Analytical identification of dynamic structural models: Mass matrix of an isospectral lumped mass model

Combining the accurate physical description of high-fidelity mechanical formulations with the practical versatility of low-order discrete models is a fundamental and open-ended topic in structural dynamics. Finding a well-balanced compromise between the opposite requirements of representativeness and synthesis is a delicate and challenging task. The paper systematizes a consistent methodological strategy to identify a physics-based reduced-order model (ROM) preserving the physical accuracy of large-sized models with distributed parameters (REM), without resorting to classical techniques of dimensionality reduction. The leading idea is, first, to select a limited configurational set of representative degrees of freedom contributing significantly to the dynamic response (model reduction) and, second, to address an inverse indeterminate eigenproblem to identify the matrices governing the linear equations of undamped motion (structural identification). The physical representativeness of the identified model is guaranteed by imposing the exact coincidence of a selectable subset of natural frequencies and modes (partial isospectrality). The inverse eigenproblem is solved analytically and parametrically, since its indeterminacy can be circumvented by selecting the lumped mass matrix as the primary unknown and the stiffness matrix as a parameter (or vice versa). Therefore, explicit formulas are provided for the mass matrix of the ROM having the desired low dimension and possessing the selected partial isospectrality with the REM. Minor adjustments are also outlined to remove a posteriori unphysical effects, such as defects in the matrix symmetry, which are intrinsic consequences of the algebraic identification procedure. The direct and inverse eigenproblem solutions are explored through parametric analyses concerning a multistory frame, by adopting a high-fidelity Finite Element model as REM and an Equivalent Frame model as ROM. Before mass matrix identification, modal analysis results indicate a general tendency of ROM to underestimate natural frequencies, with the underestimation strongly depending on the actual mass distribution of the structure. After the identification of the mass matrix and the elimination of unphysical defects, isospectrality is successfully achieved. Finally, extensions to prototypical highly massive masonry buildings are presented. The qualitative and quantitative discussion of the results under variation of the significant mechanical parameters provides useful insights to recognize the validity limits of the approximations affecting low-order models with lumped parameters.