Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions

Current work presents an exact solution for investigating the buckling, free vibration and dynamic response of plates with variable parameters in one direction. The plate has two opposite simply-supported edges subjected to in-plane loads with desired functions, whereas other edges may have various general time-dependent boundary conditions. The solution procedure is obtained by developing and generalizing Mindlin-Goodman and Frobenius methods. First, by introducing a general change of dependent variable with shifting functions, the original system with four non-homogeneous boundary conditions is transformed into a system composed of a time-independent governing differential equation and a system with four homogeneous boundary conditions. Then, assuming that the transverse displacement varies as a sinusoidal function, the governing partial differential equation of the plate is reduced to an ordinary differential equation in terms of y with variable coefficients. An exact solution is obtained as a power series to solve the resulting equation. Applying boundary conditions of two other edges yields the problem of finding eigenvalues of a fourth order characteristic determinant. Some examples are illustrated for validation with previous studies and the finite element method. Eventually, in other examples, the effect of some parameters on vibration behavior of the system is investigated.