The finite element method based on the layerwise displacement theory and the modal strain energy method is applied for the vibration and damping analysis of composite plates treated with a constrained viscoelastic layer. Through several numerical examples, the effects of stacking sequences, side-to-thickness ratios of the plate, locations of partial coverage, and the thicknesses of the viscoelastic layers on vibration and damping characteristics are investigated. The developed model is a versatile tool that can be easily used to analyze the plate with general boundary conditions and stepped configurations (change in thickness) due to a partially constrained viscoelastic layer. This is due to the fact that each 2-dimensional element can retain different degrees of freedom for the in-plane displacement through the thickness of the layers.