Existence and uniqueness of equilibrium states of a rotating elastic rod
A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, r...
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| Format: | Article |
| Language: | English |
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Wiley
1994-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171294000451 |
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| _version_ | 1832555357527867392 |
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| author | M. B. M. Elgindi |
| author_facet | M. B. M. Elgindi |
| author_sort | M. B. M. Elgindi |
| collection | DOAJ |
| description | A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α=0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α≠0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α≠0 provided λ<λc. For small α≠0 this requires the solution of a nonlinear perturbed bifurcation problem. |
| format | Article |
| id | doaj-art-c6d694419b0945299c0a75269a4b0eff |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1994-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c6d694419b0945299c0a75269a4b0eff2025-02-03T05:48:25ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251994-01-0117231532210.1155/S0161171294000451Existence and uniqueness of equilibrium states of a rotating elastic rodM. B. M. Elgindi0Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire 54702-4004, WI, USAA flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two-point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α=0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α≠0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α≠0 provided λ<λc. For small α≠0 this requires the solution of a nonlinear perturbed bifurcation problem.http://dx.doi.org/10.1155/S0161171294000451existence and uniqueness of equilibrium statesrotating rodsnonlinear eigenvalue problemsfourth order two-point nonlinear boundary value problemsSchauder fixed point theoremperturbed bifurcation problemsperturbation solution. |
| spellingShingle | M. B. M. Elgindi Existence and uniqueness of equilibrium states of a rotating elastic rod International Journal of Mathematics and Mathematical Sciences existence and uniqueness of equilibrium states rotating rods nonlinear eigenvalue problems fourth order two-point nonlinear boundary value problems Schauder fixed point theorem perturbed bifurcation problems perturbation solution. |
| title | Existence and uniqueness of equilibrium states of a rotating elastic rod |
| title_full | Existence and uniqueness of equilibrium states of a rotating elastic rod |
| title_fullStr | Existence and uniqueness of equilibrium states of a rotating elastic rod |
| title_full_unstemmed | Existence and uniqueness of equilibrium states of a rotating elastic rod |
| title_short | Existence and uniqueness of equilibrium states of a rotating elastic rod |
| title_sort | existence and uniqueness of equilibrium states of a rotating elastic rod |
| topic | existence and uniqueness of equilibrium states rotating rods nonlinear eigenvalue problems fourth order two-point nonlinear boundary value problems Schauder fixed point theorem perturbed bifurcation problems perturbation solution. |
| url | http://dx.doi.org/10.1155/S0161171294000451 |
| work_keys_str_mv | AT mbmelgindi existenceanduniquenessofequilibriumstatesofarotatingelasticrod |