Machinery's Handbook, 31st Edition
Friction Brakes
2535
Formulas for Simple and Differential Band Brakes F = force in pounds at end of brake handle; P = tangential force in pounds at rim of brake wheel; e = base of natural logarithm = 2.71828; μ = coefficient of friction between the brake band and the brake wheel; θ = angle of contact of the brake band with the brake wheel, expressed in radians (1 radian = 57.296 degrees). T P e T P e e 1 1 1 1 2 = − = − µθ µθ µθ Simple Band Brake P P
For clockwise rotation:
F
F a bT
a Pb a Pb
e
µθ
2 = =
c
m
1
e
µθ
x
−
y
For counter clockwise rotation: F a bT 1 = =
1
a
a
k
1
e
b
−
µθ
Fig. 1.
For clockwise rotation:
F a bT
1
a Pb a Pb
1 = =
a
k
1
e
µθ
−
For counter clockwise rotation: F a bT 2 = =
a
e
b
µθ
c
m
1
e
µθ
F
−
Fig. 2.
Differential Band Brake
For clockwise rotation: F =
a b T b T 2 2 1 1 −
e b e b 1 2 − − µθ µθ e b b e 1 2 1 −
a P a P
F
1
c
m
=
For counter clockwise rotation: F a b T b T 2 1 1 2 = −
µθ
a
c
m
=
− µθ In this case, if b 2 is equal to, or less than, b 1 e μθ , the force F will be 0 or negative and the band brake works automatically.
b 1
b
2 Fig. 3.
For clockwise rotation: F =
a b T b T 2 2 1 1 +
e b e b 1 2 − + µθ µθ e b e b 1 1 − + µθ µθ
a P
1
c
m
=
F
For counter clockwise rotation: F a = +
b T b T 1 2 2 1
a P
2
a
c
m
=
If b 2 = b 1 , both of the above formulas reduce to F a Pb = m . In this case, the same force F is required for rotation in either direction. e e 1 1 1 − + µθ µθ c
b 1
b 2
Fig. 4.
Example: In a band brake of the type in Fig. 1, dimension a = 24 inches, and b = 4 inches; force P = 100 pounds; coefficient μ = 0.2, and angle of contact = 240 degrees, or . 418 # θ π = =
180 240
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