Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev-Lagrangian method

A Chebyshev-Lagrangian method is proposed to handle the flexural and in-plane vibrations of plate with cutout under general boundary conditions. By setting groups of boundary springs and assigning corresponding stiffness constants to the springs, the general boundary conditions, including all the classical boundary conditions, can be readily achieved. The energy functions of the in-plane and transverse vibrations of plate with cutout are deduced. The Chebyshev expansions solutions are obtained according to the Lagrangian theory and the accuracy and reliability of current method are validated by checking eigenpairs (eigenfrequencies and eigenvectors) of present method against those available in the literature or derived from finite element software. The present method offers a unified procedure for a variety of cases since the modification of any parameter from one case to another, such as the size and location of cutout and the boundary conditions, is as simple as modifying the material properties, requiring no changes to the solution procedures. The flexural and in-plane vibrations of plate with inner cutout, corner cutout or edge cutout and under various boundary conditions are studied. It is shown that the location of cutout will obviously change the free vibration characteristics of the structure. It appears that the modal data of the in-plane vibration of plate with cutout under different boundary conditions were not previously available in literature, and might be used as a reference to other future solution methods.