Oscillation Controlling in Nonlinear Motorcycle Scheme with Bifurcation Study

Hany Samih Bauomy () and Ashraf Taha EL-Sayed
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Hany Samih Bauomy: Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
Ashraf Taha EL-Sayed: Department of Basic Science, Modern Academy for Engineering and Technology, Cairo 11439, Egypt

Mathematics, 2025, vol. 13, issue 19, 1-32

Abstract: By applying the Non-Perturbative Approach (NPA), the corresponding linear differential equation is obtained. Aimed at organizational investigation, the resulting linear equation is used. Strong agreement between numerical calculations and the precise frequency is demonstrated, and the reliability of the results acquired is established by the correlation with the numerical solution. Additionally, this study explores a new control process to affect the stability and behavior of dynamic motorcycle systems that vibrate nonlinearly. A multiple time-scale method (MTSM) is applied to examine the analytical solution of the nonlinear differential equations describing the aforementioned system. Every instance of resonance was taken out of the second-order approximations. The simultaneous primary and 1:1 internal resonance case ( Ω ≈ ω e q , ω 2 ≈ ω e q ) is recorded as the worst resonance case caused while working on the model. We investigated stability with frequency–response equations and bifurcation. Numerical solutions for the system are covered. The effects of the majority of the system parameters were examined. In order to mitigate harmful vibrations, the controller under investigation uses (PD) proportional derivatives with (PPF) positive position feedback as a new control technique. This creates a new active control technique called PDPPF. A comparison between the PD, PPF, and PDPPF controllers demonstrates the effectiveness of the PDPPF controller in reducing amplitude and suppressing vibrations. Unwanted consequences like chaotic dynamics, limit cycles, or loss of stability can result from bifurcation, which is the abrupt qualitative change in a system’s behavior as a parameter. The outcomes showed how effective the suggested controller is at reducing vibrations. According to the findings, bifurcation analysis and a control are crucial for designing vibrating dynamic motorcycle systems for a range of engineering applications. The MATLAB software is utilized to match the analytical and numerical solutions at time–history and frequency–response curves (FRCs) to confirm their comparability. Additionally, case studies and numerical simulations are presented to show how well these strategies work to control bifurcations and guarantee the desired system behaviors. An analytical and numerical solution comparison was prepared.

Keywords: NPA; motorcycle; bifurcation; Poincaré map; perturbation; stability; PDPPFF controller (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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