Multi-Control Strategies on a Cubic–Quintic Nonlinear Hybrid Oscillator with External Excitation Under Resonance Conditions

Asma Alanazy, Yasser A. Amer, Ashraf Taha EL-Sayed (), Fatma Sh. Mohammed and Taher A. Bahnasy
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Asma Alanazy: Mathematics Department, Faculty of Science, Northern Border University, Arar 1321, Saudi Arabia
Yasser A. Amer: Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44759, Egypt
Ashraf Taha EL-Sayed: Department of Basic Science, Modern Academy for Engineering and Technology, Cairo 11439, Egypt
Fatma Sh. Mohammed: Department of Basic Science, Higher Institute of Engineering and Technology, Fifth Settlement, Cairo 11835, Egypt
Taher A. Bahnasy: Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Tanta 31734, Egypt

Mathematics, 2025, vol. 13, issue 6, 1-24

Abstract: The goal of this article is to reduce the vibration of a hybrid oscillator with a cubic–quintic nonlinear term under internal and external forces in the worst resonance case. To eliminate the harmful vibration in the system, the following strategies are suggested: nonlinear derivative feedback control (NDF), linear negative velocity feedback control (LNVC), nonlinear integral positive position feedback (NIPPF), integral resonant control (IRC), negative velocity with time delay (TD), and positive position feedback (PPF). It is discovered that the PPF control suppresses vibration more effectively than typical controllers, which reduces the vibration to 0.0001 with an effectiveness of 99.92%. Moreover, the main advantages of the PPF controller are its low cost and the fast response. The multiple time scale perturbation technique (MSPT) is used to apply the theoretical methodology and obtain a perturbed response. In order to close the concurrent primary and internal resonance case, frequency response (FR) equations are also used to study and analyze the system’s stability. The MATLAB software is used to complete and clarify all numerical topics. The FR curves are examined to determine the amplitude’s subsequent impact from variations in the parameters’ values. Lastly, a comparison of the numerical and analytical solutions is performed utilizing time history. Along with comparing the impact of our PPF damper on the hybrid oscillator, earlier research is also provided.

Keywords: multiple scale perturbation technique (MSPT); linear negative velocity feedback (LNVC); positive position feedback (PPF); negative derivative feedback (NDF); resonance case; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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