E1C03 09/14/2010
15:24:55 Page 111
The overall system magnitude ratio is the product
M v
ð Þ ¼ M 1 v
ð ÞM 2 v
ð Þ. . . M H v
ð Þ
ð3:42Þ
and the overall system phase shift is the sum
F v
ð Þ ¼ F 1 v
ð Þ þ F 2 v
ð Þ þ Á Á Á þ F H v
ð Þ
ð3:43Þ
This holds true provided that significant loading effects do not exist, a situation discussed in
Chapter 6.
3.8 SUMMARY
Just how a measurement system responds to a time-dependent input signal depends on the properties of
the signal and the frequency response of the system. Modeling has enabled us to develop and to illustrate
these concepts. Those system design parameters that affect system response are exposed through
modeling, which assists in instrument selection. Modeling has also suggested the methods by which
measurement system specifications such as time constant, response time, frequency response, damping
ratio, and resonance frequency can be determined both analytically and experimentally. Interpretation of
these system properties and their effect on system performance was determined.
The rate of response of a system to a change in input is estimated by use of the step function
input. The system parameters of time constant, for first-order systems, and natural frequency and
damping ratio, for second-order systems, are used as indicators of system response rate. The
magnitude ratio and phase shift define the frequency response of any system and are found by an
input of a periodic waveform to a system. Figures 3.12, 3.13, 3.16, and 3.17 are universal frequency
response curves for first- and second-order systems. These curves can be found in most engineering
and mathematical handbooks and can be applied to any first- or second-order system, as appropriate.
REFERENCES
1. Close, C. M., and D. K. Frederick, Modeling and Analysis of Dynamic Systems, 2nd ed., Wiley,
Boston, 1994.
2. Doebelin, E. O., System Modeling and Response, Theoretical and Experimental Approaches,
Wiley, New York, 1980.
3. Ogata, K., System Dynamics, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 2003.
4. Palm, W. J., III, Modeling, Analysis and Control of Dynamic Systems, 2nd ed., Wiley, New
York, 2000.
5. Burgess, J.C., A quick estimation of damping from free damped oscillograms, Wright Air
Development Center, WADC TR 59-676, March 1961, pp. 457–460
NOMENCLATURE
a 0 ; a 1 ; . . . ; a n physical coefficients
b 0 ; b 1 ; . . . ; b m physical coefficients
c
damping coefficient ðm t
À1
Þ
f
cyclical frequency f ¼ v=2p
ð
Þ
(Hz)
k
spring constant or stiffness ðm t
À2
Þ
m
mass (m)
p(t)
pressure ðm l
À1
t
À2
Þ
t
time (t)
x(t)
independent variable
y(t)
dependent variable
y
n
nth time derivative of y(t)
Nomenclature 111
15:24:55 Page 111
The overall system magnitude ratio is the product
M v
ð Þ ¼ M 1 v
ð ÞM 2 v
ð Þ. . . M H v
ð Þ
ð3:42Þ
and the overall system phase shift is the sum
F v
ð Þ ¼ F 1 v
ð Þ þ F 2 v
ð Þ þ Á Á Á þ F H v
ð Þ
ð3:43Þ
This holds true provided that significant loading effects do not exist, a situation discussed in
Chapter 6.
3.8 SUMMARY
Just how a measurement system responds to a time-dependent input signal depends on the properties of
the signal and the frequency response of the system. Modeling has enabled us to develop and to illustrate
these concepts. Those system design parameters that affect system response are exposed through
modeling, which assists in instrument selection. Modeling has also suggested the methods by which
measurement system specifications such as time constant, response time, frequency response, damping
ratio, and resonance frequency can be determined both analytically and experimentally. Interpretation of
these system properties and their effect on system performance was determined.
The rate of response of a system to a change in input is estimated by use of the step function
input. The system parameters of time constant, for first-order systems, and natural frequency and
damping ratio, for second-order systems, are used as indicators of system response rate. The
magnitude ratio and phase shift define the frequency response of any system and are found by an
input of a periodic waveform to a system. Figures 3.12, 3.13, 3.16, and 3.17 are universal frequency
response curves for first- and second-order systems. These curves can be found in most engineering
and mathematical handbooks and can be applied to any first- or second-order system, as appropriate.
REFERENCES
1. Close, C. M., and D. K. Frederick, Modeling and Analysis of Dynamic Systems, 2nd ed., Wiley,
Boston, 1994.
2. Doebelin, E. O., System Modeling and Response, Theoretical and Experimental Approaches,
Wiley, New York, 1980.
3. Ogata, K., System Dynamics, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 2003.
4. Palm, W. J., III, Modeling, Analysis and Control of Dynamic Systems, 2nd ed., Wiley, New
York, 2000.
5. Burgess, J.C., A quick estimation of damping from free damped oscillograms, Wright Air
Development Center, WADC TR 59-676, March 1961, pp. 457–460
NOMENCLATURE
a 0 ; a 1 ; . . . ; a n physical coefficients
b 0 ; b 1 ; . . . ; b m physical coefficients
c
damping coefficient ðm t
À1
Þ
f
cyclical frequency f ¼ v=2p
ð
Þ
(Hz)
k
spring constant or stiffness ðm t
À2
Þ
m
mass (m)
p(t)
pressure ðm l
À1
t
À2
Þ
t
time (t)
x(t)
independent variable
y(t)
dependent variable
y
n
nth time derivative of y(t)
Nomenclature 111