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A Modified Timoshenko Beam Theory for Nonlinear Shear-Induced Flexural Vibrations of Piezoceramic Continua

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The application of piezoceramics excited near resonance frequencies using weak electric fields, such as in ultrasonic motors, has led to close investigations of their behavior at this state. Typical nonlinear effects, such as a softening behavior, have been observed in resonantly driven piezoceramic structures, which cannot be adequately defined by linear theories. In piezoceramic actuators the d15-effect is very attractive for the application, as the corresponding piezoelectric coupling coefficient is higher than that for d31- and d33-effects. In the present paper, the authors have modeled nonlinear flexural vibrations of a resonantly driven piezoceramic cantilever beam, which is excited using the d15-effect. First, a linear description of the problem is investigated. A modified Timoshenko beam theory is formulated and an appropriate description of the electric field is included. The exact analytical solution of the linear field equations along with the associated boundary conditions is obtained. As an alternative approach, the Rayleigh–Ritz energy method is also used to obtain the eigenfrequencies and the eigenvectors. Series comprising orthogonal polynomial functions, generated using the Gram-Schmidt method, are used in the Rayleigh–Ritz method. To ascertain the efficacy of the Rayleigh–Ritz method for piezoceramic continua, the results obtained from it are compared with the exact analytical solution. To model the observed nonlinear effects, the electric enthalpy density is extended including higher-order conservative terms. The constitutive relations are correspondingly extended and additional nonlinear damping terms are adjoined. Hamilton's principle via the Ritz method is used to derive the discretized nonlinear equations of motion of a piezoceramic cantilever beam. The eigenfunctions of the linear case are used as shape functions in the Ritz method for discretization. The approximate solution of the nonlinear equation of motion is obtained using the perturbation method. Nonlinear parameters are determined by comparing the theoretical and the experimental responses. The modeling technique and the nonlinear effects described in this paper should be helpful in optimizing the existing applications and developing new applications based on the d15-effect.