PROBLEM 1.17
KNOWN:
Data of form y = ax
b
.
FIND:
a and b; K
SOLUTION
The data are plotted below. If y = ax
b
, then in loglog format the data will take the linear
form
log y = log a
+ b log x
A more or less linear curve results with this data. From the plot, the curve fit found is
log y = 0.23
+
2x
This implies that
y = 0.59x
2
so that a = 0.59 and b = 2. The static sensitivity is found by the slope dy/dx at each value
of x.
x [m]
K(x
1
) = dy/dx
x1
[cm/m]
0.5 0.54
2.0 2.16
5.0 5.40
10.0 10.80
COMMENT
An aspect of this problem is to draw attention to the fact that many measurement systems
may have a static sensitivity that is dependent on input value. The operating principle of
many systems will determine how K behaves.
p1.17
0.1
1
10
100
0.1
1
10
X [m]
y [cm]
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View Full DocumentPROBLEM 3.4
KNOWN:
System model equation
K = 1 unit/unit
F(t) = 100U(t)
y(0) = 75 units
FIND:
y(t)
SOLUTION
(a) The solution to the system model was shown to be given by the general form,
y(t) = y
∞
+ (y(0)  y
∞
)e
t/
τ
where here,
y(0) = 75 units
y
∞
= 100 units
τ
= 0.5 s (determined from the system equation)
then,
y(t) = 100 + (75  100) e
t/0.5
units
Alternatively, by direct solution and with y(0) = 75 units,
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 Spring '08
 Shan
 static sensitivity

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