The primary resonances of a pre-deformed rotating beam model including the quadratic and cubic nonlinearities are investigated in the presence of the 3:1 internal resonance. The steady state responses of the beam are analyzed in two cases of the primary resonance with the method of multiple scales. The original dynamic equation is integrated numerically in two frequency sweep directions. The theoretical results are consistent with those obtained in the numerical simulation. The contributions of quadratic nonlinearities and cubic nonlinearities to the primary resonances behavior of the rotating beam are clarified. The frequency response curves are discussed by considering different model parameters such as the thermal gradient, the rotating speed, the damping coefficient and the gas pressure. The stability regions of coupled mode solutions are compared between the models with different nonlinearities in the case of the primary resonance of the second mode. A series of interesting findings are presented.

在存在3：1内部共振的情况下，研究了包含二次和三次非线性的预变形旋转梁模型的主要共振。使用多尺度方法在两种主要共振情况下分析光束的稳态响应。原始动态方程在两个扫频方向上进行了数值积分。理论结果与数值模拟结果一致。阐明了二次非线性和三次非线性对旋转梁的初级共振行为的贡献。通过考虑不同的模型参数（例如热梯度，转速，阻尼系数和气压）来讨论频率响应曲线。在第二模式的初级共振的情况下，在具有不同非线性的模型之间比较耦合模式解的稳定性区域。提出了一系列有趣的发现。