Investigation on the 1:2 internal resonance of an FGM blade

This paper focuses on the internal resonance analysis of a functionally graded material (FGM) blade under centrifugal and aerodynamic forces. Based on von Karman large deflection geometry and Reddy third-order shear deformation theory, the ordinary differential equations for transverse vibration of the first two modes are formulated. The parameter conditions of internal resonances under different blade length, width, thickness, speed ratio and material composition are obtained. Accordingly, in the case of 1:2 internal resonance, the amplitude responses of the first two modes of the system affected by the material component parameter, rotation perturbation amplitude, air velocity are analyzed, respectively. A variety of nonlinear phenomena such as hardening spring characteristic, the frequency band variation of the response curve and nested regions of multi-stability are obtained, which are discussed in detail through the amplitude–frequency response diagrams of single oscillator solution and dual oscillator solution. Most prominently, for the dual oscillator solution, the whole amplitude–frequency response curve can be divided into two branches according to the second-order mode amplitude: One reflects the internal resonance energy transfer mechanism, and the other has the similar law to the single oscillator solution. Moreover, it can be found that the sensitivities of the two branches to different parameters are different. This finding indicates that the energy transfer in the internal resonance can be designed by adjusting the parameters. The results obtained will help to further understand the internal resonance behavior in the process of blade rotation and provide the basis for revealing the complex nonlinear behavior.