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waveform equation, y(t). If the dynamic error is to be less than 1%, what must be the system’s time
constant?
3.19 A cantilever beam instrumented with strain gauges is used as a force scale. A step-test on the beam
provides a measured damped oscillatory signal with time. If the signal behavior is second order, show
how a data-reduction design plan could use the information in this signal to determine the natural
frequency and damping ratio of the cantilever beam. (Hint: Consider the shape of the decay of the
peak values in the oscillation.)
3.20 A step test of a transducer brings on a damped oscillation decaying to a steady value. If the period of
oscillation is 0.577 ms, what is the transducer ringing frequency?
3.21 An input signal oscillates sinusoidally between 12 and 24 V with a frequency of 120 Hz. It is
measured with an instrument with damping ratio of 0.7, ringing frequency of 1000 Hz, and static
sensitivity of 1 V/V. Determine and plot the output signal amplitude spectrum at steady response.
3.22 An application demands that a sinusoidal pressure variation of 250 Hz be measured with no more
than 2% dynamic error. In selecting a suitable pressure transducer from a vendor catalog, you
note that a desirable line of transducers has a fixed natural frequency of 600 Hz but that you have a
choice of transducer damping ratios of between 0.5 and 1.5 in. increments of 0.05. Select a suitable
transducer.
3.23 A DVD/CD player is to be isolated from room vibrations by placing it on an isolation pad. The
isolation pad can be considered as a board of mass m, a foam mat of stiffness k, and with a damping
coefficient c. For expected vibrations in the frequency range of between 2 and 40 Hz, select
reasonable values for m, k, and c such that the room vibrations are attenuated by at least 50%.
Assume that the only degree of freedom is in the vertical direction.
3.24 A single-loop RCL electrical circuit can be modeled as a second-order system in terms of current.
Show that the differential equation for such a circuit subjected to a forcing function potential E(t) is
given by
L
d
2
I
dt 2 þ R
dI
dt
þ
I
C
¼ E t
ð Þ
Determine the natural frequency and damping ratio for this system. For a forcing potential,
E(t) ¼ 1 þ 0.5 sin 2000t V, determine the system steady response when L ¼ 2 H, C ¼ 1 mF, and
R ¼ 10,000 V. Plot the steady output signal and input signal versus time. I 0
ð Þ ¼ _
I 0
ð Þ ¼ 0.
3.25 A transducer that behaves as a second-order instrument has a damping ratio of 0.7 and a natural
frequency of 1000 Hz. It is to be used to measure a signal containing frequencies as large as 750 Hz.
If a dynamic error of Æ10% can be tolerated, show whether or not this transducer is a good choice.
3.26 A strain-gauge measurement system is mounted on an airplane wing to measure wing oscillation and
strain during wind gusts. The strain system has a 90% rise time of 100 ms, a ringing frequency of
1200 Hz, and a damping ratio of 0.8. Estimate the dynamic error in measuring a 1-Hz oscillation.
Also, estimate any time lag. Explain in words the meaning of this information.
3.27 An instrument having a resonance frequency of 1414 rad/s with a damping ratio of 0.5 is used to
measure a signal of $6000 Hz. Estimate the expected dynamic error and phase shift.
3.28 Select one set of appropriate values for damping ratio and natural frequency for a second-order
instrument used to measure frequencies up to 100 rad/s with no more than Æ10% dynamic error.
A catalog offers models with damping ratios of 0.4, 1, and 2 and natural frequencies of 200 and
500 rad/s. Explain your reasoning.
114 Chapter 3 Measurement System Behavior
15:24:57 Page 114
waveform equation, y(t). If the dynamic error is to be less than 1%, what must be the system’s time
constant?
3.19 A cantilever beam instrumented with strain gauges is used as a force scale. A step-test on the beam
provides a measured damped oscillatory signal with time. If the signal behavior is second order, show
how a data-reduction design plan could use the information in this signal to determine the natural
frequency and damping ratio of the cantilever beam. (Hint: Consider the shape of the decay of the
peak values in the oscillation.)
3.20 A step test of a transducer brings on a damped oscillation decaying to a steady value. If the period of
oscillation is 0.577 ms, what is the transducer ringing frequency?
3.21 An input signal oscillates sinusoidally between 12 and 24 V with a frequency of 120 Hz. It is
measured with an instrument with damping ratio of 0.7, ringing frequency of 1000 Hz, and static
sensitivity of 1 V/V. Determine and plot the output signal amplitude spectrum at steady response.
3.22 An application demands that a sinusoidal pressure variation of 250 Hz be measured with no more
than 2% dynamic error. In selecting a suitable pressure transducer from a vendor catalog, you
note that a desirable line of transducers has a fixed natural frequency of 600 Hz but that you have a
choice of transducer damping ratios of between 0.5 and 1.5 in. increments of 0.05. Select a suitable
transducer.
3.23 A DVD/CD player is to be isolated from room vibrations by placing it on an isolation pad. The
isolation pad can be considered as a board of mass m, a foam mat of stiffness k, and with a damping
coefficient c. For expected vibrations in the frequency range of between 2 and 40 Hz, select
reasonable values for m, k, and c such that the room vibrations are attenuated by at least 50%.
Assume that the only degree of freedom is in the vertical direction.
3.24 A single-loop RCL electrical circuit can be modeled as a second-order system in terms of current.
Show that the differential equation for such a circuit subjected to a forcing function potential E(t) is
given by
L
d
2
I
dt 2 þ R
dI
dt
þ
I
C
¼ E t
ð Þ
Determine the natural frequency and damping ratio for this system. For a forcing potential,
E(t) ¼ 1 þ 0.5 sin 2000t V, determine the system steady response when L ¼ 2 H, C ¼ 1 mF, and
R ¼ 10,000 V. Plot the steady output signal and input signal versus time. I 0
ð Þ ¼ _
I 0
ð Þ ¼ 0.
3.25 A transducer that behaves as a second-order instrument has a damping ratio of 0.7 and a natural
frequency of 1000 Hz. It is to be used to measure a signal containing frequencies as large as 750 Hz.
If a dynamic error of Æ10% can be tolerated, show whether or not this transducer is a good choice.
3.26 A strain-gauge measurement system is mounted on an airplane wing to measure wing oscillation and
strain during wind gusts. The strain system has a 90% rise time of 100 ms, a ringing frequency of
1200 Hz, and a damping ratio of 0.8. Estimate the dynamic error in measuring a 1-Hz oscillation.
Also, estimate any time lag. Explain in words the meaning of this information.
3.27 An instrument having a resonance frequency of 1414 rad/s with a damping ratio of 0.5 is used to
measure a signal of $6000 Hz. Estimate the expected dynamic error and phase shift.
3.28 Select one set of appropriate values for damping ratio and natural frequency for a second-order
instrument used to measure frequencies up to 100 rad/s with no more than Æ10% dynamic error.
A catalog offers models with damping ratios of 0.4, 1, and 2 and natural frequencies of 200 and
500 rad/s. Explain your reasoning.
114 Chapter 3 Measurement System Behavior