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# Direct and Adjoint Sensitivity Analysis of Multibody Systems Using Maggi’s Equations

[+] Author Affiliations
Daniel Dopico, Yitao Zhu, Adrian Sandu, Corina Sandu

Virginia Tech, Blacksburg, VA

Paper No. DETC2013-12696, pp. V07AT10A030; 10 pages
doi:10.1115/DETC2013-12696
From:
• ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 7A: 9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control
• Portland, Oregon, USA, August 4–7, 2013
• Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
• ISBN: 978-0-7918-5596-6

## abstract

The importance of the sensitivity analysis of multibody systems for several applications is well known, concretely design optimization based on the dynamics of multibody systems usually requires the sensitivity analysis of the equations of motion. A broad range of methods for the dynamics of multibody systems include the state space formulations based on Maggis equations, nullspace methods or coordinate partitioning. Dynamic sensitivities, when needed, are often calculated by means of finite differences but, depending of the number of parameters involved, this procedure can be very demanding in terms of CPU time and the accuracy obtained can be very poor in many cases. In this paper, several ways to perform the sensitivity analysis are explored and analytical expressions for the direct and adjoint sensitivity analysis of multibody systems are presented, all of them based on Maggi’s formulations. Moreover, two different approaches to the adjoint sensitivity analysis of multibody systems are presented.

Although particularized to one formulation, the general expressions provided in the paper, are intended to be easily generalized and applied to any other formulation that can be expressed as an ODE-like system of equations, including penalty formulations.

Besides, to check the validity and correctness of the proposed equations, the solutions of all the methods proposed are compared: 1) between them, 2) with the third party code FATODE and 3) with the numerical solution using real and complex perturbations.

Finally, all the techniques proposed are applied to the dynamical optimization of a multibody system.

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