 Let us have a look at the procedure to be followed for carrying out the constraint analysis of multi-role military aircraft. First, let us have a look at the typical constraints which are specified on a military aircraft. Mainly they come from the customer requirements, they could be constraints like sustained turn rate at specified mark number and altitude. Sustained turn rate means the ability of an aircraft to turn at a particular degrees per second in the yaw plane and to maintain that particular turn rate without any loss in speed or loss in altitude. Then you have instantaneous turn rate, this is similar to the sustained turn rate but in instantaneous turn rate we are permitted to sacrifice either speed or altitude to achieve a turn rate. So this is a momentary turn rate which cannot be sustained hence it is called instantaneous whereas a sustained turn rate is something that you need to maintain. Then maximum mark number at a specified altitude is the ability to fly at that particular mark number at a given altitude. One important performance attribute of an aircraft is specific excess power that is the excess power over power required divide by the weight of the aircraft or it is also called as SEP. This SEP can be traded to achieve either acceleration or climb or both. So at a given mark number and a given altitude sometimes a specific excess power that needs to be possessed by an aircraft is explicitly specified. Turn rate while operating at a given altitude is another performance parameter. The stalling speed of the aircraft in level flight so it is 1g stalling speed at a specified weight and altitude. Takeoff and landing ground roll under ISA conditions or under ISA plus X or the off ISA design conditions. Absolute and combat ceilings once again under either ISA conditions or under specified ISA plus X conditions. These are the typical requirements which the customer specifies. There are also some airworthiness requirements which may be present and that is a function of the regulatory bodies or the airworthiness agencies which are having a civilian control over the military aircraft or the domain in which it is operating. So there is no specific requirement it is as specified. For a military aircraft it is easier to look at a master equation for constraint analysis and this master equation comes from the energy height principles. So the energy height basically is specific energy or the excess energy divided by the aircraft weight. The energy that an aircraft possesses is the summation of the potential energy and the kinetic energy that is mgh and half mv square. When you divide that by aircraft weight you get specific energy. Now w is equal to basically mg. So if you do a simple algebraic manipulation the specific energy turns out to be the units turn out to be h and v square by 2g. Now v is in meter per second and g is in meters per second square. So you know if you look at the units of the specific energy they will come in meters so that is why we call it as energy height. So the energy height is equal to h plus v square by 2g. Now the excess power that an aircraft possesses is basically equal to d by dt of the energy or the rate of change of the energy. So the excess power that is t minus d into v, t into v is the power available and v into d is the power required. So the excess power is the rate of change of the total energy and hence the specific excess power will be excess power divided by the rate of the aircraft. In other words ps which is the symbol used for specific excess power it will be p available minus p required divided by w or v into t minus d upon w which is also equal to the rate of change d by dt of h plus v square by 2g. So using this expression one can actually draw a diagram like this for various values of ps. So this particular diagram is a ps diagram at n equal to 1 and it kind of gives you some maneuver limit that the aircraft has. We will come back to this in more in more detail little bit later. So let us see how we can set up the master equation. A master equation is an equation which will be used to represent the relationship between t by w and w by s because ultimately in the constraint diagram we are going to plot t by w on the y axis and w by s on the x axis. So if we can get an equation between these two parameters then we can use that equation directly to plot the constraint diagram. So you know it can be shown that if you basically take this v in the denominator you know you will get t by w minus d by w that is this expression is equal to 1 by v dh by dt plus 1 by g v square by 2g when you differentiate you get 2 v by 2. So by cancellations you get this expression. So this is the basic master equation. Now we have to also keep in mind that this equation has got terms like t w d v dh by dt. So let us see 1 by 1 let us first look at t, t is the thrust and the thrust is not constant at all conditions. So in general you can say that t is equal to a into t0 where a is the thrust lapse ratio. This represents the change in the thrust compared to the sea level static thrust value. It depends on the density ratio sigma and the mark number or velocity as the case may be. So at any condition t will be equal to alpha times tsl note that alpha can also be more than 1 in case we are using reheat or after burner that is normally used in military aircraft during takeoff. So it is not that the value of a will be always less than 1. In general the value of a is going to be less than 1 higher altitudes. So this particular chart or this particular table helps us in determining which is the appropriate equation to be used to calculate the value of ta or the value of yeah. So ta by tsl will be alpha. So if we have for example if we look at high bypass turbofan okay we have this expression and you know you can there is a 0.1 here. If you look at turbojet for example it just shows that for turbojet or a low bypass turbofan the t available is basically equal to the tsl into rho by rho sl. So it just goes by the density ratio. So this is a simple simplification of what can be seen in real life and it helps you to find out the value of the parameter. So we take care of thrust by multiplying it by alpha. Now we look at weight. Now weight of the aircraft W is always going to be changing as we go into the flight profile. So in general weight at any segment at any point in the performance of the aircraft will be beta times WTO where beta is the weight fraction for a given constraint. Regarding drag, drag can be replaced by CD into Q into S and if we assume parabolic profile for the drag polar we can replace CD with CD naught plus K1 Cl square. Now the lift coefficient is going to be N times W. So the lift is going to be N times W therefore Cl will be equal to L upon QS which will be N times W upon QS. So this is the basic master equation that we have derived. Now what we can do is we can replace the D with CD naught plus KCl square into QS. We can replace Cl by NW by QS. We can replace W by beta times WTO and we can replace T by alpha times Tsl. So if we do that then what you get will be as follows. So T will become alpha times Tsl upon W will become beta times WTO minus D will become CD naught plus K1 Cl square into Q into S upon W will become beta times W takeoff and that will be equal to 1 by V DH by DT plus 1 by G DV by DT. Now since Cl is equal to NW by QS the Cl square here can be replaced by N square W square by Q square S square. So when you do that you will get here N square. Now W is going to be beta so beta square W0 square and divided by Q square S square. So this QS will go and it will become upon Q and S and then what you can do is you can take this alpha by beta common so long story short if you substitute in the above expression all the values and if you take out the terms common you get an expression Tsl by WTO which is the LHS is going to be beta by alpha Q by beta CD naught by WTO by S in plus K1 beta by Q whole square W0 by S plus the same RHS side. So what happens here is that on the LHS you have Tsl by WTO on the RHS you have WTO by S in the denominator here WTO by S in the numerator here and you have all the other parameters. So one advantage of this particular expression is that there is no need now for you to worry about converting the values from the ones operating or being present at the constraint value to the ground level because now you are getting directly in terms of Tsl by WTO and WTO by S through the values of beta alpha and this expression can then be used for you know calculating any of the constraints specified here except a few. Now it may be noted that the expression on the RHS 1 by V DH by DT plus 1 by G DV by DT this is nothing but PS by V because PS was DH by DT plus 1 by you know V DV by DT. So therefore during steady cruise steady cruise and level steady level cruise sorry so therefore during steady level cruise we do not have any excess power so this whole term is going to be 0. So therefore the master equation for cruising flight will convert into an expression as shown here the terms on the end will are going to be 0. So if you now plot you can see it is a quadratic relationship and you can see that on the y axis you have T by W and the x axis you have W by S for one particular condition and the line that you see represents the relationship between the T by W and W by S for some particular values of beta Q alpha n etc. Let us look at the constraint on the climb rate the rate of climb is nothing but DH by DT and therefore if I have a steady climb then there will be 0 value of DV by DT so the specified value of DH by DT at the specified climb velocity can be given in this expression to get a link between T that T by SL. So once again we have a curve where a max climb rate constraint can be shown in this form and the area below this line is going to be the infeasible area the area above this is going to be feasible. Similarly let us take a constraint on takeoff ground roll now usually specified in the terms of takeoff distance STO. So if you assume the V takeoff to be 1.2 times V stall which is the regulatory requirement then STO is equal to because VTO is 1.2 times V stall you have a factor of 1.2 square or 1.44. So therefore what will happen is that the T by WTO will be coming as a straight expression these items are going to be so for a given STO for a given altitude for a given CL max in takeoff you should be able to get a direct link between the T by W and W by S so this is going to be a linear line and the area below this line is going to be infeasible because that will not meet the specified takeoff value. Let us look at the landing distance constraint the landing speed at which you touch down is normally expected to be around you know 30% higher than the stalling speed. So with that an expression for landing distance can be obtained this 1.69 comes from 1.3 square and because during landing there is going to be some friction that is some rolling friction and some drag during landing these terms new roll and D land also appear in the calculation but in some cases one can ignore D land and L land so you get a much simpler expression and this simpler expression simply allows you to look at landing distance as a function of only the W by W by S so it will be a straight line because all the other quantities can be assumed to be constant for a given operating condition so it is basically W by S is a function of the landing distance so it puts a upper limit on W by S if I want to have a landing distance below a particular number I cannot have been loading higher than a particular number thanks for your attention we will now move to the next section.