chap85.doc
TRANSCRIPT
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Belt Friction
Let's look at a flat belt passing over a drum
Let's take a look at a differential element
Motion is assumed to be impending.dN dF
s µ =∴
The normal force is a differential force because it acts on a differential element of area.
02
cos)(2
cos
0
=
++−
−
=Σ
θ µ
θ d dT T dN
d T
F
s
x
02
sin)(2
sin
0
=+
+−
−
=Σ
dN d
dT T d
T
F y
θ θ
For small angles
cos(d θ ) = 1sin(d θ ) = d θ
0=++−−⇒Σ dT T dN T F s x µ
dN dT s
µ =
T1
P P'
β
θ
d θ
P 1 P 2
O
dN
d θ
µs
dN
T+dTT
d θ/2 d θ/2x
y
r
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02
)(2
=+
+−
−⇒Σ dN
d dT T
d T F y
θ θ
0222
=+
−
−
− dN
d dT
d T
d T
θ θ θ
02 =
θ d
dT (2 small numbers squared = 0)
Thus: θ d T dN =
Substituting:
θ µ
θ µ
d T
dT
d T dT
s
s
=
=
Integrating:
θ µ β
d T
dT
Os
T
T ∫ ∫ =2
1
BT T S
µ =− 12 lnln
Thus:
BT
T s µ =
1
2ln
OR
B seT
T µ =1
2
These formulas apply to:
1). Flat belts passing over fixed drums.
2). Ropes wrapped around a post or capstan.3). band brakes- the drum is about to rotate while the band remains fixed.
4). Problems involving belt drives. Both the pulley and belt rotate; determine whether the
belt will slip/move to the pulley.
- Should only be used if the belt, rope, or brake is about to slip.
- In the equations 2T will always be larger than 1
T . So 2T represents the tension which pulls
and 1T is the resisting tension.- β must be in radius. B may be larger than π 2 . If a rope is wrapped around a post n times,
nπ β 2= .
- If the belt, rope is actually slipping, k µ should be used.
-If no slipping occurs and is not about to slip, none of these formulas should be used.
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V-belts
Proceeding as before:
= )2/(sin
1
2 α
β µ s
eT
T
α
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Examples
1). Given: A hawser thrown from a ship to a pier is wrapped 2 full turns around a capstan. Thetension in the hawser is 7500 N; by exerting a force of 150 N on its free end, a dock worker can
just keep the hawser from slipping.
Find: a). Determine the coefficient of friction between the hawser and capstan.
b). Determine the tension in the hawser that could be resisted by 150 N force if thehawser were wrapped 3 full turns around the capstan.
a) Slipping is impending
311.0
turn
rad2turns2
150
7500ln
ln1
2
=
=
=
S
S
S BT
T
µ
π µ
µ
b)
kN7.52
150
ln
2
)2)(3)(311(.2
1
2
=
=
=
T
eT
BT
T
S
π
µ
150 N
7500 N
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2). Given: A flat belt is used to transmit the 30 ft lb torque developed by an electric motor. The
drum in contact with the belt has a diameter of 8 in and .30.0=S
µ
Find: Determine the minimum allowable value of the tension in each part of the belt if the belt is
not to slip.
lbs6.158
lbs6.68
90311.2
90
90
in12
ft1
2
in8)(30
)(
=
=
+=
+=
=−
−=
−=
Α
ΒΑ
ΒΑ
ΒΑ
T
T
T T
T T
T T
T T
r T T M
B
B B
B A
3). Given: Example problem #2, using a V-belt with 36=α
60o
40 o
A
B
60 o
TA
40 o
TB
20 o
30 o
70 o
β
AT T =
2
BT T =
1
160=β
( )
ΒΑ
Β
Α
=
=
=
T T
eT
T
eT
T s
311.2
9/83.0
1
2
π
β µ
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Find: Determine minimum allowable tension
ΒΑ
Β
Α
=
=
=
T T
eT
T
eT
T s
045.15
18sin/)9/8(3.0
)2/sin(/
1
2
π
α β µ
90
90
12
4)(30
)(
+==−
−=
−=
ΒΑ
ΒΑ
ΒΑ
T T
T T
T T
r T T M
B A