Analytical modal analysis of lumped stiffness-mass structures via a recursive static solution

Abstract

This paper examines the modal properties, including modal frequencies and mode shapes, of lumped stiffness-mass systems with one degree-of-freedom (dof) per node. For the first time, it attains recursive solutions for the modal properties of such structures with general configurations. To this end, we introduce a recursive formulation to construct the stiffness and mass matrices of structures, and transform modal analysis into equivalent static problems of ‘modified structures’, where the masses assume a role of negative-stiffness springs. The static analysis of these modified structures hinges on a family of recursive functions, named characteristic rational P-functions, which encode all modal properties of the original structures. This recursive static solution allows modal analysis with an algorithmic complexity linear to the total number of dofs. We further derive exact modal solutions for structures with uniform stiffness and mass. The analysis includes the dispersion of modal frequencies, the linearity of modal periods to structural heights, the spectral distribution and the geometrical interpretation of mode shapes. The cumulative distribution of modal frequencies exhibits a staircase shape that converges to inverse trigonometric functions for a sufficiently large number of dofs. Lastly, we validate the convergence of an infinite-dof discrete model to a continuum model.