’3. x. mm 5..» O M a... E a. .u... Pia . J tum 0 .. .0... I...“ a 3. . fl. M G 5 M We Le t...“ M. a...» w .. .1.- nm 1 , A... An. MW at .m .1 wk .. a G “a” .. mm” mark “V v. a.“ NAM mo” .7”. ,.. ”up,“ 3.. ._ .1... .2? S wry“ G «.3.» and a.) . mm Mn...“ 4m wan EH r. - A... .L f A r. N. we“ 9mm“ 3 i Y B .m. C x. D 1% Hm.“ .MWM Y. r: .1. I. W- o.‘ S E. LE, ,2: :1: ,.,;:;, H 0-169 This 1. to certify that the thesis entitled "Body Dynamics of an Automobile by Electric Analog". presented by Charles H. Single has been accepted towards fulfillment of the requirements for Ldegree in_.Elec.t.rical Ehgineerlng’ madam, Major drolkssor Date Afiq '51) L .a-_ -—— — -.-.-.—. BODY DYNAMICS OF AN AUTOMOBILE BY ELECTRIC ANALOG By CHARLES HOLLISTER SINGLE 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 1950 THESIS II III“ IV VI VII VIII IX XII XII XIII Table of Contents Section Introduction Derivation of Equations Complete Mechanical Equations Motion of Other Points on Automobile Body Development of Electric Analog Components for Electric Analog Oscillograms Comparison of Results Conclusion Appendix I Derivation of Dimensionless Groups Used Appendix II Proof That 2483 ~22; 33 Appendix III Transformation of Data Bibliography Page 12 13 15» 31 39 41. 41, . 45 4? 51 Preface This thesis establishes the mathematical equat- ions for a complex mechanical vibration problem: the motion of an automobile due to road irregularity. The solution is found by means of an analogous electric circuit. The results from the electric analog are then. compared with photographs of the automobile in motion. The thesis clearly shows that the use of electric analogy is a powerful method in the transient solution of difficult mechanical problems. Particular thanks is given to the Ford.Motor Co. for the cooperation given. Especially, to Mr. W. E. Burnett of the Ford Research Laboratories; and to Mr. R. W. Gaines, head of the Engine Test Department, for his helpful criticism of the validity of the mechanical system used to represent the automobile, and for his patience in collecting the data for the model tested. The author also wishes to express gratitude to Dr. J. A. Strelzoff of the Electrical Engineering Department of Michigan State College for his guidance in the completion of the thesis, also for his reading of the manuscript. C. H. Single. Notation - Symbols The following notations are used in this thesis: Quanity Force Displacement Spring constant Mass Viscous damping (shock absorber) Center of gravity Radius of gyration Time Voltage Charge Capicatance Elastance Inductance Resistance Conductance Current Units pound inch pound/inch poundjsecond)2 inch Pound seconds inch inch second volt coulomb farad daraf henry ohm mho ampere l Integral with respect to time ——- P First derivative with respect to time Second derivative with respect to time Symbol F', ——-D- xayszsL K. _W—— M, 5 INTRODUCTION. The problem considered in this thesis is an. extension of the type problem_covered in.the graduate course in transients by the Electrical Engineering' 1 Department;at Michigan State College . The-speci fic problem treated is the dynamics of an automobile body resulting from road irregularity. A.general vieWpoint of this motion is used;;that.is, . translation, roll, and pitch.of the car body are- included.. For the wheel assemblies the action.of these components is considered:: wheel, spring, shock absorber, and tire» A-8p801fic road condition.is used, and the resultant motion of the automobile found.. In so far as possible this motion.is solved for from three viewp points}: an analytic solution, a solution.through. electric analog, and photograph.of the actual car- model undergoing the same road irregularity. These solutions are then compared, and an evaluation.of the electric analog method for the solution of transient problems is made;. .0.......*.'.'.'.‘. .4'O'OOOQOOOOOOO*OOO-OOO‘OOOOOO'. 0-0-0-0-0‘00-0'. O... l The course covered the first eight chapters of Transients-In Linear Systems, - Gardner & Barnes. Derivation Of Equations The assumptions necessary to establish math!- ematical equations of motion for the car are to consider the components.ideal.'. That is, the.masses rigid, the springs have negligible mass.and damping, and the shock absorbers have no energy storage» These assumptions are quite accurate when.the actual- data is considereda With.these approximations, the mechanical system.representing the car 13:: Z——-—- Shock absorber (Viscous damping) Wheel Rear axle 21"“‘Tire Front wheels separately suspended Figure.l_ - 2 .. First, the motion of the body alone will be studied”. The effects of the wheel assemblies being; replaced by equivilent forces.. The general body motion can be broken.into three simple motions as.follows4; A uniform.translation.of the mass.through the center of gravity;; 2' Figure,2. A rotation.about the center of gravity in the x - z plane;f .2 And, finally, a rotation. in. the y - a plane; 2 (LG. //6) _ _____@ _ / 7 6 E; . 'JLJ 3' CD L" l L, X Figure 1+ This gives. a general movement of the car body .. £- Fa / f / / F. JH l cfi- L, —— _—-— / “a; "(r .L .. 1'7 E. 2.": T L. 2| I F51 ure 5,: F. x E With the. forces. and displacements upward con- sidered positive, the equations for the above general motion ares: ( translation.) M1320 . ~Fl -~ E2 ,1 E3 ,1 F54 (Al) (rotation in the: x. - :2 plane); (rotation in the y - a: plane); was”)? p39?! :2: ElLl 421.2 £2352 441.1 (43.7)" -1... Where hxz and byz are the radii of gyration in their respective planes, M the mass of the automobile body; displacements, dimensions, and angles as shown in figure 5a A more explicit description of the body motion can be formed by changing from the translation and rota ions to a displacement of three specific points on the car body (in the plane of the center of gravity). These three points will be directly above the first, second, and third wheel, and noted 21, 22, and z}. Knowing these three diaplacements, any other point (i.e. 24) can be found, as will be shown later. From figure 5:; 21 n-Zb “'Ll sin.¢ ~ L5 sin 0 (4} 22 3526 / L2 sin.¢ - L3 sin 0 (5) 23 g zo’/ L2 sin ¢ / L2+ sin 0 (6) Assuming sin 0 §«O (small angles of rotation), sin $3 5 ¢ ¢, and Oand so can be expressed as.functions of Zl’ 22, and z}. Thus:: zxo 3 1.2(1.1 /L4)zl /(-L1L4- L213);2 #391 flan: (L7) (CI-41 7z L2)(L3 7‘ L4)» fl :- coal / za (8) O _-_-_- ~12 )1 Z} (9) (L1 ,1 L2)? (L3 ,6 L4) Substituting (7).(8), and (9), in (1),(2), and (3). we find:: ‘ ‘ H H a 2 . m 2- . -F1 "F2 7‘53 :1 “#4 f 14152 p 211 7‘ A'l(L1L4 "" LEI-I3) p 22 (‘71., L2) (L1 imam} a...) . 2. " .ng p z3 (10) (L3 /L4l L F ,1 F /T F * F m. )2 2 mm )2 2 (11.) 31%2u43:_-u44~1 XZ pz2._ p23 . L3 {L4 L3 #154 . . . . 2 2 . . 2 2 , L131 - L2F2 /L2F3 ;_L1F4 - A(hx2) p.21 / A(hyz) p2.2 L1 ,4 L2 L1 .11.. (.12). Solving the above equationsfor F1, F2, and F3 we find: F‘l ; F4 - M[L22/(Zhy§).2] p22.l {M[(Li/L4)L('riz)2-L(.L1L4-L2L3)] p.222 (' K L )K (L. .1 HI. :1 L )E L]. 2 1. L2. 3 4 (13). - MLZL.3 p22 3 F2 = -F4 ,1 M ((1.3 #thzhyzia - 12(11th - Lalo” pazl «Ll ,4 Lei-9m, ,1 L4) (.14) - MigtgtweryaflLL/Lg)?<;hxz.)2x<.t3,zt..>2?]p222 (11.1 ,1 L2)2(L3 ,1 L412 2 ,1 ”(5L1 i‘LaHhlez '* L3('L1L4 " 12%)] P Z3 (1.1 ,1 L2).(L3 :1 L4la -5...- F = -F4 / MLQLE p221 3 (L1_/ L2)(L3 / L4) /M[(L3)(L1L4-L2L3>. ~ (hxz).2'(Ll ,1 1.2)] ua/Lyug%Lp2 % MEL; "(hlefi p223 (L3 7‘ Litla To simplify the notation, define:; Mll_ : /M(L22 {(hyz)2]; (L1 % L2)2' 1.112 :1 71M [(1% ;‘ L4)(hlz)32" L2(-L1L4 ‘ L213”; (L1_/ L2)2(L3./ L4) Ml: = 1&11 ; (Ll /L2)(L3 Kth M22 :_ {Ml(LlL4-L2L3)2 %(L1_%L2) 2(hxz:3f(t3_/L) (L1 / L2»? (L3 7‘ L4)2 ML]. )‘LzH-hxzza) - mus. 121.321 (Ll :1 L2)(L5 % L4)?- Mfi; my], (L3 1‘ Lina 3 [0 U! u y. \N \N ll This reduces (1}), (l4), and (15) to: ~ri % F4 : % Mllp221_- Mlzpezz # M13p2z§r (16) 4'2 - F 4 -_-._ -. 1.112132% ,1 Magpaza -— IVIZZDZZL). (17) F3 / F4 : /M13p321_- M23p2z2 { M33p2z3 (18) - 7-._ Lani Thn‘Bi far; only, the automobile body hash heen; taken into account, the effectarof the wheel assembliee aaaumadlaatequiVJlent;forces; Ehe-next;atepiitho solvaafor.these forcessinlterma:of the actionLQf the. various elementeéof the wheeL.aasemhix.. The fonceafi fromzfignra25;,on.the first and second connerazwene: negotive on.theecan‘body;_tha‘third;and fourthMpoaitivem Therefore; the reactive force onlthe wheel.aasemhliea; muange of oppositeesignn: one and two:positiva;;thnae and four nagatiwa.. ThiazcanybaBtLEs;aeenzinethe static casa; withgthercar body in;equilibriumb. on this:basiszthe wheel assemblies and forces are:. 2: K D2 21 K3 D3" M2 02 M3 ,3 0 K02 K03 02 ZEOZ 03 $1003 I 1F“ , I 21 Z4 K K D4 1 D1 :1 4 1: zbl 7 4 M1! Mm M4! r0 K 01 :04 01 igél O4 2004 The second and.third are quite simple, aince they are independent-euepeneionse The equatione.for the second assembly arez; F3: (nap) {1(2) zé- (Dgzn /K2).K02 AOMQQg (.19); 07 2.4132103 {Myiafz¢ (20),; F52 {Home ----(K'52)202;/(K52)z00é “ - (121),! ‘ “ Imageheral, the force caueed by the road irreg- ularity is unknown» hut.the road.diaplacement, £302? iezknown.. Iteieetherefore.logical to eleminate-Eé, ané. 2%02 by substituting equation:(20) into (19). We then have: : PM = (P2P % we - map f 32% $22? Kuaszany : 422.210.- :‘szgflugng/Dgpfxéxxbamog (23,); I - Similariy: ‘ - 1 ~13: 3. (Hip; {K3} 1} -- (Dip 7‘13) 10:1 ((24) Kara/(5093); = 4133;: 73(3)zfmjpgp‘Df/K’jfxdfinoi (25); The first and fourth-wheel.aeeembliee huet;he worked out in a manner similar to the car body, due-to the effect of mutual mesa; the-axle.. This follows a. ‘ pattern;readily'foun¢.1n the literaturegk Due to. symmetry in:axle:conetruction, the-center.of gravity for the total rear wheel maserie-at.the geometric center. O0.....0.0000......0.0.0.0000...OOOC‘OOOOOOOOOO....0.0 2; Gardner &1Barnee,.Transient34igzLinear §xstems, Page 76. - 9 - Defining the distance from.the center of gravity to the wheel L5: a/L 3, Mt - leM4/Mm and. 22 the radius of gyration, h, we have the simplified system: /?i = zi /(0)z:01 A1094 may, - (P94 #MpzoM <28) Ko4(zoo4)- (O)zl ' p2Mmzol ' (pD4 {K4)z4 /(D4p {K4 {Ko4i/p2 M4‘)zo4 (29) Where: 2 2 2L 2 2: 2: M]: -_- MT L5 ylh M4. 3 M‘E{L {h} Mm :. M .L -h. . Q: . M 5 4L5? 4L5 L 4L5 “ We are now ready to form the overall equations of the mechanical system. Replace F1, F2, F3, and F41 by their equivilents (equations (26), (22),(24), and (28), respectively) in the equations for the car body (16), (17), and (18). The resultant overall equations are given on page 12. These equations of the mechanical system include four inputs: the road irregularity to each wheel, and seven unknown displacements. Four of these displacements correspond to the four wheel motions, and. the remaining three are the points on the car body above the first, second, and third wheels. Complete Equations For Mechanical System Kolzool : (Ml'p2%Dlp/K1%Kol)zol ,.é(o)z.O2 /(o)zo3 ~(Mmp2)zo4 ~(Dlp/Kl)zl %(Oizg {(0)25 (50) 5023002 ; /(O)zol;1(Z-212p2/D,p/K271K02)202 %(0)203 /(O)Zo4 %(O)z1 ~(D2p/K2>22 HOME (31) K033005 : /(O)zol /(O)202%(M3p2/D5pfkf)/&03)ZO3 /(O)ZO4 /(O)Zl ' %(O)z2 -(D39%K3)23 (52) i 5042004 = -(Mmp2>zol %203/(M4'p2%D4p/K4/KO4)ZO1 ~(D4p%K4)zl_ /(D4p/K4322 -z5, (33) i ’ O = “(Dipfl‘flflol %(O>202 %(O)Zog —(D4pr’K4>-ZO4% E41192AD1/D4m7zK171K4] 21 43412 92/D4p71K4MQ /(i‘«ilj/D4p71K4)z3 (54) P o = 71(o)z01 -( DaprZ) 2:02 %(O)203 71(D4p/K4)zo4 ~(1412p2/D4p/K4M1/ E“122p2/( Dg/D 4>p/K2/K4] Z2 “(Ma/DAWKMZ; (55) 0 = / = (LP/Rfé )1(t) 1(t) : (Cp/G%_i_)e(t) P LP F -- V F -- I x -~ q‘ X -- e(t) V -- i v -- e M -- L M -- C D -- R D -- G-s_l_ R K. -- S a__}_ K' --_}_._ C L The choice of the analogy was not arbitrary when data from the actual automobile was.used. This will now be shown:; equations (16), (17), and (18) for the car body envolve self masses and mutual masses between the three points of displacement. One possibility is to use the F -- V analogy. -15- This circuit has the correct equations: V1 (.5 9+ [/31 L1 v4 6: 12 V2 4.?” ' -L 0. i 13 vi.;+ 2 ]fi%;§ ' V3 -r 3 °+ [1,} L323 Figure 9 - - 2 - 2 2 4 V /V4 - Lllp Q1 Ll2p Q2 / L13P q} ( l) 2 2 2 -V2-V4 :‘L12p q1 / L229 Q2 - L239 q3 (42) ,IV {v - ,IL 2 - L p2 / L 1o2 (43) 3 4 - 13p ql 23 q2 33 Q3 A further possibility that almost gives the same equations is to modify the circuit slightly: Lil-Lye-L13___JWnNWNWL, /El ijunflflhmd o L12 L13 #3 L23 4; °____{?mnn11L 1flmfinm1L_____° L22'L12’L23 L33'L13‘L21 Figure 10 - - ~ One trouble with this setup is that the sign of one of the terms is always incorrect. However, it will serve to illustrate another difficulty encountered with - 17 - the actual car data. The mutual masses.of the car body were( in one case, L11) greater than the self mass. This required the use of a negative inductance (L11‘ L12 - L13 (:0) which is impossible without using vacuum tube circuits. If a tetrode were operated in its neg? ative resistance region, and positive inductance used: in series with it, the ideal negative inductance could be approached. However, this would certainly be much more difficult than winding coils with the correct: mutual inductance (figure 9). These same problems prevent the use of the cir- cuit with F -- I since a mutual term again has the wrong sign, and in this case a negative capacitance.is needed: L m-Jb- Figure 11 The sign of the mutuals are physically possible using the circuit as shown in figure 9. It can easily. be seen from figure 12 that it is possible to wind the -18- coils such that L and L are positive while L1} is 12 23 negative. L1 L12.1 2 L 2L 13 33 Figure 12 Applying the same process to the mechanical equations for the overall mechanical system,(equation (30) through (36) we getqthe analogous electric circuit! ,——————-=Ll3,— —————— -. erW——‘1 : P—L 9____1 I-‘—L2;—<--| : I L3 Conponents For Electric Analog Once the analogous electric circuit has been found the problem becomes: what value of the electric parameters are needed, and what charge in the electric circuit corresponds to an inch displacement. Certainly, if it were convenient, with a consistent set of units, we could let ILI3'MI,IS|=|K|,|DI=|R|, and one unit of change in the electric circuit would represent one unit of displacement in the mechanical system. However, in this problem, as with many mechanical systems, an attempt to use this direct transfer of num- erical data leads to an impossible electric circuit. By use of the Buckingham Pi Theorem, a practical analogous electric circuit is generally possible.*' This theorem states that if a problem is depen- dent upon n quanities and if these n quanities can be expressed in terms of m dimensions there will be formed n - m independent dimensionless groups from combinations of the n quanities. If the equality of the dimensionless groups is maimtained between the electrical analog and the mechanical system, much greater flexibility in the select- ion of electrical components is possible, and the correct transients for the mechanical problem can be found; that is, the same degree of damping will be found in both circuits. COOOOOCOOOOCOCOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.00.0.0.0...0. * Ibid.4, page 219 -20.. In the mechanical system the n quanities are force, displacement, time, mass, viscous damping, and spring constant.. These quanities can be expressed in three fundamental terms; force, displacement, and time. So there should be three dimensionless groups. One possible arrangement of these dimensionless groups or T"s is:* “1: _%__ .11. (44) K “2: D ((+5) VB 3 all (45) Obviously, various otherjr's can be formed; the only requirement being thattthey must contain all of the n quanitiesand be independenttf To see that the invariance of these dimensionless groups in the two systemstwill yield consistent results, we need only to check a simple mechanical system and its analog. This can be done with a mechanical problem previously used (figure 6). Assume the initial conditions to be zero and apply a step function force to the mass. By Laplace Transform the resultant motion will be:*** ‘"‘°OQeeeeeeeeeeeeeeeoeeeeoeeeeeoeeeeeoeeeeeeeeeeoeeeeo * Appendix II ** Ibid 4, page 228. was Ibid 2, page 342. - 21 - (47) K - M M2 Where Id -,-_ arctan V4MK - D2-1 .-D Instead of using the direct_analogy maintain only the 1T's.. Thus: aK «4 S From thell‘s: bM -- L c =‘“ b ' _ a (48) cT -~ T- . . d «=UEM. (49) dB -- R in 8. (50) eF -- V f ' fX -- Q Then from the electric analog the resultant is: Q(t) 3 fX(t) 3 eFt 2eF “4a bMK24 aKd p . _ bM ~C1D n ' j GE 2bM “t 3* §§1_ d2D2 t bM szz c " M With M . arctan‘[4abMK - d2D -dD It can be seen that substitution of equations (48), (49), and (50) into equation (51) will reduce this equation to (47). Thus, we do have a consistent method for trans— ferring the mechanical quanities to practical electrical components.“ Data for a check on automobile dynamics by this electrical analog method was obtained from the Ford.Motor Company. The model is a 1949 four door V-8 sedan” and the data:** Masses ‘ Lbs 860.2/ in. M11 2.168 M22 1.531 M12 .55331 M13 1.807 Ml'g M4' .2130 Mm .0467? M2 : M3 .1892 Spring Constants: Lbs. In. K2 : X; 121.8 KolzKoazKango4 235 OOOOOOOCOOOOOOOO...0..OOOOOOOOOOOOOOOOOO0.0000‘...O0....- * This system is different than but consistent.with, one given in Westinghouse Engineer, March 1946; page 52. ** The data was not available in this form. For compute ation and approximation used see appendix.III . Shock Absorbers: Lb. Sec. Ins. Compression Rebound An attempt to use the direct analog would give a very impractical electric circuit. The condensers would have to be quite large. For example, C2 would be 9,210 mfd., which is extremely large. Also, coils with inductances as large as 2.842 henry cannot be wound thathave a resistance negligible with respect to the desired resistance of 2.5 ohms: Here the use of the dimensionless groups is the answer. From (48) we see thatu(b) must.be as large as Possible with respect to (a) to keep the value of (c) large.. The importance of keeping (c) large is to keep the transient frequency down below the self-resonant frequency of the inductances. Another factor requiring a large value for (c) is the actual circuit will have wiring capacitances that introduce less error at low. frequencies.. And yet, (49) demands thatqthe product of. (a) x (b) be as large as possible. This keeps the desired resistance large compared with the resistance of the coils, which is an unavoidable error.thatwmustc be introduced.. Obviously, this is a task that can and. only in a compromise.. The values of (a) and (b), that. - 24 a were used are l9.388(10)3 and 4.847(10)-3.. This gave (c) a value of .5(10)~3 and (d), 9.694.- This value of (c) makes .5(10)'3 seconds in the electric circuit to be equal to one second in the mechanical system. The resultant electrical parameters were: Inductance in mh. Condensers in mfd. L11 3 10.51 Cl_-.G4: .391 L22 -.- 7.421 02,: c: .423 L33 3 13.78 Con .0552 L12 3 2.585 L13 3 8.759 Resistances in Ohms L23 -,- 2.400 101.490 101;11(o LlaL4 = 1.032 Rl : R4_ 24.24 242.4 Ln 3 .2267 R2 : R3 101.8 143.4 ngL3 : .9171 The fact-that the shock absorbers were non-linear eliminated the analytic solution, but electrically it. was no problem. Ideally the circuit shown in figure 14. has a resistance R1 with current to the left and R2 with current to the right. Perfect rectifiers do not exist, but the desired resistance was closely approximated, using this-circuit. R FMMM-H 0 ‘ R; Figure 14 - 25 - Vacuum tube diodes were tested, but the zero voltage current and extreme non-linearity at low voltages: plus high forward impedance, eliminated their use. A germanium crystal diode with a peak surge current of 500 milliamps was found quite adequate,, The curves on pages 27, 28, and 29, show the results of this work”. Pro- bably, these resistances approach the ideal curve as well as do the actual mechanical shock absorbers» The values of the components for the analogous electric circuit of figure 13 have been determined. On this basis, the electric network was construct- ed. It is pictured on page30. From this circuit, oscillo- grams were taken to compare the motion indicated by the electric circuit with the actual motion of the automobile. The response of the car to the bump was obtained by attaching lights at the desired points and taking a time exposure of the resulting motion. The scale for these photographs is found from the reference lights, spaced two feet apart, and the speed of the car. The speed of the automobile, fifteen miles per hour, was obtained only from the speedometer, and thus, is a pose? ible source of error. 000......OOOOOOCOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOIO * The resistances and other components were tested to three significant figures on a Wheatstone Bridge. The reactances tested at 1000 cycles, which is in the mid- range of the transient frequencies. -25.. Inl< F a wu- . « .HI‘: .Uv-OM'JOI 200 KHIA‘I d 0‘:— 2(U—KU2( .mad‘: .uv-nvsfl-oz :00 muld‘l 6 Odin Z(U-2U:( ANALOGOUS ELECTRIC CIRCUIT for 1949 Ford 4 Door V-8 Sedan Top View Bottom View PHOTOGRAPHS OF BODY DYNAMICS Motion of Left Front Body and Wheel Motion of Left Rear Body and Wheel - 3o - Oscillograms In the electric circuit it is necessary to Measure the charge in the various loops. From page 22, charge is analogous to displacement, and the various displacements in the mechanical system are the unknowns. The most convenient way to measure charge is to measure the voltage across a condenser, since voltage is directly proportional to the charge stored in the condenser. From figure l3, we see that the four loop charges, corresponding to wheel displacements, can be directly measured in this way. The body displacements must be found by adding the difference in the wheel and body displacements to the wheel displacements. In this circuit 501 2 $02 = So} = 504. Since the input voltage is also proportional to this value of elastance, all voltages except across 51, $2, and 53 have the same ratio to the charges. It was therefore convenient to leave the oscillograms in terms of voltage, not charge. However, all voltages were referred to the common elastance, Son: This allows easy conversion of the voltage oscillograms to mechanical motion. The oscillograms show the motion of all points with both a positive and negative step-function "bump" applied to the left front wheel. The motion of a few . -31- points due to the same input applied to the left.rear wheel is included. With these oscillograms the motion due to square bumps can be found. This is done on pages 39 and 40 to compare with the photographs of the car.pass- ing over a similar road irregularity. The automobile dynamics.pictured on page 30 were taken as follows: Spacing of lights: 2 feet. Dimensions of bump: height 2 1/2 in. Bump input to left.wheels of automobile. Thus, the reference lights were .0909 seconds apart, and the bump applied for .035 seconds. The comparison of the photograph with the combined oscillograms is given on pages 59 and 40. / . u..‘: .IVQOM'I-o: :00 RNtA‘l ‘ n—(fln z B : -D UL} / L4)zl - (1:.3 ,1 L4)z2—J 22: C : :D [(iL/ @(Lz, / 1.4)] 2:; With A : L2(L3 ,1 L4)zl / (L1L4 - L2L3)z2 /L3(Ll {L2H} Substituting the above value for A, B, and 0, into (40) along with: x4 3 -L4 and y4 ; -L1 : Ax4 / By4 / Cz4 / D a 0 -11 [(1.1 / L2)z2 -(L1 Agni] ,1 LL VL} {L4)zl - (L3/L4)12] “24(L1 /L2)(L3 #L4) /L2(L3 /L4)zl /(L1L4 ' L2L3)32 /L3(Ll / L2)23 : o Grouping: (Ll /L2)(L3 /L4) [t1 - 22 / 23 - 24] g 0 In general: (Ll / L2)(L3 /L4) £ 0 Therefore: Z4 a 21 - z2 / 23 -45.. APPENDIX III Transformation of Model Data Given By The Ford Motor Company To Necessary Form Approximations used were recommended by Mr. R. W. Gaines, Head of the Engine Test Dept. of the Ford Research Laboratories, Dearborn, Michigan. For 1949, 4 door, V-8 sedan: Total weight * 3230 lbs. Front sprung weight 219 lbs. Rear sprung weight 301 lbs. Unloaded weight distribution of fully equiped model in terms of percent of total car weight: Front wheels 57.1% Rear wheels 42.9% Left wheels 51 % Right wheels 49 % Plane of center of gravity: 25.3 inches above road. Front spring constant: with tire 108.6 lbs./in. less tire 121.8 1bs./in. Rear spring constant: with tire 114.5 lbs./in. less tire 131.8 lbs./in. Shock absorber constant: rebound compression Front 15 1b.sec./in. 10.5 lb.sec./in. Rear 25 lb.sec./in. 2.5 lb.sec./in. e. The sprung weight consisted of the weight of the wheel, wheel assembly, shock absorbers, and springs. Two-thirds of this weight was taken as wheel mass, while the remaining weight was lumped with the car body weight, since the actual wheel weight was only 46 pounds. .. 1+7. .. Radii of gyration: not available. Wheel base length: 114 inches width: 56 inches. This data was transformed as follows: 3230 - 2/3(219 / 301)1bs. Effective car.body weight = By utilizing the moments involved, the center of gravity of the car body is located as follows: 630.8 lbs. 839.6 lbs. L4‘ % f c.g.”1L"f 1 L3* H—— L1 :1: L2 606.0 lbs. 806.6 lbs. L2 1 48.9 in. L3 2: 28.6 in. L4 2 2704 in. To test the car, there must be a driver added to above weight of the car. 69.7 lbs. 61.5 lbs. a 12.8,, in. —+ 1 M 170 lbs. 43.2 in. L 20.6 lbs 18.2 lbs. l‘53.5 in. 60.5 in. Driver's weight: 170 lbs. The location of the center of gravity is then modified to be: 64.4 in. t“ ll L2 : 49.6 in. L3 : 2934.1n. L4 : 26.6’in.. Obviously, the center of gravity is different for any other load in the automobile. The radius of gyration in the planes was not known for the car body. This is the approximation used: The weight was divided evenly atethe center of gravity, and assumed to be uniformly distributed. Thus for the y - a plane: %)/////sz§ pl fifi) I p2 _ 2883 64.4 in. 49.6 in. dI 2 de2 = pdx(x2) o L I : J. plx2dx / ‘S 2 paxedx L1, o ' I : 3.361(10)6 lb.in. h z. :.“_;_ : 3.361 10 5‘ ; 32.2 in. y M "T 3"”0'5‘13' (1'0" I“- By a similar process: hxz'= 19.6 in. 2 - 49 - The radius of gyration of the rear wheel assembly: Effective weight of rear wheel assembly: 2/3 (301) - 200 lbs. .- H I T _ ( 108) x3 28 / (2)(46)(28)2 56 3 _28 100,430 lb.in.2 ‘ I g 22.4 inches M. Using the definitions of self and mutual masses on page 7: IT h the results of page 23 are found. The mass of the car body is the weight divided by the gravitational constant. The spring constants were given in a form that. required the separation of total spring effect of both tires and springs into its components. F Kl KT‘- (K01)(K1) K01. (K01 % K1) /////77/7///” Since KT and K1 were given, K01 can easily be found. It was assumed that the tires were the same, so an average of the Kon terms was used. K ; K : 121.8 108.6 02— 03 121. " 10 e K a K = (lél.8;§ll4.5) 01 04. (13108 “\11 05 K Average = 935 lbs./in. 996.5 lbs ./1n. Ml 87203 leo/ino on ' The remaining data was in the desired form. -50- Bibliography Gardner, M.F., & Barnes, J.L., Transients in Linear Systems, Vol. I, 1942, New York, John Wiley & Sons, Inc. Johnson, W.C., Mathematical And Physical Principles of Engineering_Ana1ysis, 1st. Ed. 1944, New York, McGraw - Hill Book C0,. Inc. Westinghouse Engineer, March, 1946 American Institute of Electrical Engineers, Transactions, 1949, Vol.68, pp 661 - 4, Corbett, J.P., Summary of Transformations Useful in Constructing Analogs of Linear Vibration Problems. mu mumgmummmmmunuunm 293 O 4 9363