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zv n (i.e., smaller t). Nevertheless, for all systems with z > 0, the response eventually indicates the
steady value of y 1 ¼ KA as t ! 1.
Recall that the rise time is defined as the time required to achieve a value within 90% of the step
input. For a second-order system, the rise time is the time required to first achieve 90% of (KA À y 0 ).
Rise time is reduced by decreasing the damping ratio, as seen in Figure 3.14. However, the severe
ringing associated with very lightly damped systems can delay the time to achieve a steady value
compared to systems of higher damping. This is demonstrated by comparing the response at z ¼
0:25 with the response at z ¼ 1 in Figure 3.14. With this in mind, the time required for a
measurement system’s oscillations to settle to within Æ10% of the steady value, KA, is defined
as its settling time. The settling time is an approximate measure of the time to achieve a steady
response. A damping ratio of about 0.7 appears to offer a good compromise between ringing and
settling time. If an error fraction ðGÞ of a few percent is acceptable, then a system with z ¼ 0:7 will
settle to steady response in about one-half the time of a system having z ¼ 1. For this reason, most
measurement systems intended to measure sudden changes in input signal are typically designed
such that parameters a 0 , a 1 , and a 2 provide a damping ratio of between 0.6 and 0.8.
Determination of Ringing Frequency and Rise and Settling Times The experimental determination of the ringing frequency associated with under-damped systems is performed by applying a step
input to the second-order measurement system and recording the response with time. This type of
calibration also yields information concerning the time to steady response of the system, which includes
rise and settling times. Example 3.8 describes such a test. Typically, measurement systems suitable for
dynamic signal measurements have specifications that include 90% rise time and settling time. Adjust
Second Order Parameters.vi explores system values and response.
Determination of Natural Frequency and Damping Ratio From the under-damped system
response to a step function test, values for the damping ratio and natural frequency can be extracted.
From Figure 3.14, we see that with ringing the amplitude decays logarithmically with time towards a
steady-state value. Let y max represent the peak amplitude occurring with each cycle. Then for the first
two successive peak amplitudes, let y 1 ¼ y max
ð
Þ 1 À y 1 and y 2 ¼ y max
ð
Þ 2 À y 1 . The damping ratio is
found from
z ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 2p=lnð y 1 = y 2 Þ
2
r
ð3:18Þ
From the calculation to find the ringing frequency using Equation 3.16, the natural frequency is found
using Equation 3.17. Alternately, we could perform the step function test with y(0) ¼ KA and y 1 ¼ 0
and still use the same approach.
Example 3.7
Determine the physical parameters that affect the natural frequency and damping ratio of the
accelerometer of Example 3.1.
KNOWN Accelerometer shown in Figure 3.3
ASSUMPTIONS Second-order system as modeled in Example 3.1
FIND v n ; z
98 Chapter 3 Measurement System Behavior
15:24:53 Page 98
zv n (i.e., smaller t). Nevertheless, for all systems with z > 0, the response eventually indicates the
steady value of y 1 ¼ KA as t ! 1.
Recall that the rise time is defined as the time required to achieve a value within 90% of the step
input. For a second-order system, the rise time is the time required to first achieve 90% of (KA À y 0 ).
Rise time is reduced by decreasing the damping ratio, as seen in Figure 3.14. However, the severe
ringing associated with very lightly damped systems can delay the time to achieve a steady value
compared to systems of higher damping. This is demonstrated by comparing the response at z ¼
0:25 with the response at z ¼ 1 in Figure 3.14. With this in mind, the time required for a
measurement system’s oscillations to settle to within Æ10% of the steady value, KA, is defined
as its settling time. The settling time is an approximate measure of the time to achieve a steady
response. A damping ratio of about 0.7 appears to offer a good compromise between ringing and
settling time. If an error fraction ðGÞ of a few percent is acceptable, then a system with z ¼ 0:7 will
settle to steady response in about one-half the time of a system having z ¼ 1. For this reason, most
measurement systems intended to measure sudden changes in input signal are typically designed
such that parameters a 0 , a 1 , and a 2 provide a damping ratio of between 0.6 and 0.8.
Determination of Ringing Frequency and Rise and Settling Times The experimental determination of the ringing frequency associated with under-damped systems is performed by applying a step
input to the second-order measurement system and recording the response with time. This type of
calibration also yields information concerning the time to steady response of the system, which includes
rise and settling times. Example 3.8 describes such a test. Typically, measurement systems suitable for
dynamic signal measurements have specifications that include 90% rise time and settling time. Adjust
Second Order Parameters.vi explores system values and response.
Determination of Natural Frequency and Damping Ratio From the under-damped system
response to a step function test, values for the damping ratio and natural frequency can be extracted.
From Figure 3.14, we see that with ringing the amplitude decays logarithmically with time towards a
steady-state value. Let y max represent the peak amplitude occurring with each cycle. Then for the first
two successive peak amplitudes, let y 1 ¼ y max
ð
Þ 1 À y 1 and y 2 ¼ y max
ð
Þ 2 À y 1 . The damping ratio is
found from
z ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 2p=lnð y 1 = y 2 Þ
2
r
ð3:18Þ
From the calculation to find the ringing frequency using Equation 3.16, the natural frequency is found
using Equation 3.17. Alternately, we could perform the step function test with y(0) ¼ KA and y 1 ¼ 0
and still use the same approach.
Example 3.7
Determine the physical parameters that affect the natural frequency and damping ratio of the
accelerometer of Example 3.1.
KNOWN Accelerometer shown in Figure 3.3
ASSUMPTIONS Second-order system as modeled in Example 3.1
FIND v n ; z
98 Chapter 3 Measurement System Behavior