Barycentric Interpolation Collocation Method For Numerical Analysis Of Mechanical Vibrations

Barycentric Lagrange interpolation has merits of good numerical stability and higher computational accuracy. In this thesis, the basic properties of barycentric Lagrange interpolation are analysis firstly. The numerical algorithm of approximating continuous functions by barycentric Lagrange interpolation is given.Using barycentric Lagrange interpolation to approximate unknown function, the explicit formulations of element of differentiation matrix are constructed. The barycentric Lagrange interpolation collocation method (BLICM) for solving initial value problems of differential equation is introduced. Put the discrete algebraic equations of initial derivative conditions and algebraic equation system of governing differential equation together to form a new algebraic system. The least-square method is adopted to solve them. The numerical examples indicate that the computational accuracy of treatment of initial conditions proposed in this thesis is higher than traditional replace method.The linear vibration problems under periodic exciting force and general exciting force are numerical analysis by barycentric Lagrange interpolation collocation method. After obtaining the displacement values of vibrating, the velocities and accelerations are computing directly by differentiation matrices. Not only the displacement but also velocity and acceleration have higher accurate in barycentric Lagrange interpolation collocation method.In the procedure of barycentric Lagrange interpolation collocation method to analyze the nonlinear vibration problems, the nonlinear vibration governing equation and initial conditions are translated into a set of nonlinear algebraic equations. The nonlinear algebraic equations system is solved using Newton method to obtain the displacement of vibration. At the same time the velocity, acceleration and period of vibrating can be calculated by differentiation matrix and barycentric Lagrange interpolation, respectively.The numerical examples of linear and nonlinear vibration demonstrate that barycentric Lagrange interpolation collocation method have advantages of simple formulations, easier to program and higher computational accuracy.