超音速流中二维板的Hopf分叉

应用Hamilton变分原理建立了对边简支对边自由薄板在超音速流作用下的非线性动力学方程,其中几何非线性采用Von-Kaman几何非线性关系进行描述,气动力采用了活塞理论。然后用Galerkin方法将偏微分方程化为常微分方程,并利用Hopf分叉的代数判据得到了系统临界流速随初始载荷变化的表达式以及颤振频率随初始载荷与临界流速变化的表达式,最后用数值方法进行了验证。

HOPF BIFURCATION OF A TWO-DIMENSIONAL THIN PLATE IN SUPERSONIC AIRFLOW

The nonlinear dynamic equations of a simple supported two-dimensional thin plate in the supersonic airflow are established through the Hamiltonian variational principle. In the equation, the geometric nonlinear is expressed by the Von Karman’s thin plate theory and the aerodynamic pressure is expressed by the piston theory. The partial differential equation is turned into an ordinary differential equation by Galerkin method. A new algebraic criterion of Hopf bifurcation is utilized to achieve the analytic expression of critical flow velocity with the initial in-plane load, the flutter frequency with the initial in-plane load and the critical flow velocity. Finally, the Forth order Runge-Kutta numerical method was applied to certify the theories.