This paper presents an advanced control strategy to suppress nonlinear vibrations in a vertical conveyor system subjected to simultaneous resonance. Vertical conveyors play a crucial role in industrial applications, where stability and continuous performance are essential. However, excessive vibrations can reduce efficiency, cause mechanical fatigue, and increase maintenance costs. To describe the dynamics, the system is modeled as a multi-degree-of-freedom cantilever beam with quadratic and cubic nonlinearities under external harmonic excitations. These nonlinearities introduce complex behaviors, especially when internal and external resonances interact. Previous studies have analyzed the system using the multiple scale perturbation technique (MSPT) to investigate dynamic responses and resonance conditions. While this approach provides valuable insights, controlling nonlinear vibrations requires more effective strategies than conventional controllers. In this work, we propose a nonlinear integral negative derivative feedback (NINDF) controller, which combines first-order and second-order filters. This structure enhances stability margins, improves robustness, and ensures better vibration suppression during critical resonance states. Analytical solutions were derived via MSPT, and system stability was assessed using the Routh-Hurwitz criterion. Additionally, the system equations were integrated using the classical fourth-order Runge-Kutta (RK4) method, which provides reliable accuracy for short-term transient simulations. However, RK4 does not inherently preserve the geometric invariants (e.g., energy and phase-space structure) that are significant in nonlinear systems exhibiting internal resonance. Results demonstrate that the NINDF controller effectively reduces vibration amplitudes, particularly under 1:1 internal resonance, and achieves superior performance compared to traditional feedback methods. Hence, the proposed control strategy offers a practical and reliable tool for mitigating nonlinear vibrations in engineering systems exposed to demanding dynamic environments.
Citation: R. E. Abdullah, Rageh K. Hussein, Y. A. Amer, O. M. Khaled, Mohamed Ibrahim Attia, Asmaa M. Abd-Elal, M. N. Abd El-Salam. Nonlinear vibration reduction in vertical conveyor systems using a nonlinear integral negative derivative feedback controller[J]. AIMS Mathematics, 2025, 10(12): 28129-28150. doi: 10.3934/math.20251237
This paper presents an advanced control strategy to suppress nonlinear vibrations in a vertical conveyor system subjected to simultaneous resonance. Vertical conveyors play a crucial role in industrial applications, where stability and continuous performance are essential. However, excessive vibrations can reduce efficiency, cause mechanical fatigue, and increase maintenance costs. To describe the dynamics, the system is modeled as a multi-degree-of-freedom cantilever beam with quadratic and cubic nonlinearities under external harmonic excitations. These nonlinearities introduce complex behaviors, especially when internal and external resonances interact. Previous studies have analyzed the system using the multiple scale perturbation technique (MSPT) to investigate dynamic responses and resonance conditions. While this approach provides valuable insights, controlling nonlinear vibrations requires more effective strategies than conventional controllers. In this work, we propose a nonlinear integral negative derivative feedback (NINDF) controller, which combines first-order and second-order filters. This structure enhances stability margins, improves robustness, and ensures better vibration suppression during critical resonance states. Analytical solutions were derived via MSPT, and system stability was assessed using the Routh-Hurwitz criterion. Additionally, the system equations were integrated using the classical fourth-order Runge-Kutta (RK4) method, which provides reliable accuracy for short-term transient simulations. However, RK4 does not inherently preserve the geometric invariants (e.g., energy and phase-space structure) that are significant in nonlinear systems exhibiting internal resonance. Results demonstrate that the NINDF controller effectively reduces vibration amplitudes, particularly under 1:1 internal resonance, and achieves superior performance compared to traditional feedback methods. Hence, the proposed control strategy offers a practical and reliable tool for mitigating nonlinear vibrations in engineering systems exposed to demanding dynamic environments.
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R. E. Abdullah, Rageh K. Hussein, Y. A. Amer, O. M. Khaled, Mohamed Ibrahim Attia, Asmaa M. Abd-Elal, M. N. Abd El-Salam. Nonlinear vibration reduction in vertical conveyor systems using a nonlinear integral negative derivative feedback controller[J]. AIMS Mathematics, 2025, 10(12): 28129-28150. doi: 10.3934/math.20251237
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