Analytical Approximate Solution of Coupled Wave Equations with a Nonlinear Stiffness.

摘要

By utilizing the Euler-Lagrange equations, a set of coupled partial differential equations was derived for two crossing strings with a spring at the crossover point. This report considers three types of springs: a nonlinear softening spring, a nonlinear hardening spring, and a linear spring. The Adomian decomposition method was used to obtain an analytical approximate solution from the derived coupled wave equations. The dynamic responses of the analytical solutions for both the nonlinear softening and nonlinear hardening springs were compared to the response from the linear spring. In both the nonlinear softening and nonlinear hardening springs, higher frequency oscillations were observed at the spring location; however, when a linear spring was used, the strings did not exhibit higher frequency oscillations and only oscillated in the mode by which they were excited.