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Research Papers

Nonlinear Transverse Vibrations and 3:1 Internal Resonances of a Beam With Multiple Supports

[+] Author and Article Information
E. Özkaya1

Department of Mechanical Engineering, Celal Bayar University, 45140, Muradiye, Manisa, Turkeyerdogan.ozkaya@bayar.edu.tr

S. M. Bağdatlı

Department of Mechanical Engineering, Celal Bayar University, 45140, Muradiye, Manisa, Turkey

H. R. Öz

Department of Genetic and Bioengineering, Fatih University, 34500, B. Çekmece, İstanbul, Turkey

1

Corresponding author.

J. Vib. Acoust 130(2), 021013 (Mar 21, 2008) (11 pages) doi:10.1115/1.2775508 History: Received April 10, 2007; Revised June 14, 2007; Published March 21, 2008

In this study, nonlinear transverse vibrations of an Euler–Bernoulli beam with multiple supports are considered. The beam is supported with immovable ends. The immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion. Forcing and damping effects are included in the problem. The general arbitrary number of support case is considered at first, and then 3-, 4-, and 5-support cases are investigated. The method of multiple scales is directly applied to the partial differential equations. Natural frequencies and mode shapes for the linear problem are found. The correction terms are obtained from the last order of expansion. Nonlinear frequencies are calculated and then amplitude and phase modulation figures are presented for different forcing and damping cases. The 3:1 internal resonances are investigated. External excitation frequency is applied to the first mode and responses are calculated for the first or second mode. Frequency-response and force-response curves are drawn.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Scheme of a beam on multiple supports, general case: arbitrary number of supports

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Figure 2

(a) Three-support case, (b) four-support case, and (c) five-support case

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Figure 3

Nonlinear frequency versus amplitude for three-support case for the first five modes (η=0.1)

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Figure 4

Nonlinear frequency versus amplitude for three-support case for the first mode only (η=0.1, 0.2, 0.3, 0.4, 0.5)

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Figure 5

Nonlinear frequency versus amplitude for three-support case for the second mode only (η=0.1, 0.2, 0.3, 0.4, 0.5)

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Figure 6

Nonlinear frequency versus amplitude for four-support case for the first mode only (η1=0.1, η2=0.2, 0.4, 0.6, 0.8)

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Figure 7

Nonlinear frequency versus amplitude for four-support case for the second mode only (η1=0.1, η2=0.2, 0.4, 0.6, 0.8)

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Figure 8

Nonlinear frequency versus amplitude for five-support case for the first mode only (η1=0.1, η2–η3: 0.3–0.5, 0.5–0.7, 0.7–0.9)

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Figure 9

Frequency-response curves for three-support case for the first five modes (η=0.1)

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Figure 10

Frequency-response curves for three-support case for the first mode only (η=0.1, 0.2, 0.3, 0.4, 0.5)

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Figure 11

Frequency-response curves for three-support case for the second mode only (η=0.1, 0.2, 0.3, 0.4, 0.5)

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Figure 12

Frequency-response curves for four-support case for the first mode only (η1=0.1, η2=0.2, 0.4, 0.6, 0.8)

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Figure 13

Frequency-response curves for four-support case for the second mode only (η1=0.1, η2=0.2, 0.4, 0.6, 0.8)

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Figure 14

Frequency-response curves for five-support case for the first mode only (η1=0.1, η2–η3: 0.3–0.5, 0.5–0.7, 0.7–0.9)

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Figure 15

(a) Frequency-response curves for three support case for the first mode only for 1:3 internal resonance (η=0.347) and (b) Frequency-response curves for three support case for the second mode only for 1:3 internal resonance (η=0.347)

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Figure 16

(a) Force-response curves for three support case for the first mode only for 1:3 internal resonance (η=0.347) and (b) Force-response curves for three support case for the second mode only for 1:3 internal resonance (η=0.347)

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