We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The Wentzel-Kramers-Brillouin (WKB) method is then applied to the slow flow equations to obtain an analytic approximation.
Get full access to this article
View all access options for this article.
References
1.
Bender, C.M. and Orszag, S.A., 1999, Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag, New York .
2.
Holmes, M.H., 1995, Introduction to Perturbation Methods, Springer-Verlag, New York .
3.
Levobitz, N.R. and Pesci, A.I., 1995, “Dynamic bifurcation in hamiltonian systems with one degree of freedom,” SIAM Journal of Applied Mathematics55(4), 1117-1133 .
4.
Nayfeh, A.H. and Asfar, K.R., 1988, “Non-stationary parametric oscillations,” Journal of Sound and Vibration124(3), 529-537 .
5.
Neal, H.L. and Nayfeh, A.H., 1990, “Response of a single-degree-of-freedom system to a non-stationary principal parametric excitation,” International Journal of Non-Linear Mechanics25(2/3), 275-284 .
6.
Raman, A., Bajaj, A.K., and Davies, P., 1996, “On the slow transition across instabilities in non-linear dissipative systems,” Journal of Sound and Vibration192(4), 835-865 .
7.
Raman, A. and Bajaj, A.K., 1998, “On the non-stationary passage through bifurcations in resonantly forced Hamiltonian oscillators,” International Journal of Non-Linear Mechanics33(5), 907-933 .
8.
Stoker, J.J., 1950, Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley, New York .
