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A . ‘- ‘ 4 l I. ‘ L 0-169 Thislstocertlfgthstthe thesis entitled The application of electrical analogues and nodels to the solution of torsional vibrati on problems presented by William R. Tappo has been accepted towards fulfillment of the requirements for _M.§..._degree inJJ._ U ' Major prolessA' “ Dste “a! 21. 1953 ABSTRACT This thesis demonstrates the analogous nature of the integro— differential equations of the mechanical torsional system and the electrical circuit. The technique of establishing the equivalent electrical circuit for various torsional systems is demonstrated by means of numerous examples. A discussion of dimensionless groups leads quite naturally into electrical models. The partial solution of a complex torsional system (an in-line engine) by laborious analy- tical methods, is proceeded by an elaboration on the relative merits of the analytical versus the model type solution. The thesis clearly shows the benefits which result from the flexibility which is in- herent to model type studies. THE APPLICATION OF ELECTRICAL ANALOGUES AND MODELS TO THE SOLUTION OF TORSIONAL VIBRATION PROBEEMS By WILLIAM ROBERT mm A THESIS 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 MASTER OF SCIENCE Department of Electrical Engineering 1953 THESIS ACKNOWEEDGMENT The author wishes to express his sincere thanks to Dr. J. A. Strelzoff for the assistance and super- vision rendered throughout the writing of this thesis. The stimulus of the instruction provided by him on the undergraduate and graduate level in electro-mechanical analogies has been the inspiration for this thesis. 5.: CD ,. V He!" pa GE US II. III. IV. VI. VII. TABLE OF CONTENTS Page No. Introduction 1 Equations for Mechanical Torsional Systems 3 A. Single Degree of Freedom B. Two Degrees of Freedom C. General Equation for n Degrees of Freedom Equations for Electrical Circuits 6 A. Single Mesh Circuit B. Two Mesh Circuit C. General Equation for n Meshes Comparison of Mechanical and Electrical Equations 9 A. Method of Constructing Equivalent Electric Circuit 1. Illustrative Examples 2. Table of Analogous Quantities Models of Electrical Systems 15 A. Method of Changing Variables B. Method of Dimensional Analysis Equivalent Oscillating Systems 19 A. Equivalent Inertias B. Equivalent Elasticities Solutions for Natural Frequencies of a Hulti Cylinder Engine 24 A. Statement of Problem B. Illustrative Examples of Various Conventional Solutions 1. Holzer's Method of Successive Approximation on Mechanical and Electrical Basis VIII. IX. Page No. Application of Electrical Models to Crankshaft Studies 31 A. Description of a Typical Model B. Advantages of Model Type Solution Conclusions 33 Bibliography 34 I. INTRODUCTION Recent years have introduced many innovations into the field of en- gineering analysis. Foremost among these has been the extensive utili- zation of computers to alleviate the burdensome detail of problem solup tion and to expedite the attainment of those solutions. Computers can be divided into two classifications. the digital and the analogue. The analogue computers are the oldest. The famed mechani- cal brain at M.I.T. which had its introduction over two decades ago. is such a machine. Upon perusal of the engineering brochures and literature of today, it is hard to believe that less than a decade ago the utility of such computers was questioned. Analogue and digital computers appear in great profusion upon the pages of these publications. {As recently as l9h3, however. C. Concordia had this to say about electrical analogues and analysers. In most cases the only practical use made of it has been to intro- duce certain electrical concepts, impedance and the like as an.aid to the thinking of mechanical engineers. {Although we should not disparage the usefulness of these concepts. it is deplorable that the much more powerful use of the analogy. that is. the use of the actual electrical circuit elements connected to simulate the me- chanical circuit. and on which measurements can be directly made without laborious calculation, has not become more fashionable. Some of the complicated calculations that machines can do are even beyond human capabilities. The I.B.M. Corporations' big calculator. for instance, has com- pleted in 103 hours a Job relating to uranium fission for Prince- ton University. The same Job would have taken a flesh and blood Operator more than one hundred years. The time could not have been shortened by putting one hundred operators to work. because each part of the problem had to be done in sequence. It is intended that the pages of this thesis will convey to the reader a comprehensive picture of this rather startling revolution. In order to accomplish this. a specific area of engineering interest will be examined. This area will be that of torsional vibrations. It will be examined from several aspects. The mechanical equations and their analogous electrical equations will be presented with numer- ous examples. Following a discussion of electrical models, the solu. tion for the vibration modes and natural frequencies of an in-line en~ gins drive by laborious classical methods will be illustrated. No bet- ter case for analogue analyzers could possibly be presented in the o- pinion of this writer. II. EQUATIONS FOR MECHANICAL TORSIONAL SYSTEMS A. Single Degree of Freedom In the usual analysis of torsional vibration problems, the mechani- cal system is represented by a series of rotating inertias interconnect- ed by massless. flexible couplings.' To complete this system. which is known as an equivalent mechanical circuit, damping may be added at ap- prepriate Junctions. This is admittedly an idealization of the actual system. .A completely inelastic mass, a massless coupling, or a totally linear damping means is never physically realized. However. by applica- tion of the extensive knowledge we have of such systems. we can establish an equivalent circuit which will yield very accurate engineering results. “hen the geometrical position of the system Just defined can be 61b pressed at any instant by a single coordinate or number it is said to possess a single degree of freedom. The simplest torsional vibration system is that of a single disk on a shaft, as shown in Figure l. The equation for the motion of this system is obtained from Newton's law for angular acceleration. 1 ////// ////L Jfifffi‘fl moment of inertia in pound inches sec. angular displacement in radians. Figure 1. time in seconds. NNwN ll : torsional spring constant in inch pound per radian. By the addition of a disturbing moment and a damping moment the com- plete equation! is obtained. .0 7. 6L) : disturbing frequency in radians per second. torsional damping constant in lb. in. sec. per radian. amplitude of disturbing moment in pound inches. The disturbing moment is assumed to be sinusoidal in this example. If the moment is not sinusoidal. but varies in an irregular periodic manner. it may be resolved by means of fourier analysis into a summation of sins and cosine terms. B. Two Degrees of Freedom If the system under considerations requires two coordinates to exp press its angular displacement. the system is said to possess two degrees of freedom. Two such systems are illustrated in Figure 2 and Figure 3. LLLLL 4 i. '1 :th \ s F?” I: i 1; ~ I. ‘12. Figure 2/ Figure 3 F1 1:227 up 22‘ +1.9, ”mg a.) a gure 2 15%*P.f¥§* fi/a.—a/=a 1' 1’6, 1€_ la __ 3 11W83{J:J z“; "' 5/221 ’45! ’49, 34/ 0 Ild‘ * D/jé':“;€/ *f,/€g-€,/= 0 C. General Equation for n Degrees of Freedom When a system is possessed of n degrees of freedom, a series of n equations must be formulated to define its motion. This is shown in the equations of Figure 4 and Figure 5. 11.} [115~/ .sz/ ’ 45:1 II (£2 ____412t1__1. 3 tfl—ZZCV illjza' .Zfibv/ III/I . A/IFV/ /7///7/ £6”? flalrfi/flrarJ f/évfl/fiy'drfl ’0 ‘ Figure.h ‘PTJZ;»/ hfi-ZZ' l—I.Z;w/ J—415”1 Ada! 4£xa4___y if} —CE—L_ _ LPG—— JZZ~1 JZZV .Zgbv/ MilfaflaCJi’L/fzfl/éx-aL/f/éa'6r,/7‘/évfl[’jv -6MI/ ‘ 9 Figure 5 III. EQUATIONS FOR ELECTRICAL CIRCUITS A. Single Mesh Circuit In the usual analysis of alternating current circuits. the elec- trical system is represented by an arrangement of lumped electrical parameters interconnected by impedanceless lines. This is a very ac- curate idealization. with properly chosen circuit elements. within frequency limitations. Actually dissipationless reactors. linear re- sistors. and the complete absence of distributed capacitances are never fully achieved. The degree of an electrical system is determined by the number of independent equations needed to completely determine its behavior. By application of Kirchhoff's law. the equations for any circuit of this type may be obtained. Figure 6 and Figure 7 are illustrative of single mesh electrical circuits. inductance in henries capacitance in farads resistance in ohms amplitude of driving voltage change in coulombs current in amperes eweibmmub circular frequency in radians per second B. Two Mesh Circuit The analysis of a two mesh circuit. Figure 8. involves the assump- tion of mesh currents. In general, In is the current flowing in the Nth mesh. /2? 4M. / J 1.7;: #7317." r59, rg/qpal =0 1.9% as]? *5/4-4)=a ll lg Figure 8 C. General Equation for n Meshes In the case of the n mesh networks of Figure 9 and Figure 10. a general equation common to all meshes may be written. an,” . as a’ . aim-em) __L_. fa, fay-001v), 0 1.0.3 23/0/— 0.3,) + 53/42, — a.) Rd-I Ru Run I fri'fil/QV”0~:I) fat/QAt—gyu): 0 IC‘V" ICU ICNH L"... La: Figure 10 IV. COMPARISON OF MECHANICAL.AND ELECTRICAL EQUATIONS A. Method of Constructing Equivalent Electrical Circuit The examples of the preceding two sections have shown a correspon- dence or analogy to exist between the respective differential equations of the two. It is now possible to outline a simple criterion for deter- mining the form of an electrical circuit which is analogous to a tor- sional counterpart. The first step in this procedure is to determine the number of points in the torsional system which have independent angular displace- ments. These points are at Junctions between springs. between dampers. between springs and dampers, and always at points of inertia. The most general point for independent displacement is the last. Springs or damp- ers in series can always be resolved into equivalent systems. For each point in the torsional problem. a closed electrical mesh is drawn. In place of torques acting in parallel. voltages are drawn acting in series. Combination of these independent meshes, with due regard to common volt- ages or parameters. completes the analogy. Although the circuit Just derived is a solution for the analogy that was sought. it is not the only solution. It is common.knowledge that every circuit has an equivalent dual. Dual networks have differenp tial equations of identical form. but with mutual interchange of para- meters. The dual quantities are voltage and current. inductance and ca- pacity. and resistance and conductance. The following examples and table -10- will serve to illustrate the various relations thus far discussed. Fast: 2 5: _ fig 13 1:. I <9 I: e L 1 ® 1; +45, fang-am +14 ,/a-a,) r! /o,— 9:)0 2mm L? i: if LII. L' ‘1 13 I 1,52?*E,£0, *5/01'02/‘0 £32. .4. .u +474. ski/o. al-o \ Egg'i-Zf/A/L‘fzfi/fl/f-filflf-o Q It &+£://lg/£'/.’,/Z‘/ Jug/[4]? -///z7='0 I .— 15 if} .« Asa-a.) - a z, 5% + gin-a): o a}; ”Welt/515% a f; Iw—‘w‘b E) — m t) g) I. It '1 1's 1?. 7s 73 h' h 0 6' QC 6) 6. G. 5. I l 3 ct C; C D’ :p. D: ‘ ‘ I V T T :l- . 11o... 1,-ng ffljéffi’w %ffiffié/p-flz/= a CJt +6,l,+z:(/Ili’ "bl-:10 e , g, 41%;: +2,” 3.1/6. a} 1.5% h? a}? mfg-0,) C.” +55,;,1,/+2j&/1-/4/1) *1/91’53/‘0 ft" 1/0.: “(VJ/=0 7‘2:— [4]! f-[J/t/‘O ’13.; Ljigffl 19: 7?}Iéfl’) ‘QJflLJJtt f8f?»(i(0.~0.1£31 7f @4f1,//(,/[“//;y :0 =0 8 ' 1,39, ”1‘ [6/9, “92/ = 0 1:9,”! 1/9, -€,/ IA/Q. '62,)fl 5/6/52, 54/ - 0 1343 43/49., —5>./ -- o 1&9." * lit/5r ‘31/= 0 I, 1‘; I; II ,5] ca 7 \ A'z G I 1’ 63> L J,‘9,”+ 6/6, -9,,/ =0 L9,”! ,éIa,—a/ f/flz 9:” f/fllfl(fl7€,'€3)=0 L93” lug/6., ~m9,/= 0 [leflf é/QI‘OL) 3’ 0 44." , fife-42,) ”fife-0,) *6/41'04/ ‘ 0 140.!” f (xi/4.1-0:): 0 {7‘91”} é/04'4£)= 0 L1 ‘3 6922 66> m 1'10!” f' ELL/47’ "0:, ‘0 110‘” + 2? fol-o.) m ‘2, 47,” m + (, mag-0,)so , 44”»‘(f/Q, mm) = o T“ T“ L. T “I T "99 4.2,, ‘l’ 2,4/1,."1:)=fl c, 1,’ * zf/K—If? .«sz-zfl *Zf/‘GJ'I: ‘ 0 41,’ + zf/L'i A" = 0 c, 1/ ¥ 23/1/1157 = 0 I -1 - (,1.’+ 27/1, —1, /= a (.2.’ + 25141157 mfg/X .1- ..1 -I {’MLz/Ma’g ’13/‘0 5,, 1, ’ + zf/4”—mx,"/= 5 [MM 0r 1,19, ”f 2/9, 19.7154/9, «95M 229;"! 5, [639,7 +2/&'-a.;) + A fee, -a/»A/9, ~53) = o 2; 5,”, 269,147 “6/93'92)‘0 13559169, 5229/ {- Ag/Q‘QJ) ‘0 ’69: f 4/9; “4/5 0 4/ a; CD 7"; 4: 44"» 3/4. 10;) #575, -4>.)= 0 4,0,”!- R/Q"0. 7 *8/67; ‘0)? « 5.4/45 5/ + 5/5: 5,) - 5 1,52,”; 2’. @5’-47.’/fff/4,-@J = a 150/? 780/5 2647, * 54/4'91) ‘0 2:0: 1" Elf/47:17:} ‘0 41,2» 6/£-A/*Zf/l,'34"/= 5 44' + 6/12: 4.) + 5,0, 4,) f 26/5‘15‘7*zf&z.'114‘7= a a 15+ 4/5 4.1 +2f/4'15'7so / . i 41, 95,1, 541,1 +Zf/IZ'C-4'7? 0 fl - '__/, ;‘-, (14”‘15/414/‘0 - 13 - The identical nature of the solution for the electrical dual is not readily apparent unless a change of differentials is made. If. in the electrical solution for Figure 11, current is substituted for the time derivative of change, the following expressions are obtained. afln 7 ”fl/WW [/z/é- /z,/z]= 5 4% 7‘ a ”(DA/2‘ -/1,'/f] +Z:'[/1;/f—/z;/z]= 0 43% 54% fl- /z./z’]=5 Comparison of the two electrical solutions will establish the identity of the systems. Several features of interesttippear in the preceding six examples. Figures 12 and 15 illustrate damping prOportional to angular velocity and preportional to relative angular velocity respectively. Figure 13 illustrates a branched mechanical circuit. and Figure l“ a geared sys- tem. Figure 16 illustrates a spring to spring Junction. It is apparent from consideration of the electrical circuits that the two springs may be replaced by an equivalent spring. 0 AA 44% == Aazflf; The following table will serve to amalgamate the analogous parameters of this section. -m- MECHANICAL ELECTRICAL DUAL Moment of Inertia Inductance Capacitance .2" A C J Torsional Spring Constant Reciprocal Capacitance Reciprocal Industance / J I J=E’ 7‘4 Damping Constant Resistance Conductance I 27 7? 6-5? Angular Displacement Charge 6 4 Angular Velocity Current Voltage 69.1.3 [£031 I It ‘ Torque Voltage Current 7‘ .zs‘ 2' V. MODELS OF ELECTRICAL SYSTEMS A. Method of Changing Variables Now that the analogous electrical circuits have been obtained, we are confronted with the problem of constructing a feasible electrical model from which the desired data might be obtained. The only restric- tions on the equations thus far derived is that a consistant set of units or dimensions be used. It is evident then, that one possibility would be a model with one to one correspondence between mechanical and electrical quantities. From the viewpoint of simplicity this would have great de- sirability. But. practicality usually decrees another solution be found. One method for arriving at a practical analogous circuit is via the me- thod of changing variables. This method is really identical with the method of dimensional analysis, but it illustrates the process involved a little better. The object of this method is to put the differential equations into dimensionless form. This is accomplished mathematically by changing to dimensionless variables. The equation for a single mesh electrical circuit will serve to illustrate this method. if? if I , 4 5".“ MFA {552 =£M a)! The variable can be changed by choosing. .é’ /b=/m , 7'=w,l" In these expressions m =4? , where 9' is a unit charge, and 448.660, - 16 - wherewis the impressed frequency. Then JQ= M7660 , 5/! Therefore g? = ”7750,24; Differentiating this expression again with respect to Z" . z z 1—? g ma), 69 {iii—I):- mw, 2149; it J .17" These derivatives can now be substituted back into the original equa- tion. 2. J . /Zm1w, 62;; fW/yfiw1,&f m/b =EMZ§ET Dividing through by [ma/,2 o 434:5- L5 fizz-:5 rind“ MW Since/D and Tare dimensionless, their derivatives are dimensionless. This requires that each bracketed term in the above equation be dimension- less. Designating the parameters of the model by the subscriptm . it is apparent that the following relationships must be observed for. . fen-Wt [m1 3161-3—12: ulna“) 73"?" f2! "" ,é’ /' orm=rp,z,.=€f,€m=oé where C is an arbitrary constant. It should be noticed that this is a par- ticular case. A feature of this model is a one to one correspondence be- tween electrical and mechanical frequency. Many other possibilities are - 17 - practicable by going to higher frequencies, or by dividing by a dif- ferent quantity in expression 1. above. B. Method of Dimensional Analysis A more common method for arriving at the same results obtained above is that of dimensional analysis. This method is based on the Buckingham Theorem which states that a problem formulated in terms of n quantities. themselves expressable in terms of m dimensions. will have n - m dimen- sionless groups that can be formed by combination of the quantities. To construct a model it is necessary that the dimensionless constants of the model equal the respective constants of the prototype. In order to demonstrate the application of this theorem, we will ap- ply it to the system Just discussed. The n quantities are Q, R, C, E, L, and a) . The dimensional system used will be the E, I. T system. In this 3’3“” 4 '17, ’P'II", 1= [1"7', Caz/'5", 50:7" /7, = A‘Q‘w' 6' - /zrr/'//r/“/r‘7°jr£" a c _ _’ _ /7. = Z 4’60 2? = {II’7/’/17/‘/7 ‘51 ’ fl; * 427%.; ‘5 = {[17/717/77‘75 If these j7’groups are to be dimensionless, the sum of the exponents for each individual quantity must be equal to zero. From/7;. aha/’A'd/t'ri The first dimensionless function is, 17' =‘ [(60 Z I -18.. The other two functions obtained by the same procedure are, P —— fl'z‘z 2F ”3 = [4603 These are identical with the dimensionless groups obtained by the me- thod of changing variables. VI. EQUIVALENT OSCILLATING SYSTEMS The torsional systems thus far discussed have been characterized by several idealizations. In brief, the systems have been shown as consist- ing of concentrated inertial discs in interconnected by weightless shaft- - i C ing. Actual. practical systems are not so constructed. In order to fa- cilitate calculation. however, the complex, actual systems must be replaced by simpler, idealised, equivalent systems. Perhaps the best case for 11- A—u. - lustrating this procedure is that of the multi cylinder, reciprocating. as: internal combustion engine. The problem of establishing an equivalent system in any case can be divided into two parts: .A. Determination of equivalent inertias, and 3. Determination of equivalent elasticities. A. Equivalent Inertias Definition: the inertia of a body with respect to an axis is equal to the summation of the product of the incremental masses by the square of their distance from the axis. ,2; = //‘//m A very useful theorem in the calculation of moments of inertias of bodies is the parallel axis theorem. This theorem states. the inertia of a body with respect to any axis is equal to the inertia of the body with respect to a parallel axis through the center of gravity of the body plus the mass of the body times the square of distance between the two axes. These two theorems, individually'and in combination, make the calculation of most inertias a relatively simple matter. Almost any - 20 - conceivable engineering shape can be expressed as a combination of these elemental forms. Bodies whose contours are such as to defy analytic ex- pression can be solved graphically. or, if the part is available, by ex- perimental means. These methods may be applied to the calculation of the inertias of any of the rotating components of the crankshaft system. The only unique feature exhibited by such a system is the existance of com- ponents which are simultaneously subjected to rotation and reciprocation. The connecting rod is illustrative of such a component. This problem is usually solved by considering three-fifths to two-thirds of the total weight of the connecting rod, concentrated at crankpin radius. B. Equivalent Elasticities The determination of the equivalent elasticities of a crankshaft system does not lend itself so readily to analytical approach. A typical configuration is illustrated in the following figure. ““1i“‘*r“‘*‘ , _ f ____..-1__-_V_____. L1, L29. 7 L._____T _____ ._._ ..__..___.= __ =...‘._ “I" — + " ..-___J.___ _______. T_____.___.._1L__._ ‘ L2 LD, *— 2—4 kaph—B—r-d TbLfl—g—D— A Z ’— -21- length of Journal gk ll \ I " diameter of Journal = length of crankpin distance of center of gravity of crankpin from axis of rotation 8 depth of crankweb {Saba u = width of crankweb fl; : diameter of crankpin Jr : diameter of Journal bore I; = diameter of crankpin bore The total elasticity of this typical internal section is equal to the reciprocal sum of the individual elasticities. In practice, each contri- buting component is expressed as a length of equivalent shafting of arbi- trary diameter. The calculation of the total equivalent length 1, . of diameter D of the same torsional rigidity as the entire crankshaft sec- tion may be carried out as follows: 6 , 5 .1, .= a 5 - 22 ' = $22132: seiner?" shafting. .2} = polar inertia of the area in torsion = integral of the product of the incremental areas by the square of the distance from the axis of re- tation. 6 : Modulus of rigidity in lbs. per sq. in. /7 .v v Torsional ri 1am f = "" . . . - I s g y 0 a unit 6 ‘32 6 [p j / length of the crankshaft Journals. .4 ,, 4 C, = 32 '6 ' (fl, '1; / = Torsional rigidity of a unit length of the crankpin. Tw’f (J 8 [I = ——7-2-—- = Flexural rigidity of one crankweb. E 2 modulus of elasticity -22.. l. Journals: Let Z, = length of equivalent shaft having the same torsional rigidity as each Journal. .44 - JZ ,__ .415. .. ” c “ c, ’ Z’ c, " {,V- ,' 2. Crankpins: r Similarly, 4 = ,r. , j 3. Crankwebs: [3 : length of equivalent shaft corresponding to two crankwebs. ‘< u radius of curvature of web. a : difference in angular inclination of successive Journals due to flexive of the crankweb. 1.2,, .43 I Y , )’= M .5 = if , «9 = r (3, rw’ue' (one crankweb) ”is 29’ ”7-? c 7-W"-£ 24/? , x76-D"_ 2.355 27-6-33" .4. = Fit/"f ~32- r-w’-£ -23.. Assuming E 30,000,000 lbs. per sq. in. G) H 3, steel 12,000,000 lbs. per sq. in. [3 = .942 172' 7-. W3 The total equivalent length of one section then, [I = [’3’12 f [3 'I 4 — _£.2__ .,¢_.22__. 4. -94LP-QI - q 4 a]! 7114/3 D. -J% 29. . The derived expression above has served as a basis for several em- pirical formula capable of yielding greater correspondence with experi- mental values. For purposes of comparison, the following formula was pro- posed by B. C. Carter, "An Empirical Formula for Crankshaft Stiffness in Torsion“, Engineering, 13th July, 1928, p. 36. - ' M .758 . [e ‘ D [mt-,9," + W + %] W. Ker Wilson, in volume I of |'Practical Solution of Torsional Vibration Problems" offers the following expression: = " ”t“?!— M 1— 2(2ng .1: 1:7 .ZZV‘AK" f .ZZ'-d%" 1L '7‘IV3 ;] Although this author hasn't any experimental data wherewith an intelligent choice might be made, the latter expression has more appeal. It had a la- ter origin, and, in addition, possesses a continuity of factors lacking in the first formuha. VII. SOLUTIONS FOR NATURAL FREQUENCIES OF A MULTI CYLINDER ENGINE A. Statement of Problem The exact determination of the natural frequencies of an internal combustion engine, with all its associated cranks, rods, pistons, fly- wheel, and driven equipment, is a hopelessly complicated task. In the engineering analysis of the problem, many approximations and idealize- tions are assumed. Crankthrows have to be replaced by equivalent straight shafting. Members, such as rods, which are subjected to reciprocation in addition to rotation, must be resolved into equivalent rotating and reci- procating masses. The technique for applying Judicious Judgement in these approximations can be obtained from any good vibrations textbook. A particularly good source which has recently. l9#8, been brought up to date is Hilson, I'Practical Solution of Torsional Vibration Problems", two volumes. In the problem which is about to be considered. it will be assumed that all of this preliminary work has been completed. The equi- valent mechanical system is as shown in Figure 17. I, 7 I. z I, .2; 2; J} I. "T 1“ H W H F 19 #2 A5 I? 4*? inf! 4*? L. L. L- J LJ J a. Figure 17 - 25 - 1 é, = If; '1: " Ifr '1‘; '4‘, -"-' 675 x 106 in. lbs./radian I, .- /J.! 8/0‘ in. lbs./radian I, = ‘7’ g], g I" '1’ ,J‘ = 2.560 lbs in. 860.2 = 75,000 1b. in. sec.2 1? 4; 2 a 2#,000 lb. in. sec. While this problem may not represent the optimum in complicacy for torsional systems, it is typical of the most complex systems treated in advanced vibration textbooks. It should be noticed that damping is not considered in this simplified system. Using the methods previously de- ve10ped, the analogous electrical circuit is shown in Figure 18. To TC. "TC. Figure 18 The reader can now appreciate the considerable simplification af- fected by disregarding the systems damping. This is representative of the reactive case of cascaded four terminal pairsz. The determination of the network zeros may be accomplished by setting the determinant of the driving point impedance equal to zero. As long as the degree of the resultant frequency equation is less than four, the system can be solved for its natural frequencies rather easily. Algebraic equations of high- er degree, however, are solved for their roots by approximation methods. 1The data for this problem is taken from J. P. Den Hartog, "Mechanical Vibrations“, 3rd Edition, 1947, p.237. 2Guillemin, "Communications Networks", Volume II. - 26 - A brief explanation of the reasoning employed in the above discus- sion might be appropriate at this point. Since we have found the natu- ral frequencies for which the electrical driving point impedance is zero, we have by analogy the frequencies at which the mechanical admittance is infinite. Any excitation of the proper frequency applied anywhere in the mechanical system will, therefore, produce vibrations of infinite ampli- tude. B. Illustrative Examples of Various Conventional Solutions 1. Holzer'g method of succegpive approximation on mechanical and elec- trical basis. Several computational solutions for the system of Figure 17 will now be illustrated. The eighth order determinant will first be written, and then the solution for several frequencies by the common Holzer tabulation method will be shown. .Zkot—Ie £3 (9 6? £7 6’ C7 67 J'au‘—/fi k, 1,5, It. 0 5 5 5 5 1359/5 0 k, -15 k, 5 0 0 0 I ‘vi’ «0 (9 A5 vi;5,1 'ki 6’ 67 (9 zw‘aé 5 5 5 .6. ”,5 " lo 5 5 1‘ hi 5 5 5 5 If. ‘63,, I k. 5 15a5‘-Ah 5 5 5 5 5 It. _ 5. k, 1 5 5 5 5 5 5 45, z... 1’ - 27 - This determinant will yield an eighth degree equation in ca) . It is possible to solve this equation graphically, but the process is rather laborious. The more common method for solution is the method of successive approximations. In this process a rough guess at the solution is first made by intuitive reasoning. To find the lowest natu- ral frequency, the inertia of all six cylinders is combined with the in- ertia of the flywheel to yield a two mass system. .22 Ll —-I .2: '.fo45 f I 1% J; “-5 ’1; ’7' ‘ u_J The determinant of this system is easily solved for the natural fre- quencies. (xii); ‘147 ‘5 ‘é, 603]; “4‘7 =x£9 50, ’3 @% Substituting in the numerical data, / ' )= 25.2 rad./sec. (2% ‘ .ZZ5003f24¢m00 The determination of the lowest natural frequency can usually be made quite accurately by such crude procedure. The remaining six non» zero values are, unfortunately, less accurately defined. It is assumed that these frequencies are far removed from the resonance of the drive shaft. It follows that vibrations in the shaft can have little amplitude. The remaining modes therefore, will be in the engine itself. The ap- proximation formulae given by Wilson for these remaining frequencies is as follows. 5d"- .5183;- = / 3?: w5= 2.571; = / .432... 15. ._ 0E u L. K. o. / fly a), = .3.77fl1; == 124 rad./sec. 364 rad./sec. 595 rad./seC. 743 rad./sec. 910 rad./sec. 998 rad./sec. These rough values serve as a basis for the following method of calculation due to Holser. The explanation will be presented on an electrical basis rather than the more common mechanical. The reader may directly transcribe the analogous quantities as he reads. The nu- merical values of the inertias are placed in an inductance column and the flexibilities in a capacity column. A frequency is then estimated and the inductive and capacitive reactances calculated. A current of 1.0 amperes is arbitrarily assumed in the first inductance. The poten- tial drop is calculated and then the current in the first capacity. The current through the second inductance is equal to the original current minus the current through the capacity. The voltage across the second ‘34:; 14‘ is": H." .3 fmxtfi'fli 19‘??th I - 29 - capacity is equal to the original voltage minus the voltage drop in the second inductance. can be calculated. From this the current through the second capacity This process is continued until the voltage across the last capacity and the current through the last inductance are ob- tained. The difference between the last condenser voltage and the last inductance voltage is the voltage which must be supplied to the circuit at this point to maintain the specified currents through the various ele- ments. to maintain finite occillations. this procedure. For circuit resonance. no external voltage supply is necessary The following table will illustrate 9. 1x! col. l. [=- IX. 1.- 6' 60C 1. 3 7!: 2160 5.3] no" /.555 1.51m)" Lela/0' AVE a/o” 39.4315" 2.7-x5 ‘3 2.515 I-5/z/5 ' . 9973 1.79~/5' A16 no” 545 V5" 39,4. 15" J358/4.3 2555 (.a/a/o" .9725 1.75am" 25.35515” Aves/5" gym/5" sass/5 " 2550 Len/5' .9537 1.7mm ‘V 27.5mm” 1.4mm” 39.9w" mun/5" 2550 Len/o " .7732 5532/5 1’ Java/5" Mex/5" ”qua" /3.3 V5" 2.555 1.0/HO" .760 1.50/5" man/a" M’a W" 39421:" Ana/5" 7:; 000 /-"~5"/0‘ .997 1.5a a/o‘ nan/0‘ 25/ ~10" xs7~/5" 5:55 215550 1.38 no 5 -1556 -:.27u/5‘ , 5/ a/o‘Jr fat. ”I? Z 55"! .5 15564 [J'w‘fl ,4 1512558 I 2560 £5”: 1/5‘ /.55 o Lass/0‘ 182 no ‘ 57.5. /5 ‘ . 553 z 2555 max/0‘ .777 [.8/ -/5‘ 35.7w ‘ ”5.55 ‘ .555 3 2.555 Men/5 ‘ .972 555 n/o‘ SVJUO‘ ”Jr/a ‘ .555 9’ 2575 5525/5 ‘ .756’ 577 a/o‘ 722 x/5 ‘ 175w ‘ .51/ .5 2.575 near/o ‘ .773 /.77 am ‘ 9. 97a/5 ‘ nix/5 ‘ .513 5 2155 [52 x/o ‘ . 755 /. 751/5 ‘ I5. 775/5 ‘ ‘ 7.53/5 ‘ .0/6 7 7.5; 555 15.3 :/5 ‘ .MV 15.32 ”0‘ 5/. / -/5‘ /3J*/0‘ 5.52 5 25, 555 /2/~/5 ‘ -356 ’51}! uo‘ -/:/5‘ =1 77.1, - 3o - The preceding two tables will serve to further illustrate the iden- tical nature of the mechanical and electrical solutions. The second ta- ble was taken directly from Den Hartog, "Mechanical Vibrations". 3rd Edi- tion, p. 240. The solution for the remaining frequencies will not be at- tempted since the process is identical. It should be mentioned, however, that the remaining frequencies are not so easily isolated. In the re- ference mentioned above, Den Hartog took three trials to obtain the lowe est natural frequency, and four trials to obtain the second. The remain- ing frequencies have a tendency to cluster in the vicinity of a cut-off frequency. This accentuates the steepness of the reactance curves and necessitates many more trials to isolate the solution. To summarize this section. there are no short cuts to the desired solution. The methods of this section with slight innovations seem to be in universal use. A thorough search of the literature on the subject has revealed only one mathematical approach which gives promise of relieving some of the tedi- um connected with the solution. This is the method of Biot. the appli- cation of the calculus of finite differences. This is in essence a graphi- cal solution, but Biot, by application of various mathematical artifices, has contrived a solution function which is a smooth curve as opposed to the highly oscillatory frequency equation. The intersection of this curve with multiple values of /7 gives the desired frequencies. VIII. APPLICATION OF ELECTRICAL MODELS TO CRANKSHAFT STUDIES A. Description of a Typical Model The most effective means for investigating the torsional charac- [ teristics of crankshafts is through utilization of analogous electrical -......j._ap.': ~ 15 l c.‘ models. One midwestern manufacturer has obtained very good results with a model constructed on the electrical dual or node basis. Models of this type require a constant current suPPIY. Since the impedance of the model as constructed never exceeded 20.000 ohms, a limiting resistance of two megohms in series with the circuit was utilized to give the neces- sary regulation. Decade condensers representing the crankshaft inertias provided minute variation of these. The shaft elasticity had for its analogous electrical counterpart an iron core inductance, wound with low resistance wire, and provided with an adjustable air gap for varying coil inductance over a small range. The time in the model was chosen to have the same magnitude as in the mechanical system. For the impor- vibration modes of the particular engine being tested, it was found that the model had a frequency accuracy of approximately'lfi. and that mean deviation of measured amplitude values from the calculated values was less than 2% while the maximum deviation was about 5%. B. Advantages of Model Type Solution The results obtained from preperly designed models are, as demon— strated, very good. These results alone are not Justification for their universal adoption. Electrical models have numerous advantages which - 32 - must be weighed against their initial cost and frequency of utilization in determining their worth as an investment. The big advantage of the model is its flexibility. The mathematical solution for the important modes of an in-line engine requires several days work. If the effect of altering an inertia or rigidity is to be determined, the calculations must be repeated for each new value. Contrast this to the following ex- cerpt. The effect of changing the rigidity or inertia of various elements can be easily and quickly investigated. By adding a damping factor to the system the absolute amplitudes of vibrations can be deter- mined with the model. Provisions are made to study the effect of various firing orders on the vibration characteristics. The torque in any part of the crankshaft or to the propeller can be found. and the effectiveness of different pendulum damper tunings as well as the pendulum bob amplitudes can be determined. This is flexibility. versatility; in fact. this is versatility in a large economy-sized package about the size of a large radio chassis. WI? ' IX. CONCLUSIONS This paper has taken a specific area of engineering interest. namely that of torsional vibrations, and has attempted to demonstrate the existance of an electrical circuit, completely analogous in be- havior. for every torsion system. The restrictions which must be ob- served in constructing an electrical model have been pointed out. The derivation of the dimensionless groups, which must be held invariant in modeling, was shown by two different methods. The conventional solutia1 for the vibration modes of a typical in-line engine installation was presented in part. The use of the Holzer tabulation method on both the electrical and mechanical basis was shown. These tables served to fur— ther emphasize the analogous nature of the two systems. In its entirety, the paper illustrates a modern trend in engineer- ing calculation. The utilization of electrical analogue and digital cun- puters to facilitate engineering calculations is currently so widespread that their worth is no longer in question. The only uncertainty is where the process might stop. The companies engaged in their design and manufacture are rapidly expanding both in number and size. I think we might safely say that, when various models are placed on the market in kit form, the new era will have arrived. TL' 1. 2. 3. 9. 10. 11. 12. 13. 14. X. BIBLIOGRAPHY Den Hartog, Mechanica Vibrati ns, McGrawaHill, 3rd Edition, 1947. Timoshenko, Vibration Problemp_;p_Engineeripg, D. Van Nostrand, 1929. Wilson, Practical Solution 2; Torsional Vibration Problemp, John Wiley and Sons, Vol. I & II. l9h8. C. R. Freberg and E. N. Kemler, Elements 2; Mechanical Vibration, John Wiley and Sons, 2nd Edition, 1949. C. R. Freberg, The Solution pf,Vibration Problems py Use 2: Electrical Models, Purdue University Engineering Bulletin, Research Series No. 92, January, 19%, i- Louis A. Pipes, Electrical Circuit Analysis 2; Tortional Oscillatiaig, :_ Journal of Applied Physics, Vol. l#, P. 352-363. ** Louis A. Pipes, The Matrix Theory_p§_Tor§ipnal Oscillations, Journal of Applied Physics, Vol. 13. p. 434-44“. Maurice A. Biot, Equations pg,Finite Differences Applied ppuTorsiona; Oscillations pf Crankshafts. Journal of Applied Physics, Vol. 11, P. 530-537e Eugh.B. Stewart, Electrical Model for Investigation p§_Crankshaft Torsional Vibrations ip Lprline Engingp, S.A.E. Journal, Trans- actions, Vol. 54, P. 238-24H. C. Concordia, Network Analyzer Deterpinatioqp pf Torsional Oscillatign ‘9; Ship Drives. Marine Engineering, Vol. #8, P. 158-59. Berkeley, E. 0., Giant Brains p§_Maghines that Think, John Wiley and Sons, 1949. Guillemin, E. A., Communication Networks. John Wiley and Sons, Vol. I & II. Maa, D. Y.. A_General Reactance Theorem.for Electrical, Mechgpical, and Acoustical Systems, Procedures, IRE, Vol. 31, P. 365-65. Sokolnikoff, I. 5., Mathematical Theory pf_Elasticity, McGranHill, lst. Edition, 1946. \ w w} ' Room USE out .. , \. . I . . r' \| a ' t 1 I A . l . O ‘ A ‘ . . . L. T u . ' I s . I . V ‘ I l . a I - I V . ‘ I I v \ . ' I . w .. W I ‘ - , . I 1' . x l . l | ' I l ‘4 l L p 1‘ - v n '. 1 ' z . I l | . . i , I x . t - 1 , ’ e" . \ . I , I . I . Kw. I 3. ‘ - " — . f f I A l MIC IHIGAN STATE UNIVERSITY LIBRAR Ill llll II III II! I Iiill