hw1solnf2009 - MAE170 Homework 1 Solution Fall 2009 P1.11...

MAE170 Homework 1 Solution
Fall 2009
P1.11 The accuracy of the clock is dependent upon a constant flow from the orifice.
The flow is dependent upon the height of the water in the float tank. The
height of the water is controlled by the float. The control system controls only
the height of the water. Any errors due to enlargement of the orifice or evapo-
ration of the water in the lower tank is not accounted for. The control system
is shown in Figure 1 on the next page.
E2.21 The rotational velocity is
ω ( s ) =
2 . 5( s + 2)
( s + 5)( s + 1)
2
1
4 s
Apply the partial fraction expansion method.
2 . 5( s + 2)
4 s ( s + 5)( s + 1)
2
=
A
s
+
B
s + 5
+
C
s + 1
+
D
( s + 1)
2
0 . 625( s +2) = A ( s
3
+7 s
2
+11 s +5)+ B ( s
3
+2 s
2
+ s )+ C ( s
3
+6 s
2
+5 s )+ D ( s
2
+5 s )
and by comparing coefficients of equal powers of s on both sides, we obtain a
system of 4 linear equations in the 4 unknowns.
A + B + C = 0
7 A + 2 B + 6 C + D = 0
11 A + B + +5 C + 5 D = 0 . 625
5 A = 1 . 25
⇒
A = 0 . 25
B = 0 . 0234
C = - 0 . 2734
D = - 0 . 1563
ω ( s ) =
0 . 25
s
+
0 . 0234
s + 5
-
0 . 2734
s + 1
-
0 . 1562
( s + 1) 2
Taking the inverse Laplace transform gives
ω ( t ) = 0 . 25 + 0 . 0234 exp( - 5 t ) - 0 . 2734 exp( - t ) - 0 . 1563 t exp( - t )
We can alternatively compute