The internal resonance of a two degree-of-freedom mechanical system with all typical (quadratic and cubic) geometric nonlinearities is studied, limiting to the case of free dynamics. The Multiple Time Scale method is used to provide an analytical, closed form, approximation of the backbone curves. The cornucopia of different possible behaviours that can been obtained by varying the nonlinear stiffnesses is discussed in depth, illustrating them with some examples and verifying with numerical simulations. Some partially unexpected or relevant behaviours are highlighted, and some hints on how to exploit the proposed results to design nonlinearly tailored systems are given.

中文翻译：

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二自由度完整系统中的 1:1 内部共振：综合分析及其可能的设计开发

研究了具有所有典型（二次和三次）几何非线性的二自由度机械系统的内部共振，仅限于自由动力学的情况。多时间尺度方法用于提供主干曲线的分析、闭合形式的近似值。深入讨论了通过改变非线性刚度可以获得的不同可能行为的聚宝盆，并通过一些示例对其进行了说明并通过数值模拟进行了验证。突出显示了一些部分意外或相关的行为，并给出了有关如何利用所提出的结果来设计非线性定制系统的一些提示。