2nd order example
TRANSCRIPT
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Fifth Lecture
Dynamic Characteristics of Measurement System
(Reference: Chapter 5, Mechanical Measurements, 5thEdition, Bechwith, Marangoni, and Lienhard, AddisonWesley.)
Instrumentation and Product TestingInstrumentation and Product Testing
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Dynamic characteristics
Many experimental measurements are taken under conditions where sufficient time is available for themeasurement 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 problemusually becomes more complicated when the transientcharacteristics of a system need to be considered (e.g. a
closed loop automatic control system).
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Temperature Control
Tvin T a
v f
vin - v f
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K
H
+Input, v Output, T
A simple closed loop control system
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S ystem response
T he most important factor in the performanceof a measuring system is that the full effect of
an input signal (i.e. change in measuredquantity) is not immediately shown at theoutput but is almost inevitably subject to somelag or delay in response. T his is a delay
between cause and effect due to the naturalinertia of the system and is known asmeasurement lag.
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F irst order systems
Many measuring elements or systems can berepresented by a first order differential equation inwhich the highest derivatives is of the first order, i.e.
d x/dt , d y
/d x, etc. For example,
t f t bqdt
t dqa !
where a and b are constants; f (t ) is the input; q(t ) is theoutput
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An example of first order measurement systems is a
mercury-in-glass thermometer.
where Ui and Uo is the input and output of the
thermometer.T
herefore, the differential equation of the thermometer is:
t dt
t d T t i
o
o UU
U !
? At t T d t t d
t t d t
t d
oio
oio
UUU
UUU
!
w
1
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Consider this thermometer is suddenly dipped into a beaker of boiling water, the actual thermometer response ( Uo ) approaches the step value ( Ui)exponentially according to the solution of the
differential equation:
Uo = Ui (1 - e -t /T )
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Ui
U0(t )
U0(T )~0.632 Ui
Response of a mercury in glass thermometer to a step
change in temperature
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T he time constant is a measure of the speed of response of the instrument or system
After three time constants the response has reached95% of the step change and after five time constants99% of the step change.
Hence the first order system can be said to respond
to the full step change after approximately five t imeconstants.
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F requency response
If a sinusoidal input is input into a first order system,the response will be also sinusoidal. T he amplitude of the output signal will be reduced and the output will lag
behind the input. For example, if the input is of the
formUi(t ) = a sin [ t
then the s t ead y s t a t e output will be of the form
Uo (t ) = b sin ( [ t - J )
where b is less than a , and J is the phase lag between
input and output. T he frequencies are the same.
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Response of a first order system to a sinusoidal input
Increase in frequency, increase in phase lag (0~90)and decrease in b/a (1 ~0).
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S econd order systems
Very many instruments, particularly all those with amoving element controlled by a spring, and probablyfitted with some damping device, are of secondorder type. Systems in this class can be represented
by a second order differential equation where thehighest derivative is of the form d 2 x/dt 2, d2 y/d x2, etc.For example,
t t d t
t d d t
t d ion
o
no UU[
U\ [
U!22
2
2
where \ and [ n are constants.
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For a damped spring-mass system,
mk
n ![(in rad /s)
m
k f n T 2
1!
(in Hz)
Natural frequency
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Damping ratio
T he amount of damping is normally specified byquoting a damping ratio, \ , which is a pure number, andis defined as follows:
where c is the actual value of the damping coefficientand c c is the critical damping coefficient. T he dampingratio will therefore be unity when c = c c , where occursin the case of critical damping. A second order systemis said to be critically damped when a step input isapplied and there is just no overshoot and hence noresulting oscillation.
k m
c
c
c
c 2!!\
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Response of a second order system to a step input
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T he magnitude of the damping ratio affects the transient
response of the system to a step input change, as shown in thefollowing table.
Magnitude of damping ratio T ransient response
Zero Undamped simple harmonic motionGreater than unity Overdamped motionUnity Critical dampingLess than unity Underdamped, oscillation motion
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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 betweenthe frequency of the applied sinusoid and the naturalfrequency of the system. T he response of the systemis also affected by the amount of damping present.
F requency response
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Consider a damped spring-mass system (examples of this system include seismic mass accelerometers andmoving coil meters)
x1 = x0 sin [ t (input)
k
m
x (output)
c
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It may be represented by a differential equation
t xt k xd t
t d xc
d t t xd
m 122
!
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.T
he steady state output is
x(t ) = X sin ( [ t - J )
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F requency response of a second order system
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P hase 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 islow. T he effect of increasing damping is toreduce the amplitude at resonance.
(ii) T he resonant frequency coincides with thenatural frequency for an undamped system butas the damping is increased the resonantfrequency becomes lower.
(iii) When the damping ratio is greater than 0.707there 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 outputamplitude is very nearly constant up to afrequency of approximately [ = 0.3 [ n
(v) T he phase shift characteristics depend strongly onthe damping ratio for all frequencies.
(vi) In an instrument system the flattest possible
response up to the highest possible inputfrequency is achieved with a damping ratio of 0.707.
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Thank you