ENGN UMR3447 COMPUTER SIMULATION OF VEHICLE MOTION IN THREE DIMENSIONS 1. J. SATTINGER D. F. SMITH SPECIAL PROJECTS GROUP 9y 3T HE U N I V E R S I T Y OF M I C H I G AN May 1960 Contracts DA - 20 - 018 - ORD - 14658 and DA 20 - 018 - ORD 19635

2901 -10-T COMPUTER SIMULATION OF VEHICLE MOTION IN THREE DIMENSIONS I. J. SATTINGER D. F. SMITH May 1960 SPECIAL PROJECTS GROUP T H E U N I V ER S I O F MI C H I G A N Ann Arbor, Michigan

This project is carried on for the Ordnance TankAutomotive Command under Army Contract Numbers DA-20O018-ORD-14658 and DA-20-018-ORD-19635. University contract administration is provided to Willow Run Laboratories through The University of Michigan Research Institute. Delivered by The University of Michigan pursuant to Contract No. DA-20-018-ORD-19635. Government's use controlled by General Provision 27 of the contract which is ASPR 9-203. 1 and ASPR 9-203. 4 (Revised 15 October 1958).

The University of Michigan Willow Run Laboratories CONTENTS List of Figures........................... iv List of Tables............................. List of Sym bols............................. Abstract............. 1 1. Introduction..................... 2. Derivation of Three-Dimensional Equations.............. 2 2.1. Complete Solution of Equations of Motion 2 2.2. Simplification of Equations 5 3. Field-Test and Simulation Programs.................. 6 3. 1. Description of the Field-Test Program 6 3.2. Analog-Computer Investigation 10 3.3. Comparison of Field-Test and Simulation Results 16 4. Conclusions and Recommendations...................25 Appendix A: Equations of Motion..................... 29 Appendix B: Simplified Equations of Motion................ 45 References.................... 50 iii

The University of Michigan Willow Run Laboratories FIGURES 1. Road Configuration................... 7 2. Field-Test Setup, General View.................. 8 3. Field-Test Setup, Vehicle Interior.................. 9 4. Wheel Position and Velocity-Measuring Device........... 9 5. Field-Test Setup, Instrument-Truck Interior............ 10 6. Computer Diagram........................12,13 7. Front-Spring Force vs. Deflection.................. 15 8. Rear-Spring Force vs. Deflection.................. 15 9. Front-Shock-Absorber Force vs. Velocity............... 17 10. Rear-Shock-Absorber Force vs. Velocity............... 17 11. Calculated Forward Velocity.................... 18 12. Comparison of Gyroscopic and Centripetal Acceleration Terms....18 13. Run Number........................... 19 14. Run Number 2........................... 19 15. Run Number 3........................... 20 16. Run Number 4........................... 20 17. Run Number 5........................... 18. Static Run............................. 22 19. Vehicle Roll for Case I Run..................... 22 20. Effect of Increased Moment of Inertia.............. 23 21. Run Number 6....................2 22. Run Number........................... 26 23. Coordinate Systems....................... 30 24. Symbols for the Wheel and the Ground................ 37 25. Euler Angles.................... 42 iv

The University of Michigan Willow Run Laboratories TABLES I. Field-Test Conditions........................ 8 II. Constants Used for Simulation................... 14 III. Comparison of Actual and Simulated Periods of Oscillation....... 16 V

The University of Michigan Willow Run Laboratories SYMBOLS Coordinate Systems (see Figure 23) i, j, k Unit vectors in the body-fixed coordinate system x, y, z Coordinate axes in the body-fixed coordinate system i', j-', k' Unit vectors in the earth-based coordinate system x', y', z' Coordinate axes in the earth-based coordinate system Euler Angles (see Figure 25) A Azimuth angle E Elevation angle B Bank angle Suspension and Tire Force Functions K Suspension-spring force-vs. -displacement function for the n-th wheel C Shock-absorber force-vs. -velocity function for the n-th wheel ns f~~~0 5Shock-absorber force-vs. -displacement function for the n-th wheel ns K Tire spring force-vs. -displacement function for the n-th wheel nw C Tire damping force-vs. -velocity function for the n-th wheel nw Tire damping force-vs. -displacement function for the n-th wheel Other Symbols a Acceleration c Direction cosine between the 1 and m axes where 1 = i,, k and m = Im i', j', k' CG Center of gravity F Force Fn Vector of force acting on the body from the n-th wheel n F ni Fnj Fnk i, j, and k components of Fn g Gravity vector g Magnitude of g G Vector of the ground-reaction force on the n-th wheel G i Gnj Gnk i, j, and k components of G H Angular momentum vector of the n-th wheel H Angular momentum vector of the body J Moment of inertia J Pitch moment of inertia of the body about the axle vi

The University of Michigan Willow Run Laboratonries SYMBOLS (Continued) J, J., Jk Moments of inertia of the n-th wheel about the i, j, and k axes.though the wheel hub Joi Joi Jo Moments of inertia of the body about the i, j, and k axes through the CO K Factor by which the roll moment of inertia is increased (see Equation 1) m Mass m Mass of the q-th particle q m AMass of the n-th wheel m Mass of the body Vector of moments acting on the nt wheel M Vector of moments acting on the bodyn-th wheel M Vector of moments acting on the body o Mo Mj Mok i, j, and k components of M 01 oj ok o Poi, Pok' Pok Products of inertia of the body r Distance from the rear axle to the CG r Position vector of the q-th particle from the body's CG q r Position vector of the n-th wheel from the origin of the earth-based coordinate system r.il r p r i, j, and k components of r r Position vector of the n-th wheel's ground-contact point from the ~n~~g ~origin of the earth-based coordinate system r. i component of rn ngi nng rngi,, rng rng i', j', and k' components of rng _ngi~ngj,, rgk,' J' ^ ng r Position vector of the n-th wheel's ground-contact point from the wheel hub r Magnitude of r nW -nw r Position vector of the body's CG from the origin of the earth-based coordinate system rr,o rok i, j, and k components of r roi ro., rok i', j', and k' components of r R Radius of the wheel from hub to tire tread w T Torque T Vector of torque acting on the n-th wheel T i, T j, Trk i, j, and k components of Tn [T] Coordinate transformation matrix (see Section A. 7) V Forward velocity of the vehicle v Velocity VII

The University of Michigan Willow Run Laboratories SYMBOLS (Continued) v Inertial velocity of the q-th particle q x, yq, zq x, y, and z coordinate of the q-th particle from the body's CG x, Yn' Z x, y, and z coordinates of the n-th wheel from the body's CG xo, yO zO x, y, and z coordinates of the body's CG from the origin of the earthbased coordinate system x', y', z' X, y', and z' coordinates of th -body's CG from the origin of the ~0 0 0earth-based coordinate system z Displacement of the n-th wheel from the neutral position of the suspension oi Pitch angle of the body d. Roll angle of the body.a Coefficient of friction u;L,~ ~ Angular velocity W Vector of the angular velocity of the body oi' Woj' wok i j, and k components of w o0 oj' ok 0 W Vector of the velocity of the n-th wheel n W ni' iWnj' k i, j, and k components of on ni nj nk -n Dot ( ) or double dot ( ) over a symbol denotes derivatives with respect to time. viii

COMPUTER SIMULATION OF VEHICLE MOTION IN THREE DIMENSIONS ABSTRACT This report describes the results of a research program to develop techniques of simulating three-dimensional motion of a vehicle by means of electronic computers. In order to provide the basis for vehicle-motion simulation, a complete set of equations for a wheeled vehicle was developed, which permitted the analysis of motion in pitch, yaw, roll, bounce, surge, and sideslip. The derivation and summary of these equations is presented in this report. As a means of reducing the time and cost of computer solutions for certain restricted cases of vehicle motion, an investigation was made to determine to what extent simplifying assumptions could be made in these equations. A simplified set of equations is contained in the report for cases in which surge can be neglected and angular motions in pitch and roll do not exceed 10~. A second simplified set of equations contains the further restriction that only pitch, bounce, and roll occur. In order to establish justification for the simplified equations, an experimental program was carried out on an XM-151 military truck to determine its motion when traveling over certain road obstacles under specified conditions. The results were compared with those of corresponding runs in an analog-computer simulation. It was concluded that the simplified equations allowing only for pitch, bounce, and roll motion gave a satisfactory representation of the field-test data, but that improved correlation could be obtained by the simplified set of equations which accounted for lateral motion as well. 1 INTRODUCTION This report describes the results of a research program, undertaken by Willow Run Laboratories of The University of Michigan, to develop techniques of simulating three-dimensional'The authors wish to acknowledge the work of B. Herzog and D. Y. Liang in connection with this research program. Mr. Herzog assisted in the derivation of the complete Bet of equations of vehicle motion, as presented in Appendix A. Mr. Liang Was responsible for preparing, checking, and operating the analog-computer setup discussed in Section 3. 1

The University of Michigan Willow Run Laboratories motion of a vehicle by means of an analog computer. The work described here is a direct outit growth of previous research projects, sponsored by the Ordnance Tank-Automotive Commaihd in which methods of simulating vehicle motion in two dimensions were developed Refereftk* 1) The application of these original studies is confined to analyses involving angular mtion-of the body of the vehicle about its transverse axis (pitch), transiltional motion in a vertical direction (bounce), and translational motion in a longitudinal direction (surge). The simulation of vehicle motion by means of electronic computers provides an important method of analyzing certain types of vehicle-design problems. Examples of such problems which occur in connection with military-vehicle design are investigations of the effect of suspension-system design on ride characteristics, and of the stability of the vehicle in the presence of gun-recoil forces. The specific objectives of this program were: (1) To develop a generalized mathematical model which can be used for the investigation of design problems involving vehicle motion. The equations composing this model should represent both angular motion of the vehicle (including pitch, roll, and yaw) and translational motion (including bounce, sideslip, and surge). (2) From this general mathematical model, to obtain simplified versions of the equations suitable for design studies involving motion only in pitch, bounce, roll, and sideslip. (3) To establish the validity of these simplified equations by comparing data obtained from field tests on a real vehicle with results obtained by the use of the equations in a corresponding analog-computer simulation. The investigation described in this report has been based on a field-test program and computer simulation of a four-wheeled vehicle. However, the mathematical model has been prepared for vehicles having any number of wheels and can be applied, with some modification, to articulated vehicles. The model is also adaptable to track-laying vehicles. For certain types of problems, forces applied to the hull by the track (for example, due to accelerating or braking) significantly affect the motion; in such cases, the equations must be modified to include the effect of these forces. 2 DERIVATION of THREE-DIMENSIONAL EQUATIONS 2.1. COMPLETE SOLUTION OF EQUATIONS OF MOTION As the basis for the development of the three-dimeneidnl simulation of a vehicle, the equations of motion were derived for a wheeled vehicle wtope body is free to mowv in three 2

The University of Michigan Willow Run Laborgtories degrees of angular freedom and three degrees of translational freedom. The three quantities representing translational displacement are designated sideslip (x'), surge (o'), and bounce (z'); the three quantities representing rotational displacement are designated azimuth (A), elevation (E), and bank (B). Corresponding angular velocities are designated yaw (<k), pitch (e ), and roll (w j), respectively. The equations define the motion of the body of the vehicle and of each wheel as affected by the force of gravity, the interactions between the body and the wheels, and the reactions of the ground (including the effect of longitudinal forces produced by the engine and brakes). The complete solution of the equations of motion can provide the displacement, velocity' and acceleration as a function of time of the body and of each wheel in each of the six degrees of freedom. It can also provide a time record of the forces acting on each of these objcts. The motion of an object subject to a combination of forces and torques isgoverned by Newton's laws of motion. For each part of the vehicle (such as the body or each wheel), an equation can be written to determine linear acceleration along each of three coordinate A.xes which are fixed with respect to the earth. These three equations are of the form: F = ma where ZiF = sum of all forces acting on the object along a given axis m = mass of the object a = acceleration of the object along the given axis Similarly, for each part of the vehicle, an equation can be written to determine angular acceleration along each coordinate axis. These equations are of the form: LT = Jc where LT = sum of all torques acting on the object around a given axis J = moment of inertia of the object around the axis w= angular acceleration around the axis In order to derive velocity and position information from the above equations which define acceleration, integral equations are required of the general form: v =adt and x= vdt The body is acted on by two types of force. One is the force of gravity; the otlhr Si the summation of forces applied to the body by the suspension components of each wheel, tlat is, by the spring and shock absorber. In the case of a track-laying vehicle, additional forces on the body would be produced by the tracks themselves. In the present analysis, hlowneer, a wheeled vehicle is assumed. 3

The University of Michigan Willow Run Laboratories In addition to equations describing the motion of the body, equations are required for the motion of each wheel. It may be assumed without appreciable error that the motion of the wheel with respect to the hull is confined to a straight line parallel to the yaw axis. The motion of eash wheel is the result of the force of gravity, the reaction from the ground acting through the tire, and forces exerted by the suspension spring and shock absorber. It is convenient to write the equations of motion of the hull and of each wheel in terms of body-fixed axes, that is, a set of axes parallel to the pitch, yaw, and roll axes of the body. On the other hand, the output quantities representing translation and rotation of the body, and the profile data on the road over which the vehicle travels are best represented in terms of a set of earth-based axes. Consequently, a number of coordinate-transformation equations are required which relate displacement, velocity, and acceleration in terms of earth-based axes to corresponding quantities in terms of the body-fixed axes. In general terms, then, it is possible to say that the simulation of vehicle motion in three dimensions involves the solution of a set of equations consisting of the following types: (1) Equations representing Newton's laws of motion for both rotational and translational effects of the body and of each wheel. (2) Equations representing forces transmitted by springs, shock absorbers, and tires as functions of wheel displacement and velocity with respect to the body or to the ground. (3) Coordinate-transformation equations relating displacements, velocities, or accelerations in body axes to those in earth-based axes. The detailed derivation of the complete equations of motion of a wheeled vehicle in three dimensions is presented in Appendix A. In the derivation, the following simplifying assumptions were made: (1) The vehicle is assumed to be symmetrical about the plane of the yaw and roll axes. (2) Motion of each wheel with respect to the body is assumed to follow a straight line parallel to the yaw axis of the vehicle. (3) Each tire is assumed to make contact with the ground at a point on the straight line of travel of the wheel hub. (4) Reaction on the body of the vehicle and due to engine rotation is neglected. Using the set of equations as derived in Appendix A, rough estimates were made of the amount of analog-computer equipment which would be required to represent these equations. It was determined that, to represent the motion of a four-wheeled vehicle, the total analog equipment required would consist of approximately 160 operational amplifiers, 17 multipliers, and 16 function generators. For an eight-wheeled vehicle the amount of equipment required increases to 275 operational amplifiers, 22 multipliers, and 32 function generators. ince the time, cost, and complexity of a computer solution tncrease with the Sant/ty of equipment required, it was considered desirable to determine to what extent simpljnpftg assumpttons 4

The University of Michigan Willow Run Laboratories could lb made which would reduce the complexity of the equations and minimize the equipment required. There are numerous possibilities for making such simplifications, but it is important to avoid incurring excessive error in doing so. 2. 2. SIMPLIFICATION OF EQUATIONS In Appendix B, the equations developed in Appendix A are rewritten utilizing certain simplifying assumptions. Several simplifications are made to write the equations of Section. 1; and then one additional simplification is made to write the equations of Section B. 2. The simplifications used for the equations of Section B. 1 are as follows: (1) The results of field tests show that the forward velocity of a vehicle is very nearly constant even when the vehicle is travelling over a fairly rough road (Figure 21). Hence the equations pertaining to the forward motion of the vehicle are replaced by the proposition that the vehicle's forward velocity is constant. (2) When only small angles of motion of the body need be considered, it is possible to set the direction cosines between corresponding directions in the earth-based and bodyfixed coordinate systems equal to 1 and all other direction cosines equal to 0. Since many practical cases of vehicle motion are those in which pitch, roll, and yaw angular deviation from a mean value do not exceed 5~ or 10~, such a simplifying assumption appears to be reasonable. (3) Certain of the mathematical terms that appear in the equations as a result of working in body axes represent gyroscopic effects and centrifugal-force effects, which result from rotations of the body. It is possible by inserting typical values of these rotational velocities to establish the validity of omitting these terms. In the equations given in Section B. 2, one additional simplification is made: sidewise motion, sideslip, is omitted. This set of equations restricts the motion of the body to only three of the six degrees of freedom, specifically, to pitch, bounce, and roll. This restriction naturally limits the scope of the problems which can be investigated by means of the simplified simulation; however, many problems of interest can be studied in this manner without serious',ess of realism. For example, studies of the ride characteristics of automotive vehicles can ie cintmined to the interpretation of pitch, bounce, and roll motion, since these components of the total motion tend to be the most pronounced. Ln order to determine the feasibility of developing simplified versions of the complete system of equations, the set of equations of Section B. 2 was investigated in detail. By restricting the problem in this manner it was possible to make a computer setup requiring only 52 operational amplifiers and 16 function generators. No multipliers were required. The functions 5

The University of Michigan Willow Run LaborotQries to be represented are those of the nonlinear force-vs. -motion characteristics of the suspension components. By making the additional assumption that each of these characteristics is compoded of two or three linear segments, it is possible to accomplish the function-generation process in a simple manner, that is, by the use of additional operational amplifiers with nonlinear fedback rather than by the use of conventional function generators. The details of this computer setup are discussed in Section 3. 2. 3 FIELD-TEST and SIMULATION PROGRAMS In order to establish justification for the simplified version of the vehicle-motion equations developed in Section B. 2 as well as to gain experience in their use, an experimental program was carried out involving both a computer study and a series of field tests. Data were obtained from a computer set up for a number of runs of a Ford Motor Company XM-151 military utility trans(port truck (an experimental four-wheeled vehicle similar to a jeep) when triangular ra;,s of various sizes and spacings were traversed. In order to check the validity of this computer setup, a field-test program was conducted in which an XM-151 was instrumented by means of accelerometers, gyroscopes, and wheel-position-measuring potentiometers. Runs were made over road obstacles consisting of triangular ramps at speeds corresponding to those used in the computer tests. The field test and computer data could then be compared for what should be identical conditions to determine the accuracy of the simulation and to establish permissible simplifying assumptions. 3.1. DESCRIPTION OF THE FIELD-TEST PROGRAM The XM-151 was selected for the field-test program for the following reasons: (1) The use of a four-wheeled vehicle would require the least complication in terms of the total instrumentation required, and the magnitude of the corresponding computer setup, without limiting the validity of the results. (2) Since the Ford Motor Company had used this same type of vehicle for certain analogcomputer studies, some data concerning vehicle system parameters were available which would be useful in carrying out the program described here. The vehicle, whether real or simulated, traveled at identical speeds over a road containing identical road irregularities, and the corresponding real and simulated data in pitch, bounce, and roll motion were to be compared in order to determine hw accurately the real motion was simulated on the computer. In order to make this comxptarQO, it was decided to measure and' 6

The University of Michigan Willow Run Laboratories record the following quantities: (1) Pitch, bounce, and:'oll velocities (2) Bounce and sideslip. ccelerations (3) Pitch, roll, and yaw angles (4) Position and velocity of each wheel with respect to the body The test runs consisted of running the vehicle at various speeds along a concrete road on which were located a number of triangular obstacles. The dimensions of the obstacles and the particular arrangement of obstacles along thelpath are shown in Figure 1. EacIXarrangernent is designated by a case number. Table I shows the speeds and indicates obstacle arrangements for the field tests. The yaw, pitch, and roll angles were most easily measured by the use of nmotion-picture camera techniques. A camera was set up at a distance of about 100 feet from the side of the road and approximately in line with the road obstacles. Another camera was et up looking along the road. In order to permit measurement of the angular and translational motion of the body of the vehicle, markers were attached to it and a fence was constructed along the sidest the road to provide reference positions against which the vehicle position could be measured from frame to frame of the motion-picture record. This technique has been discussed in Case II Case I Case In FIGURE 1. ROAD CONFR~ATION 7

The University of Michigan Willow Run LabQ.rotories TABLE I. FIELD-TEST CONMITIONS Run Case Speed Number Number (fps) 1 I 12.1 2 I 16.1 3 II 10.7 4 n 15.2 5 II 22.1 6 III 16. 7 II 23.'6 greater detail in Reference 1. Figure 2 shows the setup used for the field tests.'TheiWta poles on the jeep were used to measure pitch, roll, and yaw angles, and the fenc -ii the fore* ground was used as a horizontal reference for measurements. The obstacles are,hu0h tf position for a Case III run. Angular velocities of the body were measuled by means of three rate-gyros tgidy stbcbed to the body. The sensitive axes of these threq gyros were-SD aligned with respect to the-vehide body axes as to give a continuous indication of pithb, yaw, add roll velocities.. a, m *';,'' 1,''"* A' -.~"i; "-'"e' I FIGURE 2. FIELD-TES SETUP, OEgRAL MW'\

The University of Michigan Willow Run Laboratories Accelerations were measured by means of two accelerometers which were mouoted on the frlame of thle vehlicl with their sensitive axes aligned along the vertical and longitudiial vehicle axes. Thtes accelerometers were located at the center of gravity of the body so that the accelerations nc;asllred were always those occurring at this point. Figure 3 shows the gyros.A4 accelerometers mounted in the jeep. The accelerometers are shown mounted on thqeCG marker which is above the transmission, and the gyros are mounted on the platform behind the fimont seat. This platform also served as a distribution center for the electrical cables to the. rious inst ruments. The motion of each wheel with respect to the body was measured by means of an asSembly, show:i i:l Figure 4, attached to the body frame. The assembly contained a potentiormelrQ and l tachometer for measuring wheel extension and wheel velocity, respectively. TheseSdvices could be rotated by means of a small steel cable attached to their shafts by means QOi wftely arrangement. The end of the cable was attached to the upper wishbone of the suspiwfon. Downward motion of the wheel extended the cable, and upward motion of the wheel permiittedr. iwn spring to retract the cable. This motion transmitted to the potentiometer And tacbomer p'duced the desired electrical outputs. In order to provide permanent records of the field-test data, the outputs of all AS gyros, accelerometers, and wheel-measuring devices were routed through amplifiers to a reqodgling oscillograph that had provisions for up to 14 channels of data. Synchronlzation bet-Wren t'e uscillographic data and the motion-picture data was obtained by means of an event-marker -- - t - — S ar — X.^' - XI* _... o i...... iv _.... _..i.. igy;,'e.,;'.4,. I,' "'" * * ^'- i' r:, I! *! * * \ N 9;' I * S^A.'. -.%.. ~.'J',"r ~ -,,': __: *' -' L..:..e"'';'. "''," rd i ~a' - r.-,'....' " "**'-*-'i,: FIGLRE 3. FIELD-TEST SETUP, VEHICLE FIGVRE 4*? INTERIOQ VEIYIM(a 9

The University of Michigan Willow Run LaborotorjiA system. This started the timing lines on the oscillograph at the same instant that a flas -.bulb was set off i;l view of the camera. During the test runs, the amplifiers, recording oscillograph, and power supplies were carried aboard an M4 ambulance which ran alongside the jeep (Figure 5). Interconnections be-, tween the ambulance and jeep were made by a set of electrical cables which are shown eatering the ambulance from behind the front seat in Figure 5 and are shown on the far side of the leep in Figure 3. Electric power was supplied by an engine-driven a-c generator mounted onabe front of the ambulance (Figure 2). i; -- t;',. s ^ * - V. _ - - I -;, t., -. -l *l. *. *. b.! a__-*-> - - -rr yt - y -i >. L... *-'... -.-,: FIGURE 5. FIELD-TEST SETUP, INSTRUMENT-TRUCK WNTEJIOR 3.2. ANALOG - COMPUTER INVESTIGATION The analog computer at Willow Run Laboratories was set up, using the equatiots in Section B. 2, to simulate the XM-151 military truck. This setup included 56 operational amplifier o which 8 used nonlinear feedback to represent the suspension springs and the effects of the tites leaving the ground. The computer diagram for the simulation, which was run in one-tenth reas time, is shown in Figure 6. The main component of the road-function generator is a ganged stepping switch, each gang being used for the road input to one particular wheel. The output of each step is a voltage which represents the slope of the road; each step represents 1 foot of travel along th. road. The stepping rate, which is controlled by at audio-oscillator, then rpresetits the speed of the vehicle. The vehicle speeds and road coatigurations simulated are the same as those under which field tests were run. 10

The University of Michigan Willow Run Laboratories In order to make the series of analog-computer runs, it was necessary to determine the magnitudes of the vehicle-design constants which are used as parameters of the equatiOns Table II shows the constants used and their values. The methods used to determine th/serva1aueF follow. To obtalin the spring rate of the front suspension springs,. the wheels and ibock absorbers were removed and the body was blocked up so that it was free to pivot about the rear axle. A known weight was placed directly over the front axle, and the displacement of the frontWheel hubs with respect to the body was measured. Since the two front springs are in paralll, the rate of each spring is equal to one-half this computed value. The procedure was repeated with the rear wheels to determine the constant of their suspension springs. The force-vs.-deflection curves for the front and rear springs are shown in Figures 7 and 8. The static position and positions of metal-to-metal contact in jounce and rebound were found by removing the suspension spring and measuring the wheel displacement when it was at the various positions. This was also done with the rear wheels to determine the point of contact with the rubber bump stop, In order to determine the unsprung mass of each wheel, the body was blocked up so that the wheel hung free. The shock absorber and suspension spring were removed, and a different spring of known spring rate was installed. The wheel was lifted and released, and its trdanient motion was recorded by means of the wheel-measuring potentiometer described In Section 3.1. Assuming the system to be governed by a second-order differential equation, the meaaorement of the period of oscillation permitted the calculation of the unsprung mass. A set of measurements was made to determine the distribution of the total vehicle weght among the four wheels. The sum of the individual weights on each wheel gives the total vehitle weight. By subtracting the sum of the unsprung masses at each wheel, the sprung mass wavs btained. The weight distribution in conjunction with the values of wheelbase and track width could then be used to calculate the location of the center of gravity of the sprung mass in a horizontal plane. Thus, the values of Y1, Y2, Y3, Y4, X, X2 Xg, and X4 could be Computed. The vertical position of the CG was not measured, but was available from measurements previously made by the Ford Motor Company. This particular value is not used in ths6imppitil equations studied on the computer. The moment of inertia of the body in pitch was dterlrtned by a test for which thl fehttle was tlocked up in the same manner that was used to measure the front spring rate (li~ the body was pivoted about the rear axle with the front suse1siRn springs in place). OscilatiOns were induced by dropping the front end of the vehicle, torcing it to rotate about an axis parallel to the pitch axis, with springing supplied by the front suapedion springs. Based:in the knoaw spring rate and the observed period of oscillation, the:pith Swaoeat of inertia around the resr 11

The University of Michigan Willow Run laboratories 1-00 v \ iIEE MOTION wnth RESPECT to nIDTY (Eq. I ic (,,n' for e.rkWtkeIl) --—. -- ------- I.. -'nk) ---- ^ ^ o "eo I i ~F Sl~%) _ _ _ _ _ R()OAD | I -. 7 I K S om FINCTIlON GE10 B(on f2or ~ch e 20r 250 M ns4k n RA 2 10 j ns (zn.2i.fV TII;: DEFLECTION (Eq. l c) K!2 r rn 2r v ~ n FIGURE 6. COMPUTER DIDOAM' Notes: (1) Problem ts ruo.in one-tenth time. Gain 8shOton'iategratorB is actual wgin ifth& variable through >.e integrator. (2)Y2, Y4, IXg ad X ar negptve qantitU. 12

The University of Michigan Willow Run Laboratories BODIY HWI:NC.E F(#l. 20c) -S -..00v.. K1' K]*' i ^ss^ 50 m 2 2'z 33 ~ ~ ~ a S 2 I I A BODY ROLI, (Eq. 35C) v X K (,) | I 1, nO _ I ~. K I \ 1.. 5, 0 J 2 I 3 _n o __ -I 2o n4 ns-n1 2 c J* L-Now BODY PITCH (Fq. 34c) 5Y Ks (z 50 i oj LYii 011

The University of Michigan Willow Run Laboratories TABLE n. CONSTANTS USED FOR SIMULATION* Symbol Value Description J. 461.1 lb-ft-sec2 Pitch moment of inertia 01 2 J 84. 0 lb-ft-sec Roll moment of inertia oj mO 59.1 slugs Mass of the body mI, m3 2. 56 slugs Mass of front wheel m2, m4 1. 87 slugs Mass of rear wheel 1 YQ m3.07 ft l Y Y3 3. 07 ftFore and aft distance of Y Y4 -4. 01 ft J wheel from CG 2P Y 4-4. 01 ftJ X1, X2 -1. 07 ft Transverse distance of X3, X4 1.07 ftJ wheel from CG x 3' X4 1. 07 ft 1 K s K3s 1697 lb/ft Spring constant of K 2s, K 1557 lb/ft J suspension springs 2s1557 4sb/ft J CIs' C3s 119 lb/ft/ secl Shock absorber C2s, C4 163 lb/ft/secj force v. rate K1w' K3w 8100 b/ft 38Kw ~1w00wb/ft'Tire spring rate K 2w' K4w 11,000 lb/ft Clw C3w 7.9 lb/ft/sec, r 3 Tire force vs. rate C2, Cw 8.3 lb/ft/sec *Wheel number convention: 1, left front; 2, left rear;3, right front; 4, right rear. 14

The University of Michigan Willow Run. tba orautories 1600 - ~ 31 —- 1600 - --. - 1200 800 1r ^ — -- S- ^ fJ — 1 ^ ---- l -- / - — I - j r: E t,s a C 1 y 697lb/ft ly m 3 C o o I 1-_ L - - 400- 400_ — O 0 0............... 0 0.2 0.4 0.6 0.8 1.0 0 0 0.4 06 -.0 DEFLECTION, zn (feet) DEFLECTION, z, (fet) FIGU RE 7. FRONT-SPRING FORCE VS. FPIGURE 8 REAR-SPRIRG FORCl VS DIE DE FLECTION. Front spring viewed at hub. FLECTION. Rear spring viewe4 from Mub. axle could be computed. The moment of inettia arxund tbh4 ptteh axis through the center f gravity could then be computed by means of the equation J..J -.a r o1 a 0 where J = measured moment of inertia about the axle r = distance from the pitch axis through the axle to the pitch axis through the 00 To serve as a check, this type of measurement was repeated for motion pivoting t for mtd with the vehicle blocked up on one side. To check the analog-computer setup, the tests performed on the vehicle to determine the muments of inertia were simulated on the computer. The blocked wheels of th6i vehic* — were simulated by taking a very high value for the spring constants of these wheels. Table. shows the periods of oscillation which were observed observedn the vehicle tests and the correspondtsg periods observed for the computer simulation of these tests. The o the damping constant ofugh the system due to the presee of soc absorbers and of other sources of friction was found in a test similar to that desCtbed for the moments of inertia, 15

The University of Michigan Willow Run Laborat6ries TABLE Im. COMPARISON OF ACTUAL AND SIMULATED PERIODS OF OSCILLATION Test Period of Oscillation Deviation XM-151 Computer Roll 0.525 0.520 0.95 Pitch Front Blocked 0. 525 0. 511 3.0 Rear Blocked 0.587 0.577 1.7 but with the shock absorbers in place. The front or rear axle f the vehicle was paiedin blocks, caused to oscillate, and its motion recorded by the wheel-measuring potenhiWetero Because of the high degree of damping, it was impossible to determine ALrate of ojilUattor decrement. However, by means of a technique descrtbed in Reference 2, the mpipg Constant could be determined from the shape of the overdamped transients The shock-.aborbel forcevs. -velocity curves determined by this method are shown in Figures 9 and 10, Since the damping constants for the shock absorbers are nearly the same for both postttve and Aegatie velocities, linear shock absorbers with the damping cOnstant shown tn tie ftirt quntd ot o the figures were used in the simtalation. The spring and damping characteristics of the tire were taken from data suppUsd b the Ford Motor Company. 3.3. COMPARISON OF FIELD-TEST AND SIMULATION RESULTS As the basis for evaluating the adequacy of the computer simulation using tlhiquations of Section B. 2, the results of the XM-151 field tests were compared with the computer simulation for the seven runs listed in Table L The evaluation'i t# comparison is distussed in this section. In reviewing the results, the determination of cause of any discrepanzteS betw en,eat- an simulated data will be attempted. One possible source of discrepancy is thiipreseeice *f errors in the field-test data, due to limitations in the accuracy of the data.coltetion device.aInd methods. In addition, there were difficulties in controlling'some of the conditions undr, which the field tests were run (e.g., smooth road, constant speed, and nosideslip) so that they correspond to the conditions assumed in the computer simulation. Dscrepascies due to fhede causes do not indicate any limitation of the computer repretentation. On the thher ban4 any discrepancy which is due to oversimplification of the.equations must be taken as h1itndication 16

The University of Michigan Willow Run Laboratories C400 I i / t. / L 400- --- 400- / 118.8 b/ t /see ______ I / s t/2.8bec l8b/ ft/ee' 0200- 200 p I U -4 -2 2 4 6 -4 -2 2 4 VELOCITY, in (fp) VELOCITY. a, (fpt -200- — 200 — - -200 / ____L__-. — 400- ______ _____- 178.5 lb ft sec +-400 -400 66.4 lb/ ft/sec /L66.4 lb/f/sec FIGURE 10. REAR-SHOCK-ABSORBSR ______ ____ FORCE VS. VELOCITY. Rear shoccab-orber viewed from hub. FIGL RE (9. FRONT-SHOCK-ABSORBER FORCE VS. VELOCITY. Front shock absorber viewed from hub. of the need to include additional features in the simulation. In the following discussion, some estimates of the source of discrepancies can be given, but due to limitations of time available for the analysis it has not been possible to indicate the exact reason for all differences between real and simulated data. Some general observations can be made concerning sources of discrepancies, before the individual runs are reviewed. It is believed that errors in the field-test data may range up to 0. ft for bounce (z and l), and 0. 02 rad for pitch angle (ol) and roll angle ( oj). Co - quently, discrepancies of less than these amounts may be due to experimental error and afe not significant. The assumption that the jeep moved at constant velocity was checked by using the movie data to plot distance vs. time. The data for Runs No. 1, 5, and 7 are shown in Figure 1' The fact that the slope of each curve is substantially constant indicates that this is a go6d assumption. The importance of centrifugal and gyroscopic terms in the equations was investigte4 by reading out of one of the computer runs certain products itf aular velocities which enter into such terms. The magnitude of these products is shown in Flgure 12 in comparison w1ith the angular acceleration terms, oi and oj. It can be sn' tt:the velocaty-Broduct terms are in t17

The University of Michigan Willow Run Laboratoiries 0.5 rad/ec 101 —— IRun No. 5 Run No. 7 22.06 psd/see! 22.06 fpa,~_23.6 fps 0.1 rad/see _ X- 77' E-.Ru'n No1 L1 < IGURE 11. CALCULATED 0FRADVLCT.1 rad/sec 1 fps i.oi. vV enera much smalr ta eFramese = 1 second t i o ro inr e uat i<ions b nlt te tmio s dg 2.0 rad/ 2eo 0 4 8 12 16 20 24 28 32 0o TIME IN FRAMES FROM START OF RAMP FIGURE 11. CALCULATED FORWARD VELOCITY 2.5 rad/sec2\ FIGURE 12. COMPARISON OF GYROSCOPIC AND CENTRIPETAL ACCELERATION TERMS general much smaller than the acceleration terms, indicating that the error introduced into the equations by neglecting these terms is correspondingly small. In most of the runs, the pitch angle of the vehicle throughout the run has an increasingly negative bias and the height of the body CG drifts downward slightly. This may indicate that the concrete road was not perfectly level. If the field-test data were modified to remove the bias noted, the comparison with the simulated data would be somewhat improved. The comparison for Runs Number 1 and 2 are shown in Figures 13 and 14, respectively. These runs are for Case I, in which motion occurred only in pitch and bounce. In both runs the computer data compare closely with the field-test data in terms of both magnitude and timing. For the most part, differences are within the limits of experimental error cited a~bye: The comparison for Run Numbers 3, 4, and 5 are shown in Figures 13, 14, and 15, re-. spectively. These runs are for Case II, in which one raimp wa usid, and motion was not restricted to pitch and bounce. In general, the comparison is almos as elope as for the runs of Case I, particularly with respect to displacements (Z, gpo and 9)..Although the compute data compare well with the field-test data, it was of linterA t ailtlre the data iurther in ord~r' to determine, if possible, the source of the differencei aotde 18

The University of Michigan Willow Run *LatQ tQries 1 rIld/c -_ _ ft 0.1 red "0& tI#II 0.5 ft FIGURE 13. RUN NUMBER 1. Case I, 12.1 fps.-, field test; - - - computer. I rued s _ 0.6 ft!I - 4'p 0.1 rad l FIGURE 14. RUN NUMBER 2. Case I. 16. 1 fps. -- field test; - - -. comput. 19

The University of Michigan Willow Run Laborafories "0.1 r. "'-, —-... —---,t:gl'''t I~ 0i rlk - 1 see. 0.364 red r FIGURE 15. RUN NUMBER 3. Case II, 10.7 fps. - field test; -- - - comput Oh rd/eec 0. red/ o.325 r.od/m - 0.364 red t o ___ FIGURE 16. RUN NUMBER 4. Case i, 15. 2 ps. &-fedt - - - * uter 20

The University of Michigan Willow Run Laborafories 0 XT w-q O rtwt/v^____ 0.625 Orod/" oseA 0.,36.... FIGURE 17. RUN NUMBER 5. Case II, 22.1 fps,. - - -feld test; - - = conmpuer. The first step was to determine whether the differences Wre due to static or to dynamic. inaccuracies in the equations used for the simulation. Stttc errors would be present it the equations if they did not correctly represent the spring dharaCteristiCs or 0th gpobietry of the vehicle or the road. These errors would show up as errors in pitch and roil angloi and CG height when the vehicle was standing motionless on tbi roa&.:'TO'determine whether such static' errors were present, the vehicle was allowed to stand on th iath over the road obstClJes, while angles and distances were scaled off. It was then amoved and the measurements were repeated at i-foot intervals. (In this case, the use of direct-rather than photographic I. i4e'ur ments increased the accuracy of the data.) This test was thal simmted on the O puter by running the simulation at a very low speed. The reets.tia test made on the-vehicle ftr the obstacle configuratton of Case II are shown in Ftie 18,. The computer run is not own, because it was a nearly perfect match with the vehicl dati T'ld demnhstrates that itheh;r' ulation inaccuracies are hot caused by the static error Figures 13, 14, and 15 indicate that there were ppeC. bt.ouwnts of Yaw veloity -ok) during the runs. Direct observation of the mov ing-pi' j tndimie that ild$sUp amounting to several inches also occurred during tt eCa"ii tun:' Becavue of the omissioin of the lateral forces and accelerations in the eqruatip. u -" tSlmulm 4l. vtilr-m riO{" ab'ot

The University of Michigan Willow Run LtbL^ortoroiid 9^..o Oi O'G 0.1 rad', - X _ _"'_ r l II:!f -'r ltf #;Ftotlt'19. VEHICLE ROLL FOR CASE H RUN FIGURE 18. STATIC RUN. Case U obstacle configuration. its CG while the actual vehicle in fact rolls about some other point. In the Case II runs, for example, the vehicle wheels on one side go over the ob'tacle while the off-side weels remain on level ground. Neglecting the additional effect of sideslip, the vehicle then essentiAll rols about the point where the offside wheels contact the groutnd (Figure 19). Since the aquations used for the computer simulation do not include lateral acceleration, the simulated roll will contain inaccuracies. It is possible to allow partially for the effect of lateral motion and still uUlize-the simplified equations of Section B. 2 considering that this lateral mStion modifies the value of roll moment of inertia which should be used in the equations. This can be done by the equation: J=J.+m r2 oj o where r is the distance from the CG to the point about which the vehicle rolls, whioh iwlessentially fixed in the case under consideration. To cheCk the valtdity of this apprqaCh' SRun Number 3 was repeated on the computer with a roll moment of inertia of twice the driginal vaiue, and then four times the original value (Figure 20>. With a factor of 4, the comrputer roll data check very well with the field-test data. 22

The University of Michigan Willow Run Laboiatorjes 0.5 rnd/eec WJ 4J 0 0.5 rad. oj <oJ 4J o j 2J 20) 2J 0 FIGURE 20. EFFECT OF INCREASED MOMENT OF INERTIA. Run Number 3. --- fleltd test; - - computer. If it can be shown that for a Case II run the roll equation (Equation 35b, which lncludes the effects of lateral motion) has the effect of increasing the toll moment of inertia by a factor of about 4, then it can be concluded that Equation 35b will accurately represent the vehcole roll for the more complicated cases (such as a Case HIl run) when the vehicle does not roll about a fixed point. Equation 35c, which was used in the original simulation, gave the correct roll acceleration when an increased value was used for Joj which shall be designated here as KJo: Yl XF (F) oj -KJ n. X nk1) oj n Substitutions will be made in Equation 35b so that it can be written in terms of KJ.'-The value of K obtained from Equation 35b can then be compared with the correct value of K 4: To begin with, the quantity z in Equation 35b can be ignored since It is small; this then permits the constant Z to be brought to the outside of the summation sign: 1oj = Z"T~' J pn Fni En ni (2) 3 23

The University of Michigan Willow Run Laborarorie.s Equation (18b) gives an expression for the term j Fni: F = m ri (3) EF n I:o ot The acceleration ii can In turn be given by: r. = Z iL + X 2 f4 01 0o oj n In this equation, the centripetal acceleration term X wo i lPall compared with. Z ^ Figb. e 12); hence It will be ignored. Thus, if EquatioS 3 and 4 are used, EflSaOitft rewritten:.= ~[ mz z > -r XnFn) oj J j nmo o o- Now Equations 1 and 5 can be combined to eliminate the term n XnFi1nk a 1 Z 4^- rnz (L + Ki.oj [n o o o Oj 4) From Equation 6, it is now possible to obtain an expression for KJo: KJoj J0o nmo. ('T The factor of increase to be used for Jo is then: oj Z m Z K= 1 - 0 () K, The values of the constants in Equation 8 are: Z = -1.46 feet n m = 59.1 slugs Z = 2. 00 feet J.= 84 lb-ft-sec2 oj 24

The University of Michigan Willow Run Laborotori$s.. __................ - _ - - -- ~: ~. -.........* Usi;og these values, in Equation 8, a value of K = 3. 06 is obtained. This is not quite the factor of 4 desired. However, in computing ri" only the acceleration caused by rotation was considered and the lateral acceleration of the whole vehicle caused by sideslip was neglected, This sideslip acceleration will increase K. In the equations of Section B. 1, sideslip is accounted for, so that Equation 35b will give a true representation of the vehicle roll. This bub, stantially confirms the conclusion that the accuracy of the. smnulation can be improved by accounting for lateral motion. The computer pitch data would probably more accurately match the field-test data if surge acceleration, r j, were added to the equations. However, to do this, the simplifying assumption that the vehicle has a constant forward velocity could not be used. The comparison for Run Numbers 6 and 7 are shown in Figures 21 and 22, respedtivelyi These runs are for Case III, in which two ramps were used on each sid% and motion was not restricted to pitch and bounce. Note that, because of the presence of ramps under each wheel track, the maximum values of roll are about 0. 05 rad as compared with 0.36 rad for CaoiwI runs. The difference in scales between Case II and Case III graphs should be allowedt.orAi considering how well the data compare. In general, the comparison is less satisfactory.f Case II runs than for Case I and Case II runs. In Run Number 6, the real and simulated data are in fairly good agreement for the fifr 0. 6 second. After this time, the real and simulated curves diverge noticeably. DiscQntSinuittl in some of the curves at 0.65 second indicate that both the real and simulated vehicle ma] hitting the road on the left front wheel, but the effects on the motion are different, prAeumbly because of differences in body position and velocity at the instant of contact. In Run Number 7, the field-test run for roll velocity is not consistent with that (or roll. angle; that is, integration of the roll-velocity curve does not give the roll-angle curve. Roll. velocity data were obtained from rate-gyro measurements, whereas roll-angle data were obtained from moving-picture records. This discrepancy within the rteld-te4t data ie';; difficult to arrive at conclusiins regarding the run. It appears, however, tiat the pft^-angle data tend to follow the same trend as that for Run Number 6. 4 CONCLUSIONS and RECOMMENDATIONS The experimental and analytical program described in this reprt leads to thUfV iil g conclusions: (1) A complete set of eQatlons of vehicle motion forua whealed vhle has been developed and is summarize1 in Appendix A. These qat1ona permit the ara&Sliz ot 253

The University of Michigan Willow Run Laboraotries 0.5 rad i/ec a 0.6 rcrd/e It - _ _ _ Al i 05 ra.d..S —ft-'o, -.I 1.25 rad/p c 0 > 0 6r0d5 rad FIGURE 21. RUN NUMBER (G. Case III, 16. 7 fps. = field test; - - - P computer, 0 5 red /eeO 0.5 rid___ 125 rd/ o P ec,W r~~~~~~~~~~~~~,~~J i 0.05 r;ad I&''@ \v 11 1. FIGURE 22. nUN NUMBER 7. Care liI, 23. 6 fps.. field test: - - -,i comtepV, 26

The University of Michigan Willow Run La'boratorie4* motion in pitch, yaw, roll, bounce, surge, and sideslip. Simplifying assumptfira are kept to a minimum. (2) For many purposes, the equations of Appendix A can be reduced in compilexty by making further simplifying assumptions and by restricting the number of degees of freedom. The equations of Appendix B represent such a simplification for vehicle studies in which pitch, roll, bounce, and sideslip are represented. (3) An experimental program involving the comparison of field-test data and;the analIgs computer simulation of an XM-151 military truck indicates that the equations o Section B. 2 give satisfactory results under the conditions for which they were pre, pared. However, the equations of Section B. 1 result in improved accuracy of computation and are recommended for use in computer studies. (4) Further improvements in the accuracy with which vehicle motion can be simulated by computer techniques are believed to be possible. This could be accomplished by the use of improved instrumentation methods for field testing, for example, the use of magnetic tape recorders and the adaptation of gyros, accelerometers, and other measurement devices of greater accuracy and versatility. There are possibilities for simplifying the analysis of field-test data by feeding the recorded data from the magnetic tape reproducer directly into a computer setup. (5) Additional sets of simplified equations might be developed for other restricted types of studies. The equations of Appendix A might, for example, be reducedio-a twodimensional set suitable for pitch, bounce, and surge studies. Studies of steering and handling characteristics would require the use of equations which inclide yaw, roll, and sideslip. For studies of this type, accurate representation of the forces induced by slipping of the tires would be require4 27

The University of Michigan Willow Run Laboratories Appendix A EQUATIONS of MOTION The steps given in Section 2 for developing the complete equations of motion Of a wheed - vehicle are outlined below. (1) Equations are written representing Newton's laws ofmotion for both rotational and translational effects of the body and of each wheel. (2) Relations are written which represent forces transmitted by springs, shock abspbe*9, and tires as functions of wheel displacement and velocity with respect to the body otf0. the ground. (3) Coordinate-transformation equations are written relating displacements, veloCites, and accelerations in body-fixed coordiantes to those in earth-based coordinates. A. 1. METHODS OF NOTATION The reader should be familiar with the use of vectors to describe the position of points in space and should be familiar with the vector-product operation (Reference 3, Chapter I an&f:IX) Appropriate sections of References 3 or 4 are cited when it is believed they would be usefuL The vector product is used to describe the moments on a body, and velocities and accelera'tions when a moving coordinate system is involved. The equations are written for a vehicle with a total of N wheels, individual wheels being designated by n. The summation notation, Z, used in the equatons in a condensation of the n N more complete notation, A. n=l To explain the subscript notation one can use as an example the position vector n shown in Figure 23. The bar indicates a vector quantity. The subscript n indicates that the vector quantity is associated with the n-th wheel; the subscript 0 ndcatea s that it is the position vector of the body CG. The vector T has components in i', j', and k'A rections, which are denoted as rn,, rn,, and rn,, respectively. The two coordinate systems used are also shown in Figure 22. The lower-case letters, x, y, z denote coordinates in the body-fixed coordinate systen4 rnd i, I, and k are unit vectors in this coordinate system. The primed symbols are the corr'Aprnding quantities in the eartS, based coordinate system. The capital letters Xn, Yn and Z are t ie dimensions of the vehicle. X is the lateral distce of the n-th wheel from tle body CG, and Zn is. the distance 29

The University of Michigan Willow Run Laboratories a' Position of Wbeel ne whn Spring Is at Pree Length FIGURE 2:3. COORDINATE SYSTEMS of the n-th wheel below the CG when the suspension spring is at its free length (i. e., in Its neutral position). Since the wheels are below the CG, the Z are negative quantities. The quantity z in Figure 22 is the distanc'ethat the n-th wheel has moved from its neutral n position. In the equations developed here the wheels are constrained to move only in the z direction so that the terms x and y do not appear. If it were desired to describe the suspension in more detail, it would be necessary to write an equation in xn, y n, and Zn, which would define the path of the wheel as it moved with respect to the body. A. 2. POSITION VECTORS In this section, equations are derived relating wheel and body positions, velocities, and accelerations. The vector Y shown in Figure 22 is the position of the body CG in the earthbased coordinate system. We have i' +, r k', X' y. +z (9) = roi roj r0 " X? + Yop + zok' Differentiating Equation 9: r' x o' +,y'j + o'' (10) 30

The University of Michigan Willow Run Laboratories and again differentiating: r X ^i- + yo'+ z (U) j: m ii'+yo')+'k' (i!) The vector n is the position of the n-th wheel from the origin of the earth-based system. It can be written as the sum of F plus the position of the wheel with respect to the body CG: Fn a F i+ n + + )k (12) n 0 n n n n It should be noted that the first term on the right-hand side of Equation 12 has been given in tie' earth-based coordinate system by Equation 9. while the remaining terms are in the body-fLe' coordinate system. If the components of F are desired in the body-fixed coordinates it W1iU bi necessary to apply the coordinate transformation of Section A. 7 to Equation 9 to express ro in the body-fixed coordinates. Conversely, to express F in earth-based coordinates, the last' n three terms of Equation 12 must be transformed into the earth-based coordinates. In differentiating Equation 12, the unit vectors i, j, and k have derivatives since they are not constant with respect to time. The procedure for differentiating a time-varying unit vector is developed in Reference 3, Section 12. 3, with the resulting equations: - = w xi dt o t = W oXj dk - - ar= xk (13) where wo is the angular velocity vector of the body, EW.i c w.j +w k (14) = oi oj ok The symbol x in Equation 13 is the vector-product operation which is explained in Reference 3, Chapter IX. Differentiating Equation 12 by the use of Equations 13 and 14: r r = L (Z n) - W Y] I+w- n(Z + Aj { +hi e Y - wjX 1k (15) nbj(Zn *Zn L, n 03n 11ok n Lokn 1 n t31 31

The University of Michigan Willow Run Laboratories and r' - *,;Y+s )( a2 2 ) 2 1r:r, (z+o n zn)- -' Z ok,) + TI n o ) n oi o~ wolwok n o( - -2 2 2 2 \+ X - (Z + Z) w + W. (Z Z *W X +. 1 Lok n o0 n n oj ok n n oX j \Wj ok n oinJ + Y - x + + w. Y ( w (Z +z)zlk (l6a) L oi n ol n oi okn on j ok n o n n n( A. 3. BODY TRANSLATIONAL AND ROTATIONAL EQUATIONS The translational and rotational motions of the body may now be derived using Newton's laws of motion as a basis. The vector equation for the body translation is m r = F + mg (17 00 fn o n where m = mass of the body F = vector force acting on the body from the n-th wheel n g = gravity vector Equation 17 in scalar form is m M r x = F + mgog, (1k) o oi o o Fni + m0ga' ( n 0oroj = mo Fnj mogC jk( n omork =mz =:F m*gc, (20a) or ok o o nk +moge kk' (a) n The c, term in Equation (18a) is the direction cosine of the angle between the i and k' axes. The evaluation of the direction cosine comes directly out of the coordinate-transformation matrix which is developed in Section A. 7. The vector equation for the body rotation is dH dt =Mo (21) where H = angular momentum vector 0 M = the vector of the moments acting on the body All equation numbers followed by the letter a are the final equations used in the simulation; other equations are presented only for development purposes. 32

The University of Michigan Willow Run Laborato'ries In Equations 22 through 29, an expression is derived for the left-hand side of Equation 21 in terms of the moments of inertia and angular velocities of the body. Equations 30 through 33, which follow, express the right-hand side of Equation 21 in terms of the torques and forces acting on the body from the wheels. The corresponding i, J, and k components of the Left-hand and right-hand sides of Equation 21 are then combined in EqIUtions 34a through 36a to obtain the desired rotational equations. The angular momentum can be expressed in the form q where m = mass of the q-th particle of the body q V = velocity vector of the q-ltparticle r, position vector of q-th parele from the body CG Equation 22 is a standard definition of the angular momentum and Is derived in Reference 4, Section 5. 1, along with the relation q oq Substituting Equation 23 into Eq*tion 22: o=oZ mqfi"x(ro"] J(24) When the triple vector-product operation indicated in the. braket s S performed, Equation 24 becomes: [ok r -2 )xz 9w. Y ~ -. m.C qyqX (2q: q'q qi' q q q 2 2. -',. q ln \ Jok ll:Ig )q Pojk~rq mvR (26t 33

The University of Michigan Willow Run Laboratories For ex:imple, the term Jo is the moment of inertia of the body about the i axis and- Pj is the 01 product of inertia of the body about the plane defined by the i and j axes. For most vehicles we can assume that the body is symmetrical about the (j, k) plane, giving P. = P.= 0 (27) oij oik= 0 since for every mass particle in the positive x position there is a similar particle in the negative x direction making the corresponding summations equal to zero. Substituting Eations 26 and 27 into Equation 25: o = 0J oiWoJT [JoWo- Pojk ok o k wok - Pojk (oj Differentiating Equation 28 and remembering that the T, j, and 1E vectors must also be differentiated as in Equation 13, the rate of change of angular momentum is dH dt [ oit o (Jok- J oj ok Pojik k W [oj - Pojlk +o -ok + (Jo J)oiwk + Pojkkoi' -1. + [Joit0k - Pojk J - J oj ot' PojkWoiWok]I () The Jo terms in Equation 29 are the ones usually associated with the change in angufar momen-, tum, and the terms involving the squares and products of w are centrifugal and gyroscopic terms resulting from the use of a moving coordinate systen. Working now with the right-hand side of Equation' 1: the vector moment acting On the body is the sum of the torques plus the sum of the moments acting on the body from the'heels. In vector form, o n n E, ro0, nA n n where T = the vector of the torque on the body from the n-th wheel. The quantity ( - ro) is the position vector of the n-th wheel from the body CG and can be obtained from Equation 1. The term (Fn - o)XF' is the vector representation of the moment about the body CG cause4:by 34

The University of Michigan Willow Run Laboratories the force of the n-th wheel. This vector representation of a moment is developed in Referentce 3, Section 10. 1. Equation 30 in scalar form is M = Tni +. Y F k- (Zn + Z)F (31) n n n M.7YT Y (Z + z )F. - X F (32) Moj = Tnj (Zn Zn)Fi'E a nk n n n Mok = Tnk XnF nj n YFi (3) n n n Equating Equation 29 with Equations 31 through 33, the rotational equations for the body are TT.ZYnF,- (Z 4 F'n, J (Jok-J )uo, P-wo2.) )(34a) L n ni nnk + (Zn nFnj Joioi 0 ok oj)oojwok +ojk ok oj nj + < (Z+ z)F - ) JjJ -J )) Z F P =P. j (35\) n Zn n n ni n nk oj oj ojk ok'oi ook ko olok ojk oi oj ( n YTl ~+ X F - Y F. =J', -P', +(J -J.),.i. -P.W (36a) T - ^ n Fnj Yn ni ok ok ojk oj oj 0 oi j ojk oj qk n n n A. 4. WHEEL TRANSLATIONAL AND ROTATIONAL EQUATIONS The translational and rotational equations for the wheels are written in the same manner as they were for the body. The forces acting on a wheel are gravity, g; the force from the body through the suspension, F; and the ground-reaction force on the wheel, Gn. The translational n n vector equation for the n-th wheel is m r =G - F m g (37) n n n n n and when written in scalar form it becomes: m r.=G.-F- m gc, (33a) ni = Gni ni n ik mnr = G, - Fnj + mngck (39a) mn nk = nkk k mngck, (40a) The vector equation for the n-th wheel rotation is dH -- ~= ~~~~~~~M.~ ~(41) dt n 35

The University of Michigan Willow Run Laboratories which is similar to Equation 21. By developing the right-4aWnd side of Equation 41 in the same manner that the right-hand side of Equation 21 was developed, dH dt [ni n Jnj oj + (Jni J nkniok n k (Jnj -ni)niW 42) Several substitutions have been made in Equation 42. It has been assumed that.thewheel is a disc so that the products of inertia are equal to zero: nij Pnik Pnjk= ~ and nj nk Also, since the wheels rotate with the body about the j and k axes, J. =w. and' =W n o nk ok The suspension, connecting the wheel to the body, is reither wholly a part of the wheel nor of the body. A true representation of the suspension would lead to a system with distributed parameters. A good approximation for a conventional suspension is to lump one-third of its mass with the body and two-thirds with the wheel. As a further approximation, the wheel is assumed to be a disk in the model used here. This lends simplicity to the equations and is not considered to produce noticeable errors in the simulation, particularly when the wheel mass is large compared to that of the suspension. The moment equation for the n-th wheel is developed in a manner similar to that for Equation 30 for the body: M = -T +r xG (43) n n nw n The vector r n shown in Figure 24, is the vector from the wheel center to the ground-contact point so that the term rn x G gives the moment around the wheel center caused by the groundreaction force, G. In the mathematical model developed here it is assumed that the tire conn tacts the ground at a point which lies on the k axis when this axis is extended through the wheel center (Figure 24). Since the vector r has a component only in the k direction, it can be nw written w = r- (44) For the purposes of Equation 43, one can make the further stiplification that rnw -R (45) 36

The University of Michigan Willow Run Laborafories k' / Ground - Contact r / /rPoint -rk' FIGURE 24. SYMBOLS FOR THE WHEEL AND THE GROUND The quantity R (Figure 24) is the radius of the wheel, including the tire. The minus sign appears in Equation 45 because rnW is defined as the distance from the wheel center 4owa to the ground-contact point, a negative quantity, whereas R if$ a radius which i9 a positie quantity. Substituting Equations 44 and 45 into Equation 43 and writing in scalar form: Mi -T, +RG (4 ) M -T-,T -RO G(47) Mnj = nj RwGnt M = -Tnk (48) Equating Equation 42 with Equations 46 through 48, -T niR G.=J.c. (49a) nt + w nj nt ni -Tj -RwGni = Jnoi (Jni Jnk)wi ok 50 -'T^k JnXkrok + (JnJ -Jnt)ntwo 51) 37

The University of Michigan Willow Run Laborotorine The torque T. consists mainly of the driving, braking, and bearing friction torques applied to the wheel. An equation can be written for the wheel angular velocity, w., if it is assume'tht the wheel rolls along the ground without slipping. Neglecting the effect of rotations of the bidy about the i axis, the angular velocity of the wheel is the linear velocity of the wheel center, rnj, divided by the distance from the wheel center to the groundcontact point: r. i L'(5e) nw Differentiating Equation 52:.* rnw n rw rn " nw A. 5. GROUND-CONTACT POINT In order to develop equations for the ground- reaction-forces, it is necessary to wrtUe equations which describe the point at which the tire-contacts the ground. As shown in Figure 24, the vector r is the position vector of the ground-contact point with respect to th origin ng of the earth-based coordinate system. I can be sei, that r ~r +f (5r) ng n nw An expression for the wheel position vector, r is gives by Eqltion 12, and r is givd- b Equation 44. Considering the i' component of Equation 54, r gi' + rfnl~ wk (sa) Similarly, for the j' compQllnt, rngj rnj' rnwCkj, (6) Because the vehicle can travel in the three dimensions, the gipmd over which the vehicle travels must be described by a surface, as shown in Figure 24. The equation for a surface is of the form z = f(x, y) ({7) Equations 55a and 56a are expressions for the x ai5d y of Bquattdi 57, so that th elevatio ( the ground at the wheel contact point is given by 38t(r, a i 38

The University of Michigan Willow Run Laboratories which can be determined by a survey of the land over which the vehicle travels. If tbm vebile is to travel over a prescribed path, Equation 57 reduces to a two-dimensional equatein Oi t* form z = f(x) To have a complete set of equations, it is necessary to write an equation which exz*etJ r in terms of previously defined quantities. From the geometry of Figure 24, it can bi nW seen that r = rngk' rn nk' nw c"Io Ckk, A. 6. FORCE EQUATIONS The equations previously derived have included terms representing the various forces acting on the vehicle. These forces can be described in terms of wheel displacement and velocity with respect to the body or the ground. The force Fnk is the force on the body or wheel caused by the suspension spring and shock absorber: F = K (z ) + C (z) (z) (60a) nk ns n ns n) ns n) The first term on the right-hand side of Equation 60a is the force on the body caused by the suspension spring. It is the spring force-vs. -displacement function which has the spring displacement z as the independent variable. The function C (ftn) is the suspension shockn fla n absorber force-vs. -velocitycharacteristic. The shock-absorber force may depend not only on its velocity but also on displacement; hence the term 0 n(zn). The tire exhibits spring and damping characteristics in much the same manner as the suspension. Thus, Gnk Knww n w) C nw )nw( +nw) (a) The independent variable (R + rn) and its derivative (rw) are, respectively, the amount w nw nr the tire is deflected (Figure 24), and its deflection rate. Expressions will now be derived for Gni, the sideward force exerted on the wheel by the ground. Two conditions will be considered, one in which the tire does not slip sidewise, the other in which it does. The vector rn is the position vector of the ground-contact point from the origin of the earth-based coordinate system. The i component of the correspoding acceleration vector,'ngi. will be derived as a function of forces acting on the weeL From Figure 24 r Zr or =r +r i (6) ng n nw n no w.' 39

The University of Michigan Willow Run Laboratories Differentiating this equation gives _- t - dk -r = r,ir k+r r( — ng n nw nw dt and since dk =w xk (13) dt o then 7 =n r +~ k~r (ao xk) (64) ng n nw nw (0 o Differentiating Equation 64, multiplying both sides by m n, and taking the i component: mr.=m i. + m (rm +w o r. +2w.r i) (65) n ngi n ni n ojnk 0 oiokk k o) nk Substituting for m r. from Equation 38: mr.=G.-F.+mgc.,,m(u.r + ~w t r + 2w.r) (66) n ngi ni ni n k' n oj nk + 0ok nk oj nk Two cases may be distinguished. In Case 1, G. i remains less than |iG, where t is the coefficient of friction and Gnk is the k component of ground reaction. Under this condition, the wheel will not slip sidewise. For Case 1, the foUowing conditions apply: Gni |< Gnk l (67) and ngi gni (68) so that Equation 66 becomes Gni = Fni - m (gcikf tojrnk -t oi.Wokrnk + 2w ojr) (Case 1: 69a) ni ni n ik' oj nk oi ok nk ojink + For Case 2, the wheel may slip sideways. Under this condition, assuming that the plane of the ground at the ground-contact point is normal to the k axis, the sideward ground-reaction force is equal to the vertical ground-reaction force tims the coefficient of friction. Expressed in equation form, r,.#o ngi and IG l = jG j i (Case 2: 70a) 40

The University of Michigan Willow Run Laboratories To compute G i, then, Equation 69a ts used when the condition of Equation 67 applies. When G i from Equation 69a becomes greater than Gnk j, the magnitude of Gni. s given by Equation 70a. The sign of Gni for the condition of Case 2 is the same as that given by Equation 69a, since Gni cannot change sign without going through zero under Case 1 conditions. It has been assumed either that there is no slideslip of the wheel or that, If there it sideslip, the coefficient of friction pu is a constant. It should be recognized that these assumptions are approximate only. In steering and handling studies a more detailed analysis would be required, one which uses either an analytical or empirical representation of the tire-slip angle. A. 7. COORDINATE TRANSFORMATION In the preceding equations, vectors have been given in terms of one of two different coordinate systems. It is necessary to be able to convert a vector given in one set of coordinates to its representation in the other set. For example, r is expressed by Equation 9 in the earth-fixed coordinate system; expressed in the body-fixed coordinates: r= xc +y c + ZoCk Ij o [ o ii' Yoj i oki] [X ci'j + Y0cj'j +Zckj]J + [xcilk + YOcIk + ckk]k (71a) where the c's are the direction cosines between the axes indicated in the subscript. The(direction cosines are obtained through the use of Euler angles, a method which is thoroughly discussed in Reference 4, Sections 4. 1-4. 4. The Euler angles used here are Azimuth A, Elevation E, and Bank B, as shown in Figure 25. The resulting transformation matrix is: cos A cos B - sin A sin E sin B cos A sin E sin B + sinA'cos B - cos E sin B sin E IT]= - sin A cos E cos A cos E cos A sin B + sin A sin E cos B cos A sin E cos B - sin A sin B cos E cos B (72)and 3. T' TJa}41

The University of Michigan Willow Run Laboratories x.1 y. * 8'I~~ Cii, Cij,%' x ^i' -7ry' aCk'1 x ki' Ckj, C FIGURE 25. EULER ANGLES Each of the elements of the matrix of Equation 72a is a direction cosine between an axds of one coordinate system and an axis of the other. For example, the upper left-hand element is the direction cosine between the i and i'axes, namely c11,. The complete transformation matrix may therefore be represented as, ki c cK a The rates of change of the Euler angles with respect to time are given by the following: iA Oicik +oj Ck' j * + e'o E = woi cos B + wo sin B (74a) B = %oj 42

The University of Michigan Willow Run Laboraotolies, and the Euler angles are: A = Adt E = Edt (75a) B = Bdt A. 8. SIMULATION EQUATIONS The equations to be used in the simulation (the equations with numbers followed by the letter a) are rewritten here as explicit expressions in the dependent variable. -x i J1 [~ ET + Y Ynk - L (F + Poilok (Jok - o; ok oij - J1 Tn + E nFnk-(Zn + n)Fni E- (- oi'ok o- ojook Oi' oj n )n n (oj i-I^FiCJO Jiko k3o i +- WP)(j (3()i) Wok [n T (XnFnZ nj - Y Fn + oioi -(Jo -'Joiojo (3) o mo ( ni mogikj (a) Yo mo( 7nj og jk') (Z9a) o m n y m~o =~^F.m T Egx F (l6a) n n n =nk - rok - i(Yn - OokXn) o n ok n ( 2oi t Woj )(Z + zn) (k compone't of 16a) 1 0 n n r ~ J (G -F 4m gc"t) (40a) n m k Fnk ng nkk n Fnk Kn (Zn) + C (Z)ns(zn) (60a) ~' - ~'nk ns' n%~~' X'i %ns n43

The University of Michigan Willow Run Laboratories rngi' rni' + rnwcki' (5a) r.,= r., + r c., (56a) ng, = rnj rnwCkj' rngk' f(rngi rngj) 58 r ngk' nk' nw Ckk' =K (R + r )+ Cnw( )~ (R + r ) (61a) Gnk Knw(Rnw nw) nw nw w nw Tnk Jnk ok (Jnj nii (a) r r-.-r r. nw nj nw nj (53a) ni 2 nw J.w. + T G ni ni n(49a) n) R w rnj. = X-oi(Zy + z) + oj (Z +z ) rnj'o ok n O n n ojk n n + o..X (2. + w2 y 2w. (j component of 16a) bo o) n Is ok n oi n F. =C + m ( r-t )(39a) nj = Gnj + mn(gCjk nj) T.=J.~.-(J -j Yww +RG (50a) nj njjoj -( oi Jok)^niokw + R (wGa) rn = x +,oj(Z + n) - on + WoiY (i component of 16a) ni o Oj n n Okon OiOjIn Fni = Gni + mn(gcikt - (38) 44

The University of Michigan Willow Run Laboratories Gni =F m(gc-, r +' r g + 2 ) (Case 1: 69a) ni ni n ik' ojrnk oioknk ojnk IG'i = IGnk (Case 2: 70a) xf' = x,+ Y C +z o XoC il' oji ocki0 YO = XOCiji + yOCjit + ZCkjr zo' x C + YoC + 0 k' 0o =oik' o jk' * oCi cos A cos B - sin A sin E sin B cos A sin E sin B + sin A cos B -cos E in B sin CT]= - sin A cos E cos A cos E c A in B + sin A sin E cos co A sinE B n A B cos B c B (12.) j T rj (73.) A = WiCik' WojCk' + uokCkk E Iw. cos B + ok sin B t7) 01 ok oj A = Adt E = Edt (1:5a) B =Bdt Appendx', SIMPLIIED EQUATIONS'of MOTION The equations of Appendix A were derived in a siraighrwi anner from a nmithet: atkal model of the vehicle. For nany applications, thb equat6ios o A4ppcte A canbtie Impliwte without noticeable degradation of results. bIn ectionM aB t / e*ueiattonls of Appendix A are. re written using several simplifying a8sumptions. TbMie iaatiow bouild be suficiently accurate 45

The University of Michigan Willow Run Laboratories to simulate a vehicle which is not being turned through large angles and which is moving'it constant speed over cross-country terrain which is not excessively rugged. In Section B. 2, additional simplifications are made to the equations of Secftoh B. t by eliminating the factors of sideslip forces and motion to derive the equations that were used to perform the analog-computer simulation of the XM-151. a wa* shown in Section 3. 8, when the XM-151 field-test results were compared with the stliulated results, that a better simulation could be obtained if lateral motion factors were added to the equatons used. B. 1. RECOMMENDED EQUATIONS The equations of Section A. 8 are rewritten to include the following assumptions: (1) Since the vehicle does not roll, pitch, or yaw more than about 10~, the comnpnentx of a vector are almost the same in both coordinate systems. Using a two-dimenstp1ol example: a force of magnitude F in the k direction has components in the j' and k' deviations of: k = F sin oj' + F ces 0Ei Fk' This concept can be used to simplify the equations of Appendix A, and the coordt transformation matrix, Equation 72a, can be written: 10 01l [T] = 0o 1 o0 0 t The equations in Appendix B are written with unprimed coordinates because there-to' no longer a distinction between the primed and unprimed coordinates. (2) Figure 11 shows that in the field tests the vehicle had a netrly constant forward *1ot - ity. Therefore, it is assumed here that the vehicle's forward velocity is constant: r. =V oJ This implies that the forces in the forward direction ati.ng on the vihicke are zerb* Gn= nj -.0 (3) It is assumed that the, torques caused by tbhe q^r a trationS of thiwhe* a4ret small: Tn. T = Tnj, nk 46

The University of Michigan Willow Run Labordtoriet Part of the torque, Tni, ts the driving torque, which ts zero from the conditions ol the preceding assumption. (4) The products and squares of the angular velocities are assumed to be negligibly small compared with the angular accelerations. Also, it Is assumed that the CorioliaS accelerations are smalL Expressed mathematically, 2 wQ4 W, 2wz) < < n Figure 12 shows the relative magnitude of some of these terms as obtained from the computer simulation discussed in Section 3. 2. Bice the results of any simulation will give the angular velocities, w, and angular rates, c, it ti a simple matter to 2 check the approximate magnitudes of the w and ow terms to see whether appreciable error can occur by neglecting them in the simulation. (5) All products of inertia are assumed to be zero. The equations of Section A. 8 are rewritten here with the above assumptions having been made. To distinguish this set of equations the letter a in the equation number has been replaced by the letter b. i- E YFnk (34b) l f =Z )F.- /X FA j (35b) oj J. n nk o0 j L n nni n -1,s — 1 Y F. (36b) ok J ok n n ni ok n r. =x =- F (18b) oi01 o m n o n ok o m [ Frl +lm ] (o0b) z = nk - r' -.Y *+.X (k component of 16b) n, o oi n oj N r - 1G,-F + m g] (40b) nk mn nk nk nn m Fnk = Kns(Zn)+ C ns(in)ns(zn) (60b) 47

The University of Michigan Willow Run Laboratories n Y rngk= f \+ -V ) (58b) I = r - r (59b) nw ingk nk GC =K (R + r )+C (i w)%w(R w r ) (6l1b nw w nw nw nw nw w nw ni = + Wo (Zn+z)' YokY (i component cf 16b) F.: oC.- m r. ni ni n nl Gi= F. - m.r (Case 1: 69b) ni ni n oj nk jGnil= JAGk (Case 2: tOb) B. 2. EQUATIONS USED FOR COMPARISON TESTING The equations used for the analog-computer simulation of the XM-151 (Section 3. 2) carn be obtained from the equations of Section B 1 if lateral forces and acceleration are dropped out, that is, if F. = G. = r, 0 ni ni i Under these conditions the vehicle yaw, Equation 36b, also drops out, leaving only pitch, bounce, and roll. The letter c is used in the equation numbers below to distinguish this particular set. As a guide, a brief description is given for each equation. Body Pitch w CJX? YF(34c) oi JL. F (34)n oi n Body Roll w. = - 8 X F (35) oj J. n nk oj n Body Bounce o m I- F t m g1 (20c) 0 m I lnk o o I n OLn. Acceleration of Wheel with Respect to Body z = r - - z - Y + X (16c) n nk o oin oj n 48

The University of Michigan Willow Run Laboratories Wheel Bounce i: = - -G -F +m gl (40c) n - Suspension Force F =K (z )C (i)< (z) (60c) nk ns n ns n ns(n) (60c) Terrain-Elevation Function / Y \ rag= f 1 + (58c) Tire Deflection r = r g- r (59c) nw ngk nkv Ground-Reaction Force Gk=K (R rn) Cnw(rnw)nw(Rw + rw) (61c) nk = nw ~w~ nw nw nw nw'w n49 49

The University of Michigan Willow Run Laibofatories REFERENCES 1. I. J. Sattinger, E. B. Therkelsen, C. Garelis, and V. H. Geyer, Analysis of the Suspension System of the M47 Tank by Means of Simulation Tectir qs, Report Number 2023-2-T, Engineering Research hItttute, The Un versit~yr Michigan, Ann Arbor, Mich., June 1954 (UNCLASSFtIED). 2. E. M. Grabbe, S. Ramo, and E. E. Wooldridge, Handbook Qf Atopmation, Computation, and Control, Wiley, New York, N. Y., 1958, VoL i, pp. 20-51. 3. J. L. Synge, B. A. Griffith, Principles of Mechans, 2nd ed., McGraw-Hll, New York, N.Y., 1949. 4. H. Goldstein, Classical Mechanics, Addison-Wesley, Cambridge, Mass., 1950. 50

AD Div. 11'2 UNCLASSIFIED AD Div. 1142 UCLASSIFIFD Willow Ruo Laborato-ics, U. of Michiu.i. Ann Arisr 1. Vehicles —Motion Willow Run Laboratciris. U. of Michigan, Ann Arbor 1. Vehicles-Motion COMPUTER SIMULATION OF VEIIICLE MOTION IN THF[L 2. Mathrniatical computers- COMPUTER SIMULATION OF VEHICLE MOTION UN T1llikE 2. Matlirtnitict l computers DIMENSIONS by I. J. Sattingt-r and D. F. Smith. May 60. Applications DIMENSIONS by I. J. Sattluger and D. F. Smith. May 60. Applications 50 p. inl. illus., tables, 4 refs. 1. Sattinger, I. J. and 50 p. incI illus. tables. 4 refs, 1. Sattirger. I. J- and (Technical rept. no. 2901-10-T) Smith, D. F. (Technical rept. no. 2901-1Q-T) Smath, D. F. (Contracts DA-20-018-ORD-14658 and DA-20-0I8-ORD-1)CS) 11. Ordnance Tank-Automotive (CantractS DA-20-018-ORD-46S8 and DA-20-018-ORD-19,A5) II. Ordnance Tank-Automotive Unclassified r port Command Unclassified report Cci ansI'Thlsreport describes the results of a research program to de- nL Contract DA-20-018-ORD- T report describes the results of a rieseach program to.e- m ctract DA-20-016-ORDaWtep tecbhrues of simrnratltieediOTsina nitrtiin of-a18-14D5 velap techniques of sfrnuiating three-dimensioi41 sation of a C t 0 IV. ContractDA-20-018-ORD- IV. ~~~~~~~~~~~~~~~~~~Conrac A2.1.jt) whice by meaos 4Uelectrohnc computers. In order to provide 3 vehicle by means of elcctronkic computers. In orderto Wropvide Iv 35 the-t iis tOr vehi cte-ei~on timrthatini, a ConPtUqtvset of eu- ebas for vettle moAon sirtoulatiton. a coijpisew Bet si equs- iins foWr a wheeled vehiclev li dvelope4, which permatted the tions for a wheeled vehicle was developed. which perhuited the AiAs){~ts'*-~~ Baotion i pitch, yaw. roil, bounce, surg~e, and side- analysis of motion in pitch, yaw, rolt, bounne; surge, a sioe-. Slip. Th dtrtvican #nd stminoary of t*e$e equtiuns is presented slip. The derivation and summary of these equations is presented is t e reporft- As -a meAeasoo re ttg t s of c iAcing the time and Cost of ceon- pslea adlutnotaBfow certaur restricted cases of vehicle mition, an ailer solutions for certainC sestricti case's of ve~hicle motion,,an Investigation was mtnade to determinetn twhat-extent simplifying inves tion was made to determine to what extent aimptlfyg assumptions Could be made in these "equations. Asimplified set Armeid Services assumptions could be made in these equations. A Licptifitid set Armed Serviges (over) Technical Information Agency 4creer) Technical Information Agency UNCLASSIFIED UNCLASSIFIED AD Div. 11/2UNCLASSIFIED AD Divi 11/2 Willow Run Laboratories;- V. of ichigan, Ann Arbr I. Vehicles-Motion Wtow Rus Laboratories, U.' of MichignAnn Atbor 1. VehicleV-Motton COMPUTER SIMULATION OF VEHICLE MOTifN tI? IiREE2. Mathgnnatli coniputers- COMPUTER SIMULATIQN OF VEIICLE MOJON 1N THREE 2. MathematicalcomputersiMENS=ONS y.J SatingerardI; i M 0pticattis IMENO-NS by. Satingier and D. F. Smith. My 0. Appcaions $O p. mcI. in ls*., tables. 4 ref..: Sattanger,,1. 3. and Sp. inct.ileis., tables, 4 refi. I. Sattaner.... and <1eeical 4et.'po. 2-1-10-T) Smith, D. F, (Technical rel. no. 2901-tO.T) snfet1.r. (CBoezrsunp A~040~tn.1465&ad DA-2O-0l9-OFt-1 M35 II. Ordnance Tank.Automat.ve (ContractDA-20-tS-QD-1465 %andIl)20 R ) I pr~naz 5(cs *-Aotoindtiw lscelassified report Comniand — t - a""sas* TfAT pott Q~n^B the res^ ^ a r~arch prograrto^ n Cntr -2O-(n8-ORDy Teport de3crilies th reslls $of ayesearcl progeam to e- f1 e tac 20$-4& -( T rk r~ot &ecrshes the resolts.Q Thaiesr' rgamisti-I65 eeport drusco eestsesg nin* 4 ^elogteclnndboestif lnnuiintafg thse-lmesioa mtonofa Cnract 0A 2fl ft~OPD vel~fchte by eas^ of sileetr~hrek-o muers'ion ~ordrtontpovide ^ CostrtD^A-Z0-&ih-rOft ~Yebi jie by meansof electronic^^ ftativputer In order to ptrovide ~ v tootee ltsi5 fo-r^ whfle ^nesoem-^mot att acoepplete s to deq u1tin M^s oc~vehicl-mt<dieia<tntrlaio a comoplete set of equa-i~. h; t~vbc-o^ " lns (aOra.^w heeled ehiclce was developed. which permitted theta:'61zd d *tratsis alacrtionfts pstcfs, yaw, roitL bounce, eurgq, aptd side- SinalyiaiBof motion in pitch, yaw. roll. bounice^ surge, att side*lip fthe lesrivation and summary of these eqyisticins is presented slip. The deristion atl summary of these equations is-presented ftalire rnwa. As a mesanso reducing the timae snd cost of corn- 5 hi -report:a t eans ot reducing ten trm ai cSir n-of cot-uter sohitioai for ce~ain re~rictd caa~ei of vehicle m ~tro, anWT poles autions fi<ir certain restricted caes^ of rrehaejse motion. A' i puter soution~ (r ceitisal ressd ted s.b e0poi bnvestitin wrs iade to deteermne to what extent sprenide 4nestugatton was made to drmterssturto What extent smunpltyng aeui or fbem sadel n tiese e Aiaons A sertof e tArmed Servic-s asntirptii ould beiadein teseA p Armed Services a ed v 4(over)wTechhiaal informatiod Agency (opet) Technicalhlntormaton Aeenuy ~ K<BCLA^S^iED U'"<lNCLASSIFIW?-" " + + +1

UNVERSITY OF MICHIGAN 3I0539Illlllllllllllll 63llllllllllll 3 9015 03696 6343

AD UNCLASSI FED AD UNCLASSIFI D Of equaton i contained In the report for eass la whtcl surge can be UNITERMS Of etilo s iU contand n o th or repben Ia hih tt *e can b UN'TfRMS neglected and angular mrtions un pitch ead roll do ln exceed tO1. A ctId lted and g,.mlau io tn C asr1ai do not eceed 10. A *ecoad *Impthhcd etl of temuttons contains the iurther restrlcUto that. Veticle eg d rti ~ o a ftef rerttetoe that VehleI oly ptch. bmnce. and roU occur. A alo cu pu ter bitch. Md r Ce.,..An tr Analog cu~onpater Anal eomputcr In oretr to establ4h JuatilcAton (or the lIitI fed qcations, an ex- Simoatlq Ji.ordi r to etablish jlait tftl n g the sImplifed teqtal. an ex-Simultion pIi entaAl pr atm was cartij. et on an XNl-ll miltary truck to Wleel iwer nta) prtgraar ws caryleriot on an nXl-151 ailtary tueit toWheel drtew e (its motUa whln travrun over certa rmoad otbetcle under Pitch dtWrmU it moon wtBn traengtwe oqr CetJh roaotalesi* under Pct q* actJ ls TihCe rtl pent. were r compred itl tihooe of co- Yaw *ptfhe6 0U4t:t*w e1eu~ war;etY p coaretd tWhw hd car- Yaw t M UA *PR^-Wqsptr WAP n. a WAS coOha rwee. f runs. r cet. d sn'u eon u mut*. In e,, f I coo.udedoue ^t&im aQlrwivg only ftor tpatch, lwnccb, hrb& aoitWt ro^<3tc? tSRoll OWt e y *t q M bits roll Rol maU4^bttf ta reprttat a, ol <>mrue dm gen a, Out tVaeut~ e ee *asa4 ~fyit g_ but, batS* tt l'ubM'tre ti could be r obtAlnd by t.t. mpte wt ds 8bMtdestip tLfrd cTted corrett b oained y1 th. illei Wnt d Soldfti.e.w t'e~^ct0~o.. latoral mauont ai we- tquatin eptl riCh aeouarld ^ k.qr atIm e MI' 1. Eqstaim WI tert Prlid test UNCLASSIFI)D UNCIASSi FI8 + h t t.,~''. ~,n kr -,.> _e!mc~aitw -| 1 ACLAaSSXW ^S^&y^'11^ >^fi;w^!^^AOj^ (Mt^Hw^J<^toCt~~^J^~fl~<~i mforU e~tlj~iL~I~e)A N tq ^c^^jlfl<<AM woelft,i~9^ w ^w e ~tmvelg over ertsn rPt *lk<Bttffi^^l~i^^ A 4mb ic.0004 a trapg tw a n u to-cnapater d~ulah b wa edhel4tamce'~'g~'. r>g*~mo!i$f eit A t piteih o, l^ lt fth l$trmti lm4qulttOns allowug only lor IC *int; ant sl le,4i+t~t^ #s^i w tetnct~r eipc-4ftlqtion ofju f-t0ext dj ^.rmigtipe r gave a ^tsacttay reperesettion of t1* fIld-tet data., ba ae ~f u.rttprq wreI~rod 66itd' l.".4p d by,' wiWd1'jOtir Side ip t*a t 5iprowed correlatioe coutd be obtamed by the lw pil et'd'cStetlp ~..Npns'wjri i g*matted eAalfm s*ton vtatch a(edaited ftor aterr motion As welt. I d -' F" eldS test.*. tmJL UNCLASSIFIED