In a previously reported work, generalized equations of motion were developed for the dynamic analysis of elastic mechanism systems. Emphasis was placed on the need for including vibration effects; these, until recently, had generally been neglected, primarily for reason of the complexity of the mathematical analysis of such systems. Finite element theory was used to facilitate the modeling of elastic mechanisms. In this work, Part I of two companion papers, an approach to the solution of the equations of motion is offered. Some related considerations are also discussed. Applications to the analysis and solution method are presented. To demonstrate generality, an example is selected wherein the geometrically complex follower link of a four-bar mechanism is modeled with quadrilateral finite elements. Part II of this paper presents an experimental investigation of an elastic four-bar mechanism.