Bifurcations and internal resonances in space-curved rods Original Research Article

The paper discusses a geometrically non-linear model of space-curved beams. The Timoshenko-type model is used, which includes the shear effects and rotatory inertia. The motion of the system is described by a non-linear matrix equation, which accounts for non-linearities up to the second order. To solve the non-linear problem, discretization methods based on the Galerkin method are used and the resulting non-linear eigenvalue problem is then solved by the continuation methods. The analysis of the discretized problem allows the study of the bifurcations of solutions. The post-bifurcation analysis is used to explain the phenomenon of internal resonance, which is of much importance in the prediction of the dynamic response of the analyzed systems. The approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions.