Hopf and two-multiple semi-stable limit cycle bifurcations of a restrained plate subjected to subsonic flow

This paper is mainly about the stabilities and bifurcations of a cantilevered plate with nonlinear motion constraints in an axial subsonic flow. The Galerkin method is employed to discretize the governing partial differential equation. The fixed points and their stabilities are presented in a parameter space based on qualitative analysis and numerical studies. The system loses stability by flutter and undergoes limit cycle oscillations after instability due to the nonlinearity. The stabilities of the limit cycle oscillations are addressed on the basis of the equivalent linearized method. The type of Hopf bifurcation (subcritical or supercritical) is dependent on the location of the nonlinear motion constraints. Interestingly, for some certain states the Hopf bifurcations are both subcritical and supercritical. The two-multiple semi-stable limit cycle bifurcation, which is due to the extreme point of the flutter curve, is also determined. Results of the numerical integration method sufficiently support the analytical method presented in this paper.