Asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation

SIAM Journal on Applied Mathematics

Volumn 60, Issue 2, 2000, Pages 602-632

  Boertjens, G J ^a   Van Horssen, W T ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; MATHEMATICAL MODELS; NONLINEAR EQUATIONS; OSCILLATIONS; PERTURBATION TECHNIQUES;

ASYMPTOTIC THEORY; INTERNAL RESONANCES; TWO TIMESCALES PERTURBATION; WEAKLY NONLINEAR BEAM EQUATION;

INITIAL VALUE PROBLEMS;

EID: 0034566366     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139998336766     Document Type: Article

References (24)

1. A.C. LAZER AND P.J. MCKENNA, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32 (1990), pp. 537-578.
2. A.C. LAZER AND P.J. MCKENNA, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), pp. 243-274.
3. P.J. MCKENNA AND W. WALTER, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), pp. 167-177.
4. W.T. VAN HORSSEN, An asymptotic theory for a class of initial-boundary value problems for weakly nonlinear wave equations with an application to a model of the galloping oscillations of overhead transmission lines, SIAM J. Appl. Math., 48 (1988), pp. 1227-1243.
5. W.T. VAN HORSSEN, Asymptotics for a class of semilinear hyperbolic equations with an application to a problem with quadratic nonlinearity, Nonlinear Anal., 19 (1992), pp. 501-530.
6. W.T. VAN HORSSEN AND A.H.P. VAN DER BURGH, On initial boundary value problems for weakly semilinear telegraph equations. Asymptotic theory and application, SIAM J. Appl. Math., 48 (1988), pp. 719-736.
7. G.J. BOERTJENS AND W.T. VAN HORSSEN, On mode interactions for a weakly nonlinear beam equation, Nonlinear Dynam., 17 (1998), pp. 23-40.
8. A.H.P. VAN DER BURGH, On the Asymptotic Validity of Perturbation Methods for Hyperbolic Differential Equations, in Asymptotic Analysis, Lecture Notes in Math. 711, F. Verhulst, ed., Springer-Verlag, Berlin, New York, 1979, pp. 229-240.
9. J.B. KELLER AND S. KOGELMAN, Asymptotic solutions of initial value problems for nonlinear partial differential equations, SIAM J. Appl. Math., 18 (1970), pp. 748-758.
10. J. KEVORKIAN AND J.D. COLE, Multiple Scale and Singular Pertubation Methods, Springer-Verlag, New York, 1996.
11. S.C. CHIKWENDU AND J. KEVORKIAN, A perturbation method for hyperbolic equations with small nonlinearities, SIAM J. Appl. Math., 22 (1972), pp. 235-258.
12. P.L. CHOW, Asymptotic solutions of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations, SIAM J. Appl. Math., 22 (1972), pp. 629-647.
13. R. W. LARDNER, Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing, Quart. Appl. Math., 35 (1977), pp. 225-238.
14. J.C LUKE, A perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc. London Ser. A, 292 (1966), pp. 403-412.
15. A.C.J. STROUCKEN AND F. VERHULST, The Galerkin-averaging method for nonlinear, undamped continuous systems, Math. Methods Appl. Sci., 9 (1987), pp. 520-549.
16. M.S. KROL, On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Methods Appl. Sci., 11 (1989), pp. 649-664.
17. M.S. KROL, The Method of Averaging in Partial Differential Equations, Ph.D. thesis, University of Utrecht, Utrecht, The Netherlands, 1990.
18. A.H. NAYFEH AND M. ABDEL-ROHMAN, Analysis of wind excited vibrations of cantilever beams using the method of multiple scales, J. Sound and Vibration, 144 (1991), pp. 87-93.
19. M. ABDEL-ROHMAN, Galloping of tall prismatic structures: A two-dimensional analysis, J. Sound Vibration, 153 (1992), pp. 97-111.
20. H. KAUDERER, Nichtlineare Mechanik, Springer-Verlag, Berlin, 1958.
21. L.A. SEGEL AND G.H. HANDELMAN, Mathematics Applied to Continuum Mechanics, Macmillan, New York, 1977.
22. L. WANG, D.L. BOSLEY, AND J. KEVORKIAN, Asymptotic analysis of a class of three-degree-of-freedom Hamiltonian systems near stable equilibria, Phys. D, 88 (1995), pp. 87-115.
23. D.C. PEREIRA, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Anal., 14 (1990), pp. 613-623.
24. Q.H. CHOI AND T. JUNG, On periodic solutions of the nonlinear suspension bridge equation, Differential Integral Equations, 4 (1991), pp. 383-396.