Study on the symmetries and conserved quantities of flexible mechanical multibody dynamics

Mingliang Zheng; Mingliang Zheng

doi:10.3934/math.20231430

AIMS Mathematics

2023, Volume 8, Issue 11: 27969-27982. doi: 10.3934/math.20231430

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Study on the symmetries and conserved quantities of flexible mechanical multibody dynamics

Mingliang Zheng ^{1,2
,
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1.
Faculty of Mechanical Engineering, Taihu University of Wuxi, Qianrong Road 68, Wuxi 21400, China
2.
Jiangsu Key Laboratory of Green Process Equipment, Changzhou University, Changzhou 213000, China

Received: 22 July 2023 Revised: 17 September 2023 Accepted: 26 September 2023 Published: 11 October 2023
MSC : 34C14, 37M05

In this paper, in order to provides a powerful new tool for quantitative and qualitative analysis of dynamics properties in flexible mechanical multibody systems, the symmetry theory and numerical algorithms for preserving structure in modern analytical mechanics is introduced into flexible multibody dynamics. First, taking the hub-beam systems as an example, the original nonlinear partial differential-integral equations of the system dynamics model are discretized into the finite-dimensional Lagrange equations by using the assumed modal method. Second, the group analysis theory is introduced and the criterion equations and the corresponding conserved quantities of Noether symmetries are given according to the invariance principle, which provide an effective way for analytic integral theory of dynamic equations. Finally, a conserved quantity-preserving numerical algorithm is constructed by coordinates incremental discrete gradient, which makes full use of the invariance of conserved quantity to eliminate the error consumption for a long time. The simulation results show that the deeper mechanical laws and motion characteristics of flexible mechanical multibody systems dynamics can be obtained with the help of symmetries and conserved quantities, which can provide reference for more precise dynamic optimization design and advanced control of systems.
- flexible mechanical multibody systems,
- Euler-Lagrange equations,
- symmetries,
- conserved quantities
Citation: Mingliang Zheng. Study on the symmetries and conserved quantities of flexible mechanical multibody dynamics[J]. AIMS Mathematics, 2023, 8(11): 27969-27982. doi: 10.3934/math.20231430

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Abstract

In this paper, in order to provides a powerful new tool for quantitative and qualitative analysis of dynamics properties in flexible mechanical multibody systems, the symmetry theory and numerical algorithms for preserving structure in modern analytical mechanics is introduced into flexible multibody dynamics. First, taking the hub-beam systems as an example, the original nonlinear partial differential-integral equations of the system dynamics model are discretized into the finite-dimensional Lagrange equations by using the assumed modal method. Second, the group analysis theory is introduced and the criterion equations and the corresponding conserved quantities of Noether symmetries are given according to the invariance principle, which provide an effective way for analytic integral theory of dynamic equations. Finally, a conserved quantity-preserving numerical algorithm is constructed by coordinates incremental discrete gradient, which makes full use of the invariance of conserved quantity to eliminate the error consumption for a long time. The simulation results show that the deeper mechanical laws and motion characteristics of flexible mechanical multibody systems dynamics can be obtained with the help of symmetries and conserved quantities, which can provide reference for more precise dynamic optimization design and advanced control of systems.

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