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Darboux Vector and Stress Analysis of Equi-Affine Frame

Yıl 2018, Cilt: 10, 7 - 11, 29.12.2018

Öz

A set of points that corresponds a vector of vector space constructed on a field is called an affine space associate with that vector space. We denote as affine 3-space A3 associated with IR3.
The first written sources that can be achieved about affine space curve theory are based on the 1890's when Ernesto Ces\`{a}ro and Die Schon von Pirondini lived period. From that years to 2000's there are a some affine fames used in curve theory. One of them is equi-affine frame.

The grup of affine motions special linear transformation consist of volume preserving linear transformations denoted by and comprising diffeomorphisms of that preserve some important invariants such curvaures that in curve theory as well.

In this study, we separated the matrix representing affine frame as symmetric and antismmetric parts by using matrix demonstration of the equi-affine frame of a curve given in affine 3-space. By making use of antisymmetric part, we obtained the angular velocity vector which is also known as Darboux vector and then we expressed it in the form of linear sum of affine Frenet vectors.

On the other hand, by making use of symmetric part, we obtained the normal stresses and shear stress components of the stress on the frame of the curve in terms of the affine curvature and affine torsion. Thus we had the opportunity to be able to explane the distinctive geometric features of the affine curvature and affin torsion.

Lastly, we made stress analysis of a curve with constant affine curvature and affine torsion in affine 3-space as an example.

Kaynakça

  • Blaschke, W., Differential Geometrie II, Verlag von Julius springer, Berlin, 1923.
  • Cayley, A., The algebraic structure of the orthogonal group and the other classical groups in a field of characteristic zero or a prime characteristic, J. Reine Angew.Math., 32,(1846).
  • Cesaro, E., Lezioni di Geometria Intrinseca, Napoli, Italy, 1896.
  • Ferdinand P. Beer and E. Russell Johnson, Jr, Mechanics of Materials, Second Edition, McGraw-Hill, Inc, 1992.
  • Hu, N., Affine Geometry of Space Curves and Homogeneous Surfaces, phd thesis, Hokkaido University, August, 2012.
  • James M. Gere and Stephen P. Timoshenko, Mechanics of Materials, Third Edition, PWS-KENT Publishing Company, Boston, 1990.
  • Nomizu, K. and Sasaki, T., Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
  • Salkowski, E., and Schells, W., Allgemeine Theorie der Kurven doppelter Krümmung, Leipzig und Berlin, 1914.
  • Su, B., Some Classes of Curves in The Affine space, Tohoku Math. Journ. 31,(1929),283-291.
  • Su, B., Affine Differential Geometry, Science Press, Beijing, China, 1983.
Yıl 2018, Cilt: 10, 7 - 11, 29.12.2018

Öz

Kaynakça

  • Blaschke, W., Differential Geometrie II, Verlag von Julius springer, Berlin, 1923.
  • Cayley, A., The algebraic structure of the orthogonal group and the other classical groups in a field of characteristic zero or a prime characteristic, J. Reine Angew.Math., 32,(1846).
  • Cesaro, E., Lezioni di Geometria Intrinseca, Napoli, Italy, 1896.
  • Ferdinand P. Beer and E. Russell Johnson, Jr, Mechanics of Materials, Second Edition, McGraw-Hill, Inc, 1992.
  • Hu, N., Affine Geometry of Space Curves and Homogeneous Surfaces, phd thesis, Hokkaido University, August, 2012.
  • James M. Gere and Stephen P. Timoshenko, Mechanics of Materials, Third Edition, PWS-KENT Publishing Company, Boston, 1990.
  • Nomizu, K. and Sasaki, T., Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
  • Salkowski, E., and Schells, W., Allgemeine Theorie der Kurven doppelter Krümmung, Leipzig und Berlin, 1914.
  • Su, B., Some Classes of Curves in The Affine space, Tohoku Math. Journ. 31,(1929),283-291.
  • Su, B., Affine Differential Geometry, Science Press, Beijing, China, 1983.
Toplam 10 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Yilmaz Tunçer

Yayımlanma Tarihi 29 Aralık 2018
Yayımlandığı Sayı Yıl 2018 Cilt: 10

Kaynak Göster

APA Tunçer, Y. (2018). Darboux Vector and Stress Analysis of Equi-Affine Frame. Turkish Journal of Mathematics and Computer Science, 10, 7-11.
AMA Tunçer Y. Darboux Vector and Stress Analysis of Equi-Affine Frame. TJMCS. Aralık 2018;10:7-11.
Chicago Tunçer, Yilmaz. “Darboux Vector and Stress Analysis of Equi-Affine Frame”. Turkish Journal of Mathematics and Computer Science 10, Aralık (Aralık 2018): 7-11.
EndNote Tunçer Y (01 Aralık 2018) Darboux Vector and Stress Analysis of Equi-Affine Frame. Turkish Journal of Mathematics and Computer Science 10 7–11.
IEEE Y. Tunçer, “Darboux Vector and Stress Analysis of Equi-Affine Frame”, TJMCS, c. 10, ss. 7–11, 2018.
ISNAD Tunçer, Yilmaz. “Darboux Vector and Stress Analysis of Equi-Affine Frame”. Turkish Journal of Mathematics and Computer Science 10 (Aralık 2018), 7-11.
JAMA Tunçer Y. Darboux Vector and Stress Analysis of Equi-Affine Frame. TJMCS. 2018;10:7–11.
MLA Tunçer, Yilmaz. “Darboux Vector and Stress Analysis of Equi-Affine Frame”. Turkish Journal of Mathematics and Computer Science, c. 10, 2018, ss. 7-11.
Vancouver Tunçer Y. Darboux Vector and Stress Analysis of Equi-Affine Frame. TJMCS. 2018;10:7-11.