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...

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Bibliographic Details
Published inAssociation for Engineering Education - Engineering Library Division Papers
Main Authors Prantil, Vincent C, Nagurka, Mark L
Format Conference Proceeding
LanguageEnglish
Published Atlanta American Society for Engineering Education-ASEE 23.06.2018
Subjects
Constraining
Damping ratio
Differential equations
Dynamical systems
Energy storage
Engineering education
Inertia
Order parameters
Ordinary differential equations
Resonant frequencies
Stiffness
System dynamics
Workflow
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Summary: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.