JAMP  Vol.6 No.4 , April 2018
The Study on the Cycloids of Moving Loops
Abstract: The infinite set of cycloids is created. Each cycloid of this set is defined as a movement trajectory of a point when this point circulates on the convex closed contour of arbitrary form when this contour moves rectilinearly without rotation on the plane with a velocity equal to the tangential velocity of a point on circulation contour. The classical cycloid is elements of this set. The differential equation of a cycloid set is derived and its solution in quadratures is received. The inverse problem when for the given cycloid it is necessary to fine the form of a circulation contour is solved. The problem of differential equation of the second order with boundary conditions about a bend of big curvature of an elastic rod of infinite length is solved in quadratures. Geometry of the loop which is formed at such bend is investigated. It is discovered that at movement of an elastic loop on a rod when the form and the size of a loop don’t change, each point of a loop moves on a trajectory which named by us the cycloid and which represents a circumference arch.
Keywords: Curves, Loops, Cycloids
Cite this paper: Tarabrin, G. (2018) The Study on the Cycloids of Moving Loops. Journal of Applied Mathematics and Physics, 6, 817-830. doi: 10.4236/jamp.2018.64070.

[1]   Tarabrin, G.T. (2012) Loop on Elastic Rod. Stroitelnaya Mechanika i Raschet Sooruzheniy, 3, 34-39. (In Russian)

[2]   Tarabrin, G.T. (2012) Circulations Cycloids on Moving Counters. Stroitelnaya Mechanika i Raschet Sooruzheniy, 4, 38-43. (In Russian)

[3]   Tarabrin, G. (2014) Non-Chrestomathy Problems of Mathematical Physics. Palmarium Academic Publishing, Saarbrucken. (In Russian)

[4]   Piskunov, N. (1975) Differential and Integral Calculus. Vol. 1, Mir Publishers, Moscow.

[5]   Rabotnov, J.N. (1988) Mechanics of Deformable Solid. Nauka, Moscow. (In Russian)

[6]   Timoshenko, S. (1955) Strength of Materials. D. van Nostrand Company, Inc., Princeton, New Jersey, Toronto, New York, London.

[7]   Filippov, A.T. (1990) Multiform Soliton. Nauka, Moscow. (In Russian).