On boundary damping for a weakly nonlinear wave equation

Nonlinear Dynamics

Volumn 30, Issue 2, 2002, Pages 179-191

  Darmawijoyo, ^a   Van Horssen, W T ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Asymptotics; Boundary damping; Two timescales perturbation method; Wave equation

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; DAMPING; FLEXIBLE STRUCTURES; INITIAL VALUE PROBLEMS; NONLINEAR EQUATIONS; OSCILLATIONS; OVERHEAD LINES; PERTURBATION TECHNIQUES; SUSPENSION BRIDGES;

ASYMPTOTIC APPROXIMATIONS;

WAVE EQUATIONS;

EID: 0036805123     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020473930223     Document Type: Article

References (19)

1. Keller, J.B. and Kogelman, S., 'Asymptotic solutions of initial value problems for nonlinear partial differential equation', SIAM Journal on Applied Mathematics 18, 1970, 748-758.
2. Darmawijoyo and van Horssen W.T., 'On the weakly damped vibration of a string attached to a spring-mass-dashpot system', Journal of Vibration and Control, 2002, to appear.
3. Morgül, Ö., Rao, B.P., and Conrad, F., 'On the stabilization of a cable with a tip mass', IEEE Transactions on Automatic Control 39(10), 1994, 2140-2145.
4. Krol, M.S., 'On a Galerkin-averaging method for weakly nonlinear wave equations', Mathematical Methods in Applied Sciences 11, 1989, 649-664.
5. van Horssen, W.T. and van der Burgh, A.H.P., 'On initial-boundary value problems for weakly semi-linear telegraph equations. Asymptotic theory and application', SIAM Journal on Applied Mathematics 48, 1988, 719-736.
6. van Horssen, W.T., '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 Journal on Applied Mathematics 48, 1988, 1227-1243.
7. Boertjens, G.J. and van Horssen, W.T., 'On mode interactions for a weakly nonlinear beam equation', Nonlinear Dynamics 17, 1998, 23-40.
8. Boertjens, G.J. and van Horssen, W.T., 'An asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation, SIAM Journal on Applied Mathematics 60, 2000, 602-632.
9. Conrad, F. and Morgül, Ö., 'On the stabilization of a flexible beam with a tip mass', SIAM Journal on Control and Optimization 36, 1998, 1962 - 1986.
10. Castro, C. and Zuazua, E., 'Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass', SIAM Journal on Control Optimization 36, 1998, 1576-1595.
11. Durant, N.J., 'Stress in a dynamically loaded helical spring', The Quarterly Journal of Mechanics and Applied Mathematics XIII (2), 1960, 251-256.
12. Cox, S. and Zuazua, E., 'The rate at which energy decays in a string damped at one end', Indiana University Mathematics Journal 44, 1995, 545-573.
13. Rao, B.P., 'Decay estimates of solutions for a hybrid system of flexible structures', European Journal of Applied Mathematics 4, 1993, 303-319.
14. Rao, B.P., 'Uniform stabilization of a hybrid system of elasticity', SIAM Journal on Control Optimization 33, 1995, 440-454.
15. Nayfeh, A.H., Perturbation Methods, Wiley, New York, 1973.
16. Kevorkian, J. and Cole, J.D., Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996.
17. Holmes, M.H., Introduction to Perturbation Methods, Springer-Verlag, New York, 1995.
18. Hagedorn, P. and Seemann, W., 'Modern analytical methods applied to mechanical engineering system', in Modern Methods of Analytical Mechanics and Their Applications, V.V. Rumyantsev and A.V. Karapetyan (eds.) CISM Courses and Lectures, Vol. 387, Springer-Verlag, New York, 1998, pp. 317-328.
19. Wang, H., Elcrat, A.R., and Egbert, R.I., 'Modelling and boundary control of conductor galloping', Journal of Sound and Vibration 161, 1993, 301-315.