Effect of pre-twist on the nonlinear vibration of the blades considering the bending-bending-torsion coupling

In this paper, the nonlinear dynamic behavior of the pre-twisted blades based on the Timoshenko beam theory has been investigated. The main purpose of the present work is to study the nonlinear dynamic response of uniform, rotating, cantilever pre-twisted blades. To this end, firstly, the coupled nonlinear equations of motion considering diverse parameters such as the hub radius, rotating velocity of the blade, pre-twist and pre-cone angles, shear strain, rotary inertia and warping of cross section have been derived step by step using Hamilton's principle. The obtained nonlinear partial differential equations (PDEs) of motion, which are included the bending-bending-torsion coupling terms, have been converted to the time dependent ordinary differential equations (ODEs) by applying the Galerkin partial discretization method. Finally, Euler-Gauss step by step numerical method has been employed to solve the obtained nonlinear ODEs. The obtained results illustrate that, considering the nonlinear strain terms in the formulation has significant effects on the dynamic responses of the pre-twisted blades. Furthermore, the results show that, increasing the pre-twist angle of the blade, decreases the amplitude of the blade response and the period of oscillations for bending vibration (in the Z direction) and torsional vibration. Furthermore, because of the stiffing effect of the centrifugal force, the amplitude of the blade response and the period of oscillations decrease with increasing the rotational speed. Besides, for evaluation purpose, the natural frequencies of the studied blades have been obtained by linearization of the obtained nonlinear equations of motion. The comparison of the obtained natural frequencies in this paper with those are available in the literature shows a good agreement.