INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. These are also available as one exposure on a standard 35mm slide or as a 17" x 23" black and white photographic print for an additional charge. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI University Microfilms International A Bell & Howell Information Company 3 0 0 Nortfi Z eeb Road, Ann Arbor, Ml 48106 -1 3 4 6 USA 313/761-4700 800/521-0600 Order Number 8907262 Application of suspension derivative formulation to ground vehicle modeling and simulation Maalej, Aref Younes, Ph.D. The Ohio State University, 1988 Copyright ©1988 by Maalej, Aref Younes. All rights reserved. UMI 300 N. Zeeb Rd. Ann Aibor, MI 48106 APPLICATION OF SUSPENSION DERIVATIVE FORMULATION TO GROUND VEHICLE MODELING AND SIMULATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Aref Younes Maalej, B.S.M.E, M.S.M.E, M.S.Math ***** The Ohio State University 1988 Dessertation Committee; Approved by Q Dr. Dennis A. Guenther Dr. Jack Collins AdvisoçX Dr. Chia-Hsiang Menq Department of mechanical Dr. Michael Foster Engineering Copyright by Aref Younes Maalej 1988 ACKNOWLEGEMENTS The author is very Greatfull to all who helped directly or indirictly in the completion of this work. The list includes many which makes it impossible to list all individuals in this acknowledgement statement. However special individuals merit special commendation. The author wishes to express his gratitude to professors Dennis Guenther and John Ellis for their invaluable support, assistance and encouragement throughout the entire period of the work. I extend my thanks also to the staff of Systems Engineering Associates (SEA Inc.), particularly to Betty, for their personnel support and their help in the typing of this document. Finally, I extend my appreciation to all my family members, particularly to Dad and Mom, and to my brothers in the OMAR IBN ALKHATTAB Mosque for their personnel support which was necsssry for the completion of this work. IX VITA February 3, 1963 Born - Sfax, Tunisia December, 1984 . B.S.M.E., The Ohio State University,. Columbus, Ohio 1984-1988 Research Associate Department of Mechanical Engineering The Ohio State University Columbus, Ohio March, 1986 ......... M.S.M.E., The Ohio State University, Columbus, Ohio 1986 ........................ Assistant Project Engineer Engineering Department Shell Tunirex Tunis, Tunisia 1987 . .................. Assistant Project Engineer Systems Engineering Assoc. Worthington, Ohio June, 1988 .................. M.S.Math, The Ohio State University, Columbus, Ohio 1988 ........................ Teaching Accociate Department of Mechanical Engineering The Ohio State University Columbus, Ohio PUBLICATIONS Maalej, A., " Modeling of Mechanical Friction at the Piston Cylinder INterface”, M.S. Thesis Maalej, A., Guenther, D., Singh, R., " Effect of Friction on Dynamic Response of an Actuator", NFPC Journal Vol.l, No.l, June 1988, and 42nd National Fluid Power Conference - ASME, March 1987 111 Maalej, A., " A Unified Method of Large scale System Model Reduction", Les Annales de L'ENIT, Vol.2, No.2, Juillet 1988 Maalej, A., Guenther, D., Ellis, J., " Numerical Stability of Vehicle Dynamics Simulations", LASTED International Symposium in Applied Simulation and Modeling, Texas, May 1988 Maalej, A., Guenther, D., Ellis, J., "Modeling of Tire Friction Forces and Moments" Accepted by the International Journal of Vehicle System Dynamics. FIELDS OF STUDY Major Fields: Mechanical Engineering Mathematics Areas of Specialization: System Modeling Simulation and Analysis . Mechanics, Advanced Dynamics, and Vibration Control Systems, Process Control, Digital Control, Large Scale Systems, Stochastic Processes, Signal Processing Machine Design, Advanced Mechanical Design Automation, Fluid Power Control Mathematical Methods in Engineering, Abstract and Linear Algebra, Theory of Real and Complex Functions, Integral Equations. ÏV CONTENTS _______________________________ Page ACKNOWLEDGEMENTS ...................................... Ü VITA ..................................... iii LIST OF TABLES........................................... viii LIST OF FIGURES ..................................... ix CHAPTER I. INTRODUCTION ............................ 1 1.1 General .......................................... 1 1.2 Scope and Objective .............................. 2 1.3 Research Overview ............................... 3 CHAPTER II. LITERATURE REVIEW ....................... 4 2.1 Introduction ....................... 4 2.2 Multi-body Methods in Vehicle Dynamics ......... 4 2.3 Classical Lumped Methods in Vehicle Dynamics .. 12 2.4 Review of Previous Work for the Current Research ........................................ 14 2.4.1 Vehicle Body Subsystem and Inertial Properties ................................ 14 2.4.1.1 Weight Distribution ............. 17 2.4.1.2 Center of Gravity Location ....... 17 2.4.1.3 Inertia Parameters ................ 18 2.4.2 Suspension Subsystem of the Vehicle ...... 20 2.4.3 Steering System of the Vehicle ........... 26 2.4.4 Measurement of Vehicle Parameters ....... 28 2.4.4.1 Base Structure of the SPMD ....... 32 2.4.4.2 Suspension Support Frame ......... 32 2.4. 4. 3 Wheel Pads ......................... 32 2.4.4.4 Hydraulics ......................... 34 2.4.4.5 Instrumentation ................... 34 CHAPTER III. EVALUATION OF TIRE MODELS AND FORMULATION OF SUSPENSION DERIVATIVES ............. 41 3.1 Introduction ...................................... 41 3.2 Evaluation of Tire Models ...................... 41 3.2.1 Analytical Models ................... 42 3.2.2 Empirical Models .......................... 45 3.2.2.1 Lateral Force ..................... 45 V 3.2.2.2 Longitudinal Force .............. 49 3.2.2.3 Alignment Moment ................. 50 3.2.3 Combined Braking and Steering ............ 58 3.2.3.1 Friction Ellipse Concept ......... 58 3.2.3.2 Shaping Function ................. 58 3.2.3.2 Friction Cake Concept ..... 59 3.2.4 Computation Method and Model Evaluation .. 64 3.3 Suspension Derivatives for Kinematic Suspension 71 3.3.1 Suspension Derivatives of a^2-D Suspension 73 3.3.2 Suspension Derivatives of a 3-D Suspension 83 CHAPTER IV. DEVELOPMENT OF VEHICLE MATHEMATICAL MODELS .................................. 95 4.1 Introduction .................................... 95 4.2 Lumped Parameter Vehicle Model ................. 96 4.2.1 Equations of Motion ....... 96 4.2.2 Determination of Forces and Moments ..... 100 4.2.3 Camber Effect on Forces and Moments ...... 102 4.2.4 Longitudinal and Lateral Slip ........... 104 4.2.5 Wheel Dynamics ............................ 105 4.3 Vehicle Model with Kinematic Suspension ....... 105 4.3.1 Formulation of Vehicle Model with Suspension Derivatives .................... 108 4.3.1.1 Body Motion Equations ............ 108 4.3.1.2 Suspension Kinematic Equations ... 108 4.3.1.3 Generalized Forces .............. 109 CHAPTER VI. MODEL SIMULATION '................... Ill 5.1 Introduction ........... Ill 5.2 ACSL Simulation Software ............. Ill 5.3 Numerical Stability of Vehicle Dynamics Simulations .................... 113 5.3.1 General Formulation of Ground Vehicle Dynamics .................................. 113 5.3.2 Integration Methods in Vehicle Dynamics Simulations ............................... 114 5.3.3 Open Loop Versus Closed Loop Integration . 118 5.3.4 Numerical Illustration .................. 119 5.3.5 Analytic Analysis of Simulation Stability. 121 5.3.6 Numerical Stability and Vehicle Simulation Formulation ............................... 129 5.4 Simulation Model Organization ................. 129 5.5 Vehicle Handling Maneuvers .................... 136 VI CHAPTER VI. SIMULATION RESULTS AND DISCUSSION .... 140 6.1 Introduction ...... 140 6.2 Vehicle Suspension Derivatives ................. 140 6.3 simulation Maneuver Response ........... 161 6.4 Simulation Times and Costs ..................... 182 6.5 Introduction of Developed Simulation to the SPMD Software Package ................................ 190 CHAPTER VII. CONCLUSIONS AND RECOMMENDATIONS ....... 200 7 .1 Conclusions ................................... 200 7.2