The center of elasticity concept for one-dimensional structural elements is most often understood in the context of a point in the cross-section lying between the flexural and torsional centers of the section. However, the elastic center as a fundamental property of the element itself may be discussed in terms of the displacements and loads applied to a particular point. The elastic center is the point at which an applied force produces pure translation, and an applied moment produces pure rotation about the same axis. This concept is extended to the dynamic regime by including both elastic and inertial properties, leading to a dynamic center of elasticity which is valid across a range of frequencies. Three methods are proposed to locate the dynamic elastic center of a simple lumped parameter system, revealing that the elastic center location varies greatly, even moving off to infinity at certain critical frequencies. Each method is also applied to a continuously deformable cantilevered Euler-Bernoulli beam, producing rich dynamic behavior. The three methods are shown to yield equivalent results for both systems. Finally, the results are compared with dynamic stiffness and modal analysis to obtain physical insight into the dynamic behavior of a system's elastic center.