‫Welcome back.
‫So I have rewritten this thrust derivative with respect to Roederer length here, and let's just try
‫to describe it with a random graph.
‫So on this axis, you have your Roederer length.
‫So it's this dimension here.
‫And on the Y axis, you have this DETI over döner.
‫So let's just assume that this is the function so you have some kind of our value here, which you plug
‫in here, and then you will get your DETI over D-R as a value in the Y axis.
‫So the total blade length was our Meer's.
‫So we're going to put our here.
‫So that's this one here.
‫And in fact, let me redraw the top view of the blade here.
‫So this is the Roederer length here.
‫And so the zero length both on this graph and on this top you is here.
‫So it is applied both to the graph and to this drawing.
‫And the same thing with this capital letter.
‫This are supplied to the graph and it's also applied to this top for you here.
‫And let's just take a random airfoil here.
‫And on the graph, it would be this strip here.
‫The width would, of course, be D.R.
‫And so to compute the total thrust that this blade is generating, then essentially what you have to
‫do.
‫You have to compute the area under the curve.
‫Right.
‫You will have infinitesimally thin rectangles here.
‫And this is how you would approximate the area under the curve.
‫So they would be like your T one plus D T two plus D, t three, et cetera.
‫So essentially, in order to find the total thrust generated by the blade, you have to compute the
‫area under this curve.
‫In other words, you have to integrate this function.
‫So let's say that the length from here to this red airfoil, let's say that this is R and so if you
‫have DETI over döner on the vertical axis.
‫And let's take this case here and then if you multiplied by this, these are here, then you will put
‫it here, then those D-R they will cancel out.
‫And that means that this differential area that you have in red, that's your differential thrust for
‫this airfoil.
‫And mathematically, you would do the same thing, you would put your D-R here and here, and then these
‫dealers will cancel out and now you're going to have to integrate this function.
‫And this integration looks like this.
‫So on this side, on the left side, you integrate your DETI like this from zero thrust to the total
‫thrust and this side you integrate from zero rotor length.
‫So starting from here up until the end of the blades, until the capital are roederer length, until
‫our meters, if you remember.
‫Then a mortar had two blades, right.
‫And so far we've been dealing only with one blade.
‫But in order to compute the total thrust for the Mowrer, in our case it was more one.
‫Then we would multiply this thing by two.
‫And if we do that, then we can cancel out 2s.
‫And so the left side will simply become thrust for moral one equals.
‫And now I'm going to bring Omega squared and the air density out of the integral, but I'm going to
‫leave everything else inside the integral.
‫So the total thrust for Model One equals this expression.
‫So why did I take this and this out of the integral?
‫Well, because their constant values.
‫Meaning that this omega, this angular velocity does not change as a function of Roederer length and
‫be careful.
‫Now, we are not talking about Omega Dot.
‫All right.
‫This is not the case right now.
‫We're talking about how Omega changes with respect to the length.
‫And it is not changing with respect to water length.
‫In other words, the Omega over D-R equals zero.
‫The water still rotates at Omega radians per second, regardless of which Roder length you are considering
‫or in which location on the blade.
‫You are right now.
‫And the same thing can be said for the air density.
‫Deyrolle, with respect to D-R equals zero and the units here would be kg's over meter cubed times meter
‫because D-R is in meters and the densities in kilograms over cubic meters.
‫So you can also say that it's kilograms over meters to the power of four and the units here would be
‫radians over second time meter like this.
‫So since they are constant, then we can take them out of the integral.
‫Now, the court length might not be constant.
‫If this is your model, then what if your court length is a function of the Roederer length?
‫What if it changes as you go along the blade?
‫What if it becomes wider and then narrower again?
‫So if your variable changes with respect to R, then it stays inside the integral.
‫Well, then it's obvious that your R stays inside the integral.
‫But also your S.L and CD and five variables, they will also change as a function of.
‫So, for example, if you remember then your lift coefficient was a function of alpha, the magic number
‫and the Reynold's number.
‫But all these variables can also change with respect to your Roederer length.
‫For example, if there is a twist in the blade, then your angle of attack will change and your magic
‫number will definitely change.
‫Because your Mach number, assuming that we ignore this V sub velocity, your Mach number would be Omega.
‫Times are divided by the speed of sound.
‫And so if your rotor is rotating at an Omega radians per second, then that might be constant.
‫But if you multiply by R, then obviously here this place would have a larger velocity in this direction.
‫And here you will have less in here you would have even less.
‫So if your arm becomes bigger, then your translation of the lost will become bigger, but your speed
‫of sound is not going to change because of that and because of that, your Mach number will change because
‫your denominator will be constant.
‫But since your our changes, then your entire numerator will change.
‫And since your S.L depends on your Mach number and also CWD depends on your Mach number, then they
‫will not be constant as a function of your Roder length.
‫So they have to stay inside the integral.
‫And finally, there is no guarantee that your fi angle will be constant as a function of our either.
‫It might be extremely small, but it might not be constant.
‫Everything that is not constant with respect to the length needs to stay inside the integral.
‫And so when you integrate all this and you multiply it by the air density and omega squared, then you
‫will get your thrust for your moral one.
‫But what we are going to do, we're going to take this portion here and we're going to call this portion
‫C, sub T and this C sub T is called a thrust factor.
‫And that's how the motor thrust force is usually written.
‫It's written like this thrust and in our case for model one equals.
‫And then the thrust factor times the angle of the last day of the motor squared.
‫That's how it's usually represented.
‫So the reverse thrust force is usually written as a function of Omega, as a function of the angular
‫velocity of the rudder.
‫You had this rudder, you have your Omega, and then once you have your thrust factor, then you simply
‫square your Omega and you multiply it by a thrust factor.
‫And that's when you get your thrust force.
‫And you know that whatever you get from this integration times, this air density, whatever you get,
‫it will be a number.
‫It will be some kind of constant value.
‫And that constant value that we call the thrust factor, you can either get it through theoretical calculations
‫like this or through experimentation, meaning that you find this value experimentally from some kind
‫of system identification techniques, for example, like, for example, some kind of regression analysis.
‫Where you have a lot of data points and then you will try to find some kind of middle line through data,
‫and with these techniques you can also find your trust factor.
‫And in this course and in the code, this trust factor is a given value.
‫So you will have it in the code.
‫Since in the engineering world, the trust that you get from a router is usually written like this where
‫you have a constant times Omega squared.
‫I really wanted you to understand how you can get this thing into this form.
‫I wanted you to see why you have this Omega squared.
‫It comes from the differential lift and differential drag force.
‫Where thanks to assuming that this velocity's very small, you ended up with the velocity that was Omega
‫squared, R squared.
‫And then you just separated those two, you left your R-squared inside the integral, and then you put
‫your Omega's cord out of the integral here.
‫So now, you know, it now known this in the next video, we're going to see how we can form our control
‫inputs, you one, you two and you three using this information that we have learned so far for you,
‫for we have to dig a little bit deeper.
‫This where we are going to do later.
‫So thank you very much and see you in the next video.