Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions
In this paper, exact hydrostatic particulars equations for the centre of buoyancy curve and metacentric locus curve are given for rectangular cross section using quadratic functions. Those equations have not been given for the hyperbola range of the heel angles so far, and here it is done by using b...
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| Format: | Article |
| Language: | English |
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Faculty of Mechanical Engineering and Naval Architecture
2023-01-01
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| Series: | Brodogradnja |
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| Online Access: | https://hrcak.srce.hr/file/434695 |
| _version_ | 1797773097705668608 |
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| author | Dario Ban |
| author_facet | Dario Ban |
| author_sort | Dario Ban |
| collection | DOAJ |
| description | In this paper, exact hydrostatic particulars equations for the centre of buoyancy curve and metacentric locus curve are given for rectangular cross section using quadratic functions. Those equations have not been given for the hyperbola range of the heel angles so far, and here it is done by using basic quadratic functions and their horizontally symmetric immersion shapes, with two new methods defined: 1. Rotation of basic cross section shapes, and 2. Hydrostatic cross section area complement method that uses homothety or scaling properties of emerged and immersed areas of the rectangular cross section. Observed metacentric curve for rectangle consists of semi-cubic parabolas and Lamé curve with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of those functions definition. In order to achieve above, two theorems are given: the theorem about scaling using hydrostatic cross section area complement and the theorem about parallelism of centre of buoyancy tangents with waterlines. After non-dimensional bounds are given for the existence of the swallowtail discontinuity of metacentric curve for rectangular cross section in the Part 1 of this paper, the proof of its position in the symmetry of rectangle vertex angle is given in this Part 2 of the paper, thus confirming its position from theory. |
| first_indexed | 2024-03-12T22:00:21Z |
| format | Article |
| id | doaj.art-b530182f5ac34b93a426801a3cd90a72 |
| institution | Directory Open Access Journal |
| issn | 0007-215X 1845-5859 |
| language | English |
| last_indexed | 2024-03-12T22:00:21Z |
| publishDate | 2023-01-01 |
| publisher | Faculty of Mechanical Engineering and Naval Architecture |
| record_format | Article |
| series | Brodogradnja |
| spelling | doaj.art-b530182f5ac34b93a426801a3cd90a722023-07-25T08:23:17ZengFaculty of Mechanical Engineering and Naval ArchitectureBrodogradnja0007-215X1845-58592023-01-01743174510.21278/brod74302270Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functionsDario Ban0University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, 21000 Split, CroatiaIn this paper, exact hydrostatic particulars equations for the centre of buoyancy curve and metacentric locus curve are given for rectangular cross section using quadratic functions. Those equations have not been given for the hyperbola range of the heel angles so far, and here it is done by using basic quadratic functions and their horizontally symmetric immersion shapes, with two new methods defined: 1. Rotation of basic cross section shapes, and 2. Hydrostatic cross section area complement method that uses homothety or scaling properties of emerged and immersed areas of the rectangular cross section. Observed metacentric curve for rectangle consists of semi-cubic parabolas and Lamé curve with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of those functions definition. In order to achieve above, two theorems are given: the theorem about scaling using hydrostatic cross section area complement and the theorem about parallelism of centre of buoyancy tangents with waterlines. After non-dimensional bounds are given for the existence of the swallowtail discontinuity of metacentric curve for rectangular cross section in the Part 1 of this paper, the proof of its position in the symmetry of rectangle vertex angle is given in this Part 2 of the paper, thus confirming its position from theory.https://hrcak.srce.hr/file/434695centre of buoyancy curvemetacentric curverectangular cross sectionquadratic functionsbasic geometric shapeshydrostatic area complement |
| spellingShingle | Dario Ban Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions Brodogradnja centre of buoyancy curve metacentric curve rectangular cross section quadratic functions basic geometric shapes hydrostatic area complement |
| title | Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions |
| title_full | Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions |
| title_fullStr | Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions |
| title_full_unstemmed | Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions |
| title_short | Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions |
| title_sort | re examination of centre of buoyancy curve and its evolute for rectangular cross section part 2 using quadratic functions |
| topic | centre of buoyancy curve metacentric curve rectangular cross section quadratic functions basic geometric shapes hydrostatic area complement |
| url | https://hrcak.srce.hr/file/434695 |
| work_keys_str_mv | AT darioban reexaminationofcentreofbuoyancycurveanditsevoluteforrectangularcrosssectionpart2usingquadraticfunctions |