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Research Papers

# Dynamics of Axially Accelerating Beams With an Intermediate Support

[+] Author and Article Information
S. M. Bağdatli

Faculty of Engineering, Department of Mechanical Engineering, Bartin University, 74100 Bartın, Turkey

E. Özkaya

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

H. R. Öz1

Department of Genetics and Bioengineering, Faculty of Engineering, Fatih University, B. Çekmece, 34500 İstanbul, Turkeyhroz@fatih.edu.tr

1

Corresponding author.

J. Vib. Acoust 133(3), 031013 (Mar 31, 2011) (10 pages) doi:10.1115/1.4003205 History: Received November 11, 2009; Revised September 01, 2010; Published March 31, 2011; Online March 31, 2011

## Abstract

The transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on simple supports are investigated. The supports are at the ends, and there is a support in between. The axial velocity is a sinusoidal function of time varying about a constant mean speed. Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion. Approximate analytical solutions are obtained using the method of multiple scales. Natural frequencies are obtained for different locations of the support other than end supports. The effect of nonlinear terms on natural frequency is calculated for different parameters. Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency. By performing stability analysis of solutions, approximate stable and unstable regions were identified. Effects of axial velocity and location of intermediate support on the stability regions have been investigated.

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## Figures

Figure 1

Axially moving beam with three simple supports

Figure 2

Variation of the first natural frequency with axial mean velocity for η=0.1 (—), η=0.3 (- - -), η=0.5 (-.-.), and different vf values

Figure 7

The first three mode shapes for vf=0.2, η=0.3, and v0=0.95

Figure 10

Variation of the first mode amplitude with velocity fluctuation frequency for vf=0.2, v0=1.0, η=0.1, 0.3, 0.5, σ1 (—), and σ2 (- - -)

Figure 11

Variation of the first mode amplitude with velocity fluctuation frequency for vf=0.8, η=0.5, v0=4, 4.7, σ1 (—), and σ2 (- - -)

Figure 14

Variations of stability region depending on mean speed and velocity fluctuation frequency in the second mode for vf=0.2, η=0.1 (- - -), η=0.3 (- . - .), and η=0.5 (—)

Figure 15

Nonlinear frequency versus amplitude in the first mode for vf=0.2, η=0.1 (- - -), η=0.3 (- . - .), and η=0.5 (—)

Figure 18

Nonlinear frequency versus amplitude in the second mode for vf=0.2

Figure 19

Nonlinear frequency versus amplitude in the first mode (—) and in the second mode (- - -) for η=0.1, v0=0.2, and for different flexure values

Figure 3

Variation of the second natural frequency with axial mean velocity for η=0.1 (—), η=0.3 (- - -), η=0.5 (-.-.), and different vf values

Figure 4

Variation of the first natural frequency with axial mean velocity for vf=0.2 and for different η values

Figure 5

Variation of the second natural frequency with axial mean velocity for vf=0.2 and for different η values

Figure 6

The first three mode shapes for vf=0.2, η=0.1, and v0=0.95

Figure 8

The first three mode shapes for vf=0.2, η=0.5, and v0=0.95

Figure 9

Variation of the first mode amplitude with velocity fluctuation frequency for vf=0.2, η=0.1, v0=0.2, 0.8, 1.0, σ1 (—), and σ2 (- - -)

Figure 12

Variations of stability region depending on mean speed and velocity fluctuation frequency in the first mode for vf=0.2, η=0.1 (- - -), η=0.3 (- . - .), and η=0.5 (—)

Figure 13

Variations of stability region depending on mean speed and velocity fluctuation frequency in the first mode for vf=0.8, η=0.1 (- - -), η=0.3 (- . - .), and η=0.5 (—)

Figure 16

Nonlinear frequency versus amplitude in the first mode for vf=0.8 and η=0.1

Figure 17

Nonlinear frequency versus amplitude in the first mode for vf=0.2

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