Geometric Design of Independent Suspension Linkages
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Geometric Design of Independent Suspension Linkages by David E. Kline A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in the University of Michigan 2018 Doctoral Committee: Professor Gregory M. Hulbert, Chair Assistant Professor Evgueni Filipov Associate Professor C. David Remy Professor Kazuhiro Saitou David E. Kline [email protected] ORCID iD: 0000-0001-5194-1837 Copyright c 2018 by David E. Kline. Table of Contents List of Tables v List of Figures vi Abstract xii Chapter 1: Introduction 1 1.1 Basic Terms . 4 1.2 History of the Suspension . 7 1.3 Suspension Design . 17 1.4 Research Objectives . 34 1.5 Dissertation Format . 35 Chapter 2: Literature Review 37 2.1 Wheel Motion Specification . 37 2.2 Enumeration of Suspension Architectures . 45 2.3 Linkage Dimensioning . 52 2.4 Filtering Solutions . 56 2.5 Summary . 58 Chapter 3: Wheel Kinematics 59 3.1 Mathematical Preliminaries . 59 3.2 Wheel Motion . 61 3.3 Wheel Trajectory . 67 3.4 Example Trajectory . 71 ii Chapter 4: Number Synthesis 77 Chapter 5: The R Joint 81 5.1 Synthesis . 81 5.2 Synthesis Example . 86 5.3 Analysis . 87 5.4 Analysis Example . 91 Chapter 6: The S-S Link 96 6.1 Design Equations . 96 6.2 Synthesis Example . 98 6.3 Set-Based Design of the Five S-S Link Suspension . 101 6.4 Analysis of the Five S-S Link Suspension . 108 Chapter 7: The C Joint 111 7.1 Synthesis . 111 7.2 Synthesis Example . 113 7.3 An S-S Link for the C Joint . 115 7.4 C(S-S) Example . 118 Chapter 8: The S Joint 123 8.1 Synthesis . 123 8.2 Synthesis Example . 125 8.3 S-S Links for Spherical Linkages . 126 8.4 S(S-S)2 Example . 128 Chapter 9: The R-S Link 133 9.1 Design Equations . 133 9.2 Solving the Design Equations . 136 9.3 Tie Rod Synthesis . 137 9.4 The SLA Suspension . 140 Chapter 10: The S-R Link 146 iii 10.1 Design Equations . 146 10.2 Control Blades . 148 10.3 S-S Links for the Control Blade . 152 10.4 Analysis of the Control Blade Suspension . 154 Chapter 11: The R-R Link 158 11.1 Design Equations . 158 11.2 Synthesis Example . 162 11.3 S-S Links for the Trapezoidal Link Suspension . 164 11.4 Analyzing the Trapezoidal Link Suspension . 165 Chapter 12: The S-C Link 169 12.1 Design Equations . 169 12.2 Synthesis Methods . 172 12.3 Compatible R-S & S-S Links . 175 12.4 Analyzing the MacPherson Strut Suspension . 177 Chapter 13: Discussion 181 13.1 Comparison of Selected Architectures . 181 13.2 Methods Developed . 189 13.3 Logical Extensions . 192 References 199 iv List of Tables 1.1 Market share of the front axles used by vehicles produced in 2010 [18, p. 418]; the twist beam is used in just one vehicle model. 16 1.2 Market share of the rear axles used by vehicles produced in 2010 [18, p. 418]; the De Dion axle is used in one vehicle model. 16 1.3 C5 ride and handling metrics with associated units [43]. 19 2.1 Planar links enumerated by Raghavan [35], together with their mobility. 48 2.2 Spatial links enumerated by Raghavan [35], together with their mobility. 48 4.1 The body-wheel connections. 78 4.2 Potential independent suspension architectures, organized by number of connections. Types identified as impractical are crossed out. 80 6.1 Coordinates of the five S-S link example (three significant figures). 107 9.1 Coordinates of the SLA example (three significant figures). 141 10.1 Coordinates of the three lateral S-S links used for the control blade example (three significant figures). 153 12.1 Example S-C solution (three significant figures). 174 13.1 Average and overall ranking of the example suspensions, including number of independent design variables and if steered. 188 13.2 Synthesis cases developed in this dissertation and the chapter(s) they appear in. 191 v List of Figures 1.1 A double wishbone independent suspension [59]. 2 1.2 Rack and pinion steering mechanism, where (1) steering wheel, (2) steering column, (3) rack and pinion, (4) actuated suspension link (tie rod), (5) wheel carrier [58]. 2 1.3 Cylindrical rubber bushing [56], typically pressed into suspension link and attached to vehicle body in double shear. 3 1.4 Ball joint [60], typically pressed into suspension link, with wheel carrier attached in single shear. 3 1.5 Contact patch with pressure distribution [54]. 5 1.6 Camber angle illustrated in a front view; negative shown [57]. 5 1.7 Toe angle illustrated in a top view; positive (toe-in) shown [61]. 6 1.8 Wheelbase and track illustrated [63]. 6 1.9 Ackermann's steering mechanism [1]. 8 1.10 Gearbox steering system from a Willys CJ-3B [40]. 9 1.11 Sliding pillar IFS of an early Sizaire-Naudin car [2]. 10 1.12 Ballamy swing axle IFS [2]. 11 1.13 Oldsmobile SLA IFS [2]. 11 1.14 Citroen IFS, using a torsion spring and a hydraulic shock absorber [2]. 12 1.15 MacPherson strut IFS [55]. 12 1.16 AM General Humvee featuring robust double wishbone suspensions front and rear with excellent ground clearance [28]. 14 1.17 Twist beam semi-rigid rear axle [14]. 14 1.18 Mercedes C111, circa 1969, using a five link IRS and a double wishbone IFS [62]. 15 vi 1.19 2012 Ford Focus control blade IRS; arrows indicate the control blade and three lateral links [10]. 16 1.20 Schematic of Ford's product development process as of 1998 [18, p. 451]. 18 1.21 ULSAB side view package drawing [46]. 23 1.22 ULSAB front view package drawing [46]. 24 1.23 ULSAB rear view package drawing [46]. 24 1.24 ULSAB plan view package drawing [46]. 25 1.25 Front view instant center definition [12]. 26 1.26 Side view instant center definition [12]. 27 1.27 Definition of the vertical component of the angular velocity [12]. 27 1.28 Layout of a trailing link suspension to achieve one-axis velocity speci- fication [12]. 29 1.29 Layout of a double wishbone suspension to achieve a two-axis velocity specification [12]. 29 1.30 Design evolution of the C5 rear lower control arm [45]. 31 1.31 Summarized suspension design process. 32 2.1 Example of desirable camber angle change [41]. 39 2.2 SLA geometry considered by Raghavan for roll center synthesis [38]. 40 2.3 Roll center height hRC graphical definition. 41 2.4 Joint types, reproduced from [26]. 49 2.5 Practical implementations of the turning joint with two rubber bush- ings (left), and the turning-and-sliding joint as a telescopic damper (right), reproduced from [26]. 50 2.6 Link types important for suspensions, reproduced from [26]. 50 2.7 MacPherson independent suspension, reproduced from [26]. 51 2.8 An independent suspension employing a ball joint and two rod links, reproduced from [26]. 52 2.9 Graphical synthesis of the lower control arm (LCA) and strut of a MacPherson architecture, reproduced from [30]. 53 vii 2.10 Package space considered by Raghavan for the planar suspension [36]. 56 3.1 Identifying a special orthogonal matrix (f; g; h) with the angles φ, γ; δ. 65 3.2 Suspension characteristic definitions in the side view. 71 3.3 Roll center height definition in the front view, with velocity component vc2 drawn negative to result in a positive roll center height h. 72 3.4 Camber angle specification. 73 3.5 Toe angle specification. 73 3.6 Wheel center longitudinal displacement specification. 75 3.7 Spin angle specification. 75 3.8 Wheel center lateral displacement specification. 76 5.1 Depiction of an R joint. 82 5.2 Solution for R joint from design position velocity specification. 88 5.3 Camber curve for the synthesized R joint suspension. 92 5.4 Toe curve for the synthesized R joint suspension. 92 5.5 Wheel-travel angle curve for the synthesized R joint suspension. 93 5.6 Support angle curve for the synthesized R joint suspension. 94 5.7 Roll center height curve for the synthesized R joint suspension. 94 6.1 Depiction of an S-S link. 97 6.2 Package space allotted for the S-S link synthesis example. 100 6.3 Kingpin geometry definition. 102 6.4 Plot of the back links Sb. 104 6.5 Plot of the front links Sf . 105 6.6 Plot of the toe links St. 105 6.7.