This paper aims at studying the influence of the different support types on the free nonlinear vibration frequency of a beam, both by analytical and numerical approaches and its dependence on the vibration amplitude. First, an analytical study of two cases of simply supported beams is carried out by using the nonlinear normal modes (NNM) and the multiple scale methods to obtain the analytical expressions of the nonlinear frequencies as a function of the vibration amplitude. Such results have been compared and discussed with other authors’ results. In addition, a nonlinear finite element model of these two cases showed that the analytical and numerical results are in good agreement. Additionally, several numerical studies have been carried out for different support types. A total of seven different boundary conditions have been numerically analysed and the corresponding frequency-amplitude relations have been obtained and compared. In addition, the effect of the axial inertial forces has been enhanced by adding concentrated masses at one end of the beam. It was found for instance that the hinged-hinged beam shows a hardening behaviour, which depends on the beam slenderness, or that the hinged-simply supported beam results in the largest softening behaviour. The fundamental causes of nonlinearity are, on one hand the coupling of the midline deformation and bending and, on the other hand, the coupling of axial inertial forces and bending. The significance of the first of these effects depends on the beam material and geometrical properties.

本文旨在通过解析和数值方法及其对振动幅度的依赖性，研究不同支撑类型对梁自由非线性振动频率的影响。首先，通过使用非线性法线模（NNM）和多尺度方法对两种情况下的简支梁进行分析研究，以获得作为振动幅度函数的非线性频率的解析表达式。此类结果已与其他作者的结果进行了比较和讨论。另外，这两种情况的非线性有限元模型表明，分析结果和数值结果吻合良好。另外，已经针对不同的支撑类型进行了一些数值研究。总共对七个不同的边界条件进行了数值分析，并获得并比较了相应的频率-振幅关系。另外，通过在梁的一端增加集中质量来增强轴向惯性力的作用。例如，发现铰链铰链梁表现出硬化行为，这取决于梁的细长度，或者铰链简单支撑的梁导致最大的软化行为。非线性的根本原因，一方面是中线变形和弯曲的耦合，另一方面是轴向惯性力和弯曲的耦合。这些影响中第一个的重要性取决于梁的材料和几何特性。