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So far, we spoke about single degree freedom
equations of motion right.
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Today, I will like to talk about Coupled Motions.
See, what we said earlier is that one mode
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of motion have no influence on the other mode
of motion, so we have an equation which are
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all single degree. For example, let me first
of all call this for convenience, I am going
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to like you know use index notation, or we
can call this also x 1, x 2, x 3 direction,
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like this is direction 1, 2, 3, in x 3 means
z, x 2 mean y, etcetera, as well as this motions,
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rotational motions; we will call by index
4, 5, and 6.
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In other words, let us call this xi j 1 to
6; they are the 6 modes of motion. For example,
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xi 3 equal to z heave, 4 is phi etcetera.
This is for convenience, because earlier we
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used you know like z as xi 3, phi as xi 4,
theta as xi 5, now let us use that.
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So, what we had, an equation earlier is something
like this, see m m plus say heave equation,
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if I write a z into z dot dot plus b z into
z dot C z into z equal to F z, we have this
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equation. That means, heave is as if uncoupled,
no effect of heave on any other mode of motion,
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etcetera. Now, suppose I presume that all
6 modes of motions are connected to each other,
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coupled to each other.
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How is the equation of motion look like, see
it would then appear as, let me write j k
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this is going to be ok, or maybe I may, let
me put in this way F bar I k no into sorry
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this is t, let me just write this as right
now right F k, exciting force k.
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See with what I mean is that, in other words
what happen see, it is like a matrix form,
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you end up something getting like m 1 1 plus
a 1 1, 1 dot dot plus m 1 2 sorry this is
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j you know j equal to, anyhow I think this
index we will write i j, no we will make it
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this way, other way K equal to 1 to 6, and
this we make it this as j sorry we will do
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it other round, it does not really matter,
this is k equal to 1, this will be k.
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Anyhow, I think that this is becoming basically
m 1 1 plus a 1 1 xi 1 dot plus m 1 2 a 1 2
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xi 2 and this is k only dot dot plus, like
that equal to F 1, like that we will have
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6 motion equation. In fact, it will be easier
to see while write in a matrix form.
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What would happen, we will end up getting
a matrix m plus A, added mass. What happen
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see, if I write 6 by 6 equation of motion
presuming that all 6 modes of motion are coupled,
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means one influences the other, I end up getting
an equation something like that m 1 1 plus
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a 1 1 xi 1 dot dot plus m 1 2 plus a 1 2 xi
2 dot dot, etcetera, etcetera. The question
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is that, what is the physical meaning that
is what you globally want to know now, because
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this is a kind of a, just a mathematical expression
in matrix form saying that each 6 modes of
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motion influences the other 6 modes of motion
in some sense.
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Again looking at that, if I have to lo first
term, see it going to be m 1 1, see this is
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1 to 6, 1 like that 6, m 1 1 plus a 1 1 into
xi 1 dot dot, it has 6 modes.
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Let me write one one expression, just one,
see rather we can write separately m 6 1,
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this we can break it down in two parts, plus
let me write small hand, this full thing,
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like that this is this is my first term. What
it shows here, m 1 1 plus a 1 1 xi 1 dot dot
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plus a 1 2 a 1 2 xi 2 dot dot etcetera.
In fact, this matrix of course, we will all
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know that this matrix is very simple, this
one will turn out to be simply the diagonal
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terms m m m and the three i terms, I 1 1 I
2 2 I 3 3 or I itself, because we are taking
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the c g at the origin of the c g. Therefore,
all other terms becomes 0, this is there is
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no debate on that much, because this is a
rigid body moment of inertia. In other words,
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this this this matrix M matrix is m 0 0, etcetera,
0 m 0, 0 0 m and here you end up getting I
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X X I y y I z z.
So, we do not have to worry to much on that,
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this I think we all know it, there is not
much coupling what is important is to recognize
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this one, what is meant by a i j.
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Let us now understand this, what is meant
by this, see if I go back to this or j k,
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we are having this kind of expression b j
k, a j k, c j k, this is my radiation force,
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say a j k and b j k, what is a j k, what does
it mean, this is what we want to know.
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Now, let us let us look at this, expression
of a j k and b j k. See, now remember this
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one thing, what is added mass, what we said
physically, if I took a body, if I oscillate
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at in some mode of motion, let us say I oscillate
it in j th mode or in this case well let say
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j th or k th mode of motion, say k th mode
of motion, I what that get, I get f force
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in the k direction, say f k, you oscillate
in this direction and I get a force in this
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same direction, then this was my the, or let
me put the other way along.
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Let us say that I put, I took a body in oscillated
that in the k th mode of motion. Now, obviously
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what happen, there a pressure field created
throughout, now this pressure if I integrate
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them, I get force this, but force is a vector.
So, supposing I integrate them and get a force
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in direction number k, then I get f k, but
I also can get a force in direction number
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j. So, what would happen that, I can get a
force in direction j, because of motion of
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direction in k. Because see, it is very very,
I will explain this in a physical sense, I
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am oscillating this; if I oscillate that,
there is pressure around the body; if I oscillate
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this, there is a pressure around the body.
Now, I integrate the pressure, obviously when
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I integrate the pressure, I do this p n d
s to get F. Actually, if I put k here, I get
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k, if I put n k here. In other words, depending
on which direction I take normal, I may get
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that particular that that particular direction
of the force in that direction, I think I
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need to clarify this even more.
See, take this way a longest body, this will
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very clearly explain to you, I oscillate that
this way, I oscillate it this way, what happen,
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a pressure get created for an up side. So,
there is a, when I oscillate that, let us
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say this is coming down. So, I have got acceleration
coming and here acceleration is coming down;
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after all when I do that, when I press that,
this fluid is going to give a resistant force
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or you can say other way round, this particle
particles are getting accelerated this side.
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I mean take another piece of paper, this I
am giving it, let us say also this is pitch
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motion which is actually xi 5. I have inducing
xi 5, I am giving a pitch motion about its
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origin right. When I induce that, what happens,
see what happen to this particle here, this
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particle is getting accelerated down, this
is getting accelerated down, accelerated down,
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whereas here the particles are, because this
is going up, it is going to I mean motion
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inward right. So, there is a field created
here, there is a field created here.
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Now, if I have to integral the pressures,
if I do this p and if I do n 3 d s, that means
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if I want to find out what is my force in
this direction, force in the vertical heave
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direction, because I have given a motion.
It may not be 0, why it may not be 0, because
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this and this side may not cancel each other,
because this you see, there is a net pressure
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for this part, net pressure for this part,
but they are not symmetric or enough side.
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So, since it is not symmetric, they will not
cancel out each other. So, I will not end
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up getting it a net vertical flows; in terms
of acceleration, if I if you remember we are
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talking about as if there is a mass being
attached, what would happen, if I take each
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particle and see acceleration, these particles
are accelerating down, but this is up, these
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two when I sum them up do not become 0, there
is a net acceleration in the vertical direction.
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Some small value, but non 0 value, this is
what we can call the force created in direction
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3 because of motion in direction 5. That means,
this is what is my radiation force in direction
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3 for motion in 5 and that give rise to my
added mass a 3 5 and damping b 3 5, the part
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of these, because after all added mass damping
up two parts which means, because it is not
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symmetric about all planes, I can always have
force in direction. In other words, I can
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say that I can have force or moment in direction
K due to whichever way you can interchange,
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motion in direction j that is my some kind
of F j k which gives rise to my a j k and
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b j k. Because after all, this two terms as
I said are nothing but expression of the forces.
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What we do added mass damping; we say that
that part of the force which is in face with
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acceleration is called added mass, out of
phase in acceleration or in phase of velocity
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called damping. So, these two are nothing
but the force, we are we are always saying
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that added mass and damping are nothing but
basically the two part of the same force.
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But here the important point is that, you
can understand that, if I have to keep it
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a pitch motion, I may have a heave force,
so pitch influences heave.
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Now, you see when I want to explain the total
force, what is happen is that, just these
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two examples if I give. Now, let us looking
at pitch motion or heave motion, what would
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happen, obviously the body is simultaneously
pitching.
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So, I have got one part of the, say radiation
force, I have got one part of the force is
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a 3 3 into xi 3 dot dot, this is my added
mass force in heave direction; see here, I
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am heaving it, I want to find out what is
the force in this direction, net.
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So, one part of the force is this, but remember
the simultaneously that what is having xi
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5, simultaneously at the same time. Because
of this, there is additional force of a 3
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5 into xi 5 dot dot coming, arising, so at
this is exactly what is happening. If I if
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I look that at the matrix equation, you see
here, I am looking at this 3 this 3. So, I
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have m 3 3 xi 3 dot dot plus here a 3 3 that,
but a 3 5 into xi 5 dot dot, that comes in
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if I look back.
So, essentially what would happen, if I have
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to look back at that at that in a, in terms
of j k, if I have to looking back to the first
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one that it it essentially means that, I am
looking at the all the forces in directions
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say j. But what is happening is that, j k
k dot, j k k dot, k double dot, there is force
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coming in direction j, because of motions
in direction in direction k, this is what
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is called coupling. And this of course, exists
as you have seen in heave and pick is strongly
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coupled very much. So, we will see that another
example of roll, hydrostatic force itself,
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we will see that this roll part.
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See, I have this say, I have this body, this
is my center of gravity G. Now, remember that
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let me just take this point; if I have a heel,
say I let me take this to be the reference
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point, easier to explain the concept. Suppose,
I give a heel does it, does the line go through
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this, if the heel is large? No, it goes to
somewhere else, you know that in large angle
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heel it will go through that.
What does it mean; it means that if I have
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to heel the body about this point, it will
actually undergo some kind of a upward motion,
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a heave motion. That means, roll necessarily
causing a heave, same thing is pitch. See,
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or other you see pitch and heave, that is
better to see that, now you see here; if I
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have to make it go down, heave about this
point, remember tell me will it trim? The
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answer is yes, it will trim, because the LCF
is somewhere here. So, what happen if I were
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to cause a heave, then it automatically also
trims; that means, heave is always causing
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a trim here, or rather in this case, parallel
sink is causing a trim. So, heave and trim
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are coupled, moment you introduce that, that
means when you want to push down the body,
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it also undergo the trim, this is what is
called coupling.
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So, what we are finding out in a general sense
that, when I write in a general sense, the
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coupling is looking like that, every I am
allowing every mode of motion to be coupled
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with every other mode of motion, that is of
course how I will start writing it. But now,
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you will see from practically, is it so, because
of symmetry.
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For example, added mass I said you see, let
us lets take an example of this added mass
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only again, now most ships are symmetric about
this center plane. Now, if I oscillate this
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side, say direction 1, suppose I take a body
and oscillate in the sorry this is direction
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3, will it cause a force in this direction,
why because pressure exactly symmetric, so
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what would happen a 2 3 becomes 0.
Similarly, say if I have to move in this direction,
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oscillate, will it cause any force in this
direction? No, so a 1 2 is 0. So, therefore,
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because of symmetry we would know some modes
of motions is 0, some are not 0, there is
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a strong coupling. Now, understand that almost
all ships are symmetric about center plane.
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In fact, all vehicular systems are usually
geometrically similar, external geometry similar
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about the center plane, whether you take a
automobile or aircraft or so, because you
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would always wanted to that, since the fluid
around this side and that side you want to
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be same, otherwise it is going to stir on
side.
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See you are making a body to move, obviously
you would want to be symmetric about this
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line, otherwise it is time to go on other
side right. So, therefore, they are mostly
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symmetric about center plane, but not about
other planes, there is always one plane symmetry,
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so we that causes many terms to equal 0.
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So, what happen if I have to look at added
mass matrix, you now see this again, I write
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a 1 1 a 1 2, etcetera, actually let me write
this 1, 2, 3, 4, 5, 6, and here also 1, 2.
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So, this is a matrix here right, it turns
out that 1 that is surge, three that is well
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no heave and pitch, this three are strongly
coupled. And similarly, you will find out
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I mean I am I am stating that without proof,
basically what it happen many of them see
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what would happen supposing a 1 2, etcetera.
Some of the terms are 0, what would happen
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if you write it down, there will be no influence
of this mode of motion with the first mode
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of motion, this one sway, roll. It turns out
that essentially the strong coupling exists
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between surge, heave, pitch, and sway, roll,
yaw. In other words, see this is 1, 3, 5 index
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wise, this is 2, 4, 6. So, it is any term
which is this and this, or this and this,
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that black and red are 0.
Here in other words, what would happen, if
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I want to find out this or the surge, heave,
etcetera, I simply have to delete those term,
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I will show that to you in a minute. So, because
of symmetry, it turns out that one can show
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this. In fact, what we will talk about is
most of heave and pitch coupling, because
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that is the most important coupling; what
would happen, if I want to only show heave
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and pitch coupling, just neglect that.
Let us say that I only presume as if I have
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only heave, and I have only pitch, and nothing
else is there. What would happen, I have only
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this line, the terms in this line, and then
terms in this line, and here again terms in
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this line, terms in this line. So, I only
will I consider these two and this two, everything
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else 0, then I end up getting what is known
as coupled heap and pitch equation, because
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what I am saying is that that I have only
heave and pitch, couple, nothing else is there.
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Now, remember see, what is meant by uncouple?
Suppose all modes are uncouple, there is no
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coupling, what will happen, everything every
a j k would be 0, if j is not equal to k;
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supposing I say single degree freedom equation
of motion, from here how do I derive well
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00:24:54,200 --> 00:25:00,969
I say no coupling, what does it mean, a j
k is 0 if j is not equal to k which means
182
00:25:00,969 --> 00:25:05,570
no modes of motion influence, other modes
of motion except its own modes means if I
183
00:25:05,570 --> 00:25:09,389
heave, it only give a heave force, nothing
else.
184
00:25:09,389 --> 00:25:16,389
So, when I say once again, this should be
understood if a j k is 0 for all j not equal
185
00:25:20,399 --> 00:25:27,399
to k, equal to nonzero for only j equal to
k, this is imply that all the motions are
186
00:25:34,029 --> 00:25:41,029
uncoupled. If I say that some of the j and
k are influenced, I will say that j is coupled
187
00:25:42,869 --> 00:25:48,039
with that k. So, if I want to look at heave
and pitch coupling, I will only have to have
188
00:25:48,039 --> 00:25:55,039
a 3 3 and a 3 5 as non 0 or a 5 3 a 5 5 as
non 0, everything else is 0.
189
00:25:58,009 --> 00:26:03,210
So, I can construct from this now, you see
these four equations means, I allowed all
190
00:26:03,210 --> 00:26:10,090
these to be coupled, symmetry tells me that
if you go ahead that essentially these three
191
00:26:10,090 --> 00:26:14,200
modes are coupled these 3 mode are coupled.
Now, let this surge, heave, and pitch, surge
192
00:26:14,200 --> 00:26:18,259
actually is of no interest to us, because
surge is which motion, forward motion. See,
193
00:26:18,259 --> 00:26:23,070
a ship is moving at 20 meter, 10 meter per
second, how does it matter it is going to
194
00:26:23,070 --> 00:26:27,009
make some oscillation about 10 meter, you
understand this; suppose it is going like
195
00:26:27,009 --> 00:26:33,559
10 meter per second, it is going forward and
on that mean position making some oscillation,
196
00:26:33,559 --> 00:26:38,599
not really interested much.
So, interest become heave and pitch, and similarly
197
00:26:38,599 --> 00:26:44,359
roll and yawn, and sway you will find, but
we look, let us look at heave and pitch mostly,
198
00:26:44,359 --> 00:26:45,609
so let us say heave and pitch.
199
00:26:45,609 --> 00:26:49,249
Now, if I want to write the heave and pitch
coupled equation, how it look like, you see
200
00:26:49,249 --> 00:26:56,249
the full form, it is now look like that, m
plus a 3 3, this we need to write this plus
201
00:27:15,719 --> 00:27:22,719
plus, I may call it this F, similarly we are
going to have actually this is we are writing
202
00:28:33,059 --> 00:28:40,059
sorry.
Anyhow this way, basically what you find out
203
00:28:46,169 --> 00:28:53,169
is that 3 3 and 3 5 terms are there a 3 3
xi 3 dot dot, this is actually the heave part,
204
00:28:54,359 --> 00:28:59,359
this is the part this is come for couple for
pitch. That means, this part represent the
205
00:28:59,359 --> 00:29:05,269
forces in the heave direction arising from
pitch motion, this is my pitch moment, and
206
00:29:05,269 --> 00:29:11,639
this is my pitch moment arising out of heave
motion. So, this terms a 3 5, b 3 5, etcetera
207
00:29:11,639 --> 00:29:18,639
or a j k are the coupled dump, this is how
we are getting basically equation of motion.
208
00:29:24,989 --> 00:29:31,989
Now, one interesting point that I need to
tell you is about, this added mass is, because
209
00:29:35,909 --> 00:29:41,109
we now end up getting large number of added
masses right, because it now not only those
210
00:29:41,109 --> 00:29:45,729
diagonal added masses, we also need to have
many more added masses. It turns out that
211
00:29:45,729 --> 00:29:52,729
added mass actually symmetric, that means
a j k is equal to a k j, b j k equal to b
212
00:29:55,700 --> 00:30:02,499
k j, same is with the c terms; also c the
restoring force is symmetric a j k, I fact
213
00:30:02,499 --> 00:30:06,489
restoring force 3 5 will not be existing.
So, we do not talk about it, but high the
214
00:30:06,489 --> 00:30:09,489
time it exists here.
So, what is happening is that, you end up
215
00:30:09,489 --> 00:30:16,489
having these kind of added masses, we call
this say a 3 5 will be called heave pitch
216
00:30:18,609 --> 00:30:25,609
coupled added mass. It has a unit remember,
this is what is the unit mass is ton, moment
217
00:30:28,950 --> 00:30:35,019
of inertia if I would be ton meter square,
this is going to be ton meter. So, this is
218
00:30:35,019 --> 00:30:39,519
have this is, so you cannot say it is either
mass or moment. So, you can say it is couple
219
00:30:39,519 --> 00:30:45,349
added mass, because it has unit of ton into
meter, why, because remember it is a part
220
00:30:45,349 --> 00:30:48,979
of force.
See, always you should remember this into
221
00:30:48,979 --> 00:30:54,320
this should give you a force unit, remember
this into this should give you a force unit,
222
00:30:54,320 --> 00:30:59,809
this coupled term, this has a what is the
unit of that? 1 by second square, so this
223
00:30:59,809 --> 00:31:04,249
is going to be ton meter into 1 by 7. So,
you will be able to check that it is actually
224
00:31:04,249 --> 00:31:08,759
gives a unit of moment, a force sorry not
more this thing a force.
225
00:31:08,759 --> 00:31:13,039
So, like that you will be able to find out
that essentially it has unit like that, see
226
00:31:13,039 --> 00:31:20,039
a 3 3 into xi 3 dot dot is mass into acceleration,
here it is remember it is xi 5 dot dot with
227
00:31:22,509 --> 00:31:27,489
a 3 5. So, it has got a different unit, so
this will have unit of same as this and you
228
00:31:27,489 --> 00:31:34,489
will be able to find out that how it works
So, moment of inertia is ton meter square
229
00:31:37,869 --> 00:31:44,869
mass into length square, because it is mass
moment of inertia we are talking, whereas
230
00:31:45,339 --> 00:31:51,999
this is mass only in ton, so this is in between.
231
00:31:51,999 --> 00:31:57,639
So, anyhow in other words what is happening
is that, if I have to draw this, may be I
232
00:31:57,639 --> 00:32:04,639
do not draw this matrix here, or rather if
I have to draw this matrix of a, you have
233
00:32:05,659 --> 00:32:11,789
this as ton, this is ton into meter square,
this has ton meter, this has ton meter, I
234
00:32:11,789 --> 00:32:17,599
mean you know that is obvious right, because
the way it is goes symmetric. If I have to
235
00:32:17,599 --> 00:32:24,209
look at the look at the, see here this is
actually acceleration, linear acceleration,
236
00:32:24,209 --> 00:32:29,099
this is rotation acceleration.
This part is linear acceleration means what,
237
00:32:29,099 --> 00:32:36,099
meter by second square, rotational acceleration
is 1 by second square. See, ton meter by second
238
00:32:41,019 --> 00:32:44,469
square, ton meter by second square everywhere
the same unit will come, this is ton meter
239
00:32:44,469 --> 00:32:48,089
square by second square, ton meter by second
square you know like if you look at that,
240
00:32:48,089 --> 00:32:55,089
see here this side this is ton. So, you can
see that, what I am trying to say is very
241
00:32:59,479 --> 00:33:05,320
easy to see the units, you can always see
that the units kind of match right, the units
242
00:33:05,320 --> 00:33:12,089
will be like that. Now, let us look at another
interesting point of solving this equation
243
00:33:12,089 --> 00:33:17,099
of motion, I want to go back to the first
equation of motion and I try to see how you
244
00:33:17,099 --> 00:33:18,149
can do that.
245
00:33:18,149 --> 00:33:25,149
See, I had this m j k plus a j k that is why
I wrote k and j, that is why this confusion
246
00:33:27,509 --> 00:33:34,509
came you know. I am omitting this sigma term,
because I presume that sigma term exist I
247
00:33:36,979 --> 00:33:43,979
I will tell you about this or may be we can
put them.
248
00:33:57,879 --> 00:34:04,879
Let me put this now this way, this we can
this term a little bit or I will put this
249
00:34:21,119 --> 00:34:27,929
way. See, what I wanted to say, I need to
solve this, there are six equations, how do
250
00:34:27,929 --> 00:34:34,929
I solve this. So, we have this or other sorry,
this is my motion, remember that this motion
251
00:34:41,679 --> 00:34:48,679
is a sinusoidal motion with an amplitude and
this, this is what I call complex amplitude,
252
00:34:53,349 --> 00:34:59,630
why complex amplitude because actually it
should be actually it should be something
253
00:34:59,630 --> 00:35:06,630
into cos omega t plus beta. In other words,
I think I have mentioned that before if I
254
00:35:07,160 --> 00:35:14,160
want to lo at this again, see xi k equal to
something into cos omega t minus or plus beta
255
00:35:19,190 --> 00:35:26,190
we can call it, this is actually real part
of something into e of i omega t minus i beta.
256
00:35:28,660 --> 00:35:35,660
I can bring this here, so I can call it is
real part of, so this is what we will call,
257
00:35:48,539 --> 00:35:54,710
this is also actually, this is also amplitude
into e of i omega t, this is also a complex
258
00:35:54,710 --> 00:36:00,010
number I can call, force also a sinusoidal
function, the beauty is that all are sinusoidal
259
00:36:00,010 --> 00:36:04,410
function. So, I can call it in this way with
a complex number. The reason of doing that
260
00:36:04,410 --> 00:36:09,019
is because what would happen you know is that,
now if I look at this part xi double dot k
261
00:36:09,019 --> 00:36:11,710
k, I will just go to the next one.
262
00:36:11,710 --> 00:36:18,710
So, you see if I have xi k equal to xi k bar
e of i omega t, then xi k dot equal to i omega.
263
00:36:27,369 --> 00:36:34,369
Now, I put it back to that equation what do
I get, I will get minus omega square m j k
264
00:36:43,309 --> 00:36:50,309
plus a j k e i omega t all will be common,
then we end up getting plus i omega b j k
265
00:37:01,279 --> 00:37:08,279
bar plus c j k equal to F j.
You understand that this will, it will lo
266
00:37:13,859 --> 00:37:18,440
like that with a sigma of course, there. So,
this entire thing, therefore it becomes a
267
00:37:18,440 --> 00:37:25,440
matrix into xi k equal to F j. So, what is
this, this is therefore, this gives me I write
268
00:37:37,510 --> 00:37:44,510
straightway. You know if you want to manipulate
that, what what I am trying to say, I could
269
00:37:44,619 --> 00:37:49,569
call this to be this whole thing is a complex
equation. So, it becomes something like or
270
00:37:49,569 --> 00:37:53,690
one can write this equation to be, if I if
you want it you can write it as a complex
271
00:37:53,690 --> 00:38:00,690
equation D j k into sigma k bar equal to F
j.
272
00:38:02,349 --> 00:38:09,349
This is the complex matrix, complex number,
complex number 6 by 6 equation, this is inverse
273
00:38:09,589 --> 00:38:16,210
of this into this, one line solution. So,
end up getting all those six modes of motion,
274
00:38:16,210 --> 00:38:21,029
what you get complex numbers, you end up getting
this number which is complex number, next
275
00:38:21,029 --> 00:38:27,410
line the real amplitude is equal to you know
the absolute value of this F is equal to tan
276
00:38:27,410 --> 00:38:32,369
inverse of imaginary real part. My point is
therefore you know is that, the main part
277
00:38:32,369 --> 00:38:39,309
that I am saying is that, these equations
the solution is trivial, absolutely trivial.
278
00:38:39,309 --> 00:38:46,309
In terms of solution if I know the component,
because if I assume the equation has a nature
279
00:38:48,390 --> 00:38:55,390
of sinusoid. So, if I say that it is like
an sinusoid, then this 6 by 6 such long term
280
00:38:56,220 --> 00:39:00,299
you know, if I look at that, basically there
will be 6 term 6 term. So, you have got you
281
00:39:00,299 --> 00:39:07,299
know like 6 plus 6 plus 6 plus 6 very long
expression, entire thing becomes one number,
282
00:39:09,160 --> 00:39:14,010
something into xi k, some D j k into xi k
equal to this thing.
283
00:39:14,010 --> 00:39:19,500
So, what happen that, solutions become trivial,
in a computer especially all that you have
284
00:39:19,500 --> 00:39:24,470
to do just keep on adding the terms, you end
up getting this all complex number. So, there
285
00:39:24,470 --> 00:39:29,640
is nothing to worry about solving this, when
we look at for example, our this you know
286
00:39:29,640 --> 00:39:36,640
like this couple couple sort of equations
that we are talking, same thing will happen
287
00:39:37,450 --> 00:39:44,450
as I said this couple heave and pitch equations
that we are talking about, where we have done
288
00:39:47,700 --> 00:39:53,970
that coupled pitch and heave equations, yeah
this one. If I am looking at this part, all
289
00:39:53,970 --> 00:39:58,400
that I have to do is to say this is equal
to something into cos omega t, or something
290
00:39:58,400 --> 00:40:01,670
into e i omega t, then you end up getting
all the one number.
291
00:40:01,670 --> 00:40:08,089
So, actually you end up getting only a 2 by
2 matrix, you know something absolutely you
292
00:40:08,089 --> 00:40:15,089
will end up getting this full thing into equal
to something like this.
293
00:40:19,710 --> 00:40:26,710
This expression you will end up getting something
into or rather bar; in other words something
294
00:40:39,440 --> 00:40:46,440
like you know a I am writing this in terms
of matrix a 1 1 xi rather rather this is sorry
295
00:41:00,400 --> 00:41:07,400
no no no this is 3 3, this is 3 5 something
like that.
296
00:41:16,240 --> 00:41:20,730
What I am trying to say you know, I think
I think this say I think this something into
297
00:41:20,730 --> 00:41:23,740
xi 3 plus something into xi 5 is F 3, something
into xi 5 is F 3 something into xi 5 is F
298
00:41:23,740 --> 00:41:29,269
3 is equal to sorry not xi 5, it is 3 5 equal
to this thing. In other words what what happen
299
00:41:29,269 --> 00:41:35,900
is that, it is just a 2 by 2 matrix it is
one line solution, so solution is not important.
300
00:41:35,900 --> 00:41:41,670
Now, what is important is that, how do I get
the estimate for that just like what I said
301
00:41:41,670 --> 00:41:47,339
earlier; in earlier I could get away by taking
only diagonal added masses, for coupled one
302
00:41:47,339 --> 00:41:51,920
all I have to do is add those couple, they
added masses that is all. So, there is really
303
00:41:51,920 --> 00:41:58,920
no difficulty in the solution part of it,
what is important is that you have to account
304
00:42:00,039 --> 00:42:03,059
for that, because both are influencing each
other.
305
00:42:03,059 --> 00:42:07,930
The important point is that, you must account
for the two if I want a couple mode of motion,
306
00:42:07,930 --> 00:42:13,220
but having obtained the hydro times if the
coupling, means couple added masses couples
307
00:42:13,220 --> 00:42:20,220
damping, etcetera, solution is trivial. So,
the main point once again is that, get this
308
00:42:21,529 --> 00:42:28,529
number right, it is only a one line job to
get the solution in that case, that is the
309
00:42:29,119 --> 00:42:34,049
point I am repeatedly trying to get in your
you know in your mind that, the solution becomes
310
00:42:34,049 --> 00:42:40,289
always not very important, it is invariant,
it is the terms that go in that is what is
311
00:42:40,289 --> 00:42:47,289
important, the terms that go in this one,
this one, this one, these are more important.
312
00:42:47,319 --> 00:42:52,230
If you get the numbers, you can get the solutions
and you find out in computation, it is only
313
00:42:52,230 --> 00:42:56,920
the numbers that is more important.
Now, the other thing is that, why we always
314
00:42:56,920 --> 00:43:02,670
could afford to study uncoupled motions, one
of the reason is because normally these terms
315
00:43:02,670 --> 00:43:06,779
are small in number, although it may have
some influence. Because they are small in
316
00:43:06,779 --> 00:43:13,259
number, we could sometime ignore it, of course
from also study point of view obviously, the
317
00:43:13,259 --> 00:43:19,029
uncouple modes will be the much larger, coupling
my influence there, but might may be few percentage
318
00:43:19,029 --> 00:43:23,769
depending on situation, although heave and
pitch coupling is somewhat important. Now,
319
00:43:23,769 --> 00:43:29,359
let me just put one more important thing about
combining the motions now, see we have we
320
00:43:29,359 --> 00:43:34,500
have let say we have obtained all the couple
motions we we want to.
321
00:43:34,500 --> 00:43:41,500
So, look at this point here, I have the ship
here, there is a point here, somewhere. I
322
00:43:44,970 --> 00:43:51,970
want to find out the motion of this in the
vertical direction, some point, remember here,
323
00:43:58,950 --> 00:44:03,579
now this phase thing will come in picture
that is very practical. I have got a let us
324
00:44:03,579 --> 00:44:08,809
say gun mound on a naval vessel, then I need
to find out the acceleration of this point,
325
00:44:08,809 --> 00:44:15,809
so I need to find out vertical oscillation.
So, this point is let us say a point p and
326
00:44:19,519 --> 00:44:26,519
its location is let us say distance is I mean
the coordinate of p
x b y b z b, let us say you know it is located
327
00:44:40,200 --> 00:44:47,200
at some point, 50 meter forward of the coordinate
system that is x based 50 meter, 5 meter and
328
00:44:50,319 --> 00:44:57,319
2 meter high, some location and it is what
is its motion at this point then, now I can
329
00:44:58,640 --> 00:45:05,640
always get for small amplitude motion by combining
the two, it becomes x minus y b into psi plus
330
00:45:06,710 --> 00:45:13,710
z b into theta y p becomes. See, this part
what we have done here, remember that actually
331
00:45:55,069 --> 00:46:01,609
if this was my G, I was knowing the modes
of motion of this rigid hull, how much this
332
00:46:01,609 --> 00:46:06,450
is moving that is my x here, how much it is
moving that is my y here, similarly theta
333
00:46:06,450 --> 00:46:12,890
xi and phi are the rotations.
Now, if I want to know this, it is a simple
334
00:46:12,890 --> 00:46:18,000
kind of coordinate transformation. See here,
this is going to be say say vertical direction,
335
00:46:18,000 --> 00:46:25,000
if I want to look that, vertical is going
to be z, but now because of this pitch angle
336
00:46:25,029 --> 00:46:28,210
is going to be having contribution of x b
this thing, because you know that this is
337
00:46:28,210 --> 00:46:35,210
coming down so much
and also for roll. So, what is happening you
know that, this is a very simple addition
338
00:46:41,259 --> 00:46:48,259
to find out the vertical mode of motion combining
them what do I explain that mode.
339
00:46:48,730 --> 00:46:55,730
Let us look at this in a this way, see the
heap part, if I have to find out a point here,
340
00:46:59,500 --> 00:47:06,500
if the ship is trimming, then you see amount
of this that moves out up is going to be x
341
00:47:09,150 --> 00:47:16,150
b into theta, this is the amount of its z
displacement, because of theta this thing,
342
00:47:18,670 --> 00:47:25,670
of course our theta is positive down. So,
therefore, what happen if I combine heave
343
00:47:26,130 --> 00:47:33,130
which is z and
the vertical displacement, because of trim
that is this much and also if I check rotation,
344
00:47:40,539 --> 00:47:47,539
remember this was this point is located somewhere
here. So, if I take phi, another contribution
345
00:47:48,299 --> 00:47:54,829
comes that will become y b into phi, this
is my net z point p.
346
00:47:54,829 --> 00:48:01,829
In other words, the displacement at any point
is very easily obtained by the modes of motion
347
00:48:04,569 --> 00:48:11,569
that I have already by rotation, and by you
know by rotation by linear displacement. Actually
348
00:48:13,569 --> 00:48:20,569
it is something like this, now x y z at a
point p is x y z plus in transformation matrix
349
00:48:26,700 --> 00:48:33,700
into x b y b z b, this is the general formula
for any point; in other words, x at a point
350
00:48:35,599 --> 00:48:42,599
vector b x of the central gravity plus R into
x of the point b, this is the kind of a expression
351
00:48:46,009 --> 00:48:52,190
that we know.
This is a, this has nothing to do with hydro
352
00:48:52,190 --> 00:48:59,190
dynamics, it is purely a kinematic equation,
its coordinate transformation, now looking
353
00:49:00,329 --> 00:49:03,039
back at this now I go back to this expression.
354
00:49:03,039 --> 00:49:10,039
What, why I am coming back is that see, we
end up getting, therefore z at the point p
355
00:49:10,319 --> 00:49:17,319
is equal to z minus x b theta plus y b phi.
Now, if I want to write in terms of xi, it
356
00:49:21,200 --> 00:49:28,200
is basically xi 3 minus x b into xi 5 plus
y b into xi 4. in terms of now this is the
357
00:49:36,369 --> 00:49:43,130
actual modes of motion, in terms of amplitude,
etcetera, this becomes an amplitude function
358
00:49:43,130 --> 00:49:50,130
into i omega t say complex amplitude or other,
let me put it this way if I want to find out
359
00:49:53,630 --> 00:50:00,630
this t.
See, now what happen, this expression I know,
360
00:50:13,970 --> 00:50:19,859
if I add it up remember all are sinusoidal
function all are sinusoidal function we will
361
00:50:19,859 --> 00:50:23,880
do this later on again. So, if I add this
up, it will going to be sinusoidal function
362
00:50:23,880 --> 00:50:28,339
cos omega t plus something. So, the entire
thing is going to be again another sinusoid.
363
00:50:28,339 --> 00:50:33,749
So, the question is that, this is going to
be again something into e i omega t. So, let
364
00:50:33,749 --> 00:50:37,890
us let us call this amplitude, because you
can always call that as an amplitude into
365
00:50:37,890 --> 00:50:43,190
e i omega t, if I am writing this in terms
of z p bar as a complex amplitude into e i
366
00:50:43,190 --> 00:50:48,640
omega t, each of them I am writing bar into
e i omega t, but bar represents complex amplitude
367
00:50:48,640 --> 00:50:51,490
complex amplitude.
So, what would happen if I combine them, I
368
00:50:51,490 --> 00:50:56,680
get a complex amplitude of this, now remember
this is the complex number, this is the complex
369
00:50:56,680 --> 00:51:00,999
number, this is the complex number. So, this
is very easy to do in a complex case, because
370
00:51:00,999 --> 00:51:07,249
I end up getting a complex number, this becomes
this and this number is nothing but z p bar
371
00:51:07,249 --> 00:51:14,249
which is actually z p absolute you know, I
can say this absolute into e of some i beta.
372
00:51:15,289 --> 00:51:21,390
So, I can get both of the information from
here. So, this is exactly how is one of the
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00:51:21,390 --> 00:51:27,430
way how we can combine and now here, there
is the most important point here, remember
374
00:51:27,430 --> 00:51:34,430
that this one, what is this one, this one
is xi 3 into e of i xi 3, it has a phase angle,
375
00:51:37,420 --> 00:51:42,910
remember because this is a complex number
with a phase angle, this has also a phase
376
00:51:42,910 --> 00:51:47,910
angle, now this let me call is star as the
absolute value.
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00:51:47,910 --> 00:51:54,450
The point is that, I am adding the complex
numbers which has a phase, so therefore the
378
00:51:54,450 --> 00:51:59,279
phase of these, and these, and these are very
important when I add this up. So, in other
379
00:51:59,279 --> 00:52:04,180
words, I cannot simply, I cannot say that
the ship is heaving 2 meter and pitching say
380
00:52:04,180 --> 00:52:11,180
3 degree. So, I cannot add 2 meter plus 3
degree into the distance to get this at all,
381
00:52:12,759 --> 00:52:19,759
this is what I am trying to tell, suppose
my xi my rather heave, absolute heave was
382
00:52:19,799 --> 00:52:26,799
2 meter. I am looking at a point 50 meter
in front of at the bow point which is 50 meter,
383
00:52:27,029 --> 00:52:34,029
x b is 50 meter, 50 meter ahead of mid ship,
or the center of gravity, and my maximum trim
384
00:52:34,029 --> 00:52:39,700
is say 3 degree.
I cannot say my z p is going to be 2 meter
385
00:52:39,700 --> 00:52:46,289
plus 50 into 3 degree. I have to add with
a phase in that and this is exactly where
386
00:52:46,289 --> 00:52:53,289
the phase information comes in, which are
very very important. So, this particular expression
387
00:52:53,940 --> 00:52:58,749
you have to operate in a complex domain, if
you do not want to operate in a complex domain,
388
00:52:58,749 --> 00:53:04,690
you have to operate in a the algebra will
become more complicated like, if I if I want
389
00:53:04,690 --> 00:53:09,809
to write them in terms of cos and sin, what
would happen, remember this will have something
390
00:53:09,809 --> 00:53:16,809
into cos omega t plus beta z, something into
cos omega t plus beta theta, something into
391
00:53:17,670 --> 00:53:22,410
cos omega t beta phi, add them all up, entire
thing you manipulate will be something into
392
00:53:22,410 --> 00:53:28,539
cos omega t plus beta z, much more difficult,
normally for operation point of view.
393
00:53:28,539 --> 00:53:33,829
But normally that is the reason why we always
use complex algebra, but I will end today’s
394
00:53:33,829 --> 00:53:40,829
you know the couple thing, because we will
not repeat that to just by saying that, if
395
00:53:42,470 --> 00:53:49,180
I were to consider couple modes of equation,
the form of equation remain exactly same.
396
00:53:49,180 --> 00:53:55,029
It is just that I must consider those additional
terms that you know coupling term, so called
397
00:53:55,029 --> 00:53:59,339
these terms.
In other words, I have to have all the j k
398
00:53:59,339 --> 00:54:04,839
terms when j is not equal to k, just add them
all up, practically it turns out not all j
399
00:54:04,839 --> 00:54:11,839
and k are coupled, it is only 1 3 5 and 2
4 6 of couple, primarily 3 and 5 are most
400
00:54:13,630 --> 00:54:18,630
coupled, solution is trivial, all I want to
know, have to know is that, the terms what
401
00:54:18,630 --> 00:54:24,569
is well earlier I need it to know a 3 3 a
5 5, now I need it to know I 3 5 I 5 3 in
402
00:54:24,569 --> 00:54:29,400
addition to a 3 3 a 5 5, once I know the solution
is trivial.
403
00:54:29,400 --> 00:54:34,940
Once I get this coupled solution, then also
I can also join them to get displacement at
404
00:54:34,940 --> 00:54:40,880
any other points which becomes more practically
important like this, because I am more interested
405
00:54:40,880 --> 00:54:46,759
to find out ultimately practical things like,
what is my velocity here, acceleration, etcetera.
406
00:54:46,759 --> 00:54:52,619
And I tell you this, I will just end it by
saying that once I know this kind of expression,
407
00:54:52,619 --> 00:54:56,640
I can find that velocity in acceleration by
just one line, I just have to put a dot here,
408
00:54:56,640 --> 00:55:00,779
you know i omega into this double dot here
minus omega into this.
409
00:55:00,779 --> 00:55:05,970
So, once I know one, the other two automatically
comes out. Why I am saying, because you may
410
00:55:05,970 --> 00:55:09,039
be interested to find what is acceleration
here, because that is the load coming in the
411
00:55:09,039 --> 00:55:15,529
gun mound let us say, not the displacement
find the displacement, then take double dot
412
00:55:15,529 --> 00:55:21,170
that is it. And you know that is where I will
end today, next class onwards we will be doing
413
00:55:21,170 --> 00:55:28,170
some some other topic from tomorrow, thank
you.