,
Diarah Reuben Samuel*,
Osueke Christian Okechukwu,
Ifiok Emmanuel Etuk,
Ajuwon Samuel Oreoluwa,
Jamal Abubakar Shehu,
Michael Olupinla
This study investigates the application of second-order differential equations in the modelling and simulation of engineering systems, with a mass–spring system adopted as a representative case study for vibration-based design problems. The aim is to demonstrate how analytical and numerical approaches can be integrated to accurately predict system dynamics and support engineering design decisions. The governing equations are derived from Newton’s second law and solved analytically using characteristic equation methods, while numerical solutions are obtained using the fourth-order Runge–Kutta technique. Key system parameters, including mass, damping coefficient, and spring stiffness, are defined and used to simulate system response under dynamic conditions. The results show strong agreement between analytical and numerical solutions, validating the accuracy of the computational approach. Furthermore, the simulations reveal that system performance is highly sensitive to damping and stiffness variations, which directly influence oscillation amplitude, settling time, and stability. The findings demonstrate that second-order differential equation models provide a robust framework for predicting system behaviour and optimizing engineering design. It is concluded that integrating modelling and simulation techniques into computer-aided design (CAD) environments can significantly enhance design efficiency and performance evaluation. The study recommends the incorporation of advanced simulation-driven tools and intelligent control strategies to further improve the reliability and adaptability of engineering systems.
| Published in | American Journal of Mechanical and Industrial Engineering (Volume 11, Issue 5) |
| DOI | 10.11648/j.ajmie.20261105.13 |
| Page(s) | 124-131 |
| 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), 2026. Published by Science Publishing Group |
Modelling, Simulation, Engineering Design, Mathematical Modelling, Second-Order Differential Equations
| [1] | N. Nwulu and M. Muchie, “Engineering design and mathematical modelling: Concepts and applications,” 2019, Taylor and Francis Ltd. |
| [2] | Anitha M. T, “Modeling and Simulation of Dynamic Systems,” International Journal of Research in Engineering, Science and Management, no. 12, Dec. 2022. |
| [3] | A. Ingale, “Modeling Mass-Spring-Damper System using Simscape,” Journal of Engineering Research and Application www.ijera.com, vol. 8, pp. 30–33, 2018, |
| [4] | S. O. Ejiko, S. O. Ejiko, and A. O. Filani, “Mathematical Modeling: A Useful Tool For Engineering Research And Practice,” International Journal of Mathematics Trends and Technology, vol. 67, no. 9, pp. 50–64, Sep. 2021, |
| [5] | Suresh Kumar Sahani, “A Numerical Study of Second-Order Differential Equations in Vibration Analysis of Mechanical Systems,” Membrane Technology, pp. 26–38, Sep. 2024, |
| [6] | T.-C. Hsu, S.-C. Chang, and Y.-T. Hung, “How to learn and how to teach computational thinking: Suggestions based on a review of the literature,” Comput. Educ., vol. 126, pp. 296–310, Nov. 2018, |
| [7] | A. PINTO, “Design and analysis of a mass-spring-damper system for active stabilization of a dynamic platform,” Mar. 2025, pp. 308–319. |
| [8] | J. R. G. Granada, L. F. P. Gálvez, and E. L. A. Dager, “Mathematical Modeling Using Second-Order Differential Equations: A Case Study of a Mass-Spring System,” Journal of Hunan University Natural Sciences, vol. 50, no. 7, 2023, |
| [9] | B. Hadzhamolla, A. J. Stasik, and S. A.-A. Alaliyat, “Modeling Dynamical Systems Using Neural ODEs: A Study On Scientific Machine Learning Models,” 39th ECMS International Conference on Modelling and Simulation in 2025, 2024. |
| [10] | S. Santra and J. Mohapatra, “An efficient computational approach for the solution of time-space fractional diffusion equation,” International Journal of Modelling and Simulation, vol. 43, no. 4, pp. 393–405, Jul. 2023, |
| [11] | B. Ajarmah, “A novel approach to study the mass-spring-damper system using a reliable fractional method,” Archive of Applied Mechanics, vol. 93, no. 10, pp. 3797–3808, Oct. 2023, |
| [12] | P. Frankovský, D. Hroncová, I. Delyová, and I. Virgala, “Dynamic Systems in Simulation Environment MATLAB/Simulink-SimMechanics,” American Journal of Mechanical Engineering, vol. 1, no. 7, pp. 282–288, 2013, |
| [13] | E. Martin-Martinez, R. Samsó, J. Houghton, and J. Solé, “PySD: System Dynamics Modeling in Python,” J. Open Source Softw., vol. 7, no. 78, p. 4329, Oct. 2022, |
| [14] | A. S. Olaniyan, F. R. Oyewuwo, and M. A. Akanbi, “New Higher Order Derivative Runge-Kutta Methods for Solving Ordinary Differential Equations,” International Journal of Scientific Development and Research, vol. 8, p. 1368, 2023, [Online]. Available: |
| [15] | P. Lata Sharma and A. Kumar, “Demonstration Study on Runge-Kutta Fourth Order Method by Using MATLAB Programming,” IOSR Journal of Mathematics, vol. 17, pp. 1–09, 2021, |
| [16] | L. Said and B. Latif, “Modeling and Control of Mechanical Systems in Simulink of Matlab,” in Applications of MATLAB in Science and Engineering, InTech, 2011. |
| [17] | W. Glass-Husain, “Integrating System Dynamics Models with Online Python-based Analytics,” Proceedings of the 34th International Conference of the System Dynamics Society (Delft, Netherlands), 2016. |
| [18] | Y. Antehunegn, B. Yoseph, and M. Belete, “Control of 8 DOF vehicle model suspension system by designing Second order SMC Controller,” Global Scientific Journal (GSJ), Volume 8, Issue 12., 2020. |
| [19] | S. Yaghoubi and A. Ghanbarzadeh, “Modeling and optimization of car suspension system in the presence of magnetorheological damper using Simulink-PSO hybrid technique,” Results in Engineering, vol. 22, p. 102065, Jun. 2024, |
| [20] | Katsuhiko. Ogata, Modern control engineering. Prentice Hall, 2022. |
| [21] | E. Emagbetere, P. A. Oghenekovwo, C. C. Obinabo, A. K. Aworinde, F. A. Ishola, and O. Kilanko, “Modelling and simulation of motor vehicle suspension system,” IOP Conf. Ser. Mater. Sci. Eng., vol. 1107, no. 1, p. 012092, Apr. 2021, |
| [22] | T. Xiao, “Two-Stage Approach to Parameter Estimation of Differential Equations Using Neural ODEs,” Industrial & Engineering Chemistry Research, 60(46), 2021. |
| [23] | Chen, R. T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). Neural Ordinary Differential Equations. Advances in Neural Information Processing Systems (NeurIPS), 31, 6571–6583. |
APA Style
Ihenacho, G. C., Samuel, D. R., Okechukwu, O. C., Adebanji, A. S., Etuk, I. E., et al. (2026). Modelling and Simulation for Engineering Design: A Case Study of Second Order Differential Equations. American Journal of Mechanical and Industrial Engineering, 11(5), 124-131. https://doi.org/10.11648/j.ajmie.20261105.13
ACS Style
Ihenacho, G. C.; Samuel, D. R.; Okechukwu, O. C.; Adebanji, A. S.; Etuk, I. E., et al. Modelling and Simulation for Engineering Design: A Case Study of Second Order Differential Equations. Am. J. Mech. Ind. Eng. 2026, 11(5), 124-131. doi: 10.11648/j.ajmie.20261105.13
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajmie.20261105.13,
author = {George Chikwendu Ihenacho and Diarah Reuben Samuel and Osueke Christian Okechukwu and Ajayi Samuel Adebanji and Ifiok Emmanuel Etuk and Ajuwon Samuel Oreoluwa and Jamal Abubakar Shehu and Michael Olupinla},
title = {Modelling and Simulation for Engineering Design: A Case Study of Second Order Differential Equations},
journal = {American Journal of Mechanical and Industrial Engineering},
volume = {11},
number = {5},
pages = {124-131},
doi = {10.11648/j.ajmie.20261105.13},
url = {https://doi.org/10.11648/j.ajmie.20261105.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmie.20261105.13},
abstract = {This study investigates the application of second-order differential equations in the modelling and simulation of engineering systems, with a mass–spring system adopted as a representative case study for vibration-based design problems. The aim is to demonstrate how analytical and numerical approaches can be integrated to accurately predict system dynamics and support engineering design decisions. The governing equations are derived from Newton’s second law and solved analytically using characteristic equation methods, while numerical solutions are obtained using the fourth-order Runge–Kutta technique. Key system parameters, including mass, damping coefficient, and spring stiffness, are defined and used to simulate system response under dynamic conditions. The results show strong agreement between analytical and numerical solutions, validating the accuracy of the computational approach. Furthermore, the simulations reveal that system performance is highly sensitive to damping and stiffness variations, which directly influence oscillation amplitude, settling time, and stability. The findings demonstrate that second-order differential equation models provide a robust framework for predicting system behaviour and optimizing engineering design. It is concluded that integrating modelling and simulation techniques into computer-aided design (CAD) environments can significantly enhance design efficiency and performance evaluation. The study recommends the incorporation of advanced simulation-driven tools and intelligent control strategies to further improve the reliability and adaptability of engineering systems.},
year = {2026}
}
TY - JOUR T1 - Modelling and Simulation for Engineering Design: A Case Study of Second Order Differential Equations AU - George Chikwendu Ihenacho AU - Diarah Reuben Samuel AU - Osueke Christian Okechukwu AU - Ajayi Samuel Adebanji AU - Ifiok Emmanuel Etuk AU - Ajuwon Samuel Oreoluwa AU - Jamal Abubakar Shehu AU - Michael Olupinla Y1 - 2026/09/22 PY - 2026 N1 - https://doi.org/10.11648/j.ajmie.20261105.13 DO - 10.11648/j.ajmie.20261105.13 T2 - American Journal of Mechanical and Industrial Engineering JF - American Journal of Mechanical and Industrial Engineering JO - American Journal of Mechanical and Industrial Engineering SP - 124 EP - 131 PB - Science Publishing Group SN - 2575-6060 UR - https://doi.org/10.11648/j.ajmie.20261105.13 AB - This study investigates the application of second-order differential equations in the modelling and simulation of engineering systems, with a mass–spring system adopted as a representative case study for vibration-based design problems. The aim is to demonstrate how analytical and numerical approaches can be integrated to accurately predict system dynamics and support engineering design decisions. The governing equations are derived from Newton’s second law and solved analytically using characteristic equation methods, while numerical solutions are obtained using the fourth-order Runge–Kutta technique. Key system parameters, including mass, damping coefficient, and spring stiffness, are defined and used to simulate system response under dynamic conditions. The results show strong agreement between analytical and numerical solutions, validating the accuracy of the computational approach. Furthermore, the simulations reveal that system performance is highly sensitive to damping and stiffness variations, which directly influence oscillation amplitude, settling time, and stability. The findings demonstrate that second-order differential equation models provide a robust framework for predicting system behaviour and optimizing engineering design. It is concluded that integrating modelling and simulation techniques into computer-aided design (CAD) environments can significantly enhance design efficiency and performance evaluation. The study recommends the incorporation of advanced simulation-driven tools and intelligent control strategies to further improve the reliability and adaptability of engineering systems. VL - 11 IS - 5 ER -
Mechanical Engineering Department, Pan-Atlantic University, Lekki, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Mechatronics Engineering Program, Bowen University, Iwo, Nigeria
Information