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Journal of Vibration and Acoustics | Volume 127 | Issue 4 | TECHNICAL PAPERS
TECHNICAL PAPERS

Effect of Material Coupling on Wave Vibration of Composite Timoshenko Beams

[+] Author and Article Information
C. Mei

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

J. Vib. Acoust 127(4), 333-340 (Oct 06, 2004) (8 pages) doi:10.1115/1.1924641 History: Received May 14, 2004; Revised October 06, 2004
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This paper presents the effect of coupling between bending and torsional deformations on vibrations of composite Timoshenko beams from the wave standpoint. The dispersion characteristics and the modes of vibrations are in general affected by material coupling, except those of the torsional modes at low frequencies; and higher-frequency modes are normally more sensitive to material coupling. The wave mode transition phenomenon is also investigated. It is found that like their metallic counterparts, composite Timoshenko beams also exhibit wave mode transition. Furthermore, the transition frequency is found to be unaffected by material coupling. Numerical examples for which comparative results are available in the literature are presented.
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Copyright © 2005 by American Society of Mechanical Engineers
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References

1
Rayleigh, J. W., 1945, "Theory of Sound", Dover, New York.
2
Timoshenko, S. P., 1921, “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars,” Philos. Mag., 41 , pp. 744–746.
3
Timoshenko, S. P., 1922, “On the Transverse Vibrations of Bars of Uniform Cross Sections,” Philos. Mag., 43 , pp. 125–131.
4
Weisshaar, T. A., and Foist, B. L., 1985, “Vibration Tailoring of Advanced Composite Lifting Faces,” J. Aircr., 22 (2), pp. 141–147.
5
Migunet, P., and Dugundji, J., 1990, “Experiments and Analysis for Composite Blades Under Large Deflection, Part I: Static Behavior,” AIAA J., 28 (9), pp. 1573–1579.
6
Migunet, P., and Dugundji, J., 1990, “Experiments and Analysis for Composite Blades Under Large Deflection, Part II: Dynamic Behavior,” AIAA J., 28 (9), pp. 1580–1588.
7
Abarcar, R. B., and Cunniff, P. F., 1972, “The Vibration of Cantilever Beams of Fiber Reinforced Materials,” J. Compos. Mater., 6 , pp. 504–517.
8
Teoh, L. S., and Huang, C. C., 1977, “The Vibration of Beams of Fiber Reinforced Materials,” J. Sound Vib., 51 (4), pp. 467–473.
9
Teh, K. K., and Huang, C. C., 1980, “The Effect of Fiber Orientation on Free Vibrations of Composite Beams,” J. Sound Vib., 69 (2), pp. 327–337.
10
Banerjee, J. R., 2001, “Frequency Equation and Mode Shape Formulae for Composite Timoshenko Beams,” Compos. Struct., 51 , pp. 381–388.
11
Teh, K. K., and Huang, C. C., 1979, “The Vibrations of Generally Orthotropic Beams, A Finite Element Approach,” J. Sound Vib., 62 (2), pp. 195–206.
12
Banerjee, J. R., 1996, “Exact Dynamic Stiffness Matrix for Composite Timoshenko Beams With Applications,” J. Sound Vib., 194 (4), pp. 573–585.
13
Mead, D. J., 1994, “Waves and Modes in Finite Beams: Application of the Phase Closure Principle,” J. Sound Vib.  [CrossRef], 171 (5), pp. 695–702.
14
Members of the Mathematical Handbook Editorial Group, 1977, "Mathematical Handbook", Higher Education Publications, Beijing, pp. 88, Chap. 3.
15
Wolfram, S., 1992, "Mathematica", 2nd Edition, Addison-Wesley, Reading, MA.
16
Weisshaar, T. A., and Ryan, R. J., 1986, “Control of Aeroelastic Instabilities Through Stiffness Cross-Coupling,” J. Aircr., 23 (2), pp. 148–155.

Figures

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+Uniform composite beam with unbalanced layup
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Uniform composite beam with unbalanced layup
Figure 1

Uniform composite beam with unbalanced layup

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+Definition of positive shear force, torque, and bending moment
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Definition of positive shear force, torque, and bending moment
Figure 2

Definition of positive shear force, torque, and bending moment

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+(a) Bending propagating waves with (̱) and without (ooo) coupling. (b) Bending decaying waves with (̱) and without (ooo) coupling. (c) Torsional propagating waves with (̱) and without (ooo) coupling.
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(a) Bending propagating waves with (̱) and without (ooo) coupling. (b) Bending decaying waves with (̱) and without (ooo) coupling. (c) Torsional propagating waves with (̱) and without (ooo) coupling.
Figure 3

(a) Bending propagating waves with (̱) and without (ooo) coupling. (b) Bending decaying waves with (̱) and without (ooo) coupling. (c) Torsional propagating waves with (̱) and without (ooo) coupling.

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+Wave propagation
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Wave propagation
Figure 4

Wave propagation

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+Wave reflection at a general boundary
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Wave reflection at a general boundary
Figure 5

Wave reflection at a general boundary

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+A uniform beam structure with boundaries A and B
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A uniform beam structure with boundaries A and B
Figure 6

A uniform beam structure with boundaries A and B

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+Magnitude responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)
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Magnitude responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)
Figure 7

Magnitude responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)

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+Real and imaginary responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)
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Real and imaginary responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)
Figure 8

Real and imaginary responses of characteristic polynomial of Eqs. 41 (left) and 47 (right)

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+Mode shapes. (a) Rigid body mode, (b) first coupled mode, (c) second coupled mode, (d) third coupled mode, (e) Pure torsional mode, (f) fourth coupled mode
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Mode shapes. (a) Rigid body mode, (b) first coupled mode, (c) second coupled mode, (d) third coupled mode, (e) Pure torsional mode, (f) fourth coupled mode
Figure 9

Mode shapes. (a) Rigid body mode, (b) first coupled mode, (c) second coupled mode, (d) third coupled mode, (e) Pure torsional mode, (f) fourth coupled mode

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