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15:24:57 Page 112
y
n (0)
initial condition of y
n
A
input signal amplitude
B(v)
output signal amplitude
C
constant
E(t)
voltage (V) or energy
F
force ðm l t
À2
Þ
F(t)
forcing function
G(s)
transfer function
K
static sensitivity
M(v)
magnitude ratio, B/KA
T(t)
temperature (
)
T d
ringing period (t)
U(t)
unit step function
b 1
time lag (t)
d(v)
dynamic error
t
time constant (t)
F(v)
phase shift
v
circular frequency (t
À1 )
v n
natural frequency magnitude(t
À1 )
v d
ringing frequency (t
À1 )
v R
resonance frequency (t
À1 )
z
damping ratio
G
error fraction
Subscripts
0 initial value
1 final or steady value
h homogeneous solution
PROBLEMS
Note: Although not required, the companion software can be used for solving many of these problems.
We encourage the reader to explore the software provided.
3.1 A mass measurement system has a static sensitivity of 2 V/kg. An input range of 1 to 10 kg needs to
be measured. A voltmeter is used to display the measurement. What range of voltmeter is needed.
What would be the significance of changing the static sensitivity?
3.2 Determine the 75%, 90%, and 95% response time for each of the systems given (assume zero initial
conditions):
a. 0:4 _
T þ T ¼ 4U t
ð Þ
b. € y þ 2_ y þ 4y ¼ U t
ð Þ
c. 2 €
P þ 8 _
P þ 8P ¼ 2U t
ð Þ
d. 5_ y þ 5y ¼ U t
ð Þ
3.3 A special sensor is designed to sense the percent vapor present in a liquid–vapor mixture. If during a
static calibration the sensor indicates 80 units when in contact with 100% liquid, 0 units with 100%
vapor, and 40 units with a 50:50% mixture, determine the static sensitivity of the sensor.
3.4 A measurement system can be modeled by the equation
0:5_ y þ y ¼ F t
ð Þ
Initially, the output signal is steady at 75 volts. The input signal is then suddenly increased to
100 volts.
a. Determine the response equation.
b. On the same graph, plot both the input signal and the system time response from t ¼ 0 s through
steady response.
112 Chapter 3 Measurement System Behavior
15:24:57 Page 112
y
n (0)
initial condition of y
n
A
input signal amplitude
B(v)
output signal amplitude
C
constant
E(t)
voltage (V) or energy
F
force ðm l t
À2
Þ
F(t)
forcing function
G(s)
transfer function
K
static sensitivity
M(v)
magnitude ratio, B/KA
T(t)
temperature (
)
T d
ringing period (t)
U(t)
unit step function
b 1
time lag (t)
d(v)
dynamic error
t
time constant (t)
F(v)
phase shift
v
circular frequency (t
À1 )
v n
natural frequency magnitude(t
À1 )
v d
ringing frequency (t
À1 )
v R
resonance frequency (t
À1 )
z
damping ratio
G
error fraction
Subscripts
0 initial value
1 final or steady value
h homogeneous solution
PROBLEMS
Note: Although not required, the companion software can be used for solving many of these problems.
We encourage the reader to explore the software provided.
3.1 A mass measurement system has a static sensitivity of 2 V/kg. An input range of 1 to 10 kg needs to
be measured. A voltmeter is used to display the measurement. What range of voltmeter is needed.
What would be the significance of changing the static sensitivity?
3.2 Determine the 75%, 90%, and 95% response time for each of the systems given (assume zero initial
conditions):
a. 0:4 _
T þ T ¼ 4U t
ð Þ
b. € y þ 2_ y þ 4y ¼ U t
ð Þ
c. 2 €
P þ 8 _
P þ 8P ¼ 2U t
ð Þ
d. 5_ y þ 5y ¼ U t
ð Þ
3.3 A special sensor is designed to sense the percent vapor present in a liquid–vapor mixture. If during a
static calibration the sensor indicates 80 units when in contact with 100% liquid, 0 units with 100%
vapor, and 40 units with a 50:50% mixture, determine the static sensitivity of the sensor.
3.4 A measurement system can be modeled by the equation
0:5_ y þ y ¼ F t
ð Þ
Initially, the output signal is steady at 75 volts. The input signal is then suddenly increased to
100 volts.
a. Determine the response equation.
b. On the same graph, plot both the input signal and the system time response from t ¼ 0 s through
steady response.
112 Chapter 3 Measurement System Behavior