Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance
Abstract In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-09-01
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| Series: | Journal of the Egyptian Mathematical Society |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s42787-020-00102-7 |
| _version_ | 1818448497469292544 |
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| author | M. Sayed A. A. Mousa D. Y. Alzaharani I. H. Mustafa S. I. El-Bendary |
| author_facet | M. Sayed A. A. Mousa D. Y. Alzaharani I. H. Mustafa S. I. El-Bendary |
| author_sort | M. Sayed |
| collection | DOAJ |
| description | Abstract In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω 1 ), detuning parameter (σ 1 ), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four. |
| first_indexed | 2024-12-14T20:20:27Z |
| format | Article |
| id | doaj.art-710a6d09524546c19779bec4066e3ec3 |
| institution | Directory Open Access Journal |
| issn | 2090-9128 |
| language | English |
| last_indexed | 2024-12-14T20:20:27Z |
| publishDate | 2020-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-710a6d09524546c19779bec4066e3ec32022-12-21T22:48:45ZengSpringerOpenJournal of the Egyptian Mathematical Society2090-91282020-09-0128112110.1186/s42787-020-00102-7Bifurcation analysis of a composite cantilever beam via 1:3 internal resonanceM. Sayed0A. A. Mousa1D. Y. Alzaharani2I. H. Mustafa3S. I. El-Bendary4Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia UniversityMathematics and Statistics Department, Faculty of Science, Taif UniversityMathematics Department, Faculty of Arts and Science in Baljurashi, Al-Baha UniversityBiomedical Engineering Department, Helwan UniversityDepartment of Mathematics, Faculty of Science, Tanta UniversityAbstract In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω 1 ), detuning parameter (σ 1 ), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.http://link.springer.com/article/10.1186/s42787-020-00102-7Active controlStabilityInternal resonanceBifurcation analysisPoint and periodic attractor |
| spellingShingle | M. Sayed A. A. Mousa D. Y. Alzaharani I. H. Mustafa S. I. El-Bendary Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance Journal of the Egyptian Mathematical Society Active control Stability Internal resonance Bifurcation analysis Point and periodic attractor |
| title | Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance |
| title_full | Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance |
| title_fullStr | Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance |
| title_full_unstemmed | Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance |
| title_short | Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance |
| title_sort | bifurcation analysis of a composite cantilever beam via 1 3 internal resonance |
| topic | Active control Stability Internal resonance Bifurcation analysis Point and periodic attractor |
| url | http://link.springer.com/article/10.1186/s42787-020-00102-7 |
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