Natural frequency analysis of non-uniform smart beams with piezoelectric layers, using differential quadrature method

In this paper the free vibration behavior of non-uniform beams with piezoelectric layers has been analyzed using generalized differential quadrature method (GDQM). First of all, the governing differential equations for free vibrations of this type of smart beams are derived. Then, the differential equations are discretized using GDQM and solved as an eigen-value problem to obtain natural frequencies and mode shapes. The results obtained for non-uniform beam without piezoelectric layers show good agreement with those available in literatures for different types of boundary conditions. Also, natural frequencies of a uniform beam with piezoelectric layers with three different boundary conditions are compared to those obtained from a finite element model, which is developed by ANSYS software. Good agreements are demonstrated. In addition, parametric studies are carried out to investigate the influence of some important parameters including the non-dimensional piezo-layer thickness, non-dimensional width, non-dimensional elastic coefficient, non-dimensional piezoelectric constant, and non-dimensional permittivity constant, on the natural frequencies and mode shapes of the beam lateral vibrations. The natural frequencies calculated with GDQM show accurate results in comparison to the other numerical methods. Overall, the GDQM decreases the time and computational costs.