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Let us continue our discussion on uncoupled
heave motion. In last class, we were discussing
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about the natural period and the restoring
coefficients. See the natural period here
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was written as c z by m plus a z, and of course
the natural period would be 2 pi by omega
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z is simply. Now, natural period is a extremely
important property for a oscillating system.
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To estimate this, I need to know as I can
see a restoring force and I of course also
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wanted added mass.
So, I was thinking of talking about this,
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but I think in order to get to this estimate,
we need to discuss both restoring force added
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mass, and see we discussed added mass, we
need to discuss damping. Now, let us look
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at restoring force. You see suppose, I have
the floating body here, it is at this location.
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So, at this point what happens, mass is balancing
buoyancy, weight and buoyancy are fully in
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balance. Now, whatever the weight, w and buoyancy
is fully balanced.
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What is buoyancy? Buoyancy is nothing but
the hydrostatic pressure integration over
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this wetted surface. Now, what has happened?
This is got displaced. I will draw another
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line. Let us say it has got displaced by an
amount z. How much is my total buoyancy force?
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Total buoyancy force is now the weight of
the displaced water of this entire thing,
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but of these this mean part is balanced by
the weight. So, the unbalanced force that
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acts the unbalanced part of the static force
is of course, the weight of this mass. Now,
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how much is this? This is of course, rho g
A w p, it is like parallel sinkage into z,
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assuming of course A z A w p is constant,
but strictly it should have been rho g A w
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p z dz integration over that part.
But if z is very small this distance, then
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A w p can be assumed to be constant over that,
this is exactly what we keep on doing in our
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parallel calculation, you know TPCTPI calculation
and remember, the entire theory of this equation
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of motion that we got is on the assumption
of linearity, which means that we have already
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said that all the displacements are small
in comparison to the body dimension, which
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means this part is small compared to the draft.
So, it is very logical to assume it is constant.
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In such a case, my restoring force F s is
this, but F s I I am writing as c z into z.
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First of all, you notice that this is actually
proportional to z if I of course, assume A
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w p is constant. So, restoring force F s becomes
proportional to the displacement and the constant
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of proportionality is rho g A w p, which is
my c z. So, I end up getting here that c z
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equal to rho g A w p.
What is important here to notice is that,
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it is directly in proportion to water plane
area. So, here you can see from this, that
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if water plane area was to reduce, my t z
goes up and converse; if water plane area
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is very large, my t z is small. So, one can
see that; however, I cannot still come to
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an estimate if I do not have a discussion
A z. So now, what we will do, before we go
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to t estimates, we will let us discuss this
added mass and damping and it is an extremely
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important concept because this added mass
and damping very many students, actually or
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many people have a wrong wrong conception
on. People have very misconception on added
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mass.
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So, therefore, I would expect that we discuss
this slowly. Basically, we are looking at
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here a z and b z, from the heave point of
view it can be any name, but we will talk
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generally what is added mass and damping.
Now, you see let me start this from a case
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of a accelerating body. This is the body here,
somebody the full page is fluid, there is
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no free surface as on now. Now, I begin to
oscillate. Basically, oscillation involves
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acceleration, you know if you oscillate you
are always having acceleration, it is not
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a constant motion like a ship.
So, suppose Let me say some acceleration.
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I am accelerating in this direction by some
value say a acceleration. Now, what happens?
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See. Now, you consider a fluid particle somewhere
here, pen blue or may be a better diagram
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would be to take let me let me put another
diagram, instead of that. Let us take the
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heave value, let me take a body like that
and let me let me say that I am accelerating
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this side, see acceleration if it deep water.
Right. Now, take a particle here, what happens?
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The particle also gets pushed out because
it cannot penetrate; it gets accelerated.
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Now, what happens to this particle next particle?
It also get pushed out, but to a lower extent.
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So, what would happen, there is going to be
an acceleration generated in certain amount
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of surrounding fluid. In fact, the acceleration
cannot be theoretically said where it ends,
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there is no boundary, because if suppose I
accelerate this body at say 1 meter per second
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square and it is a flat plate. This particle
will move at 1 meter per second square, but
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the next one will be may be 0.9 meter per
second square.
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So, it will gradually decay, keep on decaying.
So, there will be general acceleration induced
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on the fluid. Now, what happen? Let us look
at the force, what is the force? It is mass
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of the rigid body, mass into acceleration,
but now as I accelerate the body, I am also
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accelerating part of fluid mass. So, my total
force has become this plus some additional
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part. We do not know how much the additional
part. Now, what is happening? I can assume
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the additional part, this additional part
of the force to be some equivalent mass into
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the same acceleration. Remember, this is very
important, it is an equivalent concept.
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I presume as if there is a some kind of identified
mass accelerating at the same acceleration,
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reality wise it is not happening so. Please
understand. The fluid part there is no identified
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mass, remember. So, it is not a mass concept,
it is a force concept. Basically, I have a
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force and if I divide the force by acceleration,
I end up getting a unit which is mass and
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I am telling that mass, it is it is as if
that much of mass is attached to the body,
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the word as if is very important. So, there
is no identified mass.
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So, people think that yes, when I oscillate
there is a mass identified. No, there is simply
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a force. So, added mass therefore is a concept
of force, added mass force is equal to some
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mass into acceleration and that some mass
is what we call is added mass. This is very
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important. Now, another thing let us remember,
suppose I take a fluid particle here or let
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us say I take this particle here, now I am
accelerating. Now, you see this is a in viscid
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flow theory, added mass concept is for in
viscid flow theory. Now, if it was a viscous
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fluid, then this could have got stuck and
also accelerated for boundary layer, but;
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however, here we are not talking of that.
Here we are developing that added mass concept
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presuming fluid is not viscous. Therefore,
this this can slide fast.
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So, it is only in the normal direction, that
it is accelerated that we get. For example,
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therefore, if I take a plate here and if I
accelerate this side, what happen? There can
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be some boundary layer attached to that, but
that has nothing to do the added mass. Added
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mass is the one that is getting pushed in
the normal direction and in theoretically,
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this plate has no mass if it is a straight
line plate. So, added mass in this direction
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is going to be 0.
So, why therefore, it is important remember,
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that we sometime ask the question. In my ship
resistance, but it is also moving, but there
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is no added mass, we have never introduced,
why? Because the ship was moving at a steady
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forward speed, not accelerating. Therefore,
there is no acceleration force of the fluid.
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00:10:05,800 --> 00:10:10,920
So, there is no added mass force. This is
what we have explained to you and one of the
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reasons, why in say ship resistance, in a
if you were to plot here resistance from 0
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speed say speed goes like that, this is v,
resistance you will find out going like that
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and going like that. Why this hump is there,
because remember at this point it is accelerating,
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before it reaches steady state and because
it accelerates the net force becomes slightly
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more, which of course will diminish when you
reach a steady state. This is a say force
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or resistance whatever, with respect to time.
So, added mass force therefore, well added
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mass therefore, is a concept of force, not
a mass. Lot of people thinks it is a mass
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attached to the body. No, there is no mass
attached to the body, it is not sticking.
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First of all, it is based on potential theory.
So, it cannot stick, it is only getting pushed
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in the normal direction, if I have a body
here and I push it here, this water particle
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will get pushed this direction, not in this
direction. Because it will have a velocity,
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this this it can allow you can allow it to
slide, but you cannot allow this side; obviously,
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you cannot allow because if allowed to this
side that would amount to the fluid entering
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the fluid body you know. So, the normal relative
velocity has to be 0.
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So, this is the idea of added mass. Now, what
I said is this here in a deep water. Now,
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we will now do a little, we go to the next
level.
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Now, I have this water here, I am oscillating.
Now, if you oscillate, what happen you will
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of course, find out. Now, what we said, this
problem of oscillating by force in a calm
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water, in fact it is known to be a forced
oscillation problem. Sometime, we also call
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it a radiation problem. I am just introducing
this word, why because as you oscillate, you
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always create waves and these waves will always
radiate out from the body. In other words,
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in a planned view if you oscillate, waves
are going to go outward. You know, if you
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oscillate it, waves are going to go outward,
these are called radiated waves. Now, what
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is the force there?
Now, we look at the force part of it, what
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happen; obviously, when I oscillate it, you
see there is a particle, which are also getting
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induced. Now, in the case of deep water, the
force was just the additional mass force,
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but here what is happen, if I was to take
a vector view and if I call this to be displacement,
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say z and this to be z dot, these all of you
know I will guess that, there is a phase gap
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between in a oscillating system displacement
velocity at this thing, what happen the force
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vector turns out here happens to be somewhere
like this not in in phase with acceleration,
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but slightly behind, not in phase with acceleration.
If it is deep water, it is always in phase
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in acceleration, but here because of the wave
creation, as you give this highest force acceleration
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that is not the time when the highest force
comes.
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So, there is a phase gap. Then what we say,
this is my force; this is my F, radiation
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force and I this is what I want to find out
in my right hand side. Now, it turns out that
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by definition this part of the force that
is the part that is in phase with acceleration,
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this part we are calling added mass force
and we are calling it is added mass into acceleration
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and this part of the force, I am calling it
damping force.
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Remember again, here we are calling it potential
damping or radiation damping because we do
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not have discussed it yet in the fluid. So,
this I am calling b z into z dot. In fact,
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the negative sign, etcetera comes in, but
this is the idea. What is happening in free
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surface therefore? If there is a free surface
because of the wave creation, you remember
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wave creation means energy is getting dissipated;
no, certain energy is being taken away by
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the waves. So, what happen? this act as retardation
on the force and as a result, the phase gap
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develops and that is why, by in fluid mechanics
always a force in proportion to velocity is
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known as damping force because what happen,
it tends to lower the motion displacement
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solution wise.
So, it starts to retard it. So, you call it
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damping force, some kind of holding back.
So, here also that part of the force in proportion
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to the displacement is called sorry velocity
is called damping force and this is called
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added mass force. What happen? If you go deep
water down, this vector actually becomes like
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that because there is no damping in deep water,
no no wave creation. As you go up, more and
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00:15:15,790 --> 00:15:20,619
more wave it actually goes up down and one
can see that of course, there is a some changes
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there. see These forces are all depends on
this oscillation frequency and there is a
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very nice and simple proof for that. You see
just an intuitive proof; you take this why
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should be so? Like you know.
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Now, let me just also call damping force here
in terms of omega. Some damping force, say
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I am calling. Now, suppose I oscillate this
at a very low, means I am taking all my time
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to oscillate. What would happen to this wave?
Practically no wave will get created. You
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see that by physics. So, the that the damping
force is going to be 0.
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Now, if you do also because see this, damping
force is connected to the energy of the waves.
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00:16:06,050 --> 00:16:12,319
Now, if you that do not have any waves, as
in deep water case there is no damping force.
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Now, if you take limit of omega tends to 0
that is t equal to infinity, which means you
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are actually oscillating at a very, very slow
rate. You can take experiment yourself, take
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00:16:26,069 --> 00:16:30,470
your hand or somebody and oscillate, take
all your time you will not find any waves
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created. Then you will expect this to be very
small, actually theoretically 0. Now, you
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take the other extreme of extremely high,
you will find out if you do extremely high,
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there will be kind of ripples, but the energy
of that is going to be also 0 because lambda
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tends to 0. You see the wave that is get created,
the lambda of the wave tends to 0 at omega
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of the wave tends to infinity.
Now, energy is always over square area. So,
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you know it is like half rho g A square lambda,
that is what you are dividing by. So, if lambda
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00:17:03,929 --> 00:17:09,250
tends to 0, you actually end up having practically
no energy. This is a reason why, yesterday
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I mentioned that a large ship in a port when
there are ripples, they just does not move
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there is no energy of the waves. So, this
is also going to be 0, but in between that
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it makes wave. So, therefore, that this damping
has to go something like that, which obviously,
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tells me that damping is depending on frequency,
which basically means this graph of the force
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00:17:31,670 --> 00:17:38,670
graph. It actually swings to this side, swings
back this side. So, it depends on frequency.
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00:17:41,020 --> 00:17:45,380
So, what happen? If damping is depending on
frequency; obviously, added mass is also depending
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00:17:45,380 --> 00:17:51,020
on frequency because added mass and damping
are two part of the same force. So, what happens
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00:17:51,020 --> 00:17:58,020
in free surface phenomena radiation force,
which is added mass and damping, becomes frequency
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00:17:58,220 --> 00:18:05,220
dependant, in this also; that means, in what
it means is that a z is a z omega; b z is
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b z omega. There is nothing like, there is
the statement that added mass of this ship
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is so and so, is incorrect statement as for
as heave or modes of motions are concerned,
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you have to tell at what frequency. Normally
what has happen? You see I told you that at
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high frequency it is making ripple, practically
no energy.
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So, it is high frequency limit one can show
tends to be like same as deep water. Now,
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if you have studied vibration, you know then
you will find out that in vibration of hull,
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when we consider the added mass we always
considered the deep water limit because vibration
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frequencies are very high frequency 1 second,
2 seconds like that. So, in their practically
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00:18:49,150 --> 00:18:56,150
one takes the high frequency limit of the
added masses. That is why in vibration and
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00:18:56,250 --> 00:19:01,050
in a loosely, people talk that oh the ship
has so much added mass, but this statement
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00:19:01,050 --> 00:19:06,610
is incorrect as far as free surface is concerned
because it depends on frequency, at what frequency.
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So, therefore, if I were to find out for example,
natural period also t, which is actually having
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00:19:12,440 --> 00:19:19,440
these 2 pi this m plus a; obviously, this
has to be for that t or for that omega what
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00:19:20,380 --> 00:19:26,440
you take, but we will not go through all the
detail. We will simply try to make estimates
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00:19:26,440 --> 00:19:32,640
of that a, a and b. So, you see having said
that therefore, one of the big problems for
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00:19:32,640 --> 00:19:37,640
us becomes how do I estimate, how do I compute
added mass damping.
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Now, this a z and b z. This has been a matter
of research for very, very, very long time,
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00:19:47,440 --> 00:19:54,190
because of its historical importance and there
are numerous methods and today, we have got
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00:19:54,190 --> 00:19:59,580
very sophisticated three dimensional methods
etcetera, but before I do that again, see
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00:19:59,580 --> 00:20:06,580
remember a z, b z is function of omega, but
omega is a function of see is a function of
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00:20:08,560 --> 00:20:13,090
actually omega e because essentially omega
e is the oscillation frequency, but omega
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00:20:13,090 --> 00:20:20,090
e is a function of omega v and nu. So therefore,
for getting nu you see it is speed dependant.
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00:20:22,200 --> 00:20:28,610
It is not also only omega dependant, but speed
dependant. In the in earlier classes, we have
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seen that there are situation where two similar
omega e, a same value omega e can occur for
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00:20:34,930 --> 00:20:40,840
different combination of these two or these
three. We have seen that in following waves.
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00:20:40,840 --> 00:20:47,130
So therefore, just to make it a guess omega
e really will not tell the correct story because
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00:20:47,130 --> 00:20:51,880
the same omega e would have arise in because
of different physical phenomenon, different
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00:20:51,880 --> 00:20:58,880
actually wave systems. So, this is a function
of omega e, that is one part, but then one
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00:21:00,390 --> 00:21:05,460
has to have simple solution methods because
you cannot keep waiting for that. So, what
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00:21:05,460 --> 00:21:10,990
has happen? People have been waiting working
for that for a long time and I will mention
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00:21:10,990 --> 00:21:17,990
a practical way of estimating that now, which
is called synthesis based on strip theory.
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00:21:24,220 --> 00:21:31,220
See what happens? You know Always a two dimensional,
these are mathematical problems a two dimensional
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00:21:31,890 --> 00:21:34,260
solution is simpler than a three dimensional
solution.
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00:21:34,260 --> 00:21:41,260
Now, if I take a three dimensional ship. If
I were to take this entire thing, even if
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00:21:41,620 --> 00:21:47,560
I did not go forward speed, at 0 speed move
forward up and down, I end up having difficulty
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00:21:47,560 --> 00:21:53,840
of solving the problem; difficulty means this
becomes much more complex, complexity level
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00:21:53,840 --> 00:22:00,840
is higher, but ships are always long and slender,
typically. So, what we can do? We can presume
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00:22:01,360 --> 00:22:08,360
this ship to be something like two dimensional
section ships. Now, this I will I need to
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00:22:11,330 --> 00:22:18,250
discuss at more length here, perhaps. Now,
you see to get bigger diagram.
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00:22:18,250 --> 00:22:25,250
So, let me take a section. Now, I am oscillating
in the heave direction. Take this section
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00:22:27,500 --> 00:22:34,500
here, it is making wave. Let me take the water
plane also. Now, you see if I this section.
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00:22:41,980 --> 00:22:48,980
Now, this slope slope of this hull water line
is very small. Now, what happen? This particular
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00:22:50,330 --> 00:22:57,070
part when I oscillate, it is supposed to be
actually moving this direction. See if I oscillate
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00:22:57,070 --> 00:23:02,250
which ever direction it will be normal to
the surface, but what happen is that see that
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00:23:02,250 --> 00:23:07,450
because this slope is very small, this component
this component is much smaller compared to
215
00:23:07,450 --> 00:23:14,280
this component, this component is much smaller.
So, what happen because it is long and slender,
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00:23:14,280 --> 00:23:19,340
I can presume it to be locally two dimensional
with what it means is that, when I oscillate
217
00:23:19,340 --> 00:23:26,340
a section I can presume that the flow is contained
in this plane, the 2 D plane or rather. If
218
00:23:26,950 --> 00:23:33,950
I were to plot this this thing see, if I were
to make this x, z and y. So, this is my z
219
00:23:34,280 --> 00:23:39,310
and this is my y.
What we presume is therefore, that at a given
220
00:23:39,310 --> 00:23:46,310
section if I oscillate, as if the flow is
all in z y plane, which means that if I take
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00:23:49,880 --> 00:23:54,130
a velocity vector of fluid, it is like that.
In other words, if I take a v here of fluid,
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00:23:54,130 --> 00:24:01,130
v will have only a z component and a y component
and x component is assumed to be 0, because
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00:24:01,610 --> 00:24:06,650
it is very small. See if this is allowed,
then what happen; the ship can be assumed
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00:24:06,650 --> 00:24:12,780
to be composed of locally 2 D sections, each
one is oscillating, each one is creating a
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00:24:12,780 --> 00:24:19,780
wave in its own plane and therefore, I can
find the four added masses for each sections
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00:24:21,040 --> 00:24:27,120
two dimensional sections and add them over
the length. So, if I were calling added masses
227
00:24:27,120 --> 00:24:34,120
of 2 D section as, say a 2D z at a certain
x. If we integrate that over d x, I can get
228
00:24:34,230 --> 00:24:41,230
a 3 D z.
This approximation, that is getting a 3 D
229
00:24:41,720 --> 00:24:47,900
value by considering the problem to be a set
of 2 D problems, sections and adding them,
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00:24:47,900 --> 00:24:53,570
is what is called strip theory synthesis,
because you are assuming the ship to be composed
231
00:24:53,570 --> 00:25:00,570
of strips. See you are making a presumption
that the ship is composed of strips and that
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00:25:06,020 --> 00:25:11,740
is not illogical, practically also because
ships are long and slender. Because if you
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00:25:11,740 --> 00:25:17,250
do oscillate, you do expect; not much of excepting
of the edges, from edges some problem may
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00:25:17,250 --> 00:25:23,110
come. Excepting for the edges, you would expect
largely it is in the two dimensional plane.
235
00:25:23,110 --> 00:25:30,110
So now, this thing is more easy to compute
for 0 speed to, because what is happening?
236
00:25:32,330 --> 00:25:39,250
Now, lot of researches gone in, where oscillation
of a 2 D section where done. Now, in fluid
237
00:25:39,250 --> 00:25:46,250
mechanics or hydrodynamics you might have
learned that if I were to take a circle and
238
00:25:51,490 --> 00:25:58,350
find out oscillate this side, what is the
added mass? Most I, this is supposed to be
239
00:25:58,350 --> 00:26:01,730
a pre requisite to your course. In marine
hydrodynamics, you would have done the added
240
00:26:01,730 --> 00:26:08,110
mass of that becomes exactly same as mass
of the displaced water and in fact, at high
241
00:26:08,110 --> 00:26:12,160
frequency limit one can show that it becomes
equal to mass of this.
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00:26:12,160 --> 00:26:17,150
You know if there is a free surface here if
I what I mean if I take a semi circle, if
243
00:26:17,150 --> 00:26:24,150
I oscillate this side at a very high frequency,
then it turns out added mass is equal to this.
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00:26:24,610 --> 00:26:31,610
This is a theoretical solution obtained. In
fact, if you take any cross section, a good
245
00:26:32,690 --> 00:26:39,690
approximation of added mass becomes the mass
of that circumscribed semi circle. That is
246
00:26:40,980 --> 00:26:46,950
a good approximation, when you have no knowledge
you can always use that. Of course, this is
247
00:26:46,950 --> 00:26:53,530
only one value and this is valid strictly
speaking close to deep water, deep water meaning
248
00:26:53,530 --> 00:26:58,100
infinite fluid at high frequency like that.
There is a whole lot of discussion on that,
249
00:26:58,100 --> 00:27:02,210
we probably cannot go through.
So, what happen? There have been now charts
250
00:27:02,210 --> 00:27:09,210
available, where people have worked out. The
added mass of various sections based on certain
251
00:27:09,390 --> 00:27:12,180
sectional parameters. You see here.
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00:27:12,180 --> 00:27:16,570
Now, we will talk of ships little bit. If
you take a typical ship, in design you would
253
00:27:16,570 --> 00:27:23,570
have understood. It would have a sectional
areas S n, say n th section I am talking of.
254
00:27:31,490 --> 00:27:38,490
It has got here it is breadth b n and of course,
draft does not change, draft remains t more
255
00:27:42,760 --> 00:27:49,760
or less for all sections, if you taken.
So, I have got essentially local b x, b n
256
00:27:53,180 --> 00:28:00,180
and sectional areas. Normally, what happen?
Sectional area and the local breadth is a
257
00:28:02,380 --> 00:28:07,810
good representation of a ship section, although
you may say that there to be many section,
258
00:28:07,810 --> 00:28:13,080
but you will find out that when you do design
problem, you actually develop the lines plan
259
00:28:13,080 --> 00:28:17,900
based on those two parameters. Essentially,
you actually you know like find out the sectional
260
00:28:17,900 --> 00:28:24,900
area curve, that is S n curve and a half breadth
tan curve, that is b B n curve and based on
261
00:28:25,040 --> 00:28:30,720
the two you actually try to find out the section.
So, essentially what you do; that means any
262
00:28:30,720 --> 00:28:37,720
ship section usually is derived based on these
two primary parameters, the ships ship can
263
00:28:39,070 --> 00:28:45,330
be many. Now, what is happen; people have
worked out, here based on different such two
264
00:28:45,330 --> 00:28:50,900
parameter, that is sectional; see I can always
use a non dimensional sectional parameter
265
00:28:50,900 --> 00:28:57,900
S n by b n into t and of course, this what
is B n, beta n becomes non dimensional sectional
266
00:29:00,580 --> 00:29:04,850
area coefficient, because you always want
to non dimensional thing and if I have another
267
00:29:04,850 --> 00:29:10,640
coefficient, that is B n by t n T, this become
you can say non dimensional sectional with
268
00:29:10,640 --> 00:29:14,370
coefficient.
Now, these two parameters more or less are
269
00:29:14,370 --> 00:29:18,810
a good measure of the sectional ship. So,
what people have done earlier, what has been
270
00:29:18,810 --> 00:29:25,530
done is that based on these two sectional
ships, using certain theory the added masses
271
00:29:25,530 --> 00:29:31,340
for the 2 D sections have been worked out.
This theory etcetera fairly complex for me
272
00:29:31,340 --> 00:29:38,340
to explain here, the theory is based on things
like we call it certain transformation like
273
00:29:38,470 --> 00:29:42,730
Joukowski transformation you would have heard.
There are similar transformations, what happen
274
00:29:42,730 --> 00:29:48,500
is that theoretically one knows the added
mass of the circle. Now, it turns out that
275
00:29:48,500 --> 00:29:54,300
you can actually transform a circle to this
ship based on the parameter.
276
00:29:54,300 --> 00:29:58,820
So, like that by using transformation one
can find out added masses. So, this is certainly
277
00:29:58,820 --> 00:30:04,010
done in one case. We call Lewis form, this
transformation is also; there are number of
278
00:30:04,010 --> 00:30:11,010
transformation one is Lewis form. What it
means is that if you take a distance of certain
279
00:30:17,170 --> 00:30:23,680
area and certain breadth, you can transform
that to something like a ship like section
280
00:30:23,680 --> 00:30:25,390
you know ship like sections.
281
00:30:25,390 --> 00:30:32,390
So, what in conversely was done is that, based
on the area, based on the local beam one fitted
282
00:30:32,750 --> 00:30:39,750
that and then one ended up having this graphs
of non dimensional frequency omega square.
283
00:30:41,950 --> 00:30:48,790
Typically, it is written like that, versus
the coefficient here added mass, typically
284
00:30:48,790 --> 00:30:55,790
this is added mass coefficient. Always you
write added mass as a coefficient. Well, let
285
00:30:56,980 --> 00:31:03,980
me put it this way rather; alpha let me put
it alpha z into m z or alpha z equal to a
286
00:31:09,140 --> 00:31:15,180
z by n.
In other words, normally added mass you represent
287
00:31:15,180 --> 00:31:20,620
as how many times mass it is, rather than
just by number. You see it is a relative non
288
00:31:20,620 --> 00:31:25,310
dimensional wave; a coefficient; added mass
is a unit of mass. So, easiest way to tell
289
00:31:25,310 --> 00:31:29,660
how much is added mass is of course, trying
to tell how many times of the actual rigid
290
00:31:29,660 --> 00:31:35,970
body mass is added mass. So, you have a feel.
So, this is how it is. So, alpha z that these
291
00:31:35,970 --> 00:31:41,640
plots are available for different value of
say say different value of B n by t and this
292
00:31:41,640 --> 00:31:47,750
may be for. So, in other words if you see
basically alpha z here as function of B n
293
00:31:47,750 --> 00:31:54,750
by t and beta n and of course omega. So, this
versus this for, it is a three parameter thing.
294
00:31:55,600 --> 00:32:01,140
So, there is a family of curve available.
So, one way is that you go to this graph,
295
00:32:01,140 --> 00:32:08,100
we can go to those graph; pick up my required
values of sectional added mass. So, I will
296
00:32:08,100 --> 00:32:13,230
be knowing my sectional added masses, integrate
them and get added mass. This is one way of
297
00:32:13,230 --> 00:32:17,340
doing thing and same thing exists for damping,
exactly the same thing.
298
00:32:17,340 --> 00:32:21,990
What I am saying therefore is that see, that
one way of doing added mass is that go to
299
00:32:21,990 --> 00:32:28,990
those charts and tables, where the added mass
coefficient alpha and damping coefficient
300
00:32:28,990 --> 00:32:33,550
beta; if I call this added mass coefficient
and this damping coefficient as a function
301
00:32:33,550 --> 00:32:40,310
of; obviously, it is function of omega e,
which you of course, work on based on d beta
302
00:32:40,310 --> 00:32:46,860
and for the ship part it is based on a sectional
area coefficient beta n and B n by t. This
303
00:32:46,860 --> 00:32:53,860
is sectional area coefficient, this of course
is sectional width, breadth coefficient you
304
00:32:59,530 --> 00:33:06,530
can say; these 2 parameter. So, what happen?
Obviously, when I have to plot x, this three
305
00:33:11,050 --> 00:33:15,450
parameter; what I will do, I have this versus
this for a number of values for one of them
306
00:33:15,450 --> 00:33:18,980
for a given value of this. So, basically repeat
it, remember there was a statement I heard
307
00:33:18,980 --> 00:33:25,770
earlier. That if y was a function of x 1,
you need one graph; if it is x 1 and x 2,
308
00:33:25,770 --> 00:33:30,100
you have number of graph; if it is x 1, x
2 and x 3, then you have to repeat this number
309
00:33:30,100 --> 00:33:37,100
of graph number of times. This very obvious,
is not it? See if y is a function of x x 1
310
00:33:38,540 --> 00:33:43,140
just one parameter, one line will suffice
to define that; if it is a function of x 1
311
00:33:43,140 --> 00:33:48,160
and x 2, you will have y versus x 1 for different
value of x 2; that means, you will end up
312
00:33:48,160 --> 00:33:54,220
having number of graph. If it is y is a function
of x 1, x 2 and x 3, then these numbers of
313
00:33:54,220 --> 00:33:58,990
graph are to be repeated for different value
of x 3. So, you end up repeating. There is
314
00:33:58,990 --> 00:34:05,920
a statement that if it is 3 you require number
of pages and it is 4 you will require a book
315
00:34:05,920 --> 00:34:10,669
and 5 you require a library to represent them
because it goes one time more.
316
00:34:10,669 --> 00:34:14,879
So, here it is three. So, that is what we
are doing. You are representing this added
317
00:34:14,879 --> 00:34:19,300
mass damping as function of frequency non
dimensional for different value of beta B
318
00:34:19,300 --> 00:34:25,950
n by n for a given sectional area, repeat
that. So, obviously what happens? When I have
319
00:34:25,950 --> 00:34:32,950
a actual ship here, I will take sections I
will find out it S n, beta n value, sectional
320
00:34:34,730 --> 00:34:39,849
area values with values. So, for that I will
go to the chart, find out whatever sectional
321
00:34:39,849 --> 00:34:45,529
added mass a to z like that, I will do it
for different sections, integrate it I will
322
00:34:45,529 --> 00:34:50,049
get total added mass.
This is one simple way of doing. Of course,
323
00:34:50,049 --> 00:34:57,049
it does not answer the question about how
do I get omega e because you know it is not
324
00:34:57,730 --> 00:35:04,049
differentiating between same omega e’s arising
because of different combination of this because
325
00:35:04,049 --> 00:35:10,319
it is only a function of one parameter omega
e. This is one problem, means there is an
326
00:35:10,319 --> 00:35:14,670
approximation. The second thing is that; obviously,
here I do not have the ship moving. You imagine
327
00:35:14,670 --> 00:35:20,829
this part also that this ship is oscillating
at period 10 second.
328
00:35:20,829 --> 00:35:27,829
So, it will have some force, but second one
is moving and oscillating. So, the second
329
00:35:28,339 --> 00:35:32,049
one is the ship is moving and oscillating;
first one is just oscillating.
330
00:35:32,049 --> 00:35:39,049
So, even the both are 10 seconds; obviously,
you do not expect the physics to be same.
331
00:35:39,190 --> 00:35:43,769
So therefore, added mass has to be a function
of speed. So, for that there have been many
332
00:35:43,769 --> 00:35:50,769
elaborate theories, where a 3 D z becomes
a function of a z 3 D at 0 speed plus some
333
00:35:54,349 --> 00:35:57,180
correction terms, which is function of v,
omega, etcetera.
334
00:35:57,180 --> 00:36:02,730
So, these kinds of things are all full of
mathematics. There are many theories that
335
00:36:02,730 --> 00:36:07,410
are developed, which have worked out; what
is this correction, speed dependant correction
336
00:36:07,410 --> 00:36:14,410
we call to get to the speed dependant added
masses. This is all beyond the scope of our
337
00:36:14,910 --> 00:36:21,710
this class. It is all part of classical hydrodynamic
theories from which these are all developed.
338
00:36:21,710 --> 00:36:25,980
Whole lot of Maths, whole lot of you know
like abstract Maths are necessary in order
339
00:36:25,980 --> 00:36:31,109
to get to this and; obviously, there are different
versions because there are no close solutions.
340
00:36:31,109 --> 00:36:38,109
For example, one very popular paper, which
talks about this expression is I want to mention
341
00:36:40,170 --> 00:36:47,170
that, is called classically STF method, Salvation
Tuck Faltinson method. It is very why I am
342
00:36:49,430 --> 00:36:56,430
saying this because this was historically
recognized in our Nevelac society. Salvation
343
00:36:56,529 --> 00:37:02,589
was an American, Tuck was an Australian, Faltinson
is an Norwegian and people say this is the
344
00:37:02,589 --> 00:37:08,339
best example of international collaboration,
which actually is used even today. In any
345
00:37:08,339 --> 00:37:14,539
shipyard industry, most of the ship motions
are based on STF method of strip theory, but
346
00:37:14,539 --> 00:37:20,079
we cannot talk about it here, this is more
of a part of hydrodynamic course, but I just
347
00:37:20,079 --> 00:37:22,720
wanted to mention that you know so that you
have an idea.
348
00:37:22,720 --> 00:37:29,720
So, I am now going to go back, back to our
strip this thing. The natural period discussion
349
00:37:33,160 --> 00:37:40,160
T z part, because we need to have an idea,
physical idea regarding these values m plus
350
00:37:40,999 --> 00:37:47,999
a z by c z and I can write this to be; actually,
we can write this to be m 1 plus that coefficient,
351
00:37:50,240 --> 00:37:54,609
added mass coefficient; what do I call it?
Let me call it a bar z. Remember, a bar z
352
00:37:54,609 --> 00:38:01,609
is added mass coefficient, that is mass into
a bar z gives you; now, let further work we
353
00:38:04,160 --> 00:38:09,220
can work it out for a typical ship; let me
work it out for a typical ship. We will work
354
00:38:09,220 --> 00:38:16,220
it out for different thing, but let us work
it for typical ship. What is my mass? 2 pi,
355
00:38:18,359 --> 00:38:25,359
let me write it down basically, it is rho
into v; rho into volume. What is volume? l
356
00:38:27,789 --> 00:38:34,499
b t into c b.
So, I write l b t into c b. You agree with
357
00:38:34,499 --> 00:38:41,499
that no, because volume is length breadth
draft into block coefficient 1 plus a z bar.
358
00:38:43,190 --> 00:38:50,190
What c z? rho g A w p. What is A w p? l into
b into c w, right because c w is area coefficient.
359
00:38:56,970 --> 00:39:03,740
So, l b into c w, for a ship. I am talking
for a ship, this one is for a ship does not
360
00:39:03,740 --> 00:39:10,059
apply for offshore structure as of now standard.
So, now we can see this gets cancelled, this
361
00:39:10,059 --> 00:39:17,059
gets cancelled, this gets cancelled. So, what
do I get? 2 pi, let me write it down here
362
00:39:18,450 --> 00:39:25,450
properly. c B by c w 1 plus a z; T by g, right
see t by g stays, c b, c w, 1 plus a z.
363
00:39:44,619 --> 00:39:51,130
So, this is my typical ship formula. Let us
say, now I now want to make approximate for
364
00:39:51,130 --> 00:39:57,619
that eventually. Now, a z; let us talk about
a z, because we need to estimate a z. You
365
00:39:57,619 --> 00:40:04,619
know that typical for typical ship a z would
be almost like 1, why? I told you that if
366
00:40:05,640 --> 00:40:10,240
I want because we are making a rough guess.
See we are now trying to figure out and I
367
00:40:10,240 --> 00:40:15,140
am discussing, what is the order of T z for
a typical ship; which order? Is it 5 second,
368
00:40:15,140 --> 00:40:20,759
10 second or 5 second, 6 second or is it 25
second, 30 second or is it 100 second? I want
369
00:40:20,759 --> 00:40:25,509
to have an understanding of which range T
z varies. It is very important because if
370
00:40:25,509 --> 00:40:31,299
T z was like 1 second there is no waves of
1 second, then I do not have to worry about
371
00:40:31,299 --> 00:40:37,789
T z at all or if it is 100 second, which also
is very non-existent every day wave I do not
372
00:40:37,789 --> 00:40:41,559
have to worry about it.
Remember, I mentioned earlier; waves are mostly
373
00:40:41,559 --> 00:40:46,980
5 second to 15 second or so, mostly. So, I
need to have an estimate of that, it need
374
00:40:46,980 --> 00:40:53,089
not be very accurate. After all if I make
an error of factor of you know like large
375
00:40:53,089 --> 00:40:58,460
error in a z also, because of squaring function
the error in T z is smaller than that.
376
00:40:58,460 --> 00:41:03,450
So, we can make an approximation for a z for
this purpose. Now, I mention earlier that
377
00:41:03,450 --> 00:41:09,109
if nothing is there, you can always take this
circle, add them up and if you add it up,
378
00:41:09,109 --> 00:41:14,680
you what you will find out; essentially, a
quantity which is something somewhat close
379
00:41:14,680 --> 00:41:18,839
to the mass of the hull itself. In fact, if
the breadth is very high, suppose there is
380
00:41:18,839 --> 00:41:24,650
a barge here. If you take a barge with a section
like that; obviously, this this is much larger.
381
00:41:24,650 --> 00:41:31,180
If you take a smaller boat, this is smaller.
In other words, a flat barge would have a
382
00:41:31,180 --> 00:41:38,180
larger added mass, heave heave motion and
a narrow a frigate type of ship would have
383
00:41:39,660 --> 00:41:44,319
smaller added mass, but if even if you take
this, it might be in this case a z bar might
384
00:41:44,319 --> 00:41:51,319
be as much as 3; this may actually be may
be something like 1. So, it is in this order
385
00:41:52,829 --> 00:41:57,369
1, 2, 3. You can always work it out because
you know these things are all; you imagine
386
00:41:57,369 --> 00:42:04,369
yourself a kind of a semi circular box, mass
of that with respect to the mass of this.
387
00:42:08,749 --> 00:42:13,369
It is see what is block coefficient? Mass
of the rectangular block. What is prismatic
388
00:42:13,369 --> 00:42:18,499
area coefficient? It is area of mass of this
kind of this mid ship area throughout.
389
00:42:18,499 --> 00:42:22,160
So, this is something lower than that. You
know if you think logically, you will find
390
00:42:22,160 --> 00:42:29,160
out it is something slightly lower than your
sea prismatic. Now, sea prismatic is something
391
00:42:29,910 --> 00:42:34,670
like a sometime you know like no no not lower
sorry, the other way round. It is inverse
392
00:42:34,670 --> 00:42:41,670
of that, I I am telling opposite like one,
it is like one by c prismatic little and lower
393
00:42:43,440 --> 00:42:48,420
than that slightly. So, it is around 1.52.
So, let us take that we will take example
394
00:42:48,420 --> 00:42:55,400
of 1, as well 2, as well as 3, let us say.
Now, we will want to figure out the value
395
00:42:55,400 --> 00:43:02,400
of this; go go back to this expression of
T z here. So, we will get back to this. Now,
396
00:43:02,579 --> 00:43:09,579
see let us take a ship; let us take the kind
of ship where t may be 10 meter, this may
397
00:43:12,829 --> 00:43:18,359
be 2 or may be; let us take 2.
398
00:43:18,359 --> 00:43:25,359
So, if this 10 meter; let us do it in another
piece of paper. Now, c b by c w is what order?
399
00:43:46,480 --> 00:43:53,480
Very close to 1, we are just making an approximate
guess is almost like 1 here, this one this
400
00:43:54,289 --> 00:44:00,109
term. right This I am telling say 2 means
it becomes 3. Actually, 2 is too high, let
401
00:44:00,109 --> 00:44:06,309
me take it 1 only. So, because in real life
it become actually let me make it 2 here.
402
00:44:06,309 --> 00:44:12,829
T i said in this case 10 meter, right g is
also around 10. So, it goes off. So, how much
403
00:44:12,829 --> 00:44:19,829
it becomes, 2 pi root 2, around. How much
is this? 6 into 1.5, around 9 to 10 second.
404
00:44:25,910 --> 00:44:32,910
You see for a large ship, now take a small
barge; let me take a very small barge that
405
00:44:35,230 --> 00:44:41,619
kind of barge is that you are using to cross
a you know the how that part T is very small,
406
00:44:41,619 --> 00:44:45,680
may be 3 meter or so; however, a z is much
high because of those barges are you know
407
00:44:45,680 --> 00:44:49,749
very flat.
So, it may be 3 here. So, that will be more
408
00:44:49,749 --> 00:44:54,140
logical thing. So, in that case T z will be
now approximately say, say this is still I
409
00:44:54,140 --> 00:44:59,289
take 1, I mean it can varying. So, this may
be I will take 4 here because normally if
410
00:44:59,289 --> 00:45:05,109
you take a very high, it becomes that high.
If not 4, let us take it 3; 4 may be too high.
411
00:45:05,109 --> 00:45:12,109
So, T by g, T is how much I should take? Let
us say 3 meter. So, T by g I can take 1 by
412
00:45:12,859 --> 00:45:18,349
3, you know let me just approximately take
that, T is about 3 and half meter g is around
413
00:45:18,349 --> 00:45:25,349
9.8. So, around 1 by 3. So, how much it comes
to? It comes to 2 pi around 6 point around
414
00:45:25,739 --> 00:45:29,509
around 6 second.
If you take larger t, you may find 11 second.
415
00:45:29,509 --> 00:45:36,220
So, what I am finding, typical ships 6 to
say 11, 12 seconds, something like that; say
416
00:45:36,220 --> 00:45:42,930
12 second. This is only an estimate; remember
actual one will may be 9, whatever. Now, look
417
00:45:42,930 --> 00:45:49,930
at this 6 to 10 second. Absolutely, right
at the middle of the ocean wave that exist.
418
00:45:50,559 --> 00:45:57,349
So, you see what happen? Ships therefore,
are absolutely middle of when actual ocean
419
00:45:57,349 --> 00:46:03,660
waves exist. So, you; obviously, end up having
some waves sometime that will match to these
420
00:46:03,660 --> 00:46:10,660
and you are going to have very large heave.
You cannot avoid it by design.
421
00:46:11,499 --> 00:46:18,180
Ships, it was accepted. Why it was accepted?
Because supposing, you encounter very high
422
00:46:18,180 --> 00:46:25,180
motion, which will will talk about some problem
afterwards. What you do? You have a choice,
423
00:46:25,420 --> 00:46:31,069
I want to change omega e, because suddenly
I have hit a wave, where my omega is matched
424
00:46:31,069 --> 00:46:36,920
with my omega z, I cannot change omega z because
omega z is my property; omega e is of course,
425
00:46:36,920 --> 00:46:43,430
what I am encountering, but omega z this is
a function of v, mu and omega. So, I can of
426
00:46:43,430 --> 00:46:49,019
course, change this or change this to get
change of this right. So, I can slow down,
427
00:46:49,019 --> 00:46:53,859
well I cannot speed up normally because I
do not have the power, that I can slow down,
428
00:46:53,859 --> 00:46:57,480
I can change direction.
So, I can do a combination of the two. I can
429
00:46:57,480 --> 00:47:03,249
slightly change direction and slow down. So,
I have a choice to change my encounter frequency,
430
00:47:03,249 --> 00:47:10,249
so that I do not resonate. So therefore, even
if it is resonating, I have an option to get
431
00:47:11,630 --> 00:47:17,579
away from it you know. So, I need not worry
it so much, because I have a choice to if
432
00:47:17,579 --> 00:47:23,660
I hit a bad motion, I simply turn my head
in different direction, slow down. I will
433
00:47:23,660 --> 00:47:30,039
not have as much bad motion. So, there is
a question of avoiding excessive motion that
434
00:47:30,039 --> 00:47:31,180
is possible for a ship.
435
00:47:31,180 --> 00:47:38,180
Take an offshore structure, my offshore structure
is here. Now, I will we will you will know
436
00:47:39,279 --> 00:47:45,819
why offshore structure designs are actually
primarily guided by natural period unlike
437
00:47:45,819 --> 00:47:50,349
ships, where you are designing by you know
by carrying capacity and as a much more fundamental
438
00:47:50,349 --> 00:47:55,640
parameter. In your design courses, you will
find that natural period of a ship is never
439
00:47:55,640 --> 00:47:58,749
occurring at the primary design; you have
not even talking about it. After you have
440
00:47:58,749 --> 00:48:03,269
done you talk about how much natural period,
but you do not use. If you are you are doing
441
00:48:03,269 --> 00:48:10,269
another course, you do not even use it as
an input important parameter for your design,
442
00:48:10,910 --> 00:48:15,799
but in offshore structure you cannot do that
absolutely. Why because you see here, this
443
00:48:15,799 --> 00:48:22,799
one is staying in one place throughout, while
you have to have small heave motion therefore,
444
00:48:27,239 --> 00:48:31,099
you cannot afford a situation where some wave
will come, which will resonate, because you
445
00:48:31,099 --> 00:48:35,569
cannot get away from it.
So, what you do? By design you must choose
446
00:48:35,569 --> 00:48:42,569
a natural period T z, which is far away from
this 6 to say 12 seconds, far away. Now, if
447
00:48:42,589 --> 00:48:49,589
I want to do, we we keep on writing this all
the time. Remember, this is geometry, this
448
00:48:55,549 --> 00:49:01,150
also is geometry, but this is much more geometry.
Actually, it is geometry of water plane area,
449
00:49:01,150 --> 00:49:08,150
this one c z A w p. So, what we could do?
If I want to make it lower, I have to increase
450
00:49:10,920 --> 00:49:16,589
it. So, I become like a barge, but that is
not very practical we will find out. What
451
00:49:16,589 --> 00:49:21,599
we can do? We can make it much higher. So,
if I want to make it much higher what I should
452
00:49:21,599 --> 00:49:28,599
do, I should lower this. So, my objective
is lower c z means lower A w p.
453
00:49:29,720 --> 00:49:36,150
However, I cannot lower a z, because mass
and a z because it is a carrying capacity
454
00:49:36,150 --> 00:49:40,579
rather volume. Let me put it this way, the
mass cannot be lower, after all mass is the
455
00:49:40,579 --> 00:49:45,130
displacement of the water and it is the volume
and carrying capacity; I cannot if mass lower
456
00:49:45,130 --> 00:49:50,579
means the net available is small. see Let
us say, take that I have to store so many
457
00:49:50,579 --> 00:49:55,950
million or so much of lakhs of barrels of
oil. So, I must have so much space right.
458
00:49:55,950 --> 00:50:02,859
So, I must have so much of size. So, I cannot
reduce buoyancy or displacement, but I must
459
00:50:02,859 --> 00:50:08,019
reduce A w p. How do I do it? This is the
way we have been doing it, semi submersible
460
00:50:08,019 --> 00:50:15,019
the concept evaluation take the buoyancy carrying
units below water. So, this is water. So,
461
00:50:15,769 --> 00:50:22,769
I have this high or this high, which which
ever way you call it; support it by columns.
462
00:50:24,259 --> 00:50:31,259
So, my A w p is low, but if my A w was low.
Suppose, now question is why I do not have
463
00:50:32,839 --> 00:50:39,839
only one why I do not have only one. If I
have it one, what is my metacentric height?
464
00:50:41,910 --> 00:50:48,910
I by v; what is I, moment of inertia of this
water plane is very small. So, what I want,
465
00:50:49,390 --> 00:50:56,390
for stability point of view I want i of water
plane of course, to be high. So, how do I
466
00:50:56,690 --> 00:51:01,640
achieve it? Make small A w p, separate it
out.
467
00:51:01,640 --> 00:51:08,640
So therefore, take a water plane like that
sorry. So, I have low A w p and I have I w
468
00:51:16,069 --> 00:51:23,069
p height. So, G m is high, T z is also high,
omega z is low. This is precisely what you
469
00:51:25,489 --> 00:51:29,839
have to do. Now, we will see even a numbers,
how the numbers coming approximately? you
470
00:51:29,839 --> 00:51:36,519
know We will just talk about that numbers
in a minute. So, this is what we have understood.
471
00:51:36,519 --> 00:51:43,519
Now, you see we will put this number like
that T z, I have to go to this original formula
472
00:51:43,880 --> 00:51:50,880
because here mass plus a let me put it this
way only, here rho g A w p. We will just put
473
00:51:51,369 --> 00:51:58,200
it here. Now, take a typical semi submersible.
If you look around, you will find the mass
474
00:51:58,200 --> 00:52:04,400
may be around 20000, 25000 tons.
So, this may be around, but I will take it
475
00:52:04,400 --> 00:52:10,559
as one, So 40000; approximately I am telling.
If you take you see if you take a pontoon,
476
00:52:10,559 --> 00:52:15,529
you can also can figure out typical pontoon
you can take any book around 100 meter long
477
00:52:15,529 --> 00:52:20,479
may be 10 meter, 10 meter like that. So or
15 meter into 6 meter. So, about 10000 meter
478
00:52:20,479 --> 00:52:27,479
square volume of one pontoon, with 2 pontoon
20000 this do not contribute much; say say
479
00:52:28,200 --> 00:52:33,589
20000, let me take as an sample value. You
all agree with it 20, around 20000.
480
00:52:33,589 --> 00:52:40,319
So, if I take 20000 40000 we have running
out of time. So, this is 2 pi into say 40000
481
00:52:40,319 --> 00:52:47,319
rho g; let us put this A w p, rho g say rho
g is 10. Now, for just quickly we have to
482
00:52:48,180 --> 00:52:55,180
do that A w p is this areas. Now, let us take
very quick estimate. So, this diameters are
483
00:52:55,259 --> 00:53:00,420
10 meter. So, what is the area quickly? So,
basically pi into 10 square because all the
484
00:53:00,420 --> 00:53:07,420
four of them here, pi 10 square means 100
300, so 300 right. Work it out this, this,
485
00:53:07,440 --> 00:53:11,779
this, this, this, this 2 pi square root of;
actually, this did not come out, this will
486
00:53:11,779 --> 00:53:15,049
be a smaller than that.
What I means is that, if you did work out,
487
00:53:15,049 --> 00:53:19,960
you will end up seeing that this will be around
20 second and more or rather, well this may
488
00:53:19,960 --> 00:53:24,930
be this number may be.
It is a pi d square by 4.
489
00:53:24,930 --> 00:53:29,460
Yeah. So, yeah 4 by 3; no, pi d square by
4 into 4; there are 4 there no; 4 of them
490
00:53:29,460 --> 00:53:34,989
there. So, pi d square rho g, this is 40000,
sorry sorry I made a mistake here. So, 40000
491
00:53:34,989 --> 00:53:41,989
means 1, 2, 3, 4 40 by 3; 40 by 3 is about
10. If you work it out anyhow, about 40 by
492
00:53:43,479 --> 00:53:50,479
3 is how much about 13 square root is approximately
3 3 or 4, say 4. So, 2 pi into say 2 pi into
493
00:53:50,499 --> 00:53:57,029
3, 6 into 3, 18 see around 20 second. You
immediately see that our aim has been to make
494
00:53:57,029 --> 00:54:03,470
it 20 second and more, you cannot that is
the reason how the shape of this structure
495
00:54:03,470 --> 00:54:06,410
evolve as that.
We will discuss that tomorrow’s class little
496
00:54:06,410 --> 00:54:11,319
more on the practical side, why spar buoy
evolve the way it is? Why other offshore structure
497
00:54:11,319 --> 00:54:17,420
evolves the way it is? Why the swath hull
evolves the way it is? All of them are connected
498
00:54:17,420 --> 00:54:23,410
to making natural heave period, actually pushing
it up more than 20 second. So, you want to
499
00:54:23,410 --> 00:54:30,410
make lower water plane area, not sacrificing
on stability. This is the full principle and
500
00:54:31,329 --> 00:54:37,460
if you want to do offshore design, your starting
point has to be from natural period. So, with
501
00:54:37,460 --> 00:54:42,900
that I will stop and we will continue little
bit on this and then, go over to pitch and
502
00:54:42,900 --> 00:54:43,150
other motions. Thank you.