Stochastic autoparametric interaction in aeroelastic structures under wide band random excitation
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The nonlinear interaction of a three-degree-of-freedom structural model subjected to a wide band random excitation is examined. The equations of motion are derived by using Lagrange's equation and nonlinearity of quadratic order is considered. The nonlinearity of the structure results in four different critical regions of internal resonance conditions. These conditions represent algebraic interrelationships among the normal mode frequencies of the structure. The response statistics of the system are determined by using the Fokker- Planck equation together with a non-Gaussian closure scheme which 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 the first through fourth order joint moments of the response coordinates. The analysis is carried out with the aid of the computer algebra software MACSYMA. 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 nonlinear random vibration literature. These include multiple solutions and jump phenomena at internal detuning slightly shifted from the exact combination internal resonance condition (i.e., when the frequency of the third normal mode is greater than the sum of the first and second mode frequencies). At exact combination internal resonance the system response possesses a unique limit cycle in a stochastic sense. An attempt is made to verify the non-Gaussian closure solution by carrying out a Monte Carlo simulation of the equations of motion in the principal coordinates. The Monte Carlo simulation qualitatively supports the non-Gaussian closure solution in that it exhibits the main features of autoparametric coupling. However, the results show quantitative difference in the critical region of internal resonance.