FUNDAMENTAL STUDIES OF PASSIVE, ACTIVE AND SEMI-ACTIVE AUTOMOTIVE SUSPENSION SYSTEMS SAMY ALY HASSAN Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Mechanical Engineering July 1986 ABSTRACT The fundamental properties of various automotive suspension systems are theoretically investigated on the basis of simple vehicle models subjected to realistic inputs chosen to represent road surfaces of different qualities. The vehicle response is evaluated through a performance index representing ride comfort, dynamic tyre load and suspension working space parameters, and interpreted in the light of these individual parameters together with the implications of the suspension design for attitude control and steering behaviour. Linear analysis procedures are followed in studying the passive, active and slow-active suspension systems while suitable simulations are used for the non-linear semi-active suspension systems. Linear optimal control theory is used to determine the optimal parameters of the active and slow-active suspension systems. Semi-active suspension behaviours are evaluated on the basis of applying the optimal active parameters to each system, but the semi-active damper can only dissipate energy and switches off when external power would be needed for the system to follow the optimal active control law. Results are generated and discussed for each of these types of system and their performance capabilities are compared with each other. Conclusions concerning the practical viability of each of the systems are drawn. ii CONTENTS page ABSTRACT ........................................................ i C O N T E N T S ........................................................ ii LIST OF FIGURES................................................ iv LIST OF T A B L E S ................................................ x ACKNOWLEDGEMENTS.............................................. xii 1. INTRODUCTION ................................................ 1 2. OPTIMAL CONTROL THEORY .................................... 14 2.1. Introduction......................................... 14 2.2. General T h e o r y ...................................... 15 2.2.1. Full State Feedback ....................... 16 2.2.2. Limited State Feedback .................... 19 2.3. The Quarter Car Problem............................ 22 2.3.1. Full State Information Available . . . . 27 2.3.2. Limited State Feedback Available . . . . 28 2.4. Example Solutions .................................... 30 2.4.1. Full State F e e d b a c k ....................... 30 2.4.2. Limited State Feedback .................... 31 3. ROAD SURFACE INPUT AND SYSTEM RESPONSE ............... 35 3.1. Introduction......................................... 35 3.2. Road Surface Description.......................... 36 3.3. Linear S y s t e m s ...................................... 38 3.4. Non-Linear Analysis ................................. 44 page 4. PASSIVE SUSPENSION SYSTEM STUDIES .................... 48 4.1. Introduction......................................... 48 4.2. Two Mass System...................................... 49 4.3. Three Mass System.................................... 68 5. ACTIVE SUSPENSION SYSTEM STUDIES ....................... 77 5.1. Introduction......................................... 77 5.2. Calculations......................................... 78 5.3. Active System Results ............................ 79 6. SEMI-ACTIVE SUSPENSION SYSTEM STUDIES ............... 95 6.1. Introduction......................................... 95 6.2. Calculations......................................... 96 6.3. Semi-active System Results ....................... 100 7. SLOW-ACTIVE SUSPENSION SYSTEM STUDIES ............... 117 7.1. Introduction......................................... 117 7.2. System Model and Calculations .................... 118 7.3. Slow-active System Results ....................... 123 8. DISCUSSION OF R E S U L T S .................................... 138 9. CONCLUSIONS................................................ 148 NOMENCLATURE ................................................... 154 REFERENCES..................................................... 158 A P P E N D I X ........................................................ 162 iv LIST OF FIGURES page 2.1 Quarter car active suspension system ............... 23 3.1 Time histories of road surface profiles using different sets of phase a n g l e s ....................... 39 3.2 ISO weighting functions for squared acceleration . 43 3.3 Linear analysis ......................................... 45 3.4 Non-linear analysis ...................................... 47 4.1 Quarter car passive suspension system ............... 50 4.2 Performance and design properties of two mass passive systems, mw/mb=0.200 ......................... 54 4.3 Performance and design properties of two mass passive systems, mw/mb=0.125 ......................... 55 4.4 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.200 and SWS=1.5 c m ............................ 59 4.5 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.200 and SWS=2.0 c m ............................ 60 4.6 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.200 and SWS=2.5 c m ............................ 61 4.7 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.125 and SWS=1.5 c m ............................ 62 V page 4.8 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.125 and SWS=2.0 c m ............................ 63 4.9 Frequency responses and output mean square spectral densities of two mass passive systems, mw/mb=0.125 and SWS=2.5 c m ............................ 64 4.10 Ride comfort and dynamic tyre load variation of two mass passive systems, mw/mb=0.200 ............... 66 4.11 Ride comfort and dynamic tyre load variation of two mass passive systems, mw/mb=0.125 ............... 67 4.12 Three mass passive suspension system ............... 70 4.13 Performance and design properties of three mass passive systems, mw/mb=0.200 ......................... 73 4.14 Performance and design properties of three mass passive systems, mw/mb=0.125 ......................... 74 4.15 Frequency responses and output mean square spectral densities of two and three mass passive systems, mw/mb=0.125, ma/mw=0.20, and SWS=2.5 cm . 76 5.1 Performance and design properties of full state feedback active systems ................................. 80 5.2 Performance and design properties of limited state feedback active systems ......................... 81 5.3 Frequency responses and output mean square spectral densities of full state feedback active systems, SWS=1.5 cm .................................... 85 vi page 5.4 Frequency responses and output mean square spectral densities of full state feedback, active systems, SWS=2.0 cm .................................... 86 5.5 Frequency responses and output mean square spectral densities of full state feedback active systems, SWS=2.5 cm .................................... 87 5.6 Frequency responses and output mean square spectral densities of limited state feedback active systems, SWS=1.5 cm ............................ 88 5.7 Frequency responses and output mean square spectral densities of limited state feedback active systems, SWS=2.0 cm ............................ 89 5.8 Frequency responses and output mean square spectral densities of limited state feedback active systems, SWS=2.5 cm ............................ 90 5.9 Ride comfort and dynamic tyre load variation of full state feedback active systems .................. 92 5.10 Ride comfort and dynamic tyre load variation of limited state feedback active systems ............... 93 6.1 Performance and design properties of semi-active systems based on full state feedback active control laws with no passive d a m p e r s ............... 101 6.2 Performance and design properties of semi-active systems based on limited state feedback active control laws with no passive d a m p e r s ............... 102 vii page 6.3 Performance and design properties of semi-active systems based on limited state feedback active control laws with passive dampers .................. 103 6.4 Ride comfort and dynamic tyre load variation of semi-active systems based on full state feedback laws with no passive d am p e r s ......................... 107 6.5 Ride comfort and dynamic tyre load variation of semi-active systems based on limited state feedback laws with no passive dampers............... 108 6.6 Ride comfort and dynamic tyre load variation of semi-active systems based on limited state feedback laws with passive dampers .................. 109 6.7 Time history of the road input and the response of a semi-active system based on limited state feedback control law with no passive damper, SWS=1.5 c m ................................................ 112 6.8 Time history of the road input and the response of a semi-active system based on limited state feedback control law with no passive damper, SWS=2.0 c m ................................................ 112 6.9 Time history of the road input and the response of a semi-active system based on limited state feedback control law with no passive damper, SWS=2.5 c m ................................................ 214 viii page 6.10 Time history of the road input and the response of a semi-active system based on full state feedback control law with no passive damper, SWS=2.5 c m ................................................ 115 6.11 Time history of the road input and the response of a semi-active system based on limited state feedback control law with a passive damper, SWS=2.5 c m ................................................ 116 7.1 Quarter car slow-active suspension system .......... 119 7.2 Frequency responses of second order low-pass filters..................................................... 120 7.3 Performance and design properties of slow-active systems, spring stiffness=16000 N/m .................. 125 7.4 Performance and design properties of slow-active systems, spring stiffness=10000 N/m .................. 126 7.5 Performance and design properties of slow-active systems, spring stiffness=4000 N/m .................. 127 7.6 Performance and design properties of slow-active systems of 4 Hz bandwidth............................... 129 7.7 Frequency responses and output mean square spectral densities of slow-active systems, SWS=1.5 c m ................................................ 131 7.8 Frequency responses and output mean square spectral densities of slow-active systems, SWS=2.0 c m ................................................ 132 i x page 7.9 Frequency responses and output mean square spectral densities of slow-active systems, SWS=2.5 c m ................................................ 133 7.10 Ride comfort and dynamic tyre load variation of slow-active systems .................................... 134 7.11 Frequency responses and output mean square spectral densities of slow-active and passive systems ................................................... 137 8.1 Ride comfort and dynamic tyre load variation of different suspension systems, SWS=2.5 cm .......... 139 8.2 Ride comfort and dynamic tyre load variation of different suspension systems, SWS=2.0 cm .......... 140 8.3 Ride comfort and dynamic tyre load variation of different suspension systems, SWS=1.5 cm .......... 141 A.l Quarter car model with base parameter values . . . 165 A.2 Performance properties of passive systems as functions of stiffness and damping for condition (a) ........................................................ 167 A.3 Performance and design properties of special passive systems having SWS=4 cm for condition (a) 168 A.4 Performance properties of passive systems as functions of stiffness and damping for condition (b) 169