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The nonlinear response of a flexible structure, subjected to generally supported conditions with nonlinearities, is investigated for the first time. An analytical procedure is proposed first. Moreover, a simulation technique usually employed in static analysis is developed for confirmation. Generally, ordinary perturbation methods could analyze dynamics of flexible structures with linear boundary conditions. As nonlinear boundaries are taken into account, they are out of operation for the modal shape that is hardly to be obtained, which is the key to the analysis. In order to overcome this, nonlinear boundary conditions are rescaled and the technique of modal revision is employed. Consequently, each governing equation with different time-scales could be analyzed exactly according to corresponding rescaled boundary conditions. The total response of any point at the flexible structure will be composed by harmonic responses yielded by the analytical method. Furthermore, the differential quadrature element method (DQEM), a numerical simulation technique could satisfy boundary conditions strictly, is introduced to certify analytical results. The comparison shows a reasonable agreement between these two methods. In fact, the accuracy of the analytical method for nonlinear boundaries could be explained in theory. Based on the certification, boundary nonlinearities are discussed in detail analytically and found to play an important role in responses. Because of the important role played by the nonlinear factors in the vibration and control of the flexible structure, this paper will open the vibration analysis and numerical study of the flexible structure with nonlinear constraints.

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Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.