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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

Butler,  K. M., and Farrell,  B. F., 1992, “Three-Dimensional Optimal Perturbations in Viscous Shear Flow,” Phys. Fluids A, 4, pp. 1637–1650.
Reddy,  S. C., and Henningson,  D. S., 1993, “Energy Growth in Viscous Channel Flows,” J. Fluid Mech., 252, pp. 209–238.
Trefethen,  L. N., Trefethen,  A. E., Reddy,  S. C., and Driscoll,  T. A., 1993, “Hydrodynamic Stability Without Eigenvalues,” Science, 261, pp. 578–584.
Schmid, P. J., and Henningson, D. S., 2001, Stability and Transition in Shear Flows, Springer-Verlag, New York.
Dowell, E. H., 1995, A Modern Course in Aeroelasticity 3rd Ed., Kluwer, Dordrecht, The Netherlands.
Bamberger, Y., 1981, Mechanique de l’Ingenieur. Vol. I. Systémes de Corps Rigides, Hermann, Paris.
Benjamin,  B. T., 1961, “Dynamics of a System of Articulated Pipes Conveying Fluids I. Theory,” Proc. R. Soc. London, Ser. A, 261, pp. 457–486.
Blevins, R. D., 1991, Flow-Induced Vibration, 2nd Ed., Van Nostrand Reinhold, New York.
Naudascher, E., and Rockwell, D., 1994, Flow-Induced Vibrations: An Engineering Guide, A. A. Balkema, Rotterdam.
Bolotin, V. V., 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, New York.
Semler,  C., Alighabari,  H., and Païdoussis,  M. P., 1998, “A Physical Explanation of the Destabilizing Effect of Damping,” ASME J. Appl. Mech., 65, pp. 642–648.
Païdoussis, M. P., 1998, Fluid Structure Interactions. Slender Structures and Axial Flow, Vol. I, Academic Press, San Diego, CA.
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Figures

Grahic Jump Location
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).
Grahic Jump Location
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).
Grahic Jump Location
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).
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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).
Grahic Jump Location
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).
Grahic Jump Location
Geometry sketch for panel flutter (top), follower force (bottom left), and fluid-conveying pipe (bottom right)
Grahic Jump Location
Maximum energy amplification as a function of coupling and damping coefficient for the general damped system at Ω2=10

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