Nonlinear vibrations of kinematically exact curved beams

Abstract

The nonlinear oscillations of a kinematically exact curved beam are investigated by means of the multiple time scale method applied directly to partial differential equations of motion. A linear constitutive behaviour is assumed, and the bending strain is the (change of) mechanical curvature. A dependence of natural frequencies ${\omega }_i$ and nonlinear correction coefficients $nc{c}_i$ (describing the nonlinear behaviour of the beam) on the initial curvature $\alpha$ is investigated for the first six modes for a case of hinged–hinged boundary conditions. The occurrence of internal resonances is discussed, and a complex behaviour of the functions $nc{c}_i\left(\alpha \right)$ is illustrated in detail. A comparison is made with the results obtained by the single mode Galerkin approximation, showing that the latter yields incorrect results. Finally, the analytical solution is validated by comparing it with numerical simulations obtained by the finite element method.