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Fifth Lecture

Dynamic Characteristics of Measurement System

(Reference: Chapter 5, Mechanical Measurements, 5th Edition, Bechwith, Marangoni, and Lienhard, Addison Wesley.)

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 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

Tvin Ta

vf

vin - vf

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K

H

+Input, v Output, T

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,

tftbqdt

tdqa

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:

tdt

tdTt i

oo

ttTdt

td

ttdt

td

oio

oio

1

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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(T)~0.632i

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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Response of a first order system to a sinusoidal input

Increase in frequency, increase in phase lag (0º~90º) and decrease in b/a (1~0).

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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,

ttdt

td

dt

tdion

on

o 2

2

2

2

where and n are constants.

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For a damped spring-mass system,

m

kn

(in rad/s)

m

kfn 2

1

(in Hz)

Natural frequency

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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.

km

c

c

c

c 2

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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 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 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.

Frequency response

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Consider a damped spring-mass system (examples of this system include seismic mass accelerometers and moving coil meters)

x1 = x0 sin t (input)

k

m

x (output)

c

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It may be represented by a differential equation

txtkxdt

tdxc

dt

txdm 12

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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 = 0.1n

(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