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CLUTCHES AND BRAKES 8.27
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CLUTCHES AND BRAKES
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FIGURE 8.16 Equivalent force system on a long internal shoe.
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Torque Capacity. With a model for contact pressure variation in hand, the friction torque exerted by the shoe on the drum can be found by a simple integration: T=
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where b = width of lining. Substituting for p from Eq. (8.23) and integrating, we get T= pmax fbr 2 (cos 1 cos 2) (sin )max (8.25)
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Normal Force. To determine the actuating force and the pin reaction, it is necessary first to find the normal force P. The components of P are Px = Py =
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pbr d sin pbr d cos
(8.26)
(8.27)
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CLUTCHES AND BRAKES 8.28
MACHINE ELEMENTS THAT ABSORB AND STORE ENERGY
Again, substituting for p from Eq. (8.23) and integrating, we find Px = and Py = brpmax (cos2 1 cos2 2 ) 2(sin )max (8.29) brpmax ( 2 1 + sin 1 cos 1 sin 2 cos 2 ) 2(sin )max (8.28)
The resultant normal force P has the magnitude
2 P = (P x + P 2 )1/2 y
(8.30)
and is located at the angle p, where p = tan 1 Px Py (8.31)
Effective Friction Radius. The location rf of the center of pressure C is found by equating the moment of a concentrated frictional force fP to the torque capacity T: T = fPrf or rf = T fP (8.32)
Brake-Shoe Moments. The last basic task is to find a relation among actuating force F, normal force P, and the equivalent friction force fP. The moments about the pivot point A are Ma Mn + Mf = 0 where Ma = F Mn = Pa sin p Mf = P(rf a cos p) (8.34) (8.35) (8.36) (8.33)
Self-energizing Shoes. The brake shoe in Fig. 8.16 is said to be self-energizing, for the frictional force fP exerts a clockwise moment about point A, thus assisting the actuating force. On vehicle brakes, this would also be called a leading shoe. Suppose a second shoe, a trailing shoe, were placed to the left of the one shown in Fig. 8.16. For this shoe, the frictional force would exert a counterclockwise moment and oppose the action of the actuating force. Equation (8.33) can be written in a form general enough to apply to both shoes and to external shoes as well: Ma Mn Mf = 0 (8.37)
Burr [8.2], p. 84, proposes this simple rule for using Eq. (8.37): If to seat a shoe more firmly against the drum it would have to be rotated in the same sense as the
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CLUTCHES AND BRAKES 8.29
CLUTCHES AND BRAKES
drum s rotation, use the positive sign for the Mf term. Otherwise, use the negative sign. Pin Reaction. At this point in the analysis, the designer should sketch a free-body diagram of each shoe, showing the components of the actuating force F, the normal force P, and the friction force fP. Then the components of the pin reaction can be found by setting to zero the sum of the force components in each direction (x and y). Design. The design challenge is to produce a brake with a required torque capacity T. A scale layout will suggest tentative values for the dimensions 1, 2, a, , and . From the lining manufacturer we learn the upper limit on maximum contact pressure pmax and the expected range for values of the frictional coefficient f. The designer must then determine values for lining width b and the actuating force F for each shoe. Since the friction force assists in seating the shoe for a self-energizing shoe but opposes the actuating moment for a self-deenergizing shoe, a much larger actuating force would be needed to provide as large a contact pressure for a trailing shoe as for a leading shoe. Or if, as is often the case, the same actuating force is used for each shoe, a smaller contact pressure and a smaller torque capacity are achieved for the trailing shoe. The lining manufacturer will usually specify a likely range of values for the coefficient of friction. It is wise to use a low value in estimating the torque capacity of the shoe. In checking the design, make sure that a self-energizing shoe is not, in fact, selflocking. For a self-locking shoe, the required actuating force is zero or negative. That is, the lightest touch would cause the brake to seize. A brake is self-locking when M n Mf (8.38)
As a design rule, make sure that self-locking could occur only if the coefficient of friction were 25 to 50 percent higher than the maximum value cited by the lining manufacturer. Example 6. Figure 8.17 shows a preliminary layout of an automotive brake with one leading shoe and one trailing shoe. The contact pressure on the lining shall not exceed 1000 kilopascals (kPa). The lining manufacturer lists the coefficient of friction as 0.34 0.02. The brake must be able to provide a braking torque of 550 N m. Two basic design decisions have already been made: The same actuating force is used on each shoe, and the lining width is the same for each. Check dimension a to make sure that self-locking will not occur. Determine the lining width b, the actuating force F, and the maximum contact pressure pmax for each shoe. Solution. One way to proceed is to express the braking torque T, the normal moment Mn, and the frictional moment Mf in terms of lining width b and maximum contact pressure pmax. Then the design can be completed by equating the actuating force for the two shoes, setting the sum of the braking torques to 550 N m, and selecting the lining width b so that the maximum contact pressure is within bounds. 1. The dimension a is a = (832 + 252)1/2 = 86.7 mm = 0.0867 m
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