## 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 |
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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