Analysing panel flutter in supersonic flow by Hopf bifurcation

Document Type : Research Article

Authors

• z. Monfared ¹
• Z. Dadi ²

¹ Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

² Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran.

10.22067/ijnao.v4i2.32430

Abstract

This paper is devoted to study of a partial differential equation governing panel motion in supersonic flow. This PDE can be transformed to an ODE by means of a Galerkin method. Here by using a criterion which is closely related to the Routh-Hurwitz criterion, we investigate the mentioned transformed ODE from Hopf bifurcation point of view. In fact we obtain a region for existence of simple Hopf bifurcation for it. With the aid of computer language Matlab and Hopf bifurcation tool, flutter and limit cycle oscillations of panel are verified. Moreover, Hopf bifurcation theory is used to analyse the flutter speed of the system.

Keywords

• Panel fllutter
• Limit cycle
• Hopf bifurcation
• Routh-Hurwitz criterion
• Vibrations

20.1001.1.24236977.2014.4.2.1.1

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References

[1] Clancy, L.J., Aerodynamics, Pitman Publishing Limited, London. ISBN 0-273 01120-0, 1975.

[2] Eftekhari, S.A., Bakhtiari-Nejad, F. and Dowell, E.H. Damage detection of an aeroelastic panel using limit cycle oscillation analysis, International Journal of Non-Linear Mechanics, Vol. 58, 99-110, 2014.

[3] Feng, Z. C. and Sethna, P. R. Global bifurcations in the motion of parametri cally excited thin plate, Nonlinear Dynamics, 4, 389408, 1993.

[4] Guckenheimer, J. and Holmes, P.J. Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, 1993.

[5] Holmes, P. Bifurcations to divergence and flutter in flow-induced oscillations: A finite-dimensional analysis, Journal of Sound and Vibration, Vol. 53(4), 161174, 1977.

[6] Holmes, P. Center manifolds, normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions, Physica D2, 449481, 1981.

[7] Holmes, P. and Marseden, J. Bifurcation to divergence and flutter in flow-induced oscillations: An infinite dimensional Analysis, International Federation of Automatic Control, Vol. 14, pp. 367 384, 1978.

[8] Kovacic, G. and Wiggins, S. Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D57, 185225, 1992.

[9] Kuznetsov, Y.A. Elements of Applied Bifurcation Theory, New York: SpringerVerlag, 2004.

[10] Liu, W. Criterion of Hopf bifurcation without using eigenvalues, Journal of mathe-matical analysis and applications, 182, 250-256, 1994.

[11] Marsden, J. and McCracken, M., Hopf Bifurcation and Its Applications,Springer, 1976.

[12] Peng, L. and Yiren, y. On the stability and chaos of a plate with motion constraints subjected to subsonicflow, International Journal of Non-Linear Mechanics, Vol. 59, 28-36, 2014.

[13] Perco, L. Differential equations and dynamical systems, Springer- Verlag, 1991.

[14] Pourtakdoust, S. H. and Fazelzadeh, S. A., Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow, Nonlinear Dynamics, Vol. 32: 387404, 2003.

[15] Song, Z. G. and Li, F. M. Vibration and aeroelastic properties of ordered and disordered two-span panels in supersonic airflow, International Journal of Mechanical Sciences, , Vol. 81, 65-72, 2014.

[16] S¨uli, E. Numerical solution of ordinary differential equations, Lecture note, University of Oxford, 2013.

[17] Vedeneev, V. Panel flutter at low supersonic speeds, Journal of Fluids and Structures Vol. 29 , 79-96, 2012.

[18] Yang, X. L. and Sethna, P. R., Local and global bifurcations in parametrically excited vibrations nearly square plates, International Journal of Non-Linear Mechanics 26(2), 199220, 1990.

[19] Zhang, W. and Zhaomiao, L., Global Dynamics of a Parametrically and Externally Excited Thin Plate, Nonlinear Dynamics 24: 245268, 2001.

[20] Zhang, X. Local bifurcations of nonlinear viscoelastic panel in supersonic flow, Commun Nonlinear Sci Numer Simulat, Vol. 18, 1931-1938, 2013.