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Apart from the transducers using resistance, inductance, capacitance we have some more
transducers for measurement of displacement. I am bringing under that three different transducers
piezoelectric transducer and fiber optic displacement transducer and seismic absolute displacement
transducers. Among various things these three are widely used and now regarding piezoelectric
transducers we know that there are natural crystals, when we give vibrations they produce
charge or when we give displacement to the crystal it produces a charge or voltage. Similarly
if we give that is reversible action if you give a voltage, an AC voltage to the same
frequency it vibrates. So this is the property what is made use of in the piezoelectric transducers
we have crystals like quartz, tourmaline, Rochelle salt. These are
the three natural crystals which exhibit this piezoelectric, it's called piezoelectric property
and we have synthetic crystal also. These are natural crystals and synthetic crystals
also are available which produces the same effect and some of them is lithium sulphate,
ammonium dihydrogen phosphate and so on, some more crystals also available.
Among these three crystals the quartz is very often used, it is because it has got moderate
vibrations and cheap and available also, whereas tourmaline we get very small vibrations but
very strong and the Rochelle salt, it has got a very large amplitude of vibrations but
very weak this is very weak and the tourmaline is strong but use very small amplitude of
vibrations. So quartz is moderate vibrations and cheap and it's available hence you find
wherever we go for piezoelectric crystals we select conveniently quartz crystal unless
otherwise there is a demand for the other one.For example the natural frequency of the
crystal depends upon the thickness. Suppose this t is the thickness, the natural frequency
say omegan is equal to K by t were K is a factor decided by the cut of the crystals.
For example quartz crystal you have got, it's a hexagonal prism so it is somewhat like this
hexagonal pyramidal structures so this is the axis of the crystal and now in making
a slab the transducer is made in terms of slab. You can cut slab either here or in any
directions it's called say it is a y cut. This is x cut and xy cut like that they make
slabs and with reference to geometric axis or optical axis they cut a different slabs
and depending upon the cut its characteristics also varies. Suppose say if it is an x cut
then it will respond to the compression say here displacement we are giving it may respond
to that.
So if it is a y cut probably it will respond to shear only, the compression will not give
any signal. So like that depending upon the cut the crystal behaves and when we give AC
signal for this, now the transducer is made like this. The top surface and bottom surface
are metallic coated and we can give our supply and you find at the natural frequency of the
crystal then the whole circuit will draw the maximum current. So whatever the frequency
at which the maximum current flows through the circuit at same time vibrating this, that
is the natural frequency of the crystal. Now the natural frequency is again given by K
depending upon the type of cut, it is a constant given by the manufacturer divided by the thickness.
So when we want a higher natural frequency so up to 10 megahertz you can go for the quartz
crystal. If you want more than 10 megahertz natural frequency then the thickness becomes
too small for a very high natural frequency thickness becomes too small then under vibrating
conditions this quartz crystal fails.
In order to avoid this is done is tourmaline since it is a strongest among the available
crystals that is selected for higher than 10 megahertz natural frequency tourmaline
is selected though it gives rise to smaller amplitude of vibrations. But the crystal is
cut like this and with metallic coating on two surfaces we can take away the charge developed
in the crystal. Our displacement signal is given by compressing, this is d to the extent
it compress then you will get the output voltage that is your displacement is transduced into
a voltage that is what we want to achieve here.You find how this behaves this, now we
have got metallic plate and in between crystal which is a non-conducting material it’s
a good dielectric material. So essentially it constitutes two parallel plates with a
dielectric medium that means it is a parallel plate capacitor. So it behaves like a parallel
plate capacitor whose property we have found yesterday. In a simple circuit with one capacitor
in series with a resistance then across the resistance we take the output, that output
we find it is there only for the dynamic signal. For static displacement this charge will be
zero as per the pervious class, this single capacitor responds only for a dynamic signal.
So the displacement what we give should be of changing magnitude that is it should have
certain frequency as per our derivation yesterday, the minimum frequency should be 3.04 over
tau where tau is the time constant of the circuit and beyond that frequency alone you
will get the charge here that is the limitations. Otherwise this is crystal also is made use
of for transducing displacement into a voltage. Now the next one but what we want to see is
fiber optic displacement transducer.
It has got some special advantage in the sense here we don’t carry any voltages, only the
light is taken through fiber optic, each fiber is of the order of 20 micrometer diameter
and you will have hundreds of fibers in a bundle. This is a cable through a cable it
will be taken to an instrument where there is a photo detector and the associative instrumentation,
where you will have the reading amount of light reflected will be taken and shown as
the reading there. Suppose we want to measure a displacement d of a machine number, you
attach a flag like this and then this is a probe tip, it is fixed there. So with reference
to the probe tip, this plate will be moving to and fro this is a fixed surface against
this. What is happening? Half of the number of, now it will be if you see in the end view
this bundle you will find the glass fibers glass fibers will be arranged in this fashion
and you will find half of them will be bringing light towards the target and half of them
will be carrying the reflected light. Suppose when we say the xi zero or the displacement
zero that means it is budding against the probe tip, suppose it is budding no gap is
there xi is zero. In that situation the incoming light is not I mean is not reflected anymore.
So when there is no reflection the pickup fibers will not take up any light and that
means the photo detector inside the instrument will not receive any light then you will find
the voltage output of the instrument is zero. When xi is zero voltage output is zero. Now
as this moves away from this point the light carrying fibers we find the light beam travels in so to say with
a conical fashion and you will find depending upon the distance the area, this is the area
of reflection. The area of reflection depends upon the distance from this tip end, the more
distance means bigger area because the light will be dispersing like this. So depending
upon the area reflected light also increases so as xi increases from zero, you find reflected
light is increased and hence the pickup fiber picks more light and that goes to the photo
detector, more light comes more voltage will be produced but that is only up to a particular
distance. Later on what will happen the other light lost to the atmosphere becomes more
and more, you will find the voltage output of the instrument starts dropping down.
So you find for a particular distance you can measure that is how this is a linear range
in which the instrument can be made use of, say from zero to 1 mm or zero to 4 mm, up
to 4 mm such fiber optic displacement transducer is available. The main advantage is even if
we have the power line nearby 50 hertz; it will not induce any error. That was the case
in earlier instrumentation wherever we have a wire carrying signal, in that wire the nearby
power line also induces voltage that will be superimposed over the signal so it becomes
noise. Such a noise cannot present in such type of instrumentations, it is free from
such a magnetic disturbances. Since it is light rays passing through they are not affected
that is the main advantage.
Second advantage is it’s a non-contact type of transducer so the measuring instrument
doesn’t touch the moving member. Hence you find no loading effect in such an instrumentation.
That is a main advantage of fiber optic displacement transducer; many American firms are making
this type of transducers. Lastly we are going to see under this topic displaced measurement,
seismic absolute displacement pickup.
Now it is unique in the sense so far what you have learnt, all the transducers they
are called relative displacement transducers. That is if you consider an LVDT, LVDT is a tubular construction there
we have your core and now you find the core is given to the machine member whichever is
moving to and fro at the cylindrical portion fixed to the ground to the frame, it will
not move. So relative to this that is one of the two members of this transducer fixed
to the frame the other member is connected to the moving one. So it’s a relative to
one member, the other member is core that motion is measured by this transducer that
is our LVDT or in any other transducers in self-inductance pick up also the coils are
fixed and the flag is moving. Flag and the coils constituting transducer, one part is
fixed other part is moving but here the main difference is the whole instrument is fixed
to the body. This is the body suppose this may be the body which is moving, this may
be a table moving up and down.
So this is the instrument, as this seismic is something to do with mass. So this is the
whole instrument is mounted over the moving body in contrast to making a part of it fixed,
part of it moving that is not there that is why it is called absolute displacement. Since
the whole instrument is fixed on the whole vibrating or moving body. What is a construction?
As per the name seismic we have a mass and the mass is supported by a spring with a spring
constant Ks and a damper with a damping coefficient B, damping coefficient is Newton per unit
velocity that is Newton per meter per second. So here it is spring constant that Newton
per millimeter or Newton per meter this is kilogram and it is the one end of the spring
is fixed to the mass, other end to the frame of the instrument and the cylindrical portion
to the frame and the pistol portion to the mass and this is the frame of the instrument.
Now you find the whole instrument is fixed to the moving body whose displacement we are
interested to measure. Now the theory of function is like this, this is a construction. Theory
of functioning is, suppose the body is moving up, this is the direction in which. So for
any displacement xi there should be an acceleration also for the motion and the mass also getting
accelerated. Whatever the motion we give this also gets
accelerated, to accelerate any mass we require a force for this mass M that force can come
to the mass only through the spring and the damper. Now when the vibrating body moves
through a distance of xi, the mass moves through a distance of xM, x capital M and there may
be difference between x and xM that is what is known as xo output motion of the mass.
So the difference between these two xi minus xM that is appearing as a relative displacement
between the mass and the frame of the instrument that is xo so the xo is equal to xi minus
xM. Now the relative motion between the mass and the frame that is the compression part
or elongation in the spring, so spring constant times the xo will be the force flowing through
the spring.
Similarly corresponding to xo we have xo dot velocity so xo dot times B because B is Newton
per unit velocity. Now velocity is xo dot so B into xo dot so these are the two forces
transmitted to the mass. So K into xo plus B into xo dot is the force accelerating the
mass M that is M into xM two dot xM is the motion of the mass so this is the basic equation
using Newton’s law of motion. Now xM we know xM is equal to xi minus xo so write M
into xi to dot minus xo two dot, arranging all xo terms to one side M xo two dot plus
B xo dot plus Ks xo is equal M xi two dot and dividing whole thing by Ks the coefficient
of xo we have got this equation and now substituting the undamped natural frequency.
Undamped natural frequency omegan is equal to root of Ks by M where Ks is the spring
constant and M is the mass of the body and damping ratio now it's not damping coefficient
it is damping coefficient is d damping ratio. Let damping ratio be psi is equal to B by
two into root of Ks into M, these are two constants and D is the differential operator
we know already, D is equal to d by dt. Then substituting these terms in this equation
we arrive at this equation D squared by omegan squared plus 2 psi D omegan plus 1 into xo
is equal to D squared by omegan squared into xi. Now arranging these terms xo by xi in
terms differential operator D you receive this equation D squared omegan squared and
so on. Now to find the frequency response of this setup substitute D by i omega, D is
equal to i omega that is our standard method of finding frequency response. So by i omega
putting it, so minus i omega square whole square is equal to minus one so you have got
this equation. Now the magnitude of this, this is a complex number so magnitude of this
equation is equal to minus beta square minus you can forget also because it is a magnitude,
a root of 1 minus beta square whole square plus 4 psi square into beta squared and the
phase difference phi is equal to this minus term will give rise to 180 degree plus tan
minus 1 of 2 psi beta by beta square minus 1, these are two equations arrived from these
differential equations.
Now how the magnitude ratio varies? How the phase difference varies? Plot this for different
values of psi now here beta is equal to omega by omegan that is frequency ratio, frequency
of vibration of the body of the table to the natural frequency of the instrument root of
Ks ,they are made up of a spring constant and the mass. So that is of frequency ratio
beta is equal to omega by omegan. Now plotting those two equations magnitude ratio and phase
difference you are getting these two curves, these two curves. You can also plot versus
beta also the ratio then it becomes one. Now we find this will be one, beta is the ratio
so that is resonant condition omega by omegan is equal to one so this one here. Now you
will find when psi is equal to zero or psi is equal to 0.2 we have got; anyhow from that
equation one can find when beta is zero you find the whole thing is zero, the magnitude
ratio is zero.
So you find when beta is zero, omega is zero that is static displacement. If you give any
static displacement stop there. You don't get any output xo will be zero for any static
input of xi. That is what we understand that is when beta is equal to zero means you don’t
get any output from here and the ideal condition is we get a xo as xi that is as one that is
around one you will have only from this beta, this is your beta minimum for measurable range,
from should be around one. So you find initially zero but later on as the beta increases, it
increases and when psi is small value you have got some oscillations, some peak value
and then against stabilizes towards one. One is the ideal condition for a beta minimum
or for larger value of beta only we have got the measurement that means the instrument
can function only when the measurement can be made only when beta is greater or equal
to beta minimum.
Now beta minimum may be say here if it is one, beta minimum may be 2.5 that means omega
minimum is equal to 2.5 times omegan. Beta is omega by omegan so omega minimum is equal
to 2.5 times omegan that is natural frequency 2.5 time natural frequency of systems then
only the measurement range starts. What is the physical explanation? That is omega is
the vibrations at which the input signal is vibrating or moving and when I say at only
very high frequencies only we can measure the input displacement. That means when the
frequency is too high what happens the mass becomes more or less static. It is not able
to follow the motions; it is because frequency is very high it is not able to move. so it
becomes more or less static say xM becomes static that means xM is zero, it is unable
to follow the vibrations it is more or less become static in space. When xM is equal to
zero then we find xo is equal to xi at that time we find whatever it is xo it becomes
xi that is the physical explanation for this functioning of seismic pickup for displacement
measurement. So now omegamin is we have found omegamin is 2.5 n so omegamin we want to have
say 2.5 times omegan. If we want to have a smaller value for omegamin
because we would like to measure from lower frequency onwards then omegan should be minimum
it should be a smaller value. What is omegan? We know omegan is made up of root of Ks by
M we want this smaller valve so that we can get a smaller omegamin then either M is large,
either you make the mass of the body inside instrument large or you make a soft spring,
both the methods will lead to a smaller omegan. So to make omegan either go for soft spring
or large mass. What is the disadvantage of having a large mass here? This may give rise
to loading effect but the amplitude of vibration may be dampened. So this is not preferable
hence go for a softer spring. By having a softer spring you have a smaller omegan by
having smaller omegan you have got smaller value for omegamin so that you can use this
instrumentation at lower frequency of vibration itself. This brings to the close the different
transducers used for displacement measurement. At the end let us see a worked out problem.
So it is like this, a seismic vibration pickup has damping ratio 0.7 and the undamped natural
frequency as 8 hertz. It is mounted on a table vibrating at a frequency of 16 hertz. If the
measured displacement xo is 1.2 mm what is the amplitude of vibration of the table that
is xi and the error of measurement. What is the error of measurement? That is one part
of the problem, second part is if the mass of the seismic instrument is 0.2 kilogram
what are the values for the spring constant and the damping coefficient? So naturally
the approach for this is what is the ratio of displacement measurement xi and xo that
is our ratio of the displacements that’s magnitude ratio for the differential equation.
That is equal to beta square root of 1 minus beta squared whole squared plus 4 psi squared
beta square that is the magnitude ratio of the displacements of xo to xi.
Now here beta is given as 2, the omega by omegan; omega is 16, omegan is 8 so beta is
equal to 2 and psi is given as 0.7. So substitute for beta and psi then you will get this value
here as 0.975. So now xo is given, xo is 1.2 by xi is equal to 0.975 giving rise to xi
is equal to 1.231 millimeter that is 1.2 millimeter xo, so xi the amplitude of vibration of table
happens to be 1.213.
Now then what is the error of measurement? Definition of error is measured value minus
theoretically value by theoretically value. So error is equal to measured value is 1.2
mm, theoretical value is 1.231 by theoretically value 1.231 into 100 percentage of error so
this comes about minus 2.52 % so this is the error of measurement.
So we see within allowable limit we are able to measure, by having this seismic absolute
displacement measurement provided the vibration frequency is much larger than the natural
frequency. Here it is 2 so it is giving within that error zone. So plus or minus 3 is acceptable
so -2.52 since vibrating frequency is large then xo becomes more or less same as xi.
That’s what we have demonstrated which we saw previously. Physically we saw that it
should happen like that so it’s a demonstration. Now the mass of the body within the instrument
is given as 0.2 kilogram whatever the other things, we can easily find out by using the
equation omegan is equal to root of Ks by M. Now M is given, omegan is 8 hertz, 8 hertz
is equal to 2 phi into 8 radians per second so M is given so Ks you are getting it simply
in the equation you substitute, so Ks becomes 505 Newton per meter and now we know the damping
ratio psi is equal to B by 2 into root of Ks into M. Now psi is given as 0.7, Ks we
already found out, M we know already 0.2, so B is equal to 14 Newton per meter per second,
14 Newton per unit velocity meter per second so second comes up, so these are the value.
So that is how we demonstrate the principle of function of absolute seismic displacement
pickup that is only worth concept we demonstrated.
Now we move on to the next topic, we have completed the displacement measurement. Next
we go to velocity measurement now velocity we know there are two quantities involved,
one is length another is time because velocity is meter per second. For meter we know and
for making any measurements we need a standard without standard we cannot assign a number
to the parameter. So for displacement that unit is based on which we are going to assign
number is meter. So for meter we have already unit but now the time unit meter per second
so what is the principle of the unit for time. Second is the basic unit. So in terms of second
we assign number for duration.
Now the latest definition for second is the so many cycles of automatic resonant frequency
of cesium 133 when it resonates at its resonant atomic resonate frequency. Then so many cycles
occurs in one second 192631770, so many cycles so after radiations comes in one second, say
in one second so many cycles come. So that means 10 figures are there or 10 to power
of 10. So that means if you can have one cycle one over 10 to power of 10 of part of your
second itself can be measured. That is the beauty of this one, so one in 10 to the power
of 10 parts of a second can be measured by, we have a cesium 133 and this cycle.Anyhow
the standard for the second is the second is duration in which so many cycles of atomic
resonant frequency comes out of cesium 133 that is the basis for the second. Now the
time is in terms of seconds we can measure, that is second is the unit. Later on I mean
multiplication of second, minutes and hours are there for longer durations. So these are
the units for the velocity measurements, meter we know but velocity is measured not only
for a linear motion but also rotary motions, velocity that is so many rpm like that we
are measuring. For that what is rotary motion? Rotary motion is angular so many angles or
so many rotations. Now angle is made up of again and unit for angle is radian. Radian
is not an independent quantity but it is a derived unit, a radian is one which is suspended
by an arc equal in length to the radius.
This is radius and this also should be radius whatever is subtended. So you find it is all
length units, radian is made up of length units but we have or angle blocks so called
angle blocks available in metrology lab. So for definite angles 10 degrees, 5 degrees,
10 degrees and so on with an accuracy of plus or minus 0.1 second of arc it is available
because one degree and one degree is made up of 60 minute of arc and each minute of
arc is made up of 60 second of arc. So you have got up to 0.1 second of arc accuracy
angle blocks are there so you can build number of angle blocks to obtain any angular rotations
or any angular positions.
For any angle you can stack the angle blocks and get that angle required angle within this
accuracy. So with that we have the basis for the angular motion also though it is in terms
of length but we have got angle block themselves for giving the standards. So these are the
standards for velocity which is made up of either length or angle and second also we
have seen so the unit for the velocity measurement is there. Now the velocity measurement is
done under three circumstances. The object may be in continuous motion and that time
also we are supposed to measure velocity just like vehicle on roads vehicle on roads, we
are interested to measure the velocity at what speed we are traveling in our scooter
or car or train we are interested that velocity also is to be measured and there are also
instances that is continuous motion.
First is continuous motion and second category is vibratory motion for example a piston motion
inside a cylinder. This may be a hydraulic cylinder we give the pressure oil and the
pressure goes out. When you supply pressure oil it moves from one end to the other end
and then later on we give a supply here and this output so we switch on by using a directional
valve, you can switch on the supply here or thereby making the piston move to and fro
within the cylinder. That is called a reciprocating motions or vibratory motions and at the two
ends the velocity is zero, here it is zero here also zero, in between it attains a maximum
velocity.
So this is variation of velocity for a vibratory motion, at the two ends velocity is zero but
in between it will be varying continuously. Whereas in continuous motion vehicle moves
for a long durations at a constant speed or variable speeds and the third category is
rotating speed. An article rotates about an axis that velocity also we are interested
to measure. So these are the three circumstances in which you are asked measure velocity, continuous
motion, vibratory motion and rotating motion. Now for a measurement of continuous motion
we have got different principles and say one for continuous motion.
So first we have got an eddy current drag up tachometer it’s an instrument which measures
essentially the rotating speed of one of the wheels of the vehicle say scooter or car it
measures the rotating speed. The details you will learn little later under rotating measurement
and rotating speed times the wheel circumference is the linear velocity. So it is converted
in terms of kilometer per hour this is essentially a rotation measuring instruments with which
it is calibrated in terms of kilometer per hour by considering the circumference of a
wheel. Then second we have got the Pitot tube for the vehicles so aeroplane as well as the
ships on sea also we can use Pitot tube. You know pitot tube is used in pipes to measure
the velocity of fluid in a pipe line so this is a pipe line where it is fixed, this is
pitot tube you take this is your p stagnant pressure and we have got the static pressure,
this is p static they are connected to a manometer tube and the difference between the column in the manometer say h,
this is proportional to v square, v is the velocity of the fluid.
This is the Pitot tube principle which we will learn detailed in flow measurement but
the same thing is put in a ship like this; this will be projecting like this. So you
find now ship is moving so to say in a pipe pitot tube is stationary, fluid is moving
when we fix to a pipe the pitot tube is moving over the stationary water, this is sea water
stationary water it moves. So this is reverse of this condition, instead of fluid moving
here Pitot tube moves, the other one is stationary giving rise to same effect. So at the tip
we will have both static pressure times velocity head and the static pressure measured a perpendicular
directions and a difference between these two things as h which is here probably you
can have mercury that this head is proportional to v squared over 2 g. So from there we find
out the velocity that is the velocity with this the ship moves. You know this is by using
pitot tube and third method is we have got the accelerometers. You can also mount an
accelerometer in a ship so when it measures the acceleration of the ship, integrate that
then it gives rise to the velocity. So by measuring the acceleration, integrating that
acceleration you get the velocity also for the ship. This is third method integrating
acceleration measurement. Fourth method is Doppler effect this is what the traffic police
will use this effect to find out the velocity of any racing vehicle on a road.
Suppose on a road the maximum speed is fixed there say 30 kilometer per hour if any vehicle
travels more than that speed then police staff will stop the vehicle and fine it. So for
that purpose they use the Doppler Effect that means they have a source of light or sound
for directing it towards the moving vehicle and the reflected light, they reflect light
or sound they pick up and it is processed inside because the reflected sound or light
will have the information of the velocity of the moving body. From that they find out
the velocity of the vehicle and then if it is more than the permitted value then they
fine the driver whatever it is. So these are the four different methods which are used
nowadays to find the velocity of a continuously moving vehicle. That is in brief now we go
to the next one that is objects in vibratory motions, there we have few methods.
First method is suppose this body is there, the piston is moving within the cylinder and
you can connect an LVDT for example this is an LVDT. To measure the displacement at any
instance of the piston and then you give it to a differentiator and you get a velocity
e proportional to velocity v. So by measuring the displacement by a displacement transducer
and differentiating it we get the output voltage proportional to velocity. This is one method
differentiation or we can also integrate by mounting an accelerometer on this moving body
then integrating the acceleration we get the velocity.
So you can adopt any one of these two but the problem with a differentiation is we know
from LVDT the output voltage will have some ripples, 5% ripples because that signal comes
out of the low pass filter. That ripple when it is differentiated gives rise to peaks called
noise in the velocity measurement whereas in integration of acceleration such noise
will not be there. Hence integrating an acceleration measurement is suitable than the differentiating
a displacement measurement, so that is one method. Second method is we can find the average
velocity probably we are not interested from point to point what is velocity.
So we can measure for this whole travel but in two points whatever be the average velocity
we can measure and for that we can go for a measurement like this and the moving body
you fix it to iron strip and have in front a proximity pickup. So whenever the iron piece
moves in front of the proximity pickup if you connect the proximity pickup to a recorder
switch or recorder you will have two voltages because when IMP comes near the proximity
pickup you have pulse impulse is produced. Voltage pulse is produced that is being recorded
and the time between the two pulses is t and the distances between the two strips are d
then the velocity is equal to d by t. So this is average velocity in many instances it is
sufficient to find out the effect of roughness on the velocity or the supply pressure of
the velocity, we can use go for an average pressure. Whenever we don’t require instantaneous
pressure this average pressure measurement is sufficient.
Then next method is moving coil, moving magnet pickup, it is widely used here you find voltage
developed e is equal to b v l volt where b is the flux density of this permanent magnet
in tesla, this is in tesla and v is the velocity of, we can have either coil may be moving.
You can connect a moving member to the coil or to the magnet, anyone can be connected
to the body and now the relative velocity between the coil and the magnet is important
that is v that is meter per second and l is the length of the wire in meter then we find
the voltage developed is volt. So by this method we can find out the velocity of the
moving body. What is sensitivity of this measurement? This is output voltage, input is v so de by
dv is the sensitivity that is equal to B into L that is the flux density times the length
of the wire. So to increase sensitivity you can increase any one of these two things,
so the flux density is developed depending upon the material you have selected for the
magnet. So that you can have maximum one tesla that is available by having proper selection
of the material and length should be very large. If length is very large what happens,
the weight of the coil increases. Suppose you have connected the coil to the moving
body then it becomes a loading effect, the velocity may be dampened. So to reduce L and
having I mean we want to have smaller mass, for a given mass we want to have larger length
then reduce the cross section of the wire and have a longer length.
So when you reduce a cross section the R is increased, resistance of the wire is increased
because R is equal to row L by A, A is in the denominator so R is increased. So to measure
the voltage you require a voltmeter of very high input resistance that is a draw back.
So you cannot increase sensitivity as it is there is a limit. Otherwise this is the method
what is very often used in measuring velocity in respirating bodies. Moving coil moving
magnet pickup it is simple principle, voltage developed is given by flux density time’s
relative velocity of the coil with reference to magnet into length of the wire. So as the
coil moves up and down it cuts the magnetic lines, so voltage is produced proportional
to the relative velocity of the coil to magnet. So the other method is seismic velocity pickup.
It is same construction as the absolute seismic displacement pickup; it is similar except
we had a relative displacement we were measuring earlier. Now we have to measure the relative
velocity, it is obtained by having this equation like this.
So we have already derived xo by xi i omega is equal to d squared over, we will say d,
d squared over omegan squared by d squared over omegan squared plus 2 psi d by omegan
plus one we already derived for the displacement, same thing for mass here also it will give
rise to. Now for velocity add d on both sides then it becomes xo dot to xi dot same thing
it remains same, right hand side remains the same. Now this we know the curve for this
right hand side that is what we had beta as beta this is one. This is xo dot by xi dot
i omega that is magnitude, now we know it starts somewhat like this. So suppose this
is the one it comes like this. So same curve what we had seen for an absolute displace
measurement that means the measurement range starts from beta minimum same considerations
so only when more than beta minimum then only it can measure the input velocity in terms
of output velocity so we are measuring output velocity that is the relative motion of mass
with reference to frame that velocity is measured. Relative velocity of the mass with reference
to frame or the instrument is measured as xo dot. So for measuring xo dot I have used
here the moving coil stroke moving magnet pickup we have used. Here coil is connected
to the mass.
So relative motion of this mass with reference to the frame because this magnet is fixed
to the frame. So you can find xo dot is given by k that is voltage output of that e is equal
to K time’s xo dot. So you can substitute in terms of a xo dot by e by K so xo dot you
can write as e over k that is again one that is we are measuring the output voltage as
velocity so same considerations we should have. Here also Ks should be small so Ks is
small for having beta minimum a smaller value, here frequency range should be smaller so
that we can start from lower velocities. That is soft spring we go for the soft spring,
instead of having a larger mass we can have a softer spring so that no loading effect
at the vibrating body.
So under the same situations we can also measure the relative velocity of the body. The only
difference between the displacement velocity measurements is the instrumentation for measuring
the relative displacement earlier we have something like LVDT but here we should have
the moving coil or moving magnet that is relative velocity should be measured that is to be
interpreted in terms of the input velocity. This is the input velocity xi dot we get in
terms of xo dot that is possible only when the beta is more than beta minim this is measurable
range. So these are the various methods used for measuring the vibratory motion.