A leather belt 9 mm × 250 mm is used to drive a cast iron pulley 900 mm in diameter at 336 r.p.m.
If the active arc on the smaller pulley is 120° and the stress in tight side is 2 MPa, find the power
capacity of the belt. The density of leather may be taken as 980 kg/m3, and the coefficient of friction of
leather on cast iron is 0.35.
Answers
Answer:
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Solution. Given: t = 9 mm = 0.009 m ; b = 250 mm = 0.25 m; d = 900 mm = 0.9 m ;
N = 336 r.p.m ; θ = 120° = 120 × 180
π
= 2.1 rad ; σ = 2 MPa = 2 N/mm2 ; ρ = 980 kg/m3 ; μ = 0.35
We know that the velocity of the belt,
v = . 0.9 336 15.8 m/s
60 60
π π× × d N = =
and cross-sectional area of the belt,
a = b.t = 9 × 250 = 2250 mm2
∴ Maximum or total tension in the tight side of the belt,
T = Tt1 = σ.a = 2 × 2250 = 4500 N
We know that mass of the belt per metre length,
m = Area × length × density = b.t.l.ρ = 0.25 × 0.009 × 1 × 980 kg/m
= 2.2 kg/m
∴ Centrifugal tension,
*TC = m.v2 = 2.2 (15.8)2 = 550 N
and tension in the tight side of the belt,
T1 = T – TC = 4500 – 550 = 3950 N
Let T2 = Tension in the slack side of the belt.
We know that
1
2
2.3 log T
T
⎛ ⎞ ⎜ ⎟ ⎝ ⎠
= μ.θ = 0.35 × 2.1 = 0.735
1
2
log T
T
⎛ ⎞ ⎜ ⎟ ⎝ ⎠
= 0.735 0.3196
2.3 = or 1
2
2.085 T
T = ... (Taking antilog of 0.3196)
T2 = 1 3950 1895 N
2.085 2.085
T = =
We know that the power capacity of the belt,
P = (T1 – T2) v = (3950 – 1895) 15.8 = 32 470 W = 32.47 kW Ans.
Notes :The power capacity of the belt, when centrifugal tension is taken into account, may also be obtained as
discussed below :
1. We know that the maximum tension in the tight side of the belt,
Tt1 = T = 4500 N
Centrifugal tension, TC = 550 N
and tension in the slack side of the belt,
T2 = 1895 N
∴ Total tension in the slack side of the belt,
Tt2 = T2 + TC = 1895 + 550 = 2445 N
We know that the power capacity of the belt,
P = (Tt1 – Tt2) v = (4500 – 2445) 15.8 = 32 470 W = 32.47 kW Ans.
2. The value of total tension in the slack side of the belt (Tt2) may also be obtained by using the relation as
discussed in Art. 18.20, i.e.
1 C
2 C
– 2.3 log –
t
t
T T
T T
⎛ ⎞ ⎜ ⎟ ⎝ ⎠
= μ.θ