Approximate General Responses of Tuned and Mistuned 4-Degree-of-Freedom Systems with Parametric Stiffness

Abstract

The purpose of this study is to find approximate solutions to tuned and mistuned 4-DOF systems with parametric stiffness. In this work, the solution and stability of four-degree-of-freedom Mathieu-type system will be investigated. To find the broken-symmetry system response, Floquet theory with harmonic balance will be used. A Floquet-type solution is composed of a periodic and an exponential part. The harmonic balance is applied to the original differential equation of motion. The analysis brings about an eigenvalue problem. By solving this, the Floquet characteristic exponents and the corresponding eigenvectors that give the Fourier coefficients are found in terms of the system parameters. The stability transition curve can be found by analyzing the real parts of the characteristic exponents. The frequency content can be determined by analyzing imaginary parts at the exponents. A response that involves single Floquet exponent (and its complex conjugate) can be generated with a specific set of initial conditions, and can be regarded as a modal response. The method is applied to both tuned and detuned four-degree-of-freedom examples.