‘l \ I l! IHII m H T , l 5'1. H | W V ‘ HI 5 I ‘l 121 377 .TH_3 T'HE EVJRLUAWQN C}? §F¥ECTS Oi: TQRSEONAL WfiRfiLTEQNS Thai: for flu Dog!“ of M. 5. memem STATE COLLEGE Cl F. 93%! .1949 II'HESIS This is to C(‘rtilg that the thesis entillml THE EVALUATION OF EFFECTS OF TORSIONAL VIBRATIONS presented In] has been accelm-d tummls hIHiHnn-m ul Illt‘ requirmnents fur Masters d'—“J“"‘ in Mech. Engr. ,/ hitljur lil‘¢)ft‘SSI_D[‘ Hate m W -3- THE EVALUATION OF EFFECTS OF TORSIONAL VIBRATIONS By GORDHANBHAI FULABHAIJAT‘EL A THESIS Submitted to the School of Graduate Studiescf Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 19h9 THESIS -ACKNOWLEDGEMENT The author wishes to express appreciation to Professor George William Hobbs for his guidance and assistance, and Professor Howard Womochel for his counsel and help in making this thesis possible. 217886 (l) (2) (3) (4) (5) (5) Introduction \—~ w'r -,~_/4.‘.Jait O O O O O O O I O O O O .~ ~.--‘ --~ —< \f‘._“ . $5, 1 _’ 7’1 T3“ EVALUATIOT: cs EFSECTQ or To.» can VIE-nan is There are two ways in which the change of position of the centre of gravity or motion of mass may occur. (1) Due to variations in Speed of revolution of crankshaft. (2) By the distortion of the members of a mechanism due to elastic properties of the material of which they are made. There is distortion of crankshaft due to the forces acting upon them: The first are serious because of their ef- fects on other parts, while the second are serious because of heir effects upon themselves. Torsional vibration in an engine may likewise be due either to the change of Speed of rotating parts or to the elastic distortion of those parts. The force impulses on the pistons cause varying torque mpulses on the crank- shaft and flywheel. The Speeding up and slowing down of these parts within a cycle are due mainly to the forces with- in the engine and, therefore, the engine as a whole supplies the reacting torque. If it is not rigidly mounted, its inertia may be small enough to permit a noticeable rocking vibration. The elastic distortion of crankshaft is the most- important, however, in that at certain speeds, the rate of application of the torque impulses may coincide with the natural frequency of the cranzs4aft or with some small mul- tiple of fraction of this frequency in which the impulse adds additional energy to vibrating member causing it to oscillate with an increasing amplitude. This may continue until the part fails. Forces stin latiny torsional vibration are due (1) Pressure forces in workin: cylinder. (2) Inertia forces of reciprocating parts. (3) Gravity forces of reciprocating and unbalanced rotating parts. (4) Uneven absorption of power at the driven machinery. The damning forces are due: (1) Elastic hysteresis. (2) Driven machinery, (3) Slight slippage at the union of the two shafts line-up, (4) Surface friction between the moving and stationary parts, (5) Energy absorbed by oil film around bearings, (6) Energy transmitted by the side thrust of crankshaft, Since vibration cannot arise under the action of constant forces only, the force creating and sustaining a vibration is always a fluctuating one. Fluctuating forces may vary in magnitude only, and are then usually called re- ciprocating forces, or they may vary in direction only and are then usually called rotatin: forces. Fluctuating force that causes vibration is called an exciting force, a disturbing force or a shaking force. Spring stiffness or. spring constant K is given as the force necessarr to stretch or compress the spring one unit of length. Free Vibration: When displaced from position W [ll/Illa ? es down and force upward becomes greater, so goes up, due to momentum it continu s to (I) 30 up from center, and then due to downward force motion slows down and comes back. This is free vibration because there is no external fluctuating force. Restoring Force: Th force which tries to take W back to its original nositio :3 Friction force, which 5 t‘1 .4 ‘4 0‘ e of complex form is 'called damping. Eumber of cycles completed in one unit of time is called frequency. Frequency of free vibration is indepen- dent of amplitude but it will increase with incr (D asina \d (“F spring stiffness and with decreasing weith g“. of vibrating ma 8. This frequency of free vibration of a system is called its natural frequency and it increases as sq. root of spring stiffness K and inversely as sq. root of we :ht W. Suppose now that weight is shaken by external force. In this case, freouency of forced vibration denerds - only on that shaking force. Amplitude depends both upon 4 fluctuating force and on the ratio of its frequency to the natural frequency of the system, and wten this ratio becomes unity, the amplitude of vibration may build up to a danger- ous value. This condition is called resonance, and the pur- pose of moss vibration investigation is to avoid its occur- ence. The above mentioned vibrating system is of simplest type in-so-far as only one co-ordinate is necessary to Specify the motion of mass. This is called system of one degree of freedom. Inertia is merely a body possessinr a mass moment .‘ D of inertia J about a particular axis, and torsional stiffness, 'C', of the shaft takes place of sprin constant, 'K'. A rigid body restrained to move in two directions or to rotate about two axis is said to have two degrees of freedom. 4; Ca T G 14. ll . 1- V‘ D b 1 \ \\V \\\W \ \\ \\\\\ 1 I ’IIII l7llr47 Two degrees freedom system. lHI 1111 Four degrees of freedom fl 1 f 1 ’L \\\ I Infinite number of de~ress of freedom. as Any irregular motion of a pa rticl about some fixed position of equilibrium may be called a vibration. If the he mmer blows occur when the vibrating rod is at C and moving in direction m, vibration will obvious- ly be da.n1:ed, but if the blow s are timed to A j ,5 occur at n', the force of blows will aid in / 1’4” 1n <{;‘JG m continuin3 and amplifying the vibration. I ‘ “'D I x 1 Such a case is called synchronous vibration or resonance. Every elastic oody has a naturmd.rtriodofziona ;cmn i.e. time per cycle of movement which depends upon its mass, moment of inertia, and stiffness. Crankshaft is more complex of vibration are $3.: 0) than simple rod, and its natural erio ’d f»): U] harder to predict, but the basic i ea 1 be same. Excessive torsional vibrations in an engine cs use noise or wear on gears and auxiliary drives, and in worst cases result in a broken crankshaft. Just as a pendulum has a natural period of swinj, so the moving parts of an engine mounted on the cranksr aft, i.e. pistons, connecting rods, flywheel, have a natural period of torsional oscillation. The irre3ular turning effort dia3ram of an e1g_ine is made up of a lar 5e numb er of sine curves knoxn as harmonics, havin5 varying magnitude and frequencies. Should the period of the main forcin3 torque or of the various harmonic orders synchronize with the natural period of oscillation of the s12aft sys ten, excessive vibration of shaft will occur. Tris state is called resonance, and is avoided wherever possible. O‘\ H) Where a state 0 resonance occurs in the running Speed, the amplittdes of twist are kept down to a safe limit by stiffenin3 of parts where possible, and sometimes by vibra— tion damper, but the damper cannot be used suécessfully to enable the en3ine to run under the load at criticals. Its function is to take the e1" ne throufn criticals. . ’. A Crankshaft with a flywheel at one end forms a com- pound torsional pendulum and vibrates as such. Torsional vi- bration may be of two kinds, forced and free. When subject deflected from normal position and is then released, it exe- cutes what is known as free vibration. If the pendulum is dealt with a rapid succession of blows, it is forced to vibrate called forced vi- (J) s i [-30 at the rate of these impulses, and th bration. Such forced vibration occurs in crankshafts, but their amplitudes are small and therefore they cause trouble. Po First order of vibration when th (0 re s a single nodal point between flywheel and crank unit nearest to it. The Modal point does not vibrate but rotates at uniform Speed. All parts of crarkshafts ahead of the nodal point are then diSplaced in the same direction, while parts to the rear, to- gether with flywheel, are displaced in an Opposite direction. I First order torsional vibration of six hrow crarzshaft. \1 In case of vibration of second order, two Fourier s Series: Any periodic or recurrent f‘nction of this kind can be accurately represented by a constant mean value and a series 0; harmonics (sine curve the basic (1) function) of which he first has same period a function, and tle following save the periods equal to 1/2, 1/3, 1/4, etc., that of basic function. Amplitudes of suc- ‘ ceedinf harmonics ( vicient of suceeding terms) de- crease in a 5eneral way, although each re is not smaller than the precedi n3 one. From this explanation we see that the gas pressure torque impresses upon or nhszaft not only a succession of harmonic forces of the same frequency as its own, but also a series of harmonic forces twice, thrice, four times, etc., this frequency. The method of resolving an irrefiular periodic function nto its con: onent harnor -ics is known as harmonic analysis. Torsional vibration will depend upon the firin order of the en3ine. Frequency is between 12000 and 15000 cycles per misute. Sixth harmonic is the lowest which can cause torsional vibration of the first order in a six cylin- der engine. Such vibration in crank, havin3 frequency of free vibration of LXXK>ner minute occurs at 2x 13000/5 : 5000 T ‘ ” Lgl Q‘s. CO ‘ ' fl “‘ . ‘ "0 ' "."\ ‘- ". O.‘ ”a - ~\ ~ Tris s beyOnd oneratin, Svfieu. ine rext harmonic that may give trouble is 9th, wiicz is in resonance at an engine speed of 2 K 15000/9 = 3333 r.:.m. Resonance occurs when q = n; This vibration can be supnressed by the c; firing ordel. The next in; or tent Laraonic is 12th resonance at 2500 r.p.m. The tlird harmonics are the lowest ’J: it causin3 tor wio ial v ration o the first order, and since their *‘ ’3 *4. amplitude is only about 1/16 that of “rat harmonics, inertia torque is not a very im;ort:n factor in causing torsional Hibr tion in sulti-cylinder ersines. In aircraft en ine, since —. n - -. s v”. . ‘ b 4n M , a useful spee' ran;c is nucn less tnan in an automos usually possible to avoid tbe mosttroublesone critica a natural p riod of severe vibration w thin the desired operat- ing range. The simylest form C orie, i.e. sine or cosine function. And most vibrating systems have motions that are nearly Harmonic and may Le Iu'ritten 9.) t: x = Kcos wt. a = amplitude. wt - anyle in ra-iisns kx . é—e—y-glx W777 fix is the inertia force that resists acceleration g . x (+ to right) nd therefore‘is -ive. for —ive. value of X. This becomes +, and acts to rijht: E 3": + i}: = O S x = A cos (“ht ~I) From which acceleration found and w»: “3 W he system is given by: L If the spring snown in figure were stretched by a force equal to the weight W of vibrating mass, resulting deflection would _ W 7? I ‘ —- 8 g” {n i ‘ 1 n o 5 S= cellection This is easy, since deflecti n an be found or estimated. T TJ‘ a C9! 9 3J5 Because of twist of the shaft, a counter-clockwise torque c is exerted, and because of angular see. there will be a resisting inertia torque (actir: in -ive. direction) of J 10 for dynamic equilibruim J9+C9=O from which fn : _ _c_: T.' = 2”\l£ ---- (a) 2H J “ c By measuring natural period of oscil H $3 C 9’ Pa O ,5 0) w C+ i 3‘ CD value of g of the moment of iner ia asout the suspension axis . :: ta, C J 437-, . After free vibration has died out, there remains nly forced vibration, which is called steady state vibration. Stead state forced vibration is harmonic, and has the same frequency as the shaking force. Free vibration is the sum of n several harmonic motions 0: different frequencies. (110.3,: io--|r3,l S " m1: .3 Torsional vibration of crankshaft produces vibra- tion of the reciprocm n3 par 3, and the polar inertia of crankshaft itself is very s: all compared with tzle inertia at each cylin;er due to the motion of these reciprocating parts. Torsional analysis Ste,s: (l) alculate the amnional ri3idity o the ft between each rota tin member. (2) Calculate the moments of inertia of all the reciprocatinc and rotating V masses in the system. (3) Calculate the natL ral frequencies of torsional oscillation of the system. (4) Calculate amplitude of vibration and the resulting stresses due to resonance, ("’1 Torsional Ric‘dity L 1* 0 ll torsional rigidity of actual shaft. G = modulus of ri3idity, In: polar moment of inertia or second moment or area, L = actual len3th of shaft, D = actual dia. of shaft. Ce: torsional rigidit y of eqv ivalen sshaft, Le: equi. length of shaft, De: " dia. of shaft, 12 I c- G”: = 3:11 4 l L 32 ,4 a and Ce = I} X 7’98 But u 2 Ce :e 32 4 ‘4 gym) 9, _C_I~_:;”De L 32 Le “2 ° ._ 1+ . . Le - 4.4 (PS-’3.) D Le, then, is tie le ggth of sm ft of dia. D9 which when sub- jected to a certain torque,will twist through an angle equal to that rro'duced by sarne torque on a s1alt 0; length L and diameter D. Cranl Shef Stiffness: Reduce cranksnaft to an quiv lent lCP”tn of parallel sha: tin3 diameter of crankshaft journal. A u =aiS + .4J + 1. o; s LPG) 4-(1 4), 1.28422 (D ---:1 Zr) (3132:1135) 211v Having found he equivalent len;th of crank through, it is necessary to f'nd the len3th from the flywheel end cylinder to the flywheel. Total equiva le _ tlength from end cylinder to the last iron flywieel =i§g + L (De)Zr + 2 (De) A4 £L(D€) %ZL(D4)4311:8 2 <11 5? 6.3 The shaft is considered free to twist to a length of a quarter the dianeter of the bore, and is then co.°idered as integral with the boss, which is taken to twist to half the depth of the flange. ‘\ FE 1-19 «m g d- — '§ E o ' ‘ ‘g 3 T Jith 3 q "" “ l 6: - a; For solid crank pin and journal L = (L +.8w) +7129, 4 +l"RD4 e .i2_LL_ S i D wt) p By Carter's formula Le - D (11.5 + 5811 4 .sze .. L523?) 1% Th WT Torsional rigidity one crank depends upon the con- dition of constraint at the bearin3. Assuming that the clear- ances in the bearings are such hat free diaplacements of the cross section m-m and n—n during twist are possible, the angle of twist produced by torque moment M can be easily ob- tained. This angle consists of three parts: (1) Twist of journal (2) Twist of crash pin (3) Bending of web. Let Clbe - D54G torsional rigidity of journal, 32 Cl = D246 .torsional rigidity of crank pin, 32 . s = w T3 ‘E torsional rigidity of web, 1 In order to take into account local deformation in the web in the regions shaded in the figure due to twist, \ 14 the lengths of the journal and of the pin are taken equal to: L5 = L, + .92! and LP ‘ “P + .9v reSpy. The angle of twist O of the crank produced by a torquermoment I will then be e 3 L5 :21: + LL21- 4 gQRJ-r Cl 02 :2 In calculati-q the torsional vibration of a crank- shaft, every crank can be replaced by an equivalent shaft of uniform cross section of a torsional rigidity C. The lenrth I ; of equivalent shaft will be found from I l = 6 6 as calculated above, _6— . 0 Then length of equivalen shaft be I r A FJ I 53 I I I I A LU V L =0 t; +Lg(l-§)+g? in which - -. 2 '7 " I. = 1‘04» + *0 +3132 + LP) + 3?. + 1.2 (L: + a.) 4 03 2 c2 24 Bl 3 3 s 2 F Fl 2 LER +.3_ 2 c2 2 B in which again c3 - cBwBG 2 O o ( c + WA) which is the torsional rigidity of the web as a bar of rectangular cross section with sides W and T. Bl :ITDE IE flexible rigidity of crankpin F and Fl cross sectional areas of crank pin and of the web reSpectively. By taking LP: L5 and c1 = 02, the complete con- straint as it is seen from above equation dmiuishes the equi- P! f shaft in the ra ,2 2 v‘\ an. cf ’JO 0 valent length 0 1 T1— 0 in Another suestion to se considered is the calculatiqn D _ J. 4-" ~—. - . f“ oi inertia of novinn masses ‘ L 4 Lbss m of connecting rod is replaced in two masses ml : ( %% at crank pin and m2 : m - ( %% at cross head, where I denotes homent of Inertia of connecting rod ebou+ the centre of cross head. All other moving masses are concentra- ted in the same two points, so that finally only two masses, M and M1 are taken. For torsional considerations, all reciprocating motion must be reduced to equival-nt rotating motion. Half the weight of reciprocating parts may be considered as act- ing at crank pin. Piston = weisht of it is added to reciprocating part, Rod = partly 2/3 revolving and partly recipro- czting, H - l/2 weight of reciprocating parts of connecting rod and piston + weight of revolving parts of rod, , 2 2 N = W R k t . v 2 = ‘ £2 r s lid Journal JK HUD or o q o é = w D.“ 4 d; for hollow 5 Crank pin WK2 = d ( D22 + “2) for solid Q U D52 +'R2) for hollow II L. U .ofU + 2 + d2) for solid :3 Balance weight symmetrical about Y-Y. I = Ix-X + Iy-y I = rotational moment of inertia, IV " Iy v are found by constructins the first and ...-4’*. “'“I " second derived figures about both axis and finding radius of gyration for each. J Iatural frequency: J3 a : J 7 a T _ e. a ' Three mass system J1, J2, J3, M. of I. of 3 masses, ll: 12 lengths of connecting shafts. x be the y be the distance of node distance of node w .L. 1 TV .L‘. 2 l7 4" erm J 1. from J7 J The node is the point of reversal of twist. The period torsional oscillation of J1 about its node is gi"en by the expression: T 2 2! J1 C = torsional rifidity of X lenrth C Frequency F of mass J is F = l -_-_ Id? but C = G I} where G is modulus T +2” J1 L of riéiiity which will take as 11.8 1-: 10° #-/ square inch IP is polar moment of inertia of the sLaft in inches and L is the len th of the shaft in inches. If Jl is expressed in Lb. in.2 6 , Fl = Sq§§11.8 x lO X4383 x Ip 2n Jl X - I: I I‘! O - o4AOOEIIg VlOPTelOE per minute -— (1) J1 X and 32 = 644000323 J2 (l + l. 11-x 12- y) -- (2) F- = 644OOOJIE D r u3y -- (3) since Fl = F2 2 F3 equating the three expression give as 1 _ 13% E——1 + “—1 ) Ul-A 02 ll'fl‘ 12 - Y) 1 CCVG "- ‘ . l- . ‘1' .' «rn~e~r n- Y in terns of a, me by substitu tin: for Z in equation (2) and equatinj (2) and (l) we get two values of I. Substituting it in (l) we get two frequencies, ficult, but the system can be reduced to three mass system and results can be obtained. In a multi- cylinder en ine with flywheel, total inertia of the cylinder masses is con- sidered to act at the centre of the engine for clo.s e approx- imation. Frequency is obtained by multiplyiit the total ( W R“) Of tie CyliLiW mas 883 by .55 and considering this at centre. Exam: le: ll \) 1 H 5 i. J. I')‘ 0 U) (D O m Inertias - each cylinder ’5 - -, o r) :lywneel = 32.? # in sec.‘ First reduce to equivalent three mass system Consider the inertia as actiné at the centre i.e. between cylinder 5 and 4. Aoplv correction factor .85. Effective inertia of the cylinder masses: l I O (I) U1 0\ i‘ U 1...: C I 2.54 #,in. sec.2 1020 # in.2 : l x 11 + ( 2 x 10.23) + 8.84 = 34.80 in. 2 0J' .12 J3 #34 ‘ 141% Let X be the distance of one node from the effective cylinder mass and Y the distance of the other node fron gen- -l erator. hnfifl ° . 's-L [.15 Io-z‘s ll 19.2.3 “‘23 8’44 ‘7"; J“ 4' Single node normal elastic C&%RM#).pWGE 5 n " T ", :: __,, " V Tncn 01 a x J) u i . . Y a Jl x . 2.34 x . 0.1993x say .2x. 3; 13.23 Also i = l_ ( l + l ) Jlx 52 33.3’_ 3 ,.s5 - y) Substituting for X 1 = l ( l 4 71 ) 2.53x 32.7 31.8-x 9.05 - .2x) from which x . 23.75 in. or 47.45 in. I‘Iatural frequency . 644000 ‘ILQ JlK where J1 is tre f.n,_),.,1..' r "732 .0 .3 j E e -eccite a“ oi the cylinier masses #. in. sec. 20 one node F . 644000 J10.9 1020: 47.45 = 9560 Iv'.l: .1':. Two nodes normal elastic curve. (SF-E TR“ 7) Two node F = 544000 {To‘b 1020 x 23.75 13050 V04. 01:. The arpr 'nate figures for the two frequencies having been obtained, the exact figure must now be found by the torque summation method. “0 ethod for calculation of natural frequencies of torsional vibrations. IL the engine with a small number of cylinders and slow speed, the torsional vibration can be avoided by making shaft diameter large. (1) One type of equivalent s‘ aft arrangement is ob- tained by considering each cylinder, flywheel, and alternator as concentrated masses connected by elastic shafts having no (2) n tber type of equivalent shaft arrangement is \_- \J Obtained by aver aginr the various masses distributed along |> the shafting of insta 113 tion into a number of connected uni- form scafts, having both mass and elastici y. Eeavy flywheel may be considered concentrated at certain points of uniform shafts. here are three masses and 21 two shafts as compared with the eight masses and seven shafts when a concentrated mass system is used. The masses may be reduced from the beginning to the end, or backward from end to beginning, or from both ends up to some section. A natural frequency is obtained when the sum of the masses reduced from both ends up to any section added to mass at that section is zero. C 03 TA The effect of A, acting upon I . B, through elastic shaft C 1 L c i can be obtained at B by add- ing fraction of A to B. This sun can again be considered as the first mass, the effect of which is reduced to next concentrated mass. At this frequency the elastic forces in the shafts and the inertia forces of masses are in a state of equili- brium, so that the system when once set into motion at this frequency, will continue to vibrate indefinitely (assuming no existence of damping influence). Example: Consider six cylinder, two cycle connected to a flywheel and the alternator. ”D lo 110 1 ' 34- J I a— Weight of and oil = Weight of Center of line length 22 piston and pin, complete including cooling gear 1000 pounds. connecting rod 2 500 pounds gravity of rod occurs at 0.38 times the centre of the rod from the crank pin center line. cerll: : "I l Corr ecting rod 5 I 2 factor of journals is that of a solid cylinder about its own center line 4 9 ' Ld #.In.“ L = lensth :7“ h ‘ 30 d a diameter. weight of steel = . 2830 0#/s re inch, I = factor for crank pin is the t of solid cylinder about an axis at a is ance r, from its own center line, thus 'r 2 ~ ’ L d (g3 r2) #. in.2 3.5 8 L = length of pin. d - diameter. r crank radius, A close annloxii tion for inertia factor for webs, considering each as rectangular par Lllelon ip ed, having the same width and thickness and height to give same cross section area is given by: b d h {(b2 + h2 + (r 2) # in 2 3.533 12 Y‘ " ° b = breadth. h = height of equi. d = thickness, rectangular section. r = crank radius. [0 K»! . . I a factor are journal = 12 (3:35)4 = 4080 #. in.2 ’ 3° 2 2 2 ,, pin = 10 (11) (ll) 4 (11) = 33000 4.5 '5. Two webs 2xl4x6x22.5 .{ (_fi)2 4L22[§)2 4 (5.5)2}= 94950 3.555 12 Piston and connecting rod 112 .62 x 600 4 .5050 (.38 x 600 4 1000)}: 120,050 ‘__—#,,, Total 255,480 # in.2 J ' gégflgg = 7544 # 13.2 per inch 3+ Considering the crank hree percent stiffer than a uniform 1.. 9 8118.; t having the same section as the journal, the value of C L, may be calculated from equation. 115000 (11)4 1.03 CL Q = 17470 x 10 #. in.2 In the same way inertia factor for crank was calculated. The total inertia factor for the scavenge pump, including the shaft up to center line of the first bearing 1" v is 70000 #. in. . . 0000 1 . ' . . %ggg- - 9.9 in. correSponds to tne increase in length of main cranks due to scavenging pump. 4 FLywHEEL 2 HLTERtYRTaK d, ILD ——Z -' I» L 20 l A 67‘.“ H 45-2 ‘-<———- 34 ——-> L24 L, 3Y5" /LL 01 39’. Shlfltk ‘5, .. ’7 4O ' //Z/// 375 GCWHNKS to W 57!}, a s 204 SPOKES I i . : , ' . (5,3745 71 I‘ 4 575,0 ”01 6 3595 to. k 02 : 11 58500 (11) 28.85 588,000,000 #. in. L1 - 259.5 in. 258.5 x 7544 = 1,950,000 #. 1n.2 cl . 17470 x 105/258.5 = 67,580,000 #. in. 0 = 18.325 J1 = 3.113 CI Resulting equi. system. 588x106 #. in. O (0 ll 0.09221 L“ n 1 258.5 in. . 31 - 1950.000 #. in.2 67580000 C) H ll 0 = 30113 [0 U1 All present day trends in the application and design of these engines have been cor ducive to torsional vibration difficulties. Tendencies are totard highe “speeds and a greater number of cylinders. he higher speeds have increased the possibility of torsional vibrat ion by raising the running speeds up into the re: ion of critical Speeds. Increase in mass or fle: (ibility will lower the natural fre- quency. The holzer method is based on the principle that the sum of the inertia torques, develOped in a system because ‘ ‘V‘ tze via rations, must equal zero if the vibration is zero. H: 0 Example: A 4 c; lil*.ler engine with a flywheel runs at 1200 r.p.m. and is connected with a "enerator 0y a flexiole coupl- ing. The mass moments of inertia in in. # 860.2 are as Cylinder 31 - J2 a J3 - J4 - .55 Flywheel J5 = 25.6 Coupling hub J5 = 0.5 Generator J7 = 8.75 The equivalent lenjths referred to a 3 inch diameter steel shaft in inches are the following: U1 U1 Cranks Lei Le . 05 L = L a '- 2 e3 Qe 3 Flexible Coupling L85 : 45.5 Hub to generator 2%. 41 To determine the lowest natural free uency with the aid of Eolzer method. Before setting up the table, it is desirable to convert equivalent length to snrins scale. ‘ '—J . 2 . 4 _ _ slnce Kt = f, a; and dc - 3 in. 3 be ’3 ___I = 12 O) 47’ )‘T The values of the spring scales, then are = Kt, = 19.12 (10)° l e \_,\l (10)9 A .1 (10)b V: F, m II l__l D.) ‘4 L)! .Q :51. f7 01 n (10)0 in. #. per radian p: \ n m {.1 DJ Notes on calculation procedure: How to assume freouencv: A fai trial value can be obtained by grouping tosether the masses that have a short equi. shaft len th or scale between them (neglect coupling mass as Then J; - J1 4 J2 4 J3 4 J4 4 J5 = 27.8 f- J2 "’ J7 = 8073 and L - L e 5 = 1.105 (10)5 (D + 1%, = 85.5 in. H- +5 3 29 Kt: ll 4 la ) 2”' J1 J2 . so 1.10511015 (27.8 + 8.75) 2H 27.8 x 8.75 -{}5€90 x 6.28) 2- 155000 - 0.155 (10)‘D assumed. . o . '7 ,, : : — 3 1’ 3 ’ : 0 Item 23 25245-7105 : 4 9721313105) J':‘-§(1083.k.(17103): 533,23 ;_ L : v? : : : : 15} l .55 0.0900 1.000 C .0300 0.C§00 19 12 0.0047 2 .55 0.0905 .9953 0.0905 0.1013 17. 0.0104 3 .55 0.0908 .9849 0.0394 0.2707 19.12 0.0142 4 .55 0.0908 .,707 0.0081 0.3500 l'.l2 0.0188 5 25.30 4.2240 .9519 4.0208 4.: 96 2.10 2.0835 6 .50 0.002: -l.1345 -0.0977 4.2079 .33 1.8394 7 8.75 1.4433 ~2.9740 ~4.2939 -0.0030 filef r to above table: Second column, ine ltia as given. Seventh column, Spring cales (calculated) The value of W52 = .165 (10).5 will be as ssu umed as the natural frequency. Then fill up the table. The amplitude of the f rat di 1sk (? l) 3 always assumed to be one radian By dividing 6 column/ 7 column, we get angle of twist #3732 Radians - 0.0047 752 : 731 -0782 .-. 1 - .0047 = 0.9953 which is placed in second line of column 4. In this example torque enainder in column (6) is -0.0080 in. fi. If the correct frequency i-d b- n chosen, thi emainder would be zero. The deflexion curve may be plotted along he shaft by using the values of P in column 4, for the various disks as 81101;!an 92 T3 9‘ 95 V T6 T7 J. J. J. $\ 0.9953 0-9707 ‘4434‘ 1 It will be seen that there is only one node (located L—2.974o between disks 5 and 6). Hence the value of 0%? = .155 (lO)6 and f“ = 5880 c.p.m. is very close to the first -atural fre- quency. For torsional vibration the mode of the vibration is the same as the number of nodes, i.e. the first critical has one node, the 2nd - tw , and so on. Calculations: Holzer tabulation procedure is a widely accepted method for calculation of torsional vibration characteristics. This method can be simplified by application of the distributed mass_concept to some parts of the equivalent mass elastic system. Since with present equipment it is impractical to measure all the amplitudes and torques in entire system under all Operating conditions. Calculation, laboratory eating, and engine measurement should therefore be considered as mutually dependent techniques for guiding development. Q Fundamental calculation Torsional vibration 'mechanical network' ties, to a series of at several points and and dry' method is used. 5‘! 1 A. —A- Gene (A) a methods: Fundamental simpli (1) lumped into e (b) Exciting (2) torque vibra diagrams card. (b) .LIC-Lthem torque (3) Damping is able involves the coupled by periodic excitinv Equivalent system torque: tion r1onic Vibration, uOl‘S. ' 29 not? ods: response of a J. Shae Q Q. .5. lCCLuiOIlS Distributed factors quiva ent concentrated .1 ‘ By norma Mlcxlat ions, cal inte3rator, or test. Actual -in31e cylinder 6' 0 curves expressed a ana is of gas torque i" Y0 constructed from indicator tica l Ipression for inertia verycomplex and vari- and ring friction, electrical eddy 9 30 currents, prepeller, etc. TLe best guide is previous experience on compar- able ensi \J fi—S U] K? Calculations aids: Techniques and methods (1) Use of Eolzer table (2) Disregard of damping to avoid out of ' 3 :4: 2] U] (D components. (5) Judicious reduction of comple: systems by the use of the distributed mass with tables and curves, showing performance of simplified systems. (4) Formation of algebraic equations for balanc- ing Eolzer tables 10 t a. o the stand rd calculating machines I? of special-;.;.R. slide rule calculating A A U] \I C‘. U] (D 0 \ V C: (:1 CD board (7) Lbchanical vibrating models (8) Electrical oscillating network (9) Specialized calculating machines (10) The vectorscope (ll) Harmonic analyzer 31 ethods of calculatinr torsional vibr.tion: '1 (D r 'A *4 First, it requires he determination of a suitable mass elastic system expressed in numerical terms, having ap- proximately equivalent torsional vibration characteristics. This is called 'Equivalent system', 'mass elastic system', etc. It requires numerical evaluation of the inertia factors and stiffness fact rs for all the moving parts of he installation. After that, it requires the calculations for: (1) natural frequency (2) peak amplitudes or stresses at synchronism between natural frequency and stimulating impulses (3) forced vibration amplitudes or stresses at various frequencies of stimulating impulses Equilibrium system is obtained by considering the moving masses at each cylinder, flywheel, alternator and other Huge compact parts as concentrated masses connected .by elastic shafts having no mass. This is called 'concentra- ted mass system'. Another is obtained by averaging various masses distributed along the shtfting of the installation into a number of connected uniform shafts, having both nass and elasticity. This is called 'uniform shaft system'. A third type uses any combination of concentrated masses, elastic shafts without mass and elastic with mass called 'combination equivalent system'. K)! R) A natural frequency is defined as the frequency at which a sustained vibration of the system may occur if no damping exists, and stimulating impulses are removed. In using this Eolzer table for natural frequency calcula- tion, he initial value of unity is assumed for e for mass # l, and the rest of the values are completed. The require- ment for natural frequency is $1 P2 6 = O . Various values of P2 are tried until this condition is realized. Columns 3 and 5 hive the relat_ve amplitudes and moments of vibration at natural frequency when this occurs, 1nd the re- lative stresses due to vibration at this frequency can be calculated. Then if the actual amplitude of natural vibration at any mass is known accurately by test or calculation, the actual vibration stresses m y be computed. Calculation for peak amplitudes due to several impulses acting at various masses may be made by equating energy input = energy absorbed by damping. If the impulses are m5 cos (pt — 43) acting at the masses indicated by s, and the amplitude at the first mass is 81 sin (pt - 4d. The energy input equation is E = "91 stés where @513 the relative amplitude at the various masses given in column 3. £31555 -8{m5é-Ssin ‘92 + (£111.595 cos 4;)2 tan? = (msészll V’s émses cos a, If the magnitude of impulses are all the same srch that m s m. The energy equation is E =mfl6f8flh _ _ o where 265 = ((65 sing)“ 1' (iés COS 93)2 tan ¢ = (Elsin.%5 £95 C08 9'15 Using 0, 42 - “1, 4e - 41 ..... in place of 41, 42, 4a, the J J phase angles all refer to the instant at which the maximum value of the first impulses occurs in this case, if = ‘01 § tan -1 (é; sin (“'5- 4’1) 33, cos (4;— 41) Numerical values of m are.obtained by harmonic analysis of the applied torque curves or torque / AR curves, where A is piston area and R is crank radius. If m the mag- nitude of harmonic curve ml torque / A R curve Then m = Ale Curves for ml due to :as pressure are appended. The energy loss due to a marine prOpeller may be stated as: T _ -—2 r2 ~17pfe, ef' where e, 18 the relative amplitude of vibra- tion at the propeller as given by holzer. Average value of f is F =35 a r.p.m. Q = mean shaft torqueet the DJ .p- at which the variation occurs r.p.m. of shaft speed at which vibration occurs. Assuming torque varies with the square of r.p.m. f 3 33 20.73.31 where Q, and r.p.m‘Y full Speed. An empirical formula for the damping along the steel shafts, due to internal absorption of energy and other unknown effects is -' . . .msa 2.3 19 (did-3 - d.4;J) EQW: 6h K = 1010 (d; _ (1‘4 )2,3 23 46 = 70 (M B, 1010 d 4.9 where L is the leng h of shaft. d2 and dl outer and inner dia. I r—a :3 (+- w (D (Q vibration twisting moment i IPJ “daft correSpondini to the relative amplitude in Eolzer table. If E. - input energy for e1 - 1 KO = damping energy alon3 the shafts for 91 = l The amplitude is obtained by equating the input and damping energy as follows: . E 91 = K, 912°3 1-3 E. K, f K prepeller damping 9 1 energy for e = l Amplitude we get by E 91 = hr 912 + KO 912‘3 71" " " lo from which value of e1 may be determined. In this case K is taken as zero. e = E./K . o 1 p holzer table can be used to compute undamped forced vibrations at any frequency by using two tables, one for sine components, and one for cosine components-of im- pulses. At a atural frequency the amplitude of vibration along the system for a unit amplitude at some designed sec- tion are called the relative amplitude of natural vibration. Column 3 gives relative amplitude when 21 P2 e = O. The solution for uniform shaft equivalent system may be carried out graphically by using Lewis polar diagram. In this case if calculations at resonance are made, the energy equation for equal impulses along the uniform shaft is the same as previously stated. Where 61 is the amplitude at the beginning of the system and és are the amplitudes on the relative amplitude curve at various cylinders. The values of ésmay be measured from the polar diagram. Lewis term 213 is the same as 265 given here. The empirical formula for the dampilg along'steel shafts considered with distributed mass: I‘L:70(d2r. -d.l variable lenfith along the shaft M = is the vibration twisting moment along the shaft corr eponding to the relative amplitude curve Solution for combi1-cation type equivalen syste may be carried out by reduction method. This method reduces the masses from one end to the other or from both ends to some section in any manner convenient. Solution for the combination type equivalent system. Reduction methor of calculation makes use of follow- ing idea. F760R£ Rflfif EL A and B are the concentrated masses. C shaft without mass. Then at a given frequency, he effect of A acting on B through the elastic shaft C can be ottained at B by adding a certain fraction of A to B. This sum can again considered as the first mass, the effect of which can be duced to the _ext concentrated mass. Elastic shafts are called steps of line up. Concentrated masses are placed tween steps. be- C 2 stiffness factor # in/ radian of shaft twisting moment M(;.in.) aid deflection e (radian) I l k‘l [U {U W c!- \ L) (D O [O . ’3 a - in./sec‘ = 385.088 in./ sec.2 G = sheari-3 modulus of elasticity ( #/ squs re inch) '5 J 3 ine rti {O lactor or weifht polar moment of inertia n.1n 2) J 2 weight polar moment of .1. 3333 p: u number of typical = length of step (in. M = twisting moment (#. ‘v A '- "d N frequency constant H A ¢ n) = l§g tan ¢ n ¢NW1 'I‘l (¢n ) 3 Q nntan d n 180 lensth variable for b -(J - constant in degrees 4. 9~ constant in degress Jame H I 350 i = «C Single subj- hav n3 number of subscript. means: 01, 01, sectional polar moment 37 of inertia of cross section of inertia per unit length of a step #. in.2/in.) step in.) - frequency of vibration in vib./sec. BTTn step (in/) applying to step with mass. applying to step. amplitude of vibration in radians. amplitude of vibration in degrees. C. ect means that the symbol applies to the step Subscript made of two figure 12, 23, between steps whose numbers are included. 02, 03, beginning of step whose number is given. Ll, L, L2, L3, end of the step whose number is given, 3M1 L: 5"” .\\\\\\\\\\\Nf 4—4 04 (3 05 L A °' °i R en Thus 6e. - 9°: 3 Biz = = , ]_ 98x eOLK'H ek.K+| (*“epera ) See an,“ 2 eel“, (e number of end step) A prime mark (') means that the masses are being reduced from the beginnint. (") means masses are being reduced from end. , 3. J01 = J01 ngl = J't + J a 2. ' l n l J , - J _ J 01; " 'r'. k'K+’ J" - J (c - €,¢+I Ce-u - 09 + "e—:,c (end of'c-l) 2 HA _ H J (K-I - 9K " k-m: MJ ‘0 Interpolation formula for natural frequency n. n1 = assumed value of n, 1“ “3 \\\S // n2 = first approximation. n n: : 2nd approximation. ./ 1‘3. n = n2 4- A1 A2 .9 4A1 25 1 bl a 1.1 - n2 1“ A2 : n3 - n2 ‘3 “ "4 n \\ 04' / n1 8 first assumed value “4 W\ 4' 'n2 - first approximation \L M 5 b n3 . 2nd assumed n4 = 2nd approximation n = n 9 ( n - )ZS Al 5 + 2 ‘0 r 2 " n4 *3 Relative vibration twisting moments (l) for steps without mass ‘1 c ( e’ e' ) I"; I - :5“:- °rkfl °.K 37.20 (2) with mass 2 w . I": -(77) Cf ntxsin (th-f“) 100 e 1 Formulas for the reduction metboa of calculation: (1) For reducin: the inertia factors from the beginning toward the end (a) For steps without mass: J I Jék 1- CK . 2 l - £25_P_ CK :: (i) If the first step has no concentrated (ii) If first step has concentrated mass , step tan.Y; = %%8 Jo? I n k Jen = %%9_ it tanngk 4 '1A_) (2) For reducing inertia factors from end toward (a) For steps with mass: (1) If the last step has no Concentrated — O, .- - - lcO J tan ¢¢ n - JcT (an) (ii) If the last.step has a concentrated mass at the end or for any other step I! '0 tan “kn 4 15k): - an Jag 180 QR ”’ 41 0k 77’ wk n (b) For steps witnout mass: « u l J I JeK °" 1 — Jig P3 (3) At natural frequency: H '7' J5. K-\ + ”OK * JI-(‘l’ K = O ’ or £¥K + '%Jfli '*;&J«4 = O which is the same as l u f T. 'u Jok + J0k . O, a“ 4 Je" . O, or Egg, : J" Jéx J51: (a) For step withou me 88: £3 :- 4(la- 4 l ) Jwt Jen 3 (b) For steps with mass Y" - X' = 0 If the first step has mass and no con- centrated mass at tne beginning. {Tl ($1 n) z - an. p2 C if 1 (_J (A) Relative amplitude curve: (1) For steps without mass: .22.: = ia-lasfi 1 - at 2-" A node occurs in the Kth step ' herigmgis -ive. 80K (2) For steps with mass: 9:0(cos (nX-tYk “ k air where GK— :_-ko(cos "if“ 6&K- fipos ( z¢ n. +¥K ) 42 A node occurs in the Kth_step when -- 1 ‘ fl. r) . @1141; 1 I = 9U, L70, ""- f _ K K " ‘fi‘n '4'. + “F-t11qr m "‘D’Xt' (3) f-Lur UlO-A V IL.) .1..- L to v1- 0 TT . . is a e ( e.'K+'- OK ) I 130 K (9., K+I " 80K ) (2) :or steps wits mass: I-I.-(7r)2 C ¢ no4 c+ ‘0 rt 5‘ (D O m a CD [.1- n phase of the wave is ¢. When the wave arrives at I l , ts relative phase will be n ¢’if upon refle f V a c h. ~ .‘ s f i ction oi n + l’ theie is a cnan e in pha e oi 2 fin 4 I. Then when the wave arrives at I1 the relative phase will be np+¢n*l+n¢. If reflection at 11 causes a phase change, 201, the relative phase after one complete transit of the system will be: 2n¢ 4 2 9 + 2 91 n + 1 How if relative phase is equal to zero or some mul- tiple 0+ 3300, a so called 'stationary' or 'standing' wave 'd attern will result. in — Q A. U‘ Unier tasse seen is \<,‘ vibratinr at one of its natural frequencies and we can write at a natural frequency. 2 n 4 2 e- 429 = c or multinle of 3500 —- (A) q + l l - q 4 O .,_‘ n ¢ 4 er 4 l 4 e1 : C or multiple o: 180 -- (z) a " - sin 9. J: - l } Au pk _ {1 + ( ) gb PFC _-; :e Hive a::r_:litu;“_e of Cisc nuiber x Formulas iefinin; ¢, Gland en 4 l involve the inpedances of the system (either mechanical or electrical) and their parameter, frequency the syrin; form can be greatly simplified by introducing the F., wnicb is defined as the ratio of the driving to the frequency of one of the equal discs on cie of s i.e. F - f_ F = frequency ratio fo f = drivinf frequency f0- L-“ E 217 I It can be snown that F2 2R - l . 1 tan - - 4 - en 4 l- . E? l -1 , 2 R n + l 4) cos . l - §_ 31 = 11 a t 'I‘ R _ _ n + l ‘ *n + l 0\ 6w- 4.7 60 Iii‘:::_ ,/ 2° 4 » R.‘ ’00 /, ["11 /’/ \ ’0 ‘ ’ / I la .00, a”; [‘4‘ ~01 *9 f’“ 4.0 Fi~ures show various values of R. t may be seen that for values of R :reater than 1/2 is is ne;ative. Only absolute values are given in the chart. Curves for ¢ all have + ive. slopes while tLe curv The problem of finding the natural frequencies for a given system involves suessing at a value 0: ¢, 91, en , 1’ from curves and checkin: hese values in the equation (1). At the firs natural fr (D t r ‘uency the rijtt hand member of the equation (13) is zero; at the second frequency the right bald member is lCOO; and at the third, 3300 and so on. It is interes inf to note that F cannot ce sreater tba 4, i.e. the hi hes natural frequency of tne system is not more than 2fo. e 48 C"; I? CLUS IO}? The real difficulty in making accurate calculations for the amplitude of vibration at a natural frequency is due to the lack of knowledge and uncertainty of the damping forces. Various empirical formulas have been attacked in another way by the use of a term which K Wilson has called the "equilibrium amplitude". Consider reduction in the arithmetical work of torsional vibration. Calculation for multi-cylinder engines results from the assumption that the crankshaft and masses attacked along ts length are a uniform shaft with mass and elasticity. In a six cylinder engine, a concentrated mass system can be changed to a distributed mass. c J 5 [1C T L W \ 3‘, TA» ‘Flb ”J b {p JQ 4 (r L Q L, l, a. I J (c—‘L—j J i c/AL J <—-)- . , . Wyeth—a . .. 4 'k 9'. ’ 1:7; 25+: l¥+§ J i_L[ / f l 1 I l I I 1 ’1 r I / / l I I I I / I I I K : 34 I 4‘1: Jo.“ lvfiok' eh.) J“ b/Q"*‘ J‘VJL < I 1 I I I I l 1 I f I 1 I 1 l I I . z . L1 = 4 b 4 .§_g + a l 4 i3 2 J“ Jl 3 Ja 4 6J5 4 Jw(d - 0/2) Calculations for torsional vibration frequency of an engine and flywheel, arranged as a uniform shaft with mass and elasticity with a concentrated mass at its end are simple table for T ( ¢ n) . 180 tan ¢ n is used. "’5 n The writer prefers using the inertia factors in ”.in ., but this is not necessary. Th value of C is it is not difficult to recognize. For an example, in a marine Fl installation, consisting o: {‘0 :5 (D n“ine, flywheel, propeller P. U) CD shaftinh, and propeller, it asy to see ha the first natural frequency is determined for the most part by the fly- wheel, prOpeller, and proper shafting as a simple two mass system. It can also be seen that the second natural frequency [—1. (I) J -etermined for the most part by the flywheel and enjine r, .n a uniform heavy shaft with a concentrated mass at the end. {D U) The reduction method of calculation has been develop- - ed to take advantage of this characteristic. The reduction method used on the same equivalent system requires no more work ban the holze method. But when the reduction method is used on an equivalent system with a distributed mass for the moving parts of multi-cylinjer en;ine, considerable saving of calculations results. The reduced inertia factor from the end to the first node is 4 ive. and is greater than he original value. The_reduced inertia factor at a node is in- finite. Just after passing a node, it is minus infinity and increases throu;h zero to _ositive infinity when the next node is reached. The variety of complicated techn ques outlined above for calculating and testing torsional vibration prob- lems indicates a fundamental reason for the many uncertain- ties and disagreements which have clouded the subject. Con— siderable strides have been mafie in test instrumentation, and there are a few well established neasuring techniques. Cal- culation, however, still depends largely on the experience and discretion of the calculator. This is so because the more precise classical methods, by which it would be possible to collect and correlate the masses of data available, are extreme y laborious. Analogies are useful for analySIs in unemplored £13113. 3‘ means of analeies, an unfamiliar system may be compared with one ttat is better known. Although not generally so considered, the electrical circuit is the most common and widely exploited vibrating system. By means of analogies, the knowledge in electrical circuits may be applied to the solution of problems in mechanical .nd acous— ( tical systems. In this procedvre, the mechanical or acous- tical vibrating system is converted into the analogous elec- tric circuit. Any mechanical system can be reduced to an electrical ne work, and the problem may be solved by electri- cal circuit theory. I’ : R ale T = applied torque, o.) = angular velocity, radian/sec. r = mechanical rotational Mechanical rectilineal internal eneray: O i - m g; du h dt 7'6 II S.) O O o H v D m./sec.2 H.» p l m1 driving forcein dyneS. E3 . mass in grams, Mechanical rotational internal energy associated with moment of inertia. I = am. (or-n.)2 \’ M. of I. given by fa s I C" (D DJ (D II angular acc. rad./sec.2 l 5..J (+- H: I torque in dynes.f CIR. U1 [0 Friction, mass, and compliance govern the move- ments of physical bodies in the same manner that resistance, inductance, and capacitance govern the movement oz electri- city. L '45 . I .J... cl W“ 5 electrical rectilineal I . ¢’ 77:1 _ been rotational mechanical One degree of freedom. Principal of the conservation of energy forms one of the basic theorems in most sciences. D] K. . in magnetic field electrical T -1 L12 K.E. “ 5 K.E. stored in mass mechanical rectilineal T, a - l m x2 m - 5ms. 11.11. ‘ "' "' 2 3': : cm./sec. K.E. stored in moment of inertia " 2 T, . l I ¢2 I : gm. (Cm 10R. 2 ' radi./sec. ‘9- ll D'Alemberrt's Principal Equation for rotational system I 61% + r 01¢- + L = new“ cit R J»? 3. force Fké external applied (1) (2) (3) (4) (5) (7) (8) (9) BIBLIOGRAEHY *1 U] \N The Evaluation of the affects of Torsiona Vibra+ions 620.11 Q S 678 d S.A.E. War Engineering Board The elementary mechanical vibration 520.1123 C 561 e A.S.M;E. Journal, 1929 vol. 51, A.P.M. 51-22, By F. P. Porter 621.06 High speed combustion engines By heldt Vibration Analysis 3y Myklestad Internal combustion engine By Vincent Crankshaft design and manufacturing 621.43 3652 c Fundamental of vibration study By Kanley Ch 20.11 H279 f. Torsional vibration By Timoshenko page 239 (10) (11) (12) (13) (14) (15) (16) A.S.M.E. 1931 vol. 53, C.G.P. 53.2, Page 17 521.06 A512 t. A.S.M;3. 1932, A.P.H. 54-24 Vol. 54 AWS.LLE. 1922 vol. 44, page 653 E.gineering 1923, July, 520.5 Engineering 1937 Sept., 620.5, page 275 " ‘O 0 LI.) 3. . 1945 Methods of calculating torsional vibrations u—u‘ By b. P. Porter Dynamical Analogies 3y Olson MICHIGAN STATE UNIVERSITY LIBRARIES 0 3103 7793 31293