Transient growth of energy is known to occur even in stable dynamical systems due to the non-normality of the underlying linear operator. This has been the object of growing attention in the field of hydrodynamic stability, where linearly stable flows may be found to be strongly nonlinearly unstable as a consequence of transient growth. We apply these concepts to the generic case of coupled-mode flutter, which is a mechanism with important applications in the field of fluid-structure interactions. Using numerical and analytical approaches on a simple system with two degrees-of-freedom and antisymmetric coupling we show that the energy of such a system may grow by a factor of more than 10, before the threshold of coupled-mode flutter is crossed. This growth is a simple consequence of the nonorthogonality of modes arising from the nonconservative forces. These general results are then applied to three cases in the field of flow-induced vibrations: (a) panel flutter (two-degrees-of-freedom model, as used by Dowell) (b) follower force (two-degrees-of-freedom model, as used by Bamberger) and (c) fluid-conveying pipes (two-degree-of-freedom model, as used by Benjamin and Pai¨doussis) for different mass ratios. For these three cases we show that the magnitude of transient growth of mechanical energy before the onset of coupled-mode flutter is substantial enough to cause a significant discrepancy between the apparent threshold of instability and the one predicted by linear stability theory.

1.
Butler
,
K. M.
, and
Farrell
,
B. F.
,
1992
, “
Three-Dimensional Optimal Perturbations in Viscous Shear Flow
,”
Phys. Fluids A
,
4
, pp.
1637
1650
.
2.
Reddy
,
S. C.
, and
Henningson
,
D. S.
,
1993
, “
Energy Growth in Viscous Channel Flows
,”
J. Fluid Mech.
,
252
, pp.
209
238
.
3.
Trefethen
,
L. N.
,
Trefethen
,
A. E.
,
Reddy
,
S. C.
, and
Driscoll
,
T. A.
,
1993
, “
Hydrodynamic Stability Without Eigenvalues
,”
Science
,
261
, pp.
578
584
.
4.
Schmid, P. J., and Henningson, D. S., 2001, Stability and Transition in Shear Flows, Springer-Verlag, New York.
5.
Dowell, E. H., 1995, A Modern Course in Aeroelasticity 3rd Ed., Kluwer, Dordrecht, The Netherlands.
6.
Bamberger, Y., 1981, Mechanique de l’Ingenieur. Vol. I. Syste´mes de Corps Rigides, Hermann, Paris.
7.
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
.
8.
Blevins, R. D., 1991, Flow-Induced Vibration, 2nd Ed., Van Nostrand Reinhold, New York.
9.
Naudascher, E., and Rockwell, D., 1994, Flow-Induced Vibrations: An Engineering Guide, A. A. Balkema, Rotterdam.
10.
Bolotin, V. V., 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, New York.
11.
Semler
,
C.
,
Alighabari
,
H.
, and
Pai¨doussis
,
M. P.
,
1998
, “
A Physical Explanation of the Destabilizing Effect of Damping
,”
ASME J. Appl. Mech.
,
65
, pp.
642
648
.
12.
Pai¨doussis, M. P., 1998, Fluid Structure Interactions. Slender Structures and Axial Flow, Vol. I, Academic Press, San Diego, CA.
13.
Gebhardt
,
T.
, and
Grossmann
,
S.
,
1994
, “
Chaos Transition Despite Linear Stability
,”
Phys. Rev. E
,
50
, pp.
3705
3711
.
You do not currently have access to this content.