Active control impact analysis on shaped coupled beam structure

Hany Samih Bauomy; Hany Samih Bauomy

doi:10.3934/math.20251349

AIMS Mathematics

2025, Volume 10, Issue 12: 30732-30772. doi: 10.3934/math.20251349

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Active control impact analysis on shaped coupled beam structure

Hany Samih Bauomy ^,

Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia

Received: 29 September 2025 Revised: 06 December 2025 Accepted: 22 December 2025 Published: 29 December 2025
MSC : 70K20, 74H45

Vibration-based warning and sensing devices rely on compact nonlinear structures capable of operating reliably under combined parametric and harmonic disturbances. To address these engineering demands, we developed a two-degree-of-freedom (2-DOF) lumped-parameter model that captures the coupled dynamics of a mass-based multi-warning unit integrated into a host structure. An Ⅱ-shaped coupled beam was proposed as a practical realization of the concept, where an auxiliary "warning mass" interacts dynamically with a primary supporting beam through nonlinear oscillations. The governing nonlinear differential equations were analytically solved using the multiple-time-scale technique (MTST), with detailed emphasis on primary and internal resonance conditions ($ {\omega _d} \cong \, {\omega _1}, \, \, {\omega _e} \cong \, {\omega _2} $) that critically influence device functionality and stability. To mitigate large-amplitude oscillations and enhance robustness under strong nonlinearities, a new control law, Nonlinear Proportional-Derivative Control with Negative Cubic Velocity Feedback (NPDCVF), was introduced. This controller combines a nonlinear PD term with a cubic dissipative component to improve damping and suppress resonance amplification. Control performance was assessed using the worst-case resonance scenario of the primary mass. Comparative simulations against Integral Resonance Control (IRC), Positive Position Feedback (PPF), Proportional-Integral-Derivative (PID), and Nonlinear Integral PPF (NIPPF) showed that the NPDCVF controller achieves up to 35–60% reduction in vibration amplitude, ensures faster settling, and maintains stable operation even near low natural frequencies where conventional controllers deteriorate. Frequency-response-based stability analysis revealed the boundaries of safe operation and highlighted instability regions relevant for design considerations. Numerical simulations using MATLAB/Simulink confirmed that the NPDCVF controller significantly enhances vibration suppression and flexibility against nonlinear disturbances. These results demonstrated the potential of the proposed system and control strategy for engineering applications such as structural warning devices, precision actuators, robotic manipulators, and advanced motorcycle suspension systems.
- lumped coupled parameter,
- investigated control,
- 1: 1 internal resonance,
- perturbation,
- shaped beam
Citation: Hany Samih Bauomy. Active control impact analysis on shaped coupled beam structure[J]. AIMS Mathematics, 2025, 10(12): 30732-30772. doi: 10.3934/math.20251349

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Abstract

Vibration-based warning and sensing devices rely on compact nonlinear structures capable of operating reliably under combined parametric and harmonic disturbances. To address these engineering demands, we developed a two-degree-of-freedom (2-DOF) lumped-parameter model that captures the coupled dynamics of a mass-based multi-warning unit integrated into a host structure. An Ⅱ-shaped coupled beam was proposed as a practical realization of the concept, where an auxiliary "warning mass" interacts dynamically with a primary supporting beam through nonlinear oscillations. The governing nonlinear differential equations were analytically solved using the multiple-time-scale technique (MTST), with detailed emphasis on primary and internal resonance conditions ($ {\omega _d} \cong \, {\omega _1}, \, \, {\omega _e} \cong \, {\omega _2} $) that critically influence device functionality and stability. To mitigate large-amplitude oscillations and enhance robustness under strong nonlinearities, a new control law, Nonlinear Proportional-Derivative Control with Negative Cubic Velocity Feedback (NPDCVF), was introduced. This controller combines a nonlinear PD term with a cubic dissipative component to improve damping and suppress resonance amplification. Control performance was assessed using the worst-case resonance scenario of the primary mass. Comparative simulations against Integral Resonance Control (IRC), Positive Position Feedback (PPF), Proportional-Integral-Derivative (PID), and Nonlinear Integral PPF (NIPPF) showed that the NPDCVF controller achieves up to 35–60% reduction in vibration amplitude, ensures faster settling, and maintains stable operation even near low natural frequencies where conventional controllers deteriorate. Frequency-response-based stability analysis revealed the boundaries of safe operation and highlighted instability regions relevant for design considerations. Numerical simulations using MATLAB/Simulink confirmed that the NPDCVF controller significantly enhances vibration suppression and flexibility against nonlinear disturbances. These results demonstrated the potential of the proposed system and control strategy for engineering applications such as structural warning devices, precision actuators, robotic manipulators, and advanced motorcycle suspension systems.

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