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the abruptness and magnitude of resonance, but under-damped systems may still achieve resonance.
This region on Figures 3.16 and 3.17 is called the resonance band of the system, referring to the
range of frequencies over which the system is in resonance. Resonance in under-damped systems
occurs at the resonance frequency,
v R ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 2z
2
q
ð3:23Þ
The resonance frequency is a property of the measurement system. Resonance is excited by
a periodic input signal frequency. The resonance frequency differs from the ringing frequency of
free oscillation. Resonance behavior results in values of M(v) > 1 and considerable phase shift.
For most applications, operating at frequencies within the resonance band is undesirable,
confusing, and could even be damaging to some delicate sensors. Resonance behavior is
very nonlinear and results in distortion of the signal. On the other hand, systems having z >
0.707 do not resonate.
At small values of v=v n , M(v) remains near unity and F(v) near zero. This means that
information concerning the input signal of frequency v will be passed through to the output signal
with little alteration in the amplitude or phase shift. This region on the frequency response curves is
called the transmission band. The actual extent of the frequency range for near unity gain depends
on the system damping ratio. The transmission band of a system is either specified by its frequency
bandwidth, typically defined for a second-order system as À3 dB M(v) 3 dB, or otherwise
specified explicitly. You need to operate within the transmission band of a measurement system to
measure correctly the dynamic content of the input signal.
At large values of v/v n , M(v) approaches zero. In this region, the measurement system
attenuates the amplitude information in the input signal. A large phase shift occurs. This region is
known as the filter band, typically defined as the frequency range over which M(v) À3 dB. Most
readers are familiar with the use of a filter to remove undesirable features from a desirable product.
When you operate a measurement system within its filter band, the amplitudes of the portion of
dynamic signal corresponding to those frequencies within the filter band will be reduced or
eliminated completely. So you need to match carefully the measurement system characteristics with
the signal being measured.
Example 3.9
Determine the frequency response of a pressure transducer that has a damping ratio of 0.5 and a
ringing frequency (found by a step test) of 1200 Hz.
KNOWN
z ¼ 0:5
v d ¼ 2p 1200 Hz
ð
Þ¼7540 rad/s
ASSUMPTIONS Second-order system behaviour
FIND M(v) and F(v)
SOLUTION The frequency response of a measurement system is determined by M(v) and
F(v) as defined in Equations 3.20 and 3.22. Since v d ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
p
, the natural frequency of the
pressure transducer is found to be v n ¼ 8706 rad/s. The frequency response at selected frequencies is
computed from Equations 3.20 and 3.22:
102 Chapter 3 Measurement System Behavior
15:24:53 Page 102
the abruptness and magnitude of resonance, but under-damped systems may still achieve resonance.
This region on Figures 3.16 and 3.17 is called the resonance band of the system, referring to the
range of frequencies over which the system is in resonance. Resonance in under-damped systems
occurs at the resonance frequency,
v R ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 2z
2
q
ð3:23Þ
The resonance frequency is a property of the measurement system. Resonance is excited by
a periodic input signal frequency. The resonance frequency differs from the ringing frequency of
free oscillation. Resonance behavior results in values of M(v) > 1 and considerable phase shift.
For most applications, operating at frequencies within the resonance band is undesirable,
confusing, and could even be damaging to some delicate sensors. Resonance behavior is
very nonlinear and results in distortion of the signal. On the other hand, systems having z >
0.707 do not resonate.
At small values of v=v n , M(v) remains near unity and F(v) near zero. This means that
information concerning the input signal of frequency v will be passed through to the output signal
with little alteration in the amplitude or phase shift. This region on the frequency response curves is
called the transmission band. The actual extent of the frequency range for near unity gain depends
on the system damping ratio. The transmission band of a system is either specified by its frequency
bandwidth, typically defined for a second-order system as À3 dB M(v) 3 dB, or otherwise
specified explicitly. You need to operate within the transmission band of a measurement system to
measure correctly the dynamic content of the input signal.
At large values of v/v n , M(v) approaches zero. In this region, the measurement system
attenuates the amplitude information in the input signal. A large phase shift occurs. This region is
known as the filter band, typically defined as the frequency range over which M(v) À3 dB. Most
readers are familiar with the use of a filter to remove undesirable features from a desirable product.
When you operate a measurement system within its filter band, the amplitudes of the portion of
dynamic signal corresponding to those frequencies within the filter band will be reduced or
eliminated completely. So you need to match carefully the measurement system characteristics with
the signal being measured.
Example 3.9
Determine the frequency response of a pressure transducer that has a damping ratio of 0.5 and a
ringing frequency (found by a step test) of 1200 Hz.
KNOWN
z ¼ 0:5
v d ¼ 2p 1200 Hz
ð
Þ¼7540 rad/s
ASSUMPTIONS Second-order system behaviour
FIND M(v) and F(v)
SOLUTION The frequency response of a measurement system is determined by M(v) and
F(v) as defined in Equations 3.20 and 3.22. Since v d ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
p
, the natural frequency of the
pressure transducer is found to be v n ¼ 8706 rad/s. The frequency response at selected frequencies is
computed from Equations 3.20 and 3.22:
102 Chapter 3 Measurement System Behavior