Comparison of the four rod theories of longitudinally vibrating rods

Abstract

Dynamics of a longitudinally vibrating rod are analyzed from a wave standpoint, in which the motion of the rod is described in terms of waves that propagate through the uniform rod and are reflected and transmitted at structural discontinuities and boundaries. This study is based on the four engineering theories, namely, the elementary, Love, Mindlin–Herrmann, and three-mode theories. The propagation relations that are governed by the equations of motion are derived. The reflection and transmission relations, which are dependent upon the continuity and equilibrium conditions at structural discontinuities, are obtained. Waves generated by externally applied forces are found. The wave propagation, reflection, and transmission relations are assembled to provide a concise and systematic approach to vibration analysis of rods. Numerical examples are presented. Comparisons and recommendations are made for meaningful engineering practice.

Keywords

Longitudinal wave vibration elementary theory Love theory Mindlin–Herrmann theory three-mode theory

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References

Cremer

Heckl

Petersson

BAT

(2005) Structure-borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 3rd edn. Berlin: Springer-Verlag.

Doyle

(1989) Wave Propagation in Structures, New York: Springer-Verlag.

Graff

(1975) Wave Motion in Elastic Solids, Columbus, OH: Ohio State University Press.

Inman

(1994) Engineering Vibrations, New Jersey: Prentice-Hall, Inc.

Krawczuk

Grabowska

Palacz

(2006) Longitudinal wave propagation part I —comparison of rod theories. Journal of Sound and Vibration 295: 461–478.

Love

(1927) A Treatise on the Mathematical Theory of Elasticity, 4th edition. New York: Dover Publications.

Mace

(1984) Wave reflection and transmission in beams. Journal of Sound and Vibration 97: 237–246.

Mei

(2005) Global and local wave vibration characteristics of materially coupled composite beam structures. Journal of Vibration and Control 11(11): 1413–1433.

Mei

(2010) In-plane vibrations of classical planar frame structures – an exact wave-based analytical solution. Journal of Vibration and Control 16(9): 1265–1285.

10.

Mei

(2013) Effects of rotary inertia, shear deformation, and joint model on vibration characteristics of single-story multi-bay planar frame structures. Journal of Vibration and Control. . Epub ahead of print 29 March 2013.

11.

Mei

(2012) Studying the effects of lumped end mass on vibrations of a Timoshenko beam using a wave-based approach. Journal of Vibration and Control 18(5): 733–742.

12.

Meirovitch

(2001) Fundamentals of Vibrations, New York: McGraw-Hill Higher Education.

13.

Mindlin RD and Herrmann G (1951) A one dimensional theory of compressional waves in an elastic rod. In Proceedings of the First U.S. National Congress on Applied Mechanics, Chicago, IL, pp. 187–191.

14.

Viktorow

(1967) Rayleigh and Lamb Waves in Physical Theory and Applications, New York: Plenum Press.