基于高斯最小拘束原理的广义质量矩阵奇异性问题研究

Study on Singular Problem of the Generalized Mass Matrix Based on Gauss Principle of Least Constraints

摘要: 高斯最小拘束原理为含约束的多体系统动力学提供了一种新的建模思路：将以求解微分代数方程为主的动力学问题引入到求函数最小值的优化问题的框架中。质量矩阵奇异性问题是在常规多体系统动力学求解框架下经常遇到的难点问题。本文从建模方式出发研究了经典动力学框架下的难点问题，通过引入广义逆，建立了广义质量矩阵奇异情形下的高斯最小拘束原理，研究了针对奇异性问题的优化方法的数值求解策略。算例中分别采用了优化方法及第一类拉格朗日方程进行了建模及数值模拟。算例表明了文中方法在解决该类奇异性问题时的有效性。

Abstract: Gauss principle of least constraints provides a new modeling method for dynamics of multibody systems, which changes dynamical problem of mainly solving differential-algebra equation into the frame of solving minimum. The problem of the mass singular matrix is the difficulty in tradi-tional dynamics of multibody systems. We start with new modeling method for solving the singular problems. By introducing the generalized inverse, the Gauss principle of least constraints for matrix singular matrix and solving strategies for optimization method are established. In the example, the optimization and the Lagrange equation of the first kind are used to model and calculate. The example validates the optimization method for this kind of singular problems.