Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition
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126 Measurement and Data Analysis for Engineering and Science
FIGURE 4.8
The magnitude ratio of a second-order system responding to sinusoidal-input
forcing.
range of those that would be encountered in an actual experiment. With
this approach, the system’s dynamic errors can be quantified accurately.
Example Problem 4.5
Statement: A pressure transducer is connected through flexible tubing to a static
pressure port on the surface of a cylinder that is mounted inside a wind tunnel. The
structure of the flow local to the port is such that the static pressure, p(t), varies as
p(t) = 15sin2t,
in which t is time. Both the tubing and the pressure transducer behave as second-order
systems. The natural frequencies of the transducer, ω
n,trans
, and the tubing, ω
n,tube
,
are 2000 rad/s and 4 rad/s, respectively. Their damping ratios are ζ
trans
= 0.7 and
ζ
tube
= 0.2, respectively. Find the magnitude attenuation and phase lag of the pressure
signal, as determined from the output of the pressure transducer, and then write the
expression for this signal.
Solution: Because this measurement system is linear, the system’s magnitude ratio,
M
s
(ω), is the product of the components’ magnitude ratios, and the phase lag, φ
s
(ω), is
the sum of the components’ phase lags, where ω the circular frequency of the pressure.
Thus,
M
s
(ω) = M
tube
(ω) × M
trans
(ω)
and
φ
s
(ω) = φ
tube
(ω) + φ
trans
(ω).
Calibration and Response 127
FIGURE 4.9
The phase shift of a second-order system responding to a sinusoidal-input
forcing.
Also, ω/ω
tube
= 2/4 = 0.5 and ω/ω
trans
= 2/2000 = 0.001. Application of Equa-
tions 4.62 and 4.64, noting ζ
trans
= 0.7 and ζ
tube
= 0.2, yields φ
tube
= −14.9
◦
, φ
trans
= −0.1
◦
, M
tube
= 1.29, and M
trans
= 1.00. Thus, φ
s
(2) = −14.9
◦
+ −0.1
◦
= −15.0
◦
and M
s
(2) = (1.29)(1.00) = 1.29. The pressure signal, as determined from the out-
put of the transducer, is p
s
(t) = (15)(1.29)sin[2t −(15.0)(π/180)] = 19.4sin(2t −0.26).
Thus, the magnitude of the pressure signal at the output of the measurement system
will appear 129 % greater than the actual pressure signal and be delayed in time by
0.13 s [(0.26 s)/(2 rad/s)].
4.8 *Numerical Solution Methods
Differential equations governing a system’s response to input forcing may be
nonlinear and not have exact solutions. Fortunately, methods are available
to numerically integrate most ordinary differential equations and obtain the
system response [24]. The basic solution approach is to reduce any higher-
order ordinary differential equations to a system of coupled, first-order ordi-
nary differential equations. Then, the first-order equations are solved using
finite-difference methods. For example, the second-order ordinary differen-
tial equation
128 Measurement and Data Analysis for Engineering and Science
d
2
y(t)
dt
2
− cy(t) = d (4.65)
can be reduced to two first-order ordinary differential equations by using
the substitution dy/dt = z(t), which yields the system of equations
dy(t)
dt
= z(t) and
dz(t)
dt
= cy(t) + d. (4.66)
Two initial conditions are needed to obtain a specific solution.
The numerical solution of a first-order ordinary differential equation can
be obtained using various finite-difference methods [3]. The exact differen-
tial, dy(t)/dt = f(y, t), is approximated by a finite difference. There are
many ways to approximate dy(t)/dt. The choice depends upon the required
accuracy and computation time. A straightforward finite-difference approx-
imation for dy(t)/dt is the forward Euler expression
f(y
n
, t
n
) ≈
y
n+1
− y
n
∆t
, (4.67)
where n and n + 1 denote the n-th and n + 1-th points. Equation 4.67 leads
directly to
y
n+1
= y
n
+ ∆t f (y
n
, t
n
). (4.68)
The expression for f (y, t) is obtained from the governing first-order ordinary
differential equation. An initial condition, y(0), also is specified. This permits
the value of y
n+1
to be computed for a fixed ∆t from Equation 4.68. This
algorithm is applied successively up to the desired final time.
Other methods can be used to determine an expression analogous to
Equation 4.67. All these methods are easy to implement. The following
more commonly used methods replace f(y
n
, t
n
) in Equation 4.68 by
f(y
n+1
, t
n+1
) (4.69)
for the backward Euler method, and
f(y
n
, t
n
) + f(y
n+1
, t
n+1
)
2
(4.70)
for the improved Euler method. The improved Euler method is more ac-
curate than the forward and backward Euler methods. The fourth-order
Runge-Kutta method replacement for Equation 4.67 is
k
1
+ 2k
2
+ 2k
3
+ k
4
6
, (4.71)
Calibration and Response 129
FIGURE 4.10
The response of the system
dy(t)
dt
− 2y(t) = F(t) = 0.5 − t to forcing as
determined by the MATLAB M-file odeint.m.
where
k
1
= f(y
n
, t
n
),
k
2
= f(y
n
+
∆t
2
k
1
, t
n
+
∆t
2
),
k
3
= f(y
n
+
∆t
2
k
2
, t
n
+
∆t
2
), and
k
4
= f(y
n
+ ∆t k
3
, t
n
+ ∆t). (4.72)
The fourth-order Runge-Kutta method is more accurate than any Euler
method. It is the method used most frequently and is quite sufficient for
most numerical integrations [3].
Example Problem 4.6
Statement: A first-order system is described by the equation
dy(t)
dt
− 2y(t) = F(t) = 0.5 − t,
with the initial condition y(0) = 1. Solve this differential equation numerically and
analytically. Use four numerical methods: [1] forward Euler, [2] backward Euler, [3]
improved Euler, and [4] fourth-order Runge-Kutta. Use a step size of 0.05 s. Plot all
the results for comparison.
130 Measurement and Data Analysis for Engineering and Science
Solution: The MATLAB M-file odeint.m can be used for this purpose. The results
are presented in Figure 4.10. The fourth-order Runge-Kutta and improved Euler solu-
tions follow the exact solution closely. The backward Euler method underestimates the
response. The forward Euler method overestimates the response.
Calibration and Response 131
4.9 Problem Topic Summary
Topic
Review Problems
Homework Problems
System Basics
1, 3, 4, 5, 6, 7, 13, 14,
15, 16, 17, 19, 21, 22, 23
First-Order
2, 8, 11, 12, 18, 20
1, 2, 3, 4, 5, 8,
11, 13, 17
Second-Order
9, 10
6, 7, 9, 10, 12, 14,
15, 16, 18, 19, 20
TABLE 4.2
Chapter 4 Problem Summary
4.10 Review Problems
1. Does a smaller diameter thermocouple or a larger diameter thermocou-
ple have the large time constant?
2. The dynamic error in a temperature measurement using a thermocouple
is 70 % at 3 s after an input step change in temperature. Determine the
magnitude ratio of the thermocouple’s response at 1 s.
3. Determine the % dynamic error of a measurement system that has an
output of 3 sin(200t) for an input of 4 sin(200t).
4. Determine the attenuation (reduction) in units of dB/decade for a mea-
surement system that has an output of 3 sin(200t) for an input of
4 sin(200t) and an output of sin(2000t) for an input of 4 sin(2000t).
5. Is a strain gage in itself classified as a zero, first, second, or higher-order
system?
6. Determine the damping ratio of a RLC circuit with LC = 1 s
2
that
has a magnitude ratio of 8 when subjected to a sine wave input with a
frequency of 1 rad/s.
7. Determine the phase lag in degrees for a simple RC filter with RC = 5
s when its input signal has a frequency of 1/π Hz.
132 Measurement and Data Analysis for Engineering and Science
8. A first-order system is subjected to a step input of magnitude B. The
time constant in terms of B equals (a) 0.707B, (b) 0.5B, (c) (1 −
1
e
)B,
or (d) B/e.
9. A second-order system with ζ = 0.5 and ω
n
= 2 rad/s is subjected to a
step input of magnitude B. The system’s time constant equals (a) 0.707
s, (b) 1.0 s, (c) (1 −
1
e
) s, or (d) not enough information.
10. A second-order system with ζ = 0.5 and ω
n
= 2 rad/s is subjected to
a sinusoidal input of magnitude Bsin(4t). The phase lag of the output
signal in units of degrees is (a) −3, (b) −146, (c) −34, or (d) −180.
11. A first-order system is subjected to an input of Bsin(10t). The system’s
time constant is 1 s. The amplitude of the system’s output is approxi-
mately (a) 0.707B, (b) 0.98B, (c) (1 −
1
e
)B, or (d) 0.1B.
12. A first-order system is subjected to an input of Bsin(10t). The system’s
time constant is 1 s. The time lag of the system’s output is (a) −0.15 s,
(b) −0.632 s, (c) −π s, or (d) −84.3 s.
13. What is the static sensitivity of the calibration curve F = 250W + 125
at W = 2?
14. The magnitude of the static sensitivity of the calibration curve V =
3 + 8
√
F at F = 16 is (a) 0, (b) 1, (c) 3, (d) 4, or (e) 8.
15. What is the order of each of the following systems? (a) Strain gage, (b)
pressure transducer, (c) accelerometer, (d) RC circuit, (e) thermocouple,
(f) pitot-static tube.
16. What is the magnitude ratio that corresponds to −6 dB?
17. What is the condition for an RLC circuit to be underdamped, critically
damped, or overdamped?
18. A large thermocouple has a time constant of 10 s. It is subjected to a
sinusoidal variation in temperature at a cyclic frequency of 1/(2π) Hz.
The phase lag, in
◦
, is approximately (a) −0.707, (b) −3, (c) −45, or
(d) −85.
19. What is the sensitivity of the linear calibration curve at E =
0.5 exp (10/T ) at (a) T = 283 K, (b) T = 300 K, and (c) T = 350 K. (d)
What type of temperature sensor might result in such an exponential
calibration curve?
20. Consider a first-order system where the frequency of the sinusoidal forc-
ing function is 10 Hz and the system response lags by 90
◦
. What is the
phase lag in seconds?
Calibration and Response 133
Time (ms)
Temperature (
◦
C)
0
24.8
40
22.4
120
19.1
200
15.5
240
13.1
400
9.76
520
8.15
800
6.95
970
6.55
1100
6.15
1400
5.75
1800
5.30
2000
5.20
2200
5.00
3000
4.95
4000
4.95
5000
4.95
6000
4.95
7000
4.95
TABLE 4.3
Thermocouple Response Data
21. The signal 10sin(2πt) passes through a filter whose magnitude ratio is
0.8 and then through a linear amplifier. What must be the gain of the
amplifier for the amplifier’s output signal to have an amplitude of 16?
22. An electronic manometer is calibrated using a fluid based manometer
as the calibration standard. The resulting calibration curve fit is given
by the equation V = 1.1897P − 0.0002, where the unit of P is inches of
H
2
0 and V is volts. The static sensitivity (in V/in. H
2
0) is (a) 0.0002,
(b) 1.1897P
2
- 0.0002P , (c) 1.1897, or (d) −0.0002.
23. Determine the static sensitivity at x = 2.00 for a calibration curve having
y = 0.8 + 33.72x + 3.9086x
2
. Express the result with the correct number
of significant figures.
134 Measurement and Data Analysis for Engineering and Science
4.11 Homework Problems
1. A first-order system has M(f = 200 Hz) = 0.707. Determine (a) its time
constant (in milliseconds) and (b) its phase shift (in degrees).
2. A thermocouple held in room-temperature air is suddenly immersed
into a beaker of cold water. Its temperature as a function of time is
recorded. Determine the thermocouple’s time constant by plotting the
data listed in Table 4.3, assuming that the thermocouple behaves as
a first-order system. A more accurate method of determining the time
constant is by performing a least-squares linear regression analysis (see
Chapter 8) after transforming the temperatures into their appropriate
nondimensional variables.
3. A first-order system with a time constant equal to 10 ms is subjected
to a sinusoidal forcing with an input amplitude equal to 8.00 V. When
the input forcing frequency equals 100 rad/s, the output amplitude is
5.66 V; when the input forcing frequency equals 1000 rad/s, the output
amplitude is 0.80 V. Determine (a) the magnitude ratio for the 100 rad/s
forcing case, (b) the roll-off slope (in units of dB/decade) for the ωτ = 1
to ωτ = 10 decade, and (c) the phase lag (in degrees) for the 100 rad/s
forcing case.
4. The dynamic error in a temperature measurement using a thermometer
is 70 % at 3 s after an input step change in temperature. Determine
(a) the magnitude ratio at 3 s, (b) the thermometer’s time constant (in
seconds), and (c) the magnitude ratio at 1 s.
5. A thermocouple is immersed in a liquid to monitor its temperature fluc-
tuations. Assume the thermocouple acts as a first-order system. The
temperature fluctuations (in degrees Celsius) vary in time as T (t) =
50 + 25 cos(4t). The output of the thermocouple transducer system (in
V) is linearly proportional to temperature and has a static sensitivity
of 2 mV/
◦
C. A step-input calibration of the system reveals that its rise
time is 4.6 s. Determine the system’s (a) time constant (in seconds),
(b) output E(t) (in millivolts), and (c) time lag (in seconds) at ω = 0.2
rad/s.
6. A knowledgeable aerospace student selects a pressure transducer (with
ω
n
= 6284 rad/s and ζ = 2.0) to investigate the pressure fluctuations
within a laminar separation bubble on the suction side of an airfoil.
Assume that the transducer behaves as an over-damped second-order
system. If the experiment requires that the transducer response has
M(ω) ≥ 0.707 and |φ(ω)| ≤ 20
◦
, determine the maximum frequency
(in hertz) that the transducer can follow and accurately meet the two
criteria.
Calibration and Response 135
7. A strain gage system is mounted on an airplane wing to measure wing
oscillations and strain during wind gusts. The system is second order,
having a 90 % rise time of 100 ms, a ringing frequency of 1200 Hz, and a
damping ratio of 0.8. Determine (a) the dynamic error when subjected
to a 1 Hz oscillation and (b) the time lag (in seconds).
8. In a planned experiment a thermocouple is to be exposed to a step
change in temperature. The response characteristics of the thermocou-
ple must be such that the thermocouple’s output reaches 98 % of the
final temperature within 5 s. Assume that the thermocouple’s bead (its
sensing element) is spherical with a density equal to 8000 kg/m
3
, a spe-
cific heat at constant volume equal to 380 J/(kg·K), and a convective
heat transfer coefficient equal to 210 W/(m
2
·K). Determine the maxi-
mum diameter that the thermocouple can have and still meet the desired
response characteristics.
9. Determine by calculation the damping ratio value of a second-order sys-
tem that would be required to achieve a magnitude ratio of unity when
the sinusoidal-input forcing frequency equals the natural frequency of
the system.
10. The pressure tap on the surface of a heat exchanger tube is connected
via flexible tubing to a pressure transducer. Both the tubing and the
transducer behave as second-order systems. The natural frequencies are
30 rad/s for the tubing and 6280 rad/s for the transducer. The damping
ratios are 0.45 for the tubing and 0.70 for the transducer. Determine the
magnitude ratio and the phase lag for the system when subjected to a
sinusoidal forcing having a 100 Hz frequency. What, if anything, is the
problem in using this system for this application?
11. Determine the percent dynamic error in the temperature measured by
a thermocouple having a 3 ms time constant when subjected to a tem-
perature that varies sinusoidally in time with a frequency of 531 Hz.
12. The output of an under-damped second-order system with ζ = 0.1 sub-
jected to step-input forcing initially oscillates with a period equal to 1
s until the oscillation dissipates. The same system then is subjected to
sinusoidal-input forcing with a frequency equal to 12.62 rad/s. Deter-
mine the phase lag (in degrees) at this frequency.
13. A thermocouple is at room temperature (70
◦
F) and at equilibrium
when it is plunged into a water bath at a temperature of 170
◦
F. It
takes the thermocouple 1 s to read a temperature indication of 120
◦
F.
What is the time constant of the thermocouple-fluid system? This same
thermocouple is used to measure a sinusoidally varying temperature.
The variation in degrees Fahrenheit is given by the equation
T = 100 + 200 sin(10t).