E1C03 09/14/2010
15:24:57 Page 115
3.29 A first-order measurement system with time constant of 25 ms and K ¼ 1 V/N measures a signal of
the form F(t) ¼ sin 2pt þ 0.8 sin 6pt N. Determine the steady response (steady output signal) from
the system. Plot the output signal amplitude spectrum. Discuss the transfer of information from the
input to the output. Can the input signal be resolved based on the output?
3.30 Demonstrate for the second-order system (v n ¼ 100 rad/s, z ¼ 0.4) subjected to step function
input, U(t), that the damping ratio can be found from the logarithmic amplitude decay whether
y(0) ¼ 0 with F(t) ¼ KAU(t) or y(0) ¼ KA with F(t) ¼ ÀKAU(t). Use K ¼ 1 mV/mV and
A ¼ 600 mV. Do this by solving for the expected time response and from this response
determine the successive peak amplitudes to extract the values for the damping ratio and
natural frequency.
3.31 A force transducer having a damping ratio of 0.5 and a natural frequency of 4000 Hz is available for
use to measure a periodic signal of 2000 Hz. Show whether or not the transducer passes a Æ10%
dynamic error constraint. Estimate its resonance frequency.
3.32 An accelerometer, whose frequency response is defined by Equations 3.20 and 3.22, has a
damping ratio of 0.4 and a natural frequency of 18,000 Hz. It is used to sense the relative
displacement of a beam to which it is attached. If an impact to the beam imparts a vibration at
4500 Hz, estimate the dynamic error and phase shift in the accelerometer output. Estimate its
resonance frequency.
3.33 Derive the equation form for the magnitude ratio and phase shift of the seismic accelerometer of
Example 3.1. Does its frequency response differ from that predicted by Equations 3.20 and 3.22? For
what type of measurement would you suppose this instrument would be best suited?
3.34 Suppose the pressure transducer of Example 3.9 had a damping ratio of 0.6. Plot its frequency
response M(v) and f(v). At which frequency is M(v) a maximum?
3.35 A pressure transducer is attached to a stiff-walled catheter. The catheter is filled with saline from a
small balloon attached at its tip. The initial system pressure is 50 mm Hg. At t ¼ 0 s, the balloon is
popped, forcing a step function change in pressure from 50 to 0 mm Hg. The time-based signal is
recorded and the ringing period determined to be 0.03 s. Find the natural frequency and damping
ratio attributed to the pressure-catheter system; K ¼ 1 mV/mm Hg.
3.36 The output stage of a first-order transducer is to be connected to a second-order display stage device.
The transducer has a known time constant of 1.4 ms and static sensitivity of 2 V/
C while the display
device has values of sensitivity, damping ratio, and natural frequency of 1 V/V, 0.9, and 5000 Hz,
respectively. Determine the steady response of this measurement system to an input signal of the
form, T(t) ¼ 10 þ 50 sin 628t
C.
3.37 The displacement of a solid body is to be monitored by a transducer (second-order system) with
signal output displayed on a recorder (second-order system). The displacement is expected to vary
sinusoidally between 2 and 5 mm at a rate of 85 Hz. Select appropriate design specifications for the
measurement system for no more than 5% dynamic error (i.e., specify an acceptable range for natural
frequency and damping ratio for each device).
3.38 The input signal
F t
ð Þ ¼ 2 þ sin 15:7t þ sin 160t N
is applied to a force measurement system. The system has a known v n ¼ 100 rad/s, z ¼ 0.4, and
K ¼ 1 V/N. Write the expected form of the steady output signal in volts. Plot the resulting amplitude
spectrum.
Problems 115
15:24:57 Page 115
3.29 A first-order measurement system with time constant of 25 ms and K ¼ 1 V/N measures a signal of
the form F(t) ¼ sin 2pt þ 0.8 sin 6pt N. Determine the steady response (steady output signal) from
the system. Plot the output signal amplitude spectrum. Discuss the transfer of information from the
input to the output. Can the input signal be resolved based on the output?
3.30 Demonstrate for the second-order system (v n ¼ 100 rad/s, z ¼ 0.4) subjected to step function
input, U(t), that the damping ratio can be found from the logarithmic amplitude decay whether
y(0) ¼ 0 with F(t) ¼ KAU(t) or y(0) ¼ KA with F(t) ¼ ÀKAU(t). Use K ¼ 1 mV/mV and
A ¼ 600 mV. Do this by solving for the expected time response and from this response
determine the successive peak amplitudes to extract the values for the damping ratio and
natural frequency.
3.31 A force transducer having a damping ratio of 0.5 and a natural frequency of 4000 Hz is available for
use to measure a periodic signal of 2000 Hz. Show whether or not the transducer passes a Æ10%
dynamic error constraint. Estimate its resonance frequency.
3.32 An accelerometer, whose frequency response is defined by Equations 3.20 and 3.22, has a
damping ratio of 0.4 and a natural frequency of 18,000 Hz. It is used to sense the relative
displacement of a beam to which it is attached. If an impact to the beam imparts a vibration at
4500 Hz, estimate the dynamic error and phase shift in the accelerometer output. Estimate its
resonance frequency.
3.33 Derive the equation form for the magnitude ratio and phase shift of the seismic accelerometer of
Example 3.1. Does its frequency response differ from that predicted by Equations 3.20 and 3.22? For
what type of measurement would you suppose this instrument would be best suited?
3.34 Suppose the pressure transducer of Example 3.9 had a damping ratio of 0.6. Plot its frequency
response M(v) and f(v). At which frequency is M(v) a maximum?
3.35 A pressure transducer is attached to a stiff-walled catheter. The catheter is filled with saline from a
small balloon attached at its tip. The initial system pressure is 50 mm Hg. At t ¼ 0 s, the balloon is
popped, forcing a step function change in pressure from 50 to 0 mm Hg. The time-based signal is
recorded and the ringing period determined to be 0.03 s. Find the natural frequency and damping
ratio attributed to the pressure-catheter system; K ¼ 1 mV/mm Hg.
3.36 The output stage of a first-order transducer is to be connected to a second-order display stage device.
The transducer has a known time constant of 1.4 ms and static sensitivity of 2 V/
C while the display
device has values of sensitivity, damping ratio, and natural frequency of 1 V/V, 0.9, and 5000 Hz,
respectively. Determine the steady response of this measurement system to an input signal of the
form, T(t) ¼ 10 þ 50 sin 628t
C.
3.37 The displacement of a solid body is to be monitored by a transducer (second-order system) with
signal output displayed on a recorder (second-order system). The displacement is expected to vary
sinusoidally between 2 and 5 mm at a rate of 85 Hz. Select appropriate design specifications for the
measurement system for no more than 5% dynamic error (i.e., specify an acceptable range for natural
frequency and damping ratio for each device).
3.38 The input signal
F t
ð Þ ¼ 2 þ sin 15:7t þ sin 160t N
is applied to a force measurement system. The system has a known v n ¼ 100 rad/s, z ¼ 0.4, and
K ¼ 1 V/N. Write the expected form of the steady output signal in volts. Plot the resulting amplitude
spectrum.
Problems 115