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WORM GEARING 12.5
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WORM GEARING
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12.3 VELOCITY AND FRICTION
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Figure 12.4 shows the pitch line velocities of worm and gear. The coefficient of friction between the teeth is dependent on the sliding velocity. Representative values of are charted in Fig. 12.5. The friction has importance in computing the gear set efficiency, as will be shown.
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12.4 FORCE ANALYSIS
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If friction is neglected, then the only force exerted by the gear on the worm will be W, perpendicular to the mating tooth surface, shown in Fig. 12.6, and having the three components Wx, Wy, and Wz. From the geometry of the figure, Wx = W cos n sin Wy = W sin n W = W cos n cos
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(12.10)
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In what follows, the subscripts W and G refer to forces on the worm and the gear. The component Wy is the separating, or radial, force for both worm and gear (opposite in direction for the gear). The tangential force is Wx on the worm and Wz on the gear.The axial force is Wz on the worm and Wx on the gear.The gear forces are opposite to the worm forces: WWt = WGa = Wx WWr = WGr = Wy WWa = WGt = Wz (12.11)
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FIGURE 12.4 Velocity components in a worm-gear set. The sliding velocity is VS = VW 2 2 . (V W + V G )1/2 = cos
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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WORM GEARING 12.6
GEARING
FIGURE 12.5 Approximate coefficients of sliding friction between the worm and gear teeth as a function of the sliding velocity. All values are based on adequate lubrication. The lower curve represents the limit for the very best materials, such as a hardened worm meshing with a bronze gear. Use the upper curve if moderate friction is expected.
FIGURE 12.6 Forces exerted on worm.
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WORM GEARING 12.7
WORM GEARING
where the subscripts are t for the tangential direction, r for the radial direction, and a for the axial direction. It is worth noting in the above equations that the gear axis is parallel to the x axis and the worm axis is parallel to the z axis. The coordinate system is right-handed. The force W, which is normal to the profile of the mating teeth, produces a frictional force Wf = W, shown in Fig. 12.6, along with its components W cos in the negative x direction and W sin in the positive z direction. Adding these to the force components developed in Eqs. (12.10) yields Wx = W(cos n sin + cos ) Wy = W sin n Wz = W(cos n cos sin ) Equations (12.11) still apply. Substituting Wz from Eq. (12.12) into the third of Eqs. (12.11) and multiplying by , we find the frictional force to be Wf = W = WGt sin cos n cos (12.13) (12.12)
A relation between the two tangential forces is obtained from the first and third of Eqs. (12.11) with appropriate substitutions from Eqs. (12.12): WWt = WGt The efficiency can be defined as = WWt (without friction) WWt (with friction) (12.15) cos n sin + cos sin cos n cos (12.14)
Since the numerator of this equation is the same as Eq. (12.14) with = 0, we have = cos n tan cos n + cot (12.16)
Table 12.1 shows how varies with , based on a typical value of friction = 0.05 and the pressure angles usually used for the ranges of indicated. It is clear that small should be avoided. Example 1. A 2-tooth right-hand worm transmits 1 horsepower (hp) at 1200 revolutions per minute (r/min) to a 30-tooth gear. The gear has a transverse diametral pitch of 6 teeth per inch. The worm has a pitch diameter of 2 inches (in). The normal pressure angle is 141 2 . The materials and workmanship correspond to the lower of the curves in Fig. 12.5. Required are the axial pitch, center distance, lead, lead angle, and tooth forces. Solution. The axial pitch is the same as the transverse circular pitch of the gear. Thus px = = = 0.5236 in P 6
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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