Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

IF 2.6 2区 工程技术 Q2 MECHANICS Multibody System Dynamics Pub Date : 2024-06-27 DOI:10.1007/s11044-024-10001-9
Philipp L. Kinon, Peter Betsch
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Abstract

Mechanical systems with singular and/or configuration-dependent mass matrix can pose difficulties to Hamiltonian formulations, which are the standard choice for the design of energy-momentum conserving time integrators. In this work, we derive a structure-preserving time integrator for constrained mechanical systems based on a mixed variational approach. Livens’ principle (or sometimes called Hamilton–Pontryagin principle) features independent velocity and momentum quantities and circumvents the need to invert the mass matrix. In particular, we take up the description of rigid body rotations using unit quaternions. Using Livens’ principle, a new and comparatively easy approach to the simulation of these problems is presented. The equations of motion are approximated by using (partitioned) midpoint discrete gradients, thus generating a new energy-momentum conserving integration scheme for mechanical systems with singular and/or configuration-dependent mass matrix. The derived method is second-order accurate and algorithmically preserves a generalized energy function as well as the holonomic constraints and momentum maps corresponding to symmetries of the system. We study the numerical performance of the newly devised scheme in representative examples for multibody and rigid body dynamics.

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具有奇异和非恒定质量矩阵的多体系统的守恒整合,包括基于四元数的刚体动力学
质量矩阵具有奇异性和/或配置依赖性的机械系统会给哈密顿公式带来困难,而哈密顿公式是设计能量-动量守恒时间积分器的标准选择。在这项工作中,我们基于混合变分法,为受约束机械系统推导了一种结构保持型时间积分器。李文斯原理(有时也称为汉密尔顿-庞特里亚金原理)具有独立的速度和动量,并避免了反转质量矩阵的需要。特别是,我们使用单位四元数来描述刚体旋转。利用李文斯原理,我们提出了模拟这些问题的一种相对简单的新方法。通过使用(分割的)中点离散梯度来近似运动方程,从而为具有奇异和/或与配置相关的质量矩阵的机械系统生成一种新的能量-动量守恒积分方案。推导出的方法具有二阶精度,并在算法上保留了广义能量函数以及与系统对称性相对应的整体约束和动量图。我们在多体和刚体动力学的代表性示例中研究了新设计方案的数值性能。
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来源期刊
CiteScore
6.00
自引率
17.60%
发文量
46
审稿时长
12 months
期刊介绍: The journal Multibody System Dynamics treats theoretical and computational methods in rigid and flexible multibody systems, their application, and the experimental procedures used to validate the theoretical foundations. The research reported addresses computational and experimental aspects and their application to classical and emerging fields in science and technology. Both development and application aspects of multibody dynamics are relevant, in particular in the fields of control, optimization, real-time simulation, parallel computation, workspace and path planning, reliability, and durability. The journal also publishes articles covering application fields such as vehicle dynamics, aerospace technology, robotics and mechatronics, machine dynamics, crashworthiness, biomechanics, artificial intelligence, and system identification if they involve or contribute to the field of Multibody System Dynamics.
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