Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities
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The linear and non-linear free vibrations of a spinning piezoelectric beam are studied by considering geometric nonlinearities and electromechanical coupling effect. The non-linear differential equations of the spinning piezoelectric beam governing two transverse vibrations are derived by using transformation of two Euler angles and the extended Hamilton principle, wherein an additional piezoelectric coupling term and different linear terms are present in contrast to the traditional shaft model. Linear frequencies are obtained by solving the standard eigenvalues of the linearized system directly, and the non-linear frequencies and non-linear complex modes are achieved by using the method of multiple scales. For free vibrations analysis of a spinning piezoelectric beam, the non-linear modal motions are investigated as forward and backward precession with different spinning speeds. The responses to the initial conditions for this gyroscopic system are studied and a beat phenomenon is found, which are then validated by numerical simulation. The influences of some parameters such as electrical resistance, rotary inertia and spinning speeds to the non-linear dynamics of a spinning piezoelectric beam are investigated.
KeywordsSpinning piezoelectric beam Free vibrations Non-linear frequencies Complex modes
This work was supported in part by the National Natural Science Foundation of China (Grants 11672007 and 11832002), Beijing Natural Science Foundation (Grant 3172003), and Graduate Student Science and Technology Foundation of Beijing University of Technology (Grant ykj-2017-00045).