A Novel Methodology in Analyzing the Bifurcation, Controller, and Stability of Nonlinear Jet Engine Vibration

Abstract

Studying jet engine vibration (JEV) enhances flight safety and operational reliability through advanced detection, precision modeling, and data-driven techniques. This approach involves complex nonlinear vibration behaviors that often exceed the capabilities of conventional techniques. It facilitates early fault detection, predictive maintenance, and improved engine design. This study employs the non-perturbative approach (NPA) to examine the dynamics of a parametric nonlinear oscillatory system. The formulation is based on He’s frequency formula (HFF), which transforms a nonlinear ordinary differential equation (ODE) into an equivalent linear one. The analytical results are validated using Mathematica software (MS) (v13), showing strong agreement between the original nonlinear ODE and the corresponding linearized equation. To further explore the system behavior, bifurcation diagrams (BDs) are constructed, and the largest Lyapunov exponent (LLE) is utilized to identify stability regions and detect chaotic oscillations. The averaging method is applied to determine the critical resonance conditions and derive the frequency–response relationships; meanwhile, stability near simultaneous primary resonance is examined using the Routh–Hurwitz criterion. Finally, numerical simulations (NSs) based on the fourth-order Runge–Kutta method (RK-4) confirm the effectiveness of the positive position feedback (PPF) control strategy.