FE model of cantilever beam from Nonlinear mapping of non-conservative forces for reduced-order modelling

Non-intrusive or indirect reduced-order modelling strategies, such as the implicit condensation and expansion method, are applicable to geometrically nonlinear structures modelled using commercial finite-element (FE) software. Traditionally, the non-conservative forces acting on the structure are reduced via a linear projection onto the space spanned by the reduced modeshapes. As such, only the forces acting directly on these reduced modes can be captured, while any energy gained or dissipated by the statically condensed modes is neglected. This can lead to significant inaccuracies in the reduced-order model (ROM) predictions, which is demonstrated here using a 2-degrees-of-freedom (DOF) oscillator, and an FE model of a cantilever beam. It is shown that the non-conservative forces acting on the statically condensed modes can be captured using a nonlinear mapping of the physical DOFs into the reduced coordinates. This introduces additional terms in the reduced equations of motion, which we describe as force compensation. Excellent agreement is observed between the forced response curves of the full-order models and those of our proposed ROMs, both for the oscillator as well as the cantilever beam under different external excitation conditions (i.e. a constant-direction force and a follower force).

诸如隐式缩聚与展开法（implicit condensation and expansion method）在内的非侵入式（或间接）降阶建模策略，可应用于采用商用有限元（finite-element, FE）软件构建模型的几何非线性结构。传统研究中，结构所受的非保守力（non-conservative forces）需通过对降阶振型（reduced modeshapes）张成的空间进行线性投影来完成降阶处理。如此仅能捕捉直接作用于这些降阶模态的力，而静缩模态（statically condensed modes）所获取或耗散的全部能量均被忽略不计。这将导致降阶模型（reduced-order model, ROM）的预测结果出现显著误差，本文通过两自由度（degrees-of-freedom, DOF）振子与悬臂梁的有限元模型验证了该问题。研究表明，通过将物理自由度非线性映射至降阶坐标，可捕捉作用于静缩模态的非保守力。该方法会在降阶运动方程（equations of motion）中引入额外项，我们将其定义为力补偿（force compensation）。在不同外部激励条件下（即恒定方向力与随从力（follower force）），无论是振子还是悬臂梁，全阶模型（full-order models）的受迫响应曲线（forced response curves）与本文提出的降阶模型的受迫响应曲线均呈现出极佳的一致性。