Natural frequencies of an inhomogeneous square thin plate

Plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. For free vibration frequencies of a plate, which thickness and/or Young's modulus depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a square plate with parameters linearly or parabolically depend on the radial coordinate, are examined. The double frequencies of square plates with similar edge support of all edges are of special interest, since any variation of the thickness or stiffness causes some loss of symmetry one may expect the split of double frequencies. The asymptotic formulas permit to determine, which of two equal unperturbed frequencies corresponding to wave numbers n and m increases faster with the small parameter. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.