Dynamics of a Piezoelectric Restrained Nanowire in an Elastic Matrix

This study presents an analytical approach for vibration analysis of piezoelectric nanowires. First, the piezoelectric nanowire is fixed at both ends with elastic springs that do not move. At the same time, the piezoelectric nanowire is enclosed in an elastic medium. For this elastic medium, the Winkler foundation model is considered. The equation of motion of the problem is generated using natural and force boundary conditions, equations of equilibrium, and energy methods. Fourier series are preferred as the axial displacement function. This function is substituted into the discretized equation of motion and the Fourier coefficient is determined. The Fourier coefficient is substituted into the force boundary conditions and an eigenvalue problem of size 2 × 2 is formulated. This eigenvalue problem is the most general and has not been presented in the literature. When high values are assigned to the spring parameters, embedded boundary conditions are reached. Conversely, when the spring parameters are assigned very small values, the solutions can be made for the free end.