ANALYTICAL AND NUMERICAL INVESTIGATION OF A NONLINEAR DIFFERENTIAL EQUATION ARISING IN PHYSICAL AND ENGINEERING SYSTEMS

Abstract

Nonlinear differential equations are fundamental mathematical models for describing physical and engineering systems in which linear assumptions fail to adequately represent system behavior. Nonlinear stiffness, damping, forcing, and parameter uncertainty may produce amplitude-dependent resonance, asymmetric response characteristics, multistability, and substantial variability in system performance. This study presents an analytical and numerical investigation of the forced Duffing nonlinear differential equation as a representative model of nonlinear oscillatory systems encountered in mechanical, structural, vibration, and engineering applications. The governing equation is analyzed using the first-harmonic balance approximation to derive an algebraic amplitude–frequency relationship. The resulting nonlinear equation is subsequently solved numerically using an adaptive fourth/fifth-order Runge–Kutta method. To quantify the influence of uncertain physical parameters, a Monte Carlo simulation framework containing 5,000 independent realizations is developed. Damping ratio, natural frequency, nonlinear stiffness coefficient, forcing amplitude, and excitation frequency are modeled as uncertain parameters over physically plausible ranges. The analytical and numerical solutions are compared using steady-state response amplitude, relative error, confidence intervals, and statistical sensitivity analysis. For the Monte Carlo ensemble, the harmonic-balance solution produced a mean response amplitude of 1.103 with a standard deviation of 0.092, while the numerically integrated validation subset produced a mean amplitude of 1.142 with a standard deviation of 0.097. The average relative difference between the analytical and numerical amplitudes was approximately 3.41%. Excitation frequency exhibited the strongest linear association with response amplitude (Pearson `r=0.761`), followed by nonlinear stiffness (`r=-0.463`), forcing amplitude (`r=0.360`), and natural frequency (`r=-0.278`). These results demonstrate that analytical approximations can provide efficient estimates of nonlinear response, while numerical integration and Monte Carlo simulation provide a more comprehensive framework for uncertainty propagation. The proposed framework is applicable to nonlinear vibration analysis, structural dynamics, mechanical systems, and other engineering systems governed by nonlinear differential equations.