Critical relationships in nonviscous systems with proportional damping

Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers

“…This scalar equation, together with the eigenvalue problem D(s)u = 0, makes up the system of equations for solving the critical curve. We find, in this equation, the two particular cases that have been already solved in the literature: (a) purely viscous damping, studied by Papargyri-Beskou and Beskos [10], and (b) proportional nonviscous damping, investigated by Lázaro and García-Raffi [26]. Both cases deserve some comments before addressing the proposed strategy to solve the general case of nonproportional nonviscously damped systems.…”

“…It is clear that if the system is proportional, then g ik (s) ≡ 0, 1 ≤ i, k ≤ n, and such an equation will depend on s and on the given mode φ j . Proportional systems admit closed-form analytical solutions for critical curves, as proved in reference [26]. However, the presence of u in Equation (38) requires some assumptions to reach approximate solutions.…”

“…Indeed, if g ij (s) = φ T i G(s)φ j = g jj (s) δ ij , then it follows that Equation ( 45) is no longer an approximation, but that both sides of the equation are irrefutably equal. Furthermore, as shown by [26], the eigenvalue problem can be decoupled using the undamped modal space, and the critical curves arise as exact closed forms. The graphical realization of such curves matches perfectly with the overdamped regions, and overlappings represent regions with several overdamped modes.…”