Machinery's Handbook, 31st Edition
2362
Polygon Shafts Strength of Polygon Connections: In the formulas that follow, H w = hub width, inches H t = hub wall thickness, inches M b = bending moment, lb-inch M t = torque, lb-inch Z = section modulus, bending, in 3 = 0.098 D M 4 / D A for three sides = 0.15 D I 3 for four sides Z P = polar section modulus, torsion, in 3 = 0.196 D M 4 / D A for three sides = 0.196 D I 3 for four sides D A and D M . See table footnotes. S b = bending stress, allowable, lb/in 2 S s = shearing stress, allowable, lb/in 2 S t = tensile stress, allowable, lb/in 2 For shafts,
M t (maximum) = S s Z p M b (maximum) = S b Z
For bores,
H K S H M minimum t t w t = ^ h in which K = 1.44 for three sides except that if D M is greater than 1.375 inches, then K = 1.2; K = 0.7 for four sides. Failure may occur in the hub of a polygon connection if the hoop stresses in the hub exceed the allowable tensile stress for the material used. The radial force tending to expand the rim and cause tensile stresses is calculated from , . tan D n B M 113 2 Radial Force lb max I t = + ^ h This radial force acting at n points may be used to calculate the tensile stress in the hub wall using formulas from strength of materials. Manufacturing: Polygon shaft profiles may be produced using conventional machining processes such as hobbing, shaping, contour milling, copy turning, and numerically con trolled milling and grinding. Bores are produced using broaches, spark erosion, gear shapers with generating cutters of appropriate form, and, in some instances, internal grinders of special design. Regardless of the production methods used, points on both of the mating profiles may be calculated from the following equations: X D I 2 --- e + α cos e n ( α) cos α cos – ne n ( α) sin α sin – =
D I 2 --- e +
α sin e n ( α) cos α sin – ne n ( α) sin +
α cos
=
Y
In these equations, α is the angle of rotation of the workpiece from any selected reference position; n is the number of polygon sides, either 3 or 4; D I is the diameter of the inscribed circle shown on the diagram in the table; and e is the dimension shown on the diagram in the table and may be used as a setting on special polygon grinding machines. The value of e determines the shape of the profile. A value of 0, for example, results in a circular shaft having a diameter of D I . The values of e in the table were selected arbitrarily to provide suitable proportions for the sizes shown.
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