‘éVALUATION '3? 'E’ORSIONAL VtakAHGNS EN CRANKSRAFTS fi‘hesis $96 $1“ mm» a! M. 5. WCHiGAN STAYE COLLEGE K. Bahadu: 5':th 51949. This is to certify that the thesis entitled EVALUATION OF TORSION“.- VIBRATIOHS IN W8 presented by K. Bahadur Singh has been accepted towards fulfillment of the requirements for ML degree in Mr. Major professor EVALUATION OF TORSIUKAL VIBRATIONS v. In C; LXI-TIL“) HAFT S 5‘! I. BAHADUR SIKiH w Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of Departnent of Mechanical Enfiineering 1949 ‘1" H Kim! ACKNOWLEDGMENT The writer wishes to eXpress his sincere appreciation to Mr. George W. Hobbs, Professor of mechanical Engineerin;,for his interest, supervision and helpful subdesticne in tris study. \\ a“ 21(991 PREFACE The object of this thesis is to study some methods for the evaulation of torsional vibration effects in crankshafts. The study includes calculation of natural frequency, amplitude and nominal stresses at resonant fre— quency. Four widely used methods for these calculations have been included. The choice for the use of one, or another method depends upon the designer and the accuracy required. The Holzer's method is the simplest and most commonly used. For application of any one method it is essential to reduce the actual system into an equivalent one, which simplifies the calculations. In some places empirical formulas have been used, which have been established by experience of previous identical assemblies. After theoretical calculations are finished the crankshafts are checked into actual installa- tions to verify for the accuracy of calculations. For preparing this thesis help has been taken from the various articles of the journals described in the bibliography at the end. Chm-kUMH INDEX Introduction . . . . . . . . . The Equivalent System . . . . Porter's Method, . . . . . . . Holzer Method I. . . . . . . . Electric-mechanical Analogy . Simplified Method USed by Chrysler Corporation. . . . . . . . . Calculation Aids . . . . . . 55 6O d) 0 X c+ '6 II II II II II LIST OF SYMBOLS Stiffness factor (lb.inch/rad.) gravitational acceleration a 32.2 ft./sec.2 : 386. inch/sec.2 Shearing modulus of elasticity (lb./sq.inch) Inertia factor or weight polar moment of inertia (1b.inch2) Sectional polar M.I. of cross section of shaft (in.4) weight polar M.I. per unit length of a step with mass (lb.inch2/inch) Number of a typical step length of a step (inch) Twisting moment (1b.-inch) Frequency of vibration in vibrations per second Frequency constant . 2Nn time variable (sec.) length variable (inches) amplitude of vibration in radians amplitude of vibration in degrees INTRODUCTION With the increase of size and velocity in modern machines the analysis of vibration problems has become more important in mechanical engineering problems. The automotive industry especially is much concerned about the crankshaft vibration problem. The general term “vibration” is given to the effect produced by periodic forces. As such, the subject of vibration is essentially the study of oscillatory motion of the machine and the forces that create the motion. In an engine, the moving parts mounted on the crankshaft, e.g., piston, connecting rod, flywheel, etc., have a natural period of torsional oscillations Just as a pendulum has a natural period of swing. The irregular turn. ing effort diagram of an engine can be resolved into a number of sine and cosine curves known as harmonics, having various magnitudes and frequencies. If the period of these harmonics coincides with the natural period of oscillation of the shaft, serious vibrations result. This state of coincidence is called fresonancef and is avoided whenever possible. For at resonance the vibration amplitude of the system is maximum and the resulting stresses in the system similarly have maximum values. If this condition is allowed to con- tinue for some time, there are chances of crankshaft failure by breaking. Several methods are utilized in dealing with the torsiOnal vibration problem, e.g., the analytical treatment used for determining torsional vibrations with and without aid of actual test measurements, instrumentation, stress determination procedures, etc. When dealing with torsional vibration of a crankshaft, it is always better to consider the whole system consisting of the driven machinery. The initial step in the analytical procedure used for this pur- pose is the determination of moment of inertias of the mov- ing parts and the stiffness values for the connections between these parts. Thereby the system is reduced to "Equi- valent Elastic System"; the procedure will be described fully later. The steps following are: 1. Determination of natural frequency 2. Determination of amplitudes and nominal stresses at resonant frequencies 3. Determination of forced amplitudes and nominal stresses at any frequency There are two important methods for the calcula- tion of natural frequencies, torques and stresses at various sections of the elastic system. In the first method, the torques and stresses are in relation to unit angular deflec- tion at some given section along the shafting of the actual system; usually the last disc is chosen to have unit angular deflection. The results thus obtained are used for comput- ing probable amplitudes arising from harmonic torque im- pulses occurring under resonant vibration conditions. The factors used are found from actual test results of previous comparable installations. The second important method involves calculations which include the determination of total damping energy. Appreciable errors are introduced due to difficulties in- volved in predicting damping characteristics. Both of these methods include various assumptions involved in the choice of representative indicator card and harmonic torque curves. In general, the methods are good for lower harmonics. Forced amplitudes at non-resonant frequencies can also be calculated, but the procedure is very complicat- ed and the amount of labor is so great that the calculations are made only in Special cases. In all the discussions given here, only nominal stresses are considered. Nominal stress due to torsional vibration is defined as torque at some section divided by section modulus. But this does not include effects of stress concentration, residual stress, or other working stresses. It is evident that the life of the shaft is dependent on the conditions of adverse stress, which is resultant influ- ence of numerous stress causing conditions, e.g., localized stress due to torsional vibration, bending due to gas pres- sure and inertia, misalignment, axial vibrations, and resi- dual stresses resulting from the manufacturing process. “It is important to have some knowledge of the material and the mode of fabrication, which includes influence caused by forging, flow lines, heat treatment, work hardening, size effects, the degree of uniformity and grain size, the sur- face finish, included impurities, and residual stresses. The localized stress due to torsional vibration is commonly taken to be the maxE shear stress occurring at 4 a small area. This is obtained from the resultant nominal stress and an appropriate concentration factor evaluated by calculations or eXperimental procedure. The shaft is statically strained under conditions stimulating those to which the resultant nominal stress is related; the stress is determined by extensometer reading or by brittle lacquers. The study of instruments used for measuring amp- litudes of torsional vibration has not been included in the discussion. The results obtained from the measurement of steady state oscillatory motion of the elastic system are used for calculation of nominal stresses. The nominal stress is determined by multiplying each harmonic component of the amplitude at the given section of the elastic system by nominal stress per unit angular deflection at that section. Even the resultant nominal stress due to torsion, basically obtained from quotient of resultant vibratory torque at some given section and section modulus, can be determined by direct measurement of bonded wire strain gage. The stress concentration and measurements are not included in the discussions, although they have great bearing on the subject and could not be ignored by a designer. First, some of the techniques are listed for estimating nominal stresses. Harmonic analysis of gas torque curves are not discussed separately. Then some of the means for reducing the actual system to equivalent elastic system are given. Concentrated mass, equivalent shaft and 5 combination equivalent systems are given. For equivalent length of crank throws there are only some empirical formu- las. At the end, a simple equivalent three mass system is given with formulas for finding natural frequency. The next discussion includes four methods describ- ed for calculating natural frequency, peak amplitudes and nominal stresses: I 1. Porter method 2. Holzer method 3. Mechanical-electrical analogy 4. Simpler method used at Chrysler Corporation First the installation is reduced to equivalent system. Porter has given some equivalent systems and tables for calculation of those systems. The derivation of the formulas used is given and then the formulas for obtain- ing various quantities are listed. In the end, some of the tables are given with brief discussion of procedure. Holzer's method is described as the torque summa- tion method. Helzerlstable, which is used for natural fre- quency calculation, is given. This is a trial method in 'which some value is supposed for net. frequency and calcula- tions carried on. ‘With this method is also included a com- plicated discussion of forced torsional vibration with «damping. This treatment was introduced by J. P. Den Horthog ‘as an extension of Holzer method. mechanical-electrical analogy is given for the :reason.that more is known.about electrical circuits than 6 about complicated mechanical systems. Some idea has been given for electro-static and electrc-magnetic analogies. The formulas have been derived from a simple circuit and then generalized. A simplified method devised by L. M. Ball is used at Chrysler Corporation for torsional vibration calcu- lations. The method employs a number of useful graphs pre- pared for this purpose. In conclusion, some of the calculation aids which are really useful in this tedious calculation work are given in summary form. t Now some of the techniques for calculating nominal stress are reviewed: In torsional calculation problems, it is customary to resort to approximations and simplifying assumptions in the practical cases. Experience is a comidsrable factor in obtaining trustworthy results. However, it is the calcula- tion work which makes it possible to understand and evaluate the vibration test records, since with present equipment it is impractical to measure all the amplitudes and torques in the entire system under all Operating conditions. Calcula- tion, laboratory testing, and engine measurements should therefore be considered as mutually dependent techniques guiding development. They serve as means of reducing cut and.try engine endurance testing. The degree of refinement tattempted and techniques adopted depend upon the nature of :problem. For instance, full scale fatique tests are desir— vable on heavily stressed parts, but are practical only if parts are mass produced and available in quantity; on large units in which quantity produced is small, safety factors must be increased and more theoretical and model work is desirable. mechanical vibration involves the response of.a 'mechanical network‘ of inertias, coupled by shaft flexi-v bilities, to a series of complex periodic exciting torques applied at several points with different phase angles. The general mathematical solutions are well established, but the amount of arithmetical labor involved for exact results is prohibitive. .A cut and try method is usually used to establish salient points, using simplifying assumptions which have been proved satisfactory. .A general method is outlined below: 1. Fundamental Simplifications: (a) Equivalent system - Distributed factors lumped into equivalent concentrated factors. (1) Inertias - By normal calculations, mechnical integrator, or test (ii) Elasticities: Crank throws by empirical formulas of Timoshenko, Carter, etc., adjusted by experience. Shafting and couplings, etc. by calculation and test, with allowances for discontinuities and connections. (b) Exciting torques - actual single cylinder torque - variation curves expressed as Fourier series 2. (1) (11) Harmonic analysis of gas torque diagrams, constructed from indicator card. These are checked whenever possible by figuring back from actual vibration test results on similar engines. mathematical expression for inertia torque variation, expressed as Fourier series. (0) Damping factors - Very complex and variable Depend upon shaft material and stress, bear- ing, distortion, piston and ring friction, coupling absorption, electrical eddy currents, prepellers, etc. Best guide is experience on previous similar comparable engines. Critical Speeds: — Relative amplitudes and torque distribution at resonance also determined by this calculation step. (a) Natural frequency - From equivalent system; there may be several modes. (1) (11) (iii) (iv) Porter's method Holzer tables (1) Calculating machines (2) G.M. special slide rule mechanical models Electrical models (b) Speed of resonance with varicus harmonics -— best presented by plotting natural frequencies and harmonics on a Speed frequency diagram 3. Relative importance of critical speeds: - (a) Relative size of the various harmonics of the torque diagram (b) Effectiveness of the cylinders along the elastic line (c) Star diagrams for various harmonics showing the effect of firing order and vee-angle (d) Combined effect of these 4. Non-Resonant and 'Roll' Amplitudes: - Same methods as for natural frequencies, but with applied har- monic torques included in the system. 10 WW: It is always simpler for calculation to reduce the actual system to a equivalent one. The methods of re- duction are: ' l. Concentrated mass, consisting of point mass moments of inertia connected by massless springs. 2. Uniform shaft system, obtained by averaging the various masses distributed along the shaft- ing into a number of connected uniform shafts having both mass and elasticity. 3. Combination equivalent system, using combina- tion of concentrated masses, elastic shafts without mass and elastic shaft with mass. The inertias of engine parts are averaged to uniform shaft with flywheel and other isolated heavy masses attached to shaft with relative- ly short length, as concentrated masses attached by massless springs. If there is a short length of shaft B between the uniform shaft A and concentrated mass C, which has a smaller inertia factor (Jw) per unit length than A, in this case, the length of B is changed to have the same stiffness as A. c - B Then sufficient inertia is deducted from C to make the value of Jw over B equal to its value for A. 11 Equivalent mass moment of inertia - J g m?“2 = r’2 --- (1) uncut Jo= Jc + :11): --- (2) Connectigquod. The lower end of the connecting rod rotates with the crank-pin while the upper part recipro- cates with the wrist-pin or crosshead Rotating part of connecting rod = (l-K) w Reciprocating part of connecting rod = Kw where W a weight of connecting rod K a distance of c.g. from crank-pin centre total length of connecting rod Reciprocating Earts. W: a weight of reciprocating parts. EQuivalent rotating weight = W (l 9 R2) --- (3) 2 8L2 Crankshafts. No definite formula can be given for the crank-shaft as it has a variety of shapes. The journals are cylinders rotating about their c.g. and crank pins are cylinders rotating about an axis at a distance equal to crank-throw. This can be transferred by Be. «.13 (2). then all the individual inertias are summed up. Another simple method for finding M.I. of crank- shaft is with comparison to a round disc. The disc of known weight and dimensions is hung from a piano wire and the time for one oscillation is noted. Then the crankshaft is also put in the same position and period of oscillation again noted. M.I. of crankshaft is prOportional to the square of ratio of time. 12 Equivalent Elasticities. Equivalent shaft lengths — - L L 32 C. a. . L119: 8 Le 32 4 4 ° G .7‘De = g, .781) But C :1 Ge 0 o i; 3 L 32 Le = L‘+%§J4 Where Ce, Le and D6 are torsional rigidity, length and die. of equivalent shaft Crankshafts - Several empirical formulas are given to calculate the equivalent length of crank-throw. It is usual to re- duce the crank-throw to equivalent length of parallel shaft- ing. Any dia. may be selected, but it is convenient to choose die. of crankshaft journal. Reference Figure l. ggpter's Formula: ‘ Le . (Ls~+'0.8w) + f L s + l R De4 _ - t it... sis—WT For hollow crankpins and journals: 4 L 4 . 1 RD 4 Le . “(1)34 -asg) * 7? pggs win? 8 Geiger's Formula: L6 = (Ls 4 0.4?!) + 0.773 (R—z.D3.) is + u \ (D (Lp 1 0.4W) Wkgg ' J 13 Where J a Polar M.I. of crankpin g 1:.Dp4 32 JS 3 Polar M.I. of crankshaft a 11,.Ds4 _ 32 Z =0for1L.l.6-l.3 3:12-032 Ds . Ds Z : 0.4 for X’- : 1.49 i = 0.84 Ds Ds Very often the values of Z are between 0 and 0.4 and the mean equivalent length for the two values should be used. Heldt's Formula - _ Le = 2L3 + 0.4w . 1.096 Lp Essie 4; . DpE-dpn 1.284s D54 - as4 w Timoshenko: L6 = (L8 4 0.9W) 4 (LP 4 0.9W) (DEA-d84) § (Dpfi'dpx) 4 o .9312 D34 -d m3 Wilson's - Le - (Ls + 0.4Ds) + (Lp + O.4Ds)( Ds4-dg4)-+- (Dp ‘dp J [R—O.2(Ds 4 1310)] De4 - dg4 WT3 gemanenko - Le = (Ls . 0.6 .Ds) + (0.8Lp + 0.2 2 DS) R *4 +4 4 4 De -ds + R De -ds DPI'dPI 3 “B- wnere ¢ = angle of twist : / 23 ‘w EL. Ls l4 Carter's formula is the simplest and gives 3% error in frequency calculation. .All the formulas seem to be similar in form but they differ slightly. The Zemanenko's formula is the latest and gives more accurate results. This is chosen from the result of 55 tentative formulae and also accounts for overlapping. In the flanges the shafts are considered to twist to a length of quarter the die. of bore and rest 2 integral with the boss, which is taken to twist to half the depth of the flange. For tapered shafts: “=3. (fliflfiiifi) d2 d22 d2} where l = length of shaft and d1 and d2 are die at ends. Sudden changes in shaft dia. - If d1 and d2 are dis of two sections than d2 is effectively replaced by d1 over length in calculating stiff- ness factor. ' V for (11 1.5 . “A: 0.1 d2 ....> ‘12 3; between 1.5 and l 3.15 between O.ld and 0. d2 l5 Three_Mass System: Three mass system is the simplest and gives ap- proximate values. This method is not complete in itself, but helps in giving approximate values of natural frequen- cy for trial in Holzer method. % t a J;oc-al El 145,... -. -0 x and y are distances of nodes N1 and N2 from J1 and J3 respectively. The period of oscillation of J1 about its node is given by J p. u, The frequency fl of mass Jl is f : l = l. .9 "' (2) 1 t 2RJJ1 But C : 635 L fl : _6_g I GJ: v.p.m. if J1 is expressed in lb.-inch2 r1 = g9 G,Js 285 --- (3) 21K J}..x Similarly f2 = fig G.J. 3865 -—- (4) 2R J2 l l . . WI-XXM-Y) and f3 2 £9 G°JS .386 2-“ J3 o y --- (5) EXpressing y in terms of x we have y : J1 x. J 16 Substituting the value of y in (4) and equating to (3) or (5) the value of x is calculated, the higher value being the single node frequency representing oscil- lations of J2 and J3 against J1, while the lower value gives oscillation of J2 against J1 and J3, causing a two node vibration. The complicated system can be reduced to three mass system. For instance, for 3 to 6 cylinder, close approximation is obtained by multiplying WE? of cylinder masses by 0.85 and taking them as acting at the centre of the engine, while for 7 and 8 cylinders or more the factor is 0.80. The second mass is flywheel and the third usually the driven machinery, e.g., pump, impeller or generator, etc. This method helps in getting on the track as it gives a nearer approach for the solution. 1? Porter's method for DeterminingNatgggl' FrequencI. First the system is reduced to equivalent system: 1. Concentrated mass system 2. Uniform shaft system 3. Combination equivalent arrangement These methods are discussed elsewhere in the 'paper. The masses may be reduced from beginning to the end or backward from end or from both ends up to some section. A.natural frequency is obtained when the sum of masses reduced from both ends up to any section added to the mass at that section is zero. .At this frequency, the elastic forces in the shaft and the inertia forces of the masses are in a state of equilibrium so that system when set into motion will continue to vibrate indefinitely if Vthere is no damping. K. means number of typical steps, and character- istics of Step No. l are obtained by putting K n l, and so on. C a stiffness factor M - 0.9. or c = g (lb,-inch) e rad. where M - twisting movement cg : constant in degress defined by equation (17) ‘Y = constant in degress defined by equation (11) qb~- constant for step with mass defined by equation (1) 18 UV: constant defined by equation (2) T (4>n) . 1,89 . tan4>n 1<4bn TO (V11) =D'é1ined ($3 57?? (20) The values of T (4>n) and To (Vn) are tabulated by F. P. Porter. For a circular shaft of constant dia. C.G.Jng. K'.d4 . _1_ 2 32 L G = shearing modulus of elasticity : 11,800,000. lbs./sq. inch for steel Subscripts, eg., 12, J2, 02, etc., refer to length, inertia and stiffness of Step No. 2. Numbers differing by units, e.g., Ol, 12, 45, etc., refer to between those steps. (') Prime mark means effect of masses from be- ginning has been reduced to a certain section and (") means reduced back. Subscripts It, £3, 13 . . . 01, 02, 03 . . . etc., mean end and beginning of steps. z 360 . __._;r__ = 18.325 .1 (sec.deg.) -— (1) 4: I; go w/g Some equations used in the solution: 8: 386 -- (2) Let us consider an irregular shaft vibrating torsionally in S.H.M. If a concentrated mass occurs at the beginning of the system having inertia factor 351 and is connected to another section.A by elastic shaft having 19 no mass, the twisting moment due to inertia is: M 3 J01 . deco; --.. (2) 8 F432 Since its s.1~:.m. M . - 39; p2 901 s where p = 2th twisting moment in the shaft is M. ‘31- (6,1 - 901) -- (3) Two values of M must equal for equilibrium 2;; =1 -gpz e01 881 If a concentrated mass is to be placed at Section A.to give the same twisting moment at the section, we must have inertia factor such that M = 3‘11. d29 ‘ at? 8 I a 2 ' 3.3.1. P 911 = °1<911 f' 901) s 8 Thus fin = 1 1 = 901 --- (4) - O 9a a, "R: J01 052 O1 Let us consider a similar case with the exception that the shaft is uniform and has mass. For equilibrium it is necessary that the sum of all twisting moments due to inertia forces of the masses on one side of any section be equal to internal resisting moment of the shaft, as only external forces assumed to be in action are inertia forces. 20 The twisting moment due to inertia at the beginning is 2 - ) M01 = .49.; a so; = - g9; £601 -- <5 8 dt 3 Resisting moment of the shaft at any section Ml : 0111 ‘31 --- (6) dxl Hence at any point between the concentrated mass and the beginning of the section A In .«J ’ TJ14 102"’1 ‘1’“ ‘ .9; P2901 = “in c 81 8 dx Differentiating w.r.t.x 2 _ J1 p2 91 = Cl ‘1 . d 9! 8-11 or d2el _ _ Jl.p2 e1 dx2 - gel 112 .2 The general solution of the differential Eq. is 6 - a cos ( J1 p x +-b) l - 8.01 21 The values of a and b, the integration constants is evaluated by the end conditions. For computations it is easier to deal with angles in degrees so equation for 91 in degrees. 601 gee'l cos (+ln.3i<_i + Y'l) --- (7) where 4=ln 3 360 J}- .n. 061; and Y'l are all 3 801 angles in degrees. 21 Differentiating 91 in (7) and putting in (6) gives M.l twisting moment at any section. Putting x = o in M1 gives M01 equating the two values. tan Y'l‘ : 1%?)- 4;, n. g? --- (8) If concentrated mass is to be replaced at section A to give the same twisting moment as by M1, when x I ‘1 , the inertia factor must be such that I Mm = ill. P2 91.1 8 Formulae necessary for the solution of natural frequency as given by F. P. Porter: 1. Reducing the inertia factor from beginning toward and (a) For steps without mass} J'lk = J'ok To (W"0k.n. ) --- (9) (i) If no concentrated mass at the beginning J'll - J1 T. 0’1 n ) --- (10) (ii) Concentrated mass at the beginning or for any other step I tan Y’k I ¢k~n°Jok ’ u—.—._.. .. .. J'lk - 57.30 Jk 4szn .tan «Pk.n+4"k) --- (11) Considering from end toward beginning the steps are the same except 1>is replaced by O and ( .) by (") 2. 22 At natural frequency: ' J“ J - O or J 1,k-1 * ok +’ k-l,k - ' t n ' J 1k 1 J O k11+ Jk’k1l . o —-- (12) which is same as J'ok 1 J'ok = 0, or J'Ik . J"1k = 0, or J'ok = J" k J'lk 37%; --- (13) At natural frequency of the system y" -‘Y' - 0. The relative amplitude curve of vibration follows immediately from the above solution. The amplitude is re- presented by 9 (rad.) or 9° (degrees); however, the amp- litude at the end of one step is equal to that at the be- ginning of the next step. 911 = 902, 912 g 603 o o 0 91k = eo,k‘l 000 o o o 918 = 90,2... l --- (14) Relative amplitude is represented by 9 with sub- script of the step as a function of x. Relative amplitudes at natural frequency: (a) (b) Step without mass a n ' elk : J 0k 3 J 0k 3 1 n 3 90k J'zk J"zk T° (9)°k° ) 730,41! n) --- (15) For steps with mass e°k = «k cos (ékm . x +Yk) --- (16) 2k 23 o o where 6 ok zock cos Yk, and 6 1k : dk COS (¢k.n 1Yk) ""' (17) The relative mflnetion twisting moment for various sections of system, therefore, are: - (a) For steps without mass M = 0k (eo,k4l -‘90’k) 2 ok §?T3 (Goo,k+l-eoo,k) --- (18) (b) For steps with mass ‘ M = - (1.38.6)2 C n¢atsin (Q1153; +Y) -- (19) The functions 2 To (W .n) :L1- 5-1- (W-n)2 J ‘1 - 8 ‘ -l = [1- 0.102297 (vn)?] --- (20) 180 T (4> n) g *7; . tan4>n --- (21) when (¢>n) is angle in degrees. Peak Amplitudes:- Energy fitnput due toga number of equal impulses of different phases given by m.cos (pt -Vg) acting at various points from beginning of step with mass is 2 E a .m.e° 5? f%5 ol ZELB., If impulses are unequal in magnitude 2 o .- E: K o e 'X‘ZHIfl T35 01 where 52 is theaé- constant for relative ampli- tude of curve for 6°01 = 1. 24 If impulses having same magnitude but different phases occur in two different steps then, 2 o _ E:]:RO m echdfi Vibration stress in any portion can be determined at fibre distance x. .M =.§ I x Empirical formulas for damping along steel shafts due to internal absorption of energy and other unknown effects: (a) For step without mass = 70:324°3-d 54 3) 1 Exec 3 (eel)2 3 10 (d24 _ dl4)2'3 where E is vibration.twisting moment in the shaft for relative amplitude curve for 601 = l (b) For steps with mass = (601)2°3 700(d24.3fi 4 3) 1% ~ 16101 x2 “I sing°3(£%§ +Y) dx X1 d2 -d1 ) where, 110 = - (R2) C¢n3< for relative amplitude curve 180 With 601 = 10 Some of the tabular forms used in the finding of the solution follow. For Reducing From Beginning Towards End 3 First Step'Without mass (1) <2) <3) <4) <5) ' . _ H_T _ 2 J'li J' 2 ‘1 L”°1 n ° ( ) (3).Jol (4)312 k3? Step Without mess (a) ' (b)' (c) (d) ' 11 TO (a) J. J. l ‘P ok‘ 1k o,k+l n' J_C__._Ok (b)oJ'Ok (C) a, Jk9k+l k ' w: - First Step With Mass - J01 = 0 (l) '(2) ”(3) ' (4) (5) n 451-“ T'(2) J'21 J'02 EE step With Mass For lst Step ifJolqp'O and For Any Other . Step (3) (b) (c) * - (d) > (e) (f) (a) ‘ g I fi.n4 y'k Jrlk J! 'P . (3)“7 0k Yk : t a) o,k+l k n 57.3 Jk tan ‘l(b53)‘ (c) an( 57~3Jk (f)¢Jk k+l (e?) ’ (ET 1,3 26 A similar process is used for reducing from end towards beginning. If value of Jk,kol of a concentrated mass at N.F. is required, it may be obtained from Eq.3 (12) by first solving f°r_J'Ik and J'o,k+l by means of tabular form, then . . ' . Jk,k+l' ‘ (J 1k * J'o,k.1) For the value of stiffness factor Ck without mass for a given frequency, the following table is used after solving for J'ok and J"1k. (a) , .._ (b) (a) ‘ (b) (o) ‘ (d) (a)' (b) (c) (d) _J' k .5- (b) 2 Ck ank -1 (b 2 C 31%; Tea“) {5] J” F312; T0(a) [3)] '15 Four Special equivalent systems for solution directly from tables and charts given by F. P. Porter: Reference Figure 2. N2(l) n = _:__ /3-°1 (J01 * J12) “ .. Jol°Jl2 - 1 To"1 J - 1 To“1 {Lg “*w*“ (‘ —Q;).’ “7—“ (-J ) V11 J12 Vol 01 Reference Figure 2. Iv3(2) -1 l J 1 (1st or 2nd frequency) 27 Reference Figure 2. NQ(3) qbln correSponding to $79.3... and J12 may be read J J 1 1 $1.11 ¢1 Reference Figure 2. NINA») from the chart, then n : (1st. frequency). ¢1.n correSponding to £23.; and “2% may be read J1 2 from chart, than n £133 c”1 . For more complicated systems, it happens that some parts of the system correSpond to either of the simpler forms, for example, a marine installation. First, N.F. is determined mostly by flywheel, propeller shafting and prOpeller correSponding to #1, and second, N.F. by flywheel and engine corresponding to #2. Taking advantage of the above characteristic, we may reduce the inertia factors from either end for assumed frequency (n1) until the remain- ing system corresponds to (l or 2) having greater influence on frequency. Now a first approximation (n2) for N.F. is obtained. Using this value and repeating the process we can obtain (n3) as 2nd approximation. If the simpler system does have influence on frequency n1, he and n3 will form a converging series with numbers lying alternately on either side of actual frequency. M. n1 = Assumed value of n \\\ “3 n2 . First approximation /// ‘“ n; = Second approximation V\¢<; Alf-'nl'n2 andA2=n3-n2 28 Let hl and h2 be the ratios such that “2"1‘1 A1 I1 assuming h . hl . h2 (its appro. true when nl,.n2, n3 do not diverge) n2 1 h41 : n3 - h132 'h.A2 41+fl2 - - n zne * A142 Al‘Mz Then n can be found by above formula. It is not necessary in using the first appr. (n2) in repeating the calculation 2nd time. a. m. n1 : First assumed value \ \“3 \ “4 n2 a First approximation \ h—— or /21 n3 g Second assumed value \ M \ 43 n4 . Second approximation M1 “t A]. 3 n1 " n2 and, A23 n4 " n3 named-A1111 =n4 - 112112” assuming hl : hg : ho n2 a. hA13n4- hA2 0.0 h: 114-212 151 fiAe so that 29 Another procedure to solve for n is to reduce inertia factors from each end to a step that has mass, and with several assumed values of n, to find the value that makes v“ - Y' o. In this case linear interpolation be- tween n and y" - y” can be used. In dealing with systems shown in No. (5), Figure 22—4- The solution may be made by reducing from beginning or from end as shown in Tables.A and B reapectively. If the way of procedure is not directly apparent, J12 may be removed. If the value ofv¢>l.n is less than 90°, Table.A is to be used for frequency and if 4>1.n)>9O°, Table B is to be used. If 451.n.= 90°, then let freq. is given by 29_ since J12 is at the node and has no effect on this fre- 4b quency. When the first frequency is obtained, the 2nd fre- quency is found by Table B. _The relative amplitude curves for the first and second frequency may be obtained by Eq.E§ (15). (15). and (17). (l) ,. (2) (3) (4) (5) (6) (7) (8) n 4> .n T.(2) J' J' J' -u" $2 1 El 02 -__Q§ ‘ n. (assumed) (3).J1 (4)§J12 J23 To(6) 12 Table B __ F (1) (2) (3)‘ -(4) (5) (6) (7) (8) I, n V” .n To(2) J"- J" J" -1 (ll 12 02 . 11 ‘__!'.]_- 11: (Mule 4 l 30 Tables for solution of form shown in No. (6) Figure 2. Table C Reducing from beginning up to Step No. 2 and solving appr. value of n for Form No.(l) (1) (2) (3) (4) (5) I z n ¢1.n (2) J01 Yl‘ ¢l.n +Y1‘ (assumed) 57-3-Ji tan-1‘3) (2) + (4) (6) (7) (s)‘ (9) (10) iii) J'11 3' 2 afJe 1 0 tan (,) . J (g) ° - ° To” (9) n: itrl D 57 3 1(2) (7)*J12 J23 9’12 Table D Reducing from end back to Step No. l and solving I by finding‘Yia -Y1’ = o. r . ,_ (l) (2) (3) _i4) (5) (6) n V4231 To(2) .1502 , Jnll 4>l.n I (3HJ23 (4) + 312 _;g:) (8) (9) (10) (11) (12) - 6 4) o Y”): II?”-! I. H“- I, 57.3:J 1 g: 1 Y1 (5)'J01 Y1 Y1 Yi 1 tan (7) (8){‘(6) .57.3.J1 tan'l(lO) (9)(11) 31 ggiger Method. In the calculation of natural frequency, Holzer method is the most widely used, this method being common due to its simplicity in calculations and understanding. The first step of the procedure as usual is to find out M.I. of rotating parts and reduce the system to a simpler form containing discs. It is always easier to reduce the installation to concentrated mass equi- valent system. Then trials are started with a certain guessed value of natural frequency. The guess may be made on the basis of previous experience of some comparable in- stallation, or by computation of a simple three mass system to which all the assemblies can be reduced easily. This method is also recommended by S.A.E. War Engineering Board for calculation of torsional vibration characteristics. Some of the formulae used are given below: Principle equations for disc n in a system: Th : Jn 6n p2 -—- (1) Total torque Tn acting on a shaft n n Tn : P22 J 9 --- (2) l . ' n Angle of twist Aen, n-l : €111 _-_ (3) Cn Amplitude 9n of disc n n 6:} . 9n_1 an E J p26 Cn-l 32 o - Holzer tabulation method used the standard table shown below: 1 2 A 5 6 Y 2 2 2 ‘ZJpZG Mass No. J J.p e Jp 9 ZIP e c c 1 2 3 e _- .. p : frequency constant ; 2Tn where n :. frequency In using the Holzer's table for natural frequency calculation the initial value of e for Step No. l is as- sumed unity. In the first line, column 7 gives the value of 61 -k62, which, whensubtracted from column 3 value, gives line 2 value in column 3. Second line value in column 5 is the sum of lst line column 5 and line 2, column 4. The -0. requirement for a natural frequency is i J .p2.9. various values of p2 are tried until this condi- tion is realized. When this occurs, the columns 3 and 5 give the relative amplitudes and moments of vibration at N.F. Calculations for the Peak.Amplitudes: If the impulses are ms cos (pt-Vg) acting at masses indicated by s and the amplitude of first mass is elsin(pt-v0, the peak amplitudes may be calculated by equating the energy 33 input and energy absorbed by damping. The energy EqE. is E 51(91st 53 where 53 is relative amplitude as given in column 3 of Holzer table Ems'és . [(Emsés sin ((8)2 + (Ems-es cos 9’8)2 tantr: Ems—6.8 sin [3% Zmaéscos Vs If the impulses are all the same such that m3 3 m, the energy qu' is E :‘Km 61 ESE .. fi- ‘ 2 -- 2 where 293 . (is,3 sin vs) + (29., cos Ira) tan W a 263 sin 9:9, 213 cos Vs The values of m are obtained from torque AR curve given by F. P. Porter in A.S.M;E. Trans., 1943, A933. Empirical formula for damping along steel shafts due to internal absorption of energy . k = 70(d24'-3-ai4'3> Z 5712'} 912'3= 7___TO “2'3 e 2.3 1010(d24_d14) 1010d2 '9 1 if d1 : 0. For distributed mass: x k = 70 ((124.3 _ (114.3) 912.3 I 2 $12.3 d2“ 1010(a24 - d14) 2.3 X1 where h. = vibration twisting moment corresponding to re- lative amplitude in Holzer table. If E0 2 input energy at 91 = l and KO . damping energy, than E091 g K0912-3 34 Undamped forced vibration at any frequency can be computed by using two tables, one for the sine components and one for cosine components of the impulses. Each table has to be started with unknown amplitude x for e in column 3, line 1. Then the components of impulses are added to column 5 wherever they occur. Finally, the sine and cosine components of resulting vibration are determined by equating column 5, line e to zero and solving for x. The peak ampli- tudes are calculated, using the empirical formula given above for damping along steel shafts. ‘ Med Torsignal Vibrations with napping. (Extension of Holzer Method) - In the calculation of free and forced vibration of torsional system with small damping, Holzer's method is widely used. It consists of determining the shapes and fre- quencies of the free vibrations, disregarding damping. The damping is subsequently taken into consideration by equating its energy dissipation during one cycle to the work done by exciting force during that cycle. Obviously, this method is satisfactory only when the natural frequency and partic- ularly the vibrating shape of the system are practically in- dependent of damping, but in most practical casesit is true. For an engine system with a large number (six or more) of identical cylinders, it is advantageous to consider the engine as a uniformly distributed elasticity and inertia. Holzergiablefgr Eggped System - The discussion is based on the usual engine idealization where the moving parts of each crank are 35 replaced by an equivalent flywheel and an equivalent stiff- ness. The damping is assumed to be linear and can consist of two parts. (a) Dash pot between each flywheel and ground (b) Dash pot between each flywheel and one or two neighboring flywheels See Figure 3. Co and C1 are external and internal damping con- stants. C : Torsional damping constant K . Torsional Spring constant P and Q : Angular diaplacement functions of x 9 3 Angle of torsional displacement n a Number of discs n,(n—l) = Running number of discs 1 is used for per unit length i and 0 represent inner and outer J": imaginery quantity .-. J- l Newton's equation for motion of the n22 disc is: Jnen . Kn (en-enn). 131-12.14.14) + on. 6n amen-gm) + Cn—l: i (én‘9n-1) .-.- o m (1) Holzer method consists in finding the end torque necessary to cause a steady state forced vibration of an assumed frequency a)at unit amplitude at the other end. With damping, the steady state motion at various discs is still harmonic at frequency69 but they no longer have same phase from disc to disc. an is An sintJt 4 311 cos 0t --- (2) 36 This can be put as 0n .-.- 15.1 t {En én amounts tomultiplcation by {.0 and 9n to multipli- cation by -‘32. Substituting the values of an and On in Eq.3°(l) we have ("1.11.32 't Jano)9 n ‘(Kn 4 JOCmH9n 'on-tl) * (Kn-l " Hat-1.1) (On—9nd) = o --- (3) This equation differs from that of undamped only in that 11102. is replaced by (Jn‘ba - Jocno) and Kn by (KH 4 (foam). The computations are similar to undamped case -except that the numerical figures in the table are complex. In undamped case the end torque or 'Remainder Torque" is a real number; 1.6., it is in phase with motion at other end. But here it is complex quantity, having components in phase and quadrature to the motion. ~Inundamped case for certain frequencycb, the end torque becomes zero; i.e., the system can have steady state vibration without external excitat- ion. 'With damping this is no longer true. The and torque never becomes zero, but for certain values of “)it becomes ming. and we may define these as damped natural frequencies. System with ‘Unifornilpgstripnted Inertia, Elastgcity andeamping: _ . The differential qu. for torsional vibration of a shaft with uniformly distributed inertia, elasticity or internal or external damping is 37 J16+ 0100 a K10“ 4 0119" --- (4) where dots mean. ‘3 and dashes mean 3 as usual. tt 1x The angle will not have the same time phase along the length of shaft so we may assume the solution of the form 9 = P(X) sin «3t 40(X) cos 0’0 --- (5) P and Q are real functions of x. This repree sents a forced vibration of frequency d3the forcing torque to be applied at the end. Substituting (5) in (4) and putting the terms prOportional to sinth and cos¢3t equal to zero: J1021> + KlP'f + clooa- an OR" 0 ) ) ---(6) O Jlo2a + Kla" - 61002 + char" These are linear differential equations in P(x) and Q(x). P 2 Poepx , Qaaoepx and subst.in (6) (lez + K1192) Po 1- (010‘)- 011P2)90 .0 ) q (7) (-cloo. shape», .(Jloé‘ + gent, =o ) In this set of equation in P0 anon the determin- ant must be zero or 2 2 2 (ch32 + Klpe) = - (coco .. 0110p ) which is equation of 4th degree having 4 roots . . , 2 P1,2,3,4 = .4: l “J10 1’ J‘Jcio --- (8) K1 1 3&3011 38 or we can write Pl,2,3,4 ; + a + jb. The values of a and b can be found semigraphically in the complex plane diagram A,fi_ b2 =.[(-liIsa 4 cliclofl?2+(klleQ+chl£v3)2 +- .% (li102 - cliclocJe) k12 + 0112o2 "' (9) 8.2 CA L: K1010 “)4" chlit‘O} K1? + 0119 “:2 '2b " From (7) {g = 01143132 - Clog) o ___ for value of p s a + 3b, p = -a - 3b P0 = + J 3'; , For p : a - 3b, p : - a o Jb; fig 3 - j The P andeeceme P = Cle1a+3b)x + Cee(-a-Jb)x + 038(a-jb)x+ C4e(-a+jb)x e('a"3b)x+3 03e(a-Jb)x+3 C4e(-a4jb)x Q - Jole(a*3b)x_30 2 --- (10) This can be written as P = Cos bx (A coshax 4 B sinh ax) +gJ sin bx (C cosh ax +-Dsinh ax) Q -_-, Sin bx (B cosh ax 4 A ainh ax) - J cos bx I (D cosh ax f-C sinh ax) P and Q must be real while A, B, C, D may be .imaginary or real, so making either A = B = O or C = D a O 6 _-_- (A cosh ax 4-B sinh ax) cos bx sith 4r (A sinh ax 4 B cosh ax) sin bx cosdt --- (ll) 39 A.& B are determined by and conditions. X 6 : 0. e' z o andéB: 1 (Cosh ax cos bx sith + sinh ax sin bx cosa3t) --- (12) The torque M (x) in general along the shaft is M (x) = K1 9' (x) - on e'(x) --- (13) In the case of 2322% end at x g 0 M (x) . SintJt [(Kl a- 09C11 b) sinh ax cos bx-(Kl b+0011 a) cosh ax sin bx]+ coth [(Kl b +2011 a)sinh ax cos bx - (K1 a -“JCli b) cosh ax sin bx] --- (14) For the case of damped end at x = 0. M (x) = sincJt l:(Kla -03011b) coshax cos bx - ' 8. «t b (Klb {‘AClia) cosdt sinh ax sin bx] 4 [(Klb +JDC11a) Ja 2 + b E cosh ax cos bx + (Kla -0011b) sinh ax sin bx]---(15) It is to be noted that all equations (l2), (l3), (l4) and (15) are in real quantities fit for computation. The following example illustrates the method and shows the comparison with Holzer method: if we consider six equal inertias _ . . J :103 lb.in.sec.2 coupled by five shafts each K =107 having dash pots CO and Ci as well. The engine can be considered as a uniformly dis- tributed beam and can be treated by Eqn. (l2), (l4) and (8) by taking the length of one crank throw as unit length we have Jl z 103 K = 107 and col = 104 011 = 2x104 1 fl : 6 Assuming a frequency of 1000, we find in graphical construction and computation in the complex plane from Eqn. (8) a1 = 1.916 bl - 0.355 By substituting the values in Eqn. (12) and using table for hyperbolic and trigometric functions, the shape of 9 (x) can be plotted. Sample Holzer Table 41 J02. 3 deg (cl-(32‘: _1tOCo)9 No. r66 6 ‘lUQ, 1 1.00-0.3163 1,00 - 1.00-0.3163 2 1,oo-o.3163 0,902+o,0583< 0.914-o.2#73 3 l;00-03316j 0.7i5+o.1053 O.769-O.1203 4 ‘1.oo-o.3163 'wo,454+o.1913 0.515+0.0473 5 1.00-0.3163 0,142+o.2743 0.228+O.229J 5 1.00-0.3153 -0.194+o.3363 -o.osg+o.388y ‘No. 2/106 E—‘fagfls-i- K+2j¢o°i 1 1.00-0.3163 10,00+o.632} 0.098-o.0383 2 1,914-0.5033 10,004..6323 0,187-0.0683 3 2.663-0.6833 io,oo+o.6323 0.251-o.os53 4 3,178-0,6363 io,00+o,6323 0.312-0.0833 5 3,406-0.4073 io.oo+o.6323 0.336—0.0623 6 3.317-0.0083 --- --- 42 mechanical-Electrical Analogies. = Electra-motive force (E.M¢F.) volts z Current(amps) 8 Capacity (farad \ . Inductance(henries) g Impedancetohms) th‘OHt-EI : Resistance(phms) The rest of the terms are defined in the table for comparison. An analogy is valuable to the extent that it per- mits a knowledge of one field to be applied to another field. Since much more is known of the characteristics of electrical circuits than of certain.kinds of mechanical systems, it is often valuable to discuss a mechanical system in terms of its electrical analogies. Two electrical-mechanical analogies are usually proposed: (1) The conventional electro-static or older mechanical-electrical analogy, (2) Firestone's mobility (electro-magnetic). Before going into details, it will be better to make it clear that the conventional analogy is based upon f E Z and v = z and the mobility system on the form I = E g 12 and v = fz. Ebbility method gives the exact ideas of the words 'through' and 'across', for instance, force through and E.M;F. across. Conventional method gives incom- pleteness in the mechanical analogue of Kirchhoff's Laws. 425 Secondly, where the elements are additive in electrical system, they show the Opposite for mechanical system, 6.5., behaviour of impedances and resistances in series and paral- lel circuits. These points are not given with the idea that a comparison of the two analogies is going to be given, but just to point out the advantage of one method over the other. And now some idea of the analogies will be given in brief. (1) Electro-Static or Conventional Analogy: The quantities of inductance and capacitance were illustrated as analogous to mechanical quantities of mass and compliance. Electromotive force (E.M;F.) took its name from the similarity to a mechanical force, and current to velocity. Now energy stored in the magnetic field of induc- tance L is 1/2 L I2, and kinetic energy (.K.E.) of a mass m is 1/2 m v2. The inductance tends to prevent change of cur- rent by generating back E.M.F. of magnitude L Q3. Just as dt mass prevents change of velocity by producing a reacting dv. ET Capacity in an electrical circuit plays a part force of magnitude m similar to compliance C of a spring in mechanical circuit. Energy stored in the electrostatic field of condenser is 1/2 CE2, corresponding to energy 1/2 Cf2 stored in a spring by force f, and a condenser will hold a charge C.E. prOpor- tional to E.M}F., while a spring undergoes a diaplacement Cf proportional to applied force. 44 The analogies were formerly based on the simi- larity of the differential equations of motion in the two systems as expressed in terms of displacement and charge. 2 . an d Q ”at at? dt 0 For an inductance (L.), a capacitance (c) and re- sistance R in series, where Q is the charge flowing and E is the applied voltage. 2 + 3t MEL—J25 T‘Yg'ac‘ +95. sfe-J dt dt Q1 1-, +301: or Mdt 4Yv+cfvdt -_-_fe Steady state solution being: I : E Q .4 i : E R43(L- 1 Z V: f I. - =2 y+JWm-.L) Z :30 For a mass M, a linear Spring (compliance C) and a dashpot Y in parallel where x measures the displacement of the mass and F is the applied force. Here the velocity of a mechanical element which is a relative quantity has been represented by the absolute quantity of current. This is a misnomer; it may be more logical to represent the velocity by voltage, as both of them are relative terms. Force may be better represented by current. However the mechanical elements in parallel have common velocity difference across them, while the electrical 45 elements in series have common current through them and vice versa: As will be shown later in the table, the laws of addition of impedances are not analogue. (2) Mobility Method — Electromagnetic. Capacity plays the role of electrical mass. Energy stored in the electro-static field of the condenser is 1/2 0E2 corresponding to K.E. showed in a mass 1/2 m v2, and the condenser tends to prevent change of E.M;F. by ab- sorbing a current of magnitude 01%? just as mass tends to prevent change of velocity by producing a force of magnitude H -%% . Similarly, inductance plays a part of electric stiffness; e.g., energy in magnetic field of inductance L g 1/2 LJ2 and energy stored in Spring 3 1/2C f2. In- ductance stores a voltage impulse of (fE.dt) of magnitude LI proportional to current while spring undergoes displace- ment of prOportional to applied force. The differential equation is: Ldeo. +R. Q + LzEeJOt at2 dt 0 05.2 +Yg-Q +p__ =Vej‘bt sincefggf dt dt m E =C : J (3:) (g 386 inch/sec.2) Admittance of resistor = 1/y , Next the elements of A are obtained, eg., Yll or Y32, etc. Then by application of (10) the values of a can be found. The answers are merely amplitudes of respec- tive velocities with phase displacement from the exciting force, 51 Appligation of the Analogy: In applying the analogies it is to be remembered that all springs are assumed to follow Hooke's Law and all mechanical resistance (friction or viscdhty) is directly 'proportional to the velocity between the two surfaces, or points under consideration. If both of these conditions are not satisfied, the analogy is not valid. Next, all forces are assumed to be sinusoidal, so that mechanical elements exhibit a simple harmonic motion. If a force is not sinusoidal, it must be represented by the real part of JQt the rotating vector e wherefik: 2R f is the angular frequency. The ohms law is commonly used in the solution of electrical circuits, but when the circuit becomes compli- cated, the Kirchhoff's Laws are used. (1) The vector sum of all the currents going into a point is equal to the current generated at that point. (2) The vector sum of all the voltage drOps around any closed 100p isequal to the voltage generated in the 100p. These laws are stated in mechanical system as: Force law 3 The sum of all the forces acting on any junction_ point is zero. Velocity Law The sum of all the velocities across the structures included in any closed mechanical circuit is Zero. 52 Unfortunately, the direct application of Kirchhoff's Laws always leads to more currents than are necessary, i.e., dependent variables are introduced. In the mechanical system it means more parameters than the degrees of freedom. Electrical engineers are familiar with 'Mesh-Circuit' analysis and the exactly similar, but in, verted system of 'Junction Analysisf. The latter one leads to the solution of any circuit in terms of voltages at cer- tain junctions referred to ground, and this is especially simple when there is current generator in the circuit. Al- though any voltage generator may be replaced in any circuit by an equivalent current generator, a circuit will often have fewer independent parameters when solved in terms of junction voltages, and will lead to lower order determinants than would a mesh current solution. In the application of electro-magnetic method, the junction solutions are particularly advantageous, not only because they are dealing with current generators, but because the velocity (voltage) of all masses is referred to ground, and because elements in parallel (common voltage) are more common than elements in series (common current). The circuits are made more simple by the introduc- tion of the term admittance ‘Y which is l_, the advantage 2 . being that admittances add directly in parallel circuit, 9.5. 1" - + *— ’72; " Y‘Y1+Y2 Y; NIH :51)“ l .22 53 But in series l. = 1 Y la.+-l_ +.li + --- Y1 Y2 Y3 However, mechanical elements usually occur in parallel. The first step in solving the vibration problem is to set up the equivalent circuit, being careful to refer all the masses to the ground. For each junction of mechan- ical elements other than the common or ground, we have a variable, viz., velocity of that point. The velocities of the junctions are then obtained by 6y: g; 1k Aki k = l 4; where (v-z l, 2, 3....n) Generally each junction will be a terminal of a mass, so that we have the velocities of all masses. To find force we apply I I e‘! where e is the relative velocity between the two points. This method is just as convenient with torsional vibrations as with linear vibrations. The same symbols are used, the analogy being torsional. I - torque through an element in lb.-inch z = torsional mobility in red/sec./1b.-inch ‘Y tors ional admittance 1b .-in/md/sec . M.I. in slug - inches (.g lb.inch2) 8 t : torsional compliance in rad/lb.-in. y : torsional resistance or responsiveness in rad/see./1b.-inch E = angular velocity across an element in rad./sec. 54 An equivalent circuit for a typical crankshaft is shown in Figure 5 at the end. The inertias are taken equal to simplify the problem, otherwise the actual values for K and I may be taken and the circuit solved. Often the mechanical systems involves gears. If the gear ratio is g from shaft ‘1 to 22, then torque 12 is 11, and displacement 62 is g.el; then it may be shown that S torque transferred through gears is 2 11 = 6.12 I 8.92 Y2 = 8 elY2 and Yl=52Y2 1l 91 55 Sigplified Method for Torsional Vibration Calculations - Used at Chrysler Corporation. The torsional vibration system of an in—line engine is usually considered as comprising a number of discs, con- nected in series through torsional springs. The springs are all equal and all the discs except the end ones are equal. This simplifies the natural frequency calculations and the 'shape of the elastic curves, by the use of charts described later. By means of few auxiliary calculations, the method can be extended to systems having additional springs and discs connected at each end. In figure 6 on the last sheet there are shown n Springs, all being equal and having stiffness - K. ' All the discs between J1 and Jn + l have equal If the let disc be given a sinusoidal angular de- flection, a torsional disturbance will travel along the system from left to right. It can be considered as a wave of tor- sional deflection travelling back and forth through the system with reflection at both ends. If we assume that the change in phase of the wave is 9!: in progressing from one disc to another, when wave arrives at J its phase will be ncfi. n-tl’. If upon reflection at Jn’a 1’ there is a change in phase of 2€3n + 1, then when the wave arrives back at J1, the relative phase will be n¢>+ 29n ,, 1 + IMP. 55 If reflection at J1 causes a phase change 291, the relative phase after one complete transit of system is 2n4>+2 911*14291. Now, if this relative phase is equal to zero or some multiple of 360°, a so-called ‘stationary' or ’standing' wave pattern will result. Under these conditions the system is vibrating at one of its natural frequencies. we can write at natural frequency: 2n¢>+ sen . 1 + 291 0 or multiple of 360° -- (1) \ l and ’ px = sin [61 + (x-l)4>] -- (3) where Bx . relative amplitude of disc number x. n4>+ e l .* e 0 or multiple of 180° .. (2) n 4 These relations are derived by converting the system of above figure in equivalent electrical system and applying long line theories and electric filters. They can also be derived on the basis of mechanical impedanceyby MNA. Biot. ,The formulas defining gb , 61 and en ,, l involve the impedance of the system (electrical or mechanical), and their form can be simplified by introducing the parameter F, which is ratio of driving frequency to frequency of one of equal discs on one of the springs, 9.5., F: where f a driving frequency 0 3;. 1 3 f0 ' 2Fr. J and F . frequency ratio Fqfil 57 It can be shown that: tanell-E‘Z'l -_.. (2+) tan 9 ; ' -— ' 1 n41 Fe 23n,1 - l --- (5) cos¢ =l-E ---(6 2 where R1 = J1 3‘ Rn+1=Jn+l J In the graph l, 6 and e 1 en *_ l ‘ elateml 0.010 ’ 6 ' 48 ' ‘-55.2 ' 427.0 -82.2 0.015 7 56 -49.0 -22.8 -7l.8 0.020 8 64 -45.2 -20.0 -65.2 0.025 9‘ 72_ -42.0 -18.0 -70.0 0.021 8.1 64.8 -44.8 -19.5 ~64.3 1 —— —.- —---..-..- So the value of F2 can be taken at 0.0215, and we can write .I0.0215 - ~ .°. :1 =[0.0215 x f0 H: 1H: 0 59 The values of K and J being known, we can find out fl first natural frequency. In the above example the elastic curve is given by: Disc. ‘ l ‘ 2 3 4 e1 +£efg5p .44.21' .36.1 -28 -l4.9 - Bx : 0,596 0.620 0.469 0.255 - Px 1.00 0.890 0.674 0.371 0.696 Disc. 5 6' 7 8 9 91 * 9nel 11.6. -4.2 4.6 12.5 20.6 -‘px 0.207 0.0706 -o.o772 -o.2lo -o.35 ‘ PX ' 0.297 0.0101 -0.0ll -0302 -0304 .. . . e II.. L it). v. . .. . l . . . 2m, . ”.71.. . . r . _ .0 .ol .ufivlr J‘flo. e! p. _ _ . . . . P H .rt'nfln. . 2 .. . . w . L .911». .L.-. 633 . , . . . ..J 1.1.4.4. , y. . a D S E E R G E D «9.31" p‘ - mu” -,x 4 i. . i 1.? .5: . L a q ,...... 1.. 72¢ v . foul... 0'20 ’7 I A 0.26 o. . 0. 30 0-38 o-M 6O Calculation Aids: The amount of labor involved in calculations of torsional vibration problems suggest the use of some sort of aids to simplify them. Considerable strides have been taken in test instrumentation, and there is a growing trend towards the use of standard terminology and a few well- established measuring techniques. Calculation, however, de- pends largely on the eXperience and discretion of the calcul- ator. This is because more precise classical methods are ex- tremely laborious. As a result, the analysts are forced to abridge and approximate calculations to suit their particu- lar situations. Evidently there is a definite need for cal— culation methods and aids which would demonstrate enough speed and accuracy worthy of general adoption. The following few methods and techniques are worthy of mention: 1. The use of Holzer Table to select the conditions of interest rather than solving completely the general equations. Formation of algebraic equations for balanc- ing some types of Holzer tables to eliminate cut-and-try method. I Use of standard calculating machines for the Holzer tabulation to obtain high accuracy and standardized methods while maintaining high speeds. The work may be ar- ranged on some machines to combine Operations and reduce the number of columns in the table. 61 2. Disregard of damping to avoid out-of—phase components. . 3. Judicious reduction of complex systems, especially by the use of distributed mass with tables and curves. 4. Use Of special G.M;R. slide-rule calculating board. This computer is designed to perform Holzer calcula- tion sequence and present result in curve forms. Addition is performed vertically and multiplication horizontally. ‘ The constants and scale factors are entered in cross slides, and normal Holzer assumptions and calculations are made. The individual amplitude and torque values are located at the proper points on the work sheet by scribing the tops of the vertical sliders. It has been found that more than ord- inary slide rule accuracy is obtained because copying down and re-entering of numbers is minimized. The computer is used both for resonant and forced vibration calculations. 5. mechanical vibrating models - These are excel- lent for demonstration, but are not easy to use in quanti- tative work. Electrical oscillating networks - These have been used with success notably at.Allison and General Electric. It must be quite elaborate and precise for success- ful general use. it 7“ 6. Specialized calculating machines, e.g., M.I.T. Integrator and the Harvard I.B.M. machine. The 'Vectorscopef for adding vectors as is necessary in studying the firing orders. 62 7. Harmonic analyzers are available in several forms for studying torque diagrams and actual vibration waves. The latter are frequently analyzed directly by feeding the output of an electrical torsiograph directly into an elec- trical wave analyzer. Boo'k: BIBLIOGRAPHY .— Vibration Problems in Engineering, S. Timoshenko ‘Elementary Mechanical Vibrations, Elements of mechanica1:Vibrations, Vibration Analysis, Fundamentals’of Vibration Study, Dynamigal Analo ies,_ Austin H. Church C. R. Freberg and E. N. Kemler myklestead Manley, R. G. H. P. Olson Mechanical Vibrations, Crankshaft manufacture, mechanical Vibrations, Evaluation of Effects of Torsional Vibrations, 'W. T. Thomson Smith J. P. Den Horthog S.A.E. war Engineering Board Torsional Vipration, W..A. Tuplin I3ubiicatidns: l. 2. S. A..E. Jr., Nov. 1920, pp. 418, Critical Speeds of Torsional Vibrations, F. 2. Lewis Trans. A..S. H.2E., 1922, Vol.44, Torsion in Crankshafts, S. Timoshenko Trans. A. S .M.LE., 1925, V01. 47, p. 619, Torsional Stress Concentration of Circular Cross-Section and Variable Diameter, L. S. Jacobson Automotive Industries, V01. 54, No. 23, June 10,1926, pp. 957, Lower Harmonics of Gas Pressure Are Principal Factors and Not Inertia Forces . Inst. Elect. Eng. Jr., Vol.65, No. 36, Dec. Theopy of TorsionalfOscillations 1926, pp.76 Trans. .A. S. M.iE. paper A. P. M., 50- 8 and A. P.M. 50-14, Vol.50,1928, The Range and Severity of Torsional Vibrations in Diesel Engines, F. P. Porter Engineering, Julyl}, 1928, Experimental Formula for Crankshaft in Stiffness in Torsion, B. C. Carter 10. ll. 12. 13. 14. 15. .LO. 1?. .LO. 49. 64 Engineering, Nov. 1, 1929, Stiffness in Multi-Throw Crankshafts, Constent Trans. A..S. M.IE. paper.A. P. 11., 51-22,V01. 51,1929, Practical Determination of Torsional Vibration in An Enpine Installation Which May be Supplied to a Two Mass System, F. P. Porter Mechanical world, Vol. 88, No. 22-83, 1930, pp. 322 Inghgnmgning Theoretical Natural Period of Vibration by Calculation and Graphical Method Trans. A. S R..E., paper O.G.P. 53-2,Vol. 53,1931, A Simple method for the Calculation of Natural Frequencies of Torsional Vibration _, F. P. Porter. Engineering, Vol. 131, Nos. 3397 and 3411, 1931, pp. 259, Attempt to deduce equationfi for reducinr onl le th, stiffness and inertia Automotive Industries, V01. 65, July 25, 1931, pp. 118, Critical Speeds are not affected but amplitude reduced in V-engines with angle less than 220 n Trans..A.S.M;E. paper.A.P.M. 54-24, 1932, V01. 54, Vibration during acceleration through critical SPQEQ F. M. Lewis . a. s. H. 13., June 6,1932, Mathematical and Graphical Analysis of Vibration Characteristics of Crankshafts Jr. Accoustical Soecity of America, Jan. 1933, Analoqy Between Electrical and Mechanical Systems, P.A. Firestone . .. . _ Engineering, Sept. 10, 1937, pp. 275, The Torsional Rigidity of Crankshafts. H.A. Tuplin Trans..A.S.MgE., 1943,pp. A933, Harmonic Coefficients Engine Torque Curves, Portis J.A.S.A., Jan. 1943, ApplicationS'of_Limitationgo: . Electrical and Mechanical Analogy, John Miles 20. 21. S.A.E. Meeting, Jan. 8412, 1945, paper presented by F. P. Porter on methods for calculating torsional vibrations S .A..E., V01. 54, Ray 1946, pp. 238, Electric Equipment or Investirating Crankshaft Torsional Vibrations 6b 22. Engineer's Digest, Vbl. 3, No. 11, Nov., 1946, pp. 537, New Formula for Calculation of Torsional Oscillations 43. 5.AWE., Dec. 1946, pp. 173, Torsional Vibrations ang Fracture o; Crankshafcts Simplified Methods for 3% Accuracz - 24. Engineer's Digest, vol, 4, No. 2, Feb. 1947, Graghioal getermination of Frequencies of Torsional and:E;exural Vibragigns . is J]; ‘2' “Ila.- ' I. 10!“ m_ ~‘ 5“" ‘° ' Q J. «s. Q L 4.5.5-2.: . 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