A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation
An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engi...
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/16 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832589093920309248 |
|---|---|
| author | Asma Alanazy Galal M. Moatimid T. S. Amer Mona A. A. Mohamed M. K. Abohamer |
| author_facet | Asma Alanazy Galal M. Moatimid T. S. Amer Mona A. A. Mohamed M. K. Abohamer |
| author_sort | Asma Alanazy |
| collection | DOAJ |
| description | An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He’s frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system’s behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior. |
| format | Article |
| id | doaj-art-088aea657d32406b94270c140db0610b |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-088aea657d32406b94270c140db0610b2025-01-24T13:22:09ZengMDPI AGAxioms2075-16802024-12-011411610.3390/axioms14010016A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and BifurcationAsma Alanazy0Galal M. Moatimid1T. S. Amer2Mona A. A. Mohamed3M. K. Abohamer4Mathematics Department, Northern Border University, Rafha 91431, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Ain Shams University, Cairo 11566, EgyptDepartment of Mathematics, Faculty of Science, Tanta University, Tanta 31527, EgyptDepartment of Mathematics, Faculty of Education, Ain Shams University, Cairo 11566, EgyptDepartment of Engineering Physics and Mathematics, Faculty of Engineering, Tanta University, Tanta 31734, EgyptAn examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He’s frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system’s behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior.https://www.mdpi.com/2075-1680/14/1/16cantilever beamHe’s frequency formulationnon-perturbative approachnonlinear Mathieu equationbifurcation diagramsPoincaré maps |
| spellingShingle | Asma Alanazy Galal M. Moatimid T. S. Amer Mona A. A. Mohamed M. K. Abohamer A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation Axioms cantilever beam He’s frequency formulation non-perturbative approach nonlinear Mathieu equation bifurcation diagrams Poincaré maps |
| title | A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation |
| title_full | A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation |
| title_fullStr | A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation |
| title_full_unstemmed | A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation |
| title_short | A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation |
| title_sort | novel procedure in scrutinizing a cantilever beam with tip mass analytic and bifurcation |
| topic | cantilever beam He’s frequency formulation non-perturbative approach nonlinear Mathieu equation bifurcation diagrams Poincaré maps |
| url | https://www.mdpi.com/2075-1680/14/1/16 |
| work_keys_str_mv | AT asmaalanazy anovelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT galalmmoatimid anovelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT tsamer anovelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT monaaamohamed anovelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT mkabohamer anovelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT asmaalanazy novelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT galalmmoatimid novelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT tsamer novelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT monaaamohamed novelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation AT mkabohamer novelprocedureinscrutinizingacantileverbeamwithtipmassanalyticandbifurcation |