Vibration suppression and Hopf bifurcation control in a nonlinear rotating pendulum using time-delayed acceleration feedback

Khalid Alluhydan; M. N. Abd EL-Salam; Khalid Alluhydan; M. N. Abd EL-Salam

doi:10.3934/math.2026876

AIMS Mathematics

2026, Volume 11, Issue 7: 21654-21673. doi: 10.3934/math.2026876

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Research article

Vibration suppression and Hopf bifurcation control in a nonlinear rotating pendulum using time-delayed acceleration feedback

Khalid Alluhydan ^{1
,
,},
M. N. Abd EL-Salam ²

1.
Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh 12372, Saudi Arabia
2.
Basic Sciences Department, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia

Received: 29 May 2026 Revised: 26 June 2026 Accepted: 30 June 2026 Published: 21 July 2026
MSC : 34C15, 34C46, 34F15, 70H03, 74G10

The nonlinear vibration characteristics of a rotating pendulum subjected to harmonic excitation were examined in this work. To mitigate excessive oscillations and improve dynamic stability, a negative delayed acceleration feedback control scheme was implemented. Approximate analytical solutions were derived near the primary resonance using the method of multiple scales, resulting in nonlinear modulation equations that govern the amplitude and phase evolution. The steady-state response and its stability were analyzed through eigenvalue-based linearization techniques. Numerical simulations employing a fourth-order Runge–Kutta algorithm were performed to verify the analytical predictions and to explore the influence of key system parameters. The effects of detuning, excitation amplitude, nonlinear modulation strength, control gain, and time delay on the system response and stability were systematically investigated. The results demonstrate that time delay significantly alters stability boundaries through Hopf bifurcations, while appropriate tuning of the delayed feedback enhances vibration suppression. A good agreement between analytical and numerical results confirms the applicability of the perturbation approach and highlights the effectiveness of delayed acceleration control in suppressing nonlinear oscillations.
- rotating pendulum,
- time-delayed acceleration control,
- primary resonance,
- Hopf bifurcation
Citation: Khalid Alluhydan, M. N. Abd EL-Salam. Vibration suppression and Hopf bifurcation control in a nonlinear rotating pendulum using time-delayed acceleration feedback[J]. AIMS Mathematics, 2026, 11(7): 21654-21673. doi: 10.3934/math.2026876

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Abstract

The nonlinear vibration characteristics of a rotating pendulum subjected to harmonic excitation were examined in this work. To mitigate excessive oscillations and improve dynamic stability, a negative delayed acceleration feedback control scheme was implemented. Approximate analytical solutions were derived near the primary resonance using the method of multiple scales, resulting in nonlinear modulation equations that govern the amplitude and phase evolution. The steady-state response and its stability were analyzed through eigenvalue-based linearization techniques. Numerical simulations employing a fourth-order Runge–Kutta algorithm were performed to verify the analytical predictions and to explore the influence of key system parameters. The effects of detuning, excitation amplitude, nonlinear modulation strength, control gain, and time delay on the system response and stability were systematically investigated. The results demonstrate that time delay significantly alters stability boundaries through Hopf bifurcations, while appropriate tuning of the delayed feedback enhances vibration suppression. A good agreement between analytical and numerical results confirms the applicability of the perturbation approach and highlights the effectiveness of delayed acceleration control in suppressing nonlinear oscillations.

References

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