A new algorithm for the shape design of turning tool
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The cutting tool design algorithm that is used for many years is based on the principle of parametric equations. A turning point needs multiple calculations. The principle is tedious, difficult to grasp, and the calculation error is large. This article describes a new algorithm. 1 Necessity of shaping the turning tool design The contour of the workpiece is represented in its axial section. The profile of a round turning tool is also represented in its axial section, while the profile of a prismatic turning tool (including a flat forming turning tool) should be represented in its normal section. The following first discusses whether the shape of the forming lathe is exactly the same and only the direction of the asperities is reversed. As shown in Fig. 1, let r0 be the minimum radius on the workpiece and ri be the radius of any turning point on the workpiece. Then, the contour depth of the workpiece at the turning point is AB=ri-r0, and point B is processed by point C on the turning tool. C for AB parallel line turning tool turning flank at point E, because the projection of point E to AB line is between AB, and the projection of point C to AB line is also between AB, so there is CE ≤ AB (1 ) Only when the forming tool nose angle gf is 0°, the point C coincides with the point B, the point E coincides with the point A, and CE=AB, but in general (gf>0°), CE>AB. Set the depth of the tool profile at this point to Ti. From Fig. 1, it can be seen that Ti=CEcosaf (2) forms the turning angle af>0° of the turning tool, so Ti<CE (3) comprehensive formulae (1), (3) can be obtained Ti< AB means that in any case, the tool depth is not greater than the depth of the workpiece. Therefore, it is necessary to design the contour of the shaping tool according to the workpiece profile and the shape of the turning tool, such as the front and rear corners. 2 New algorithm for the design of the turning tool's truncated design When designing the turning tool, the straight part of the workpiece profile is calculated only for the turning point, and then the corresponding point on the tool can be connected with a straight line to form the blade shape. For the arc section of the workpiece profile, take the apex of the arc (the lowest point of the concave arc or the highest point of the convex arc) and the two ends as the design points, determine the corresponding three points on the tool, and then set the circle according to the three points. In principle, the corresponding three points on the tool make a circular arc edge; for left and right asymmetric arcs, the left and right endpoints and the midpoint can be calculated. The forming tool profile (truncated) representation is related to the tool body. The prism forming turning tool is characterized by the depth Ti (i=1, 2, 3,..., N; T0=0) of the relatively highest point of each turning point on the tool. The round body forming turning tool is characterized by the radius of each turning point Ri (i=1,2,3,...,n) on the tool, the maximum radius is represented by R, and the radius R is selected according to the depth of the workpiece before calculation. The new algorithm uses the trigonometric principle to determine the relationship between Ti and ri for prismatic turning tools or Ri and ri for circular turning tools. Before the calculation, regardless of the type of body, the following preparatory work shall be done first: Known conditions: the minimum radius of the workpiece r0, the radius of the rest of the turning point ri (ri>ri); the front angle gf of the shaping tool, the rear angle af; The maximum radius of the knife R. Calculate the fixed parameter: the distance from the centerline of the workpiece to the plane where the rake face of the turning tool is located is h=r0singf The distance from the tool nose to the workpiece axis observed on the rake face is a=g0cosgf. Calculate any turning point ri on the workpiece. The depth profile of the tool observed on the rake face is bi=(ri2-h2)1⁄2-a (4) The following analysis is performed on the prismatic turning tool and the circular turning tool. Prism turning tool in △ACE, ∠CAE=90°-gf-af, ∠AEC=90°+af, according to sine theorem CE = bi sin(90°-gf-af) sin(90°+af) Simplify to CE= cos(gf+af) bi cosaf Substitution (2) into equation (2) yields Ti= cos(gf+af) bicosaf=bicos(gf+af) cosaf Substituting equation (4) into the above equation, ie The calculation formula for the cutting depth of prismatic turning tool is Ti=[(ri2-h2)1⁄2-a]cos(gf+af). (5) The circular turning tool is shown in Fig. 2. In △ACO1, according to the cosine theorem Bi2+R2-Ri2 = cos(gf+af) 2Rbi From which Ri=[R2+bi2-2Rbicos(gf+af)]1⁄2 can be solved (6)

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The equation (4) and the formula (6) are immediately related to the relation between the radius of any point of the circular turning tool Ri and the corresponding turning point ri on the workpiece, where bi can be regarded as an intermediate variable. Substituting equation (4) into equation (6) yields a functional formula between Ri and ri, but this function is too complex, so general calculations are still appropriate using equations (4), (6), ie Bi=(ri2+h2-a)1⁄2 Ri=[R2+bi2-2Rbicos(gf+af)]1⁄2 (7) 3 Design Example Workpiece As shown in Figure 3, try the new algorithm to find the depth of each point of the prism. Ti, round turning tool radius Ri. Known conditions: gf=16°, af=12°, maximum radius of round turning tool R=20mm, free tolerance on the workpiece is calculated according to IT12. Solution: The IT12 tolerance of the basic size 10 is 0.15mm The IT12 tolerance of the basic size 14 is 0.18mm The radius of the turning point on the workpiece is determined: r1=(6+0.05/2)/2=3.0125mm r0=r1-1=2.0125 Mm r2=r1=3.0125mm r4=r3=(10-0.15/2)/2=4.9625mm r6=r5=(14-0.18/2)/2=6.955mm Calculating the fixed parameter: h=h0singf=2.0125×sin16 °=0.55472mm h2=0.554722=0.30771mm2a=r0cosgf=2.0125×cos16°=1.93454mm cos(gf+af)=cos(16°+12°)=0.88295 Calculate the profile depth of each point of the prismatic tool. A, cos (gf + af) into (5) to get the formula is Ti = [(ri2-0.30771) 1⁄2-0.193454] × 0.88295 Prism turning tool points profile depth T2 = T1 = [(3.01252-0.30771) 1⁄2-0.193454]×0.88295=0.906mm T4=T3=[(4.96252-0.30771)1⁄2-0.193454]×0.88295=2.646mm T6=T5=[(6.9552-0.30771)1⁄2-0.193454]×0.88295=2.646mm Calculation of a circle The radius of the turning point Ri will be substituted into equation (7) for h2, a, R, cos(gf+af) to be calculated as bi=(ri2-0.30771)1⁄2-0.193454
Ri=(400+bi2-35.31790bi)1⁄2 Each intermediate variable bi is b2=b1=(3.01252-0.30771)1⁄2-0.193454=0.1265mm b4=b3=(4.96252-0.30771)1⁄2-0.193454=2.9969mm b6=b5= (6.9552-0.30771)1⁄2-0.193454=4.9983mm The radius of each turning tool is R2=R1=(400+1.06252-35.3179×1.0265)1⁄2=19.100mm R4=R3=(400+2.99692-35.3179×2.9969)1⁄2 =17.411mm R6=R5=(400+4.99832-35.3179*4.9983)1⁄2=15.762mm 4 Conclusions Using the traditional algorithm, an edge turning tool on a turning point requires 4 operations, and a round turning tool needs 7 operations. The algorithm requires only one and two operations, respectively, and its workload is 25% to 29% of the traditional algorithm. It should be pointed out that this new algorithm has a clear concept, is simple, easy to grasp, and has high calculation accuracy. It does not need to calculate trigonometric functions in the calculation process. It is of practical value.

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