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Unformatted text preview: Homework #8 ME 576 1. 1 kW=1.34 HP
2
∴ 2 HP=
= 1.492 kW
1.3
(a). Back induced e.m.f. during idling ( ⎛ 1 min ⎞
rad
Vi = kb⋅n = 1 (V ⋅ s )(600 rpm) ⎜
⎟ 2π
sec
⎝ 60 sec ⎠
= 20π ⋅ (V ) ) Power consumption during idling
P = VΑ ⋅ J Α
= ( RA ⋅ J A + Vi ) I A = 1, 492 RA I A2 + (20π ) I A 1, 492 = 0
∴ IA = (10π ) −10π ± 2 + 10 × 1492 10 = 9.47(Amp). For speed control, IA is maintained at a constant level and so is Tmax. ( Tmax = KT ⋅ I A = 1 kg ⋅ m A ) ⋅ 9.47( A) = 9.47kg ⋅ m = 92.8 N − m.
Maximum acceleration is attained when the motor starts running 0 TM = J M ωmax = 0 d ω (t )
+ Bω (t ) + TL
dt Tmax
Jm ) ( m2
9.47( kg ⋅ m ) 9.47 × 9.8 kgm ⋅ sec 2
=
= 0.1( kg ⋅ m 2 )
0.1 ( kgm ⋅ m 2 ) ( = 928 rad sec 2 ) = 147.7 ( rev sec )
2 (b). During idling, the power generated by the motor is consumed to overcome this dissipative energy due to damping P = Tm ⋅ ω
0 0 ⎛ dw (t )
⎞ = ⎜ Jm
+ Bω ( t ) + TL ⎟ ω (t )
dt
⎝
⎠
1, 492(W )
P
.
B=
=
= 0.38 N ⋅ m
2
2
rad / sec
ω (t )
( 20π ) (c). VA = RA I A + Vi
= RA I A + K bω (t )
∴ ω (t ) = VA  RA I A
Kb 250 − 10 × 9.47
= 155.3 rad
= 1, 483 rpm sec
1
Pmax = VA max ⋅ I A max ωmax = = 250 × 9.47 = 2,368W
= 3.17 HP 2. (a) VA = RA ⋅ I A + Vi When motor starts rotation, Vi = 0 ∴ RA = 200
= 10(ohms ) 20 at ω (t) = 600 rpm 50(V )
min
×1
600( rpm) × 2π rad
rev
60
sec = 0.796 V ⋅ sec. ∴ kb = ( ) ( ) For constant torque up to 1000 rpm, IA must be maintained at [email protected] V A = RA ⋅ I A + k b ⋅ ω ( t ) = 10 × 20 + 0.796 ⋅ 1000 x 2π 60 = 283V (b) w(t)α
1000 : 1 φ
1 φ1000 1000 = = 1500 : 1 φ1500 φ1500 φ1000
φ1500 1000
=
= 0.667
φ1000 1500 1500
∴ 33.3% (c) 0
d ω (t )
+ Bw (t ) + TL
dt
Tmax = K T ⋅ I A max Tm = J m = ⋅1 × 20 = 20 N ⋅ m. ∴ ωmax =
= Tm  TL
Jm
20 − 5( N ⋅ m )
10 ( kg ⋅ m 2 ) = 1.5 rad sec 2 3. R r = 10 Ω L r =0.06 H Vf =220V n=1, p=3 ( ωp =60 × 2π rad sec = 376 rad sec
ωf = ) ωp = 60 × 2π rad
sec
n
2
p ⋅ nV sR r
Tm =
ωp ( R r 2 + s2ωp 2 L r 2 ) = 3 × 1 × (220)2 ⋅ (10) ⋅ s
(60 × 2π ) ⎡102 + s2 ⋅ (60 × 2π ) 2 × (0.06) 2 ⎤
⎣
⎦ = 38.5s
1, 452,000 ⋅ s
=
2
37, 699 + 192, 884 s 1 + 511 s2 s=1 ωm
ω
⋅ n = 1− m
ωp
ωp or ωm =(1s)ωp ...
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This document was uploaded on 12/28/2011.
 Fall '09

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