Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions

The existing studies of nonlinear vibration of elastic structures are usually focused on straight structures with homogeneous linear boundaries. Differently, this paper investigates the nonlinear transverse vibrations of a slightly curved beam with nonlinear boundary conditions. By using the generalized Hamilton's principle, the governing equation with geometric nonlinearity is obtained for the dynamics of the curved beam. A method of dealing with nonlinear boundaries is proposed, which is considered as a nonlinear concentrated force at the boundary. The normal modes and natural frequencies of the curved beam are determined using two different hypothetical modes based on the derived system. The harmonic balance method in combination with the pseudo arc-length method is employed to obtain the primary resonance response and 1/2 super-harmonic resonance response of the slightly curved beam. It is found that the initial curvature plays a significant role in the characteristics of the nonlinear vibrations of the curved beam. With an increase of the initial curvature, the nonlinear characteristics of softening and hardening types can coexist in the steady-state amplitude-frequency response. Moreover, the results show that the initial curvature can induce 1/2 super-harmonic resonance. Furthermore, it is also found that the nonlinear boundary has a significant influence on the nonlinear vibration of the curved structure. Therefore, the obtained results provide useful information for further studying the nonlinear vibrations of the curved beam with nonlinear and time-dependent boundary conditions.