A theory of solid friction damping of mechanical vibrations is presented that is based on a solid friction mathematical model previously proposed by the author. A summary and improved description of the general analytic features of the solid friction model are given as necessary background for the theory. The Coulomb friction damped oscillator is analyzed to establish an approach to the treatment of a simple friction damped oscillator. The approach then is generalized to treat a more general model of friction where the author's model is used to describe friction force primarily as a function of displacement. The solid friction damped oscillator studied is a wire pendulum where solid friction enters via inelastic flexing of the wire at the support. Theoretical results are generalized to be applicable to other types of oscillators and other sources of solid friction. An expression for the decay rate of the oscillation amplitude envelope of an unforced oscillator is derived. The decay rate and an equivalent linear damping ratio are determined for several values of an exponent parameter in the solid friction model.