Principal internal resonances in 3-DOF systems subjected to wide-band random excitation

Abstract

The autoparametric interaction of two normal modes in a three-degree-of-freedom (3-DOF) structural model subjected to a wide-band random excitation is investigated. The analysis deals primarily with the structure response in the neighborhood of three different internal resonance conditions. The Fokker-Planck equation approach, together with a non-Gaussian closure scheme, is used. The analysis is carried out with the aid of the computer algebra software MACSYMA, and leads to 69 differential equations in the first through fourth order moments of the response co-ordinates. Contrary to the Gaussian closure solution, the non-Gaussian closure scheme yields a strictly stationary response. The Gaussian closure scheme fails to predict the system response when two non-adjacent modes are internally tuned. According to the non-Gaussian solution, the autoparametric interaction is found to be sensitive to a relatively high level of excitation spectral density only if the first and second or the first and third normal modes are internally tuned. However, for the case of second and third normal mode interaction, the non-linear response is critical to lower excitation levels. The random responses of the three cases are characterized by energy exchange between the interacted modes. The sensitivity of autoparametric interaction to a certain excitation level is mainly dependent upon the system dynamic properties.