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measuring reciprocating velocity we have seen last time a seismic mass mounted over a spring
and damper used as velocity for the purpose of measuring reciprocating velocity. Now this
is another version, this modified seismic velocity pickup. Here the conventional mass
is replaced by the coil because in the earlier setup we had the mass here and spring. This
is damper and to measure the relative velocity of the mass we had separately the moving coil,
moving magnet pickup like that.
Now the moving coil weight itself is taken as mass here for this instrumentation so that
is the only difference. So this was our earlier instrumentation. Now this mass and the coil
mass or this is our permanent magnet and this permanent magnet is here as it is and the
coil mass is taken as the seismic mass here. Seismic mass is the coil itself and for spring
you have got the whole mass is suspended from your spring and for damping purpose this whole
volume is filled up with oil preferably silicon oil which doesn’t decompose in course of
time.
So damping is from the filled oil, for the spring you have got this is ks, for the mass
the coil itself. So the same vibrating body the whole instrument is fixed to the vibrating
body, bottom is the vibrating body a table or whatever it is, whose velocity we are interested
to measure. So bolt it and as it vibrates more than the natural frequency of this mass
spring system say twice more than that then you will find the relative motion of the mass
with reference to the frame that is now the permanent magnet itself forms part of the
frame fixed to that, so any relative motion we have got a voltage output. So this voltage
output will represent when the frequency is much more than the natural frequency the input
velocity itself that's what you have learnt yesterday.So this is one modified version
of the earlier type of seismic pickup where we have separately a mass but the mass and
coil are combined together in this setup. So it is cheaper in the sense and simple in
construction. So this completes our measurement of reciprocating velocity. We are seeing three
types of velocities one is continuous motion, second reciprocating motion and third one
is velocity of rotating objects. This is simple to measure hence we find large number of methods
are available for measuring rotating speeds rpm or rotation per second, rotation per hour
all these things are available in plenty. We will see few of them.
The first one is the fly ball sensor; it is used since long time especially in power plants.
The turbine speed is control by having this fly ball mechanism. This fly ball sensor it
senses the speed of the turbine shaft and gives signal for the control of the speed.
So you find this fly ball mechanism that is if m is the mass of the fly ball then centrifugal
force equal to m omega squared r proportional to omega squared, omega being rotating speed.
So this force is transmitted to this collar, this is the collar which is mounted free on
the rotating shaft. Rotating shaft rotates at omega so the part of the force through
the linkage the centrifugal force is transmitted to this collar. So the collar try to raise
and in the process it is compressing a spring so proportional to the force it gets compressed
and that distance is this is our output signal xo.
So now if you plot the xo versus omega you have got some nonlinear characteristics that
is because of the quadratic equation comes into the picture but the operating speed of
the turbine will be somewhere in the middle of this linear portion of the curve. So the
speed will be varying along this then immediately if speed falls down then the governor, it
will be sensed when speed falls this comes closer and then some linkage will be actuated
and I mean turbine speed steps will be taken to increase the turbine speed. That is for
power plants operation this is widely used. Still in some other tachometers fly ball mechanism
are still used old type this is one of the conventional methods. Now we have got the
modern methods of measuring the shaft rotation but there in the rotating shaft itself we
have connected suitably the fly balls. This is the shaft where you have connected but
now we find in the rotating shaft we fixed one reflecting tape this is supplied by the
manufacturer. So you just cut a small piece and fix it and it will have its own adhesive.
You can just paste it on the shaft and this is based upon the photoelectric speed probe
but even before that we have learnt this proximity pickup. We already learnt proximity pickup,
suppose to the rotating shaft a gear is fixed. This is a gear so a gear is fixed and near
about the rim of the rotating gear we can fix a proximity pickup, this is a proximity
pickup. Whenever a tooth comes in front of the proximity pickup one pulse is given, suppose
there are 10 teeth then 10 pulses will be given and this is connected to a universal electronic counter where it can
be counted per second. So per second so many pulses will be there that divided by the ten
will give the rpm per second. You can select a different base one second or 10 seconds
whatever it is. Suppose if you have selected once again whatever be the pulse it will be
counted and that if you divide by 10 that will give you the rotating speed per second
that is the inductance principle.
The principle here is inductance principles; the inductance varies whenever a tooth comes.
So when you don’t have any gear anything like that then what you can do, you can substitute
gear by a screw. Now the shaft is there we can substitute it by a screw head just by
a screw head and now put your proximity pickup near the screw. Whenever the screw head comes
in front of proximity pickup one pulse will be made. In all these methods we are measuring
the pulse that’s why we can call it pulse counting method of speed measurement. There
we are measuring, when the reluctance, this is actually reluctance variations under inductance
principles when the reluctances varies we have got one pulse and in another method we
can have light source instead of the inductance principle you can have light source. In this
probe tip we have got light source then it contains two chambers at the top chamber now
we have the lamp and the bottom one we have got the photocell and in the rotating shaft
we fix a tape like reflecting tape and the light falls on the tape and it gets reflected.
Whenever the tape comes in front of the probe tip you get one pulse. This photocell will
be forming part of an electronic circuit so one pulse electric voltage pulse will be produced.
In case you have a disk already in or let us look in another way of measuring this using
lamp and photocell is fix a disk to the rotating shaft like this with the number of holes say
4 or 6 holes.Whenever a hole comes in front of this say this measuring station lamp is
there this side, other side photocell when the hole comes light passes through the hole
and it falls on the photocell and photocell will be forming a part of electronic circuit
and one pulse will be produced. So whenever a hole comes it will be producing one pulse.
In such measurements say either in the proximity pickup using proximity pickup inductance principle
or reluctance variation or in the reflecting type photocell or this is we call it obstruction
type photoelectric speed probe here. All these things are part of pulse counting method of
speed measurement, the error in all these methods, error is plus or minus one pulse
and how do you say plus or minus one pulse for that we just see here as an example, say
consider the disk where the hole is there as measuring station when the hole comes the
one pulse is produced.
Suppose the shaft starts rotating at this station A and it starts rotating in the direction,
it comes and stops at station B just before the measuring station then it has completed
almost one rotation but since at no time this was in front of the measuring station no pulse
is produced. So you find theoretically it is rotated for one rotation but no measurement
actually measured is zero so error is 1 minus zero plus error, this a plus 1 error, plus
one pulse is error and in another way suppose the hole started just in front and stopped
immediately after it passes through the measuring station. So it is registered as one, measured
as one pulse or one rotation because only one hole means one rotation, one pulse represents
one rotation and really if the size is small it may rotate at very few degrees only. So
in limit it may be zero so zero measured is minus one, actual rotation zero so it is minus
error so plus or minus here one rotation is the error.
Similarly if you have 4 holes so one fourth of the rotation will be the error. So as the
number of holes increases error comes down, this is the advantage of having more number
of holes. So that is the pulse counting method and next is we have DC and AC tachometer that
is in a DC generator the voltage developed is proportional to the rotor speed. So the
voltage output of the DC generator is calibrated in terms of rotation that is in terms of rpm.
So it is then called a tachogenerator, that is simple.
The ordinary DC generator is used to measure the rpm of the rotor; the voltage output we
know is proportional to rpm of the rotor. So the voltage is calibrated in terms of rpm.
Now in the AC tachogenerator what is done is, it is induction two phase squirrel cage
induction motor that can be used for measuring speed. What is done is there are 2 coils at
electrically 90 degrees and separate them out and in one of the coils you give the supply
voltage say at particular frequency say 5 kilohertz for example and in the other winding
you will get the output proportional to speed as per this diagram. Suppose omega varies
starts at a high omega and falls to zero and rotates in the negative direction then in
that case the eo output from the other one will be modulated wave like this. So again
if you have only one direction rotation we can simply use an AC voltmeter to give the
magnitude of the rotation or rotating speed or if it is rotating in both the direction
then you will get the direction by using a phase sensitive demodulator and low pass filter
what we have learnt earlier.
So that is the AC tachogenerator, if it rotates on both directions anticlockwise and clockwise
then the voltage output should be given to phase sensitive demodulator and so on that
is AC tachogenerator. The last method is the eddy current tachometer which is used in all
vehicles mostly used in vehicles say a scooter or lorries and all places they are using this
principle and it gives the speed of vehicle, the driver can note at any instant while he
drives what is the speed of the vehicle or bus. There what is done is from one of the
four wheels through a flexible shaft the rotation is taken to a permanent magnet.
Permanent magnet is situated within a copper cup this is a copper cup actually you have
to join this, it’s not open, this rim. It’s open but you will see the rim as per the drawings
principle. So when it rotates the magnetic lines are cutting the copper cup so copper
cup is a good container so electrical lines are there, so eddy currents are there and
due to eddy current we have the magnetic lines that magnetic lines interact with the rotating
magnetic lines we have got a torque. So torque is produced in the copper cup, this torque
is absorbed by the spiral spring while it turns that is what we have the signal flow
diagram is we have omega as the input signal that is given to the magnet plus the copper
cup and then you have the torque. Torque is produce from omega, this torque is taken by
the spiral spring and gives rise to an angular rotation theta, so theta will be the rotation
in the shaft. The shaft will be forming a part of the cup and when it rotates a pointer
is attached to the shaft of this cup and we find the scale is just perpendicular to the
board. So when the pointer rotates it moves over the scale which is calibrated in terms
of earlier known speed, you can write 0, 100, 200 like that rpm and later on unknown speed
you can just use it. So this is the principle of eddy current tachometer often used. Now
since it is rpm if you multiply with the circumference of the tire then it will be kilometer per
hour, you can use kilometer per hour that’s how it is a calibrated.
Next we will see the acceleration measurement. So far we have seen displacement measurement,
velocity measurement and now the last acceleration measurement. All these three forms part of
motion measurement. So dv by dt is acceleration or d squared x by dt is acceleration and here
also we are using the absolute seismic pickup for accelerations.
So we put the instrument, all seismic pickup as per the displacement as for the velocity,
here also the whole instrument is put on the body whose acceleration we are measuring.
So it is absolute measurement, so the same instrument what we have seen earlier mass,
spring, Ks and damper it contains and it is fixed to the vibrating body, by bolted to
vibrating body the same acceleration now we are measuring by xo here that theory is obtained
like this. We already derived x0 by xi difference in terms differential operator from the Newton’s
law we derived earlier for the displacement measurement, same equation I have written
here.
Now bringing d square this side xo by d squared xi that is nothing but xo by xi two dot and
calling K as 1 by omegan squared then you will get as xo by xi two dot K and substituting
d by i omega and finding out the magnitude for the frequency response you get this equation
for the magnitude ratio xo by k by xi two dot and the phase difference will be pi is
equal to tan minus 1 of 2 psi beta by beta square minus one where beta we know already
omega by omegan the frequency ratio. Now you find the right hand side of the these two
equations is the same as what we have derived for the second order systems, frequency response
of second order systems under dynamic response of say instruments we have learnt already.
So the same curves are plotted here xo by K by xi two dot and this is beta that is frequency
ratio and same curve when omega, the resonant condition beta is equal to one resonant conditions,
psi is equal to zero is infinity and for other values it attains peak value here. Now we
know beta maximum for this bandwidth this is called the bandwidth, it is starting from
beta is equal to zero to beta maximum that is say for 5% deviation this one is the ideal
condition. So as beta increases it deviates so maximum deviations 5% error or 2% error
accordingly we get beta maximum. So zero to beta maximum will be the error. So whatever
the error you know we have to put it here, suppose 2% error means so psi is equal to
0.8 and 2% error means 0.98 you have to put and find out what is the beta value for a
given psi is 0.8. We can find the beta value that is beta maximum that is how for any given
error 2% error means the magnitude ratio will be 0.98.
Suppose psi is equal to 0.2 and error is 2% means for 0.2 the error is positive side so
you have to put 1.02 here and then solve this equation. So for putting the magnitude ratio
you should be careful by noting down the value of psi and accordingly take it. If the psi
is less than say 0.7 you should put positive side or higher values it is negative. Its
smaller values are negative so it should be less than one for smaller psi more than one
you have to put error is positive, measured is more than the theoretical one. So that’s
how the bandwidth is fixed but now we find beta maximum suppose beta maximum is equal
to say a given example may be beta maximum 0.8. That is omega maximum by omegan is equal
to 0.8 so omega maximum is equal to 0.8 times omegan zero to omega where this is the bandwidth
we want to have more bandwidth so omegan should be more.
For having more omegan, what is omegan? Omegan is equal to root of Ks by M this is what we
have learnt earlier. So for having large omegan we should have large spring constant so it
is a hard spring. So for measuring the velocity and displacement by using seismic pickup we
should have a shaft spring but for the acceleration measurement, instrument is same but this spring
what we have to select should be a hard spring. That is the main difference in design of seismic
pickup for displacement, velocity and acceleration. For acceleration purpose we should select
a harder spring, for displacement and velocity measurement you should a have a soft spring.
So this should be hard and mass to have a large omegan mass should be small there that
also possible. We can have smaller mass it will not give rise to any loading effect so
generally the harder spring is selected.
So that is the bandwidth and phase difference we know as within the bandwidth for psi is
equal to 0.6 or 0.8 more or linear. So this is what we can accept so this is the basic
principle with which the acceleration measurement is made but making in this fashion it is little
bit combustion. So you have got simplified versions of the accelerometer that actual
construction actually available in the market accelerometers. They have this one of the
constructions here that is first we are seeing the potentiometer type accelerometer just
a mass suspended at both sides by springs and instead of having damper the volume is
filled with silicon oil for damping purpose and the relative motion of the mass is measured
by a potentiometer circuit. Here we have omegan or fn the natural frequency is around 100
hertz because there is some resistance for the wiper to move over the resistance it's
part of the potentiometer circuit. So the relative motion is measured by potentiometer
as an output voltage as you have learnt earlier how the potentiometer works. Instead of potentiometer
we can also use LVDT so this point we give it to the core and the cylindrical constructions
will be fixed to the casing and you can take the output. So this is LVDT either potentiometer
or LVDT we can use to measure the relative motion of the mass with reference to the frame
of the instrument.
If you use LVDT resistance for motion is small then the omegan goes up to 300 hertz, higher
omegan means we have got higher band width that is the advantage. So another type is
the cantilever type and here the xi two dot is measuring direction is this one, here also
it is this one xi two dot measurement direction always in the instrument accelerometer they
write the measurement direction. So for this type suppose it moves this way due to inertia
it tries to an earlier, so it bends inertia force due to inertia force when it is accelerated
it bends. When it bends then you will find a stress and strain are produced near about
the fixed end of the cantilever and that strain is converted into an electrical signal by
using strain gauges.
So Strain Gauge Bridge and all it will built, you can use carrier frequency amplifier which
we have learnt already. Two strain gauges will be forming two adjacent terms of the
bridge network and the whole instrumentation is same, carrier frequency of about oscillator,
excitation, the modulated wave, amplified then demodulated and low pass filter and then
the signal comes out. All those things we have learnt I am not repeating those things.
So that a final reading will be calibrated in terms of accelerations. How it is calibrated?
Suppose when we want to calibrate, suppose this is basic instrument put it on the table
and then you will find g will be acting, g is the acceleration due to gravity.
So due to the g the mass is converted in terms of weight so this weight acts over this spring
and deforms a distance and the distance is measured by the relative displacement pickup
and the reading of the instrument, we write it as plus g and then now tilt the instrument
just 90 degree you tilt it, now when it is fixed like this the g acts along the axis
of the spring. This is the axis of the spring, the helical spring when it acts like this
then it is plus g that is spring is being compressed.
Now tilt 90 degree then this spring will come in this direction and no force will be acting
along the axis of the helical spring. So we will find a zero reading so it's zero, now
tilt another 90 degree so it will fix upside down, this will be top and this will be bottom.
That means the g will be acting opposite direction and we will have minus g that is how three
readings are obtained in an accelerometer. You can check any accelerometer by mounting
in these three ways.
First as per the direction g along the measurement direction that you will have plus g and then
tilt 90 degree zero, another 90 degree upside down that means minus g. So you can measure
on both sides I mean plus g and minus g in both directions. So this calibration is mainly
meant for horizontal motion when the motion takes place horizontal perpendicular to the
g direction the calibration is valid. It is because when the motion is vertical, you have
to be careful. In case it falls freely down what will happen. What is the weight of the
mass, weight of any freely falling body has got zero weight. So when it is falling freely,
acceleration is g but what is shown is zero. So for vertically downward motions what are
the instruments shows you have to add 1 g, that you have to be careful.
If the motion is against g then you have to subtract 1 g so for vertical motion you have
to be careful otherwise whatever we calibrated by using g that is valid for horizontal motion.
When it is stationary it will be shown g, that is no motion it is g for vertical motions
when we go up then it’s already one is there that we have to subtract. That’s why for
falling down body we have to add 1 g and for moving up you have to subtract 1 g from the
reading. Otherwise the calibration is valid for horizontal motion. So that is calibration
that is how all these instrument are calibrated.
Now that is cantilever type, here we have got 300 hertz omegan as we have, if you have
LVDT whatever the same. Instead of LVDT we have strain gauge measurement. Now this type
of construction is suitable for micro manufacturing, this type of construction is valid. We have
got the mechatronic components the so called silicon accelerometer. So the whole accelerometer
may be within few mm square it can be accommodated that is called micro manufacturing.
That is of type lithography or and all these photo etching principles are made use of.
From silicon crystal itself the whole mass and the lever everything is made and the strain
gauge also will be made up of silicon and that is amenable for the micro manufacturing,
this construction rather than the other type. That is the specialty of this cantilever type
manufacturing and another type is reluctance principle is made use of.
So reluctance type of accelerometer. That is the resistance for the flow of magnetic
lines that is made use of. Now the magnetic lines are created by the excitation winding,
probably this can vary from 5 kilohertz to 20 kilohertz depending upon the frequency
of the… Now this is the direction of motion for this xi two dot, this mass is here and
it is supported by the four leave springs, these are the four springs.
Suppose the acceleration in this direction then the mass tries to occupy the earlier
position due to inertia force, so it will deform like this. The springs will be deforming
like this that means the gap between the e stamping at the left hand side and the mass
will reduce, this side gap will increase. So correspondingly you will find here we have
got more number of lines created so magnetic path will be taking, this is iron, this also
is made up of irons and e stamping is made up of iron. So we will find in this number
of magnetic lines will increase and to the same extent number of magnetic lines in this
e stamping will reduce. So the two secondary coils are in series opposition so the voltage
developed here and subtracted from the voltage developed here, net voltage comes here and
this voltage is a modulated one. It will have the frequency same as around supply frequency
and the amplitude of this signal will be proportional to the deflection here. So the deflection
is proportional to our inertial force, inertial force is proportional to our acceleration
hence we find eo is proportional to the acceleration but it is a modulated signal, since acceleration
take place in both the ways. This modulate signal you have to give it to phase sensitive
demodulator and the low pass filter.
If you want you can use amplifier also and then low pass carrier I mean demodulator phase
sensitive demodulator and then filter all those circuits will be there that I have not
shown, we understand. That is something like LVDT also we are making measurement, afterwards
the signal processing is known. So that is the reluctance type of accelerometer, the
inductance principle is made use of.
The next type of accelerometer is piezoelectric accelerometer which is very widely used in
measuring shaft and vibrations because it has got very high undamped natural frequency.
So it is widely used type of accelerometer where you find say mass is there. Now the
relative measurement of mass we are using earlier types potentiometer LVDT and so on.
The reluctance principle here we are using the piezoelectric crystals that is say for
example quartz crystal is made use of to measure the displacement. We have learnt as one of
the methods is using a quartz crystal that is what is used here for measuring the relative
motion of the mass with reference to the frame of the seismic pickup.
There is one problem if we want to measure the acceleration in both the directions, the
crystal can take only one direction stress, it can be only compress it cannot take any
tensile load. To overcome that difficulty what is done is the crystal is always kept
under compression by having a spring, this is a spring lee spring suitably bent and by
tightening the nut, we can have different compression in the crystal.
Suppose motion is upwards then the compression is will increase and motion is downwards acceleration
downwards, compression will reduce that’s all. The crystal will never go into tensile
strain that is advantage of initial compression. Initially compress it later on you can use
for both directions, in one direction compression is increased, in another direction compression
will reduce. So to facilitate that we have got this special set up with the spring and
nut so it is initially assembled under compression. So now when it is subjected to any acceleration
it produces a charge, the crystal and that is taken out and connected to a charge amplifier,
the output can be read there. There is another draw another difficulty in this accelerometer.
We know accelerometer it has got bandwidth from zero to some omega maximum that we know
in all the other types of accelerometers what we have learnt earlier, potentiometer and
cantilever type and all but here in piezoelectric accelerometer this static measurement is not
possible. It is because we know the piezoelectric crystal for displacement measurement can be
only for dynamic measurements.
That is omega minimum is equal to the same as the single capacitor principle 3.04 over
tau this is what we have learnt already for a single capacitor displacement transducer.
The tau is the time constant of that capacitor circuit or piezoelectric, we know it is metallic
coating and from there it is taken. So in between we have got this insulating material
so it’s a typical capacitor. So same thing holds good so omega minimum should be is equal
to 3.04 over tau, tau is the electronic circuit for this piezoelectric crystal that time constant
tau. So it is sometimes omega minimum may be 10 hertz below 10 hertz you cannot use
this instrument that is why it is used in measuring shocks. What is shock? Shock contains
continuously varying accelerations, so varying acceleration up to this frequency that is
minimum frequency it can measure below than that we cannot use this one and what is the
higher range.
Here mostly you will find for the omegan the natural frequency of this is equal to root
of Ks by M. That is generally the equation, now the M is the mass of the seismic mass
but Ks the spring constant of the crystal is normally very high. It is of the order
of 9 into that Ks, Ks for the piezoelectric crystal is 9 into 10 to the power of 8 to
it varies 63 into 10 to the power of 8 Newton per meter. Such a high value it has got, so
omega natural frequency is also very high. So it is of the order of 5 kilohertz this
is a bandwidth. So it will be omegan will be still may be of this order but another
problem what you have get is the psi is more or less zero. When it is psi zero in the earlier
case you remember xo by k divided by xi two dot for psi is equal to zero, this is beta
for this is one so psi is equal to zero it is going like this. This psi is equal to zero
curve, so range is for 1% or 2% it immediately goes up, range is beta maximum, this should
be a small value.
So you have to substitute in that equation this is xo by k divided by xi two dot is equal
to one over root of one minus beta square whole square plus 4 psi square beta square,
in that suppose you can permit an error of say 5% then you have to give 1.05, 5% because
psi is equal to zero so this is zero then from this you will get a beta maximum for
psi is equal to 0.5 you have got approximately 0.2. Beta maximum is 0.2, 5% error when psi
is equal to zero beta maximum comes around 0.2 because this also 0.04 5% error only for
a capacitor circuit or piezoelectric crystal. So for 5% 0.02 that means into omegan so the
beta maximum may be 0.2 so that is equal to the omega maximum is equal to 0.2 into omegan.
Natural frequency is a high value and it gives rise a bandwidth with which we get a bandwidth
of 10 hertz to 5 kilohertz. So it may be around say 25 it may be omegan may be 25 kilohertz
so then it gives rises to 5 kilohertz, omegan is large will be 25 kilohertz. So even though
we have got beta maximum B is of very small value 0.2 but still you have got measurements
range of bandwidth very large it is because that omegan is very large.
So we can go up to 5 kilohertz whereas in the earlier cases omegan itself is of the
order of 300 and 0.8 times equal to 240. So 240 hertz will be maximum hence they cannot
be used for measurement of shock where acceleration is varying at very high frequencies, you cannot
use it. So in such instances you have to go only for this piezoelectric accelerometer.
Next type is servo accelerometer so all these instruments up to here they are open loop
systems, their accuracy is say plus or minus 1% of the full scale whereas if you want to
have 0.1% accuracy or uncertainty then you should go for necessarily closed loop instruments
and this servo accelerometer is one such closed loop accelerometer. Here the one type basic
construction it is being sketched here so coil is there in between permanent magnet
that is actually it’s a torque coil it is called and that is mounted on 2 pivot bearings
with its axis. To the axis we have got this mass which is attached to a lever connected
to the axis of the coil and at the other end we had a plate which is moving in between
two inductive pickups.
We know inductive pickups we can measure motion up to plus or minus 0.6 mm or plus or minus
1 mm. Actually they will be very close but I have drawn like this just for explanation
purpose and to the whole axis we have connected spiral spring and the output of the displacement
pickup this is a displacement pickup, inductive pickup given to the bridge network and then
the current comes through a constant resistor, the voltage drop is taken as output signal
and when i goes through the coil then this coil is put in between permanent magnet a
torque is produced. So this is the construction, how it functions we can see in this signal
flow diagram, the signal flow diagram of the servo accelerometer say when xi is there when
that is xi is in this direction perpendicular to this of arm xi two dot.
Suppose the xi is towards stress then the inertial force will be there, trying to keep
it in earlier position inertia force. So you have a torque, this inertia force is acting
at a distance r from the axis of the coil then you will find a torque is produced. The
inertial force due to this acceleration, inertial force is felt by the mass and that force is
acting at a distance r gives rise to a torque Ti and this supported system. The coil and
assembly is supported at the two pivots, this is pivot bearing bottom and top we have got
pivot bearing so it can rotate about this axis. So a torque is applied and there is
another torque coming from the current flow that is called Tb when this torque is larger
than this feedback torque then you will find that is the difference between the two torques
is converted into an angular rotation theta by the spring. That is how the it is called
Te is equal to Ti minus Tb so difference between this torque is taken up and that is converted
into the angular rotation theta and theta times this length of this lever gives a displacement
in front of this two inductive pickups and when it moves then we we know how inductive
pickups are functioning and it is amplified, it is a carrier frequency amplifier again.
So the bridge output, the amplifier gives a current output of i. So then this i goes
through the coil then a torque is produced in opposite direction trying to bring back
the mass to its original position. So how much current required to bring it back, it
is a measure of the torque acting on this that again of inertia mass, inertia force
that again in terms of the accelerations.
So proportional to acceleration the current is maintained here so that this is always
brought back or near about the original positions. Required current, it is a signal that current
flowing through a constant resistor gives rise to a voltage drop eo. So now eo is our
output signal xo that is eo is our output signal so input signal is xi two dot and output
signal is eo. So it is a closed loop system, all advantages of the closed loop system is
there. In open loop system if the temperature rise the spring constants will come down then
it will show higher reading but to bring back also we require a smaller current that’s
why the error is eliminated in any closed loop systems. Hence closed loop systems are
higher or generally more accurate than the open loop systems up to 0.1 or even less than
that we can obtain depending upon the elements you have selected.