A Highly Accurate Collocation Method for Linear and Nonlinear Vibration Problems of Multi-Degree-Of-Freedom Systems Based on Barycentric Interpolation

Abstract A highly accurate collocation method based on barycentric interpolation (BICM) is proposed for solving linear and nonlinear vibration problems for multi-degree-of-freedom systems in this article. The mathematical model of the linear and nonlinear vibrations of multi-degree-of freedom systems is the initial value problem of the linear and nonlinear differential equations. The numerical solution of the linear differential equations can be directly solved by BICM. The numerical solution of nonlinear differential equations can be solved as following: Firstly, the nonlinear governing equation is converted to linear differential equation by assuming the initial function. Secondly, the linear differential equations are discretized into algebraic equations by using barycentric interpolation differential matrices. Thirdly, the numerical solution can be calculated by iteration method under given control precision. Finally, the numerical solution of calculation examples by using barycentric Lagrange interpolation iteration collocation method (BLIICM) and barycentric rational interpolation iteration collocation method (BRIICM) are compared with the analytical solution. Numerical results illustrate the advantages of proposed methodology are efficient, fast, simple formulations, and high precision. Comparing with BLIICM, BRIICM can still maintain its computational stability when dealing with a large number of nodes, especially the equidistant nodes.