Nonlinear Response of Shallow Arches at Veering

Nonlinear interactions in a simply supported shallow arch with a torsional spring at one end are investigated. The underlying one-to-one internal resonance is activated in the vicinity of veering of the frequencies of the lowest two modes and results from the nonlinear stretching of the arch centerline. The investigation is carried out considering a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing nonlinear responses are studied using force-response and frequency-response curves and bifurcation-analysis tools. The internal and external resonance detunings are varied to construct the system response curves. A frequency island generation is manifested in the frequency-response curves. The arch dynamical response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations. Furthermore, the phenomenological appearance of mode localization is here discussed in the context of nonlinear interactions