A Lie group variational integrator in a closed-loop vector space without a multiplier

As a non-tree multi-body system, the dynamics model of four-bar mechanism is a differential algebraic equation. The constraints breach problem leads to many problems for computation accuracy and efficiency. With the traditional method, constructing an ODE-type dynamics equation for it is difficult o...

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Bibliographic Details
Published in:Mechanical sciences (Göttingen) 2024-03, Vol.15 (1), p.169-181
Main Authors: Bai, Long, Xia, Lili, Ge, Xinsheng
Format: Article
Language:English
Subjects:
Algebra
Analysis
Closed loops
Design
Differential equations
Dynamics
Feedback control
Geometry
Integrators
Iterative methods
Kinematics
Lagrange multiplier
Lie groups
Linear algebra
Mathematical analysis
Matrix algebra
Methods
Multibody systems
Parallelograms
Runge-Kutta method
Vector spaces
Vectors (mathematics)
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Summary:As a non-tree multi-body system, the dynamics model of four-bar mechanism is a differential algebraic equation. The constraints breach problem leads to many problems for computation accuracy and efficiency. With the traditional method, constructing an ODE-type dynamics equation for it is difficult or impossible. In this exploration, the dynamics model is built with geometry mechanic theory. The kinematic constraint variation relation of a closed-loop system is built in matrix and vector space with Lie group and Lie algebra theory respectively. The results indicate that the attitude variation between the driven body and the follower body has a linear recursion relation, which is the basis for dynamics modelling. With the Lie group variational integrator method, the closed-loop system Lagrangian dynamics model is built in vector space, with Legendre transformation. The dynamics model is reduced to be the Hamilton type. The kinematic model and dynamics model are solved using Newton iteration and the Runge–Kutta method respectively. As a special case of a crank and rocker mechanism, the dynamics character of a parallelogram mechanism is presented to verify the good structure conservation character of the closed-loop geometry dynamics model.
ISSN:2191-916X
2191-9151
2191-916X
DOI:10.5194/ms-15-169-2024