The present research is concerned with the free vibrations and energy transfer of a vibrating gyroscope, which is composed of a flexible beam with surrounded piezoelectric films in a rotating space. The governing equations involve nonlinear curvature, and rotary inertia of an in-extensional rotating piezoelectric beam is obtained by using the transformation of two Euler angles and the extended Hamilton principle. The gyroscopic effect due to the rotating angular speed is investigated in the frame of complex modes based on the invariant manifold method. The effects of angular speed, initial values, and electrical resistance to the nonlinear natural frequencies of a rotating piezoelectric beam are studied by both linear and nonlinear decoupling methods. The results reveal that the rotation causes one nonlinear frequency to bifurcate into a pair of frequencies: one forward and one backward nonlinear frequencies. The variation of the frequency with the angular speed is used to measure the angular speed. Finally, the energy transfer due to nonlinear coupling under 1:1 internal resonance condition and the energy transfer due to the linear gyroscopic decoupling are investigated.

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