov- 
 erned by an ordinary ball-governor, will be kept at a uni- 
 form speed by the governor so long as the average resistance 
 to be overcome by the engine remains constant ; but when- 
 ever any of the work is taken off, the speed of the engine 
 will be increased to a higher rate, corresponding to the 
 diminished work, and at this faster speed the engine will 
 then run uniformly under the mastery of the governor so 
 long as the work continues without further alteration. This 
 arises from the fact that the degree of opening of the steam- 
 valve is directly controlled by the angle to which the gov- 
 ernor-balls are raised by their velocity of revolution, the 
 steam-valve being moved only by a change of speed, and 
 consequently by a change of the angle of suspension of the 
 governor-balls ; whence it follows that a larger supply of 
 steam for overcoming any increase of work can be obtained 
 only in conjunction with a smaller angle of the suspension- 
 rods of the governor-balls, and consequently with a slower 
 speed, and that a larger angle of the ball-rods, and con- 
 sequently a higher speed, must be attained in order to reduce 
 
OF THE STEAM-ENGINE. 153 
 
 the supply of steam for meeting any reduction of work to 
 be done by the engine. 
 
 (3.) The governor must be sensitive i. e., quick to act. 
 This result is usually attained in the centrifugal governor by 
 giving to the balls a speed much greater than that of the 
 engine, so that a slight variation of speed in the engine is 
 multiplied in the governor many times. 
 
 A high speed, however, is attended with the disadvantage 
 of rapid wear, and, in the case of an ordinary governor, 
 wear such as to admit of any lost motion is attended with 
 much trouble to the engineer and sudden variations of speed 
 in the engine. 
 
 (4.) The governor must have power, which means an 
 even and sure motion of the valve notwithstanding the 
 almost unavoidable defects of workmanship, such as the 
 sticking of the valve or the binding of the valve-stem 
 through careless packing of the stuffing-box. In the ordi- 
 nary governor this power is sought to be obtained either by 
 a high speed, the defects of which have already been pointed 
 out, or by means of very heavy balls, which results in a very 
 cumbersome and large machine, besides adding largely to 
 the expense. 
 
 Thus we see that not only is the speed of the steam-engine 
 entirely different with different loads, but also that with a 
 constant load the speed varies between limits which are de- 
 termined by the sensitiveness of the governor and is at no 
 time regular. 
 
 The necessity of a very sensitive governor is done away 
 with by the use of a properly proportioned fly-wheel. 
 
 The use of the governor to determine the point of cut-off, 
 as shown in the Corliss engines, if the fly-wheel be of the 
 proper weight and size, is attended with great regularity of 
 speed and economy of steam. Siemens' chronometric gov- 
 ernor, in which first the inertia of a pendulum and after- 
 
154 THE RELATIVE PROPORTIONS 
 
 ward hydraulic resistance were used as a point d'appui to 
 move the valve from, produces a very regular speed of en- 
 gine (Proceedings of the Institution of Mechanical Engineers, 
 January, 1866), but is too costly for general use. 
 
 Marks' isochronous governor (patented), in which the 
 motion of the valve precedes any change of speed in the 
 governor-balls, or. as since altered, in the hydraulic cup, 
 subserves the same purpose, and is much cheaper than the 
 former. (Journal of the Franklin Institute, May, 1876.) 
 
 A vast number of forms of governor of varying merit 
 have been invented, this portion of the steam-engine appear- 
 ing to be the most attractive to, and the most considered by, 
 mechanics and engineers. 
 
 The only test of beauty and elegance of design in an en- 
 gine is fitness and perfect proportion to the stresses placed 
 upon the various parts. 
 
 The severest simplicity of design should be adhered to. 
 Every pound of metal, where it does not subserve some use- 
 ful purpose, every attempt at mere ornament, is a defect, and 
 should be avoided. 
 
 The well-educated engineer should combine the qualities 
 of the practical man and the physicist ; and the more he 
 blends these together, making each mould and soften what 
 the other would seem to dictate if allowed to act alone, the 
 more will his works be successful and attain the exact object 
 for which they are designed. 
 
 The steam-boiler and its construction will be found to be 
 very thoroughly treated in Professor Trowbridge's Heat and 
 Heat Engines, in Wilson's Steam-boilers, and in The Steam- 
 Engine, by Professor Rankine. 
 
 (71.) Note on the Taper of Connecting-Rods. The 
 
 following remark, page 62, needs qualification : 
 
 " It is customary to make round connecting-rods with a 
 
OF THE STEAM-ENGINE. 155 
 
 taper of about one-eighth of an inch per foot, from the 
 centre to the necks, which should be of the calculated di- 
 ameter. Experiment does not show an increased strength 
 from a tapering form." 
 
 Passed Assistant Engineer C. H. Manning,* in a pleasant 
 correspondence with the writer, differed from him, deeming 
 it a better method to taper connecting-rods from the crank- 
 pin end to the cross-head end, as " experience had shown 
 that connecting-rods usually failed at the crank-pin." 
 
 Led by this discussion to make a more thorough inves- 
 tigation into the stress upon connecting-rods due to their 
 own inertia than he had before deemed necessary, the writer 
 submits the results, hoping they may be of interest to engi- 
 neers engaged in the designing of mechanism. 
 
 The connecting-rod, if of wrought iron, must be considered 
 either, 1st, as a short column, tending to rupture by crushing, 
 or 2d, as a long column, tending to rupture by buckling. The 
 tendency to fail in tension can be neglected, as being much 
 less than in the two cases mentioned. 
 
 In the first case it is obvious that tapering will not add 
 to its strength if we neglect the stress due to its inertia and 
 weight. 
 
 In the second case, theoretically, if we disregard the stress 
 in flexure due to the inertia and weight of the connecting- 
 rod, the increase in its diameter will be a maximum at its 
 centre. (See Weisbach's Mechanics of Engineering, Sec. iv., 
 Art. 267.) 
 
 The connecting-rod, if of steel, may be considered, 1st, as 
 a tension-rod, tending to fail in tension, or 2d, as a long 
 column, tending to rupture by buckling. The ability of 
 steel to withstand a much greater stress in compression than 
 in tension avoids the necessity of considering it as a short 
 
 * Instructor in steam engineering, U. S. Naval Academy, Annap- 
 olis, Md. 
 
156 
 
 THE EELATIVE PROPORTIONS 
 
 column, failing by crushing. In both cases the inertia and 
 weight of the rod are disregarded. 
 
 These are the general conditions which have controlled 
 the mechanical engineer in the consideration of the propor- 
 tions of wr ought-iron or steel connecting-rods. In addition, 
 it has been customary, in order to meet an unknown stress 
 in flexure due to the inertia of the rod, to increase the 
 diameter of the rod at the middle, or latterly to give an 
 increased diameter at the crank-pin, and taper from this to 
 the smallest dimension of the rod at the necks or neck, this 
 increase being purely empirical. 
 
 FIG. 30. 
 
 Another correspondent says : " Where iron and steel are 
 used, the figures adopted as constants must vary greatly, 
 especially with steel. The value of E ( modulus of elas- 
 ticity) varies enormously;" with which opinion the writer 
 agrees perfectly, only regretting that the very basis of 
 all correct calculations is thus taken from us, and we are 
 forced to be contented with results which have a reasonable 
 probability of correctness if we assume average values for 
 the safe stresses per square inch and the moduli of elasticity 
 of the material with which we are dealing. 
 
 If a straight line, A B, Fig. 30, have its extremities A 
 and B respectively caused to move reciprocally upon the 
 
OF THE STEAM-ENGINE. 157 
 
 straight line EAH and in the perimeter of the circle 
 BFG, any point upon the line AB as d will trace an 
 approximate ellipse as d H e F. 
 
 If, now, we let the length of the line AB = l; the radius 
 CB of the circle = r ; the variable distance of the point d 
 from the point A = Ad = x ; we have the semi-major axis 
 of this ellipse hF=hH=r, and the semi-minor axis hd = 
 
 y ' 
 
 he = -r, and the radius of curvature of the osculatory circle 
 
 I 
 
 at the vertex d of the semi-minor axis, 
 
 R = -r. (203) 
 
 x 
 
 The maximum resistance due to its inertia of any small 
 mass m at the end B of the rod A B, to motion in the direc- 
 tion B C, is equal to its centrifugal force, 
 
 mv 2 
 
 (204) 
 
 in which v = the linear velocity of the point B in feet per 
 second. 
 
 (For a demonstration of this refer to a paper by F. A. P. 
 Barnard, Transactions of the American Institute.) 
 
 Observing now that in the position shown in the figure 
 every point on the line A B as d is moving with a velocity 
 
 v in an arc of an osculatory circle whose radius is R = -r, 
 
 x 
 
 equation (203), we have the means of determining the re- 
 sistance due to the inertia of each element of mass m. 
 For the point d it would be 
 
 IT-IT (205) 
 
 for the point A, since x = 0, R = , and the resistance of a 
 
 14 
 
158 
 
 THE RELATIVE PROPORTIONS 
 
 mass w would be = 0. We thus see, equation (205), that 
 the resistance of each element of mass due its inertia to 
 motion in a direction at right angles to E A C is directly 
 proportional to its distance =x from the point A. 
 
 If, now, for the line A B we substitute a rod of uniform 
 cross-section =F, we have 
 
 F r dx 
 
 (206) 
 
 in which Y = weight per cubic unit, and # = 32.2 feet per 
 second, F being taken in square units. 
 
 Substituting this value of m in equation (205), and in- 
 tegrating between the limits x = and = , we have, denoting 
 the whole resistance of the rod A B by P, 
 
 glr 
 
 xdx 
 
 or since the weight of the rod G = Ffl, 
 
 (207) 
 
 And since this load upon the rod due to the resistance of its 
 own inertia increases uniformly from the end A to the end 
 B, the rod can, with sufficient approximation, be supposed 
 
 FIG. 31. 
 
 L9 
 
OF THE STEAM-ENGINE. 159 
 
 in the condition of a horizontal beam loaded with a tri- 
 angularly-shaped load, as shown, Fig. 31, in the cross- 
 hatched portion A KB. 
 
 For a vertical engine we can neglect the weight of the 
 connecting-rod, considering only the stress due to its inertia. 
 
 For the moment of flexure of any cross-section at a dis- 
 tance x from the extremity A we have, letting c = the load 
 per unit of length at the point K t 
 
 (208) 
 
 which becomes a maximum for # = l/|f = 0.578/, at which 
 point, therefore, the maximum cross-section of the connect- 
 ing-rod should be placed. 
 
 In horizontal engines it is necessary to take into consid- 
 eration the weight of the rod, as well as its inertia ; the 
 weight of the rod may be regarded as a uniformly dis- 
 tributed load always acting in one direction. 
 
 For the cross-section, at a distance x from the extremity 
 A, the moment of flexure is, letting G = whole weight of 
 rod = rl, 
 
 "lf~W f 
 
 which is a maximum for 
 
 But we have, from equation (207), 
 
 1 fir 
 = *p in which g = acceleration of 
 
 velocity of the crank-pin, J5, in feet per second, and r = the 
 
 C 1 fir 
 - = *p in which g = acceleration of gravity, v = the linear 
 
160 THE RELATIVE PROPORTIONS 
 
 radius of the crank in feet. Substituting this value in equa- 
 tion (210), we have 
 
 which shows that as the velocity of the crank-pin increases 
 the value of x approaches more nearly to / V\ = 0.578Z, but 
 can never quite equal it. 
 
 Referring to equation (209), and taking P = zero, we find 
 
 the maximum value of x = 0.5 
 
 2i 
 
 We thus see that the greatest moment of flexure of any 
 connecting-rod lies between the limits 0.5/ and 0.578, mea- 
 sured from the point A. 
 
 It is of interest further to note that the crank-pin takes 
 one-half the stress due to the weight of the rod, and two- 
 thirds of the stress due to the inertia of the rod. 
 
 The end, A, takes one-third of the stress due to the 
 inertia of the rpd, alternately increasing and decreasing 
 the stress upon the guides to this amount, and one-half the 
 stress due to its weight. 
 
 In engines having a large number of revolutions per 
 minute P becomes worthy of notice. In slow-moving en- 
 gines it is very small, and may be neglected. 
 
 From these considerations the writer is of the opinion 
 that the failure of connecting-rods at the neck nearest the 
 crank-pin, if they are properly proportioned, is probably due 
 to the crank-pins being out of truth, or to the seizing of 
 heated boxes on the crank-pin, rather than to the weight 
 or inertia of the rods themselves. 
 
 The assumption of a connecting-rod of uniform cross- 
 section is manifestly incorrect, and only serves to prove 
 generally a principle which should be recognized. 
 
 In the case of a practical application, graphical methods 
 
OF THE STEAM-ENGINE. 161 
 
 taking cognizance of the usual changes in cross-section 
 must be used for a series of approximations. (Vide Du 
 Bois's Graphical Statics, or Reuleaux, Der Constructeur.') 
 
 CHAPTER XV. 
 
 (72) The Limitations of the Steam-Engine. Per- 
 haps there is no more unsafe proceeding in science than to 
 attempt to predict the limitation of the development of any 
 of its results. Yet the theory of the steam-engine has so 
 far been developed as to seem to permit us to mark out the 
 boundaries of its progress with tolerable accuracy. 
 
 At least, we trust that a full criticism of its present de- 
 ficiencies, and a deduction of its limitations from known 
 laws, will interest such of our readers as are engaged in 
 the improvement of this machine. 
 
 In this lecture we will consider the steam-engine alone, 
 as it is in no wise responsible for the economy of the boiler 
 which supplies the steam, or for the losses incurred in con- 
 veying the steam from the boiler to the engine. 
 
 The boiler should be considered apart from the engine ; 
 and it is well to incidentally remark here that the perform- 
 ances of the best types of boiler leave but little room for 
 improvement so far as evaporation is concerned. 
 
 It is rather in the utilization of the steam after it reaches 
 the engine that we must look for progress, and in this point, 
 we take it, lies the value of a thorough investigation of the 
 machine itself. 
 
 The well-known formula for the horse-power of a steam- 
 engine * contains three factors, which can be varied at will 
 
 * = (HP] : in which P= mean effective pressure per square 
 
 33000 
 
 14* 
 
162 THE RELATIVE PROPORTIONS 
 
 the mean effective pressure, the volume of the cylinder, 
 and the number of strokes per minute. 
 
 Concentration of power in a small space is the greatest 
 attribute of the steam-engine ; it is its power to concentrate 
 the strength of thousands of horses in the space of an ordi- 
 nary room that renders the steam-engine so useful and in- 
 dispensable to us. 
 
 For this reason we cannot go on increasing the volume 
 of the steam-cylinder indefinitely without a proportionate 
 increase in power ; and it is only necessary to remark to 
 mathematicians, with regard to the proportions of the cylin- 
 der, that that cylinder whose stroke equals its diameter con- 
 tains the maximum of volume with the minimum of con- 
 densation-surface.* 
 
 The importance of reducing condensation-surface is be- 
 coming more and more appreciated among engineers, and 
 shortening the stroke will prove less of a bugbear when the 
 laws of inertia, as applied to the reciprocating parts of a 
 steam-engine, are better understood. 
 
 Shortening the stroke has the further advantages of short- 
 ening the space passed over by the piston-head during a 
 given number of strokes, and consequently of reducing the 
 wear on the piston-head, as well as of rendering the engine 
 more compact for a given power. 
 
 The following formula gives for any assumed horse-power, 
 mean effective pressure, and number of strokes per minute, 
 
 inch ; L = length of stroke in feet ; A = area of piston in square inches ; 
 N= number of strokes per minute; (Jf?P) = indicated horse-powers; 
 
 = V= volume of cylinder in cubic feet. 
 144 
 
 * Should the internal condensation of cylinder-surface prove to be 
 directly proportional to the time of exposure, as well as to the area 
 exposed, the stroke should be twice the diameter; but the writer 
 doubts the practical accuracy of this statement in high-speed engines. 
 
OF THE STEAM-ENGINE. 163 
 
 the required common diameter and stroke of cylinder, in 
 inches : 
 
 <* = 79.59/' ( 
 
 Concentration being attained, the next important consid- 
 eration that arises is economy of steam ; and this is the battle- 
 ground upon which the struggle for supremacy among our 
 engine-builders is to-day being practically fought out, while 
 the physicists are quite as hotly engaged in a dispute as to 
 the relative theoretical merits of steam-, hot air-, and electric- 
 engines, into which we will not enter. 
 
 Practically, we think that, as far as the steam-engine is 
 concerned, the impossibility of supplying to the steam in 
 the cylinder any appreciable amount of heat by means of 
 steam or hot-air jackets is acknowledged by all, and the 
 jacket is only expected to keep the cylinder warm and pre- 
 vent it from abstracting heat from the steam inside the 
 cylinder. 
 
 The jacket proving imperfect, we must find other means 
 for preventing condensation by cooling ; and they are these : 
 Diminution of condensation-surface, as already stated above, 
 and shortening of the duration of the time for condensation, 
 which means an increase in the number of strokes of the 
 piston. The effect upon the structure of the engine of in- 
 creased rotative speed is not injurious to the engine, as so 
 positively stated by many engineers ; but, if attempted, rapid 
 speed requires a far greater knowledge of the dynamics of 
 the steam-engine than has as yet been applied to it by many 
 constructors, and will almost inevitably shake to pieces a 
 faultily designed structure. 
 
 The length of the crank-pin and shaft-bearings must also 
 increase directly with the number of revolutions made, while 
 their diameter, within ordinary limits as to pressure and 
 speed, has no appreciable effect upon the heating of the 
 
164 THE RELATIVE PROPORTIONS 
 
 bearing ; but when we consider the practical fact that the 
 eccentrics, possibly because of greater speed of rubbing sur- 
 faces, are always sources of large frictional losses, it would 
 seem best to make the diameters of all bearing journals just 
 as small as the pressures allowable upon the surfaces of 
 contact will permit. Much yet remains to be determined 
 with regard to the laws controlling bearings. 
 
 It has been shown conclusively that the coefficient of 
 friction is much affected and reduced by the state of the 
 rubbing surfaces, by the method of lubrication, and by the 
 quality of the lubricant. 
 
 Perfect, almost mirror-like, surfaces reduce the coefficient 
 of friction far below the 3 to 5 per cent, formerly stated by 
 Morin as a minimum, provided a continuous lubricating 
 apparatus is used with good sperm oil ; and it will be found 
 that a costly lubricant, such as sperm oil, is cheapest where 
 care is exercised not to waste in its application.* 
 
 The important influence of the number of turns upon the 
 length of bearings has not been clearly enough understood 
 to prevent accidents from heating of the crank-pins and 
 bearings, which is one of the most annoying mishaps which 
 
 * Let Z = the length of the crank-pin, in inches. 
 " L= " stroke, in feet. 
 
 " ( HP} = the indicated horse-power. 
 
 Then, for marine propeller engines a practical formula, giving safe 
 lengths, would be 
 
 ^0.622 
 
 One-fourth of this length will serve safely for side-wheel or stationary 
 engines. 
 
 For locomotive engines practice seems to prove that i = .013cP is a 
 safe value, in which d = diameter of the steam -cylinder, in inches. 
 
 For crank-shaft bearings, letting R = the whole maximum weight 
 on bearings, and N= the number of strokes per minute, we have 
 / = . 000005 RN. 
 
OF THE STEAM-ENGINE. 165 
 
 can befall an engine, and at times requires the utmost vig- 
 ilance of the engineer for its prevention. 
 
 The great difficulty attendant upon making bearing and 
 shaft press uniformly throughout the length of the journal 
 has almost forced the use of Babbitt or soft metal for high- 
 speed engines ; and where proper care is taken in propor- 
 tioning the results have justified its use. 
 
 Crank-pins shrunk in by heating the crank are apt to be 
 out of truth, no matter how carefully the work has been 
 done, as heat is apt to warp the crank. 
 
 Pins should be forced in cold, or the crank and pin be 
 forged in one piece, as in the case of most small double 
 cranks. 
 
 High rotative speed, while increasing the power of the 
 steam-engine, also renders it more compact and diminishes 
 the weight of the fly-wheel necessary to obtain regularity 
 of speed. 
 
 Rotative speed has, however, its limitations, independ- 
 ently of the trouble sometimes caused by the heating of 
 bearings. 
 
 For instance, it can be shown that if the safe stress in 
 tension upon cast iron equals 1800 pounds per square inch, 
 a cast-iron ring can be revolved safely at a maximum lineal 
 speed of about 8000 feet per minute, which at once places a 
 limitation upon the rotative speed of the fly-wheel, and 
 demands reduction of its diameter as the rotative speed is 
 increased. 
 
 Let JV=the number of revolutions per minute. 
 " D = the diameter of cast-iron ring, in feet. 
 We have 
 
 or . DN-254G. 
 
 This formula would limit the diameter of the fly-wheel, 
 if used, of a pair of engines to make 600 revolutions per 
 
166 THE RELATIVE PROPORTIONS 
 
 minute, lately proposed by the distinguished engineer, Mr. 
 Charles T. Porter, to about 4 feet. 
 
 Of course, as the tensile strength of the material of 
 which the fly-wheel is made is increased, the lineal speed 
 at which it can be driven is increased. 
 
 If we take 5000 pounds per square inch as the safe ten- 
 sile strength of the steel tire of a locomotive, we find its 
 lineal speed to be limited to about 2^ miles per minute, or 
 150 miles per hour, and it only remains for our engineers 
 to provide boiler-capacity and road-bed adapted to such 
 speeds in order that they may be realized by a properly 
 constructed locomotive. 
 
 This speed would shorten the time between New York 
 and Philadelphia (90 miles) to 36 minutes. 
 
 High rotative speed necessitates special precautions as to 
 the weight of the reciprocating parts. 
 
 If we impose the condition that the initial and final 
 stresses upon the crank-pin be equal, we must make the 
 centrifugal force of the reciprocating parts (piston, piston- 
 rod, cross-head, and connecting-rod), supposed to be concen- 
 trated at the crank-pin, equal to half the difference of the 
 initial and final steam-pressures upon the whole piston-head. 
 
 High rotative speed has also demanded a change iii the 
 form of the connecting-rod, which was formerly largest at 
 its mid-length, but has of late years been made largest at 
 the crank-pin end ; a consideration of the stress upon it due 
 to its own inertia would place the point of greatest stress 
 at 0.578 of its length from the cross-head end; but the 
 accidental stresses due to the twisting of the crank-pin in 
 its boxes recommends as a measure of safety that the rod 
 be made largest at the crank-pin neck, and tapered down 
 from that to the cross-head. 
 
 In a properly proportioned engine of any assumed horse- 
 power the only method of reducing the weight of the moving 
 
OF THE STEAM-ENGINE. 167 
 
 parts is to increase the number of strokes per minute ; length- 
 ening the stroke will not do it. 
 
 The points of interest with regard to the volume of the 
 steam-cylinder and the rotative speed of the engine having 
 been touched upon, there remains to be considered the mean 
 effective pressure of the steam. 
 
 In the Journal of the Franklin Institute for June, 1880, 
 the writer has shown, under the assumption, approximately 
 correct, that the steam-expansion curve is an equilateral 
 hyperbola, that the point of cut-off giving greatest econ- 
 omy of steam, but not of money, for steam-engines, is 
 determined by dividing the absolute back pressure by the 
 absolute initial pressure of the steam. 
 
 Indicators do not give such curves, even if in perfect 
 order, unless the steam is a little wet, but the approxima- 
 tion is sufficiently close for all practical purposes. 
 
 A more accurate way would be to say that the most 
 economical point of cut-off is attained when the final steam- 
 pressure equals the back pressure ; but our first statement is 
 more convenient in form, and will serve our purpose with all 
 the practical accuracy necessary. 
 
 Let P= the mean effective pressure of steam in pounds 
 
 per square inch. 
 " P b = the initial absolute pressure of steam in pounds 
 
 per square inch. 
 " B = the back absolute pressure on piston in pounds 
 
 per square inch. 
 " e = the fractional part of the stroke at which the 
 
 steam is cut off. 
 
 " F=the volume of the steam-cylinder in cubic feet. 
 Then we have 
 
 r fi 
 
 P=eP b \\ +nat. log- -B (212) 
 
 L 1 
 
 Let (HP) = the indicated horse-power. 
 
 j 
 
168 THE RELATIVE PROPORTIONS 
 
 Let $ = the specific volume of the steam for a pres- 
 sure, P b . 
 
 " TF"=the weight of water used per horse-power. 
 We have 
 
 33000 ' 
 
 in which L = the length of stroke in feet. 
 
 N = the number of strokes per minute. 
 
 A = the area of the piston-head in square inches. 
 Therefore 
 
 f -tl -v 
 
 VN 
 
 log - -B\ 
 
 C J 
 
 (J?P)=- L J : X H4, -(213) 
 
 33000 
 
 since 144 V=LA. 
 
 The quantity of water used during one minute is 
 
 > ( 214 > 
 
 o 
 
 62J pounds being taken as the weight of one cubic foot of 
 water, and S representing the specific volume of the steam 
 for the pressure P 6 . 
 
 Therefore, the weight of water used per horse-power is 
 
 144 
 33000 , 
 
 143236 
 
 S eP b l-mat log - B . (215) 
 
 j) 
 
 If in this last formula we substitute for e, we obtain 
 
OF THE STEAM-ENGINE. 169 
 
 the theoretical minimum quantity of water to be evaporated 
 per horse-power, and 
 
 W __ 14323 _ 6220.3 
 
 For one-horse-power per hour we would have, multiply- 
 ing by 60, 
 
 60 W = 873218 . (217) 
 
 PS log 
 
 As, for instance, for a non-condensing engine having a 
 gauge-pressure of 45 pounds, and cutting off at \ the 
 stroke, we find the minimum limit of evaporation of water 
 required to be about 24 pounds of water per horse-power 
 per hour. 
 
 Mr. Charles E. Emery has experimentally realized as low 
 as 39 pounds of water per horse-power per hour with the 
 given pressure in small non-condensing engines. 
 
 If in the last formula we regard the product of the pres- 
 sure and the specific volume, P b S, as constant, which it is 
 with a considerable degree of approximation, we observe 
 
 T> 
 
 that the economy of steam varies directly as the log -A 
 
 JB 
 
 showing that both Watt and Oliver Evans were partially 
 right in their attempts to increase the economy of the steam- 
 engine. Watt was right in perfecting more and more the 
 vacuum obtained, and Evans in increasing the steam-pres- 
 sure at the boiler. Neither of them were wholly astray, 
 but were soon met and their progress "stopped by the slow 
 increase of the economy beyond certain limits, and by the 
 practical difficulties arising from surface-condensation in 
 the pursuit of a partially apprehended law. 
 
 15 
 
170 THE RELATIVE PROPORTIONS 
 
 It is evident that we are limited as to the steam-pressure, 
 and that a great increase of that, or a great decrease of 
 back pressure by reason of more perfect vacuums, will 
 render the point of cut-off so early, the mean effective- 
 pressure so low, and the condensation so great that the 
 necessary increase of volume in the steam-cylinder will 
 prevent that compactness and concentration of power so 
 desirable in the steam-engine ; and, although it has already 
 been shown that high rotative speed aids in rendering the 
 engine compact, we are also limited in that direction. 
 
 Regularity of speed is an imperative condition in all en- 
 gines used to drive high-speeded machinery; under such 
 circumstances regularity becomes the first requisite, and to 
 it all other considerations must be subordinated. 
 
 Mr. G. H. Corliss, with his automatic cut-off engine, was 
 the first to satisfactorily solve this problem. 
 
 With a constant steam -pressure and a constant load an 
 engine will run, under the control of a throttling governor, 
 with an approximation to regularity determined by the sen- 
 sitiveness of the governor. With a throttling governor 
 every change of the steam-pressure, every change in the 
 load, causes a change in the speed of the engine, no matter 
 how well proportioned it may be. 
 
 The laws regulating the centrifugal governor are such as 
 to prevent it from ever becoming a very perfect mechanism, 
 for the engine and governor must go fast in order to go slow, 
 and vice versd. With an automatic cut-off governor the 
 speed will be held with greater regularity under all steam- 
 pressures and loads within the engine's capacity, the ap- 
 proximation to perfection attained being determined by the 
 proportions of the fly-wheel and the sensitiveness of the 
 governor. 
 
 Governors in which the inertia of the balls is utilized to 
 check any attempted change of speed, without an apprecia- 
 
OF THE STEAM-ENGINE. 171 
 
 ble change of speed in the governor itself, will ultimately 
 supersede the centrifugal governor, which is full of radical 
 defects. 
 
 Valve-motions actuated by a weight or spring released 
 automatically by the centrifugal governor are comparatively 
 so slow in their motions as to prevent high rotative speeds 
 and short strokes. This form of valve-motion has been 
 elaborated with great skill and ingenuity on such engines > 
 as the Corliss, Harris-Corliss, Keynolds-Corliss, Wheelock, 
 and Brown, which, so long as long stroke and slow rotation 
 is adhered to, leave but little to be desired on the score of 
 regularity of average speed or of perfection of workmanship. 
 
 To a different class belong the Porter- Allen and the Buck- 
 eye engines : in these engines, which are the exponents of 
 high rotative speeds and short strokes, the cut-off of the 
 engine is actuated by means of a link or an eccentric, whose 
 position is regulated by means of a centrifugal governor, 
 and, whatever the speed, the valves must keep pace with 
 the engine in its motions. 
 
 A large number of valve-motions are in use on the various 
 forms of locomotive, marine, and stationary engines through- 
 out the civilized world, and it is perhaps worth our while to 
 practically note the important points of some of the older 
 forms. 
 
 The plain hollowed slide-valve is the most commonly used 
 on the rougher forms of engines, because of its simplicity ; 
 as, however, the earliest point at which it can cut off steam 
 without choking the exhaust is about f of the stroke, it is 
 not economical with the high pressures and large engines 
 of the present day or with the vacuums ordinarily attained 
 in condensing engines. 
 
 When a reversing gear is used the Stephenson link-motion 
 is commonly used, where simplicity of mechanism rather 
 than economy of steam is a point to be gained. 
 
172 THE RELATIVE PROPORTIONS 
 
 With open rods it has a variable lead, decreasing from 
 the dead point of the link ; with crossed rods it has a varia- 
 ble lead, increasing from the dead point of the link. 
 
 For crossed rods the position of the piston is nearly con- 
 stant at the opening of the steam-ports ; this is a decided 
 advantage. Crossed rods permit a slightly greater length 
 of eccentric rods. 
 
 Fastening the link to the ends of the eccentric rods at a. 
 point behind the centre line of the link, as is usually done 
 in American practice, introduces irregularities into the 
 point of cut-off, but does not affect the lead. 
 
 When the ends of the eccentric rods are fastened to the 
 link at points on the centre line of the link, this error does 
 hot occur, but larger eccentrics are required, which are 
 sometimes inconvenient. 
 
 The mode of suspension of the link is far more important 
 than is usually supposed, as imperfect suspension not only 
 introduces serious errors in the valve-motion, but also affects 
 the durability of the construction by producing excessive 
 slip in the block. 
 
 Where a constant lead is desired, using different angles 
 of advance for the two eccentrics will produce a constant 
 lead in one direction of motion, but ruins the action of the 
 valve-motion in the opposite direction. 
 
 The Gooch link-motion is difficult to fit into an engine, 
 because of the distance required between the centre of the 
 crank-shaft and the end of the radius-rod ; it is very easy 
 to handle, and should therefore be used where frequent re- 
 versing is necessary, as in the cases of yard locomotives and 
 hoisting-engines, even though requiring considerable trouble 
 to fit in. 
 
 If a constant lead be a great advantage, this link-motion 
 has this advantage. The suspension-link for the link proper 
 should be made as long as possible, and the suspension-link 
 
OF THE STEAM-ENGINE. 173 
 
 for the* radius-rod should be attached as closely as possible 
 to the link-block. The tumbling-shaft should be placed on 
 the opposite side of the link from the crank-shaft. 
 
 Allan's link-motion, having a straight link, can be more 
 easily fitted up than motions having a curved link ; it has 
 all the disadvantages of the Gooch link-motion, but, unlike 
 it, has not a constant lead, and has a greater number of 
 parts. Particular attention must be paid to the relative 
 lengths of the tumbling-shaft arms for simultaneously 
 raising and lowering the link and radius-rod. The Allan 
 link-motion does not handle as easily as the Gooch. 
 
 Heusinger von Waldegg's link-motion is the easiest of 
 all to handle ; it can be so arranged as to have very little 
 slip in the block, but cannot be easily attached to all forms 
 of cross-head. It is the simplest reversing gear yet men- 
 tioned, and has a constant lead. 
 
 Pius Fink's link-motion is the simplest reversing gear in 
 existence which at the same time can be adjusted as a cut- 
 off; it has a constant lead, but a limited range of action,, 
 and in this point is inferior to all the others, as well as 
 being irregular in its variation from its calculated positions. 
 However, if carefully designed, it will do good work, and is 
 very easy to handle and very durable. 
 
 The fact that it is impossible to avoid choking the ex- 
 haust, and the consequent compression of the steam, thereby 
 reducing the power of the steam-engine whenever the at- 
 tempt is made to cut off early in the stroke with a single 
 valve, has led engineers to devise double valve-motions 
 which will effect an early cut-off of the steam, independ- 
 ently of the motion of the hollow of the slide-valve rel- 
 atively to the exhaust. 
 
 To this type belong the valve-motions of the Porter- Allen* 
 
 * The Porter-Allen valve-motion is essentially Pius Fink's. 
 15* 
 
174 . THE RELATIVE PROPORTIONS 
 
 and Buckeye engines, both of which show a wide range of 
 action for motion in one direction, and great simplicity of 
 structure. 
 
 The Gonzenbach link-motion, with its double steam- 
 chest and excessively complicated outside gear, is now 
 almost obsolete ; it will not cut off sharply and well, ex- 
 cept for very early points, and that, too, only for motion in 
 one direction ; and when properly arranged for motion in one 
 direction gives a double admission of steam for motion in 
 the opposite direction ; it is difficult to handle, and its com- 
 plication renders it very liable to break down. 
 
 The Polonceau link-motion, while it has not the double 
 steam-chest or as much complication in its outside valve- 
 gear, cannot cut off steam later than ^ of the stroke, and 
 is therefore not very well adapted to general locomotive use, 
 where economy is sacrificed to concentration of power. It 
 is not very difficult to handle, but, like the Gooch, presents 
 some difficulty in adapting to the narrow limits offered by 
 a locomotive. 
 
 Meyer's link-motion seems to have met with the greatest 
 favor from builders of marine and stationary engines, as it 
 presents a wide range for motion in one direction, and a 
 greater range when used as a reversing motion than any of 
 the others. The screws which move the expansion-blocks 
 are liable to rust or wear, and cause trouble; the expan- 
 sion-valve should have separate eccentrics to work best ; it 
 is intricate in its construction, and liable to break down. 
 It is also difficult to handle, and absorbs much power unless 
 the distribution-valve is balanced. 
 
 An ingenious modification of the Meyer valve-motion is 
 the Ryder cut-off, in which a cylindrical expansion-valve, 
 whose edges approach each other after the manner of an 
 isosceles triangle wrapped around a cylinder, is moved 
 around its axis by means of the governor, so as to have 
 
OF THE STEAM-ENGINE. 175 
 
 the same effect as that of increasing or decreasing the dis- 
 tance between the expansion-blocks of the Meyer cut-off. 
 
 In order that we may make ourselves clearly understood, 
 we recapitulate : 
 
 A really good steam-engine should possess the following 
 qualities : Concentration of power, economy of steam, regular- 
 ity of speed, simplicity of design, and durability of construction. 
 
 We have endeavored to mark the limitations of the first 
 three attributes of a good engine ; the last two must be 
 worked out in the shop. 
 
 We are perhaps too bold in our prediction, but it seems 
 to us that the engine of the future will have a cylinder 
 whose stroke is equal to its diameter; its speed will be 
 limited by the tensile strength of its revolving parts ; its 
 bearings will be of great length ; its cut-off will be deter- 
 mined, approximately at least,* by the ratio of the back 
 pressure to the initial pressure; its valve-motion will be 
 rigidly in connection with the crank-shaft ; its reciprocating 
 parts will be so proportioned as to put a nearly constant 
 pressure on the crank-pin ; its fly-wheel, if it has one, will 
 be proportioned so as not to admit of more than a prede- 
 termined variation of speed during one stroke ; its governor 
 will no longer be centrifugal, but a revolving pendulum, 
 almost isochronous, and holding the engine, with all neces- 
 sary approximation, within fixed limitations, to a constant 
 speed. 
 
 * For reasons for varying cut-off see Lecture XVI., on The Cheapest 
 Point of Cut-Off. 
 
176 THE RELATIVE PROPORTIONS 
 
 CHAPTER XVI. 
 
 (73) The Cheapest Point of Cut-Off. In the Journal 
 of the Franklin Institute for June, 1880, the writer published 
 a brief paper, determining, in an approximate way, from a 
 purely dynamical point of view, the most economical point 
 of cut-off for steam engines.* This inquiry confined itself 
 entirely to the ratio of the indicated horse-power to the steam 
 used, and provoked an amount of criticism in German, Eng- 
 lish, and American serials which was as surprising to him 
 as the misapprehension of the limitations of the paper, to- 
 gether with the way in which the intent of the writer was 
 gratuitously assumed. 
 
 As a question of finance the subject becomes more com- 
 plicated, for the engine-owner asks not, How can I save the 
 most steam ? but, All the circumstances being taken into 
 consideration, how can I get the useful work which I re- 
 quire most cheaply? 
 
 So far as the delivery of useful work by the engine alone 
 is concerned, a method has been given by Professor Ran- 
 kine, and elaborated by himself as well as others; but, 
 perhaps because of insufficiently profound study of the 
 question from a financial point of view, or because the 
 subject was hardly deemed worthy of his thought, the 
 question was never exhausted by him or his followers. 
 
 The ship-owner says, How can I obtain the power to 
 drive the propeller most cheaply? 
 
 The mill-owner says, How can I obtain the power to 
 drive the mill-stones most cheaply? and the shop-owner 
 says, How can I obtain the power to drive my machinery 
 most cheaply ? and, as they use the engine for the purpose 
 of making money, wish to have it designed for that pur- 
 
 * Appended to this Lecture. 
 
OF THE STEAM-ENGINE. 177 
 
 pose, and care nothing at all for purely scientific considera- 
 tions. 
 
 It is to these men that the writer will endeavor to make 
 reply, giving to a most perplexing question, involving many 
 considerations, at least an approximately correct reply, and 
 indicating a method which, by elaboration and a detailed 
 consideration of the thermodynamic questions raised, will, 
 we trust, enable an engineer to reach an economy of useful 
 power as yet not knowingly obtained by other means than 
 an inspection of the Profit and Loss account at the end of 
 the year. 
 
 In "The Limitations of the Steam-Engine" the writer 
 has stated as the five points of an engine : (1) concentra- 
 tion of power, (2) economy of steam, (3) regularity of 
 speed, (4) simplicity of design, and (5) durability of con- 
 struction. 
 
 When we do not restrict ourselves to economy of Nature's 
 forces, economy of steam becomes economy of money. 
 
 The following assumptions are made and particulars must 
 be understood in the discussion which follows. 
 
 The expansion-curve of steam is assumed to be, with suf- 
 ficient practical accuracy, an equilateral hyperbola. 
 
 The steam made by the boiler is to the steam shown by 
 the indicator diagram as 4 to 3. This, certainly, is not cor- 
 rect under all circumstances, but is an approximation de- 
 rived from the experiments under favorable conditions upon 
 the Reynolds-Corliss, the Harris-Corliss, and the Wheelock 
 engines at the Millers' Exhibition, Cincinnati, June, 1880 
 (Report of J. T. Hill, pages 77 and 79), and is nearly the 
 same for both condensing and non-condensing engines. Those 
 in possession of more accurate experimental data can sub- 
 stitute other ratios in each case. 
 
 A percentage of the true stroke must be added at each 
 end of the sketch which is made to allow for the clearance, 
 
178 THE RELATIVE PROPORTIONS 
 
 which must be determined. The cost of all charges upon 
 the engine and machinery is taken in steam for the sake of 
 convenience ; and this proceeding is perfectly proper, since 
 money and steam are convertible. 
 
 For the present all reference to the saving of fuel result- 
 ing from the diminished number of heat-units required to 
 increase the pressure of steam is premeditated ly omitted, 
 because we are practically limited by expense to low press- 
 ures in ordinary cases. 
 
 The constant charges which come upon engine-boilers and 
 machinery are as follows : 
 
 (1) Wages of attendants upon engine, boilers, and ma- 
 chinery. 
 
 (2) Interest upon cost of engine, boilers, and machinery. 
 
 (3) Depreciation of engine, boilers, and machinery. 
 
 (4) Repairs to engine, boilers, and machinery. 
 
 (5) Cost of lubrication of engine, boilers, and machinery. 
 
 (6) Taxes and insurance upon engine, boilers, and ma- 
 chinery. 
 
 (7) Interest upon cost of shelter and room for engine, 
 boilers, and machinery. 
 
 Many other charges may exist which the writer has not 
 mentioned, and some of the charges mentioned are not 
 applicable in cases where methods of charging cost may 
 differ from that of works and factories engaged in the 
 production of some staple article or of a ship where the 
 engine serves only for propulsion. In every case the dis- 
 tribution of cost must form an individual problem. 
 
 The only variable charge when the engine is already 
 established is the cost of the steam i. e., fuel and water. 
 When engaged in the design of an engine and plant most 
 of the constant charges may be regarded as variables, but 
 not according to any uniform law, and must be considered 
 
OF THE STEAM-ENGINE. 179 
 
 as separate problems which must be solved from known pre- 
 cedents which vary in different localities. 
 
 Let P = the mean effective steam-pressure in pounds per 
 
 square inch. 
 " C = the constant charges in dollars and cents for 
 
 any assumed time. 
 " P b = the absolute initial pressure in the cylinder in 
 
 pounds per square inch. 
 " B =the absolute back pressure in the cylinder in 
 
 pounds per square inch while the exhaust-port 
 
 is open. 
 " e = the fraction of the volume at which steam is cut 
 
 off. 
 
 " V =the volume of the steam -cylinder. 
 " c = the factor of the volume of steam proportional to 
 
 the ratio of the constant charges to the cost of 
 
 steam for any assumed time. 
 " b = the fraction of the volume at which compression 
 
 begins, being measured from the opposite end 
 
 from which e is measured. 
 " k = the fraction of the volume allowed for clearance. 
 
 We can then write : 
 
 Useful work PV 
 
 Cost of work in steam 
 
 B 4 .B 
 
 - e+c _6__ 
 
 b\l 
 
 . (218) 
 
 But P=eP 6 (l + nat. log -] - B\ 1 -6/1-nat. log-) . (219) 
 
 This value of P could be much more accurately determined 
 by the careful use of an indicator. 
 
 Substituting in equation (218), we have 
 
180 THE RELATIVE PROPORTIONS 
 
 eP 6 (l+nat. log ^-) 
 
 Useful work_ 
 Costof work~4 C 
 3 D 
 
 (1 \ / b \~\) B 
 
 1-fnat. log 1 B 1 ol 1 nat. log -- I I > o 
 
 I e+ |eP 6 (l+nat. log ^ ) B [l 6^1 nat. log |j] l~bp- 
 
 Differentiating with respect to e and seeking a maximum, 
 we have 
 
 e = 
 
 The natural logarithm can be obtained by multiplying 
 the common logarithm by 2.3026, and this transcendental 
 equation must be solved by a series of approximations, be- 
 ginning with an assumption that 
 
 nat. log - = nat. log -, (222) 
 
 1 -b(l -nat. log - 
 
 and substituting the nearer value of - again in the second 
 
 6 
 
 member of equation (221), and so on until two successive 
 values of e nearly agree. 
 
 As logarithms do not vary rapidly, the approximations 
 required to obtain all the accuracy justified by the data or 
 realizable in practice will be few. 
 
 Perhaps how to deduce the value of c is not clear. 
 . Determine the constant charges, in dollars and cents, 
 upon engine and machinery for one day. Regardless of 
 the power, determine approximately, from the first term 
 only of the second member of equation (221), the most 
 economical point of cut-off for steam alone. 
 
OF THE STEAM-ENGINE. 181 
 
 With the following formula determine the weight of water 
 required per horse-power per hour : 
 
 W= weight of water per H.-P. per hour. 
 S = specific volume of steam for pressure P b . 
 4 859380e 
 
 < r 
 
 1 + nat. log- 
 
 6 
 
 (See " The Limitations of the Steam-Engine.") 
 
 With this determine the cost of fuel and water for the 
 
 required horse-power per day, remembering that is an 
 
 assumed quantity which, at the best, is only approximately 
 
 correct. 
 
 The value of c given in formula (220) is deduced as 
 
 follows : 
 
 De 
 
 c:e::C: 
 
 \ 1+nat. log - 
 
 L !l 
 
 Therefore, 
 
 ' (224) 
 
 An inspection of equation (221) reveals many interesting 
 facts. 
 
 We observe that an increase of the initial pressure or a 
 diminution of the back pressure renders the cut-off of steam 
 earlier. Also that a diminution of the back pressure has 
 much more influence than an increase of the initial pressure. 
 
 If we have no compression \b = 0], equation (221) becomes 
 
 B 
 
 -See (223) D = 
 
 S 
 
 16 
 
182 
 
 TME RELATIVE PROPORTIONS 
 
 If with these conditions we substitute in equation (223), 
 we have 
 
 ^4 859380 
 
 Pfr nat. log 
 
 and for any assumed initial pressure we can say, 
 
 1 
 
 W = constant: 
 
 nat. log 
 
 The following tabulation will approximately show the 
 relative economy of successively increased expansions with 
 
 the cheapest point of cut- 
 off per horse-power per 
 hour. At what particu- 
 lar point this gain is met 
 and annulled by the 
 losses due to internal 
 condensation cannot be 
 said with our present ex- 
 perimental knowledge. 
 The demand for concen- 
 tration of power, which 
 is the greatest attribute 
 of the steam-engine, will 
 
 B 
 
 <=-- 
 
 1 
 
 Saving. 
 
 Per cent saved. 
 
 natlog^ 
 
 
 
 
 Succes- 
 
 Of the 
 
 
 
 
 sive 
 
 first 
 
 
 
 3.222 
 
 
 vols. 
 
 volume. 
 
 
 
 2.096 
 
 1.226 
 
 37 
 
 37 
 
 
 
 1.661 
 
 0.435 
 
 21 
 
 49 
 
 
 
 1.431 
 
 0.230 
 
 14 
 
 56 
 
 
 
 1.286 
 
 0.145 
 
 10 
 
 60 
 
 
 
 1.182 
 
 0.104 
 
 8 
 
 63 
 
 
 
 1.107 
 
 0.075 
 
 6 
 
 66 
 
 
 
 1.048 
 
 0.059 
 
 5 
 
 69 
 
 T 
 
 7T 
 
 1.000 
 
 0.048 
 
 5 
 
 70 
 
 ft 
 
 0.961 
 
 0.039 
 
 4 
 
 70 
 
 s 
 
 0.927 
 
 0.034 
 
 3 
 
 71 
 
 TV 
 
 0.898 
 
 0.029 
 
 3 
 
 72 
 
 TV 
 
 0.873 
 
 0.025 
 
 2* 
 
 73 
 
 not permit us to logically accept these results, as they would 
 give very slight initial pressures for condensing engines or 
 require impossible points of cut-off with present engines. 
 A consideration of equation (223) will point out to us sev- 
 eral methods of escape. The specific volume of steam 
 steadily decreases as its pressure increases ; therefore for a 
 fixed point of cut-off and back pressure the economy of 
 steam effected is greater as the pressure is greater. A re- 
 duction of the back pressure effects a rapid increase in 
 
OF THE STEAM-ENGINE. 
 
 183 
 
 saving, since it is multiplied by ($) the specific volume, 
 usually a large quantity. 
 
 Very few condensing engines have good vacuums, but for 
 the purpose of giving approximate results we will neglect 
 the back pressure. Equation (223) then becomes 
 
 constant 
 
 W= 
 
 SP> 
 
 r 
 
 l + nat. log 
 
 IT 
 
 If the point of cut-off (e] is fixed, we can still effect some 
 saving by increasing the steam pressure, since the product 
 SP b is a slowly increasing quantity as the initial pressure is 
 increased. 
 
 If, again, we fix upon some particular value of P b in the 
 preceding equation, it becomes 
 
 1 
 
 W = constant 
 
 1 -I- nat. log - 
 
 Tabulating the results of insertion of decreasing values 
 of e, we have 
 
 These rudely approxi- 
 mate figures which, 
 however, are probably 
 quite as accurate as any 
 that can be realized in 
 practice show the im- 
 perative need that ex- 
 ists for experimental 
 demonstration of the 
 causes of condensation 
 in the cylinder, and the 
 application of a remedy for it before further intelligent 
 progress can be made in the economical use of steam. A 
 comparison of the two tables shows that in the absence of 
 
 e 
 
 1 
 
 Saving. 
 
 Per cent saved. 
 
 1 + nat. log j 
 
 
 
 
 Succes- 
 
 Of the 
 
 
 
 
 sive 
 
 whole 
 
 1 
 
 1 
 
 
 vols. 
 
 volume. 
 
 i 
 
 0.591 
 
 0.409 
 
 41 
 
 41 
 
 1 
 
 0.476 
 
 0.115 
 
 19 
 
 52 
 
 i 
 
 0.419 
 
 0.057 
 
 12 
 
 58 
 
 
 
 0.383 
 
 0.036 
 
 9 
 
 62 
 
 I 
 
 0.358 
 
 0.025 
 
 7 
 
 64 
 
 
 0.339 
 
 0.019 
 
 5 
 
 66 
 
 | 
 
 0.324 
 
 0.015 
 
 4 
 
 68 
 
 * 
 
 0.312 
 
 0.012 
 
 i| 
 
 69 
 
 A 
 
 0.303 
 
 0.009 
 
 3 
 
 70 
 
184 THE RELATIVE PROPORTIONS 
 
 condensation, expansion could probably be carried further 
 with profit in the former than in the latter case. The ex- 
 pansion necessary under existing conditions to effect the 
 greatest saving of steam per horse-power has already been 
 exceeded in many cases. Besides expansion, the condensa- 
 tion certainly is a function of the temperatures and of the 
 surfaces exposed, and of the conductivity of the surfaces, 
 as well as of their time of exposure, and perhaps of the 
 relative time of exposure of the interior surfaces of the 
 steam-cylinder. It should be remembered that an increase 
 of horse-power is the only way to decrease the constant 
 charge per horse-power. 
 
 We have obtained but 13 per cent, of the power in coal, 
 and even now we are at the limits of commercial economy 
 in the use of the steam-engine. 
 
 With our present types of boilers and engines it does not 
 pay to use coal more economically. It is cheaper to waste 
 the 87 per cent, of the power of the coal than to go on in- 
 creasing the cost of the engine in the endeavor to save coal. 
 
 Of course if we wish only to conserve Nature's forces, 
 and disregard the money it costs to produce a diminished 
 consumption of coal, we can, and probably will, succeed in 
 obtaining larger results. 
 
 But money derives its value from the labor of humanity, 
 and for this reason should be saved in preference to Nature's 
 forces. 
 
 There are some avenues of escape open to us from this 
 dilemma, and I will mention them. 
 
 Taking the best type of engines of to-day as a starting- 
 point, we must depart in the following directions : 
 
 We do not particularly need to increase the efficiency of 
 the boiler as an evaporator, but we must increase its ability 
 to withstand pressure without increasing its cost. 
 
 We must decrease the condensation inside of the steam- 
 
OF THE STEAM-ENGINE. 185 
 
 cylinder by using a non-conducting surface or by superheat- 
 ing to a small extent. 
 
 We must decrease the friction of the engine and of the 
 machinery of transmission to the point where the useful 
 work is delivered. 
 
 We must produce better vacuums in the condenser, and 
 diminish its cost. 
 
 We must diminish the cost of the engine. 
 
 We must diminish the cost of the attendance on engines, 
 boilers, and machinery, and of lubrication. 
 
 We must increase the durability of engines, boilers, and 
 machinery. 
 
 Coal is too cheap even now to admit of increased economy 
 of it at the cost of increased outlay for plant and attendance. 
 We would be saving coal without saving money, or rather 
 spending more money in the difference in constant charges, 
 such as interest, deterioration, and attendance are, than we 
 would save in the difference in the coal-bill. 
 
 The reason why automatic cut-off engines have produced 
 such favorable results is this : The condition in every case 
 is that a point of cut-off shall be determined, and then the 
 engine designed for some certain power with that cut-off. 
 The demands of the shop or mill will not permit such a 
 condition of uniformity to be fulfilled ; and here the automatic 
 cut-off steps in, and, with the added advantage of its carefully 
 regulated speed, uses less steam, although not so economically, 
 per horse-power per hour, whenever the demand for power 
 is lessened ; and so at the end of the month a real saving is 
 apparent in the coal-bill, although the engine may not have 
 cut off at the point of least cost for a good part of the time. 
 
 The engine-owner must have an engine which is adequate 
 to the greatest demand for power which may be made upon 
 it, but he should not choose an engine which would cut-off 
 too early under its average load. 
 
 16* 
 
186 THE RELATIVE PROPORTIONS 
 
 Economy of Steam Alone without Compression. [From 
 
 J. F. I., June, 1880.] 
 
 Let P = the mean pressure in pounds per square inch. 
 " P 6 = the initial pressure in pounds per square inch 
 
 (absolute). 
 " B = the back pressure in pounds per square inch 
 
 (absolute). 
 " e =the fractional part of the stroke at which the 
 
 steam is cut off by the valve-motion. 
 " V = the volume of the steam-cylinder. 
 " E = the economy. 
 
 Now, it will further be acknowledged that the economy 
 increases directly as the work done during one stroke and 
 inversely as the steam used during one stroke. Therefore 
 we have 
 
 PV P 
 
 *-%'* (225) 
 
 We have further the well-known formula for the mean 
 effective pressure of steam used expansively, supposed to 
 expand according to Mariotte's law: 
 
 r fi 
 
 P=eP 6 \1+L- \-B. (226) 
 
 L e ] 
 
 While this does not accurately represent the law of ex- 
 pansion of steam, it does it with close approximation for all 
 practical purposes. Substituting the value of P from equa- 
 tion (226) in equation (225), we have 
 
 E=P b 1-t-i- --. (227) 
 
 L ll 6 
 
 Differentiating with respect to e, and seeking the max- 
 imum, we find it to be 
 
 e-2-. (228) 
 
OF THE STEAM-ENGINE. 187 
 
 CHAPTER XVII. 
 
 (74) The Errors of the Zeuner Diagram as Applied 
 to the Stephenson Link-Motion.* I. INTRODUCTION. 
 The mathematical elegance of Professor Gustav Zeuner's 
 Treatise on Valve- Gears is due to the fact that he has shown 
 that the equation representing the distance of a slide-valve, 
 controlled by an eccentric or by means of a link, is in all 
 cases with greater or less approximation the polar equation 
 of a circle. Deservedly, his work has met with a most 
 gratifying acceptance from all intelligent engineers, as not 
 only being the most correct, but also the only method 
 which, without the aid of models or templates, enables 
 the practitioner to devise and study any desired form of 
 valve-gear. 
 
 As shown in the drawing (Fig. 32), upon the top of a 
 standard behind the section of the cylinder a pulley of 
 equal size with the crank-shaft was connected with the 
 crank-shaft by means of a steel saw-band running upon 
 thin gutta-percha strips glued to the shaft and pulley sur- 
 faces. This steel band was kept very taut by means of a 
 stretching pulley about the middle of its length. 
 
 Upon the end of the pulley-shaft and just back of the 
 valve a drawing-board was so attached as to permit a pen- 
 cil, attached to the valve and kept pressed against the paper 
 
 * The drawings for this chapter were made by Mr. G. H. Lewis, a 
 graduate of the department of Dynamical Engineering, and used by 
 him as a part of a thesis. 
 
 The mathematical treatment is my own. Mr. Lewis's drawings 
 have been somewhat added to, in order to give graphical methods of 
 determining the errors of the diagram. I am indebted to Mr. Lewis 
 for many ingenious and thoughtful suggestions and much accurate 
 and painstaking work in tracing the diagrams. 
 
188 
 
 THE RELATIVE PROPORTIONS 
 
 stretched upon the 
 drawing - board, to 
 trace the curve, 
 showing the distance 
 of the valve from its 
 central position. 
 
 Had the drawing- 
 board been attached 
 directly to the crank- 
 shaft, and a rod hav- 
 ing a pencil in the 
 end been attached 
 to the link-block or 
 any point on the 
 valve or valve-stem, 
 and carried back to 
 the centre of the 
 board, it would have 
 been more service- 
 able for scientific 
 
 purposes, as 
 
 elimi- 
 
 nating some of the 
 possible sources of 
 error due to the im- 
 perfections of the 
 model. 
 
 This model was 
 constructed of iron, 
 brass and mahog- 
 any, and every pos- 
 sible precaution was 
 taken to obtain rigid- 
 ity and avoid shrink- 
 age ; it was construct- 
 
OF THE STEAM-ENGINE. 
 
 189 
 
 ed full size from the dimensions stated by Zeimer in his 
 Treatise on Valve -Gears, page 78. 
 
 Eccentricity = r = 2.36 inches. 
 
 Angular advance = 3 = 30. 
 
 Length of the eccentric-rods = I = 55.1 inches. 
 
 Half length of the link = c = 5.9 inches. 
 
 Outside lap = e = 0.94 inches. 
 
 Inside lap = i = 0.27 
 
 Open eccentric-rods and equal angles of advance were 
 taken. The link was so attached to the eccentric-rods as 
 to permit the link-block to be placed immediately in front 
 of the ends of the eccentric-rods ; in other words, so that 
 the variable distance u of the link-block from the centre of 
 the link could at its maximum be made equal to the half 
 length of the link c. 
 
 This form of link is shown in Fig. 33 a. 
 
 FIG. 33. 
 
 (a) 
 
 \ 
 
190 THE RELATIVE PROPORTIONS 
 
 The diagrams taken upon this model clearly showed that 
 some greater sources of error existed than the so-called 
 "Missing Quantity" of Zeuner. 
 
 Acceptance of authority is a great preventive of advance- 
 ment of knowledge, and it will be our task to show clearly 
 what points have been overlooked by Professor Zeuner, with, 
 we hope, the result of making even more clearly understood 
 this construction, so simple in its mechanism and so intricate 
 in its action. 
 
 II. THE SIMPLE SLIDE-VALVE. CONSIDERATION OF 
 THE MISSING QUANTITY IN THE SIMPLE SLIDE-VALVE. 
 SETTING VALVE FOR EQUAL LEADS EQUIVALENT TO AL- 
 TERING THE LAPS OF THE VALVE. For the sake of sim- 
 plicity let us first consider the simple slide-valve. 
 
 On page 11 of his Treatise on Valve- Gears, Zeuner gives 
 for the distance of a simple slide-valve from its centre of 
 motion , 
 
 ' = rsm(w + d')+ sin (2# + w) sin w. (229) 
 
 The first term of the second member of this equation is the 
 polar equation of a circle, with the origin in its circumfer- 
 ence and its diameter forming an angle equal to <5; with 
 the axis of ordinates OY (see Fig. 34) w is the angle 
 which the crank forms with the axis of abscissas OX. 
 All of this can readily be understood from the explana- 
 tions given in the book. 
 
 It is with the second term of the second member (" the 
 missing quantity ") that we shall have particularly to deal, 
 for Zeuner has considered it as inappreciable in most cases ; 
 which is not practically true, for many cases occur in which 
 of necessity the eccentric-rods are comparatively short. 
 
 Dr. Zeuner fixes the central position of the slide-valve by 
 taking the mean of the two positions of the valve when the 
 
OF THE STEAM-ENGINE. 191 
 
 crank is on its dead points. He does this on the assumption 
 that the valve will be set for equal leads ; which is always the 
 proper method. 
 
 This central position differs from the true central position 
 
 r 2 cos 2 <5 
 by a quantity = (230), for the true central point of 
 
 the valve travel is a mean between the extreme positions of 
 the valve, and farther away from the crank-shaft, a distance 
 equal to the above-stated quantity ; therefore at one extreme 
 
 7* 2 COS 2 S 
 
 the valve's distance from Zeuner's centre = r + , and 
 
 r 2 cos 2 d 
 at the other extreme = r . 
 
 If now we can convert the missing quantity into a func- 
 tion of the theoretical valve distance from its centre for 
 equal leads (Zeuner's centre), we can much more conve- 
 niently lay down the irregular curve of the valve-circle for 
 the case of a short eccentric-rod. 
 
 According to the diagram, = rsin (w + <5) (231). Page 
 43, Z. T. V. G.* the missing quantity is given as 
 
 z - [cos 2 d - cos 2 (w + <?)], (232) 
 
 Zv 
 
 T \ S* -O ^ Sin2 ( W + ^) ^OQON 
 
 or 2 = _(cos 2 <5-l) + |p S (233) 
 
 or 2lz = r 2 (cos 2 d-l) + r 2 sin 2 (w + d). (234) 
 
 Letting 0=r 2 (cos*<5-l) and substituting for its value, 
 we have 
 
 e -2&--C, (235) 
 
 the equation of a parabola whose ordinates are the theoret- 
 ical travels of the valve from its centre of motion, and 
 whose abscissas are the missing quantities for the same. 
 
 * Abbreviation of Zeuner's Treatise on Valve-Gears. 
 
192 THE RELATIVE PROPORTIONS 
 
 The radius of curvature of this parabola at its vertex 
 = % the latus rectum or parameter, and is equal to /, the 
 length of the eccentric-rod, and we can substitute an arc 
 of a circle with the radius I for this parabola without ap- 
 preciable error. 
 
 For the travel = o, 
 
 z = - T sin 2 5. (236) 
 
 cyi ' nj 
 
 Forf = r, 
 
 z= i cos 2 d. (237) 
 
 For 2 = 0, 
 
 * 2 > * 2 / V\ ^OQQ"\ 
 
 Therefore w = o. 
 
 That is to say, the " missing quantity " disappears on the 
 dead points, since the valve is actually set for different leads. 
 
 To lay down the actual curves of valve travel, the " miss- 
 ing quantity " being taken into account. 
 
 Fig. 34. With a radius L and the centre O describe 
 an arc L L to intersection L with the diameter of the valve 
 circle P . At the point O, and at right angles with P , 
 draw the indefinite line O Z. 
 
 With a radius of compass = I, and with the centre on the 
 line O Z, describe through L and Lj the arc Q L L^. The 
 ordinates to this arc from the line O P measure in quantity 
 and direction the values of the " missing quantity," which 
 must be added to or subtracted from the theoretical radius 
 vector in order to obtain the true curve of the motion of 
 the valve. 
 
 Fig. 34, for the purpose of showing an extreme case, has 
 been laid down to scale as follows : 
 
 Eccentricity = r = 2 inches. 
 Angular advance = 3 = 30. 
 
OF THE STEAM-ENGINE. 
 
 FIG. 34. 
 
 193 
 
 H, H, R 
 
 H* H H, R| 
 
 Outside lap = e = 0.82 inches. 
 Width of port = a = 0.75 inches. 
 Length of eccentric-rod = I = 8 inches. 
 The construction for the missing quantity, for the sake of 
 clearness, has all been added in heavy lines. 
 The effect of the " missing quantity/' when considerable 
 17 
 
194 THE RELATIVE PROPORTIONS 
 
 enough to be noticed, is, when the piston-head moves toward 
 the crank-shaft, the cylinder being at the right hand, 
 
 (1) To delay slightly the pre-admission of steam. 
 
 (2) To increase the over-travel. 
 
 (3) To hasten the cut-off of the steam (very slightly). 
 
 (4) To hasten the compression of the steam. 
 
 (5) To hasten the release of the steam. 
 
 When the piston-head moves away from the crank-shaft, 
 
 (1) To hasten the pre-admission. 
 
 (2) To diminish the over-travel. 
 
 (3) To delay the cut-off (very slightly). 
 
 (4) To delay the compression. 
 
 (5) To delay the release. 
 
 A glance at the diagram at once reveals the fact that 
 equalizing the lead very nearly equalizes the cut-off. 
 
 It is only when the valve is set for equal extreme travels 
 from the centre that different laps are required. No atten- 
 tion has been paid to the variation in position of the piston 
 due to the obliquity of the connecting-rod. 
 
 III. THE PISTON'S POSITION. The effect of the obliquity 
 of the connecting-rod is to keep the piston nearer to the 
 crank-shaft when it is moving away from it, and to draw it 
 closer to the crank-shaft when it is moving toward it, than 
 it would be if the connecting-rod was constantly parallel to 
 the centre line of the cylinder. 
 
 At the dead points, the connecting-rod being in the centre 
 line of the cylinder, this action ceases. 
 Letting w = angle of the crank, 
 " R = radius " 
 
 " L = the length of the connecting-rod, 
 we would have, if the connecting-rod were constantly par- 
 allel to the centre line of the cylinder, for the space passed 
 over by the piston-head = S, 
 
 ; (239) 
 
OF THE STEAM-ENGINE. 195 
 
 and when we take the obliquity of the connecting-rod into 
 consideration, 
 
 T N A. VL 1 R* sin 2 w\ t 
 -L}ll ----- 1. (240) 
 
 Then for the difference d between the two positions we 
 have 
 
 (241) 
 
 or, expanding, 
 
 7O2 
 
 d = -- sin 2 w, approximately. (242) 
 
 2Z/ 
 
 Fig. 34. The positions HI to H 7 can be corrected by lay- 
 ing down in the opposite direction from the cylinder, from 
 the points as already found, the values of d. 
 
 It will be observed that the equation for d is the equation 
 of a parabola whose se,mi-latus rectum is equal to L. Fur- 
 ther, for w = o or 180 d=o. If for this parabola we sub- 
 stitute the osculatory circle of a radius L to its vertex, we 
 are practically close enough. 
 
 If now with a radius of compass = L, with one point in 
 M and the other on the line Y O bisecting the cylinder, we 
 describe the arc K 2 K 3 , we have, with sufficient approxima- 
 tion, the desired parabola. 
 
 Taking off for the position O R of the crank the distance 
 JRT=Rsmw, and laying it off from M to S, we have the 
 correction S U of the position of the piston H, which, if we 
 consider the cylinder at the right-hand side, should be laid 
 off to the left of H, giving the true position of the piston- 
 head at H'. 
 
 Thus we can lay down graphically the actual positions 
 of the piston and the true distances of the slide-valve from 
 its centre of motion, when set for equal leads for every posi- 
 tion of the crank and for any proportions of the mechanism. 
 
196 THE RELATIVE PROPORTIONS 
 
 For the sake of emphasis we again repeat : Different laps 
 are not necessary when the valve is set for equal leads, when 
 the piston position is disregarded. 
 
 Altering the laps will alter the leads. If the piston 
 position is regarded and the alteration in the leads is dis- 
 v egarded for the sake of a very accurate cut-off, the lap 
 should be shortened on the side toward the crank and 
 lengthened on the side away from the crank. These 
 amounts can be determined from the diagram. 
 
 It is only in the case of a very short connecting-rod that 
 such a procedure is necessary ; short eccentric-rods do not 
 require it. 
 
 IV. THE STEPHENSON LINK-MOTION. ERROR DUE TO 
 AN IMPERFECT MODE OF ATTACHING THE LINK TO THE 
 ECCENTRIC-KODS. On pages 56-98 of Z. T. V. G. the Ste- 
 phenson Link-Motion is very fully treated for both open and 
 crossed rods, and for both forms of link shown in Fig. 33 a 
 and b, no distinction being made between them. 
 
 In Fig. 33 a it will be observed that the rods are attached 
 on the concave side at the points C and d, introducing an 
 error which we will next endeavor to determine. 
 
 Fig. 35 is the Zeuner diagram carefully laid down for the 
 dimensions already given of the model, on which was used a 
 form of link shown in Fig. 33 a. 
 
 The method of making the slide-valve describe its own 
 diagram has already been explained. It is only necessary 
 to add that as the drawing-board turns synchronously with 
 the crank, the valve-circles (curves) will both be on the 
 same side of the origin instead of on opposite sides, as 
 drawn for the sake of clearness in Fig. 34. 
 
 The object of making the link of the form Fig. 33 a is 
 twofold : first, to reduce the eccentricity ; second, to enable 
 us to place the valve wholly under the control of one eccen- 
 tric-rod. 
 
OF THE STEAM-ENGINE. 
 FIG. 35. 
 
 197 
 
 -* 
 
 Fig. 36 is a centre-line sketch of Fig. 33 a, similarly let- 
 tered. It will be observed that as the suspended link sweeps 
 
 17 * 
 
198 THE RELATIVE PROPORTIONS 
 
 to and fro with a scythe-like motion, the line K C forms an 
 angle with the horizontal line K N 1? which is approximately 
 equal to the angle a, which the chord of the link forms with 
 the vertical. 
 
 The value of sin a is given with very close approximation 
 on page 61, equation (11), Z. T. V. G. 
 
 As our only object is to point out an error which can be 
 
 avoided, we will make use of the principal term of this 
 quantity, and take 
 
 sin a = - cos d sin w. (243) 
 
 c 
 
 Let us denote the missing quantity due to this error by z l = 
 NN lt its effect being to keep the link closer to the crank- 
 shaft except where it equals zero. 
 
 Let KC=q: 
 
 NN, = z l = q (1 - cos ) - 2/1 - ^ 1 - *- cos 2 d sin 2 w \ (244) 
 
 or, expanding the quantity under the radical, and neglecting 
 terms containing greater than the second power of the cir- 
 cular functions, we have 
 
 z^^-cos^sin 2 . (245) 
 
 2c 3 
 
OF THE STEAM-ENGINE. 199 
 
 For w = 90 this quantity is a maximum, and for w = it 
 is equal to zero. That is, it does not appear in the lead 
 when the valve is set for equal leads, but it does attain its 
 maximum near the point of usual cut-off, and is particularly 
 pernicious there and at the point of exhaust closure. The 
 reason that it has remained unperceived hitherto is probably 
 because it does not appear in the lead. 
 Transposing, we have 
 
 0/.2 
 
 (246) 
 
 q cos 2 3 
 
 The equation of a parabola whose ordinates are rsmw 
 
 c 2 
 
 and whose abscissas are z it its semi-latus rectum is - 
 
 q cos o 
 
 which is also the radius of curvature of the osculatory circle 
 to its vertex. 
 
 A moment's reflection will convince the reader that the 
 error due to Zeuner's missing quantity is inappreciable 
 (where of any consequence) in the present case. See Fig. 
 34 and explanation. 
 
 To determine the error z 1? through O (Fig. 35), with a 
 centre on O X produced, describe an arc of a circle T V 
 
 with a radius = - . With as a centre and the radius 
 q cos 2 3 
 
 r describe an arc S T U. Draw any position of crank as 
 O Z to intersection Z with the arc S T U. Parallel to O X 
 draw through Z the line Z Q. The distance PQ - the error, 
 which can be laid off both inside and outside the theoretical 
 valve-circle, as at pfpb. In the model q = 3 inches, 3 = 30, 
 and c = 5.9 inches. 
 Therefore 
 
 c 2 
 
 15.47 inches, 
 q cos 2 3 
 
 which is the radius of the arc V Q T. 
 
200 
 
 THE RELATIVE PROPORTIONS 
 
 Laying down after the manner described the arcs blc fh, 
 we have the corrected circles for the valve-motion at the 
 IV grades. These arcs are laid down for the neighborhood 
 of the point of cut-off only. 
 
 This most pernicious error can be avoided by the use of 
 
 FIG. 37. 
 
 the link (Fig. 33 5), although a larger eccentric is required, 
 and, therefore, it is sometimes difficult to fit into confined 
 
OF THE STEAM-ENGINE. 201 
 
 spaces. Certainly it is of great importance to avoid so 
 faulty a construction if it be possible. 
 
 FIG. 38. 
 
 Figs. 37 and 38 are diagrams automatically traced by the 
 working model. 
 
 To avoid the errors due to suspension, the link-block was 
 clamped in the link for each grade, and the link, therefore, 
 swung upon the rocker-shaft arm. 
 
 To a,void the errors due to the " lost motion," the valve- 
 circles were traced twice by reversing the direction of the 
 motion, and the mean between the two circles traced with 
 
202 THE RELATIVE PROPORTIONS 
 
 pen and ink by hand. The difference was very slight, if 
 any at all. 
 
 It will be seen that these actual valve-curves verify with 
 great accuracy the corrected valve-circles (Fig. 35) for the 
 fourth grade. 
 
 Similar corrections can be made for each grade of link if 
 desired. 
 
 The letters / and b refer to the direction of motion of 
 the piston-head, forward (/) meaning toward the crank- 
 shaft, and backward (6) meaning away from the crank- 
 shaft, a rocker-shaft intervened reversing the direction of 
 motion of the valve. 
 
 When the form of link shown (Fig. 33 6) is used, the 
 increased eccentricity required will increase the "missing 
 quantity " given by Prof. Zeuner, and it must therefore be 
 guarded against, particularly in extreme cases. 
 
 Cases may occur when it will prove advantageous to at- 
 tach the eccentric-rods to the link at points nearer its centre 
 than the extreme limits of the travel of the link-block, but 
 special pains should be taken to place the centre of the pin- 
 joint on the central arc of the link ; this method of attach- 
 ment, however, will result in increasing the slip of the link- 
 block. 
 
 V. SLIP OF THE LINK-BLOCK. Zeuner gives two cases 
 of the suspension of the link, by means of a hanger attached 
 at the centre of the chord of the link and at the bottom of 
 the link ; in the first case the upper end of the hanger 
 should theoretically move in an arc of a circle which has 
 for a radius the length of the eccentric-rod, and whose 
 centre is above the centre line a distance equal to the 
 length of the hanger. The lower end of the hanger 
 should be attached at the centre of the link and on the 
 central arc of the link, thus placing the origin of the arc 
 of suspension at a horizontal distance equal to the length 
 
OF THE STEAM-ENGINE. 
 
 203 
 
 of the eccentric-rod from the centre of the crank-shaft. 
 Fig. 39 will render this clear. J lt and not J, should be 
 the point of attachment of the hanger, and L, not K 1? 
 
 FIG. 39. 
 
 should be the centre of the arc in which the upper end 
 of the hanger E should move. 
 
 "We can thus avoid increasing the slip by the quantity 
 J J x tan a in one direction, and decreasing it by the same 
 amount in the other direction. 
 
 Fig. 40 shows the second method of suspension of the 
 link by a hanger attached to the bottom. 
 
 Both of these methods are fully explained by Zeuner, 
 and the reader is referred to his work for further details. 
 
 It will be perceived that the hanger is but a rude ap- 
 proximation to a parallel motion, used only because of its 
 simplicity and lightness when the link-block is forced to 
 move to and fro in a straight line. 
 
 When the link-block is attached to the end of a rocker- 
 shaft arm, as is commonly the case for American locomo- 
 
204 
 
 THE RELATIVE PROPORTIONS 
 
 tives, if the hanger is made the same length as the rocker- 
 shaft arm, there will be no slip when the link-block coincides 
 with the point of suspension ; the slip for other positions of 
 the link-block will be due to the angular position of the 
 link. 
 
 Assuming, when the block is forced to move in a straight 
 line, that some method has been adopted to force the point 
 of suspension of the link to move in an at least very close 
 
 FIG. 40. 
 
 approximation to a straight line, and, further, that when 
 the link-block moves in an arc of a circle of a given radius 
 the hanger is of the same length as this radius, we can 
 consider the slip as due only to the angular position of 
 the link. 
 
 Of course these conditions cannot always be fulfilled, but 
 it is best to know what ought to be done, even if we cannot 
 exactly do it. 
 
 Fig. 41 shows the two positions of the links K K x and 
 K,K 3 , for which the slip is zero ; and letting 8 equal the 
 
OF THE STEAM-ENGINE. 205 
 
 amount of slip for all other positions, if we suppose the 
 hanger attached to the middle point of the link, and u the 
 distance of the link-block from that point, we will have 
 
 s = (seca-l) w , (247) 
 
 or, since the angle a is always very small, 
 
 S = u(l-COSa)=u(l-Vl-BUL*a); (248) 
 and substituting for sin a its value, and expanding and neg- 
 
 FIQ.4L 
 
 lecting all terms containing higher powers than the square 
 of the circular functions, we have 
 
 = cos o sin* 10. 
 
 (249) 
 
 We thus see that the effect of the slip does not appear in 
 the lead, but, being a maximum for w = 90 or 270, will 
 affect the points of cut-off and exhaust closure. 
 
 Increasing the angular advance diminishes the slip, as 
 also does increasing the length of the link. The tendency 
 of the slip is to increase the travel of the valve by an 
 amount V : 
 
 . (260) 
 
 18 
 
206 
 
 THE RELATIVE PROPORTIONS 
 
 This amount is very small for a well-proportioned valve- 
 gear, but it increases directly as the distance u from the 
 point of attachment of the hanger to the link. 
 
 FIG. 42. 
 
 u=o 
 
 When the link is suspended from the bottom the value w 
 must be replaced by (c + u). 
 
 We thus see that for general usage at all points suspend- 
 ing the link at the middle is the best, while if one particular 
 point is intended to be constantly used, and the other points 
 only exceptionally, it is best to attach the hanger to the link 
 at that point. 
 
 If the tumbling-shaft arm cannot be made equal to the 
 length of the eccentric-rods (and for obvious reasons it 
 rarely can be so proportioned), the centre of the tumbling- 
 shaft must be so placed as to make an arc, struck with its 
 arm as a radius, intersect the theoretical arc at the point or 
 points of greatest usage. 
 
 From what has been said about the position of the arc of 
 suspension, it will readily be perceived that its length on 
 either side of the horizontal line E K! (Figs. 39 and 40) is 
 determined by the point of attachment of the hanger to the 
 link. 
 
OF THE STEAM-ENGINE. 
 
 207 
 
 The link-motion on which experiments were made with a 
 view to testing the correctness of these results had the fol- 
 lowing dimensions : 
 
 Length of eccentric-rods = I = 18 inches. 
 Kadius of eccentricity = r = 1 " 
 Length of link = 2c = 6 " 
 
 Angular advance d = 30. 
 Open rods. 
 These results verify the above theory only in a qualitative 
 
 FIG. 43. 
 
 *-> *e 
 
 way, as the upper end of the hanger was not always kept on 
 the true arc of suspension. 
 
 The link being suspended after the manner described, a 
 pencil was fixed in the link-block, and the block success- 
 ively fixed at different grades, the pencil being allowed to 
 trace on a paper back of it the curves of slip. 
 
 CASE I. The link suspended at the centre of its arc. Fig. 
 
208 
 
 THE RELATIVE PROPORTIONS 
 
 42 shows that the slip of the block increases both ways from 
 its centre, as had been predicted. The arc for u = o is the 
 standard with which the other curves must be compared. 
 CASE II. The link being suspended at the centre of Us 
 chord. Fig. 43 is inserted merely for the purpose of show- 
 ing an imperfect mode of suspension. All of the slip-curves 
 are bad, and at no point is there any cessation of the slip. 
 
 FIG. 44. 
 
 CASE III. The link suspended at the bottom. Fig. 44 re- 
 veals the fact that the slip becomes very great for the upper 
 end of the link so great as to seriously affect the distribution 
 of the steam. The lower half of the link only can be relied 
 upon for accurate work. 
 
 CASE IV. The link suspended halfway between the bottom 
 and centre. Fig. 45 shows a better average result than any 
 of the others, and is undoubtedly the best mode of hanging 
 
OF THE STEAM-ENGINE. 
 
 209 
 
 the link when the grade u = - c is to be generally used. 
 Viewed from a practical point, slip is of great importance, 
 being the cause of the wear upon links, which soon unfits 
 
 FIG. 45. 
 
 them for accurate work. Great pains are taken to reduce 
 this wear by case-hardening the links or using steel in the 
 place of wrought iron. 
 
 A proper mode of suspension is the most important point 
 to be attained when the durability of the link-motion is under 
 consideration. 
 
 18* 
 
TABLES. 
 
 THE following four tables, .condensed from the fourth 
 section of Weisbach's Mechanics of Engineering, which has 
 formed the basis of this work, are inserted as a means of 
 ready reference for ordinary problems in the strength and 
 elasticity of materials. The same notation as that used by 
 Weisbach is retained, in order to avoid confusion in refer- 
 ring to his work. 
 
 211 
 
212 THE RELATIVE PROPORTIONS 
 
 TABLE V. 
 
 (75) Elasticity and Strength of Extension and Com- 
 pression. (Arts. 201-214.) 
 
 (Art. 204.) To find increase or decrease in length under a strain 
 of extension or compression. 
 
 Where the weight of the body under strain is not considered 
 
 X = the amount of the extension in inches. 
 P= the weight acting in pounds. 
 
 Z = the length of the body acted upon in inches. 
 
 F= the area of the cross-section in square inches. 
 J=the modulus of elasticity in pounds per square inch. 
 
 A Pl 
 ~ FE' 
 
 Let (7= the weight of the body under strain. 
 
 (Art. 207.) Where the weight of the body under the strain is also 
 taken into account, %=- * 
 
 (Art. 205.) To find the proof-strength of a body to be submitted 
 to strain. 
 
 T =the proof-strength for extension per square inch in pounds. 
 jr t = the proof-strength for compression per square inch in pounds. 
 R = the ultimate strength for extension per square inch in pounds. 
 Ji = the ultimate strength for compression per square inch in pounds. 
 
 For a pull, P= FT. For a thrust, P l = FT V 
 
 To find the ultimate strength of a body to be submitted to strain. 
 
 To tear the body asunder, P= FK. To crush it, P l = 
 
OF 
 
 STEAM-ENGINE. 
 
 213 
 
 S- B 
 > * 
 
 *S 
 
 gg 
 
 E 5 
 
 I 
 
 5' of 
 
 ** 
 
 Pfy 
 5 
 
 B 
 
 L. 
 
 s I 
 i? 
 
 I & 
 
 15 
 
 }i 
 ? ^ 
 
 > 
 
 S 9 
 M 
 
 r 
 
 i 
 
 <rs 
 Is 
 & | 
 
 1 
 I 
 
 R 
 
 f 
 
 o 
 
 2- "a 
 
 .S, 
 
 * 
 
 i: 
 
 g 
 
 
 2? . 
 
 I 
 
 1 
 
 * 
 
 11 
 
 p'S"5' 
 
 r 
 
 ^00 
 
 .- 
 
 3W& 
 Art. 217 
 
 
 d I * B 
 
 IIHII: 
 
 11 fa 
 
 
 
 |t.| 
 
 iP 
 
 y ^ s 
 
 * ^ 
 
 M~'S. ^'* s 
 
 US. p*_ 
 
 rt) M * 
 
 
 !i 
 
 K 
 
 H 
 
 4 
 I? 
 
 gs 
 
 H 
 
 l&sg 
 
 nl 
 
 IB 
 
 *c - 
 
 S" f 
 
 |P } 
 
 p* to 
 
214 
 
 THE RELATIVE PROPORTIONS 
 
 TABLE VII. 
 
 (77) The Elasticity and Strength of Torsion. (Arts. 
 262-265.) 
 
 Notation. P= the load in pounds, a = the lever-arm of the load 
 in inches. T= the modulus of proof-strength for shearing in pounds 
 per square inch. TF the measure of moment of torsion. e = the 
 greatest distance in inches of any element of the cross-section from 
 the neutral axis. 
 
 Z=the length in inches submitted to stress. 
 d=the diameter of round shafts in inches. 
 6 = the length of one side of a square shaft in inches. 
 a = the angle through which the body is twisted in degrees. 
 
 Form of body. 
 
 Proof-strength. 
 
 Angle of torsion. 
 
 For any form of cross- \ 
 
 TW 
 
 
 section j 
 
 JL = . 
 
 
 
 OM5 
 
 See Art. 264. 
 
 Cast iron (Art. 263). 
 
 
 
 o- 0.0002053^. 
 
 
 rPT 
 
 d* 
 
 For a solid round shaft... 
 
 P= 0.1963.- 
 a 
 
 See Art. 264. 
 
 Wro't iron (Art. 263). 
 p i 
 
 
 
 a = 0.0000648^-. 
 
 
 
 Cast iron (Art. 263). 
 
 
 
 PnJ 
 
 
 3 
 
 a = 0.0001211^. 
 
 For a solid square shaft.. 
 
 P= 0.2357 .- 
 a 
 
 See Art. 264. 
 
 Wro't iron (Art. 263). 
 pi 
 
 
 
 a = 0.0000382^-. 
 
OF THE STEAM-ENGINE. 
 
 215 
 
 TABLE VIII. 
 
 (78) The Proof-Strength of Long Columns. (Arts. 
 265 to 270.) 
 
 / = the length of the column in inches. 
 d = the diameter of the column in inches. 
 
 When the length of columns is so increased as to cause rupture by 
 first bending and then breaking across (buckling). 
 The following formulae will apply approximately : 
 
 Method of adjustment. 
 
 Force necessary to 
 rupture by buck- 
 ling. 
 
 Remarks. 
 
 Column fast at the lower end, ") 
 load applied at the upper 1 
 end, which is free to move [ 
 sideways .. J 
 
 p-()V* 
 
 See Art 265 
 
 W and E are the 
 same as in the 
 case of flexure. 
 
 
 
 
 Column not fixed at either ~\ 
 end, but neither end free [ 
 to move sideways J 
 
 4P. 
 
 See Art 266 
 
 
 
 
 
 Column fixed at both ends, ~| 
 and not free to move side- > 
 
 16P. 
 See Art. 266. 
 
 According to Hodg- 
 kinson's experi- 
 ments, we have 
 
 
 
 only 12P. 
 
 (Art. 266.) For a solid cylindrical pillar not fixed at either end, 
 but neither end free to move sideways, whose diameter is d and whose 
 length is I, we must take the formulae for buckling, in preference to 
 that for compression. 
 
 In the case of cast iron when j = 9 or a larger number. 
 In the case of wrought iron when - = 22 or a larger number. 
 

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 r 
 
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 '4-V 
 
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