Effects of static loads on the nonlinear vibration of circular plates

Maritime structures in water always experience a mean hydrostatic pressure. This paper investigates the nonlinear vibration of a clamped circular thin plate subjected to a non-zero mean load. A set of coupled Helmholtz–Duffing equations is obtained by decomposing the static and dynamic deflections and employing a Galerkin procedure. The static deflection is parameterized in the linear and quadratic coefficients of the dynamic equations. The effects of the static load on the dynamics, i.e. stiffening, asymmetry and softening, are investigated by means of the numerical solution of the coupled multi-mode system. An analytical solution of the single-mode vibration near primary resonance is derived. The analytical solution provides a theoretical explanation and quick quantification of the influence of the static load on the dynamics. The numerical and analytical results compare well, especially for lower values of the static deflection, confirming the effectiveness of the analytical approach. The proposed analysis method for plate vibration can be applied to other structures such as beams, membranes and combination forms.