Transient Response of Tapered Elastic Bars

Abstract

Exact solutions are obtained for a model of the longitudinal displacement along an elastic tapered bar due to a force applied at its blunt end. A formula for velocity amplification is given; it specifies the velocity of the pointed end of the bar shortly after it feels the influence of the force. For a bar with an exponentially decreasing cross-sectional area, the velocity is magnified by twice an exponential function of length. This result has applications in the design of piezoelectric drills. In addition, we discuss the differences between the motions of rigid and elastic bars during the transient before one complete reflection of the wave induced by a force applied to an end of the bar. In this regime, force is proportional to velocity for elastic bars with constant cross-sectional areas. While the force–velocity relationship is more complicated for tapered elastic bars, their exact relationship is determined.