Formulas for Fundamental Natural Frequency of Plane Periodic Truss
This study considers a plane statically determinate truss with double lattice structure and without a lower chord. Well-known versions of this design are Fink and Bollman trusses. Two methods are used to derive the analytical relationship of the lower limit of the fundamental frequency with the numb...
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2023-12-01
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| Series: | Structural Mechanics of Engineering Constructions and Buildings |
| Subjects: | |
| Online Access: | https://journals.rudn.ru/structural-mechanics/article/viewFile/37677/23117 |
| _version_ | 1797338157513965568 |
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| author | Mikhail N. Kirsanov |
| author_facet | Mikhail N. Kirsanov |
| author_sort | Mikhail N. Kirsanov |
| collection | DOAJ |
| description | This study considers a plane statically determinate truss with double lattice structure and without a lower chord. Well-known versions of this design are Fink and Bollman trusses. Two methods are used to derive the analytical relationship of the lower limit of the fundamental frequency with the number of panels in the periodic structure. It is assumed that mass of the truss is concentrated at its joints (nodes). The nodes vibrate vertically, and the number of degrees of freedom coincides with the number of nodes. The stiffness analysis of the truss is performed using the Maxwell - Mohr method. The forces in the elastic elements and the reactions of the roller and pin supports are calculated by the method of joints depending on the size of the truss and its order of periodicity. The system of linear equations is solved using the inverse matrix method. The Dunkerley method of partial frequencies is used to calculate the lower limit of the fundamental frequency. For a series of solutions obtained for trusses with different number of panels, the common term in the sequence of solution formulas is found by induction using Maple software. The solution coefficients have polynomial form in the number of panels of order not higher than the fifth. The solution is compared with an approximate version of the Dunkerley method, in which the sum of the terms corresponding to partial frequencies is calculated using the mean value theorem. The closeness of the frequency obtained by the two analytical methods to the numerical frequency spectrum solution is shown by particular examples. An approximate version of the Dunkerley method has a simpler form and an accuracy comparable to the original Dunkerley method. |
| first_indexed | 2024-03-08T09:25:59Z |
| format | Article |
| id | doaj.art-6fc6493108b8482ea8fe86faeadde3c1 |
| institution | Directory Open Access Journal |
| issn | 1815-5235 2587-8700 |
| language | English |
| last_indexed | 2024-03-08T09:25:59Z |
| publishDate | 2023-12-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Structural Mechanics of Engineering Constructions and Buildings |
| spelling | doaj.art-6fc6493108b8482ea8fe86faeadde3c12024-01-31T08:58:52ZengPeoples’ Friendship University of Russia (RUDN University)Structural Mechanics of Engineering Constructions and Buildings1815-52352587-87002023-12-0119655155910.22363/1815-5235-2023-19-6-551-55921014Formulas for Fundamental Natural Frequency of Plane Periodic TrussMikhail N. Kirsanov0https://orcid.org/0000-0002-8588-3871National Research University “MPEI”This study considers a plane statically determinate truss with double lattice structure and without a lower chord. Well-known versions of this design are Fink and Bollman trusses. Two methods are used to derive the analytical relationship of the lower limit of the fundamental frequency with the number of panels in the periodic structure. It is assumed that mass of the truss is concentrated at its joints (nodes). The nodes vibrate vertically, and the number of degrees of freedom coincides with the number of nodes. The stiffness analysis of the truss is performed using the Maxwell - Mohr method. The forces in the elastic elements and the reactions of the roller and pin supports are calculated by the method of joints depending on the size of the truss and its order of periodicity. The system of linear equations is solved using the inverse matrix method. The Dunkerley method of partial frequencies is used to calculate the lower limit of the fundamental frequency. For a series of solutions obtained for trusses with different number of panels, the common term in the sequence of solution formulas is found by induction using Maple software. The solution coefficients have polynomial form in the number of panels of order not higher than the fifth. The solution is compared with an approximate version of the Dunkerley method, in which the sum of the terms corresponding to partial frequencies is calculated using the mean value theorem. The closeness of the frequency obtained by the two analytical methods to the numerical frequency spectrum solution is shown by particular examples. An approximate version of the Dunkerley method has a simpler form and an accuracy comparable to the original Dunkerley method.https://journals.rudn.ru/structural-mechanics/article/viewFile/37677/23117plane trussdunkerley methodfundamental frequencyanalytical solutionnatural vibrationperiodic structureinduction methodmaxwell - mohr method |
| spellingShingle | Mikhail N. Kirsanov Formulas for Fundamental Natural Frequency of Plane Periodic Truss Structural Mechanics of Engineering Constructions and Buildings plane truss dunkerley method fundamental frequency analytical solution natural vibration periodic structure induction method maxwell - mohr method |
| title | Formulas for Fundamental Natural Frequency of Plane Periodic Truss |
| title_full | Formulas for Fundamental Natural Frequency of Plane Periodic Truss |
| title_fullStr | Formulas for Fundamental Natural Frequency of Plane Periodic Truss |
| title_full_unstemmed | Formulas for Fundamental Natural Frequency of Plane Periodic Truss |
| title_short | Formulas for Fundamental Natural Frequency of Plane Periodic Truss |
| title_sort | formulas for fundamental natural frequency of plane periodic truss |
| topic | plane truss dunkerley method fundamental frequency analytical solution natural vibration periodic structure induction method maxwell - mohr method |
| url | https://journals.rudn.ru/structural-mechanics/article/viewFile/37677/23117 |
| work_keys_str_mv | AT mikhailnkirsanov formulasforfundamentalnaturalfrequencyofplaneperiodictruss |