Dynamics of hierarchical beam lattice structures by an exact reduced-order dynamic-stiffness model

• Published in: Thin-walled structures 2023-03, Vol.184, p.110496, Article 110496
• Main Authors: Russillo, Andrea Francesco, Failla, Giuseppe
• Format: Article
• Language: English
• Subjects: 2D hierarchical beam lattice; Dynamic condensation; Dynamic-stiffness model; Modal response; Model order reduction
• ISSN: 0263-8231; 1879-3223
• DOI: 10.1016/j.tws.2022.110496

An exact reduced-order dynamic-stiffness model is proposed to study the dynamics of two-dimensional hierarchical beam lattices. Upon modelling every member of the lattice by a single exact two-node beam element, the model retains a few primary nodes of the unit cell and involves an exact dynamic condensation of the nodal degrees of freedom of the lattice structure internal to the unit cell and connected to the primary nodes. It is shown that the modes of the reduced-order model exhibit two orthogonality conditions, with dynamic-stiffness terms pertinent to the condensed lattice structures. The orthogonality conditions are the basis to decouple the motion equations and derive the exact modal response to external loading, in both time and frequency domains, under the assumption of proportional damping. The modal response is obtained in simple and elegant analytical form for the whole lattice, including the condensed lattice structures internal to the unit cells. Moreover, it can be calculated very expeditiously thanks to an exact representation of the eigenfunctions based on frequency dependent shape functions, which applies regardless of geometry and complexity of the hierarchical lattice structure. Numerical results confirm the exactness of the proposed reduced-order dynamic-stiffness model by comparison with standard finite-element solutions in ABAQUS. •Reduced-order dynamic-stiffness model is proposed for 2D hierarchical beam lattices.•Degrees of freedom within unit cells are removed by dynamic condensation.•Frequencies/modes are computed from global reduced-order dynamic-stiffness matrix.•Analytical modal responses are derived in time and frequency domains.•Frequencies/modes and modal responses are obtained exactly for the whole lattice.