The frequency response of a rectangular cantilever plate vibrating in a viscous fluid

The fluid reaction on a wide rectangular cantilever plate vibrating in a viscous fluid is constructed by superposition of a potential flow solution and an asymptotic correction built from the Navier–Stokes equations for an incompressible fluid with nonlinear terms neglected. Using the Wiener–Hopf method, a simple analytic expression for the fluid reaction is obtained for high values of the dimensionless parameter ββ, which is the frequency times the length of the plate squared divided by the kinematic viscosity. Fluid and structure motion are then coupled when solving the equation of motion of the plate with a fluid reaction expressed in terms of weighted averages of the velocity of the plate. The frequency response is then calculated for plates of different sizes and materials and the results are shown for a gas and a liquid.