When an aircraft flies, the wings (and the horizontal tails to some extent) support the weight of the whole aircraft. The rest of the aircraft just hangs on these "lifting surfaces". Of course the wings and tails themselves have weight. On most aircraft, the wings contain most of the fuel. A large aircraft like a Boeing 747 might have 250,000 lbs of fuel when it takes off on an intercontinental journey.

We can use Newton's Laws of Motion to calculate the acceleration of an aircraft, and thus to decide how the forces on the aircraft must be balanced to make it go in a desired direction.

 

 

 

 

Force = Rate of Change of Momentum.

Usually there are several forces acting on any object. For example, a rocket is pushed forward by the thrust from its engines, it is being pulled back by air drag, and it being pulled towards the center of Earth by Earth's gravitational attraction. In this case, we must add all these forces together, taking their directions into account, and find the "net" or "resultant" force. This, then is what is equal to the rate of change of momentum.

 

Since Momentum = mass*velocity, the above becomes

Force = mass*(rate of change of velocity) + velocity*(rate of change of mass)

Force = (mass)*(acceleration)

Note: Example of mass changing: A rocket-powered vehicle, whose mass decreases as propellant is dumped out the back end. Strictly, the mass of an aircraft keeps decreasing too, as it consumes fuel. However, this occurs pretty slowly compared to the rate at which a rocket consumes fuel.

Force and acceleration are vectors: they have magnitude and direction.

If two vectors  and  are equal, i.e., 

then,  and  . In other words, two vectors can be equal only if their corresponding components are equal. Using this, we can rewrite the vector equation relating force and acceleration as a set of "scalar" component equations, one along each direction. This is very simple, as soon as we define a set of conventions for defining positive directions.

Coordinate system

We usually use the "right-hand rule". Hold out your right index finger with your thumb pointing up and your middle finger perpendicular to both your thumb and index finger. If moving along the thumb is positive x, and moving along the index finger is +y, then moving along your middle finger is positive z. The other way to look at this is the right-handed thread of a bolt (the usual way a bolt or wood-screw operates: if you rotate from x to y, you move along +z (you are also considered to be "rotating about the z-axis"). If you rotate from y to z, you move along +x.

 

In terms of thiscoordinate system fixed to the earth, the airplane moves along +X if it moves to the right. It moves along +Y is if it moves into the screen. Then, positive movement along z is downwards. The weight, for example, acts along -z, towards the center of the earth. The lift acts perpendicular to the flight direction (in precise terms, the lift is taken as acting perpendicular to the direction of the air coming towards the lifting surface, as we will see later). The drag acts opposite to the flight direction, or along the direction of the air coming at the aircraft. The thrust, in this figure, is shown acting along the direction in which the aircraft is flying. The angle between the flight direction and the horizontal (X) axis is shown as q.

The "Freestream Vector  ". This is equal and opposite to the flight velocity.

Lift L is perpendicular to the freestream vector  (but it may be up, down or sideways).

Drag D is parallel to  .

Thrust T is along the thrust direction (modern fighter aircraft have "thrust vectoring nozzles" which can point the thrust in arbitrary directions. Usually, commercial aircraft have the thrust vector pointing forward, almost along the aircraft fuselage, but they also have "thrust reversers" which can point the thrust backwards.

Weight acts towards the center of the earth (or whatever the closest massive heavenly body is). The two component equations are:

Along X, lets add the forces and equate the sum to the rate of change of momentum:

Along Z:

This is an example of how we figure out the acceleration, and the balance between forces along each direction. The total acceleration vector of the aircraft is given by the components along the x,y and z directions. In the case above, there is no acceleration along the y-direction because there is no net force along that direction.

Lets consider some special cases:

 

Case 1:

In Straight and Level Steady Flight, where all the accelerations are zero, Lift = Weight, and Thrust = Drag.

 

L = W

T = D

 

Case 2: When the aircraft is flying almost level  ;

Acceleration is zero, but  .

From the X-momentum equation,

From the Z-momentum equation,

 

Now consider the coordinate system again, and lets look at the components of the velocity vector, the big U with the arrow on top of it. It can be expressed as two components, the component u along the X-axis, and w along the Z-axis, as shown. We want to relate the Rate of Climb, which is w, to the forces acting on the aircraft, using the equations which we have developed.

Dividing the X-momentum equation by the Z-momentum equation, we get:

If the angle q is small (as is usual under a routine climb condition where one is not in any desperate hurry), the value u is fairly close to the magnitude of the veliocity vector, U. Then, approximately,

If w>0, then the aircraft is climbing. From these we note:

1) If the lift is greater than the weight, then the aircraft will accelerate upwards.

2) If the thrust is greater than the drag, the aircraft can climb if the thrust acts at an angle to the flight direction. So there are different ways of achieving the same result. Let's consider sideward forces.

, the centrifugal force. Note that  is the centripetal force: the force directed towards the center.  is the radial acceleration. By Newton's 3rd Law of Motion, the centrifugal force is the reaction, which is equal and opposite to the centripetal force.

Note: To pull tighter turns, (i.e., smaller R), at a given value of U,  must be made larger. If we are not to lose height during this turn,  must be as large as W.

 

If the lift on one wing is changed relative to the other, the aircraft tends to roll, and this causes a turn, as seen above.

If the lift (side force) on the vertical tail is changed, the aircraft tends to yaw. Then the aircaft must roll to avoid sideslipping.

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Table 1: Direct Access to the Sub-Disciplines of Aerospace Engineering