Development and analysis of a multi-link suspension for racing applications W. Lamers DCT 2008.077 Master’s thesis Coach: dr. ir. I.J.M. Besselink (Tu/e) Supervisor: Prof. dr. H. Nijmeijer (Tu/e) Committee members: dr. ir. R.M. van Druten (Tu/e) ir. H. Vun (PDE Automotive) Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Group Eindhoven, May, 2008 Abstract University teams from around the world compete in the Formula SAE competition with prototype formula vehicles. The vehicles have to be developed, build and tested by the teams. The University Racing Eindhoven team from the Eindhoven University of Technology in The Netherlands competes with the URE04 vehicle in the 2007-2008 season. A new multi-link suspension has to be developed to improve handling, driver feedback and performance. Tyres play a crucial role in vehicle dynamics and therefore are tyre models fitted onto tyre measure- ment data such that they can be used to chose the tyre with the best characteristics, and to develop the suspension kinematics of the vehicle. These tyre models are also used for an analytic vehicle model to analyse the influence of vehicle pa- rameters such as its mass and centre of gravity height to develop a design strategy. Lowering the centre of gravity height is necessary to improve performance during cornering and braking. The development of the suspension kinematics is done by using numerical optimization techniques. The suspension kinematic objectives have to be approached as close as possible by relocating the sus- pension coordinates. The most important improvements of the suspension kinematics are firstly the harmonization of camber dependant kinematics which result in the optimal camber angles of the tyres during driving. The suspension is designed to have a steady ride height during cornering which causes the suspension to operate in the intended region. The driver feedback is improved by means of the suspension kinematics and steering wheel forces. The vehicle characteristics are validated with a dynamic vehicle model. Reference: vehicle dynamics, kinematic suspension design, tyre models, multi-body vehicle models, numerical optimization ii Preface The last challenge of my study is the master’s thesis. After careful consideration about the thesis subject, I started late 2006 with the development of a racing suspension for the University Racing Eindhoven team. It is a real challenge to design the complete vehicle dynamic characteristics of a vehicle, one which is not easy to find externally. And even so important: the design will be build and tested in practise which gives a complete picture of the design proces. The first choice in the proces was to analyse tyre behaviour. Mainly because the tyres are a very fun- damental aspect of vehicle dynamics. I did this partly at the Automotive department of TNO which is located in Helmond, the Netherlands. Here I had the opportunity to fit the tyre measurement data on the latest tyre model of TNO Automotive: Delft-Tyre. Therefore I want to thank Antoine Schmeitz for helping me doing this. The next subject was to analyze the steady state vehicle behaviour of a racing vehicle and to find out what the influence of basic vehicle parameters is on the vehicle performance. It appeared to be quite an extensive job, but a very useful one. The last and major part of the thesis was to design the suspen- sion itself. This was a really interesting subject. When I look back to the thesis I have to say that I had a very pleasant time doing this, while working in a truly unique environment: the University Racing Eindhoven team! Then I want to thank the people who have helped and supported me during my master’s thesis. Igo Besselink for his theoretical input and coaching, Henk Nijmeijer for his supervising role, Roell van Druten and Hans Vun for participating in the committee and the University Racing Eindhoven team where I could fulfill this challenge. Than I want to thank my family for their support and finally my girlfriend Patricia van Dongen for her support, patience and the corrections she suggested for this report. Willem-Jan Lamers, May 2008 iii Contents Abstract ii Preface iii Sign conventions and symbols 1 1 Introduction 5 1.1 Background . 5 1.2 Objective and thesis outline . 6 2 Modeling a racing tyre 8 2.1 Introduction . 8 2.2 Tyre measurement data . 9 2.3 Fitting the measurement data . 10 2.4 Validation of the tyre model . 12 2.5 Tyre choice . 15 3 Steady state vehicle behaviour 19 3.1 Introduction . 19 3.2 Load distribution . 19 3.3 Two track roll axis vehicle model . 21 3.4 Base line vehicle . 23 3.5 Model objective . 25 3.6 Numerical optimization . 26 3.7 Calculation sequence . 27 3.8 Other model applications . 30 3.9 Pure cornering results . 31 3.10 Combined cornering/driving results . 34 3.11 Optimization of the steering angles . 36 3.12 Optimal tyre inclination angle . 38 4 Kinematic suspension design 39 4.1 Introduction . 39 4.2 Multi-link layout . 39 4.3 Design tools . 41 4.4 Kinematic suspension model . 41 4.5 Suspension kinematic characteristics . 44 4.6 Numerical optimization . 53 4.7 Dynamic vehicle model . 56 4.8 Suspension collisions . 58 iv CONTENTS CONTENTS 5 Suspension design considerations and results 60 5.1 Introduction . 60 5.2 Allowable suspension settings . 60 5.3 Contact patch pressure fluctuation . 61 5.4 Pitch attitude . 63 5.5 Roll attitude . 64 5.6 Tyre orientation target . 66 5.7 Driver feedback . 70 5.8 Suspension forces . 74 5.9 Initial suspension settings . 75 5.10 Double wishbone comparison . 76 6 Conclusions and recommendations 79 6.1 Conclusions . 79 6.2 Recommendations . 80 Bibliography 82 A Two track roll axis model equations 84 B Optimization algorithm 87 C Suspension coordinates 91 D Quarter car model derivation 94 v CONTENTS CONTENTS Sign conventions Sign conventions often cause communications problem, therefore will the ISO 8855 [1] sign conven- tion be used throughout this report. The most important definitions are depicted in figure 1, for a complete overview see [1]. on cti ire g d in iv Dr Figure 1: ISO 8855 sign conventions ISO defines the vehicle axis system as a right-handed orthogonal axis system fixed at the center of gravity of the vehicle. The x-axis is parallel to the road surface and pointing forwards, the y-axis is also parallel to the road surface and pointing to the driver’s left. The z-axis is pointing upwards, normal to the road. Throughout the report several angles are used. 12 different rotations sequences can be defined. Only 2 are used, one for the chassis rotation sequence and one for the wheel/tyre rotation sequence. Table 1 shows the rotation sequences for the chassis and wheel starting from the world coordinate system. Rotation order Produced chassis angle Produced tyre angle First yaw ( ) steer angle (δ) Second pitch (θ) inclination angle (γ) Third roll (φ) wheel rotation angle (!) Table 1: Rotation sequence The wheelbase [l] is defined as the distance between the center of the tyre contact point of the two wheels on the same side of a vehicle projected on the x-axis. The track [b] is defined as the distance between the centers of tyre contact points of the two wheels of an axle projected on the yz-plane. The steer angle is defined as the rotation of the wheel around the positive z-axis according to the axis definition. 1 CONTENTS CONTENTS The tyre inclination angle [γ] is defined positive when the tyre is inclined by positive rotation around the x-axis of the vehicle. The camber angle is defined positive as an angle between the global z-axis and the wheel plane when the top of the wheel is inclined outward relative to the vehicle body. More specific definitions used in this report are given when needed. Symbols α tyre side slip angle [deg] β vehicle slip angle [deg] γ tyre inclination angle [deg] δ steer angle [deg] θ chassis pitch angle [deg] κ longitudinal tyre slip [-] µ friction coefficient [-] ξ dimensionless damping coefficient [-] σ kingpin inclination angle [rad] τ castor angle [rad] φ chassis roll angle [deg] vehicle yaw angle [deg] _ yaw velocity [rad/s] ! wheel rotation angle around the y-axis [deg] !i angular velocity [rad/s] !δ steering velocity [rad/s] a vector pointing to the instant center and anti center 2 ax longitudinal acceleration [m/s ] 2 ay lateral acceleration [m/s ] b track width [m] cφ roll stiffness [Nm/rad] 2 CONTENTS CONTENTS d steering axis vector ds damping coefficient [Ns/m] f degrees of freedom [-] frf ride frequency [Hz] g gravitational acceleration [9.81 m/s2] h centre of gravity height [m] hra height between the roll axis and the centre of gravity [m] k spring stiffness [N/m] l wheelbase [m] m vehicle mass [kg] n castor offset [m] nτ castor offset at wheel centre [m] p brake force distribution [-] r yaw rate [rad/s] ri axis vector rs scrub radius [m] rc wheel centre offset [m] t total track width [m] u longitudinal vehicle speed [m/s] v lateral vehicle speed [m/s] vi velocity vector w wheel load lever arm [m] Ax longitudinal acceleration [g] Ay lateral acceleration [g] ACf front anti center ACr rear anti center D point where the virtual steering axis intersects with the road 3 CONTENTS CONTENTS Fx longitudinal tyre force [N] Fy lateral tyre force [N] Fz vertical tyre load [N] H transfer function 2 Iy rotational inertia [kgm ] ICf front instant center ICr rear instant center Mx overturning moment [Nm] My rolling resistance moment [Nm] Mz self aligning moment [Nm] MR motion ratio [-] R cornering radius [m] V vehicle speed [m/s] W weighting factor [-] WTR wheel base - track ratio [-] 4 Chapter 1 Introduction "We’re all on the limit, the car is on the limit, the human being is on the limit,..