The role of higher modes in the chaotic motion of the buckled beam. I. II
We use the generalized Melnikov's method to study the system of modal equations of a buckled beam with an external excitation. It is shown that the critical function of damping, forcing amplitude, and frequency for the first component of the Melnikov vector for the system of higher modal approximations and for a single modal Duffing's equation are related. The direct calculation of the stable and unstable perturbed manifolds is used to establish the necessary condition for chaotic motion. It is shown that the higher components of the modified Melnikov vector do not change the critical condition obtained from consideration of only the first component. Analysis of the Lyapunov exponents of the system demonstrates the difference between necessary and sufficient conditions for steady-state chaos. Finally, the additional consideration of the non-hyperbolic modes shows that they can be neglected while developing the condition for the intersection of the stable and unstable manifolds.
scientific article; zbMATH DE number 1076568