Bifurcation analysis for vibrations of a turbine blade excited by air flows

A reduced three-degree-of-freedom model simulating the fluid-structure interactions (FSI) of the turbine blades and the oncoming air flows is proposed. The equations of motions consist of the coupling of bending and torsion of a blade as well as a van der Pol oscillation which represents the time-varying of the fluid. The 1:1 internal resonance of the system is analyzed with the multiple scale method, and the modulation equations are derived. The two-parameter bifurcation diagrams are computed. The effects of the system parameters, including the detuning parameter and the reduced frequency, on responses of the structure and fluid are investigated. Bifurcation curves are computed and the stability is determined by examining the eigenvalues of the Jacobian matrix. The results indicate that rich dynamic phenomena of the steady-state solutions including the saddle- node and Hopf bifurcations can occur under certain parameter conditions. The parameter region where the unstable solutions occur should be avoided to keep the safe operation of the blades. The analytical solutions are verified by the direct numerical simulations.