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2.24 Sketch representative waveforms of the following signals, and represent them as mathematical
functions (if possible):
a. The output signal from the thermostat on a refrigerator.
b. The electrical signal to a spark plug in a car engine.
c. The input to a cruise control from an automobile.
d. A pure musical tone (e.g., 440 Hz is the note A).
e. The note produced by a guitar string.
f. AM and FM radio signals.
2.25 Represent the function
e t
ð Þ ¼ 5 sin 31:4t þ 2 sin 44t
as a discrete set of N ¼ 128 numbers separated by a time increment of (1/N). Use an appropriate
algorithm to construct an amplitude spectrum from this data set. (Hint: A spreadsheet program or the
program file DataSpect will handle this task.)
2.26 Repeat Problem 2.25 using a data set of 256 numbers at dt ¼ (1/N) and dt ¼ (1/2N) seconds. Compare
and discuss the results.
2.27 A particular strain sensor is mounted to an aircraft wing that is subjected to periodic wind gusts. The
strain measurement system indicates a periodic strain that ranges from 3250 Â 10
À6 in./in. to
4150 Â 10
À6 in./in. at a frequency of 1 Hz. Determine:
a. The average value of this signal.
b. The amplitude and the frequency of this output signal when expressed as a simple periodic
function.
c. A one-term Fourier series that represents this signal.
d. Construct an amplitude spectrum plot for the output signal.
2.28 For a dynamic calibration involving a force measurement system, a known force is applied to a
sensor. The force varies between 100 and 170 N at a frequency of 10 rad/s. State the average (static)
value of the input signal, its amplitude, and its frequency. Assuming that the signal may be
represented by a simple periodic waveform, express the signal as a one-term Fourier series and create
an amplitude spectrum from an equivalent discrete time series.
2.29 A displacement sensor is placed on a dynamic calibration rig known as a shaker. This device
produces a known periodic displacement that serves as the input to the sensor. If the known
displacement is set to vary between 2 and 5 mm at a frequency of 100 Hz, express the input signal as a
one-term Fourier series. Plot the signal in the time domain, and construct an amplitude spectrum plot.
2.30 Consider the upward flow of water and air in a tube having a circular cross section, as shown in Figure
2.26. If the water and air flow rates are within a certain range, there are slugs of liquid and large gas
bubbles flowing upward together. This type of flow is called ‘‘slug flow.’’ The data file gas_
liquid_data.txt with the companion software contains measurements of pressure made at the wall of
a tube in which air and water were flowing. The data were acquired at a sample frequency of 300 Hz.
The average flow velocity of the air and water is 1 m/s.
a. Construct an amplitude spectrum from the data, and determine the dominant frequency.
b. Using the frequency information from part a, determine the length L shown in the drawing in
Figure 2.26. Assume that the dominant frequency is associated with the passage of the bubbles
and slugs across the pressure sensor.
76 Chapter 2 Static and Dynamic Characteristics of Signals
13:35:23 Page 76
2.24 Sketch representative waveforms of the following signals, and represent them as mathematical
functions (if possible):
a. The output signal from the thermostat on a refrigerator.
b. The electrical signal to a spark plug in a car engine.
c. The input to a cruise control from an automobile.
d. A pure musical tone (e.g., 440 Hz is the note A).
e. The note produced by a guitar string.
f. AM and FM radio signals.
2.25 Represent the function
e t
ð Þ ¼ 5 sin 31:4t þ 2 sin 44t
as a discrete set of N ¼ 128 numbers separated by a time increment of (1/N). Use an appropriate
algorithm to construct an amplitude spectrum from this data set. (Hint: A spreadsheet program or the
program file DataSpect will handle this task.)
2.26 Repeat Problem 2.25 using a data set of 256 numbers at dt ¼ (1/N) and dt ¼ (1/2N) seconds. Compare
and discuss the results.
2.27 A particular strain sensor is mounted to an aircraft wing that is subjected to periodic wind gusts. The
strain measurement system indicates a periodic strain that ranges from 3250 Â 10
À6 in./in. to
4150 Â 10
À6 in./in. at a frequency of 1 Hz. Determine:
a. The average value of this signal.
b. The amplitude and the frequency of this output signal when expressed as a simple periodic
function.
c. A one-term Fourier series that represents this signal.
d. Construct an amplitude spectrum plot for the output signal.
2.28 For a dynamic calibration involving a force measurement system, a known force is applied to a
sensor. The force varies between 100 and 170 N at a frequency of 10 rad/s. State the average (static)
value of the input signal, its amplitude, and its frequency. Assuming that the signal may be
represented by a simple periodic waveform, express the signal as a one-term Fourier series and create
an amplitude spectrum from an equivalent discrete time series.
2.29 A displacement sensor is placed on a dynamic calibration rig known as a shaker. This device
produces a known periodic displacement that serves as the input to the sensor. If the known
displacement is set to vary between 2 and 5 mm at a frequency of 100 Hz, express the input signal as a
one-term Fourier series. Plot the signal in the time domain, and construct an amplitude spectrum plot.
2.30 Consider the upward flow of water and air in a tube having a circular cross section, as shown in Figure
2.26. If the water and air flow rates are within a certain range, there are slugs of liquid and large gas
bubbles flowing upward together. This type of flow is called ‘‘slug flow.’’ The data file gas_
liquid_data.txt with the companion software contains measurements of pressure made at the wall of
a tube in which air and water were flowing. The data were acquired at a sample frequency of 300 Hz.
The average flow velocity of the air and water is 1 m/s.
a. Construct an amplitude spectrum from the data, and determine the dominant frequency.
b. Using the frequency information from part a, determine the length L shown in the drawing in
Figure 2.26. Assume that the dominant frequency is associated with the passage of the bubbles
and slugs across the pressure sensor.
76 Chapter 2 Static and Dynamic Characteristics of Signals