Primary resonance suppression in spur gear systems using hybrid proportional and fractional-order derivative displacement feedback control

Zhongyang Su; Zhoujin Cui; Hanlin Huang; Zhongyang Su; Zhoujin Cui; Hanlin Huang

doi:10.3934/math.2026077

AIMS Mathematics

2026, Volume 11, Issue 1: 1857-1877. doi: 10.3934/math.2026077

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Primary resonance suppression in spur gear systems using hybrid proportional and fractional-order derivative displacement feedback control

1.
Jiangsu Maritime Institute, Nanjing, 211170, China
2.
School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing, 210013, China

Received: 08 November 2025 Revised: 28 December 2025 Accepted: 14 January 2026 Published: 20 January 2026
MSC : 34A08, 70K42, 93C95

This paper investigated primary resonance suppression in nonlinear spur gear systems using a hybrid proportional and fractional-order derivative displacement feedback (P-FDDF) controller. The dynamic model of the system was established through a second-order non-autonomous differential equation incorporating time-varying meshing stiffness, backlash, and external excitations. The amplitude-frequency response equation of primary resonance was derived via the multiple scale method, while Lyapunov stability theory was employed to analyze the stability of steady-state solutions. Numerical analyses examined the effects of meshing damping, load fluctuations, meshing stiffness variations, and control parameters on resonance characteristics. Time history responses and phase diagrams demonstrated that the P-FDDF strategy achieves simultaneous resonant amplitude suppression and frequency tuning. The fractional-order component's frequency-weighting and memory properties enhance adaptability to complex nonlinear dynamics induced by time-varying meshing stiffness and backlash, establishing the P-FDDF as a reliable solution for gear system vibration control.
- hybrid control,
- primary resonance suppression,
- spur gear systems,
- fractional-order control,
- multiple scale method,
- amplitude-frequency response
Citation: Zhongyang Su, Zhoujin Cui, Hanlin Huang. Primary resonance suppression in spur gear systems using hybrid proportional and fractional-order derivative displacement feedback control[J]. AIMS Mathematics, 2026, 11(1): 1857-1877. doi: 10.3934/math.2026077

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Abstract

This paper investigated primary resonance suppression in nonlinear spur gear systems using a hybrid proportional and fractional-order derivative displacement feedback (P-FDDF) controller. The dynamic model of the system was established through a second-order non-autonomous differential equation incorporating time-varying meshing stiffness, backlash, and external excitations. The amplitude-frequency response equation of primary resonance was derived via the multiple scale method, while Lyapunov stability theory was employed to analyze the stability of steady-state solutions. Numerical analyses examined the effects of meshing damping, load fluctuations, meshing stiffness variations, and control parameters on resonance characteristics. Time history responses and phase diagrams demonstrated that the P-FDDF strategy achieves simultaneous resonant amplitude suppression and frequency tuning. The fractional-order component's frequency-weighting and memory properties enhance adaptability to complex nonlinear dynamics induced by time-varying meshing stiffness and backlash, establishing the P-FDDF as a reliable solution for gear system vibration control.

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