Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 2: Perturbation Analysis

[+] Author and Article Information
F. Pellicano

Dipartimento di Scienze dell’Ingegneria, Università di Modena e Reggio Emilia, Modena, I-41100 Italy

M. Amabili

Dipartimento di Ingegneria Industriale, Università di Parma, Parma, I-43100 Italy

A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

J. Vib. Acoust 122(4), 355-364 (May 01, 2000) (10 pages) doi:10.1115/1.1288591 History: Received March 01, 1999; Revised May 01, 2000
Copyright © 2000 by ASME
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Amabili,  M, Pellicano,  F., and Vakakis,  A. F., 2000, “Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells. Part 1: Equations of Motion and Numerical Results,” J. Vibr. Acoust., 122, pp. 346–354.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, Berlin.
Wiggins, S., 1990, Introduction to Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.
Amabili,  M., and Pellicano,  F., and Païdoussis,  M. P., 1998, “Nonlinear Vibrations of Simply Supported, Circular Cylindrical Shells, Coupled to Quiescent Fluid,” J. Fluids Struct. 12, No. 7, pp. 883–918.
Evensen, D. A., 1967, “Nonlinear flexural vibrations of thin-walled circular cylinders,” NASA TN D-4090.
Ginsberg,  J. H., 1973, “Large amplitude forced vibrations of simply supported thin cylindrical shells,” J. Appl. Mech., 40, pp. 471–477.
Chen,  J. C., and Babcock,  C. D., 1975, “Nonlinear vibration of cylindrical shells,” AIAA J., 13, pp. 868–876.
Nayfeh,  A. H., and Raouf,  R. A., 1987, “Non-linear oscillation of circular cylindrical shells,” Int. J. Solids Struct., 23, pp. 1625–1638.
Arnold, V. I., 1978, Mathematical Methods of Classical Mechanics, Springer Verlag, New York.
Chirikov,  B. V., 1979, “A Universal Instability of Many-Dimensional Oscillator Systems,” Phys. Rep., 52, No. 5, pp. 263–379.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear oscillations, Wiley, New York.
Kunitsyn,  A. L., and Tuyakbayev,  A. A., 1992, “The Stability of Hamiltonian Systems in the case of Multiple fourth-order Resonance,” J. Appl. Math. Mech., 56, No. 4, pp. 572–576.
Lichtenberg, A. J., and Lieberman, H. A., 1983, Regular and Stochastic Motion, Springer-Verlag, Berlin.
Brjuno,  A. D., 1971, “Analytical form of differential equation,” Trans. Moscow Math. Soc., 25, pp. 131–288.
Semler,  C., and Païdoussis,  M. P., 1996, “Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe,” J. Fluid Struct., 10, pp. 787–825.
Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B., and Wang, X., 1998, “Auto 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont),” Concordia University, Montreal, Canada.
Wolfram, S., 1996, The Mathematica Book, 3rd ed., Cambridge University Press, Cambridge, UK.
Press, H. W., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., 1986, Numerical Recipes, Cambridge University Press, New York.


Grahic Jump Location
Response amplitude versus the level of the modal excitation; exact resonance condition. Perturbation results: – stable solutions; [[dashed_line]] unstable solutions. Results of direct integrations: “○” companion mode participation; “×” absence of companion mode. (a) amplitude a1; (b) amplitude a2; (c) amplitude a3; (d) amplitude a4.
Grahic Jump Location
Response amplitude and phase versus the modal excitation frequency for f1=0.01. Perturbation results: – stable solutions; [[dashed_line]] unstable solutions. (a) Amplitude a1; (b) amplitude a2; (c) amplitude a3; (d) amplitude a4; (e) phase ϑ1.
Grahic Jump Location
Poincaré map for ω/ω1n=0.99 and f1=0.01 showing a limit cycle. (a) (A1n,Ȧ1n)-plane; (b) (B1n,Ḃ1n)-plane; (c) (A10,Ȧ10)-plane; (d) (A30,Ȧ30)-plane.
Grahic Jump Location
Response A1n in presence of the limit cycle condition: ω/ω1n=0.99 and f1=0.01. (a) Time history showing amplitude modulation; (b) spectrum of the time response.
Grahic Jump Location
Jump from an unstable steady oscillation to the stable one: ω/ω1n=1 and f1=0.01
Grahic Jump Location
Response amplitude and phase versus the external point excitation frequency for f1=0.01,f4=0.003,f2=f3=0. Perturbation results: –  stable solutions; [[dashed_line]]  unstable solutions. (a) amplitude a1; (b) amplitude a2; (c) amplitude a3; (d) amplitude a4; (e) phase ϑ1.



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