The present paper studies the dynamical behavior of the moving viscoelastic plate which is in contact with barotropic compressible viscous fluid with finite depth. The time harmonic lineal located forces act on the plate. The linearized Navier-Stokes equations are employed for describing the fluid flow, however, the field equations and relations of the theory of visco-elastodynamics are employed for describing the plate motion. The analytic-numerical method based on the Fourier transform is employed for the solution to the related mathematical problems. Under concrete numerical investigations the viscoelasticity of the plate material is described by Rabotnov's fractional exponential operators, and dimensionless rheological parameters are introduced which characterize the long-term values of the mechanical properties and the characteristic creep time of the plate material. The main aim of the numerical investigations is determination of how these dimensionless rheological parameters influence the frequency responses of the stress and velocities on the interface plane between the fluid and viscoelastic plate. Analyzing these results, corresponding conclusions are drawn. In particular, it is established that the character of the influence of the rheological parameters on the stress and velocities depends significantly on the vibration phase of the external forces.