Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights

This study presents a systematic benchmarking of numerical methods for solving ordinary differential equations (ODEs) applied to damped single-degree-of-freedom (SDOF) vibration systems. Ten solvers—including Runge–Kutta variants, Adams–Bashforth–Moulton, Rosenbrock, and Backward Differentiation Formula (BDF)—were evaluated under both non-stiff and stiff conditions by varying mass, damping, and stiffness parameters. Analytical solutions were used as references to quantify global error, convergence behavior, and computational efficiency. High-order adaptive solvers, such as Verner’s and Runge–Kutta 7/8, consistently achieved the highest accuracy, reducing global error by up to 15% compared with the classical Runge–Kutta (ODE45) method. Implicit methods, including Rosenbrock and BDF, demonstrated superior stability in stiff and highly damped cases. In contrast, low-order approaches, particularly the Trapezoidal rule, exhibited the largest errors, exceeding 30% in oscillatory regimes. The results confirm that solver performance is problem-dependent, emphasizing that no single algorithm is universally optimal. Beyond the technical contributions, this study introduces a pedagogical framework that allows engineering students to visualize solver trade-offs, quantify numerical accuracy, and interpret computational efficiency. The educational integration strengthens conceptual understanding of numerical methods and supports data-driven solver selection in vibration analysis and related engineering applications.