### AIRCRAFT PERFORMANCE

### Fixed Wing Aircraft

### TAKE OFF AND LANDING

### Take Off Performance

Take off performance can be predicted using a simple measure of the acceleration of the aircraft along the runway based on force equilibrium.

The forces involved will be,

**T** – Thrust of propulsion system pushing aircraft along
runway.

**D **– Aerodynamic Drag of vehicle resisting the aircraft
motion.

**F** – Rolling resistance friction due to the contact of
wheels or skids on the ground.

During take-off run the imbalance in these forces will produce an acceleration along the runway.

where **dV/dt** is the acceleration along the runway and **m**
is the mass of the vehicle.

**Rotation
Velocity, V _{R}**

The procedure for take-off will be that the vehicle will
accelerate until it reaches a safe initial flying speed. The pilot
can then rotate the vehicle to an attitude to produce climb lift and
it will ascend from the ground. The determination of this safe flying
speed or rotation speed, **V**_{R}, is a critical
factor in determining take-off performance.

Take-off rules vary slightly depending on the aircraft category. Small commuter aircraft should be considered as meeting FAR 23 rules, transport category aircraft should comply with FAR 25 rules.

__Small commuter aircraft :__

For safety reasons **V**_{R} is usually
determined as being **1.1 × V**_{STALL} or **1.05
× V**_{MIN CONTROL}

which ever is greater. Stall speed, **V**_{STALL},
is the lowest speed that the aircraft can be flown before the airflow
starts to separate from wings as the angle of attack becomes too
great. The wing is assumed in this case to be in take-off
configuration or "clean".

It can be calculated based on knowledge of the aircraft take-off
configuration and hence the maximum achievable lift coefficient
**C**_{L}**(max)**. As shown in the previous
section , to maintain level flight the lift produced must equal the
weight, hence the stall speed can be calculated as,

Minimum control speed, **V**_{MC} is a more
complex calculation and requires knowledge of the stall
characteristics of the tailplane and elevator. For conventional
aircraft there is only a small difference between **V**_{R}
calculations based on stall speed or minimum control speed.

As well as rotation speed there are other safety considerations as shown in the following Figure.

**V**_{1}- Abort decision speed. Below this
speed the take-off can be safely aborted. After this there will not
be sufficient runway length to allow the aircraft to decelerate to a
stop.

**V**_{2} – Safe climb speed. **V**_{2}
must be no less than **1.2 * V**_{stall}. Below
this speed aircraft cannot attain sufficient climb rate.

__Transport Aircraft :__

**V**_{R} must not be less than **V**_{1}

**V**_{R}
must be greater than **1.05 * V**_{MC}

**V**_{R}
must be set so that aircraft achieves **V**_{2}
before reaching a height of **35ft** above the runway surface.

Aircraft must climb at a minimum gradient to avoid obstacles at the end of the runway. With engine failure on multi-engined aircraft, this speed should still be achievable.

### Thrust

The thrust of gas turbine or turbofan engines will be relatively constant during take-off. A good assumption is to use the manufacturer's values for maximum static thrust for take-off calculations.

The thrust of a propeller driven aircraft can be found from the given shaft horsepower data for the engine and the use of the equations using propeller efficiency given in the previous section.

It is critical to correctly estimate the propeller efficiency for
the particular aircraft velocity along the runway. At **V=0** the
efficiency is **0** so the above equation makes no sense. At **V=V**_{R}
the efficiency will be in the range 50% to 80% depending on the type
of propeller system used and the thrust value at this point will be
easy to obtain. In practice, the thrust obtained throughout the
take-off roll is roughly constant so this end point value is a good
approximation from **V=0** to **V=V**_{R}

### Drag

The resistance to motion due to the air viscosity will give a drag of

where **C**_{D} can be considered constant and
calculated using the formula shown in the previous section

Although Drag Coefficient is constant, Drag will increase in proportion to the square of velocity.

### Rolling Resistance

The friction between aircraft and runway will be proportional to the normal force exerted by the aircraft on the runway.

The normal force will be the difference between Weight of aircraft
and Lift, the friction coefficient will be typically of a magnitude
of **0.02** for a standard tarmac runway.

**Average
acceleration and distance to rotation**

The rate of change of velocity can be predicted at any point on
the take-off roll by substituting results for **T**, **D** and
**F** into the initial equation for **dV/dt**. The subsequent
velocity at any point can be found by integrating this resulting
equation and the distance traveled found by then integrating the
velocity.

Typically acceleration will be dominated by the drag component as thrust, weight and friction are relatively constant during this period. This leads to the result shown where acceleration is inversely proportional to velocity squared.

Due to the quadratic nature of acceleration change, an average value, $({dV}/{dt})_{avg} = a↖{-}$ , can be used for the take-off run. This average acceleration can be found at the point where,

This average acceleration can be used to simplify calculations and
the take-off run can be calculated as an equivalent constant
acceleration over the complete period of time (**t**_{R})
taken to get from **0** to **V**_{R}. For a
constant acceleration take-off calculation,

Rearranging leads to a relatively simple calculation to predict distance to rotation point.

### Obstacle Clearance Distance

From the rotation point, the end of the runway can be defined by the requirement to clear a 35ft obstacle at the end. During rotation it can be assumed that any residual excess thrust is absorbed in overcoming the lift induced drag as the aircraft begins to climb. Acceleration reduces and a constant flight speed during this climb phase can be assumed. The distance along the ground from rotation point to obstacle clearance point with thus be,

This distance estimate will require knowledge of the climb gradient which can be calculated by using the methods in the following section on Climb and Descent.

### Take-Off (Balanced) Field Length

The required length of runway will be the sum of the distance
required to get to rotation speed and the extra length required to
clear a 50ft obstacle or the extra length required to allow for rapid
braking if the pilot decides to abort take-off at the decision speed
**V**_{1}.

This length will typically be considerably longer than the
distance required to achieve rotation (flying) speed. A rough
approximation is that runway total length is around 2 x **s**_{1}

Distance to **V**_{1} can be calculated in a
manner similar to that shown for **V**_{R}. The
calculation of braking distance will require knowledge of the maximum
braking friction coefficient that can be generated by the aircraft.
This information should be available from manufacturer's data.
Braking distance calculations should also be done without any
assumption of reverse thrust from engines as during a take-off abort,
engine power may not be available.

### Landing

The landing run can be calculated in a similar fashion to the take off distance. The aim is again to minimise the distance.

The touch down velocity should be approximately the stall speed of the aircraft in landing configuration. This will be achieved by a pitch manoeuvre during the flare portion of the approach which which increase drag an decelerate the aircraft to minimum flying speed.

The deceleration on the landing roll from **V**_{TD}
to **V**_{0} will be accomplished by braking and
reverse thrust. This can be solved by the average acceleration
approach that was used to estimate the take-off roll.

The negative acceleration or deceleration value will be based on friction coefficient for maximum braking and the value of reverse thrust (if available).