A non-perturbative methodology for a cantilever-beam dynamical system with bifurcation and negative derivative feedback controlling

Asma Alanazy; A.T. EL-Sayed; F. T. El-Bahrawy; Asma Alanazy; A.T. EL-Sayed; F. T. El-Bahrawy

doi:10.3934/math.2025795

AIMS Mathematics

2025, Volume 10, Issue 8: 17832-17867. doi: 10.3934/math.2025795

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A non-perturbative methodology for a cantilever-beam dynamical system with bifurcation and negative derivative feedback controlling

1.
Department of Mathematics, Faculty of Science, Northern Border University, Arar 1321, Kingdom of Saudi Arabia
2.
Basic Science Department, Modern Academy for Engineering and Technology, Elmokattam, Cairo 11439, Egypt

Received: 26 April 2025 Revised: 19 July 2025 Accepted: 22 July 2025 Published: 06 August 2025
MSC : 34A34, 37N35, 70J99, 70K20, 74H10

The existence study explored a cantilever-beam dynamical system with bifurcation and negative derivative feedback to understand and regulate nonlinear phenomena in flexible constructions, improving stability and performance in engineering applications. The nonlinear transverse vibrations of a cantilever beam mechanism were investigated under initial resonance conditions. A detailed clarification of the non-perturbative method (NPM) was presented to introduce a novel approach in handling nonlinearities systems. Simply, NPM transformed the weakly nonlinear oscillator into an equivalent linear differential equation (ODE). The theoretical results obtained through NPM were validated using numerical simulations in MATLAB. Moreover, NPM facilitated the stability analysis of the system, which was not previously attainable via traditional approaches. Significantly, NPM resolved limitations encountered in previous methods, offering a more strong solution framework. The system was completely different in the second part of the study, since it only had two degrees of freedom (2DOF) controlled through an individual input. The study outlined the bifurcation and control strategy of the 2DOF's response vibrations. Negative derivative feedback (NDF) was used to control the system's operation to minimize the harmful vibrations it produced. To identify several bifurcations inside the system, bifurcation characterization was conducted on an examined figure, considering two varied values of the controller gain. The fundamental objective of this study was to explore the transformation of nonlinear ODEs into linear ones and assess the efficacy of the control approach and bifurcation analysis in stabilizing cantilever beams. The multiple time-scales method (MTSM) was utilized for analyzing the controlled linear equivalent algorithm. Several diagrams illustrated the prototype's sturdy construction. To mitigate damaging vibrations in the system, the NDF was recommended. Predicted solutions were validated through numerical simulations using the fourth-order Runge-Kutta method (RK4), demonstrating an excellent correlation. The stability and steady-state amplitude of nonlinear patterns were analyzed both before and after applying control. Additionally, frequency response curves (FRCs) and optimal system configurations were evaluated under various controller and system parameter settings.
- nonlinear oscillations,
- bifurcation,
- Poincaré map,
- multiple time-scale method,
- stability analysis,
- NDF controller,
- non-perturbative method
Citation: Asma Alanazy, A.T. EL-Sayed, F. T. El-Bahrawy. A non-perturbative methodology for a cantilever-beam dynamical system with bifurcation and negative derivative feedback controlling[J]. AIMS Mathematics, 2025, 10(8): 17832-17867. doi: 10.3934/math.2025795

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Abstract

The existence study explored a cantilever-beam dynamical system with bifurcation and negative derivative feedback to understand and regulate nonlinear phenomena in flexible constructions, improving stability and performance in engineering applications. The nonlinear transverse vibrations of a cantilever beam mechanism were investigated under initial resonance conditions. A detailed clarification of the non-perturbative method (NPM) was presented to introduce a novel approach in handling nonlinearities systems. Simply, NPM transformed the weakly nonlinear oscillator into an equivalent linear differential equation (ODE). The theoretical results obtained through NPM were validated using numerical simulations in MATLAB. Moreover, NPM facilitated the stability analysis of the system, which was not previously attainable via traditional approaches. Significantly, NPM resolved limitations encountered in previous methods, offering a more strong solution framework. The system was completely different in the second part of the study, since it only had two degrees of freedom (2DOF) controlled through an individual input. The study outlined the bifurcation and control strategy of the 2DOF's response vibrations. Negative derivative feedback (NDF) was used to control the system's operation to minimize the harmful vibrations it produced. To identify several bifurcations inside the system, bifurcation characterization was conducted on an examined figure, considering two varied values of the controller gain. The fundamental objective of this study was to explore the transformation of nonlinear ODEs into linear ones and assess the efficacy of the control approach and bifurcation analysis in stabilizing cantilever beams. The multiple time-scales method (MTSM) was utilized for analyzing the controlled linear equivalent algorithm. Several diagrams illustrated the prototype's sturdy construction. To mitigate damaging vibrations in the system, the NDF was recommended. Predicted solutions were validated through numerical simulations using the fourth-order Runge-Kutta method (RK4), demonstrating an excellent correlation. The stability and steady-state amplitude of nonlinear patterns were analyzed both before and after applying control. Additionally, frequency response curves (FRCs) and optimal system configurations were evaluated under various controller and system parameter settings.

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