Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters

Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant ones and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young's modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a plate with parameters quadratically or exponentially depending on the radial coordinate, are examined. The effect of the small perturbation parameter on the behavior of frequencies is also analyzed under special conditions: i) for a plate, the mass of which is fixed, if the thickness is variable and ii) for a plate with the fixed average stiffness, if Young's modulus is variable. Finally, effects of the boundary conditions and values of the wave numbers on the corrections to frequencies are studied. 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 and the numerical results of other authors.