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TECHNICAL PAPERS

Transient Growth Before Coupled-Mode Flutter

[+] Author and Article Information
P. J. Schmid, E. de Langre

Laboratoire d’Hydrodynamique (LadHyX), École Polytechnique, F-91128 Palaiseau, France

J. Appl. Mech 70(6), 894-901 (Jan 05, 2004) (8 pages) doi:10.1115/1.1631591 History: Received August 13, 2002; Revised June 12, 2003; Online January 05, 2004
Copyright © 2003 by ASME
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References

Figures

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General undamped system with Ω2=1.1 and a/ac=0.9. Optimal energy amplification versus time (top, solid line) and energy amplification for four random initial conditions of unit energy (top, dashed lines). Maximum energy amplification versus the coupling coefficient (bottom). The dashed curve (bottom) represents the function 1/(1−(a/ac)2). The continuous curve (bottom) represents both the maximum of G(t) over time and the upper bound given in Eq. (9).
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General undamped system with Ω2=10 and a/ac=0.9. Energy amplification versus time (top) and maximum energy amplification versus the coupling coefficient (bottom). The dashed curve represents the function 1/(1−(a/ac)2). The continuous curve represents both the maximum of G(t) over time and the upper bound given in Eq. (9).
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General damped system with Ω2=10 and a/ac=0.9. For damping a coefficient of b=0.1, (top) and a damping coefficient of b=1 (bottom).
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General damped system with Ω2=10 at criticality. Energy amplification versus time for b=1 (top), and maximum energy amplification versus damping coefficient (bottom). The dashed curve represents the asymptotic behavior ∼1/b2.
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Maximum energy amplification as a function of coupling and damping coefficient for the general damped system at Ω2=10
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Geometry sketch for panel flutter (top), follower force (bottom left), and fluid-conveying pipe (bottom right)
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Energy amplification for undamped panel flutter with a/ac=0.9 versus time (top), maximum energy amplification versus coupling coefficient (bottom). The dashed curve represents the asymptotic behavior 1/(1−(a/ac)2). The continuous curve represents both the maximum of G(t) over time and the upper bound given in Eq. (9).
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Energy amplification for undamped follower force problem with a/ac=0.9 versus time (top), maximum energy amplification versus coupling coefficient (bottom). The dashed curve represents the asymptotic behavior 1/(1−(a/ac)2). The continuous curve represents both the maximum of G(t) over time and the upper bound given in Eq. (9).
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Energy amplification for the fluid-conveying pipe problem with a/ac=0.999 versus time (top), maximum energy amplification versus coupling coefficient (bottom). The dashed curve represents the asymptotic behavior 1/(1−(a/ac)2). The top curve represents the square of the condition number of the eigenvector matrix and acts as an upper bound on the maximum energy amplification.

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