Liquid motion in cylindrical containers with elastic covers under external excitation

The coupled motion of liquid with an elastic plate or membrane cover in a cylindrical container under external excitation is investigated. Unlike self-oscillation problem at a natural frequency, the problem is fully transient, and it is first converted from the time domain to the s-domain through the Laplace transform. For each given s, velocity potential for the fluid flow and cover deflection are obtained through the Bessel-Fourier series. The solution in the time domain is then obtained through the inverse Laplace transform with respect to s. When doing so analytically, it is necessary to find singularities of the integrand in the entire complex plane s. It is shown that these singularities are only on the imaginary axis, corresponding precisely to the natural frequencies of the system and the excitation frequencies. This allows that the final solution to be obtained explicitly, which gives insight how the motion behaves. Extensive results are presented for the time history of the cover deflection and the energy components under various external excitation, including tank motion and external pressure on the cover. The frequency components of the solutions are analysed both at resonance and off-resonance. The energy transfer into the system from external forcings and its redistribution during vibration within the system are analysed.