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TECHNICAL PAPERS

Transient Response of Tapered Elastic Bars

[+] Author and Article Information
Carmen Chicone

Department of Mathematics, University of Missouri, Columbia, MO 65211carmen@math.missouri.edu

Michael Heitzman

Department of Mathematics, University of Missouri, Columbia, MO 65211heitzman@missouri.edu

Z. C. Feng

Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211fengf@missouri.edu

J. Appl. Mech 74(4), 677-685 (Jul 22, 2006) (9 pages) doi:10.1115/1.2424719 History: Received February 02, 2006; Revised July 22, 2006

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.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The velocity response versus time of the elastic tapered bar with l=1 and c=8 (computed using a finite difference scheme applied to the PDE 1), with cross-sectional area A(x)=e−4x in the top panel and A(x)=e−8x in the bottom panel, is depicted for the quadratic prescribed input displacement u(0,t)=f(t)=2t2

Grahic Jump Location
Figure 2

The force (−ρc2ux(0,t)) versus time (computed using a finite difference scheme applied to the PDE 1) at the blunt end of the elastic tapered bar (l=1, ρ=1, and c=8) with cross-sectional areas A(x)=eβx, for β∊{−1,−4,−8} is depicted for the quadratic prescribed input displacement u(0,t)=f(t)=2t2

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