Chapter 1 presents topics that form the foundation of structural dynamics by introducing the equations of motion for one-degree-of-freedom systems. Both damped and undamped free vibrations are discussed and solution techniques for both types of problems are presented. Solution procedures for forced harmonic vibrations (damped and undamped) are introduced. Forced periodic loading conditions are presented using Fourier series and the response of general periodic forces. The work performed by external forces for materials with damping and for elastic–plastic as well as viscoelastic materials is presented. Additionally, the response of a system to general forcing functions using Dirac delta (or impulse) functions is discussed. The effect of arbitrary forcing functions on the solution to single-degree-of-freedom systems is also considered. These considerations include step, time varying external, exponentially decaying, and asymptotic-step forcing functions. In addition, the responses of a system to slowly applied constant forces as well as rectangular, triangular, and half-cycle sine impulses are discussed. The chapter concludes with discussions of integral transformations. The application of Fourier and Laplace transformations to viscoelastic relations and materials with damping as well as Dirac and Heaviside functions are introduced.