Reduction Methods for Nonlinear Vibrations of Spatially Continuous Systems with Initial Curvature

Reduction methods and the resulting models for studying nonlinear vibrations of shallow monodimensional continuous systems are discussed. Primary resonances of the first mode of buckled beams and suspended cables are investigated. The convergence of the relevant Galerkin-reduced models with variation of the nondimensional buckling level (buckled beam) and the elasto-geometric parameter (cable) is analyzed. For low values of the control parameter, one-dof models (with the first relevant linear eigenmode) are sufficiently accurate, whereas, for higher values of the parameter (above the first crossover), three- or higher-dof models (with the lowest relevant symmetric eigenmodes) are the minimum reduced-order models that can capture qualitatively the symmetric planar dynamics of the original systems. The major modification of the mode shapes of the lowest symmetric modes with respect to the initial nonlinear static shape, due to crossover and snap-through-type bifurcations, is highlighted as the key mechanism for the breakdown of low-dimensional models.