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JAN 30 1968 Efgineering Llbf \NT O ECJW .' J IS1 1967 4-1P-RA RIE-: OPTIMIZATION OF THE RANDOM VIBRATION CHARACTERISTICS OF VEHICLE SUSPENSIONS by ERICH KENNETH BENDER S.B., Massachusetts Institute of Technology (1962) S.M., Massachusetts Institute of Technology (1963) M.E., Massachusetts Institute of Technology (1966) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June, 1967 Signature of Author ., Department of Mechanical Engineering, ') /) Wfril 27, 1967 Certified by Thesfs Su4'sor, /K/J Accepted by ....... ...... Chairman, Departktnital'Commit 4 'ee on Graduate Students .\ST. OF TECHN 010 JAN 30 1968 EfGINEERING Llbf. OPTIMIZATION OF THE RANDOM VIBRATION CHARACTERISTICS OF VEHICLE SUSPENSIONS by ERICH KENNETH BENDER Submitted in partial fulfillment of the require- ments for the degree of Doctor of Science at the Massachusetts Institute of Technology, June, 1967. ABSTRACT Some of the fundamental limitations and trade-offs regard- ing the capabilities of vehicle suspensions to control random vi- brations are investigated. The vehicle inputs considered are sta- tistically described roadway elevations and static loading vara- tions on the sprung mass. The vehicle is modeled as a two-degree- of-freedom linear system consisting of a sprung mass, suspension and an unsprung mass which is connected to the roadway by a spring. The criterion used to optimize the suspension characteristics is the weighted sum of rms vehicle acceleration and clearance space required for sprung mass-unsprung mass relative excursions. Two ap- proaches to find the suspension characteristics which optimize the trade-off between vibration and clearance space are considered. The first, based on Wiener filter theory, is used to synthesize the optimum suspension transfer function. Mechanization of this function is discussed. The second approach, using a computer parameter search, -2- consists of finding optimum parameter values for a fixed configura- tion suspension. Since the above optimization is without regard to the wheel-roadway interaction a constraint on vibration-clearance design charts is developed to insure that wheel hop is not excessive. The minimum rms force required to maintain adequate wheel-roadway contact is also established to provide an estimate for upper vehicle speed limitations. Some of the potential capabilities of a suspen- sion which reacts to roadway irregularities before the vehicle reaches them are investigated. This preview control concept is studied for suspensions using Wiener filtering and for a suspension that may be easily mechanized. Finally, to illustrate the use of general design charts established in the first few sections of the thesis, a specific numerical example is carried out. The results of this thesis show that under certain circumstances active suspensions can substantially reduce the clearance space and sprung mass vibration levels over that obtainable by passive spring-shock absorber suspensions. For a given clearance space, preview control may be used to reduce vibration by as much as a factor of sixteen as compared with optimum non-preview suspensions. Thesis Supervisor: Dean C. Karnopp Title: Associate Professor of Mechanical Engineering -3- ACKNOWLEDGMENT The author is indeed grateful to those who have contributed to the development of this thesis, He wishes to thank particularly the mem- bers of his thesis committee, Professors Dean C0 Karnopp, Igor L. Paul, and Nathan H. Cook for their advice and careful review of the rough draft of this manuscript0 They have provided thought-provoking suggestions and have shown a genuine interest in this undertaking0 The author is especially grateful to Professor Karnopp, the thesis committee chairman, who, through many fruitful discussions, has helped to crystalize much of the work discussed here, Thanks are also given to Misses Patricia Perry and Mary Swan for their typing and to Mr0 James Jackson for the computer programs he wrote0 Special gratitude is extended to Mrs. George S, Mead for the fine job she has done in typing the final copy of this thesis and to the au- thor's wife, Cornelia, not only for her typing but for the encouragement she has given and the personal sacrifices she, as a student's wife, has made0 This thesis was sponsored in part by the Department of Com- merce under contract C-85-65 and by the Division of Sponsored Research of M0 1.T. TABLE OF CONTENTS Section Page ABSTRACT 0 0 0 0 0 0 a a 0 0 2 ACKNOWLEDGMENT 0 0 0 0 0 0 0 0 0 0 0 0 4 LIST OF ILLUSTRATIONS . 0 0 0 0 0 0 0 0 0 0 0 7 NOMENCLAURE 0 0 0 0 0 0 0 0 o 0 0 0 a 0 9 INTRODUCTION 0 0 0 0 0 0 0 0 a 0 0 a 0 0 0 11 1.1 Trends in Ground Transportation . 0 0 0 0 0 11 1.2 Suspension Requirements 0 0 0 a 0 0 0 0 13 0 1.3 State of the Art , 0 0 0 0 0 0 0 14 1.4 Optimization of Suspension Vibration Characteristics 0 0 0 0 0 a 0 0 17 2 SYSTEM MODEL AND PERFORMANCE CRITERIA a 0 0 0 0 20 2.1 Vehicle Disturbances 0 0 0 0 0 20 0 2,2 Vehicle Dynamics 0 0 0 0 0 0 24 0 2,3 Criteria and Constraints 0 0 0 0 0 0 0 26 3 OPTIMUM VIBRATION CONTROL 0 C 0 0 C a 0 C 0 31 3.1 Synthesis of Optimum Suspensions C e 0 0 0 0 32 3,11 Derivation 0 0 0 0 C, 0 0 0 0 0 0 32 3o12 Mechanization of Synthesized Suspension 0 0 0 0 0 0 0 a 0 0 37 3o143 Sensitivity Analysis .0 0 0 0 0 0 42 3.2 Analytical Optimization of Fixed Configuration Suspensions 0 0 0 0 0 0 a 45 3,3 Use of Design Chart , 0a 0 0 0 0 0 0 0 49 4 WHEEL-ROADWAY INTERACTION 0 0 0 51 4,1 Design Chart Constraint 0 0 0 0 0 0 0 0 0 52 4,2 Minimum RMS Sprung Mass Force 0 0 0 0 0 0 6 55 5 OPTIMUM LINEAR PREVIEW CONTROL 0 0 0 0 0 0 0 0 0 61 5,l Zero Preview 0 0 0 0 0 0 0 e o o 62 5,2 Infinite Preview c, e ,0 o e o 65 5,3 Finite Preview C ( eC o e o 67 5,4 Step Responses C C 0 e e o 73 5,5 Mechanization e o o 78 Section Page 6 NUMERICAL EXAMPLE 0 0 0 0 0 a 0 0 0 0 0 0 85 6ol Use of Design Charts 0 0 Q 0 0 0 0 0 0 85 6.2 Investigation of Several Specific Systems o . , ( . o e 0 88 6.2,J Frequency Responses and Power Spectral Densities o 0 0 88 6.2.2 Analog Computer Simulation 0 o 0 0 88 7 RESULTS, CONCLUSIONS, AND SUGGESTIONS FOR FURTHER STUDY 0 0 0 0 0 0 0 0 0 0 o 0 0 95 7Ll Results and Conclusions 0 a 0 0 0 7,2 Further Study , , 0 0 0 0 0 0 0 0 0 97 REFERENCES o o a 0 0 100 Appendix A DERIVATION OF OPTIMUM SYNTHESIZED SUSPENSION TRANSFER FUNCTION W(s) . 0 0 0 0 0 o a 0 104 B DERIVATION OF EXPRESSION FOR CLEARANCE SPACE FOR OPTIMUM SYNTHESIZED SUSPENSION 0 0 0 o a 109 C MINIMIZATION PROGRAM 0 6 0 0 o 0 0 111 j0 M 114 D EVALUATION OF I = - tfl e 0 e 0 0 0 2J'j (04+1)2 E ANALOGY BETWEEN DETERMINISTIC AND RANDOM PROCESSES 0 C 0 0 0 0 0 0 0 0o 0o 0e 0e 0o 116 F EVALUATION OF lim 0n x Go e' " J * 0 118 BIOGRAPHICAL NOTE Coo 00 0 0 0 0 0 0 120 =6- LIST OF ILLUSTRATIONS 1. Representative Roadway Mean Square Elevation Spectral Densities CC C 0 0 o C 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 a 0 0 0 22 2, Typical Runway Profile o o o o o o o o o o o o a a a 0 0 a 0 0 23 3. Vehicle - Roadway Configuration o , , 0 0 0 0 .* 0 0 0 0 0 0 25 4, Suspension Deflection Quantities , 0 0 0 0e 0 0 0 29 5. Block Diagram Showing the Relation of Vibration and 0a 0 00 00 00 00 Relative Displacement to the Roadway Elevation 0 0 0 33 0 0 0 0 0 0 6. Acceleration - Suspension Design Chart 0 0 0 0 a 0 0 38 0 0 0 0 0) 0 7. Schematic Diagram of Active Suspension , 0 0 0 a 39 8. Active Suspension System Block Diagram , 0 0 0 0 0 6 a e 41 9. Optimum Active System Parameter Values 0 0 0 0 a 0 0 0 . 0 a 0 0 43 10. Roots of Optimum Synthesized Transfer Function 0 0 0 0 0 0 0 0 0 46 11. Wheel - Roadway Excursion for Optimum Suspensions Specified by Figure 6 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 o 0 e 54/ 12. Minimum RMS Force Required to Maintain Wheel - Roadway Contact 99,9% of the Time 0 0 0 . 0 0 0 0o o 0 0 a 0o 59 13. Preview Suspension System a) Schematic b) Block Diagram .. ., 0 0 00 0 0 0 0 . e o o * , 63 14.
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