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3.39 A signal suspected to be of the nominal form
y t
ð Þ ¼ 5 sin1000t mV
is input to a first-order instrument having a time constant of 100 ms and K ¼ 1 V/V. It is then to be
passed through a second-order amplifier having a K ¼ 100 V/V, a natural frequency of 15,000 Hz,
and a damping ratio of 0.8. What is the expected form of the output signal, y(t)? Estimate the
dynamic error and phase lag in the output. Is this system a good choice here? If not, do you have any
suggestions?
3.40 A typical modern DC audio amplifier has a frequency bandwidth of 0 to 20,000 Hz Æ 1 dB. Explain
the meaning of this specification and its relation to music reproduction.
3.41 The displacement of a rail vehicle chassis as it rolls down a track is measured using a transducer
(K ¼ 10 mV/mm, v n ¼ 10 000 rad/s, z ¼ 0.6) and a recorder (K ¼ 1 mm/mV, v n ¼ 700 rad/s, z ¼ 0.7).
The resulting amplitude spectrum of the output signal consists of a spike of 90 mm at 2 Hz and a
spike of 50 mm at 40 Hz. Are the measurement system specifications suitable for the displacement
signal? (If not, suggest changes.) Estimate the actual displacement of the chassis. State any
assumptions.
3.42 The amplitude spectrum of the time-varying displacement signal from a vibrating U-shaped tube is
expected to have a spike at 85, 147, 220, and 452 Hz. Each spike is related to an important vibrational
mode of the tube. Displacement transducers available for this application have a range of natural
frequencies from 500 to 1000 Hz, with a fixed damping ratio of about 0.5. The transducer output is to
be monitored on a spectrum analyzer that has a frequency bandwidth extending from 0.1 Hz to 250
kHz. Within the stated range of availability, select a suitable transducer for this measurement from
the given range.
3.43 A sensor mounted to a cantilever beam indicates beam motion with time. When the beam is deflected
and released (step test), the sensor signal indicates that the beam oscillates as an underdamped
second-order system with a ringing frequency of 10 rad/s. The maximum displacement amplitudes
are measured at three different times corresponding to the 1st, 16th, and 32nd cycles and found to be
17, 9, and 5 mV, respectively. Estimate the damping ratio and natural frequency of the beam based on
this measured signal, K ¼ 1 mm/mV.
3.44 Write a short essay on how system properties of static sensitivity, natural frequency, and damping
ratio affect the output information from a measurement system. Be sure to discuss the relative
importance of the transient and steady aspects of the resulting signal.
3.45 Burgess (5) reports that the damping ratio can be approximated from the system response to an
impulse test by counting the number of cycles, n, required for the ringing amplitudes to fall to within
1% of steady state by z ¼ 4:6=2pn
ð
Þ . The estimate is further improved if n is a noninteger.
Investigate the quality of this estimate for a second-order system subjected instead to a step function
input. Discuss your findings.
3.46 The starting transient of a DC motor can be modeled as an RL circuit, with the resistor and inductor in
series (Figure 3.25). Let R ¼ 4 V, L ¼ 0.1 H, and E i ¼ 50 V with t(0) ¼ 0. Find the current draw with
time for t > 0
þ .
3.47 A camera flash light is driven by the energy stored in a capacitor. Suppose the flash operates off a 6-V
battery. Determine the time required for the capacitor stored voltage to reach 90% of its maximum
energy ð¼
1
2 CE
2
B Þ. Model this as an RC circuit (Fig. 3.26). For the flash: C ¼ 1000 mF, R ¼ 1 kV, and
E c (0) ¼ 0.
116 Chapter 3 Measurement System Behavior
15:24:57 Page 116
3.39 A signal suspected to be of the nominal form
y t
ð Þ ¼ 5 sin1000t mV
is input to a first-order instrument having a time constant of 100 ms and K ¼ 1 V/V. It is then to be
passed through a second-order amplifier having a K ¼ 100 V/V, a natural frequency of 15,000 Hz,
and a damping ratio of 0.8. What is the expected form of the output signal, y(t)? Estimate the
dynamic error and phase lag in the output. Is this system a good choice here? If not, do you have any
suggestions?
3.40 A typical modern DC audio amplifier has a frequency bandwidth of 0 to 20,000 Hz Æ 1 dB. Explain
the meaning of this specification and its relation to music reproduction.
3.41 The displacement of a rail vehicle chassis as it rolls down a track is measured using a transducer
(K ¼ 10 mV/mm, v n ¼ 10 000 rad/s, z ¼ 0.6) and a recorder (K ¼ 1 mm/mV, v n ¼ 700 rad/s, z ¼ 0.7).
The resulting amplitude spectrum of the output signal consists of a spike of 90 mm at 2 Hz and a
spike of 50 mm at 40 Hz. Are the measurement system specifications suitable for the displacement
signal? (If not, suggest changes.) Estimate the actual displacement of the chassis. State any
assumptions.
3.42 The amplitude spectrum of the time-varying displacement signal from a vibrating U-shaped tube is
expected to have a spike at 85, 147, 220, and 452 Hz. Each spike is related to an important vibrational
mode of the tube. Displacement transducers available for this application have a range of natural
frequencies from 500 to 1000 Hz, with a fixed damping ratio of about 0.5. The transducer output is to
be monitored on a spectrum analyzer that has a frequency bandwidth extending from 0.1 Hz to 250
kHz. Within the stated range of availability, select a suitable transducer for this measurement from
the given range.
3.43 A sensor mounted to a cantilever beam indicates beam motion with time. When the beam is deflected
and released (step test), the sensor signal indicates that the beam oscillates as an underdamped
second-order system with a ringing frequency of 10 rad/s. The maximum displacement amplitudes
are measured at three different times corresponding to the 1st, 16th, and 32nd cycles and found to be
17, 9, and 5 mV, respectively. Estimate the damping ratio and natural frequency of the beam based on
this measured signal, K ¼ 1 mm/mV.
3.44 Write a short essay on how system properties of static sensitivity, natural frequency, and damping
ratio affect the output information from a measurement system. Be sure to discuss the relative
importance of the transient and steady aspects of the resulting signal.
3.45 Burgess (5) reports that the damping ratio can be approximated from the system response to an
impulse test by counting the number of cycles, n, required for the ringing amplitudes to fall to within
1% of steady state by z ¼ 4:6=2pn
ð
Þ . The estimate is further improved if n is a noninteger.
Investigate the quality of this estimate for a second-order system subjected instead to a step function
input. Discuss your findings.
3.46 The starting transient of a DC motor can be modeled as an RL circuit, with the resistor and inductor in
series (Figure 3.25). Let R ¼ 4 V, L ¼ 0.1 H, and E i ¼ 50 V with t(0) ¼ 0. Find the current draw with
time for t > 0
þ .
3.47 A camera flash light is driven by the energy stored in a capacitor. Suppose the flash operates off a 6-V
battery. Determine the time required for the capacitor stored voltage to reach 90% of its maximum
energy ð¼
1
2 CE
2
B Þ. Model this as an RC circuit (Fig. 3.26). For the flash: C ¼ 1000 mF, R ¼ 1 kV, and
E c (0) ¼ 0.
116 Chapter 3 Measurement System Behavior