Research Papers

Sensitivity-Based, Multi-Objective Design of Vehicle Suspension Systems

[+] Author and Article Information
Alfonso Callejo

Universidad Politécnica de Madrid,
Madrid 28031, Spain
e-mail: alcallejo@gmail.com

Javier García de Jalón

Applied Mathematics Professor
Universidad Politécnica de Madrid,
Madrid 28031, Spain
e-mail: javier.garciadejalon@upm.es

Pablo Luque

Transport Engineering Professor
Ingeniería e Infraestructura de los Transportes,
Universidad de Oviedo,
Gijón 33204, Spain
e-mail: luque@uniovi.es

Daniel A. Mántaras

Transport Engineering Professor
Ingeniería e Infraestructura de los Transportes,
Universidad de Oviedo,
Gijón 33204, Spain
e-mail: mantaras@uniovi.es

1Present address: Research Scholar, National Institute for Aviation Research, Wichita State University, Wichita 67260, Kansas.

2Corresponding author.

Manuscript received January 15, 2014; final manuscript received October 17, 2014; published online February 11, 2015. Assoc. Editor: Rudranarayan Mukherjee.

J. Comput. Nonlinear Dynam 10(3), 031008 (May 01, 2015) (9 pages) Paper No: CND-14-1022; doi: 10.1115/1.4028858 History: Received January 15, 2014; Revised October 17, 2014; Online February 11, 2015

This article deals with the dynamic response optimization of mechanical systems, based on the computation of independent state sensitivities. Specifically, the dynamic behavior of a coach is analyzed in detail so as to improve its response in terms of handling and ride comfort behaviors. To that end, the coach is modeled as an 18DOF multibody system, whose equations of motion are posed using an efficient dynamic formulation based on Maggi's equations. Next, a direct-automatic differentiation approach for the computation of independent state sensitivities is applied. This allows one to quantify the effect of 19 design parameters on the vehicle dynamic response and to compute the design sensitivities or objective function gradients. Finally, handling and ride comfort objective functions are defined and are used to carry out a multi-objective suspension design optimization process, improving the vehicle response by 70% in an effective yet automatic way.

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Bestle, D., and Eberhard, P., 1992, “Analyzing and Optimizing Multibody Systems,” Mech. Struct. Mach., 20(1), pp. 67–92. [CrossRef]
Pagalday, J., and Avello, A., 1997, “Optimization of Multibody Dynamics Using Object Oriented Programming and a Mixed Numerical-Symbolic Penalty Formulation,” Mech. Mach. Theory, 32(2), pp. 161–174. [CrossRef]
Etman, L. F. P., Van Campen, D. H., and Schoofs, A. J. G., 1998, “Design Optimization of Multibody Systems by Sequential Approximation,” Multibody Syst. Dyn., 2(4), pp. 393–415. [CrossRef]
Gonçalves, J. P. C., and Ambrósio, J. A. C., 2005, “Road Vehicle Modeling Requirements for Optimization of Ride and Handling,” Multibody Syst. Dyn., 13(1), pp. 3–23. [CrossRef]
Thoresson, M., Uys, P., Els, P., and Snyman, J., 2009, “Efficient Optimisation of a Vehicle Suspension System, Using a Gradient-Based Approximation Method, Part 1: Mathematical Modelling,” Math. Comput. Modell., 50(9–10), pp. 1421–1436. [CrossRef]
García de Jalón, J., Callejo, A., and Hidalgo, A. F., 2012, “Efficient Solution of Maggi's Equations,” ASME J. Comput. Nonlinear Dyn., 7(2), p. 021003. [CrossRef]
Ambrósio, J. A. C., Neto, M. A., and Leal, R. P., 2007, “Optimization of a Complex Flexible Multibody Systems With Composite Materials,” Multibody Syst. Dyn., 18(2), pp. 117–144. [CrossRef]
Chang, C. O., and Nikravesh, P. E., 1985, “Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method,” ASME J. Mech. Des., 107(4), pp. 493–498. [CrossRef]
Haug, E. J., 1987, “Design Sensitivity Analysis of Dynamic Systems,” Computer Aided Optimal Design: Structural and Mechanical Systems, Springer, Berlin, Germany, pp. 705–755.
Krishnaswami, P., and Bhatti, M. A., 1984, “A General Approach for Design Sensitivity Analysis of Constrained Dynamic Systems,” ASME Paper No. 84-DET-132. [CrossRef]
Serban, R., and Freeman, J. S., 1996, “Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems,” The 1996 ASME Design Engineering Technical Conferences and Computers in Engineering Conference, Irvine, California, August 18–22, pp. 18–22.
Maly, T., and Petzold, L. R., 1996, “Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems,” J. Appl. Numer. Math., 20(1–2), pp. 57–79. [CrossRef]
Dopico, D., Sandu, A., and Sandu, C., 2014, “Direct and Adjoint Sensitivity Analysis of ODE Multibody Formulations,” ASME J. Comput. Nonlinear Dyn., 10(1), p. 011012. [CrossRef]
Wang, X., Haug, E. J., and Pan, W., 2005, “Implicit Numerical Integration for Design Sensitivity Analysis of Rigid Multibody Systems,” Mech. Des. Struct. Mach., 33(1), pp. 1–30. [CrossRef]
Brüls, O., and Eberhard, P., 2008, “Sensitivity Analysis for Dynamic Mechanical Systems With Finite Rotations,” Int. J. Numer. Methods Eng., 74(13), pp. 1897–1927. [CrossRef]
Banerjee, J., and McPhee, J., 2013, “Symbolic Sensitivity Analysis of Multibody Systems,” Multibody Dynamics, Computational Methods in Applied Sciences, Vol. 28, J.-C.Samin, and P.Fisette, eds., Springer, Amsterdam, The Netherlands, pp. 123–146.
Griewank, A., and Walther, A., 2008, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia, PA.
Callejo, A., Narayanan, S. H. K., García de Jalón, J., and Norris, B., 2014, “Performance of Automatic Differentiation Tools in the Dynamic Simulation of Multibody Systems,” Adv. Eng. Software, 73, pp. 35–44. [CrossRef]
Pacejka, H., 2005, Tyre and Vehicle Dynamics, Elsevier, Oxford, UK.
Callejo, A., 2013, “Dynamic Response Optimization of Vehicles Through Efficient Multibody Formulations and Automatic Differentiation Techniques,” Ph.D. thesis, Universidad Politécnica de Madrid, Madrid, Spain.
Gutiérrez-López, M. D., Callejo, A., and García de Jalón, J., 2012, “Computation of Independent Sensitivities Using Maggi's Formulation,” The 2nd Joint International Conference on Multibody System Dynamics, Stuttgart, Germany, May 29–June 1.
García de Jalón, J., and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody Systems, Springer-Verlag, NY.
Walther, A., and Griewank, A., 2010, “A Package for the Automatic Differentiation of Algorithms Written in C/C++,” https://projects.coin-or.org/ADOL-C
Mastinu, G., Gobbi, M., and Miano, C., 2006, Optimal Design of Complex Mechanical Systems—With Applications to Vehicle Engineering, Springer, Berlin, Germany.
Griffin, M. J., 2007, “Discomfort From Feeling Vehicle Vibration,” Veh. Syst. Dyn., 45(7–8), pp. 679–698. [CrossRef]


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Fig. 1

(a) Front and (b) rear suspension systems

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Fig. 2

(a) Air spring and (b) damper forces

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Fig. 3

(a) Steering function and lateral acceleration in the step steer test, (b) steering function in the DLC maneuver

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Fig. 4

Sensitivity of the roll angle with respect to the relaxed length of the rear air springs in the DLC maneuver (handling)

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Fig. 5

Sensitivity of the vertical position with respect to the stiffness of the rear air springs in the speed-bumps maneuver (comfort)

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Fig. 6

Coach dynamic maneuvers: (a) DLC maneuver, (b) speed bumps test

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Fig. 7

(a) Design variable and (b) objective function spaces

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Fig. 8

Objective functions history

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Fig. 9

Design parameter history

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Fig. 10

Initial versus optimized responses: (a) load transfer in the double lane-change maneuver, (b) vertical acceleration in the speed-bumps test




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