Nonlinear vibrations of plates in axial pulsating flow

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long and wide rectangular plates subjected to pulsatile axial inviscid flow. The flow is set in motion by a pulsating pressure gradient. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The plate is assumed to be periodically simply supported in both in-plane directions with immovable edges and the flow channel is bounded by a rigid wall. In this way the system under study is a finite rectangular plate with conditions on the fluid and the plate boundaries coming from the periodicity of the infinite system. The equations of motion are obtained based on the von Kárman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on the potential flow theory. The resulting Lagrange equations of motion of the coupled system contain quadratic and cubic nonlinear terms and are studied by using a code based on the pseudo-arc-length continuation and collocation scheme. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. It has been found that the presence of positive transmural uniform pressure and small pulsation frequency would destroy the pitchfork bifurcation (divergence) that flat plates exhibit when subjected to uniform flow. Moreover, in case of zero uniform transmural pressure numerical results show a hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighborhood of the plate's fundamental frequency. On the contrary, a softening type behavior is presented when a uniform transmural pressure is introduced.