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The dynamics of atoms or molecules adsorbed on a metal surface, and excited by collisions with an atomic beam, are treated within a theory that includes energy dissipation into lattice vibrations by means of a frequency and temperature dependent friction function. The theory provides dynamic structure factors for energy transfer derived from collisional time correlation functions. It describes the relaxation of a vibrationally excited atom or molecule within a model of a damped quantum harmonic oscillator bilinearly coupled to a bath of lattice oscillators. The collisional time correlation function is generalized to include friction effects and is applied to the vibrational relaxation of the frustrated translation mode of Na adsorbed on a Cu(001) surface, CO on Cu(001), and CO on Pt(111), following excitation by collisions with He atoms. Results for the frequency shift and width of line shapes versus surface temperature are in …