This chapter highlights the fundamental behavior of free and forced vibrations of a single-degree-of-freedom system. It discusses several integration techniques such as unit step function and DuhameGs integral, and examines the effects of damping and various forcing functions on dynamic response. The motion equation is derived on the basis of Newton’s second law for which the solution is based on the integration of differential equations. A study of the dynamic analysis of structures may begin logically with an investigation of elementary systems. An understanding of the dynamic behavior of elementary systems is essential for the practising engineer as well as for the student who, with the aid of high-capacity computer programs, intends to use matrix methods for the solution of structural dynamics problems. The behavior of dynamic response is derived from the differential equation of motion for the purpose of observing the relationship between the amplification factor and the ratio of the forced period to the natural period.