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TECHNICAL PAPERS

Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures

[+] Author and Article Information
C. Mei

Department of Mechanical Engineering, The University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128cmei@umich.edu

B. R. Mace

Institute of Sound and Vibration Research,  University of Southampton, Highfield, Southampton SO17 1BJ, UK.brm@isvr.soton.ac.uk

J. Vib. Acoust 127(4), 382-394 (Dec 17, 2004) (13 pages) doi:10.1115/1.1924647 History: Received June 08, 2004; Revised December 17, 2004

This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.

Copyright © 2005 by American Society of Mechanical Engineers
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References

Figures

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Figure 11

Free response of a uniform Timoshenko beam

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Figure 12

Forced response of a clamped free uniform Timoshenko beam

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Figure 13

Frequency responses of a uniform beam based on Timoshenko (̱) and Euler Bernoulli (…) beam theory

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Figure 14

Free response of a stepped Timoshenko beam

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Figure 15

Forced response of a stepped Timoshenko beam

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Figure 16

Frequency responses of a stepped Timoshenko beam

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Figure 1

Free-body diagram and geometry of a beam element (1)

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Figure 2

Wave propagation

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Figure 3

Wave scattering at a point support

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Figure 4

Wave reflection at a general boundary

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Figure 5

Wave reflection and transmission at change of section

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Figure 6

Beams jointed at right angle

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Figure 7

(a) Reflection coefficients at a pin joint (̱ r11 and r12, … r21, and r22); (b) transmission coefficients at a pin joint (Modulus: ̱ t11 and t21, … t12, and t22; Phase: ̱ t11, ̱.̱.̱.t21, … t12, and ̱ ̱ ̱ t22)

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Figure 8

(a) Reflection coefficients at a step change with wave incident from the right side ( ̱ r11, ̱.̱.̱. r21, … r12, and ̱ ̱ ̱ r22); (b) transmission coefficients at a step change with wave incident from the right side ( ̱ t11, ̱.̱.̱.t21, … t12, and ̱ ̱ ̱ t22); (c) Reflection coefficients at a step change with wave incident from the left side ( ̱ r11, ̱.̱.̱. r21, … r12, and ̱ ̱ ̱ r22); (d) transmission coefficients at a step change with wave incident from the left side ( ̱ t11, ̱.̱.̱. t21, … t12, and ̱ ̱ ̱ t22)

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Figure 9

(a) Transmission coefficients at a right-angle joint with longitudinal incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components); (b) reflection coefficients at a right-angle joint with longitudinal incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components); (c) transmission coefficients at a right-angle joint with bending propagating incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components); (d) reflection coefficients at a right-angle joint with bending propagating incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components); (e) transmission coefficients at a right-angle joint with bending transition incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components); (f) reflection coefficients at a right-angle joint with bending transition incident wave ( ̱ bending propagation, ̱.̱.̱. bending transition, and … longitudinal propagation components)

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Figure 10

Point excitations

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