Nonlinear vibration analysis of composite blade with variable rotating speed using Chebyshev polynomials

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A composite cantilever plate model with variable high speed rotating blade subject to transverse aerodynamic force and centrifugal force is established. Based on the third-order shear deformation theory, von Karman large deformation theory and Hamilton's principle, the nonlinear partial differential governing equations of motion are established for the rotating blade with variable rotating speed. The two-degree-of-freedom nonlinear ordinary differential equations of motion are obtained by using Galerkin method with Chebyshev polynomials for the rotating blade. The influences of different structural parameters on the natural frequencies of the blade with variation of the rotating speed are investigated. The method of multiple scales is applied to obtain the averaged equations with the primary parametric resonance-1/2 subharmonic resonance and the relationship of 1:2 internal resonance. Numerical simulations are performed to portray the frequency-response curves and the complex nonlinear dynamic behaviors of the rotating blade by discussing the effect of the aerodynamic force. The effects of the varying rotating speed, the centrifugal force, the pre-twist angle and the pre-setting angle are taken into account on the nonlinear dynamics of the rotating blade. It is observed from the frequency-response curves that the rotating blade exhibits the hardening nonlinear behaviors as well as the jumping phenomena. The bifurcation diagrams, the phase portraits and the waveforms are utilized to illustrate the complex nonlinear dynamic behaviors of the rotating blade, such as the periodic, the quasi-periodic and the chaotic motions.