Nonlinear Dynamics of Systems under Interacting Parametric and External Excitations and Cubic Nonlinearity

Abstract

This paper presents an in-depth analysis of the dynamic response of the nonlinear forced Mathieu equation, encompassing a cubic nonlinearity as well as linear damping. The study focuses on a 2:1 frequency ratio between parametric and external excitations. Utilizing the Method of Varying Amplitudes, approximate analytical solutions for the response of the system are derived. This encompasses the exploration of jump phenomena, bifurcation analysis, and the potential for multiple stable solutions. Numerical simulations are undertaken to establish the reliability of the analytical analysis. The analytical results demonstrate internal stable and unstable solutions, influenced by critical parameters including external excitation amplitude, initial phase angle, and damping coefficient. To establish the reliability of the analytical analysis, numerical simulations obtained from direct integration of the equation of motion are undertaken, showing good agreement with the theoretical results.