A two-degree-of-freedom model for a two-beam structure with nonlinear joints is presented and used to investigate the nonlinear responses of the system to a primary resonance of its first two modes in the presence of three-to-one internal resonance. By using the equilibrium conditions between the beams and the joints, the joint dynamic characteristics included linear and nonlinear torsional stiffness are introduced into the model of the system. The first two global mode functions of the two-beam structure, which can be obtained by the global mode method, are used to formulate the nonlinear dynamic model of the system and the specific linear torsional stiffness that result in three-to-one internal resonance is mainly considered. The method of multiple scales is employed to obtain the governing equations of the amplitudes and phases for the two-degree-of-freedom nonlinear dynamical system under the three-to-one internal resonance condition. Based on the modulation equations obtained by the method of multiple scales, the approximation solutions are derived and compared with those obtained by the numerical integration method. Through the cases of primary resonance of the first two modes with the three-to-one internal resonance condition, the frequency–response and force-response curves are plotted for investigating the nonlinear dynamic behavior in the two-beam structure connected with nonlinear joints.

提出了具有非线性节点的两梁结构的两自由度模型，并使用该模型研究了在存在三对一内部共振的情况下系统对其前两种模式的一次共振的非线性响应。 。通过利用梁与节点之间的平衡条件，将包括线性和非线性扭转刚度的节点动力学特性引入到系统模型中。可以通过全局模式方法获得的双梁结构的前两个全局模式函数，用于公式化系统的非线性动力学模型和导致三对一内部共振的特定线性扭转刚度主要考虑。在三对一内部共振条件下，采用多尺度方法获得了两自由度非线性动力系统的幅值和相位控制方程。基于通过多尺度方法获得的调制方程，推导近似解并将其与通过数值积分方法获得的近似解进行比较。通过前两个模态的一次共振以及三对一的内部共振条件，绘制了频率响应曲线和力响应曲线，以研究与非线性节点相连的双梁结构的非线性动力学行为。得出近似解并将其与通过数值积分方法获得的近似解进行比较。通过前两个模态的一次共振以及三对一的内部共振条件，绘制了频率响应曲线和力响应曲线，以研究与非线性节点相连的双梁结构的非线性动力学行为。得出近似解并将其与通过数值积分方法获得的近似解进行比较。通过前两个模态的一次共振以及三对一的内部共振条件，绘制了频率响应曲线和力响应曲线，以研究与非线性节点相连的双梁结构的非线性动力学行为。