采用多尺度法进行直接摄动构造一端转动弹性约束浅拱发生2：1内共振时的非线性模态。在线性频率/模态及形函数中考虑弹性约束边界的影响，分析约束刚度不同时浅拱非线性模态的频率/幅值及时空效应。研究结果表明：转动弹性约束的存在使得频谱上出现模态转向，模态与形函数不对称；内共振非线性系统存在单模态运动和耦合模态运动；转动弹性约束的存在导致最低2阶模态之间的2：1内共振被激发；非线性模态的频率/幅值具有显著的非线性效应，且不同约束刚度时动力形态呈现不同的时空效应。%The nonlinear normal modes of a shallow arch with one end rotationally elastically constrained in case of two-to-one internal resonance were constructed by using the multi-scale method and direct perturbation. The effects of elastic constraint boundaries were considered in the linear frequencies, modes and shape functions. Then the nonlinear frequencies/amplitudes and space-time evolutions of nonlinear normal modes in shallow arch with different rotational stiffnesses were investigated. The results show that there are veering between two adjacent modes in frequency spectrum and asymmetries in the modes and shape functions due to the existence of rotational stiffness. The internally resonant nonlinear system has both single-mode and couple-mode motions. Also the two-to-one internal resonance between the two lowest modes is activated because of the elastic constraint stiffness in the boundary. Moreover, the frequencies/amplitudes in modulation equations exhibit remarkable nonlinearities, and the dynamic configuration of the nonlinear normal modes for different rotational constraints display various space-time evolutions.
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