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Research Papers

Coupling of In-Plane Flexural, Tangential, and Shear Wave Modes of a Curved Beam

[+] Author and Article Information
B. Kang1

Mechanical Engineering Department,  Indiana University - Purdue University Fort Wayne, Fort Wayne, IN 46805-1499kang@engr.ipfw.edu

C. H. Riedel

Mechanical Engineering Department,  Lawrence Technological University, Southfield, MI 48075-1058riedel@ltu.edu

1

Corresponding author.

J. Vib. Acoust 134(1), 011001 (Dec 22, 2011) (13 pages) doi:10.1115/1.4004676 History: Received August 17, 2010; Revised May 12, 2011; Published December 22, 2011; Online December 22, 2011

In this paper, the coupling effects among three elastic wave modes, flexural, tangential, and radial shear, on the dynamics of a planar curved beam are assessed. Two sets of equations of motion governing the in-plane motion of a curved beam are derived, in a consistent manner, based on the theory of elasticity and Hamilton’s principle. The first set of equations retains all resulting linear coupling terms that includes both static and dynamic coupling among the three wave modes. In the derivation of the second set of equations, the effects of Coriolis acceleration and high-order elastic coupling terms are neglected, which leads to a set of equations without dynamic coupling terms between the tangential and shear wave modes. This second set of equations of motion is the one most commonly used in studies on thick curved beams that include the effects of centerline extensibility, rotary inertia, and shear deformation. The assessment is carried out by comparing the dynamic behavior predicted by each curved beam model in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and mode shapes, and frequency responses. In order to ensure the comparison is based on accurate results, the wave propagation technique is applied to obtain exact wave solutions. The results suggest that the contributions of the dynamic and high-order elastic coupling terms to the response of a thick curved beam are quite significant and that these coupling terms should not be neglected for an accurate analysis of a thick curved beam with a large curvature parameter.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a planar curved beam with a constant radius of curvature

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Figure 2

Dispersion relations of a curved beam with for κ=3 (a) a=0.1 and (b) a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively. The markers F, T, and S represent the flexural, tangential, and shear wave modes, respectively.

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Figure 3

Positive branches of the frequency spectra of a curved beam with κ=3 for (a) a=0.1 and (b) a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively. The markers F, T, and S represent the flexural, tangential, and shear wave modes, respectively.

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Figure 4

a-κ parameter plane demarcated by the conditions that characterize the wave motion of the curved beam model. Solid and dashed curves correspond to Models 1 and 2, respectively. The thick solid and dashed curves represent the condition for ωc1=ωc2 for Models 1 and 2, respectively.

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Figure 5

Mode shapes of a simply supported curved beam for κ=3 and θ0=90∘; (a) 1st mode for a=0.1; (b) 4th mode for a=0.1; (c) 1st mode for a=0.4; (d) 4th mode for Model 1 and 4th and 5th modes for Model 2 when a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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Figure 6

Mode shapes of a clamped-clamped curved beam for κ=3 and θ0=90∘; (a) 1st mode for a=0.1; (b) 4th mode for a=0.1; (c) 1st mode for Model 1 and 1st and 2nd modes for Model 2 when a=0.4; (d) 4th mode for Model 1 and 4th and 5th modes for Model 2 when a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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Figure 7

Mode shapes of a clamped-free curved beam for κ=3 and θ0=90∘; (a) 1st mode for a=0.1; (b) 4th mode for a=0.1; (c) 1st mode for a=0.4; (d) 4th mode for a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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Figure 8

Frequency responses of a simply supported curved beam for κ=3, θ0=90∘, θ*=45∘, and ξC=20∘; (a) flexural, (b) tangential, (c) shear wave mode for a=0.1; (d) flexural, (e) tangential, (f) shear wave mode for a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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Figure 9

Frequency responses of a clamped-clamped curved beam for κ=3, θ0=90∘, θ*=45∘, and ξC=20∘; (a) flexural, (b) tangential, (c) shear wave mode for a=0.1; (d) flexural, (e) tangential, (f) shear wave mode for a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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Figure 10

Frequency responses of a clamped-free curved beam for κ=3, θ0=90∘, θ*=45∘, and ξC=20∘; (a) flexural, (b) tangential, (c) shear wave mode for a=0.1; (d) flexural, (e) tangential, (f) shear wave mode for a=0.4. Solid and dashed curves correspond to Models 1 and 2, respectively.

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