Vibrations of Viscoelastic Plates with Attached Concentrated Masses

In this paper, a thin viscoelastic plate with attached masses, which are widely used in modern engineering and construction is considered. The natural and forced linear vibrations of round viscoelastic plates with attached concentrated masses is investigated. To obtain the equations of motion of the mechanical system, the plate equation with operator coefficients and generalized Dirac functions is used. The problem is reduced to the joint solution of integro-differential equations in partial derivatives and the system of ordinary integro-differential equations of the corresponding dimension. To solve the obtained systems of integro-differential equations, the method of separation of variables, the freezing method, the methods of ordinary differential equations, the Gauss method and the Mueller method are applied. The analytical solution of the problem of linear vibrations of circular viscoelastic plates of constant cross-section with attached concentrated masses is obtained. On the basis of numerical results it is established that point attachment of masses leads to a decrease of real and imaginary parts of the natural frequencies up to 20 . It is found that by attaching masses to the plate the number of frequencies will not change. In the case when the mass is attached to the plate with massless elements, then the number of frequencies increases, while the first frequency decreases.