Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller

EL-Sayed, Ashraf Taha; Hussein, Rageh K.; Amer, Yasser A.; Mahmoud, Sara S.; Abu Alrub, Sharif; Bahnasy, Taher A.

doi:10.3390/act14090427

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Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller

by

Ashraf Taha EL-Sayed

^1,*

,

Rageh K. Hussein

²

,

Yasser A. Amer

³,

Sara S. Mahmoud

³,

Sharif Abu Alrub

² and

Taher A. Bahnasy

⁴

¹

Department of Basic Science, Modern Academy for Engineering and Technology, Elmokattam, Cairo 11439, Egypt

²

Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia

³

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44759, Egypt

⁴

Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Tanta 31734, Egypt

^*

Author to whom correspondence should be addressed.

Actuators 2025, 14(9), 427; https://doi.org/10.3390/act14090427 (registering DOI)

Submission received: 3 August 2025 / Revised: 25 August 2025 / Accepted: 29 August 2025 / Published: 31 August 2025

(This article belongs to the Special Issue Editorial Board Members’ Collection Series: Nonlinear Control and Dynamics for MEMS)

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Abstract

This study investigates the effect of a nonlinear saturation controller (NSC) on the van der Pol–Mathieu–Duffing oscillator (VMDO). The oscillator is a single degree of freedom (DOF) system. It is driven by an external force. It is described by a nonlinear differential equation (DE). The multiple-scale perturbation method (MSPT) is applied. It gives second-order analytical solutions. The first indirect Lyapunov method is used. It provides the frequency–response equation. It also shows the stability conditions. Internal resonance is included. The analysis considers steady-state responses. It studies simultaneous primary resonance with a 1:2 internal resonance (

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MDPI and ACS Style

EL-Sayed, A.T.; Hussein, R.K.; Amer, Y.A.; Mahmoud, S.S.; Abu Alrub, S.; Bahnasy, T.A. Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller. Actuators 2025, 14, 427. https://doi.org/10.3390/act14090427

AMA Style

EL-Sayed AT, Hussein RK, Amer YA, Mahmoud SS, Abu Alrub S, Bahnasy TA. Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller. Actuators. 2025; 14(9):427. https://doi.org/10.3390/act14090427

Chicago/Turabian Style

EL-Sayed, Ashraf Taha, Rageh K. Hussein, Yasser A. Amer, Sara S. Mahmoud, Sharif Abu Alrub, and Taher A. Bahnasy. 2025. "Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller" Actuators 14, no. 9: 427. https://doi.org/10.3390/act14090427

APA Style

EL-Sayed, A. T., Hussein, R. K., Amer, Y. A., Mahmoud, S. S., Abu Alrub, S., & Bahnasy, T. A. (2025). Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller. Actuators, 14(9), 427. https://doi.org/10.3390/act14090427

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Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller

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