Free vibration of elastically restrained flexural-shear plates with varying cross-section

This paper presents an analytical approach to determining natural frequencies and mode shapes of non-uniform flexural-shear plates with line translational spring and rotational spring supports and line masses under action of axial forces. The governing differential equation for vibration of a non-uniform flexural-shear plate under axial forces is established first. It is shown that it is possible to separate a flexural-shear plate as two beams for free vibration analysis, one is a flexural beam, and the other is a shear beam. The natural frequency of the plate is equal to the square root of the square sum of the two natural frequencies of the two beams, and the mode shape of the plate is the product of the corresponding two mode shapes of the two beams. In this paper, power functions and exponential functions are adopted for describing the distributions of mass and stiffness along the height of the plate as well as the axial forces acting on the plate. The exact solutions for free vibrations of non-uniform flexural-shear plates for several cases that are important in engineering practices are derived. A numerical example shows that the calculated results are in good agreement with the experimental data and it is convenient to apply the proposed method to free vibration analysis of elastically restrained flexural-shear plates with varying cross-section.