Home > LIBRARY > JOURNALS > CURRENT_JOURNALS > CODEE > Vol. 19 (2025)
Publication Date
9-6-2025
Keywords
inverse problems, damped oscillations, parameter estimation, finite element methods, undergraduate differential equations
Disciplines
Mathematics | Ordinary Differential Equations and Applied Dynamics
Abstract
We investigate the inverse problem of identifying damping and stiffness parameters in one-dimensional damped oscillatory systems governed by second-order differential equations. Focusing on mass–spring–damper models, we analyze the qualitative behavior of solutions across underdamped, critically damped, and overdamped regimes, and derive explicit conditions for parameter recovery based on time-domain observations such as equilibrium crossings and turnaround points. Two numerical estimation methods are developed and compared: a finite-difference least-squares approach based on central difference approximations, and a finite element formulation derived from a variational framework using piecewise linear basis functions. Computational experiments using synthetic data assess the accuracy, stability, and noise sensitivity of both methods. Our results demonstrate that while both approaches perform well in noise-free settings, the finite element method offers greater robustness in the presence of measurement noise or coarse sampling. In addition to its theoretical and numerical contributions, this study serves as a pedagogical resource for integrating differential equations, numerical methods, and inverse modeling in undergraduate mathematics or engineering curricula.
Recommended Citation
Crooks, Gracie and Çetinkaya, F. Ayça
(2025)
"Analytical and Numerical Approaches to Parameter Estimation in Damped Oscillatory Systems,"
CODEE Journal:
Vol. 19, Article 3.
Available at:
https://scholarship.claremont.edu/codee/vol19/iss1/3
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