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The dynamic stability of viscoelastic plates of variable stiffness is analyzed. The deflections are described by partial integro-differential equations of motion. The Bubnov-Galerkin method based on monomial and polynomial approximation of deflections is used to reduce the problem to ordinary integro-differential equations with time as an independent variable. These equations are solved numerically using a singular kernel. A numerical problem-solving algorithm based on this method is described. A number of new mechanical effects are revealed [ABSTRACT FROM AUTHOR]