Tool surface curvature analysis Indented coordinate system refers to the cutting coordinate system of the base bevel gear such as the milling cutter. For example, the origin Oi of the wheel blank coordinate system i:[Oi,ji,ki] is located at the top of the cone, and the big end of the theoretical tooth line is located at jiOiki. In the plane; the origin Oc of the tool coordinate system c:[Ocic,jc,kc] is located on the split-cone busbar, the å¹ºvector is perpendicular to the split-cone busbar in the tool axis direction; the spatial fixed coordinate system: [Oi,j,k] The origin O coincides with Oi, the tool axis is always in the jOk plane, j is parallel to jc; the distance between the tool center Oc and O is Rc, and the diameter of the Oc is Rc=Rc(sinii cosik)(7) when the tool is located When Rc is located, the angle between ji and i axis is e=c/sini, and the transformation matrix between each coordinate system is Mi0, M0c.
The tool surface equation and its tangential and normal vector tool axis are rotated around the axis to form the tool surface r(c): r(c)=(xc, yc, zc) the tool surface formed by the CD profile of the tool profile (c) For xc=rkcosk skrk-rv;yc=-rksink skrkcos;zc=rksink skrksin(8)The tool surface is a curved surface with k as the parameter, and its parameter curve direction is its main direction, corresponding to the slanting vector in the main direction. For g(c)i1,g(c)i2, in order to ensure that the large and small wheels are in the same direction at the meshing point, the first main direction g(c)11 of the surface of the small wheel cutter 1 takes the positive direction of k, the large wheel The first principal direction g(c)21 of the tool surface takes the negative direction of k, and the second principal direction g(c)i2 of the curved surface of the large and small wheel cutters takes the tangent vector and the å¹ºvector of the positive tool surface. g(c)i1=(-1)(i-1)r(c)kr(c)k;g(c)i2=r(c)r(c);n(c)i=g(c) I1g(c)i2, i=1, 2(9).
The tool surface curvature analysis sets the tool surface (c) The principal curvatures in the two main directions are K(c)i1 and K(c)i2 respectively. The tool axis truncation profile equation is calculated according to the curvature of the plane curve. The principal curvature K(c)i1K(c)i1=(-1)(i-1)B(1 T0) CRktank(1 T20)3/2i=1 of the plane of revolution (c) along the direction of g(c)i1, In Equation 2(10), C=sec2k tanksk(r2v r2k)rv(rv-rk)2; T0=tank-sk(rv rk)rv(rv-rk); B=T0tank by MouniÃ¨re theorem, tool rotation The principal curvature K(c)i2 of the surface (c) along the g(c)i2 direction is K(c)i2=(-1)iKcos0i=1, where 2(11) where K is over g(c)i2 vertical The curvature of the circle intersecting the contour of the tool axis and the contour K=1rksink skrk0 is the angle between the normal of the contour and the circular plane, 0=arcsinD2D21 D2 coefficient D1, D2 see  tool surface from Euler formula (c) The normal and short-range torsion angles in the angular direction with g(c)i1 are K(c)ni=K(c)i1cos2 K(c)i2sin2; (c)gi=0.5(K(c) i2-K(c)i1)sin2(12)
Conclusion According to the differential geometry theory, the surface equation of the finger milling cutter is determined, and the principal curvature of the tool and the calculation method of the tangent vector in the main direction are derived. According to the principle of equal-base bevel gear and the theory of conjugate surface, the characteristic parameters and tool of the tool surface are utilized. Compared with the relative motion of the wheel blank, the general calculation of the principal curvature of the equal-base bevel gear tooth surface, the tangential vector and the tooth surface å¹º normal vector in the main direction when the tooth profile and the tooth line shape are modified are derived. The foundation lays the foundation for the analysis of meshing characteristics, shape modification and strength calculation of equal-base bevel gears.
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