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We will continue our talk on this uncoupled
pitch and heave motions.
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Let me look at this estimate for added moment
of inertia in pitch and also the damping,
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that is basically del I y y and b theta. See
just like heave, if I have this ship here,
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and if I took a section, and if I have my
sectional added masses was a z 2 d, was my
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sectional
heave added mass and of course, you can also
have that similar damping, I can always call
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this.
Remember in previous class we mention about
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strip theory, that the ship was made in a
strip and oscillating. Now, here what happen,
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see heave and pitch are so correlated, that
if I were now oscillating this way, it basically
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mean sections are oscillating this way. So,
what happen as far as pitch is concerned,
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this has an sectional added mass and what
is my pitch moment of inertia that is delta
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I y y, it simply becomes integration of x
square a z 2d into dx over the length. Of
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course, you have to take the x measure from
the origin of the system C G. So, you see
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this how easily it can be found out.
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I will show this picture in this form again
here, because it is easier to see a two dimensional
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forms. So, I have a section here, let us say
this much is my added mass, body is rotating
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like that. So, what I am doing is that this
of added mass, square of that integrating
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over it. So, you see what is happening is
therefore, that if I, well I can estimate
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pitch moment of inertia also from sectional
heave added mass and similarly, the damping
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part. And if you recall, we talk yesterday
about different mean of estimating that one
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of the simplest one was also of course, that
as I said that if I were to take these a sectional
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b by a z 2d, if nothing is available I will
simply take it to be mass of this semi circle,
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how much is this? pi B square by 4 into half.
And of course, I can integrate that if I do
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over dx I get a good estimate that is a very
rough estimate. In fact, theoretical this
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is known as high frequency limit
Theoretically which means, this is strictly
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speaking valid in deep water which happens
to be at high frequency limit, never mind
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that part, but it is a good estimate. It is
this estimate, that is used in Bareilly’s
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formula as somebody was asking, one of the
student was asking in vibration, because vibration
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is a high frequency phenomena, and when you
take a hull you vibrate it, what is the period?
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One second or so, is there any wave of one
second No. Waves are mostly six, seven, ten
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etcetera.
So, 1 second is really high frequency, so
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you could use that. So, here also we can make
a very rough case, if not as I mention in
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yesterdays class from that Luis form there
are graphs and charts, where omega verses
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those sectional, you know like this values
are given for different values of beta n and
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B n by T, etcetera, various graphs are given.
The one that we have talked yesterday, there
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are charts available of this average as a
function of sectional coefficient beta n,
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then with coefficient B n by T and of course,
omega e. We have talked about that there are
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charts available; it is only a simple matter
of going to the chart, taking up for relevant
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value of that, finding out this.
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So, in one short, you can actually get always
added mass in heave and added moment of inertia,
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very simple. You simply make a like typically
hydrostatic type, I have got x here stations,
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I end up getting this a z 2d, then I have
so called, if I want to use Simpson multiplier,
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then I get this one, this gives me by sigma
I end up getting here a z 3d, and here I use
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x square into a z 2d, again I use Simpson
multiply, again I get that, this will give
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me. So, essentially in one table I can get
both of them, very simple, you know like straight
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forward. So, heave and pitch are really related
to each other, you can get from one table
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in any assignment on any problem.
But these, having said that these are only
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approximate formulas. See the subject of this
dynamics like in water wave resistance, evolved
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was last so many years. Today of course, we
have got much more sophisticated programs,
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three dimensional c f d kind of course, to
find out the same thing. But, I am talking
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of practical estimate when you go to shipyard,
when you have to make a quick estimate, when
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you are doing a first design, you are not
going to run probably a code very, you know
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expensive code. However, for offshore structure
running such code have become important, it
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has become a routine industry affair, where
you do run available codes to find out these
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things. Anyhow, so, this is only an estimate,
I am focusing on, so that you get a rough
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idea as I mention repeatedly that rough case.
Somebody came to your company, you are in
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a shipyard and try to tell you, give me a
guess of the natural period, etcetera. You
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can quickly use this added mass and moment
of inertia, but this formula that is the high
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frequency formula, just this mass is slightly
better than using a k y equal to 1 or some
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factor, slightly more accurate. So, you can
always get to that.
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So, these are all various levels of approximation
that you should know. So, this is the same
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thing will apply to damping, exactly damping
is also available in this way, like damping
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function of sectional coefficient, sectional
with and frequency.
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See, there is another very crucial thing that
I want to tell, now from physical point of
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view, of the behavior. Remember, when this
goes like that, this is my theta, or motion,
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or z in, and this is my omega. We said that
this low frequency part, is the low frequency
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part dominated by hydrostatic force, this
is my high frequency part, this is the damping
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part. Let us look at the pitch how it looks
like in this case, see low frequency wave
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is like that. What happens you know, if you
look at that, you will find out that the pitch
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is exactly in phase with this, that mean it
is always aligning itself like that. The low
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frequency part one can show, it will be in
phase with exciting moment, exciting moment
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is actually, with phase with this low that
makes sense, because you see why I do not
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have to go through a math’s to tell you.
You can always use your own kind of logic
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to find out. See, it is taking very small
frequency means long time, so I have enough
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time. So, when suppose water waves, suppose
this shape is there, the water wave become
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like that, It has there enough time for the
hydrostatic to balance itself and make it
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align to, because this align position of course,
is where my hydrostatic force exactly balances
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Since the frequency is low means, period is
high. I have enough time for the body to adjust
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itself to the local slope, because only in
the local slope my dynamic force, my dynamic
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moment becomes 0. See, after all for equilibrium
in hydrostatic equilibrium, what I need total
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hydrostatic force must be 0 moment must be
0, in this case that happens. So, this is
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why it follows that typically, remember this
makes perfect sense. Remember here heave,
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this is what I will talk now, what happen
to heave? Heave is also following this, remember
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if you go up and down it is exactly here,
you stand here, you find same local draft
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or same free boat, which mean it is simply
going up and down another way.
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But, now I will tell that there is a interesting
phase gap between the two motions, which I
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will come in a moment. But, before that let
us talk about this side, what will happen?
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Just the opposite you will end up getting
the ship, you try to see that it is slightly
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this way, which is more dangerous. Obviously,
here you find out there is a tendency, because
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there is a phase gap for water to go on top
of the deck, and here go to come out and slant.
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So, you see here, because here when the water
slope is rising, my ship is going down, right?
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So, there is a kind of opposite phase. So,
one again the same thing is proven that situation
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where dynamics very important, becomes physically
more significant in a sense, because that
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is more dangerous. After all, you do not want
water to come on your take. In fact, you will
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find out many of the small boats of course,
not only for pitch reason, the bow is very
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much like very much up.
Of course, this is for the wave raises wave,
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but also for pitch to some extent. So, this
is another extreme of that. So, we must realize
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this part, where inertia is dominating, typically
my exciting moment and my response are out
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of phase typically, or with respect to wave
slopes, and as a result when water is rising,
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ship is tending go down. In this case, the
bow is going to go down when this is going
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to rising. So, here there is a phase gap.
Now, let us look at this phase thing which
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is more easily seen, the relative phase between
heave and pitch, in this case. We can, I want
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to show you that, you see when maximum, let
us take an example of lower frequency. What
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we want to, I want to tell you is that, the
maximum heave and maximum pitch, usually do
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not occur at the same time and which is good
for us, because if it did occur at the same
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time for a given motion, you have got a very
large motion.
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Now, how do I see that? See, let us take eta,
so A cos curve. So, eta equal to A cos, what
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is eta x equal to d eta by dx? that gives
you minus, right, minus A K. This is right
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K x minus omega t. So, this will be minus
K sign means, it will look like, let us look
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another pen minus sign will look like this.
This is right. So, this is my eta x, this
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of course, is t this is my this is my eta,
this is right. Now, you see what is happening,
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motion in low frequency, I am having my heave
given by this blue line, in phase with blue
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line and that is heave is like that, this
is my heave .Where is my pitch? my pitch is.
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So, you can see that there is a phase gap
between heave and pitch. When my heave is
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maximum, my pitch is 0, when my heave is 0,
my pitch is maximum. You see that phase gap
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between the two, this is important because
when I want to find out what is happen when
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you are standing at some point, phase is important,
I cannot add all the maximums together. See,
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I am standing at this location, we will discuss
this later on. Ship goes down also ship comes
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down.
Supposing at the same time the amount it went
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down maximum, it came down maximum. It would
have been much lower, but that is not happening,
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usually do not happen, why because there is
a phase gap. The concept of phase is therefore,
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very very very important, if I was to write
this as z equal to z A cos omega t, this will
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become theta A cos omega t plus a phase. Remember
what we have done, we are always measuring
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the phase with respect to input web signal
in our usual study, with respect to input
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web. In fact, what we do is that you see,
that if I always say that, if I had eta equal
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to A e power of minus i omega t you know.
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Let us say this at X equal to 0 then, I am
writing my z to be z A e of minus i omega
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t into e of minus i beta z. This is all of
course, real part, what is this? This is nothing,
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but cos of, we can say omega t actually, here
I wrote plus, we can write plus also, you
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can say this way because it is beta z minus
cos means, both are this thing and theta is
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going to be real part of theta a.
What I found, now relative gap between these
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two is beta z minus beta theta. This is my
phase gap between z and theta, but normally
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what I am doing? I am always finding between
this two, and you will find out in later solution
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I will tell, that we actually, we write this
way, I will write this to be real part of
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z a bar e of minus omega t, when this is equal
to z e of i beta z.
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This term is known as complex amplitude, it
is a complex number. Because you see, if I
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were making this, this part and this part
combining and I call that z a bar; obviously,
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that number is a complex number z a bar. So,
this signal is a sinusoidal signal in time
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with this amplitude, but this is complex amplitude.
So, for manipulation part as you will find
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out later on, for solution purpose we actually,
determine complex amplitude because complex
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amplitude tells me directly, phase and absolute
value.
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So, concept of complex amplitude is also important,
we will see that eventually in when we actually
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do the calculation. Complex amplitude tells
me that not only this value, also this gap
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that is where it happens, the phase gap. Anyhow,
my point of course, is not so much to go into
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complex amplitude right now, but to tell you
that there is a phase gap, phase is extremely
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important just by telling me that I have this
heave and so and so pitch so and so, the story
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is not complete. You must also say when does
it occurs, that is important. As we have seen
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in this case, yes pitch is so and so. Let
us say, here the ship is pitching by 3 degree,
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but when does 3 degree occur? Well, let us
say heaving by 4 meter maximum, is it pitching
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3 degree when it is going 4 meter down or
no? This is what I need to also know, because
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without knowing that I cannot really find
out the actual history of the ship motion.
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So, phase is the extremely important concept
in any motion that you have to understand,
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without phase, it is not completed at all.
Now, I will like to get back to some means
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of determining added mass
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Once again I go back to added mass, because
it is important. Can anybody give me a guess
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about how can experimentally determine added
mass, why I am asking I will tell you, how
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do I do that? You see, people will say that
many people do not agree all theories, they
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00:19:52,300 --> 00:19:55,530
want an experimental verification, experimental
measurement.
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Now, resistance test, very simple, you have
tore the model, just figure out how much is
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00:20:02,350 --> 00:20:07,380
drag, As far as measurement is concern you
have measured it, you just tore the model,
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find out the drag and what we debated, is
not how to measure it, how to extra politic.
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That was our main debate scaling and all that.
Similarly, for if I want a ship motion, maybe
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I will send a wave and I will find out how
much it is moving, but added mass is an interim
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concept, somewhat. You do not see it, how
do I measure it? that what my question is.
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But having said that, added mass, I said yesterday
is basically a force concept there is nothing
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like identified mass it is added mass force.
So, you can immediately understand that the
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00:20:45,760 --> 00:20:49,440
measurements system must consist of measuring
some kind of force.
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So, that is 0.1. 0.2 is that, see I now that
this T equal to natural period equal to, like
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2 pi into mass plus added mass by c. Now,
here I know this for the geometry, I know
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this for the geometry, I have to only know
this added mass. So, if I were to take a body
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and just push it down, it might go like that.
So, I can figure out this period, and I can
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say well I know this anything, I know I can
find out this.
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But, there is something wrong, what is wrong?
What I found out this is only for this period.
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Remember this a is the function of frequency,
see I have no choice on the frequency variant,
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let us say it is a 11 second, scale down maybe
3 second. So, what I found a, is only for
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3 second.
So, in order for me to find it out added mass
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which is over all frequency, I have to conceive
a different experiment, very important to
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understand that. So, this is what is called,
what we are going to talk to little bit is
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called forced oscillation test in only one
mode heave I will tell, forced oscillation
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test. In fact, we can do that in our tank
with the dynameters we have, provided we have
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a force mechanism.
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00:22:16,060 --> 00:22:23,060
What we do here, this is my tank here, say
water tank. So, let us put the water like
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this, and I have the body here. What I do
is that, see I want to give it a particular
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oscillation motion z, or say z a e i omega
t, I am controlling it, omega and z a. I can
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have a mechanism, some kind of circular mechanism
by which I can push it down. Then what I do
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by an instrument here, I measure the force
that is been required to push it down, the
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00:23:19,270 --> 00:23:24,480
force that is coming
So, what I do is that, remember this force
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00:23:24,480 --> 00:23:31,480
oscillation test for a given non frequency
omega and for an oscillation amplitude a,
188
00:23:32,060 --> 00:23:39,060
which should be sinusoidal, I am measuring
F z. Once again, let me tell you this I have
189
00:23:42,840 --> 00:23:49,840
this body, I oscillate it. Oscillation frequency
is decided by me, like I want to do the test
190
00:23:50,630 --> 00:23:57,630
for this frequency, ram number 1 omega equal
to 0.1 to 0.15 like that. I decide a amplitude
191
00:23:59,780 --> 00:24:04,880
1 centimeter or whatever, these are my input
But, this is the interesting point; see I
192
00:24:04,880 --> 00:24:10,470
am doing that of course, with shape of a pressure
field here, so therefore, I have to walk against
193
00:24:10,470 --> 00:24:14,360
the water, so there is a force coming and
that is what I measure by an equipment here,
194
00:24:14,360 --> 00:24:18,720
so I measure that force. Now, from this I
can find out, you see this is called forced
195
00:24:18,720 --> 00:24:24,840
oscillation test, I can find out the added
mass of damping, how we will tell? You see
196
00:24:24,840 --> 00:24:31,840
here the interesting point is that, I we will
just work it out my z here equal to z a, I
197
00:24:34,230 --> 00:24:41,230
will write it again e power of, well we can
do it plus or minus does not matter a real
198
00:24:41,960 --> 00:24:48,960
part is implied
Now, F z that I measure will becomes F an
199
00:24:50,040 --> 00:24:57,040
amplitude, let me call it o e power of i omega
t plus beta F. You see why this because remember
200
00:25:01,790 --> 00:25:08,790
that, how I get this beta a, I oscillated
that, oscillation is this.
201
00:25:10,770 --> 00:25:17,770
This is what I have given right, input data.
This is my z a, if this my period t, this
202
00:25:18,910 --> 00:25:25,910
is my 2 pi by omega agreed. See, I have oscillated
that, so I know of course, z a period 2 pi
203
00:25:29,410 --> 00:25:35,260
by omega, etcetera. What did I measure? The
force history, remember history. So, force
204
00:25:35,260 --> 00:25:42,260
looks something like this, this is output,
this is what I measure, the equipment have
205
00:25:43,280 --> 00:25:47,350
measured me this force.
So, the equipment that have measured me this
206
00:25:47,350 --> 00:25:53,090
force at every second I plot that, when I
plot that it likely to be side to be sinusoidal,
207
00:25:53,090 --> 00:25:59,910
actually for that purpose I would do some
kind of filtering, feeding, etcetera, but
208
00:25:59,910 --> 00:26:05,990
it will be taken as sign curve. What do I
get of that, the amplitude here, that is my
209
00:26:05,990 --> 00:26:12,990
this and I will know my phase, remember that
is important this phase.
210
00:26:15,050 --> 00:26:19,280
This picture may not be correct, but the point
is that, I would know that both these and
211
00:26:19,280 --> 00:26:26,280
these from the measurements. So, I have fate
this and I have fate this, what I found out
212
00:26:26,840 --> 00:26:33,840
is this and this, this is always a harmonic
test. So, I once again repeat this is input,
213
00:26:38,220 --> 00:26:45,210
this is for a given omega output is measure,
what I measured is F 0 and beta F, now from
214
00:26:45,210 --> 00:26:50,610
there how do I find added mass? This is what
I have put in and what I measure. Now, how
215
00:26:50,610 --> 00:26:54,240
do I find added mass, we will go to that.
216
00:26:54,240 --> 00:27:00,080
So, we now we have to go back to this equation
of motion for heave m plus, we will work it
217
00:27:00,080 --> 00:27:07,080
out slowly, sorry sorry, not m plus a or rather
we may cut it again a z, see in this case
218
00:27:21,500 --> 00:27:27,610
I have got F 0
Let me write in a complex number only, i omega
219
00:27:27,610 --> 00:27:34,610
t e i, you can just call beta, remember that
this and this we have found out, this F 0
220
00:27:36,410 --> 00:27:43,410
and this, Of course, z equal to z a e i omega
t. So, what is z dot equal to, here I have
221
00:27:49,260 --> 00:27:54,600
written plus actually, this plus minus, remember
again the way you proceed really does not
222
00:27:54,600 --> 00:28:00,100
matter. You can always write this as minus
plus something else, instead of sin cos it
223
00:28:00,100 --> 00:28:04,270
it the answer will be the same thing.
I am only showing therefore, the principle
224
00:28:04,270 --> 00:28:08,620
of how we go proceed. So, here I wrote plus,
I could have written minus also, does not,
225
00:28:08,620 --> 00:28:13,660
did not matter. Because both of them are cos
omega t, remember real part. So, this is become
226
00:28:13,660 --> 00:28:20,660
I omega, how much is z dot dot; minus omega
square. So, what happens to this equation
227
00:28:30,720 --> 00:28:37,720
minus omega square m plus a z, z a e I omega
t, that is right, because z z a comes a e
228
00:28:47,020 --> 00:28:52,390
i comes here, minus omega square z double
dot this is okay, know.
229
00:28:52,390 --> 00:28:59,390
Next term is plus I omega z a, b z that is
okay, b z into i omega z a this thing. The
230
00:29:09,580 --> 00:29:16,580
next one is c z into z a e i omega t this
is equal to F 0 e i omega t e i beta. Now,
231
00:29:26,130 --> 00:29:33,130
you can delete all this. So, I end up getting
minus omega square m plus i omega, we just
232
00:29:58,750 --> 00:30:05,750
see this also, i omega z a b z plus c z this
is see this you all agreed with that minus
233
00:30:23,370 --> 00:30:30,370
omega m plus a z z a plus i omega b z z z
a b z c z z a equal to F 0 e i beta that is
234
00:30:32,260 --> 00:30:37,770
F 0 cos beta plus i F 0 sin beta. So now,
obviously, from this equation real part is
235
00:30:37,770 --> 00:30:42,140
real part, imaginary part is imaginary part.
So, you end up getting the two equation from
236
00:30:42,140 --> 00:30:46,190
which you can find this thing
So, what I end up getting is minus omega square
237
00:30:46,190 --> 00:30:53,190
n plus a z, z a plus c z equal to F 0 cos
beta and well i omega z a, b z equal to i
238
00:31:05,690 --> 00:31:12,690
F 0 of course, from here, this is 2, say this
is 1 from two, I get straight forward b z
239
00:31:16,710 --> 00:31:23,710
equal to F 0 sin beta by omega z a, and from
one I can get a z, well I can get a z as let
240
00:31:29,910 --> 00:31:36,910
us see how F 0 cos beta minus c z, z a divided
by omega minus, omega square z a minus m,
241
00:31:45,450 --> 00:31:51,970
something like that. See, this divided by
minus that gives you this, you divide by omega
242
00:31:51,970 --> 00:31:57,770
square z a, then take m on that side
I mean, you can work it out whichever way.
243
00:31:57,770 --> 00:32:02,040
So, what I means is therefore, that this is
what is called forced oscillation test, which
244
00:32:02,040 --> 00:32:09,040
is what you could do to find out basically
like added mass experimentally. This is a
245
00:32:12,820 --> 00:32:18,220
very important test, why I mention to you
this now in the class is because, usually,
246
00:32:18,220 --> 00:32:25,220
this is not so commonly known to people, because
many test you do for design purpose is to
247
00:32:27,510 --> 00:32:30,450
directly find the output, but here you are
finding the added mass
248
00:32:30,450 --> 00:32:34,940
See added mass is just a quantity, which would
be useful tomorrow, if you want to predict
249
00:32:34,940 --> 00:32:40,610
these motions, but you are not measuring ship
motion directly, remember. Like resistance
250
00:32:40,610 --> 00:32:47,610
your measuring directly, somebody can figure
out, so many kilo Newton is the total distance,
251
00:32:48,190 --> 00:32:55,190
meta centric height may be measuring sometime,
you may be k g you are measuring inclining
252
00:32:55,360 --> 00:33:01,290
test. But, here added mass, if you go to industry
and say I want to measure added mass, people
253
00:33:01,290 --> 00:33:05,170
will not know about it.
So, therefore, this experiment for added mass
254
00:33:05,170 --> 00:33:09,950
measurements force oscillation is not usually,
commonly known. That is why I am trying to
255
00:33:09,950 --> 00:33:15,360
tell you, and you can also see here that experimental
measurements here are not so direct, you have
256
00:33:15,360 --> 00:33:20,790
to make certain assumption, you measure something
and go through certain equation to get something
257
00:33:20,790 --> 00:33:27,530
ultimately. That is very important and this
also involves inherently a phase concept,
258
00:33:27,530 --> 00:33:31,090
you have to measure phase without, suppose
beta was in fact, you will see beta is 0,
259
00:33:31,090 --> 00:33:37,560
b z is 0, that is what will happen in high
frequency.
260
00:33:37,560 --> 00:33:41,660
You will find that beta will become 0. In
fact, one can again find out the same concept
261
00:33:41,660 --> 00:33:48,660
as I mention, that this is z z dot z dot dot,
this is my F, this is my beta, this is my
262
00:33:51,970 --> 00:33:58,970
F, the kind of like how much it is lagging
or leading. One can relate to that in principle.
263
00:34:00,540 --> 00:34:07,540
So, this is what is about is added mass. Now
of course, you are doing a scale test, you
264
00:34:07,690 --> 00:34:11,750
are not doing a full scale, you do not have
a full scale ship doing a test. So, what is
265
00:34:11,750 --> 00:34:18,750
the scale factor? Another point, that i should
understand, scaling factors, when I do a measurements.
266
00:34:29,720 --> 00:34:36,510
Added mass is scale at what? See, added mass
is proportion to mass.
267
00:34:36,510 --> 00:34:43,510
So, obviously, it will scale lambda q, tell
me how, with what will scale? This is important,
268
00:34:46,630 --> 00:34:53,630
I want to tell you this, see maybe before
that we should go to time part. Omega square
269
00:34:56,360 --> 00:35:03,360
is g k and T square is how much? T square
will be 2 pi square by omega square. T square
270
00:35:11,200 --> 00:35:18,200
is going to be 2 pi square by omega square
2 pi square by g into k, 2 pi square by g
271
00:35:20,280 --> 00:35:24,900
k. Whatever, I mean its opposite, see what
I am trying to say it is opposite, T square
272
00:35:24,900 --> 00:35:31,900
is listing it to lambda and omega square is
in proportion to 1 by lambda. Now, lambda
273
00:35:33,140 --> 00:35:39,750
is wavelength is scaling well, wavelength
here is lambda here, let me call it L w, otherwise
274
00:35:39,750 --> 00:35:46,670
you will have a confusion. See, what I am
saying once again if you understand, omega
275
00:35:46,670 --> 00:35:53,670
square is g k; that means, omega square is
inversely proportion to wavelength L w, you
276
00:35:55,110 --> 00:36:00,710
agree with that, omega square is inversely
proportion to wavelength. So, T square is
277
00:36:00,710 --> 00:36:07,250
linearly proportion to wavelength.
Now, wavelength; obviously, scales at lambda,
278
00:36:07,250 --> 00:36:14,250
so frequency will scale at what, 1 by root
lambda and T will scale at root lambda, this
279
00:36:18,250 --> 00:36:24,230
is important. That means, if natural period
is 10 for actual body, one is to 10 scale
280
00:36:24,230 --> 00:36:31,230
body would have a period of one second 1 by
root 10, please understand this very importantly.
281
00:36:33,560 --> 00:36:40,560
If a natural pair of a ship is 10 second,
it is one is to 100 scale model would or should
282
00:36:41,230 --> 00:36:48,230
have natural pair of 1 second. If the frequency
there are, is 10 second means 2 pi by 10,
283
00:36:49,480 --> 00:36:54,930
means about h 0.6. Here it is going to be
2 pi by 1 6.
284
00:36:54,930 --> 00:37:00,540
So, frequency is much faster but period is
much smaller, but they are not linear square
285
00:37:00,540 --> 00:37:05,350
root. Why I am saying this, because in b z
I need that. Now, let us work out how do I
286
00:37:05,350 --> 00:37:12,350
scale b z? Anybody knows, you see how do I
scale remember b z into z dot, this is a force,
287
00:37:16,990 --> 00:37:22,640
this scales at lambda cube, because force
scales at lambda cube. Now, z, what is z dot?
288
00:37:22,640 --> 00:37:29,640
d z by dt. How does it scale at? Lambda by
T is root lambda, that is a scaling at root
289
00:37:31,950 --> 00:37:38,950
lambda. So, what should b z scale at? Absolutely.
So, b z scaling at lambda power of 2.5, everybody
290
00:37:42,120 --> 00:37:49,120
agrees. b z should scale lambda power 2.5,
very simple to work out. What about c z, immediately
291
00:37:54,080 --> 00:38:01,080
one can tell, because c z is first of all,
exactly, because it is c into z. So, this
292
00:38:01,790 --> 00:38:05,330
is; obviously, this is lambda. So, this must
be lambda square.
293
00:38:05,330 --> 00:38:10,070
So, I will just organize and write it now.
Therefore the scaling factor becomes the a
294
00:38:10,070 --> 00:38:17,070
z is lambda q, b z or damping, once again
we will come to next level. These are all
295
00:38:19,330 --> 00:38:26,330
linear motion 2.5, c z equal to lambda square,
omega of course, scale with root lambda, 1
296
00:38:30,900 --> 00:38:37,330
by root lambda and T scale with root lambda.
So, this linear you please work out yourself
297
00:38:37,330 --> 00:38:44,330
what will happen to dy y and I b theta and
c theta, because they are moment there, what
298
00:38:45,990 --> 00:38:50,140
would happen to moment, force into distance?
another L get’s multiplied
299
00:38:50,140 --> 00:38:55,940
So, you will end up getting this part, it
is not going to be just 4 because moment of
300
00:38:55,940 --> 00:39:01,100
inertia mass into distance square. So, I will
leave it to you as an exercise to work it
301
00:39:01,100 --> 00:39:05,910
out, you can work it out from elementary example,
but this look at c theta straight forward,
302
00:39:05,910 --> 00:39:12,910
because it is o g into V into G M L. So, this
is lambda cube, this is lambda, so lambda
303
00:39:12,990 --> 00:39:18,160
4.
So, you will see this lambda 5, lambda 4.5,
304
00:39:18,160 --> 00:39:24,940
why this not 4? Because, remember it is theta
dot, dot not z dot, dot. There is one lambda
305
00:39:24,940 --> 00:39:31,940
gets out from that side for moment because
the motion here is theta which is actually
306
00:39:32,310 --> 00:39:39,310
only h there is not l there in other words
I tell you I into theta dot dot this is my
307
00:39:40,430 --> 00:39:47,430
lambda 4, but this lambda 5 and this is 1
by lambda, gives me lambda 4
308
00:39:47,800 --> 00:39:52,700
So, you can work it out this scaling. Very
important to understand the scaling, because
309
00:39:52,700 --> 00:39:59,700
I do a test, I do not know how to scale up,
I have no really meaningless thing. So, this
310
00:40:02,760 --> 00:40:09,760
brings me to some simple kind of a problem
type, I thought I will spend some time to
311
00:40:14,360 --> 00:40:21,360
tell you about some very simple problems.
Let us see this, I have a ship, I am standing
312
00:40:30,530 --> 00:40:30,910
here.
313
00:40:30,910 --> 00:40:37,910
Now, I find out that at a speed of V, equal
to say 18 knot, when waves are coming opposites,
314
00:40:44,850 --> 00:40:50,640
I find some regular wave field, I am heading
into that and I find I have excessive heave
315
00:40:50,640 --> 00:40:57,640
motion. I have the ship parameters given,
please understand I have the ship parameter,
316
00:41:02,580 --> 00:41:09,580
this is known, all the ship parameters is
known.
317
00:41:15,360 --> 00:41:22,290
I have a ship of course, I now the hydrostatic
parameters unknown. I find out that in some
318
00:41:22,290 --> 00:41:29,290
wave condition, waves are coming, heading
on at a certain speed 18 or some given speed,
319
00:41:29,330 --> 00:41:36,330
I have excessive heave motion. The question
is, can I guess the length of the waves? Wave
320
00:41:36,490 --> 00:41:43,490
field, is regular wave field; that means,
what will be the lambda? problem is very simple,
321
00:41:44,690 --> 00:41:51,040
because excessive motion means; obviously,
wave frequency, exciting frequency omega is
322
00:41:51,040 --> 00:41:56,390
equal to natural frequency omega z; that means,
and this is equal to of course, we know root
323
00:41:56,390 --> 00:42:03,300
over of c by m plus a of course, I have to
make some estimate of a, without that you
324
00:42:03,300 --> 00:42:10,300
cannot proceed. So, how do I go, I can estimate
a, I of course, now c and m from hydrostatic
325
00:42:10,650 --> 00:42:15,640
and mass inertia.
So, what will happen from there, from this
326
00:42:15,640 --> 00:42:20,860
relation, I will be able to find out my; well
I know the natural period, so I know what
327
00:42:20,860 --> 00:42:25,640
is my omega e. Once I know omega e I can go
back to the relation another equal to omega
328
00:42:25,640 --> 00:42:31,220
into 1 minus omega V by G cos mu, mu is given
here. So, I can find out omega, once I know
329
00:42:31,220 --> 00:42:38,220
omega, I can find out lambda.
So, if you do that, I thought say this problem,
330
00:42:39,020 --> 00:42:46,020
I will give it to you with the numbers here,
see this ship is there, let us say displacement
331
00:42:49,070 --> 00:42:56,070
of 20000 ton assume added mass to be equal
to mass, which means added mass coefficient
332
00:42:57,510 --> 00:43:04,380
a z bar or what, I do not remember what I
took a z bar as 1, you take it as that, you
333
00:43:04,380 --> 00:43:11,380
assume that.
What happen in the A w p, well actually, sometime
334
00:43:12,220 --> 00:43:17,580
what will happen will be L and B and del C
w p, etcetera. Say, L w p is also given it
335
00:43:17,580 --> 00:43:24,580
can be, maybe 1500, it is 100 into 1800 meter
square or something.
336
00:43:27,410 --> 00:43:34,410
Something like that, find out the lambda for
eighteen knots? So, let me organize this and
337
00:43:38,550 --> 00:43:41,560
write it properly, otherwise probably you
are getting this thing.
338
00:43:41,560 --> 00:43:48,560
So, I have displacement of 20000 ton, actually
let me put the other way round. L of say 150
339
00:43:48,960 --> 00:43:55,960
meter, B of maybe 20 meter which is of course,
no. Let me 20, 30 meter, I do not want to
340
00:43:59,160 --> 00:44:06,160
make a distinct is 5, 28 meter I make it,
because 30 makes it L by T 5 is too short,
341
00:44:08,350 --> 00:44:15,350
25 makes it L by B of 6 so 25 is good number.
T of 10 meter, let me out it deep vehicle,
342
00:44:22,710 --> 00:44:29,710
C B of 0.75, C w p of 0.78 say this all given
to you, assume a z bar to be equal to 1, wave
343
00:44:47,890 --> 00:44:54,890
heading 180 degree, head waves excessive at
18 knots in regular wave field.
344
00:45:27,820 --> 00:45:34,820
What is the, remember, please remember that
this is forward ship problem. So, the omega
345
00:45:46,760 --> 00:45:53,760
you find out is omega e, omega e is equal
to the natural period, natural frequency.
346
00:45:54,950 --> 00:45:59,810
Omega e is not omega is equal to, remember,
that I mention here also. It is this some
347
00:45:59,810 --> 00:46:05,030
people make this mistake, instead of omega
e you say omega equal to natural period and
348
00:46:05,030 --> 00:46:09,040
then.
In other words, you find out the natural period,
349
00:46:09,040 --> 00:46:13,990
let us say natural period happens to be, let
us say something like 10 second. So, you think
350
00:46:13,990 --> 00:46:19,690
ten second has omega and find out the wave
length, Some people do that it is the because
351
00:46:19,690 --> 00:46:24,970
this is omega e. So, from omega e I have to
first find out omega and then find out what
352
00:46:24,970 --> 00:46:31,140
is the wave length for that omega.
So, I am repeating this because this is the
353
00:46:31,140 --> 00:46:36,370
common mistake people make, that is one common
mistake people make, and the other common
354
00:46:36,370 --> 00:46:42,360
mistake people always make is in delta I y
y and I y y part in pitch, because people
355
00:46:42,360 --> 00:46:48,340
always take I y y for a real. If I give a
ship dimension and all that, estimate I will
356
00:46:48,340 --> 00:46:55,050
come to that, they always take I y y to be
the moment of inertia for water wave. That
357
00:46:55,050 --> 00:47:01,780
is another mistake that is commonly made by
people. So, this is a very simple problem
358
00:47:01,780 --> 00:47:07,240
we can do and we can manipulate with that
in various ways, because it need not be a
359
00:47:07,240 --> 00:47:13,120
like a head waves, it can be some 45 degree
waves, it can be falling wave, but essentially
360
00:47:13,120 --> 00:47:17,980
it does excessive motion.
The word excessive motion means, it is resonating.
361
00:47:17,980 --> 00:47:24,980
Now, let us look at another one that I have
here, see there is an estimate of added mass,
362
00:47:26,810 --> 00:47:31,920
we will just do this based on some kind of
a simple calculation, we have another maybe
363
00:47:31,920 --> 00:47:33,340
5 minutes.
364
00:47:33,340 --> 00:47:40,340
So, here it says I have a wooden log of uniform
cross section or let us take a barge, I have
365
00:47:43,970 --> 00:47:50,970
a barge here of uniform cross section, which
can be any uniform of side B with B length
366
00:47:59,080 --> 00:48:06,080
L and it is statistical stable. Now, here
once again the question I ask was, what is
367
00:48:08,100 --> 00:48:15,100
my T z and T theta? Estimate that. What you
see, then I said that use your engineering
368
00:48:21,220 --> 00:48:25,210
judgment, I mean I am trying to leave it to
you, use your engineering judgment for that,
369
00:48:25,210 --> 00:48:32,210
for estimates
So, you understand now that T z I require
370
00:48:32,810 --> 00:48:38,640
to estimate, what added mass the other things
are all known, this is the L B T, etcetera,
371
00:48:38,640 --> 00:48:42,800
mass and all is know, see this is the square
bar L B, because I would have given also is
372
00:48:42,800 --> 00:48:48,480
L B and T and probably C B. So, mass and all
is known in this problem.
373
00:48:48,480 --> 00:48:55,480
Now, for T z I need and estimate of a z, please
do that. How would do it? Well I said when
374
00:48:58,220 --> 00:49:05,020
nothing is here, simply take this sectional
added mass to be B square and this case is
375
00:49:05,020 --> 00:49:08,910
very simple, because the uniform cross section
and B is constant throughout. So, I have B
376
00:49:08,910 --> 00:49:15,910
square by pi B square by 8 into L becomes
a z, you understand that sectional added mass
377
00:49:20,550 --> 00:49:26,280
is pi B square by 8 integrate of length is
L, that is one part.
378
00:49:26,280 --> 00:49:32,610
What is my added moment of inertia and moment
of inertia? For t theta I need two things,
379
00:49:32,610 --> 00:49:39,610
I need I y y as well as I need delta I y y,
remember. Let us talk of delta I y y first,
380
00:49:40,470 --> 00:49:47,470
how much it is? It is a uniform cross section.
So, I had actually pi B square by 8 into x
381
00:49:51,100 --> 00:49:54,970
square integration from minus L by 2 into
plus L by 2 dx this is agree.
382
00:49:54,970 --> 00:50:01,970
Because, take a next section, this turns x,
sectional added mass is B square by 8 pi into
383
00:50:07,260 --> 00:50:11,760
x square, integrate that. Actually, it will
be x cube by 3 and if you work it out, you
384
00:50:11,760 --> 00:50:18,760
will find out x cube by 3 L by 1 by root 12
or something, very easy. What about this?
385
00:50:20,210 --> 00:50:27,210
If you again, you have to make an estimate,
realistic estimate. Make K y y to be maybe
386
00:50:28,510 --> 00:50:35,510
0.25, I would not stick to only 0.25 suppose,
you are making 0.30, make it 0.24. No hard
387
00:50:38,650 --> 00:50:43,290
and first rule, but try to make a realistic
estimate.
388
00:50:43,290 --> 00:50:50,290
If you do that, you will end up getting basically
a estimate of this numbers and that can be
389
00:50:54,340 --> 00:51:01,340
of by 10 20 percent, not a big deal, but at
least you get a good estimate I have still
390
00:51:02,680 --> 00:51:07,810
a few minutes for this and I want to touch
upon just, not a problem one more thing.
391
00:51:07,810 --> 00:51:14,510
This relation between this T z and T theta
from design point of view. You see what would
392
00:51:14,510 --> 00:51:21,510
be your objective of design would you like
them to be same or different; obviously, full
393
00:51:22,340 --> 00:51:27,690
aim for us will be to make sure that there
are separated out because if they are not
394
00:51:27,690 --> 00:51:32,900
separated out I am going to have when the
accelerating heave or accelerating pitch.
395
00:51:32,900 --> 00:51:37,950
So, always in turns out see this always around
8 to 12 seconds this also 8 to 12 second.
396
00:51:37,950 --> 00:51:42,440
So, one of the thing that designer needs to
worry and we have seen it practical experience
397
00:51:42,440 --> 00:51:49,440
is that there is a design is bad you actually
end up having this two very close by and these
398
00:51:50,310 --> 00:51:56,840
two very close by means a very bad design.
So, if supposing it happens you will try to
399
00:51:56,840 --> 00:52:01,800
see how can I separate out even if you separate
by one or two second is good enough, but you
400
00:52:01,800 --> 00:52:07,560
must separate out that is an important thing
it has happen to practical ships that exist
401
00:52:07,560 --> 00:52:14,040
where heave and pitch periods are closed by.
And therefore, everybody says ten on hundred
402
00:52:14,040 --> 00:52:21,040
to be on the boat because it is. So, bad motion
and one of the thing that we could we will
403
00:52:21,510 --> 00:52:28,510
see again that we will see in a role motion
later on and that an interesting inverse case
404
00:52:29,520 --> 00:52:35,170
occurs you actually might have to a raise
of central gravity to lower meta centric height
405
00:52:35,170 --> 00:52:40,970
because you see G M say in off shore structure
we have seen that period are in proportional
406
00:52:40,970 --> 00:52:47,050
to 1 by root G M.
Well L or T in the case off shore structure
407
00:52:47,050 --> 00:52:53,610
same we will talk about that in role. So,
what happen if it is too high you have a very
408
00:52:53,610 --> 00:52:59,369
high means this low, but I want to have this
low therefore, I have to go basically I must
409
00:52:59,369 --> 00:53:06,369
reduce it this we will see later on, but this
for for a ship please remember this that unfortunately
410
00:53:09,640 --> 00:53:15,800
my heave and pitch periods happen to be very
close to everyday waves it is sometimes known
411
00:53:15,800 --> 00:53:22,800
as everyday waves you know it is always 6
to 12 seconds 15 seconds waves and as a results
412
00:53:23,170 --> 00:53:29,330
at times resonance is unavoidable and therefore,
I should try to design where I do not resonate
413
00:53:29,330 --> 00:53:32,240
together. So, that I have really really bad
time.
414
00:53:32,240 --> 00:53:37,590
So, therefore, I should design. So, that heave
and pitch are slightly pushed out by a few
415
00:53:37,590 --> 00:53:43,210
seconds that is important. So, with that we
will stop and next class we will now shift
416
00:53:43,210 --> 00:53:44,600
to roll motion thank you.