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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