Structural modal interaction with combination internal resonance under wide-band random excitation

In this paper the non-linear interaction of a three degree of freedom structural model subjected to a wide-band random excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response statistics. With reference to the combination internal resonance of the summed type the system response is analyzed by using the Fokker-Planck equation approach together with a non-Gaussian closure scheme. The non-Gaussian closure is based on the cumulant properties of order greater than three. As a first order approximation the scheme yields 209 first order differential equations in first through fourth order joint moments of the response co-ordinates. The analysis is carried out with the aid of the computer algebra software MACSYMA. The response statistics are determined, numerically in the time and frequency (internal detuning) domains. Contrary to the Gaussian closure scheme, the non-Gaussian closure solution yields a strictly stationary response in addition to a number of complex response characteristics not previously reported in the literature of the area of non-linear random vibration. These include multiple solutions and jump phenomena (jump and collapse in the response mean squares) at internal detuning slightly shifted from the exact internal resonance condition. At exact internal resonance the system response possesses a unique limit cycle in a stochastic sense. The regions of multiple solutions are defined in terms of system parameters (damping ratios and non-linear coupling parameter) and excitation spectral density level.