For computational dynamics problems, direct time integration and mode superposition are the most widely employed approaches. Of the two, direct time stepping methods have been quite popular in many commercial codes because of their various inherent advantages. However, for numerous engineering applications, mode superposition techniques continue to be the choice of many analysts, especially for linear systems and for long time responses. To date, much progress has been made in the development and understanding of direct time integration methods for computational dynamics problems encountered in engineering. Besides investigations encompassing accuracy and stability properties of these time integration schemes, the related developments include single step, multi-step, mixed and/or variable time integration features and the like. Unlike past practices customarily employed for computational dynamics, the present paper introduces a new VIrtual-Pulse (VIP) time integral methodology which offers several attractive and effective features. The proposed method possesses a unique solution methodology of its own, thus avoiding the need to employ any existing time stepping methods. The present paper primarily serves to lay down the fundamental developments towards establishing the theoretical basis via new perspectives for subsequent applications to general computational structural dynamics. However, for expository purposes, attention is confined only to linear structural dynamic systems in this paper; and Rayleigh damping is considered here, although non-Rayleigh damping can be readily permitted but is not described here. As a consequence, the characteristics of the methodology include: 1) an explicit unconditionally stable representation, 2) second-order accuracy, 3) much less algorithmic damping than direct time integration methods advocated in existing commercial codes; if an undamped system is considered, the algorithmic damping is zero, 4) the relative period error is zero, and 5) self-starting features. Theoretical details of the developments, the stability and accuracy characteristics are described. Illustrative numerical examples are presented to validate the proposed methodology for computational dynamic problems.