OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 Fig. 4 shows the result of developing the milling cutter generating surface with parameters d = 30 mm, j = 90° and l = 20° according to equation (3). As a result of specifying the angle l ≠ 0, the generating surface of the milling cutter is a unipolar hyperboloid of rotation, characterized by the variability of the values of the principal radii of the surface curvature along the rotation axis, where its minimum value is reached at points (0, q), for all q ∈ [0; 2p]. The equation of the surface machined with the translation tool feed movement along axis X0 of the part and a set value of the milling cutter rotation by the angle ξ (during oblique milling) is developed on the basis of the shaping equation { } { } { } 1 5 6 0( , , ) ( ) ( ) ( ) ( , ) f r x t A x A A r t q = x −q q (4) where x is the parameter of the milling cutter travel along axis X0; { } 1 ( ) A x is the matrix that specifies the travel of the milling cutter along axis X 0: { } 1 1 0 0 0 1 0 0 ( ) 0 0 1 0 0 0 0 1 x A x = ; { } 5 ( ) A x is the matrix of the milling cutter rotation along the direction of translation feed movement by angle value ξ: { } 5 cos 0 sin 0 0 1 0 0 ( ) sin 0 cos 0 0 0 0 1 A x x x = − x x ; { } 6 ( ) A −q is the matrix that specifies the rotation of the milling cutter: { } 6 cos ( ) sin ( ) 0 0 sin ( ) cos ( ) 0 0 ( ) 0 0 1 0 0 0 0 1 A −q −q −q −q −q = . To take into account the connection of the envelope of the form q = q(x,t), based on equation (4) we compose and solve the equation with respect to parameter q: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) 0 ( , , ) ( , , ) ( , , ) r x t r x t r x t i j k x x x r x t r x t r x t i j k t t t r x t r x t r x t i j k ∂ q ∂ q ∂ q ∂ ∂ ∂ ∂ q ∂ q ∂ q = ∂ ∂ ∂ ∂ q ∂ q ∂ q ∂q ∂q ∂q (5) which makes it possible to represent equation (4) as a bivariate function: { } { } { } ( ) ( ) 1 5 6 0( , ) ( ) ( ) ( , ) , ( , ) f r x t A x A A x t r t x t = x −q q . (6) Fig. 4. Milling cutter generating surface
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