A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods
The exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.1155/2017/8186976 |
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| author | Tamer Ahmed El-Sayed Said Hamed Farghaly |
| author_facet | Tamer Ahmed El-Sayed Said Hamed Farghaly |
| author_sort | Tamer Ahmed El-Sayed |
| collection | DOAJ |
| description | The exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational elastic supports. The beam start, end, and intermediate equations are assembled using the present normalized transfer matrix (NTM). The advantage of this method is that it is quicker than the standard method because the size of the complete system coefficient matrix is 4 × 4. In addition, during the assembly of this matrix, there are no inverse matrix steps required. The validity of this method is tested by comparing the results of the current method with the literature. Then the validity of the exact stepped analysis is checked using experimental and FE(3D) methods. The experimental results for stepped beams with single step and two steps, for sixteen different test samples, are in excellent agreement with those of the three-dimensional finite element FE(3D). The comparison between the NTM method and the finite element method results shows that the modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape. Meanwhile, the deviation decreases when a beam step location coincides with a straight portion in the mode shape. |
| format | Article |
| id | doaj-art-099496303418420794947909186f8764 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-099496303418420794947909186f87642025-02-03T05:46:28ZengWileyShock and Vibration1070-96221875-92032017-01-01201710.1155/2017/81869768186976A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) MethodsTamer Ahmed El-Sayed0Said Hamed Farghaly1Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat-Elzaton, Cairo, EgyptDepartment of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, P.O. Box 11718, Helmeiat-Elzaton, Cairo, EgyptThe exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational elastic supports. The beam start, end, and intermediate equations are assembled using the present normalized transfer matrix (NTM). The advantage of this method is that it is quicker than the standard method because the size of the complete system coefficient matrix is 4 × 4. In addition, during the assembly of this matrix, there are no inverse matrix steps required. The validity of this method is tested by comparing the results of the current method with the literature. Then the validity of the exact stepped analysis is checked using experimental and FE(3D) methods. The experimental results for stepped beams with single step and two steps, for sixteen different test samples, are in excellent agreement with those of the three-dimensional finite element FE(3D). The comparison between the NTM method and the finite element method results shows that the modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape. Meanwhile, the deviation decreases when a beam step location coincides with a straight portion in the mode shape.http://dx.doi.org/10.1155/2017/8186976 |
| spellingShingle | Tamer Ahmed El-Sayed Said Hamed Farghaly A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods Shock and Vibration |
| title | A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods |
| title_full | A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods |
| title_fullStr | A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods |
| title_full_unstemmed | A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods |
| title_short | A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods |
| title_sort | normalized transfer matrix method for the free vibration of stepped beams comparison with experimental and fe 3d methods |
| url | http://dx.doi.org/10.1155/2017/8186976 |
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