Machinery's Handbook, 31st Edition
120
COMPOUND ANGLES Compound Angles
Referring to the Formulas for Compound Angles on page 121, in Fig. 1 is shown what might be considered as a thread-cutting tool without front clearance. A is a known angle in plane y - y of the top surface. C is the corresponding angle in plane x - x that is at some given angle B with plane y - y . Thus, angles A and B are components of the compound angle C . Example Referring to Fig. 1 : Angle 2 A in plane y - y is known, as is angle B between planes x - x and y - y . It is required to find compound angle 2 C in plane x - x . Solution: Let 2 A = 60 and B = 15; then
tan C = tan A × cos B = tan 30° × cos 15° tan C = 0.57735 × 0.96592 = 0.55767 C = 29° 8.8 ′ 2 C = 58° 17.6 ′
Fig. 2 shows a thread-cutting tool with front clearance angle B . Angle A is one-half the angle between the cutting edges in plane y - y of the top surface, and compound angle C is one-half the angle between the cutting edges in a plane x - x at right angles to the inclined front edge of the tool. The angle between planes y - y and x - x is, therefore, equal to clearance angle B . Example Referring to Fig. 2 : Find the angle 2 C between the front faces of a thread- cutting tool having a known clearance angle B that will permit the grinding of these faces so their top edges form the desired angle 2 A for cutting the thread. Solution: Let 2 A = 60 and B = 15; then
30 ° tan 15 ° cos ---------
A tan B cos ------
0.57735 0.96592 ----------
C tan
=
=
=
tan C = 0.59772 C = 30° 52 ′
2 C = 61° 44 ′ In Fig. 3 is shown a form-cutting tool in which the angle A is one-half the angle between the cutting edges in plane y - y of the top surface; B is the front clearance angle; and C is one-half the angle between the cutting edges in plane x - x at right angles to the front edges of the tool. The formula for finding angle C when angles A and B are known is the same as that for Fig. 2. Example Referring to Fig. 3 : Find the angle 2 C between the front faces of a form-cutting tool having a known clearance angle B that will permit the grinding of these faces so their top edges form the desired angle 2 A for form cutting. Solution: Let 2 A = 46 and B = 12; then
23 ° tan 12 ° cos ---------
A tan B cos ------
0.42447 0.97815 ----------
C tan
=
=
=
tan C = 0.43395 C = 23° 27.5 ′
2 C = 46° 55 ′ In Fig. 4 is shown a wedge-shaped block, the top surface of which is inclined at com pound angle C with the base in a plane at right angles with the base and at angle R with the front edge. Angle A, in the vertical plane of the front of the plate, and angle B , in the vertical plane of one side that is at right angles to the front, are components of angle C .
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