Approximate General Responses of Multi-Degree-of-Freedom Systems with Parametric Stiffness

Abstract

In the work presented, the solutions and stability of multi-degree-of-freedom Mathieu-type systems are investigated. An approach combining Floquet theory with harmonic balance is used to find the system response. The assumed Floquet-type solution consists of a product between an exponential part and a periodic part. The periodic part is approximated with a finite number of harmonics, and without making further assumptions, this solution is directly applied the original differential equations of motion. A harmonic balance analysis results in an eigenvalue problem. The characteristic exponents are the eigenvalues and the corresponding eigenvectors provide the Fourier coefficients of the harmonic part of the solution. By examining the solutions of the eigenvalue problem, the initial conditions response, frequency content, and stability characteristics can be determined. The approach is applied to two and three DOF examples. For a few parameter sets, the results obtained from this method are compared to the numerical solutions.