Nonlinear dynamics and bifurcation analysis of a PPF-controlled broadband Piezoelectric energy harvester

Moamen Wafaie; Rageh K. Hussein; Ashraf Taha EL-Sayed; Fatma Taha El-Bahrawy; Moamen Wafaie; Rageh K. Hussein; Ashraf Taha EL-Sayed; Fatma Taha El-Bahrawy

doi:10.3934/math.2026534

AIMS Mathematics

2026, Volume 11, Issue 5: 12977-13007. doi: 10.3934/math.2026534

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Nonlinear dynamics and bifurcation analysis of a PPF-controlled broadband Piezoelectric energy harvester

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

Received: 19 January 2026 Revised: 07 April 2026 Accepted: 20 April 2026 Published: 11 May 2026
MSC : 34A34, 37N35, 70J99, 70K20, 74H10

In this paper, we present a comprehensive analytical–numerical investigation of the nonlinear dynamics and bifurcation behavior of a broadband piezoelectric energy harvester (PEH) integrated with a Positive Position Feedback (PPF) control strategy. The novelty of this work lies in the development of a unified framework that captures the coupled effects of primary and internal resonances in the presence of active control. The governing nonlinear equations of motion were solved analytically using the second-order Multiple Scales Perturbation Technique (MSPT), yielding accurate approximate solutions and frequency–response relationships. The proposed approach provided a rigorous analytical treatment of the controlled nonlinear system. It revealed the underlying mechanisms governing vibration suppression and stability enhancement. Furthermore, bifurcation analysis was conducted to identify multistability, jump phenomena, and transitions between stable and unstable solutions as system and control parameters vary. The analytical predictions were validated through numerical simulations using MATLAB and MAPLE, showing excellent agreement in frequency–response curves, time histories, and phase portraits. The results demonstrated that the PPF controller significantly improves the system's dynamic performance by suppressing nonlinear oscillations, mitigating jump phenomena, reducing settling time, and stabilizing the response, leading to more stable and favorable operating regimes for effective energy harvesting. Overall, this study provides new insights into the design and control of nonlinear piezoelectric energy harvesting systems with potential applications in self-powered sensing and smart vibration control systems.
- nonlinear dynamics,
- bifurcation analysis,
- multiple scales method,
- stability analysis,
- positive position feedback control,
- Piezoelectric energy harvesting
Citation: Moamen Wafaie, Rageh K. Hussein, Ashraf Taha EL-Sayed, Fatma Taha El-Bahrawy. Nonlinear dynamics and bifurcation analysis of a PPF-controlled broadband Piezoelectric energy harvester[J]. AIMS Mathematics, 2026, 11(5): 12977-13007. doi: 10.3934/math.2026534

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Abstract

In this paper, we present a comprehensive analytical–numerical investigation of the nonlinear dynamics and bifurcation behavior of a broadband piezoelectric energy harvester (PEH) integrated with a Positive Position Feedback (PPF) control strategy. The novelty of this work lies in the development of a unified framework that captures the coupled effects of primary and internal resonances in the presence of active control. The governing nonlinear equations of motion were solved analytically using the second-order Multiple Scales Perturbation Technique (MSPT), yielding accurate approximate solutions and frequency–response relationships. The proposed approach provided a rigorous analytical treatment of the controlled nonlinear system. It revealed the underlying mechanisms governing vibration suppression and stability enhancement. Furthermore, bifurcation analysis was conducted to identify multistability, jump phenomena, and transitions between stable and unstable solutions as system and control parameters vary. The analytical predictions were validated through numerical simulations using MATLAB and MAPLE, showing excellent agreement in frequency–response curves, time histories, and phase portraits. The results demonstrated that the PPF controller significantly improves the system's dynamic performance by suppressing nonlinear oscillations, mitigating jump phenomena, reducing settling time, and stabilizing the response, leading to more stable and favorable operating regimes for effective energy harvesting. Overall, this study provides new insights into the design and control of nonlinear piezoelectric energy harvesting systems with potential applications in self-powered sensing and smart vibration control systems.

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