Chaotic Vibrations of a Shallow Arch with Initial Configuration of Catenary

Analytical results are presented on chaotic vibrations of a shallow arch with initial configuration of catenary. The shallow arch is simply supported at both ends and is subjected to periodic lateral force. Equations of motion are reduced to ordinary differential equations of multiple-degree-of-freedom system by the Galerkin procedure. To clarify effects of sag-to-span ratio on chaotic motions, first, steady-state responses are calculated by the harmonic balance method. The chaotic responses are obtained by numerical integration and are examined by Poincaré projection, the maximum Lyapunov exponent and bifurcation diagrams. For the arch with comparatively small sag-to-span ratio, chaotic vibrations are generated mainly in the regions of subharmonic resonance of 1/3 order and of ultra-subharmonic resonance of 3/2 order, accompanied by the lowest-symmetric-mode of vibration. As the sag-to-span ratio increases, chaotic responses are generated from the subharmonic resonances both of 1/2 and 1/3 orders. Furthermore, chaotic motion appears within a wider range of exciting frequency and the number of modes that contribute to the chaos increases. Period doubling bifurcation was mainly observed in the bifurcation from periodic responses to chaotic ones. Hopf bifurcation was also found for the arch that satisfies the condition of internal resonance.