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

John Cannon; Tadeh Zirakian

doi:doi:10.11648/j.ajce.20251305.15

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

John Cannon, Tadeh Zirakian^*

Published in American Journal of Civil Engineering (Volume 13, Issue 5)

Received: 4 October 2025 Accepted: 14 October 2025 Published: 30 October 2025

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Abstract

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.

Published in	American Journal of Civil Engineering (Volume 13, Issue 5)
DOI	10.11648/j.ajce.20251305.15
Page(s)	304-312
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Numerical Methods, Ordinary Differential Equations (ODEs), Runge-Kutta Methods, Stiff Equations, Computational Efficiency, Single-Degree-of-Freedom (SDOF) Systems, Engineering Education

References

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APA Style

Cannon, J., Zirakian, T. (2025). Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. American Journal of Civil Engineering, 13(5), 304-312. https://doi.org/10.11648/j.ajce.20251305.15

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ACS Style

Cannon, J.; Zirakian, T. Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. Am. J. Civ. Eng. 2025, 13(5), 304-312. doi: 10.11648/j.ajce.20251305.15

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AMA Style

Cannon J, Zirakian T. Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. Am J Civ Eng. 2025;13(5):304-312. doi: 10.11648/j.ajce.20251305.15

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@article{10.11648/j.ajce.20251305.15,
  author = {John Cannon and Tadeh Zirakian},
  title = {Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights
},
  journal = {American Journal of Civil Engineering},
  volume = {13},
  number = {5},
  pages = {304-312},
  doi = {10.11648/j.ajce.20251305.15},
  url = {https://doi.org/10.11648/j.ajce.20251305.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20251305.15},
  abstract = {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.
},
 year = {2025}
}

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TY  - JOUR
T1  - Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights

AU  - John Cannon
AU  - Tadeh Zirakian
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.ajce.20251305.15
DO  - 10.11648/j.ajce.20251305.15
T2  - American Journal of Civil Engineering
JF  - American Journal of Civil Engineering
JO  - American Journal of Civil Engineering
SP  - 304
EP  - 312
PB  - Science Publishing Group
SN  - 2330-8737
UR  - https://doi.org/10.11648/j.ajce.20251305.15
AB  - 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.

VL  - 13
IS  - 5
ER  -

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Author Information

John Cannon

Department of Mechanical Engineering, California State University, Northridge, USA

Contact Email
Tadeh Zirakian

Department of Civil Engineering and Construction Management, California State University, Northridge, USA

Contact Email

http://orcid.org/0000-0003-2392-3601

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APA Style

Cannon, J., Zirakian, T. (2025). Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. American Journal of Civil Engineering, 13(5), 304-312. https://doi.org/10.11648/j.ajce.20251305.15

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ACS Style

Cannon, J.; Zirakian, T. Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. Am. J. Civ. Eng. 2025, 13(5), 304-312. doi: 10.11648/j.ajce.20251305.15

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AMA Style

Cannon J, Zirakian T. Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights. Am J Civ Eng. 2025;13(5):304-312. doi: 10.11648/j.ajce.20251305.15

Copy | Download

@article{10.11648/j.ajce.20251305.15,
  author = {John Cannon and Tadeh Zirakian},
  title = {Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights
},
  journal = {American Journal of Civil Engineering},
  volume = {13},
  number = {5},
  pages = {304-312},
  doi = {10.11648/j.ajce.20251305.15},
  url = {https://doi.org/10.11648/j.ajce.20251305.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20251305.15},
  abstract = {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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Benchmarking Numerical Solvers for Damped Single-Degree-of-Freedom Vibration Systems: Technical Evaluation and Educational Insights

AU  - John Cannon
AU  - Tadeh Zirakian
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.ajce.20251305.15
DO  - 10.11648/j.ajce.20251305.15
T2  - American Journal of Civil Engineering
JF  - American Journal of Civil Engineering
JO  - American Journal of Civil Engineering
SP  - 304
EP  - 312
PB  - Science Publishing Group
SN  - 2330-8737
UR  - https://doi.org/10.11648/j.ajce.20251305.15
AB  - 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.

VL  - 13
IS  - 5
ER  -

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