Nonlinear vibrations of spring-supported axially moving string

Abstract

In this study, multi-supported axially moving string is discussed. Supports located at the ends of the string are simple supports. A support located in the middle section owns the features of a spring. String speed is assumed to vary harmonically around an average rate. Hamilton’s principle has been used to figure out the nonlinear equations of motion and boundary conditions. These equations and boundary conditions are dimensionless. Considering the nonlinear effects caused by the string extensions, nonlinear equations of motion are obtained. By using multi-timescaled method, which is one of the perturbation methods, approximate solutions have been found. The first term in the perturbation series causes the linear problem. With the solution of the linear problem, exact natural frequencies have been calculated for different locations of the supports on the middle, various spring coefficients and, with the spring support in the middle of the different location, different spring coefficient and axial speed values. Nonlinear terms on second order add correction terms to the linear problem. Effect of nonlinear terms on the natural frequency has been calculated for various parameters. The cases when the changing frequency of speed is equal to zero, close to zero and close to two times of the natural frequency have been analyzed separately. For each case, the stable and unstable areas in the solutions have been identified by stability analysis.