Asymptotics for a system of nonlinearly coupled wave equations with an application to the galloping oscillations of overhead transmission lines
Author:
W. T. van Horssen
Journal:
Quart. Appl. Math. 47 (1989), 197-219
MSC:
Primary 35C20; Secondary 35L70
DOI:
https://doi.org/10.1090/qam/998096
MathSciNet review:
998096
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Abstract: In this paper an asymptotic theory for a class of initial-boundary value problems for systems of weakly and nonlinearly coupled wave equations is presented. The theory implies the well-posedness of the problem in the classical sense and the asymptotic validity of formal approximations on long time scales.
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C. J. Myerscough, A simple model of the growth of wind-induced oscillations in overhead lines, J. Sound Vibration 28, 699–713 (1973)
C. J. Myerscough, Further studies of the growth of wind-induced oscillations in overhead lines, J. Sound Vibration 39, 503–517 (1975)
H. H. Ottens and R. K. Hack, Results of an exploratory study of the galloping oscillations of overhead transmission lines (in Dutch), Report NLR TR 80016 L of the National Aerospace Laboratory NLR, The Netherlands (1980)
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A. Simpson, Wind-induced vibration of overhead power transmission lines, Sci. Prog. Oxford 68, 285–308 (1983)
O. Vejvoda, Partial differential equations: time-periodic solutions, Martinus Nijhoff Publishers, The Hague, 1982
C. G. A. van der Beek and A. H. P. van der Burgh, On the periodic wind-induced vibrations of an oscillator with two degrees of freedom, Nieuw Arch. Wisk. 2, 207–225 (1987)
J. G. Besjes, On the asymptotic methods for non-linear differential equations, J. Mécanique 8, 357–372 (1969)
S. C. Chikwendu and J. Kevorkian, A perturbation method for hyperbolic equations with small non-linearities, SIAM J. Appl. Math. 22, 235–258 (1972)
W. Eckhaus, New approach to the asymptotic theory of nonlinear oscillations and wave-propagation, J. Math. Anal. Appl. 49, 575–611 (1975)
J. P. den Hartog, Mechanical vibrations, 4th ed., McGraw-Hill, New York, 1956
W. T. van Horssen and A. H. P. van der Burgh, On initial-boundary value problems for weakly semi-linear telegraph equations. Asymptotic theory and application, SIAM J. Appl. Math. 48, 719–736 (1988)
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, 1227–1243 (1988)
H. M. Irvine and T. K. Cauchey, The linear theory of free vibrations of a suspended cable, Proc. Roy. Soc. London Ser. A 341, 299–315 (1974)
J. B. Keller and S. Kogelman, Asymptotic solutions of initial value problems for nonlinear partial differential equations, SIAM J. Appl. Math. 18, 748–758 (1970)
J. Kevorkian and J. D. Cole, Perturbation methods in applied mathematics, Springer-Verlag, New York, 1981
C. J. Myerscough, A simple model of the growth of wind-induced oscillations in overhead lines, J. Sound Vibration 28, 699–713 (1973)
C. J. Myerscough, Further studies of the growth of wind-induced oscillations in overhead lines, J. Sound Vibration 39, 503–517 (1975)
H. H. Ottens and R. K. Hack, Results of an exploratory study of the galloping oscillations of overhead transmission lines (in Dutch), Report NLR TR 80016 L of the National Aerospace Laboratory NLR, The Netherlands (1980)
J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Appl. Math. Sci. 59, Springer-Verlag, New York, 1985
A. Simpson, Wind-induced vibration of overhead power transmission lines, Sci. Prog. Oxford 68, 285–308 (1983)
O. Vejvoda, Partial differential equations: time-periodic solutions, Martinus Nijhoff Publishers, The Hague, 1982
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© Copyright 1989
American Mathematical Society
