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

REVIEWThe two-revolution cyclical surface of the type P g 2(u,v) is createdby translation of the circle c 2 (R 2 , r 2 ) along the curve k g 2 at anangular velocity w 2 m 1 v, where the circle is always in the plane(n 2 , b 2 ) if the index g t, or in the plane (t 2 , b 2 )ifg n, or inthe plane (t 2 , n 2 )ifg b and its centre is the point R 2 ∈ k g 2. Wewill create it so that the circle c 02 determined by the vector functionc 02 (u) (0, r 2 cos u, r 2 sin u, 1) if g t, or c 02 (u) (r 2 cos u,0, r 2 sin u, 1) if g n, or c 02 (u) (r 2 cos u, r 2 sin u, 0, 1) if g bwe will transform into the circle c 2 in the coordinate system (R 2 ,t 2 , n 2 , b 2 ) using the matrix M 2 (w 2 ) by equations (5) (Fig. 4). Thevector function of the cyclical surface of the type P g 2(u,v) isP g 2(u,v) r 2 (v) c 02 (u) . M 2 (w 2 ),u ∈ 0, 2π. v ∈ 0, 2π. (9)Fig. 2 Creation the curves k 1 , k g 2, k g 3 h and their trihedronswhere r 2 (v) is the vector function of the curve k g 2 determinedby (6).Fig. 3 Combinations of the curves k 1 , k t 2, k t 3 t , k 1 , k n 2, k n 3 n and k 1 , k b 2, k b 3 b .3. Vector functions of cyclical surfaces of the typeP 1 (u,v), P g 2(u,v), P g 3 h (u,v)The cyclical surface of the type P 1 (u,v) is created by translationof the circle c 1 (R 1 , r 1 ) in the plane (n 1 , b 1 ) along the curvek 1 at an angular velocity w 1 v. We will create it so that the circlec 01 determined by the vector function c 01 (u) (0, r 1 cos u, r 1 sin u,1) will be transformed into the circle c 1 in the coordinate system(R 1 , t 1 , n 1 , b 1 ) using the matrix M 1 (w 1 ) expressed by equations (5)(Fig. 4).isThe vector function of the cyclical surface of the type P 1 (u,v)P 1 (u,v) r 1 (v) c 01 (u) . M 1 (w 1 ),u ∈ 0, 2π. v ∈ 0, 2π. (8)where r 1 (v) is the vector function of the curve k 1 determined byequation (1). This surface is surface of torus.The three-revolution cyclical surface of the type P g 3 h (u,v) iscreated by translation of the circle c 3 (R 3 , r 3 ) along the curvek 3 at an angular velocity w 3 m 2 m 1 v, where the circle is alwaysin the plane (n 3 , b 3 ) if the index h t, or in the plane (t 3 , b 3 ) ifh n, or in the plane (t 3 , n 3 ) if h b and its centre is the pointR 3 ∈ k g 3 h . We will create it so that the circle c 03 determined the byvector function c 03 (u) (0, r 3 cos u, r 3 sin u, 1) if h t, orc 03 (u) (r 3 cos u, 0, r 3 sin u, 1) if h n, or c 03 (u) (r 3 cos u, r 3sin u, 0, 1) if h b we will transform into the circle c 3 in the coordinatesystem (R 3 , t 3 , n 3 , b 3 ) using the matrix M 3 (w 3 ) by equations(5). The vector function of the cyclical surface of the typeP g 3 h (u,v) isP g 3 h (u,v) r 3 (v) c 03 (u) . M 3 (w 3 ),u ∈ 0, 2π. v ∈ 0, 2π. (10)where r 3 (v) is the vector function of the curve k g 3 h determined by(7).74 ● COMMUNICATIONS 3/2008