Computerized implementation of the multiple scales perturbation method using Mathematica

Computers and Structures

Volumn 76, Issue 5, 2000, Pages 565-575

  Khanin, Raya ^a   Cartmell, Matthew ^a   Gilbert, Anthony ^b  

^a UNIVERSITY OF GLASGOW   (United Kingdom)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); MATHEMATICAL MODELS; ORDINARY DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; VIBRATIONS (MECHANICAL);

MULTIPLE SCALES METHOD; SOFTWARE PACKAGE MULTIPLESCALE.M;

STRUCTURAL ANALYSIS;

EID: 0033735095     PISSN: 00457949     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7949(99)00184-4     Document Type: Article

References (30)

1. Ashworth R.P., Barr A.D.S. The resonances of structures with quadratic inertial nonlinearity under direct and parametric harmonic excitation. Journal of Sound and Vibration. 118:(1):1987;47-68.
2. Bux S.L., Roberts J.W. Nonlinear vibratory interactions in systems of coupled beams. Journal of Sound and Vibration. 104:(3):1986;497-520.
3. Cartmell M.P., Lawson J. Performance enhancement of an autoparametric vibration absorber by means of computer control. Journal of Sound and Vibration. 177:(2):1993;173-195.
4. Cartmell M.P., Roberts J.W. Simultaneous combination resonances in a parametrically excited cantilever beam. Strain. 23:(3):1987;117-126.
5. Cartmell M.P. Introduction to Linear, Parametric and Nonlinear Vibrations. 1990;Chapman and Hall, London.
6. Corless R.M., Jeffrey D.J., Monagan M.B., Pratibha. Two perturbation calculations in fluid mechanics using large-expression management. Journal of Symbolic Computation. 23:(4):1997;427-443.
7. Franciosi C., Tomasiello S. The use of Mathematica for the analysis of strongly nonlinear two-degree-of-freedom systems by means of the modified Lindstedt-Poincare method. Journal of Sound and Vibration. 211:(2):1998;145-156.
8. Gilbert A.D. A geometric approach to two-timing. Math. Proc. Camb. Phil Soc. 87:1980;167-178.
9. Jordan D.W., Smith P. Nonlinear Ordinary Differential Equations. 1977;Clarendon Press, Oxford.
10. HaQuang N, Mook DT, Plaut RH. A nonlinear analysis of the interactions between parametric and external excitations. Journal of Sound and Vibration 1987;425-39.
11. Haxton R.S., Barr A.D.S. The autoparametric vibration absorber. Journal of Engineering for Industry. 71:1972;119-125.
12. Mook D.T., HaQuang N. The influence of an internal resonance on nonlinear structural vibrations under combination resonance conditions. Journal of Sound and Vibration. 104:1986;229-241.
13. Nayfeh A.H. Perturbation methods. 1973;John Wiley and Sons, New York.
14. Nayfeh A.H., Mook D.T. Nonlinear Oscillations. 1979;John Wiley and Sons, New York.
15. Nayfeh A.H. Interaction of fundamental parametric resonances with subharmonic resonances of order one-half. Journal of Sound and Vibration. 96:(3):1984;333-340.
16. Nayfeh A.H. The response of nonlinear single-degree-of-freedom systems to multifrequency excitations. Journal of Sound and Vibration. 102:1985;403-414.
17. Othman A.M., Watt D., Barr A.D.S. Stability boundaries of an oscillator under high frequency multicomponent parametric excitation. Journal of Sound and Vibration. 112:1987;249-259.
18. Plaut R.H., HaQuang N., Mook D.T. Simultaneous resonances in nonlinear structural vibrations under two-frequency excitation. Journal of Sound and Vibration. 106:1986;361-376.
19. Rahman Z., Burton T.D. Large amplitude primary and superharmonic resonances in the Duffing oscillator. Journal of Sound and Vibration. 110:1986;363-380.
20. Rand R.H., Armbruster D. Perturbation Methods, Bifurcation Theory and Computer Algebra. 1987;Springer, NY.
21. Sanchez N.E. The method of multiple scales: asymptotic solutions and normal forms for nonlinear oscillatory problems. Journal of Symbolic Computation. 21:(2):1996;245-252.
22. Schielen R.M.J. An amplitude equation for the nonlinear analysis of river beds. Cohen A.M. Computer Algebra in Industry. 1993.
23. Schmidt F., Tondl A. Nonlinear Vibrations. 1986;Cambridge University Press, Cambridge.
24. Stoker J.J. Nonlinear Vibrations in Mechanical and Electrical Systems. 1992;John Wiley and Sons, New York.
25. Thomsen J.J. Vibrations and Stability. Order and Chaos. 1997;McGraw-Hill.
26. Vakhidov A., Vasiliev N.N. A new approach for analytical computation of Hamiltonian of a satellite perturbed motion. Journal of Symbolic Computation. 24:(6):1997;705-710.
27. Watt D., Cartmell M. An externally loaded parametric oscillator. Journal of Sound and Vibration. 170:1994;339-364.
28. Wolfram S. The Mathematica Book. 1996.
29. Wu W.Z., Mote C.D. Parametric excitation of an axially moving band by periodic edge loading. Journal of Sound and Vibration. 110:1986;27-39.
30. Yu P. Computation of normal forms via a perturbation technique. Journal of Sound and Vibration. 211:(1):1998;19-38.