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‫Welcome back.

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‫So I have rewritten this thrust derivative with respect to Roederer length here, and let's just try

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‫to describe it with a random graph.

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‫So on this axis, you have your Roederer length.

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‫So it's this dimension here.

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‫And on the Y axis, you have this DETI over döner.

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‫So let's just assume that this is the function so you have some kind of our value here, which you plug

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‫in here, and then you will get your DETI over D-R as a value in the Y axis.

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‫So the total blade length was our Meer's.

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‫So we're going to put our here.

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‫So that's this one here.

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‫And in fact, let me redraw the top view of the blade here.

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‫So this is the Roederer length here.

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‫And so the zero length both on this graph and on this top you is here.

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‫So it is applied both to the graph and to this drawing.

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‫And the same thing with this capital letter.

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‫This are supplied to the graph and it's also applied to this top for you here.

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‫And let's just take a random airfoil here.

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‫And on the graph, it would be this strip here.

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‫The width would, of course, be D.R.

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‫And so to compute the total thrust that this blade is generating, then essentially what you have to

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‫do.

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‫You have to compute the area under the curve.

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‫Right.

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‫You will have infinitesimally thin rectangles here.

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‫And this is how you would approximate the area under the curve.

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‫So they would be like your T one plus D T two plus D, t three, et cetera.

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‫So essentially, in order to find the total thrust generated by the blade, you have to compute the

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‫area under this curve.

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‫In other words, you have to integrate this function.

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‫So let's say that the length from here to this red airfoil, let's say that this is R and so if you

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‫have DETI over döner on the vertical axis.

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‫And let's take this case here and then if you multiplied by this, these are here, then you will put

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‫it here, then those D-R they will cancel out.

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‫And that means that this differential area that you have in red, that's your differential thrust for

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‫this airfoil.

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‫And mathematically, you would do the same thing, you would put your D-R here and here, and then these

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‫dealers will cancel out and now you're going to have to integrate this function.

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‫And this integration looks like this.

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‫So on this side, on the left side, you integrate your DETI like this from zero thrust to the total

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‫thrust and this side you integrate from zero rotor length.

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‫So starting from here up until the end of the blades, until the capital are roederer length, until

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‫our meters, if you remember.

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‫Then a mortar had two blades, right.

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‫And so far we've been dealing only with one blade.

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‫But in order to compute the total thrust for the Mowrer, in our case it was more one.

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‫Then we would multiply this thing by two.

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‫And if we do that, then we can cancel out 2s.

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‫And so the left side will simply become thrust for moral one equals.

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‫And now I'm going to bring Omega squared and the air density out of the integral, but I'm going to

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‫leave everything else inside the integral.

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‫So the total thrust for Model One equals this expression.

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‫So why did I take this and this out of the integral?

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‫Well, because their constant values.

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‫Meaning that this omega, this angular velocity does not change as a function of Roederer length and

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‫be careful.

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‫Now, we are not talking about Omega Dot.

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‫All right.

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‫This is not the case right now.

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‫We're talking about how Omega changes with respect to the length.

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‫And it is not changing with respect to water length.

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‫In other words, the Omega over D-R equals zero.

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‫The water still rotates at Omega radians per second, regardless of which Roder length you are considering

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‫or in which location on the blade.

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‫You are right now.

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‫And the same thing can be said for the air density.

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‫Deyrolle, with respect to D-R equals zero and the units here would be kg's over meter cubed times meter

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‫because D-R is in meters and the densities in kilograms over cubic meters.

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‫So you can also say that it's kilograms over meters to the power of four and the units here would be

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‫radians over second time meter like this.

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‫So since they are constant, then we can take them out of the integral.

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‫Now, the court length might not be constant.

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‫If this is your model, then what if your court length is a function of the Roederer length?

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‫What if it changes as you go along the blade?

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‫What if it becomes wider and then narrower again?

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‫So if your variable changes with respect to R, then it stays inside the integral.

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‫Well, then it's obvious that your R stays inside the integral.

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‫But also your S.L and CD and five variables, they will also change as a function of.

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‫So, for example, if you remember then your lift coefficient was a function of alpha, the magic number

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‫and the Reynold's number.

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‫But all these variables can also change with respect to your Roederer length.

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‫For example, if there is a twist in the blade, then your angle of attack will change and your magic

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‫number will definitely change.

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‫Because your Mach number, assuming that we ignore this V sub velocity, your Mach number would be Omega.

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‫Times are divided by the speed of sound.

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‫And so if your rotor is rotating at an Omega radians per second, then that might be constant.

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‫But if you multiply by R, then obviously here this place would have a larger velocity in this direction.

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‫And here you will have less in here you would have even less.

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‫So if your arm becomes bigger, then your translation of the lost will become bigger, but your speed

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‫of sound is not going to change because of that and because of that, your Mach number will change because

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‫your denominator will be constant.

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‫But since your our changes, then your entire numerator will change.

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‫And since your S.L depends on your Mach number and also CWD depends on your Mach number, then they

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‫will not be constant as a function of your Roder length.

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‫So they have to stay inside the integral.

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‫And finally, there is no guarantee that your fi angle will be constant as a function of our either.

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‫It might be extremely small, but it might not be constant.

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‫Everything that is not constant with respect to the length needs to stay inside the integral.

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‫And so when you integrate all this and you multiply it by the air density and omega squared, then you

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‫will get your thrust for your moral one.

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‫But what we are going to do, we're going to take this portion here and we're going to call this portion

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‫C, sub T and this C sub T is called a thrust factor.

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‫And that's how the motor thrust force is usually written.

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‫It's written like this thrust and in our case for model one equals.

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‫And then the thrust factor times the angle of the last day of the motor squared.

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‫That's how it's usually represented.

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‫So the reverse thrust force is usually written as a function of Omega, as a function of the angular

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‫velocity of the rudder.

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‫You had this rudder, you have your Omega, and then once you have your thrust factor, then you simply

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‫square your Omega and you multiply it by a thrust factor.

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‫And that's when you get your thrust force.

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‫And you know that whatever you get from this integration times, this air density, whatever you get,

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‫it will be a number.

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‫It will be some kind of constant value.

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‫And that constant value that we call the thrust factor, you can either get it through theoretical calculations

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‫like this or through experimentation, meaning that you find this value experimentally from some kind

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‫of system identification techniques, for example, like, for example, some kind of regression analysis.

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‫Where you have a lot of data points and then you will try to find some kind of middle line through data,

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‫and with these techniques you can also find your trust factor.

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‫And in this course and in the code, this trust factor is a given value.

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‫So you will have it in the code.

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‫Since in the engineering world, the trust that you get from a router is usually written like this where

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‫you have a constant times Omega squared.

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‫I really wanted you to understand how you can get this thing into this form.

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‫I wanted you to see why you have this Omega squared.

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‫It comes from the differential lift and differential drag force.

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‫Where thanks to assuming that this velocity's very small, you ended up with the velocity that was Omega

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‫squared, R squared.

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‫And then you just separated those two, you left your R-squared inside the integral, and then you put

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‫your Omega's cord out of the integral here.

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‫So now, you know, it now known this in the next video, we're going to see how we can form our control

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‫inputs, you one, you two and you three using this information that we have learned so far for you,

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‫for we have to dig a little bit deeper.

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‫This where we are going to do later.

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‫So thank you very much and see you in the next video.