THE CLOSED-FORM SOLUTION FOR THE FORCED VIBRATION OF NON-UNIFORM PLATES WITH DISTRIBUTED TIME-DEPENDENT BOUNDARY CONDITIONS

The closed-form solution for the forced vibration of a non-uniform plate with distributed time-dependent boundary conditions is obtained. Three Levy-type solutions for a plate with different boundary conditions are studied. The two-dimensional system is transformed so that it becomes a one-dimensional one. By taking a general change of the dependent variable with shifting functions, the one-dimensional system is further transformed so that it becomes a system composed of a non-homogeneous governing differential equation and four homogeneous boundary conditions. The self-adjointness and the orthogonality condition for the eigenfunctions of the further transformed system with elastic boundary conditions are derived. Consequently, the method of separation of variables can be used to solve the transformed system. The shifting functions expressed in terms of the four fundamental solutions of the transformed system, instead of the fifth degree polynomials taken by Mindlin–Goodman, are derived. The physical meanings of these shifting functions are explored. Its application to the vibration control of a non-uniform plate with boundary inputs is investigated.