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Research Papers

Aeroelastic Stability of Axially Moving Webs Coupled to Incompressible Flows

[+] Author and Article Information
Merrill Vaughan, Arvind Raman

Dynamic Systems and Stability Laboratory, School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088

J. Appl. Mech 77(2), 021001 (Dec 08, 2009) (17 pages) doi:10.1115/1.2910902 History: Received September 22, 2003; Revised January 21, 2007; Published December 08, 2009; Online December 08, 2009

The aeroelastic flutter of thin flexible webs severely limits their transport speeds and consequently the machine throughputs in a variety of paper, plastics, textiles, and sheet metal industries. The aeroelastic stability of such high-speed webs is investigated using an assumed mode discretization of an axially moving, uniaxially tensioned Kirchhoff plate coupled with cross and machine direction flows of a surrounding incompressible fluid. The corresponding aerodynamic potentials are computed using finite element solutions of certain mixed boundary value problems that arise in the fluid domain. In the absence of air coupling, the cross-span mode frequencies tightly cluster together, and the web flutters via mode coalescence at supercritical transport speed. Web coupling to an initially quiescent incompressible potential flow significantly reduces the web frequencies, substantially modifies the mode shapes, and separates the frequency clusters, while only marginally affecting the flutter speed and frequency. The inclusion of machine direction base flows significantly modifies the web stability and mode shapes. Cross machine direction flows lead to the flutter with vanishing frequency of very high cross-span nodal number modes, and the unstable vibration naturally localizes at the leading free edge. These results corroborate several previous experimental results in literature and are expected to guide ongoing experiments and the design of reduced flutter web handling systems.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

A schematic depicting the translating web with uniform uniaxial tension and coupling to base fluid flows

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Figure 2

Convergence of modal frequencies with respect to the number of basis functions used in the discretization. (a) Four cross-span modes are included as the number of in-span modes is varied. Symmetric cross-span modes 00 (×) and 10 (+) are calculated at c=1.0000027 and antisymmetric cross-span modes 01 (○) and 11 (◇) are calculated at c=1.00000354. (b) One in-span mode is included as the number of cross-span modes is varied at Vy=0.0686 near divergence instability.

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Figure 3

(a) Contour plot of mass-normalized ψ00 basis function, (b) corresponding aerodynamic potential generated on the web surface, z=0, and (c) corresponding aerodynamic potential generated at x=0

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Figure 4

Frequency clustering of a stationary, uniformly tensioned, aerodynamically uncoupled web as a function of web stiffness to tension ratio ε

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Figure 5

Modal frequencies of the m=0 cluster versus Λ. Λ=40 corresponds to the web system considered in this paper.

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Figure 6

Eigenfunctions of the stationary web system. (a) 00 mode shape at Λ=0. (b) 00 mode shape at Λ=40. (c) 01 mode shape at Λ=0. (d) 01 mode shape at Λ=40. (e) 10 mode shape at Λ=0. (f) 10 mode shape at Λ=40. The mode shapes are normalized in this graph such that the maximum amplitude is set to 1.

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Figure 7

(a) Modal frequencies of the in vacuo (Λ=0) web system versus nondimensional transport speed. (b) Modal frequencies of the incompressible air-coupled (Λ=40) web system. For both plots: symmetric cross-span modes (×) and antisymmetric cross-span modes (○).

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Figure 8

Eigenvalues of the web system just before, at, and after the nondimensional critical transport speed: (a) imaginary part of in vacuo (Λ=0) web system eigenvalues, (b) imaginary part of incompressible air-coupled (Λ=40,000) web system eigenvalues, (c) real part of in vacuo (Λ=0) web system eigenvalues, and (d) real part of incompressible air-coupled (Λ=40000) web system eigenvalues. Mode numbers are labeled on the plot.

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Figure 9

First flutter mode shape for Λ=40, c nonzero, Vx=Vy=0. Nine time instances during 1cycle of oscillation are shown.

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Figure 10

(a) Variation in critical transport speed, transport speed of gyroscopic stabilization, and flutter versus and Λ (b) variation in modal frequency at the onset of flutter versus Λ

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Figure 11

Eigenvalues of the stationary web system versus fluid axial flow velocity; (a) imaginary part of eigenvalues and (b) real part of eigenvalues. Mode numbers are labeled on both plots.

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Figure 12

First flutter mode shape for Λ=40, c=Vy=0, Vx nonzero. Nine time instances during 1cycle of oscillation are shown.

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Figure 13

First occurrences of 00 mode divergence; 00 mode gyroscopic restabilization; 00,10 modal coalescence and flutter, 01 mode divergence; 01 mode gyroscopic restabilization; and 01,11 modal coalescence and flutter for Λ=40, c=Vy=0, Vx nonzero

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Figure 14

Eigenvalues and eigenfunctions of the stationary web versus fluid cross flow velocity at an artificially high bending stiffness to tension ratio ε=5×10−6; (a) imaginary part of eigenvalues, (b) real part of eigenvalues, (c) eigenfunction of the ninth lowest frequency at Vy=0, (d) eigenfunction of the eight lowest frequency at Vy=0.008, (e) eigenfunction of the third lowest frequency at Vy=0.051, and (f) eigenfunction of the lowest frequency at Vy=0.055. Mode numbers are labeled on plot (a).

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Figure 15

First flutter mode shape for ε=5×10−6, Λ=40, c=Vx=0, Vy nonzero. Nine time instances during 1cycle of oscillation are shown.

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Figure 16

First occurrence of divergence and flutter as bending stiffness to tension ratio ε and nondimensional cross flow Vy are varied

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