TakeOff and Landing

7/28/2019 TakeOff and Landing

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TakeOff and Landing

This section will present the theory of takeoff and landing for conventional aircraft.

Conventional aircraft would be any aircraft with a main gear, a nose gear and a singlesource of thrust at some angle of incidence it. Therefore, "conventional" could includesome aircraft that are considered STOL (Short Takeoff Or Landing). One could derive

equations that are more complex for a VSTOL (Vertical or Short Takeoff Or Landing).

Takeoff Parameters

Let us define the following forces, distances, angles and coefficients as depicted in thefollowing drawing.

Dbw = Drag of the aircraft body and wing - along the aircraft flight path axis

During the ground roll, the flight path will be parallel to the runway

Dt = Drag of the aircraft tail - acts along the aircraft flight path {this term is often

lumped into the body drag for aircraft without a T-tail}.

L1 = Lift of the wing - acts perpendicular to the flight path

L2 = Lift of the tail - also acts perpendicular to the flight path

Wt = gross weight - acts through the center of gravity of the aircraft.

Fn = net thrust acting parallel to the flight path. {We will however include a term

perpendicular to the flight path}

F1 = Load on the nose gear.

F2= Load on the main gear.

X1= Distance from the nose gear to the aircraft center of gravity.

X2 = Distance from the main gear to the aircraft center of gravity.

XL1 = Distance from the center of gravity to action point of the wing lift (Mean

Aerodynamic Chord)

XL2= Distance from the wing lift point to the tail lift action point.

Z1 = Height of the body axis of the aircraft above the ground plane.


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Z2= Height of the tail center of lift and drag above the aircraft body axis.

= Aircraft pitch attitude.

= Runway slope.

Not shown on the drawing (to avoid clutter) is gross thrust (Fg) and the engine inlet drag

(Fe).

Using the above diagram, we can formulate the equations of motion for the aircraft

during the ground roll. The equations are the same for either a takeoff or a landing.

Forces are in pounds; speeds in feet per seconds and angles are in degrees.

1. Requiring the summation of forces in the X-axis to be zero. ( note: Here thepositive X-axis direction is along the runway to the left).

it = Thrust Incidence Angle

D = Total Aerodynamic Drag

Frw= Total Runway Resistance

Fex= Excess Thrust


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with:

where:

= Coefficient of friction associated with the nose wheels.

= Coefficient of friction associated with the main wheels.

Also,

where:

Nx= Longitudinal Load Factor

g0 = 32.174 feet/second2

Vg = Ground Speed

Collecting terms:

1. Requiring the summation of forces in the Z-axis to be zero. ( note: The positive

Z-axis direction is perpendicular to the runway and pointing towards the top right

of the page.)

2. Requiring the summation of moments about the Y-axis to be zero. ( note: The Y-

axis in this case is perpendicular to the page and coming out of the page.) Wewill take moments about the main wheels, since the aircraft will pitch about the

main wheels during the takeoff or landing ground roll. We will ignore any pitch


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dynamics during the ground roll. We will ignore any moment caused by the

vertical component of gross thrust.

What we now have is three equations with three unknowns for purposes of simulating atakeoff or landing ground roll. It is assumed that one has a thrust and drag model for the

lift, drag, gross thrust, and engine drag terms in the above equations.

The three unknowns are the two normal forces on the wheels (F1andF2) and the excess

thrust (Fex). Of course, the primary parameter of interest is the excess thrust from whichwe can compute the derivative of ground speed. Once we have the excess thrust, we can

differentiate the ground speed derivative to obtain speed and distance versus time.

Collecting the three equations:

Rearranging the equations:

We will define the terms in the square brackets in each one of the equations asA1,A2 and

A3, respectively. Then we can rewrite the three equations in matrix form as follows:


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Here the matrix equation is of the form C F = A, where the bold letters represent vectorsand C is the three-by-three matrix. We need to solve forF.

During the course of flight test, we measure excess thrust (Fex). However, the thrust and

drag may be unknown, or at least not known precisely. Therefore, we may need to iteratebetween the above equation and the solution of the above equation. TheA1 term is thrust

minus drag minus the runway component of weight.

The above matrix relationship can be solved by multiplying both sides by the inverse of

the square matrix, C, namely C-1, as long as the determinant of C is not zero. Thus

Developing a Takeoff Simulation

Usually, the designer will provide an initial estimated model for lift and drag as a

function of angle of attack ( ). Normally, is zero during the ground roll and that is

why it was not included in the above general equations. The thrust incidence angle, it , isalways usually either zero or small. Only the most precise simulations will typicallyaccount for a separate tail and body drag, so we can ignoreDt, the drag of the tail, in

many cases. Accounting for tail lift and drag becomes more important when modeling

braking performance to determine the load distribution on the main gear and the nosegear.

For takeoff performance, a value of 0.015 is usually assumed for the rolling coefficient of

friction ( ). In addition, a point mass model will be assumed with all the forces acting

through the center of gravity of the aircraft. With these assumption, the three equationsreduce to two equations, namely:

where F=Frw.


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Combining the above two equations by substituting forFleads to:

The above equation can be used in two ways:

(a) first, to solve for excess thrust, i.e.,

or, (b) second, to solve for thrust minus drag, i.e.,

We know (or assume values for) the other variables.

From the first equation, we can compute the excess thrust during the ground roll of the

aircraft. One would be provided models for net thrust drag and lift. The drag and lift

models would be in the form of drag and lift coefficients versus angle of attack. Typicalmodel formulations are as follows:

where:

M= Mach number

H= Pressure altitude

Ta = Ambient temperature

= angle of attack

hAGL= Aircraft wing height above ground level


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The fuel flow is required in order to account for weight change during the takeoff. The

parameterhAGLis needed to account for ground effect. The above are just typical modelforms. They may also include Reynolds number terms. In addition, the engine is usually

not at 100% thrust at brake release so a thrust spool up factor needs to be supplied.

Ground Effect

The following plot is typical of a relationship defining the decrease in drag due to lift inground effect.

A very simplified model that approximates an F-16 in military thrust was created toillustrate takeoff simulation. The model constants and equations are as follows:

S= 300. = reference wing area (feet2)

b = 35. = wing span (feet)

AR = 4.0 = b2/S= Aspect Ratio

hw = 5.0 = height of wing above ground while aircraft on the ground (feet)


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Wts = 25,000. = start gross weight (pounds)

Fno = 10,000. = thrust at zero Mach (pounds)

Fnslope= 5,000 = slope of thrust vs. Mach (pounds)

KFn0 = 0.65 = thrust factor at zero time

= 2.0 = thrust time constant (seconds)

We assume that the overall thrust factor increases from its zero time value via theformula

and ifKFn >1, thenKFn= 1.

The equation for the net thrust for this model becomes:

and the weight of the fuel can be related to the thrust by:

where sfc = thrust specific fuel consumption.

The percent of out of ground effect drag is computed from (see previous graph)

under the condition thatXGE=1.0 , ifXGE >1.0.

With the following parameters defined, namely,

Clmin = 0.05 = lift coefficient corresponding to minimum drag

Cdmin = 0.0500 = minimum drag coefficient

then the drag coefficient (Cd) is computed as follows:
http://www.allstar.fiu.edu/aero/TO&Land.htm#ground%20effecthttp://www.allstar.fiu.edu/aero/TO&Land.htm#ground%20effect


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With the definition of the initial pressure altitude for the runway, for example,

HC = 2,300 feet = initial pressure altitude

then the ambient pressure ratio ( ) is as follows:

and we may relate it to the standard day value at sea level by

where:

Pa = ambient pressure, and

PaSL = ambient pressure standard day sea level = 2116.117 pounds/foot2

Using the aspect ratio and the angle of attack, then the lift coefficient (CL) is as follows:

The angle of attack is held to zero during the ground roll until a rotation speed is reached.

This rotation speed (in this simulation example) is at a calibrated airspeed of 100 knots.Upon reaching the rotation speed, the typical takeoff will rotate to some given angle of

attack. Then, that angle of attack is held until the aircraft generates enough lift such that

lift is greater than the weight and the aircraft lifts off the runway. The angle of attack

profile used in this example simulation is as follows:

where we assume that


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The angle of attack ( ) is limited to a pre-determined value. In the example that value

is 13 degrees. In the numerical integration, 13 degrees angle of attack is reached at 143knots calibrated airspeed. The lift first exceeds weight at airspeed of 156 knots. The

aircraft (or the simulated aircraft) will lift off the ground when lift is greater than the

weight.

Lift and drag are computed as follows using the mach number definition and ambient

pressure ratio:

Finally, the last terms in our model are for the runway resistance. We will assume zero

runway slope and a runway coefficient of friction, = 0.015. Then,

under the condition thatFrw= 0. forL > Wt since the aircraft will become airborne.

Now, the equation (previously derived above):

will reduce to

Also, the longitudinal load factorNx was related to the excess thrust through the totalgross weight by

where
http://www.allstar.fiu.edu/aero/TO&Land.htm#excess%20thrusthttp://www.allstar.fiu.edu/aero/TO&Land.htm#excess%20thrust


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During the ground roll, the h-dot term is zero. During the air phase, the normal load

factor equation is used, i.e.,

where

and = flight path angle.

From theNX , NZ andequations we can integrate to find ground speed (Vg) andgeometric height (h). All of the forces, however, are functions of airspeed and pressurealtitude. We have assumed a standard atmosphere for temperature (in degrees Kelvin)

We can now find the true air speed using

where:

Vt= true airspeed

Vw = wind speed

if the wind speed were nonzero. We will assume wind speed equals zero in the example.

The following equations relating the aircraft's calibrated velocity, VC, to other

parameters are given here for the sake of completeness, but whose derivation may be

found elsewhere:

From the speed of sound and the mach number:


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where

The compressible dynamic pressure, qC, may be calculated, knowing the ambient pressureand the mach number, by

The calibrated velocity of the aircraft with respect to a standard sea level day is thengiven by

A plot of thrust, drag plus the runway resistance terms and excess thrust versus

calibrated airspeed is shown in the following:


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The following plot is a blow-up of just the drag from just before rotation to about 60 feet

above the ground. This illustrates the changes in slope of drag versus speed as the fixedalpha is achieved and as the aircraft is no longer in ground effect. Of course, these are

idealized computer simulations so one would not see such clearly defined effects in flighttest data.

We can numerically integrate the equations to provide a plot of distance versus calibrated

airspeed or height versus calibrated airspeed. By scaling the distance by a factor of 100,one can present both the longitudinal and vertical distance on one plot as follows.


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Units for y axis: 100 feet for distance (blue), seconds for time (purple)

and feet for altitude (green)

Effect of Runway Slope

Using the pseudo-F-16 model, the effect of runway slope, the values of time and distanceas a function of runway slope (in degrees) is shown in the following table. The average

acceleration is computed as follows:

where:

t = time at liftoff (seconds)

d = distance at liftoff (feet)

Slope

distance time acceleration % from zero

-1.0 3001 22.6 11.75 4.52%

0.0 3131 23.6 11.24 0%


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0.5 3164 24.0 10.99 -2.29%

1.0 3247 24.6 10.73 -4.56%

2.0 3403 25.8 10.22 -9.06%

As can be seen, the effect of runway slope for this particular model is about 4.5% per

degree of runway slope. For a typical light aircraft, the effect of runway slope is at leasttwice that amount, due to the much smaller thrust to weight ratio of the typical light

aircraft. Although the percentage change in acceleration is about the same for a positive

or negative runway slope, one must take into account the fact of having a negative

absolute rate of climb at liftoff for a negative slope runway. For instance, for a liftoff at100 knots ground speed with a negative 1.0 degree slope runway, the absolute rate of

descent is about three feet/second. The rate of climb (or descent) with respect to the

horizontal plane is given by:

Effect of Wind on Takeoff Distance

Again using the same pseudo F-16 model, the following plot illustrates the effect of wind

Y axis: percent change in liftoff distance


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Idle Thrust Decelerations

To assist in the development (or verification) of a Takeoff and Landing simulation, idle

thrust decelerations may be performed. One would accelerate the aircraft on the runwayto some high airspeed. Then, cut the throttle to idle and allow the aircraft to freely

decelerate. We can solve for drag (D) in the above equation

and then putD into coefficient form.

where:

and

CD = drag coefficient

q = incompressible dynamic pressure

= density

Lift and drag coefficients are discussed in the lift and drag section.

A more convenient form for the drag coefficient has been presented previously in the

takeoff simulation portion of this section, namely:

Landing

Braking Performance

Using the same model that was used for takeoff portion of this page, one can see the

effect of braking coefficient upon stopping performance. The thrust has been set to a

constant 600 pounds, representing idle thrust. Minimum drag coefficient has beenincreased from 0.0500 to 0.0700 to account for additional drag devices (such as spoilers)

activated during braking. For the following plot, the coefficient of friction has been set to
http://www.allstar.fiu.edu/aero/lift_drag.htmhttp://www.allstar.fiu.edu/aero/lift_drag.htm


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a constant 0.35; this is a typical dry runway value. The initial ground speed was 130

knots for a calibrated airspeed of 126 knots. The gross weight has been reduced to 20,000

pounds, more representative of aircraft weight for landing speeds.

For a dry runway, the coefficient of friction, , is typically on the order of between 0.25

and 0.50.


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For wet runway conditions, the braking coefficient, , is much less than for a dry runway.

This is especially true at high speed where hydroplaning may occur. Hydroplaning is

where the wheels ride on a film of water and never contact the runway. The following

plot is of the braking coefficient computed from braking tests with the F-15C in 1977 atEdwards Air Force Base. The test was on a wet runway, with the water applied using

water tankers. The data points were average values of the actual data and the line was a

4

th

order polynomial curve fit of the data points.


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A warning is appropriate for using curve fits in simulations. Invariably the data will not

extend to the full range of the desired simulation. Using the curve fit beyond the range ofits data should be avoided by use of limits. A limit would be where the curve fit value (y)

would take on some pre-determined constant value if the x value exceeds the highest (or

lowest) value used in the curve fit. The limits that will be used in applying the curve fitwill be the curve fit values at the extreme points. These are as follows.

Using the above curve fit (with its limits) for the modelleads to the following graph.


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The simulation for our wet runway model produces a total distance of 6718 feet. This

compares to a distance of 2241 for our dry runway model using a constant of 0.35.

That's a factor of three times longer for a wet runway. That's typical, but as the saying

goes, "your results may vary".

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TakeOff and Landing