Suppressing Nonlinear Resonant Vibrations via NINDF Control in Beam Structures

Yasser A. Amer, Rageh K. Hussein, Sharif Abu Alrub, Ahmed S. Elgazzar, Tarek M. Salman, Fatma Mousa and M. N. Abd El-Salam ()
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Yasser A. Amer: Mathematics Department, Science Faculty, Zagazig University, Zagazig 44511, Egypt
Rageh K. Hussein: Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Sharif Abu Alrub: Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Ahmed S. Elgazzar: Mathematics Department, Faculty of Science, Arish University, Arish 45516, Egypt
Tarek M. Salman: Mathematics Department, Faculty of Science, Arish University, Arish 45516, Egypt
Fatma Mousa: Basic Sciences Department, Higher Technological Institute, 10th of Ramadan City 44634, Egypt
M. N. Abd El-Salam: Basic Sciences Department, Higher Technological Institute, 10th of Ramadan City 44634, Egypt

Mathematics, 2025, vol. 13, issue 13, 1-18

Abstract: In this paper, a unique method for controlling the effects of nonlinear vibrational responses in a cantilever beam system under harmonic excitation is presented. The Nonlinear Integral Negative Derivative Feedback (NINDF) controller is used for this purpose in this study. With this method, the cantilever beam is represented by a three-DOF nonlinear system, and the NINDF controller is represented by a first-order and second-order filter. The authors derive analytical solutions for the autonomous system with the controller by utilising perturbation analysis on the linearised system model. This study aims to reduce vibration amplitudes in a nonlinear dynamic system, specifically when 1:1 internal resonance occurs. The stability of the system is assessed using the Routh–Hurwitz criterion. Moreover, symmetry is present in the frequency–response curves (FRCs) for a variety of parameter values. The results show that, when compared to other controllers, the effectiveness of vibration suppression is directly correlated with the product of the NINDF control signal. The amplitude response of the system is demonstrated, and the analytical solutions are validated through numerical simulations using the fourth-order Runge–Kutta method. The accuracy and reliability of the suggested approach are demonstrated via the significant correlation between the analytical and numerical results.

Keywords: beam; MTSPM; NINDF; simultaneous resonance case; FREs; IRC; NDF; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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