In this article, a powerful computational methodology, named as barycentric rational interpolation iteration collocation method (BRICM), for obtaining the numerical solutions of nonlinear vibration problems is presented. The nonlinear vibration problems are governed by initial-value problems of nonlinear differential equations. Given an initial guess value of the unknown function, the nonlinear differential equations can be transformed into linear differential equations. By applying barycentric rational interpolation and differential matrix, the linearized differential equation is discretized into algebraic equations in the matrix form. The latest solution of nonlinear differential equation is obtained by solving the algebraic equations. The numerical solution of nonlinear vibration problem can be calculated by iteration method under given control precision. Then, the velocity and acceleration can be obtained by differential matrix of barycentric rational interpolation, and the period of nonlinear vibration is also computed by BRICM. Some examples of nonlinear vibration demonstrate the proposed methodological advantages of effectiveness, simple formulations, and high precision.