The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of the phase space. The second method is the modal derivative approach, and more specifically, the quadratic manifold defined in order to derive a second-order nonlinear change of coordinates. Both methods share a common point of view, willing to introduce a nonlinear mapping to better define a reduced-order model that could take more properly into account the nonlinear restoring forces. However, the calculation methods are different and the quadratic manifold approach has not the invariance property embedded in its definition. Modal derivatives and static modal derivatives are investigated, and their distinctive features in the treatment of the quadratic nonlinearity are underlined. Assuming a slow/fast decomposition allows understanding how the three methods tend to share equivalent properties. While they give proper estimations for flat symmetric structures having a specific shape of nonlinearities and a clear slow/fast decomposition between flexural and in-plane modes, the treatment of the quadratic nonlinearity makes the predictions different in the case of curved structures such as arches and shells. In the more general case, normal form approach appears preferable since it allows correct predictions of a number of important nonlinear features, including the hardening/softening behaviour, whatever the relationships between slave and master coordinates are.

该贡献的目的是比较最近提出的两种方法，以便为几何非线性结构建立有效的降阶模型。第一种方法依赖于正常形式理论，该理论允许人们获得坐标的非线性变化，以表达基于相空间的基于不变性的跨度中的降阶动力学。第二种方法是模态导数方法，更具体地说，是定义二次流形以便导出坐标的二阶非线性变化。两种方法有共同的观点，愿意引入非线性映射以更好地定义降阶模型，该模型可以更适当地考虑非线性恢复力。然而，计算方法不同，二次流形方法的定义没有嵌入不变性。研究了模态导数和静态模态导数，并强调了它们在二次非线性处理中的独特特征。假设慢速/快速分解可以理解这三种方法如何趋于共享等效属性。虽然它们对具有特定形状的非线性以及弯曲模式和平面模式之间明显的缓慢/快速分解的平面对称结构给出了适当的估计，但是对二次非线性的处理使得在诸如弓形和弧形等弯曲结构的情况下，预测有所不同。贝壳。在更一般的情况下，