Abstract
An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presented in this paper. The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodes and Legendre nodes. Based on Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula are obtained by comparison with Pade approximation. Newton iteration method is used during the solution process. Taking the planar two-link manipulator system as an example, the results of L-stable method presented are compared for different number of nodes in the time interval and the step size in the simulation, and also compared with the classic Runge-Kutta method, A-stable method, Radau IA, Radau IIA, and Lobatto IIIC methods. The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions.
Highlights
The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions
The stability of the numerical solution of differential equations is one of the important indicators to judge the quality of a numerical method
The typical form of multibody system dynamic equations is differential-algebraic equations (DAEs), which consist of ordinary differential equations and algebraic constraint equations
Summary
The stability of the numerical solution of differential equations is one of the important indicators to judge the quality of a numerical method. Butcher [6, 7] studied the nonlinear stability of Runge-Kutta method by using nonlinear equations y = f(y) as model equations, proposed B-stable and algebraic stability of the Runge-Kutta method, and established the algebraic stability criteria for general linear methods. Hairer [10] pointed out that the L-stable numerical method will introduce damping effect, but it is very effective for suppressing high-frequency oscillation, and some scholars have proposed implicit single-step and multistep methods based on L-stable for solving rigid equations, but these methods are based on ordinary differential equations. The typical form of multibody system dynamic equations is DAEs, which consist of ordinary differential equations and algebraic constraint equations. The general form of L-stable method for multibody system dynamics DAEs is established. Based on the equidistant nodes, Chebyshev nodes, and Legendre nodes, the concrete construction process is given and the planar twolink manipulator system verified and compared
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