Non-linear limit cycle flutter of a plate with Hertzian contact in axial flow

Abstract

This paper is aimed at the nonlinear flutter of a cantilevered plate with Hertzian contact in axial flow. The contact effect is modeled as a nonlinear spring force with both square and cubic nonlinearity. The fluid force is considered as the sum of two parts, one is the reactive fluid force due to plate motion and the other is the resistive fluid force independent on plate motion. The reactive fluid force is derived by solving the bound and wake vorticity with the help of Glauert’s expansions, and the resistive force is evaluated in terms of drag coefficient. The governing nonlinear partial differential equation of the system is discretized in space and time domains by using the Galerkin method. Results show that the plate loses its stability by flutter and then undergoes limit cycle motions due to the contact nonlinearity after instability. The present fluid model is reliable and shows good agreement with other theories archived. A heuristic analysis scheme based on the equivalent linearization method is developed for the analysis of bifurcations and limit cycles. The Hopf bifurcation is either supercritical or subcritical, which is closely dependent on the contact location. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. When the plate experiences limit cycles, with the increasing dynamic pressure there firstly appear the lock-in motions; and then the quasi-periodic motions show up as a breaking of limit cycle by inclusion of a secondary significant frequency with an irrational value of 14π of the dominant limit cycle frequency. Finally the plate undergoes dynamic buckling characterized by quasi-periodic divergence when the dynamic pressure is relatively large.