Work in progress: Connections between first-order and second-order dynamic systems - Lessons in limit behavior

There exists a unique relationship between the natural frequency and damping ratio of a lumped-parameter second-order dynamic system and the time constants of equivalent first-order systems. These first-order systems result in the limit of vanishing stiffness or inertia, with the system then capable of storing only a single type of energy. To emphasize the correspondence of first-order-like behavior with storage of primarily one type of system energy, a pair of two degree-of-freedom systems, one inertia-dominated and one stiffness-dominated, are presented. Although the governing ordinary differential equations are second-order, these systems are overdamped only. In studying limiting behaviors, the paper raises the question of what it means for a system that can never be underdamped to possess a natural frequency. The paper shows that expressions for natural frequency and damping ratio can be explicitly written in terms of pairs of time constants that arise naturally from the limiting process. The analysis is presented in a way that is amenable to undergraduate engineering students in courses in system dynamics.