simulation of torsion moment at the wheel set of the railway vehicle ...

Tatiana Olejnikova *TWO AND THREE-REVOLUTION CYCLICAL SURFACESThe creation of two-revolution and three-revolution cyclical surfaces is presented in the paper. Classification and vector equations of thesurfaces are given. The surfaces are created by translation of the circle along the curves and its centre is on the curve. The curves are createdby revolution of a point about any edge of the trihedron of the previous curve and this trihedron moves simultaneously along this curve. Allspecific forms of surfaces are illustrated in figures visualized in Maple.1. IntroductionThe point S 1 revolves about the coordinate axis z with angularvelocity w 1 in the distance d 1 from the origin of the coordinatesystem (O, x, y, z). For every value of the angle w 1 there exists onlyone position of the point R 1 and the trajectory of this point R 1 isthe curve k 1 (circle). The trihedron (R 1 , t 1 , n 1 , b 1 ) defined in everypoint R 1 ∈ k 1 is determined by tangent, principal normal andbinormal of the curve k 1 . The point S 2 revolves at an angularvelocity w 2 about any axis of the coordinate system, which is identicalwith the trihedron (R 1 , t 1 , n 1 , b 1 ) of the curve k 1 , in the distanced 2 from the origin of this coordinate system which is movingsimultaneously along the curve k 1 . For every value of the angle w 2there exists only one position of the point R 2 . The trajectory ofthis point R 2 is the curve k g 2, where g t, n, b. The trihedron (R 2 ,t 2 , n 2 , b 2 ) in every point R 2 ∈ k g 2 is determined by the tangent,principal normal and binormal of the curve k g 2. The point S 3 revolvesabout any axis of the coordinate system identical with the trihedron(R 2 , t 2 , n 2 , b 2 ) of the curve k g 2 at an angular velocity w 3 inthe distance d 3 from the origin of this coordinate system which ismoving simultaneously along the curve k g 2. For every value of theangle w 3 there exists only one position of the point R 3 . The trajectoryof the point R 3 is the curve k g 3 h , where g, h t, n, b. Thetrihedron (R 3 , t 3 , n 3 , b 3 ) in every point R 3 ∈ k g 3 h is determined bythe tangent, principal normal and binormal of the curve k g 3 h .The surface of the type P 1 (u,v) is created by translation of thecircle c 1 (R 1 , r 1 ) along the curve k 1 , the surface of the typeP g 2(u,v) is created by translation of the circle c 2 (R 2 , r 2 ) alongthe curve k g 2 and the surface of the type P g 3 h (u,v) is created bytranslation of the circle c 3 (R 3 , r 3 ) along the curve k g 3 h . Theindex g t, n, b determines that the point S 2 revolves about thetangent t 1 , or principal normal n 1 or binormal b 1 of the curve k 1and the index h t, n, g determines that the point S 3 revolvesabout tangent t 2 , principal normal n 2 or binormal b 2 of the curvek g 2.REVIEW2. Vector functions of the curves k 1 , k2, g k3 ghLet the curve k 1 be a circle created by revolution of the pointS 1 S 1 (d 1 , 0, 0, 1) about the axis z of the coordinate system (O,x, y, z) at an angular velocity w 1 v and k 1 is determined by thevector functionr 1 (v) (x k1 (v), y k1 (v), z k1 (v), 1) S 1 . T z1 (w 1 ) (d 1 cos v, s 1 q 1 sin v, 0,1), v ∈ 0, 2π. (1)The matrix T z1 (w 1 ) represents the revolution of the point S 1about the coordinate axis z given by (5) (3 rd matrix for i 1),where the parameter q 1 1 determines the right-turned or leftturnedrevolution movement of the point (Fig. 1, i 1, j z) [3].We will define the trihedron (R 1 , t 1 , n 1 , b 1 ) of the curve k 1 in everypoint R 1 ∈ k 1 by the tangent t 1 , principal normal n 1 and by binormalb 1 with their unit vectors t 1 (v), n 1 (v), b 1 (v) by equations (2),(3), (4) for i 1, v ∈ 0, 2π1 dr1t^vh1 = ^ati, bti,ctih= =h1i dv(2)1 dx ^vh kidy ^ki vh dz ^ki vh dr1= e , , o , h1i =h dv dv dv dv1i21 dr1n^vh1 = ^ani, bni,cnih= =2h2i dv22221 dx^ ki vh dy^ ki vh dz^ki vh dr1= f , , p , h222 2i=2h2i dv dv dv dv1b,. (4)h t v n v h t v n vi = _ ^ hi # ^ihi3i = ^ hi # ^ hi3iThe curve k2 g is created by revolution of the point S 2 in thedistance d 2 from the origin of the coordinate system (O, x, y, z)about any coordinate axis x, y, or z through the angle w 2 into thepoint S 2 (Fig. 1, i 2 for , j x, y, z). Angular velocity w 2 m 1 v(3)* Tatiana OlejnikovaInstitute of Technology, Economics and Management in Civil Engineering, Civil Engineering Faculty, Technical University of Kosice, Slovakia,E-mail: tatiana.olejnikova@tuke.sk72 ● COMMUNICATIONS 3/2008