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Wave Propagation in Linear Planar Beams

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In Chap. 10 we derived the equations governing the dynamic response of planar beams. A simpler set of equations emanated from the linearization of the equations of motion. Further, we introduced assumptions to yield Timoshenko, Rayleigh, and Bernoulli–Euler beam theories. Timoshenko beam theory is the most complete, including rotary inertia effects and shear deformation. Rayleigh beam theory neglects shear deformation but includes rotary inertia. Bernoulli–Euler beam theory neglects both shear deformations and rotary inertia. While these differences are usually not very important in the context of static response, they have significant consequences in dynamics. In this chapter we explore the phenomenon of wave propagation in beams through classical solutions of the linearized equations of motion of these three theories.

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Change history

  • 01 January 2022

    This book was inadvertently published without updating the following corrections:


  1. 1.

    This use of λ here is not to be confused with the normalized wavelength for the analysis of a train of waves in an infinite beam. The context should make the usage clear.

  2. 2.

    Note that λL = 0 is a solution, but it also a trivial solution. Also, the negative multiples of π are solutions, but these are redundant because \(\sin (-\lambda L)\,{=}\, {-}\sin {}(\lambda L)\), which can be absorbed into D4.

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Hjelmstad, K.D. (2022). Wave Propagation in Linear Planar Beams. In: Fundamentals of Structural Dynamics. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89943-1

  • Online ISBN: 978-3-030-89944-8

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