Nonlinear longitudinal vibrations of a rod

This paper deals with the longitudinal vibrations of a nonlinear rod. The nonlinearity is strong and of cubic type. The motion of the rod is described by a second-order strong nonlinear partial differential equation. The exact numerical solution is obtained. Four approximate analytical methods for solving the differential equation are developed: the eigenmode solutions are denoted by applying the known space or time distribution functions, respectively, the method based on invariant manifolds, and the method for optimizing the two-mode eigenfrequency solution. The analytical and numerical solutions are compared. The amplitude-time-position diagrams and amplitude-frequency diagrams are plotted.