Chaotic behavior analysis of a hanging wire system with NIPPFC controller

This study presents a dynamic analysis for a model of a suspended cable subjected to both static and dynamic loading conditions. To enhance stability and reduce undesired vibrations, particularly in the resonance region, the nonlinear integral positive position feedback controller with time delay (NIPPFC) is implemented. The differential equation (DE) for a hung cable is developed using the Hamilton variation approach (HVA) and solved by the multiple-scales approach (MSA) in to appropriate order to get the approximate solutions (AS). By comparing the AS to the determined numerical solutions (NS) using the Runge–Kutta method of fourth order (RK-4), the accuracy of the AS is confirmed. The modulation equations (ME) are obtained by exploring solvability criteria and resonance instances. Graphical representations of the time histories and frequency responses of the derived solutions are presented using MATLAB software and Wolfram Mathematica 13.2. Additionally, a graphical analysis is performed on the temporal histories of the achieved solutions, both with and without control. Resonance curves are also utilized to explore stability analysis and steady-state solutions. Furthermore, the model’s nonlinear dynamics are investigated through bifurcation analysis and visualized using bifurcation diagrams to identify critical parameters leading to qualitative changes in system behavior. The Poincaré map is utilized to study periodic and chaotic oscillations, providing insights into the system’s long-term stability and the emergence of chaotic dynamics. The significance of the acquired results lies in their contribution to understanding and mitigating dynamic instabilities and chaotic behavior in flexible structures under varying operational conditions. Furthermore, a comparative analysis with conventional controllers, such as the linear negative displacement feedback controller (LDFC), linear negative velocity feedback controller (LVFC), and negative cubic velocity feedback controller (CVFC), showed that the NIPPFC significantly enhances damping. The amplitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> was reduced up to 99.44%, with the best result achieved at time delay <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.004</mml:mn> </mml:mrow> </mml:math> . This model is widely applied in the analysis and design of flexible structures such as suspension bridges, transmission lines, and cable-supported systems. In addition, it helps engineers analyze how structures respond to static and dynamic forces, such as weight, wind, earthquakes, or moving loads. By monitoring displacements, it predicts critical behaviors like vibrations, deflections, or resonance, preventing structural instability or failure. It also aids in vibration control and optimization to improve safety and performance.