 Hello, let us have a look at how the VN diagram is calculated for a long range transport aircraft. As is the usual in this course, we have taken Boeing 787 Dreamliner as our reference aircraft. Let us look at the color scheme in this presentation which is also standard, you must be used to it by now. The general instructions are given in brown color. If there are any values which are specified in any standard document or any online source which is taken as a reference, they will be shown in the black color. The values which are assumed or which are given in the database are in the light blue color. Wherever you do some calculations, there is a hint that it is in red color and also there is a pause button after that. So I would request if you really want to learn to do these calculations properly, it is best that when you see the pause button, you can pause the video, do the required calculations and then proceed further. The values which are calculated are going to be shown in the dark blue color and any comparison is going to be in the green color. But in this presentation, of course, we do not have anything to compare. Let us first start with a quick recap about what is a VN diagram. At this point, I suggest if you have not watched the video clips regarding VN diagram, it is a good idea to first go and watch them and then do the tutorial. But those of you who have already watched the video clips, here is a quick recap about a VN diagram. It is basically a structural operating envelope of the aircraft. In this diagram, we plot the limit load factor versus the airspeed. But this is applicable only for symmetric maneuvers in the xy plane. We use equivalent airspeed on x axis. The reason for that is very clear because we want to use the same diagram for various altitudes. Usage of equivalent airspeed allows that to happen. On the y axis, we have the vertical load factor experienced by the aircraft, which is the total vertical force acting divided by the aircraft weight. This is the largest in magnitude among the three load factors nx, ny, nz. So, this is actually nz, but since we will talk only about one load factor, we are not going to use the subscript z. So, when we say load factor n, we actually mean the vertical load factor nz. And on this particular diagram, the regulatory bodies have suggested that we should superimpose some gust load factors. Let us look at the steps in the construction of a VN diagram. So, here is the VN diagram. So, first we normally calculate the value of Vs, which is the stall speed in level flight. It is also called as 1g stall speed. And while calculating this number, you use load factor nz equal to 1 and V equal to V stall. And all along this line from O to A, the CL is actually equal to CL max at various speeds. So, therefore, this area on the left of this line is the positive stall area. So, we cannot have sustained flight in this area because the aircraft is going to stall. So, in a way, this gives you an operating limitation that you cannot operate for a long time on the left of the line OA. The next important point is the point D, which corresponds to the negative load factor area, usually when you have inverted flight. So, here what we do is we calculate the point D as the y-axis value will be the most negative permitted value of the load factor that is minus n max and the corresponding speed V, depending on what is the lift coefficient maximum permissible in inverted flight. So, just like we got this point Vs for 1g level flight, this is for you know maximum, this is for the maximum negative. The next point is point A, which is a very important point, it is called as the point of corner speed because the speed corresponding to this point is the corner speed. At this point, there are two things which are applicable, the CL is equal to CL max because we are along this line. And also the NZ is equal to positive NZ max because we are also on this line. So, this is the intersection of the positive maximum vertical load factor and the smallest speed because it corresponds to the highest value of CL. This speed should be as low as possible because corresponding to this speed, you have the tightest turn and the fastest turn rate. So, this is called as a corner speed, this is obtained by the intersection between this line and the maximum permissible load factor line. The next point of interest is the point corresponding the VN diagram due to the cruise speed. So, cruise speed is either specified or taken as a multiple, multiplied fraction of the stall speed 1.25 or 1.5. So, here the load factor here is going to be the maximum value and the speed is going to be the maximum permitted cruise speed. The next important point corresponds to the point where you have the maximum dynamic pressure or half rho V square into N. So, this corresponds to the point which is having the design diving speed and also the NZ max. So, this particular point gives the maximum loading on the aircraft. Therefore, flight on the right hand side of this line is not permitted from the structural safety considerations. So, corresponding to point B, there is a point C on the negative load factor. This again corresponds to the most highest negative value of the dynamic pressure. So, because it is at design diving speed and the maximum negative value of the load factor. And then on that we superimpose the gust lines. So, first this is a gust line due to a positive gust starting from level flight. So, that means N equal to 1 and as you will see very soon the delta NZ because of gust which is going to be the number above this line is proportional to V. So, it will be a straight line up to VC. Similarly, there will be another straight line up to the highest speed at which you are allowed to fly inverted. In this diagram or in this sketch it is shown that this speed is lower than the speed VC for the positive flight, but it could also be the same it depends on the specifications of the aircraft. And then up to design diving speed, we have been asked to also plot the delta NZ both in the positive direction and in the negative direction as shown here. And finally, we just join these lines from cruise to design diving speed both in the top of the graph and the bottom of the grass. So, therefore, if the dark lines shown earlier correspond to the limit maneuver VN diagram and if the these lines correspond to the limit gust diagram. So, if you superimpose these two, this particular curve, the outermost curve is going to be the limit combined envelope. Now, let us start doing the calculations for our aircraft Boeing 787-8, but before we proceed, we need to recall some of the important data and specifications of this aircraft. First, let us look at some aircraft related specifications. This aircraft is designed for a maximum gross weight of 476,000 pound, which translates to 215912 kilograms. Wing area is 3870 square feet, which translates to 359.53 square meters. The wing aspect ratio is 10.58. The sweep of the maximum thickness line is 30 degrees. This information will be needed for calculating the value of the lift curve slope, which is needed in the estimation of the additional load factor due to gust. Maximum positive lead coefficient is 1.91 and we do not have the value of maximum negative lead coefficient. So, we will assume it to be 1.00. Remember, this data is not available, so we have just assumed it. As far as the operational relative parameters are concerned, this aircraft has a cruise Mach number of 0.85. Its design cruise speed as specified in the document is 393 knots equivalent airspeed, which can be converted by multiplying by 0.51444 to get meters per second. If you want to do a more accurate conversion, you can multiply the knots with 1852 to convert it into meters and then divide by 3600 to convert hours into seconds. Similarly, the design diving speed is given as 426 knots EAS, which converts to 219 meters, 0.5 meters per second. The maximum positive load factor that this aircraft is designed to sustain is specified in the sample analysis as 2.54. And we do not have any information regarding what is the maximum negative load factor. So, as specified by the airworthiness agencies, if the designers do not specify the maximum negative load factor, it can be assumed to be at least half of the maximum positive load factor. So, please note that the maximum negative load coefficient of 1 and maximum negative load factor of 1 point minus 1.27 are assumed values in this particular analysis. Let us look at the 425 regulations for gusts. So, these regulations are explained in detail in 14 CFR 25.341 titled gusts and turbulence loads. So, there is a huge amount of information mentioned in that including things like how to look for changes in the gust velocity etc. We will take a very simple approximation and we will only look at specification of the vertical gust velocity up to the design cruise speed VC. So, we need to take the gust speeds to be at least 56 feet per second equivalent airspeed at sea level. They can be reduced to 44 feet per second at the rate of 15,000 feet and at higher altitudes, gusts are actually very, very weak. So therefore, the maximum velocity specified is only 20.86 feet per second equivalent airspeed. So, in between you can always interpolate to the value that you want. As far as the specification of the gust velocity for the up to the design diving speed, it is considered to be just half of the above values. So, this information can be graphically also indicated like this, where you can see that at sea level, you have the highest requirement of gust value, gust speed of 56 feet per second, which is supposed to be constant up to 15,000 feet and then from there it is expected to reduce linearly. If you look at the specifications at 15,000 feet, it is 44 up to 15,000 and then again it is reduced linearly and if you look at the design diving speed, the value is 20.86 up to 15 up to 15,000 feet and then again it is to be linearly reduced. In our case, we are going to look mainly at the situation at sea level. So, therefore, we are only going to be concerned with this velocity 56 feet per second EAS at sea level and of course, the value up to design diving speed, which will be half of this or 28 feet per second. Let us look at the calculations at sea level. So, this is our V-end diagram for reference. First, let us find out the value of stall speed at n equal to 1, that means the 1G level stall speed. So, this can be estimated by a simple formula 2W divided by rho SCL max, where you have to multiply the value of W with G to convert the value into Newton's per meter square from kg per meter square or to convert the weight W in kilograms to Newton's and we have to use CL max positive, which is 1.91. So, at sea level conditions, if you replace these symbols with the appropriate values, you should be able to get the estimate of the stall speed. At this stage, I would request you to pause the video and do this calculation. The value turns out to be 70 meters per second. So, the 1G level stall speed for this aircraft is 70 meters per 70.9 meters per second under sea level ISA conditions. Next, we look at the corner speed. Remember, for the corner speed, we both have CL equal to CL max positive and load factor n will be equal to this n max, which is 2.54. So, you can see that the value for corner speed will come from the same equation except W will be replaced by NZ max times W. So, therefore, what we can do is we can just assume it to be V star will be assumed to be or calculated to be just root of n max times VS, which already is known to us. So, it will be root of 2.54 times 70.9. Please pause the video, do this calculation. The value turns out to be 113 meters per second. Next, we look at what is the value of velocity when n is the maximum negative value. That means, we want to get the value of V corresponding to this particular point. So, at that point again, the formula is the same, only we replace n with n max negative and we replace CL with CL max negative. Be careful about the signs because we are going to have a square root. So, because I know that in the numerator we have minus 1.27, the load factor and the denominator we have minus 1. Therefore, I have just avoided putting the minus sign and I am getting the value directly. At this stage, please pause the video and have a look at the calculation. So, V at minus n is going to be 97.8 meters per second. These three values, please remember because we will need them in the future when we plot our V-n diagram. So, this is our limit maneuver diagram or the limits imposed because of the maneuvers in the study XZ plane. So, the first number that we have already calculated is this particular point which corresponds to the stalling speed in 1G level flight. So, n is equal to 1 and V as we just now calculated is 70.9 meters per second. The next point is the point that corresponds to minus 1, the value of stalling speed at minus 1G. So, what is the minus 1G level flight stalling speed in inverted flight? So, this one will be also along the same line and the only thing is n will be equal to minus 1. So, for that there is a corresponding value of 97.8 meters per second. The next point is the corner speed which we have just now calculated corresponding to maximum NZ 2.54 and also along this line. So, the intersection of these two lines that is V star the corner speed, the value is the coefficients of this particular point are going to be for the x axis, it will be 113.0 that is the corner speed value and for the y axis it is going to be 2.54. Next point is the design cruising speed point. So, this particular the coordinates of this point if you see here they will correspond to cruising speed which is 190 and the value of NZ here is maximum. So, for VC we have this particular point. The next point is point B which corresponds to the maximum design diving speed and also the maximum load factor. So, that would be 219.5 and plus 2.54 corresponding to B there will be a similar point C which will have the same design diving speed value 219.5 but NZ is going to be now the maximum negative which is minus 1.27 and finally we have the point D where we have the value of velocity at which the load factor has the maximum negative value of minus 1.27. So, you are on this particular line and on this line CL will be maximum CL permitted for inverted flight or in the negative condition maximum negative CL and NZ will be the maximum negative permitted load factor. So, this point is analogous to the corner speed but it is in inverted flight. So, this is the limit maneuver diagram which we have already plotted for this particular aircraft. Let us now look at how to estimate the additional gust load factor and how additional limits get imposed in the VN diagram due to these additional gust loads. For that recall from the previous video on VN diagram we have already shown there that the additional load factor due to a gust delta NG is equal to A into KG into VG into VC into rho into SW divided by 2 times W. Here KG is the gust alleviation factor which is defined as 0.88 times mu G upon mu G plus 1 where mu G is a factor that depends upon the W and S or the wing loading lift curve slope and the mean aerodynamic chord C bar. So, we notice that larger the wing loading W by S smaller the lift curve slope and smaller the chord larger is the value of mu G and then that adds in the calculations here. So, now the value of A that is needed here for the calculation of lift curve slope and also here for the calculation of mu G is the DCL by D alpha of the aircraft which can be shown to be 2 pi into A upon 2 plus root of 4 plus A square beta square bracket 1 plus tan square delta M by beta square bracket close. So, where this delta M beta A are aggravated parameters beta is basically 1 minus the cruise mark number. So, in this case VG here is the specified vertical gust speed which we just now saw we see is the cruise speed or the or the diving speed depending on which condition we calculate rho is the density of air which is used in the calculations here. Then W is the aircraft weight that is this W and S is the wing area many people use W by S directly in this particular calculation. So, in that case S comes below and this becomes 2 into W by S and similarly mu G becomes 2 upon 2 W by S by AC bar. Then the value A here is the wing aspect ratio as I mentioned to you delta M is the sweep of the maximum thickness chord line and C bar is the mean wing chord. So, out with this information we will now first calculate the lift curve slope for this aircraft. Then using that we will calculate the value of mu G. Once we know mu G we will calculate the value of kg and once we know kg we can put kg here in the expression and get delta NG. So, the gust load factor would be 1 plus minus delta NG due to gusts because gusts are always assumed to act on a level flight which means you already are flying at a load factor of 1 and either on that there is a vertical downward gust or a vertical upward gust depending on that we use either plus or minus. Let us estimate the lift curve slope of this aircraft. So, the formula for that I have just reproduced from the previous slide. Now beta is 1 minus MCR square which is 1 minus 0.85 square. So, please calculate this number it turns out to be 0.5268. So, now that we know beta we can actually calculate the lift curve slope by putting all the numbers in this big formula. So, I would like you to pause the video and do this calculation and then compare with the value that I got. The value that I got is A is equal to 6.327. This is actually per degree because 30 is 30 degrees here. Now under the information let us now estimate the gust load factor. First we examine the gust load factors at the design cruising speed. So, you know that mu G is equal to 2W by A SWC bar. A has just been calculated as 6.327 SW is given 359.53 C bar is given 6.465 W is given 21592 kilograms. So, just multiply by 2. So, let us pause here and calculate the value. The value is 23.93. Moving ahead the specified value for the gust speed at sea level was 56 feet per second. You divide by 3.28 to convert it to meters per second and kg which will be used in the calculation for delta NG. Once you know the value of mu you can get the gust elevation factor kg which will be 0.88 times mu G by mu G plus 1 or 0.88 times 23.93 upon 24.93. Please calculate this value this number comes out to be 0.7204. So, what it means what it means is that vertical gust acting on this aircraft the sharp vertical gust will actually be reduced by a factor of around 28 percent to 0.7204. So, the additional load factor created because of a vertical gust is going to be reduced by a factor of around 28 percent because of the value of kg which is an aircraft specific parameter. It depends upon the wing loading, it depends upon the lift curve slope and the span chord of the aircraft. Now, NG the load factor due to gust will be acting in level flight. So, it will be 1 plus minus delta NG or 1 plus minus this whole expression and what we have done here is we have just reproduced AS 6.3272 kg as 0.7204 VG as 17.07 VC remains as it is rho is 1.225 NSW is 359.53 and on the bottom we have replaced W by W into G. So, just to check out the units. So, please calculate the value of this whole expression. So, it will be 1 plus a constant times VC that number turns out to be 0.008091. So, additional gust would be this number times the cruise speed and that will add or subtract from 1 because it will act in the level flight either as a vertical upward gust in which case you will add it or a vertical downward gust in which you will subtract it. Similarly, we have to apply the conditions for the design driving speed VD. Remember formula remains the same except that instead of VC now we will have VD in the numerator. And also we have to be careful that the value of the gust speed VG is also going to change because you know that VG is specified as half of the value at C at the cruising speed. So, the value of VG will be half it will be 8.54 meters per second as compared to 17.07 meters per second. So, substituting the values in the formula delta NG equals 6.327 which is A these are the other parameters like area, weight, NG, etc. I would like you to pause and calculate the value of delta NG the value comes to be 0.004053 times VD. So, therefore, NG the gust load factor due to a vertical and downward gust acting at the design driving speed would vary as the speed. So, from 0 to VD with this particular formula 1 plus minus 0.004053. So, notice that this NG versus V will be a straight line and it will go it will start from VC equal to 0 it will become 1 and when VC equal to whatever value is there for the design for the design gust speed it will add a number to that. So, it will be a straight line with origin from the point 1 comma 0 comma 1. Similarly, here so let us look at the limits gut envelope. So, the first equation is NG equal to 1 plus minus 0.00818091 VC this is for the vertical gusts acting up to the cruising velocity when the gust speed is 56 feet per second. So, from V equal to 0 to V equal to VC we will just get 2 lines like this where this point will correspond to you know it will correspond to 0 comma 1 level flight and as we changes as per this expression the NZ value will be 1 plus 0.008.91 VC and this line will correspond to the value when you have VC when you have from 0 to VC when you have 1 minus 0.008091 times VC. Similarly, if you look at the line for the additional load factor up to the design diving speed. So, we notice that once again we get 2 lines which start from the same point 0 comma 1 one of them increases the other of them decreases this increases because this is equal to 1 plus a number which is never going to become less it is only going to only increase and this is going to be 1 minus a number. So, that will depend upon the value. So, you can see that even when you are at design diving speed you still have some vertical load factor and then finally what we do is we just join these from straight lines from VC to VD on positive and from VC to VD on negative. So, this is the limit gust envelope that means if you are in a if you are flying an aircraft and if there is a gust acting on the aircraft in level flight the aircraft can go into any of these areas. So, the aircraft has to be strong enough to withstand the loads that come when the operating point is within this particular area. So, on this I have just superimposed the previous VN diagram. So, we see that there is not any problem because of gust no additional material comes out I mean the gust envelope is almost completely inside the general maneuver envelope. So, in this case it turns out that the gusts are not imposing any additional limits. But that is not true always that is not always true. So, here is our limit combined envelope here is our limit combined envelope in which the dotted lines represents the additional load factor imposed due to the vertical gusts which are known to be or taken to be 56 meters feet per second for up to design cruising speed and half of that up to design diving speed both in the positive axis and in the negative axis. So, thank you so much for your attention. I would like to acknowledge the contribution of Dimitri Simbos for providing the piano sample data which we have used in this calculation Professor Mohamed Saderay for his excellent textbook on aircraft performance analysis in which this particular VN diagram calculation procedure is very nicely and lucidly explained and I have taken it from there and also some of the sketches that you have seen in this presentation. And last but not the least I would like to thank Namanuddin for help in creating this tutorial and plotting the VN diagram. Thank you.