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BELT DRIVES 14.9
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BELT DRIVES
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Because (m 1) Fu = F 2 m = exp w 180 (14.19)
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w becomes greater, until the belt slips on the pulley with the smaller arc of contact when w = . When Ff = F2, there are no usable forces; that is, F 2 = F 1 = Fu = 0. In this case, no torque can be transmitted. If belt velocity v is increased further, the belt runs off the pulley. The maximum force in the belt sides is given by Fmax = F1 = F 2 + Fu + Ff (14.20)
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With only the centrifugal forces acting, the belt is in equilibrium. They do not act on the pulleys at all. Hence, the shaft load Fw of a belt drive results from only the usable forces F 1 and F 2 in the belt sides (Fig. 14.3): Fw = F 2 1 F22 2F1F cos 2 (14.21)
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FIGURE 14.3 Equilibrium of forces.
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The force rating = Fu = (m 1) Fw
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2m cos
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defines the minimum shaft tensioning force required for peripheral force production as a function of the friction coefficient and the arc of contact . The rated output K = Fu /F 1 = 1 1/m defines the peripheral force Fu which can be produced by the permissible force F 1 as a function of the friction coefficient and the arc of contact . The reduction in rated output with decreasing arc of contact is defined by an angular factor c , based on = 180 , that is, a speed ratio of i = 1. The tensions in a homogeneous belt result from the forces acting in the belt and the belt cross section, A = bs. For multiple-ply belts, these tensions can be used only as theoretical mean values.
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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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BELT DRIVES 14.10
POWER TRANSMISSION
Bending of the belt around the pulley produces the bending stress b. This stress can be calculated from the elongation of the belt fibers with respect to the neutral axis: l = (r + s) r + = s s = 2 2 (14.22) (14.23)
l s/2 s s = = l (r + s/2) 2r + s d s Eb d
b = Eb
The strain increases with decreasing pulley diameter d. For practical design purposes, b is not taken into consideration, since belt life depends much less on b than on the bending frequency. The maximum stress is in the tight side of the belt at the beginning and end of the arc of contact, i.e., the points where it passes onto or off the smaller pulley (Fig. 14.4): max = 1 + f + b = F1 s + v2 + Eb A d (14.24)
The safety stress depends on the bending frequency and the smallest pulley diameter as well as on the material and the construction of the belt as indicated by the manufacturer. With z = number of pulleys, the bending frequency is given by fb = vz l (14.25)
FIGURE 14.4 Stress distribution.
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.
BELT DRIVES 14.11
BELT DRIVES
The maximum power transmission capacity of a belt drive can be determined as follows: The power transmission capacity P = Fuv = n Av equals zero if the belt velocity v either equals zero or reaches a maximum at which the belt safety stress limit is approached by the centrifugal and bending stresses alone, so that 1 = 2 = n = 0 Then
2 zul = f + b = vmax + b
(14.26)
from which the maximum belt velocity can be calculated as follows: vmax = zul b (14.27)
Optimum power transmission is possible only at the optimum belt velocity vopt within the range of v = 0 and v = vmax. It depends on the belt safety stress and is given by vopt = vmax zul b = 3 3 (14.28)
In theory, this equation applies to all flexible connectors, under the assumption of zul (belt safety stress) [or Fzul (allowable load)] being independent of belt velocity. Since zul decreases with increasing belt velocity, though, the stress and power transmission capacity diagrams are as shown in Fig. 14.5.
14.1.4 Arrangement and Tensioning Devices Because of their good twistability, flexible connectors are suited for drives with pulleys in different planes and nonparallel shafts of equal or opposite directions of rotation. Since the outer fibers of a twisted flat belt or synchronous belt are strained more than the center fibers, stress is higher there, resulting in the reduction of the belt power transmission capacity. Figures 14.6 and 14.7 show several belt drives with pulleys in different planes. Note that for drives with crossed belts (Fig. 14.6), endless belts have to be used, in order to avoid damage. For half- or quarter-turn belt drives (Fig. 14.7), the side of delivery must lie in the plane of the mating pulley. By the use of step (cone) pulleys, different speed ratios may be obtained (Fig. 14.8). Pulley diameters have to be selected to ensure equal belt lengths on all steps. The belt rim running onto the larger diameter of a cone pulley (Fig. 14.9) has a higher velocity than the opposite rim. Thus, the following belt portion is skewed and then runs onto a larger diameter. The drive is balanced when the bending moment due to the bending deformation of the belt is compensated by the skew of the belt side running off (Fig. 14.10).
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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