Abstract
In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, a, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly. fits the sawtooth in the limit at alpha=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at alpha=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.
Original language | English |
---|---|
Pages (from-to) | 635-652 |
Number of pages | 18 |
Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences |
Volume | 366 |
Issue number | 1865 |
Early online date | 13 Aug 2007 |
DOIs | |
Publication status | Published - 28 Feb 2008 |
Keywords
- Melnikov method
- piecewise linearization
- saddle-like singularity
- homoclinic-like orbit