Nonlinear free damped vibrations of a rectangular plate described by three nonlinear differential equations are considered when the plate is being under the conditions of the internal resonance two-to-one. Viscous properties of the system are described by the Riemann-Liouville fractional derivative. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the fractional derivative is represented as a fractional power of the differentiation operator. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent.