The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simulation free, in the sense that it allows to pass from physical coordinates (FE nodes) to normal coordinates, describing the dynamics in an invariant-based span of the phase space. The number of master modes for the ROM is not a priori limited since a complete change of coordinate is proposed. The underlying theory ensures the quality of the predictions thanks to the invariance property of the reduced subspace, together with their curvatures in phase space that accounts for the nonresonant nonlinear couplings. The method is applied to a beam discretised with 3D elements and shows its ability in recovering internal resonance at high energy. Then a fan blade model is investigated and the correct prediction given by the ROMs are assessed and discussed. A method is proposed to approximate an aggregate value for the damping, that takes into account the damping coefficients of all the slave modes, and also using the Rayleigh damping model as input. Frequency-response curves for the beam and the blades are then exhibited, showing the accuracy of the proposed method.

中文翻译：

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通过范式直接计算有限元非线性结构降阶模型的非线性映射

详细介绍了使用有限元 (FE) 方法离散化的几何非线性结构的三阶范式的直接计算。该过程允许定义非线性映射，以便根据不变流形理论推导出准确的降阶模型 (ROM)。所提出的减少策略是直接的且无需模拟，因为它允许从物理坐标（有限元节点）传递到法线坐标，描述基于不变的相空间跨度中的动力学。ROM 的主模式数量不受先验限制，因为建议完全改变坐标。由于缩减子空间的不变性，基础理论确保了预测的质量，连同它们在相空间中的曲率，这些曲率解释了非谐振非线性耦合。该方法应用于用 3D 元素离散化的梁，并显示其在高能量下恢复内部共振的能力。然后研究风扇叶片模型并评估和讨论由 ROM 给出的正确预测。提出了一种近似阻尼聚合值的方法，该方法考虑了所有从模式的阻尼系数，并且还使用瑞利阻尼模型作为输入。然后展示了梁和叶片的频率响应曲线，显示了所提出方法的准确性。然后研究风扇叶片模型并评估和讨论由 ROM 给出的正确预测。提出了一种近似阻尼聚合值的方法，该方法考虑了所有从模式的阻尼系数，并且还使用瑞利阻尼模型作为输入。然后展示了梁和叶片的频率响应曲线，显示了所提出方法的准确性。然后研究风扇叶片模型并评估和讨论由 ROM 给出的正确预测。提出了一种近似阻尼聚合值的方法，该方法考虑了所有从模式的阻尼系数，并且还使用瑞利阻尼模型作为输入。然后展示了梁和叶片的频率响应曲线，显示了所提出方法的准确性。