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Input optimization for flexible multibody systems using the adjoint variable method and the flexible natural coordinates formulation

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Abstract

This work discusses a sensitivity information methodology for input optimization of flexible multibody systems. Mechanisms often need to perform desired motions, and especially in the case of underactuated flexible systems, the required drive motion or torque are nontrivial to determine. In this work we present an optimization approach which exploits the Adjoint Variable Method (AVM) in combination with the Flexible Natural Coordinates Formulation (FNCF) for obtaining the sensitivity information. The Adjoint Variable Method has the advantage of obtaining accurate sensitivity information at a computational cost that is relatively independent of the number of model parameters. FNCF combines the advantageous properties of the Floating Frame of Reference Formulation (FFRF) and the Generalized Component Mode Synthesis (GCMS) methods for which the equations of motion are characterized by a constant mass and stiffness matrix, and quadratic constraint equations. This work shows how the specific structure of the motion equations obtained through FNCF reduces the complexity of the Adjoint Variable Method as the required simulation derivatives for input optimization can be readily obtained and are of limited order. The approach is illustrated for an input optimization case on a flexible slider–crank multibody model, where the objective is to construct an input torque signal such that the crank’s angular velocity matches a predefined reference signal.

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Notes

  1. \(\widetilde{\pmb \omega} = \widetilde{ \begin{bmatrix} \omega _{1}& \omega _{2}&\omega _{3} \end{bmatrix}} ^{\mathrm{T}}= \begin{bmatrix} 0& -\omega _{3} & \omega _{2} \\ \omega _{3}& 0 & -\omega _{1} \\ -\omega _{2}& \omega _{1} & 0 \end{bmatrix} \)

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Acknowledgements

Internal Funds KU Leuven are gratefully acknowledged for their support. This research was partially supported by Flanders Make, the strategic research center for the manufacturing industry.

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Authors and Affiliations

  1. Division LMSD, Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, Box 2420, 3001, Leuven, Belgium

    Simon Vanpaemel, Martijn Vermaut, Wim Desmet & Frank Naets

  2. Flanders Make@KU Leuven, Leuven, Belgium

    Simon Vanpaemel, Martijn Vermaut, Wim Desmet & Frank Naets

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  1. Simon Vanpaemel
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Vanpaemel, S., Vermaut, M., Desmet, W. et al. Input optimization for flexible multibody systems using the adjoint variable method and the flexible natural coordinates formulation. Multibody Syst Dyn 57, 259–277 (2023). https://doi.org/10.1007/s11044-023-09874-z

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