The eigenvalue problem for a circular plate with an attached mass is considered. The receptance method and the finite element method are both used to solve the problem. The eigenvalue equation obtained with the receptance method allows for both numerical and asymptotic solutions. By means of the method of a small disturbance, the simple formulas for eigenvalues are evaluated and the accuracy of these formulas is estimated. The asymptotic formulas describe the influence of the attached mass, the stiffness of a spring and damping on eigenvalues. The results of the computations show that eigenvalues obtained with the different methods show good agreement.