Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity

The nonlinear structural–acoustic problem considered in this study is the large amplitude free vibration of a rectangular elastic plate backed by a cavity. Very few classical solutions for this nonlinear structural–acoustic problem have been developed, although there are many for nonlinear plate or linear structural–acoustic problems. Thus, the main contributions of this study paper include (1) a concise multi-acoustic single structural modal formulation that is derived from two coupled partial differential equations representing the nonlinear structural free vibration and the acoustic pressure induced and (2) the approximate elliptical integral solution that is obtained by solving one residual equation only, and well agrees with that obtained from a harmonic balance finite element analysis. It is found that the natural frequency convergences with the increase in the numbers of acoustic modes and harmonic terms, and the effects of vibration amplitude, air cavity depth, and aspect ratio on the nonlinear natural frequency are also examined.