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TECHNICAL PAPERS

Multi-Objective Optimization of the Handling Performances of a Road Vehicle: A Fundamental Study on Tire Selection

[+] Author and Article Information
Carlo Miano, Massimiliano Gobbi, Giampiero Mastinu

Department of Mechanical Engineering, Politecnico di Milano (Technical University), Piazza Leonardo da Vinci, 32, 20133 Milan, Italy

J. Mech. Des 126(4), 687-702 (Aug 12, 2004) (16 pages) doi:10.1115/1.1759359 History: Received January 01, 2003; Revised December 01, 2003; Online August 12, 2004
Copyright © 2004 by ASME
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References

Figures

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Single track vehicle model
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Lateral forces transmitted by the tires at front (F) and rear (R) axle as function of the lateral slip angle α
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Lateral slip at front and at rear axles (αFR) as a function of sideslip body angle β and yaw velocity ψ̇
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Phase plane plot, μFR. Vehicle running at v=25 m/s, a): δ=0°<δ̃ and b): δ=7°>δ̃(δ̃ is given by Eq. (23)). Stable equilibrium point ‘o’ is given by Eq. (8) and by Eq. (9). Stable equilibrium point ‘+’ is given by Eq. (24) and by Eq. (25). Vehicle data are reported in Appendix B.
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Phase plane plot, μFR. Vehicle running at v=25 m/s, a): δ=0°<δ̃ and b): δ=7°>δ̃(δ̃ is given by Eq. (23)). Stable equilibrium point ‘o’ is given by Eq. (8) and by Eq. (9). Unstable equilibrium points ‘* ’ are given by Eq. (29) and by Eq. (30). Vehicle data are reported in Appendix B.
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Peak response time, ψ̇max and ψ̇ss definition with reference to the yaw velocity response
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Example of Pareto-optimal set boundary obtained in design variables domain for a multi-objective optimization problem with two design variables. □ is a Pareto-optimal solution taken from an initial approximated Pareto-optimal set. ○ are the solution found by applying the method presented in Sec. 2.2.
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Pareto-optimal set plotted in the design variables domain. Linear vehicle model. The Pareto-optimal set is defined by symbol ‘+’. Boundaries have been computed analytically. Vehicle data are given in Appendix B. γ is defined by Eq. (46). The time responses for cases 〈1〉 to 〈7〉 are plotted in Fig. 10.
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Time responses to a step steering angle input of cars (design solutions) denoted by numbers 〈1〉 to 〈7〉 in Fig. 9. The symbol • indicates the peak response time. The forward speed considered is v=30 m/s. The solution 〈1〉 has the best Tψ̇ and Kβ2 (see Table 3, min F1,F2), the solutions 〈3〉 and 〈4〉 have the best Oψ̇ (see Table 3min F3) and the solutions 〈1〉, 〈5〉, and 〈4〉 have the best ψ̈/δsteady(0) (see Table 3min F4).
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Pareto-optimal set plotted in the design variables domain for the linear vehicle model as in Fig. 9. The arrows indicate decreasing values of each objective function, see Table 2. Vehicle data are given in Appendix B.
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Influence of the vehicle speed on the Pareto-optimal set. Linear vehicle model as in Fig. 9, a): v=15 m/s, b): v=20 m/s, c): v=25 m/s and d): v=35 m/s. The Pareto-optimal set is defined by symbols ‘+’. Vehicle data are given in Appendix B.
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Pareto-optimal set plotted in the design variable domain. Nonlinear vehicle model: μR=2.0 and μF=2.2. Influence of speed and of steering input angles, a): v=25 m/s, δ=6°, b): v=35 m/s, δ=4°, and c): v=35 m/s and δ=6°. The Pareto-optimal set is defined by symbols ‘+’. Vehicle data are given in Appendix B.
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(Left) Pareto-optimal set plotted in the design variable domain. Nonlinear vehicle model, μR=2.2 and μF=2.0.(v=25 m/s and δ=6°). The Pareto-optimal set is defined by symbols ‘+’. (Right) Trends of the objective functions (the arrows shows decreasing directions of the objective functions). Vehicle data are given in Appendix B.
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(Left) Pareto-optimal set plotted in the design variables domain. Nonlinear vehicle model, μR=2.2 and μF=2.0.(v=35 m/s and δ=4°). The Pareto-optimal set is defined by symbols ‘+’. (Right) Trends of the objective functions (the arrows shows decreasing directions of the objective functions). Vehicle data are given in Appendix B.
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(Left) Pareto-optimal set plotted in the design variables domain. Nonlinear vehicle model, μR=2.2 and μF=2.0.(v=35 m/s and δ=6°). The Pareto-optimal set is defined by symbols ‘+’. (Right) Trends of the objective functions (the arrows shows decreasing directions of the objective functions). Vehicle data are given in Appendix B.
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Pareto-optimal set (gray area) in the design variable domain as a function of CαFlF/l and CαRlR/l. Data denoted by ‘○ ’ were derived from the literature 2424344, data denoted by ‘▵ ’ were available to the authors and measured from by a number of tire manufacturers (see Table 4).
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Solution of the problem (47) ‘o’ for different values of the parameters εOβ and εT. Different values of these parameters determine the shape of the feasible domain of the problem (47). The union of points ‘o’ obtained for the problem (47) solved for different values (εOβT) allows to obtain the Pareto-optimal set of the problem considered.

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