DOI: 10.11648/j.ijiis.20261503.11 ISSN: 2328-7683

Modelling and Simulation of Engineering Design, A Second-Order Differential Equations in Mechanical and Electrical Systems

George Ihenacho, Diarah Samuel, Osueke Okechukwu, Samuel Ajayi, Evoh Emeng, Oluwasade Kehinde, Olaomi Agnes
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This paper presents a rigorous and systematic investigation into the fundamental role of second-order ordinary differential equations (ODEs) in the modelling, analysis, and design of engineering systems. The study is anchored on two canonical and cross-disciplinary case studies: the quarter-car suspension system and the series RLC circuit. For each system, the governing differential equations are derived from first principles using Newtonian and Kirchhoffian formulations, respectively, followed by complete analytical solutions obtained via the characteristic (auxiliary) equation framework.The classical dynamic response includes underdamped, critically damped, and overdamped are comprehensively analyzed, with explicit linkage to practical engineering design criteria. In particular, damping ratio targets are contextualized within industry standards, where passenger vehicle suspensions typically operate within ζ ≈ 0.3–0.4 to ensure ride comfort, while high-performance systems adopt ζ ≈ 0.65–0.70 to achieve improved transient response characteristics. The forced harmonic response is further examined, with emphasis on resonance behavior, amplitude amplification, and stability considerations, highlighting its critical implications for structural integrity and failure prevention, as exemplified by the Tacoma Narrows Bridge collapse. A unified cross-domain analytical framework is established, demonstrating that mechanically and electrically distinct systems are governed by mathematically analogous second-order ODEs. This analogy enables the transfer of insights and design strategies across engineering domains. Furthermore, numerical solutions obtained using the Euler method and the fourth-order Runge–Kutta (RK4) algorithm are systematically benchmarked against exact analytical solutions. Convergence and error analyses confirm the superior accuracy of RK4 with global truncation error of order O(h⁴), compared to the first-order accuracy O(h) of the Euler method. The paper concludes by highlighting key design implications and emphasizing the transformative role of simulation-driven engineering in accelerating system development, optimizing performance, and reducing reliance on costly experimental prototyping.

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