Mixed Lagrangian formulation for modeling structures with clutched inerter devices

Abstract

Inerters (ID) and Clutched Inerter Devices (CID) are a novel technology with demonstrated seismic control potential. However, the inherent nonlinearity and discontinuity of the clutching phenomena in CIDs can pose significant challenges for their accurate numerical modeling. In general, conventional existing methods either oversimplify the physics involved or are sensitive to the step size and thus are inherently unstable, demanding excessive numerical resources. Most relevant studies to date have focused on small-scale systems with a limited number of inerters and have used simplified models due to the lack of analysis tools. At the same time, the Mixed Lagrangian Formulation (MLF), has proven to be a powerful tool for simulating non-smooth dynamics phenomena. This paper presents an alternative way of modeling the behavior of CIDs in both MLF and conventional finite element method. We put forward an original formulation of the inerter element, clutching behavior, and the inerter-related dissipation model, as well as their associated computational scheme in MLF and the equivalent construction in FEM. The newly proposed CID element in MLF is then implemented and validated through three examples, including a single degree of freedom system, a multi 10-storey moment resisting frame (MRF), and a 10-storey self-centering concentrically braced frame (SC-CBF) with multiple rocking sections. The results are compared to those from existing models used for clutching inerter and to the proposed FE model. Finally, the advantages of using the MLF framework and salient characteristics of the structures equipped with clutched inerters are discussed. The modeling strategy proposed in this work empowers researchers to simulate structures with a larger number of degrees of freedom, equipped with a considerable amount of inerter-based devices, with reduced effort and improved computational performance.