Machinery's Handbook, 31st Edition
2132 MEASURING SCREW THREADS If the thread is cut in a lathe to the standard angle as measured in the axial plane, Case 2 applies in determining the pin size W and the overall measurement M . In solving all problems under Case 1, angle A n used in Formulas (3) and (3a) equals one- half the included angle of the milling cutter. When Case 2 applies, angle A n for milled threads also equals one-half the included angle of the cutter, but the cutter angle is reduced and is determined as follows: tan tan cos A A B n # = The included angle of the cutter or the normal included angle of the thread groove = 2 A n . Examples 1 and 2, which follow, illustrate Cases 1 and 2. Example 1 (Case 1): Take, for example, an Acme screw thread that is milled with a cutter having an included angle of 29 degrees; consequently, the angle of the thread exceeds 29 degrees in the axial section. The outside or major diameter is 3 inches; the pitch, 1 ∕ 2 inch; the lead, 1 inch; the number of threads or “starts,” 2. Find pin size W and measurement M . Pitch diameter E = 2.75; T = 0.25; L = 1.0; A n = 14.50 ° tan A n = 0.258618; sin A n = 0.25038; and cos A n = 0.968148. . . . . . . . . . . . . . tan B B W M 31416 275 10 0115749 66025 0968148 0 25 0 993368 0 25651 2 75 0 25651 1 0 25038 3 0707 inch inches # # # ° = = = = = = + + = ^ h Note: This value of M is only 0.0001 inch larger than that obtained by using the very accurate involute helicoid Formula (4) discussed on the following page. Example 2 (Case 2): A triple-threaded worm has a pitch diameter of 2.481 inches, pitch of 1.5 inches, lead of 4.5 inches, lead angle of 30 degrees, and nominal thread angle of 60 degrees in the axial plane. Milling cutter angle is to be reduced. T = 0.75 inch; cos B = 0.866025; and tan A = 0.57735. Again use Formula (3) to see if it is applicable. tan A n = tan A × cos B = 0.57735 × 0.866025 = 0.5000; hence A n = 26.565 ° , making the included cutter angle 53.13 ° , thus cos A n = 0.89443 and sin A n = 0.44721. . . . . . . . . W M 089443 0 75 0 866025 0 72618 2 481 0 72618 1 0 44721 3 532 inch # # = + = ^ inches = = + h Note: If the value of measurement M is determined by using the following Formula (4) it will be found that M = 3.515 + inches; hence the error equals 3.532 − 3.515 = 0.017 inch approximately, which indicates that Formula (3) is not accurate enough here. The applica tion of this simpler Formula (3) will depend upon the lead angle and thread angle (as pre viously explained) and upon the class of work. Buckingham Exact Involute Helicoid Formula Applied to Screw Threads.— When extreme accuracy is required in finding measurement M for obtaining a given pitch diameter, the equations that follow, although somewhat cumbersome to apply, have the merit of pro - viding a direct and very accurate solution; consequently, they are preferable to the inde terminate equations and successive trial solutions heretofore employed when extreme pre cision is required. These equations are exact for involute helical gears and, consequently, give theoretically correct results when applied to a screw thread of the involute helicoidal form; they also give very close approximations for threads having intermediate profiles. Helical Gear Equation Applied to Screw Thread Measurement: In applying the helical gear equations to a screw thread, use either the axial or normal thread angle and the lead angle of the helix. To keep the solution on a practical basis, either thread angle A or A n , as the case may be, is assumed to equal the cutter angle of a milled thread. Actually, the
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