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TECHNICAL BRIEFS

# Lateral Vibration of Two Axially Translating Beams Interconnected by a Winkler Foundation

[+] Author and Article Information
Mohamed Gaith

Faculty of Engineering, Al-Isra Private University, Amman, Jordan

Sinan Müftü1

Department of Mechanical Engineering, Northeastern University, Boston, MA 02115s.muftu@neu.edu

Note that the critical speed expression given in Ref. 21 for a traveling Timoshenko beam can be shown to be identical to $νc$ given by (2) after letting rotary inertia, $ρI=0$, and shear stiffness $G→∞$.

1

Corresponding author.

J. Vib. Acoust 129(3), 380-385 (Sep 13, 2006) (6 pages) doi:10.1115/1.2732353 History: Received January 04, 2006; Revised September 13, 2006

## Abstract

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. The natural frequencies and associated mode shapes are obtained. The natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at a critical velocity and a critical tension; and, divergence and flutter instabilities coexist at postcritical speeds, and divergence instability takes place precritical tensions. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams are presented.

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## Figures

Figure 1

First four complex mode shapes for a traveling system with nonequal mass densities, Rm=0.6; the real (solid line) and imaginary parts (dashed line) for K=100, ν=5, μ=10, and Rp=Rs=1

Figure 2

Imaginary and real parts of the nondimensional natural frequencies of a system with identical properties, as a function of translation speed ν, K=100, μ=10, and Rm=Rs=Rp=1

Figure 3

Nondimensional natural frequencies as a function of the nondimensional axial tension parameter μ for K=100 and ν=5

Figure 4

Nondimensional transport speed ν as a function of nondimensional axial tension parameter μ for K=100 and Rs=Rp=1 and different mass ratios Rm

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