The Method of Finite Differences in Solutions Concerning Circular Plate

Abstract:

Finite difference method in solving classic problems in theory of plates is considered a standard one [1], [2], [3], [4]. The above refers mainly to solutions in right-angle coordinates. For circular plates, for which the use of polar coordinates is the best option, the question of classic plate deflection gets complicated. In accordance with mathematical rules the passage from partial differentials to final differences seems firm. Still final formulas both for the equation (1), as well as for border conditions of circular plate obtained in this study and in the study [3] differ considerably. The paper describes in detail necessary mathematical calculations. The final results are presented in identical form as in the study [3]. Difference of results as well as the length of arm in passage from partial differentials to finite differences for mixed derivatives are discussed. Generalizations resulting from these discussions are presented. This preliminary proceeding has the purpose of searching for solutions to technical problems in machine building and construction, in particular finding a solution to the question of distribution of load along contact line in worm gearing.

• Periodical: Key Engineering Materials (Volume 490)
• Pages: 305-311
• DOI: https://doi.org/10.4028/www.scientific.net/KEM.490.305
• Online since: September 2011
• Authors: Henryk G. Sabiniak
• Keywords: Circular Plate, Finite Difference, Worm Teeth

References

[1] S. Timoshenko, S. Woinowsky-Krieger - Theory of plates and shells (Polish edition), Arkady, Warsaw 1962.

[2] Łukaszewicz S.: Concentrated load of plates, shields and shells. (Polish edition), PWN. Warsaw 1976.

[3] Kączkowski Z.: Static calculations. (Polish edition), Arkady. Warsaw 1980.

[4] Kovalenko A.D.: Wheel plates with variable thickness. (Russian language). GIFML. Moscow 1959.

[5] Benda J.: Calculations of wheel plates of finite differences method. (Czech language). Notebook of Technical High School in Brno, nr 4, pp.175-189, 1964.

[6] Sabiniak H.G.: Load distribution along the contact line between two bracket plates, treated as models for teeth of cylindrical and globoidal toothed gears. (German language). PhD dissertation Technical University in Chemnitz, 1992.