freq analysis from Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy

Nonlinear dynamic analysis of complex engineering structures modelled using commercial finite element (FE) software is computationally expensive. Indirect reduced-order modelling strategies alleviate this cost by constructing low-dimensional models using a static solution dataset from the FE model. The applicability of such methods is typically limited to structures in which (a) the main source of nonlinearity is the quasi-static coupling between transverse and in-plane modes (i.e. membrane stretching); and (b) the amount of in-plane displacement is limited. We show that the second requirement arises from the fact that, in existing methods, in-plane kinetic energy is assumed to be negligible. For structures such as thin plates and slender beams with fixed/pinned boundary conditions, this is often reasonable, but in structures with free boundary conditions (e.g. cantilever beams), this assumption is violated. Here, we exploit the concept of nonlinear manifolds to show how the in-plane kinetic energy can be accounted for in the reduced dynamics, without requiring any additional information from the FE model. This new insight enables indirect reduction methods to be applied to a far wider range of structures while maintaining accuracy to higher deflection amplitudes. The accuracy of the proposed method is validated using an FE model of a cantilever beam.采用商用有限元（Finite Element, FE）软件建模的复杂工程结构，其非线性动力学分析的计算成本往往极高。间接降阶建模策略通过利用有限元模型的静态解数据集构建低维模型，以此缓解此类计算成本问题。此类方法的适用范围通常局限于满足以下两类条件的结构：(a) 非线性的主要来源为横向模态与面内模态间的准静态耦合（即薄膜拉伸效应）；(b) 面内位移量有限。我们的研究表明，第二项限制条件源于现有方法中默认面内动能可忽略不计这一前提假设。对于固支/铰支边界条件下的薄板与细长梁等结构，该假设通常合理；但对于自由边界条件的结构（如悬臂梁），该假设不再成立。本文基于非线性流形（nonlinear manifolds）的概念，阐明了如何在无需额外调用有限元模型信息的前提下，将面内动能纳入降阶动力学模型之中。这一全新的理论见解使得间接降阶方法能够应用于更广泛的结构类型，同时可在更大的挠度幅值范围内保持较高的分析精度。本文通过悬臂梁的有限元模型对所提方法的分析精度进行了验证。