Modified Runge_kutta Method for Solving Nonlinear Vibration of Axially Travelling String System

In this paper, based on the classical Fourth-Order Runge-Kutta method, the modified FourthOrder Runge-Kutta method is presented for solving nonlinear vibration of axially travelling string system, that is to solve time varying and nonlinear differential equations. The classical Fourth-Order Runge-Kutta method can only be used to solve first-order linear differential equations. Its main idea is to calculate the value of the function for the state equation four times and then take a linear combination of these values, which increases the order of the truncation error for the numerical method, so as to improve the accuracy of the method. The modified Fourth-Order Runge-Kutta method is a novel method considering the nonlinear term of differential equations, it applies to whatever nonlinear equations that can be transformed into first order matrix differential equation and has no restrict of the order of the original equations. Compared with other methods, it avoids the matrix inversion and has high precision and simplicity. Then a time-varying nonlinear governing equation of an axially travelling string system will be considered, by setting different parameters, the corresponding magnitude of the nonlinear term of the governing equation is different. The modified Fourth-Order RungeKutta method and the Newmark-Beta method are used respectively to solve the time-varying nonlinear governing equation of the string model. Good agreements are observed in the results by the two methods, which proved the validity and effectiveness of the modified FourthOrder Runge-Kutta method and also by setting different parameters indicated that the proposed method is suitable for the differential equations both with strong and weak nonlinear terms.