非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

Supercritical as Well as Subcritical Hopf Bifurcation in Nonlinear Flutter Systems

研究了二元机翼非线性颤振系统的Hopf分岔.应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程.研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质.用等效线性化法和增量谐波平衡法验证了所得结果.

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated with the flow speed as a bifurcation parameter. The center manifold theory and complex normal form method were used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.