Fifth Lecture

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Instrumentation and Product Testing
Fifth Lecture Dynamic Characteristics of
Measurement System (Reference Chapter 5,
Mechanical Measurements, 5th Edition, Bechwith,
Marangoni, and Lienhard, Addison Wesley.)
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Dynamic characteristics Many experimental
measurements are taken under conditions where
sufficient time is available for the measurement
system to reach steady state, and hence one need
not be concerned with the behaviour under
non-steady state conditions. --- Static
cases In many other situations, however, it may
be desirable to determine the behaviour of a
physical variable over a period of time. In any
event the measurement problem usually becomes
more complicated when the transient
characteristics of a system need to be considered
(e.g. a closed loop automatic control system).
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Temperature Control
vin - vf
vin
Ta
T
vf
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K
Output, T
Input, v

-
H
A simple closed loop control system
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System response The most important factor in the
performance of a measuring system is that the
full effect of an input signal (i.e. change in
measured quantity) is not immediately shown at
the output but is almost inevitably subject to
some lag or delay in response. This is a delay
between cause and effect due to the natural
inertia of the system and is known as measurement
lag.
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First order systems Many measuring elements or
systems can be represented by a first order
differential equation in which the highest
derivatives is of the first order, i.e. dx/dt,
dy/dx, etc. For example,
where a and b are constants f(t) is the input
q(t) is the output
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An example of first order measurement systems is
a mercury-in-glass thermometer. where ?i
and ?o is the input and output of the
thermometer. Therefore, the differential
equation of the thermometer is
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Consider this thermometer is suddenly dipped into
a beaker of boiling water, the actual thermometer
response (?o) approaches the step value (?i)
exponentially according to the solution of the
differential equation ?o ?i (1- e-t/T)
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?i
?0(T)0.632?i
?0(t)
Response of a mercury in glass thermometer to a
step change in temperature
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The time constant is a measure of the speed of
response of the instrument or system After three
time constants the response has reached 95 of
the step change and after five time constants 99
of the step change. Hence the first order system
can be said to respond to the full step change
after approximately five time constants.
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Frequency response If a sinusoidal input is
input into a first order system, the response
will be also sinusoidal. The amplitude of the
output signal will be reduced and the output will
lag behind the input. For example, if the input
is of the form ?i(t) a sin ? t then the steady
state output will be of the form ?o (t) b sin
(? t - ? ) where b is less than a, and ? is the
phase lag between input and output. The
frequencies are the same.
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Increase in frequency, increase in phase lag
(0º90º) and decrease in b/a (10).
Response of a first order system to a sinusoidal
input
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Second order systems Very many instruments,
particularly all those with a moving element
controlled by a spring, and probably fitted with
some damping device, are of second order type.
Systems in this class can be represented by a
second order differential equation where the
highest derivative is of the form d2x/dt2,
d2y/dx2, etc. For example,
where ? and ?n are constants.
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For a damped spring-mass system,
Natural frequency
(in rad/s)
(in Hz)
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Damping ratio The amount of damping is normally
specified by quoting a damping ratio, ?, which is
a pure number, and is defined as
follows where c is the actual value of the
damping coefficient and cc is the critical
damping coefficient. The damping ratio will
therefore be unity when c cc, where occurs in
the case of critical damping. A second order
system is said to be critically damped when a
step input is applied and there is just no
overshoot and hence no resulting oscillation.
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Response of a second order system to a step input
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The magnitude of the damping ratio affects the
transient response of the system to a step input
change, as shown in the following table.
Magnitude of damping ratio Transient
response Zero Undamped simple harmonic
motion Greater than unity Overdamped
motion Unity Critical damping Less than unity
Underdamped, oscillation motion
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Frequency response
If a sinusoidal input is applied to a second
order system, the response of the system is
rather more complex and depends upon the
relationship between the frequency of the applied
sinusoid and the natural frequency of the system.
The response of the system is also affected by
the amount of damping present.
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Consider a damped spring-mass system (examples of
this system include seismic mass accelerometers
and moving coil meters)
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It may be represented by a differential equation
Suppose that xl is a harmonic (sinusoidal) input,
i.e. xl xo sin ? t where xo is the amplitude
of the input displacement and ? is its circular
frequency. The steady state output is x(t) X
sin (? t - ? )
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Frequency response of a second order system
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Phase shift characteristics of a second order
system
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Remarks (i) Resonance (maximum amplitude of
response) is greatest when the damping in the
system is low. The effect of increasing damping
is to reduce the amplitude at resonance. (ii)
The resonant frequency coincides with the
natural frequency for an undamped system but as
the damping is increased the resonant frequency
becomes lower. (iii) When the damping ratio is
greater than 0.707 there is no resonant peak but
for values of damping ratio below 0.707 a
resonant peak occurs.
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(iv) For low values of damping ratio the output
amplitude is very nearly constant up to a
frequency of approximately ? 0.3?n (v) The
phase shift characteristics depend strongly on
the damping ratio for all frequencies. (vi) In
an instrument system the flattest possible
response up to the highest possible input
frequency is achieved with a damping ratio of
0.707.
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Thank you