Category MECHANICS. OF FLIGHT

There is one particular force that we are all familiar with; it is known as the force due to gravity. We all know that any object placed near the earth is attracted towards it. What is perhaps less well known is that this is a mutual attraction like magnetism. The earth is attracted towards the object with just as great a force as the object is attracted towards the earth.

All objects are mutually attracted towards each other. The force depends on the masses of the two bodies and the distance between them, and is given by the expression:

Gm, m,

F =—— LJ.

d2

where G is a constant which has the value 6.67 X ICG11 N m2/kg2, m1 and іщ are the masses of the two objects, and d is the distance between them. Using the above formula you can easily calculate the force of attraction between two one kilogram masses placed one metre apart. You will see that it is very small. If one of the masses is the earth, however, the force of attraction becomes large, and it is this force that we call the force of gravity. In most practical problems in aeronautics, the objects that we consider will be on or relatively close to the surface of the earth, so the distance d is constant, and as the mass of the earth is also constant, we can reduce the formula above to a simpler one:

F = m X g

Fig 1A Weight and thrust The massive Antonov An-255 Mriya, with a maximum take-off weight of 5886 kN (600 tonnes). The six Soloviev D – 18T turbofans deliver a total maximum thrust of 1377 kN.

where m is the mass of the object and g is a constant called the gravity con­stant which takes account of the mass of the earth and its radius. It has the value 9.81 m/s2 in the SI system, or 32 ft/s2 in the Imperial or Federal systems.

The force in the above expression is what we know as weight. Weight is the force with which an object is attracted towards the centre of the earth. In fact g is not really a constant because the earth is not an exact sphere, and large chunks of very dense rock near the surface can cause the force of attraction to increase slightly locally. For most practical aeronautical calculations we can ignore such niceties. We cannot, however, use this simple formula once we start looking at spacecraft or high-altitude missiles.

Weight is an example of what is known as a body force. Body forces unlike mechanical forces have no visible direct means of application. Other examples of body forces are electrostatic and electromagnetic forces.

When an aircraft is in steady level flight, there are two vertical forces acting on it, as shown in Fig. 1.3. There is an externally applied force, the lift force provided by the air flowing over the wing, and a body force, the weight.

In the above example, of the accelerating glider, the force applied to one end of the rope by the aircraft is greater than the air resistance acting on the glider at the other end. As far as the rope is concerned, however, the force it must apply to the glider tow-hook must be equal to the air resistance force plus the force required to accelerate the glider. In other words, the forces on the two ends of the rope are in equilibrium (as long as we ignore the mass of the rope). The extra force that the rope has to apply to produce the acceleration is called an inertia force.

As far as the rope is concerned, it does not matter whether the force at its far end is caused by tying it to a wall to create a reaction or by attaching it to a glider which it is causing to accelerate, the effect is the same – it feels an equal and opposite pull at the two ends. From the point of view of the glider, however, the situation is very different; if there were a force equal and oppo­site to the pull from the rope, no acceleration would take place. The forces on the glider are not in equilibrium.

Great care has to be taken in applying the concept of an inertia force. When considering the stresses in the tow-rope it is acceptable to apply the pulling force at one end, and an equal and opposite force at the other end due to the air resistance plus the inertia of the object that it is causing to accelerate. When considering the motion of the aircraft and glider, however, no balancing inertia force should be included, or there would be no acceleration. A free-body diagram should be drawn as in Fig. 1.2.

This brings us to the much misunderstood third law of Newton: to every action there is an equal and opposite reaction. If a book rests on a table then the table produces a reaction force that is equal and opposite to the weight force. However, be careful; the force which is accelerating the glider produces a reaction, but the reaction is not a force, but an acceleration of the glider.

In the case of the glider mentioned above, what would happen if the pilot of the towing aircraft suddenly opened the engine throttle? The pulling force on the tow-rope would increase, but at first the aerodynamic resistance on the glider would not change. The forces would therefore no longer be in equilib­rium. The air resistance force is still there of course, so some of the pull on the tow-rope must go into overcoming it, but the remainder of the force will cause the glider to accelerate as shown in Fig. 1.2 (overleaf), which is called a free – body diagram.

This brings us to Newton’s second law, which says in effect that if the forces are not in balance, then the acceleration will be proportional to force and inversely proportional to the mass of the object:

a = F/m

,________ ^

Pull applied Aerodynamic

by towing resistance

aircraft force

1000 N

Fig 1.2 Forces not in equilibrium

where a is the acceleration, m is the mass of the body, and F is the force. This relationship is more familiarly written as:

F = m X a

If two tug-of-war teams pulling on a rope are well matched, there may for a while be no movement, just a lot of shouting and puffing! Both teams are exerting the same amount of force on the two ends of the rope. The forces are therefore in equilibrium and there is no change of momentum. There are,

however, other more common occurrences of forces in equilibrium. If you push down on an object at rest on a table, the table will resist the force with an equal and opposite force of reaction, so the forces are in equilibrium. Of course, if you press too hard, the table might break, in which case the forces will no longer be in equilibrium, and a sudden and unwanted acceleration will occur.

As another example, consider a glider being towed behind a small aircraft as in Fig. 1.1. If the aircraft and glider are flying straight and level at constant speed, then the pulling force exerted by the aircraft on the tow-rope must be exactly balanced by an equal and opposite aerodynamic resistance or drag force acting on the glider. The forces are in equilibrium.

Some people find it hard to believe that these forces really are exactly equal. Surely, they say, the aircraft must be pulling forward just a bit harder than the glider is pulling backwards; otherwise, what makes them go forward? Well, what makes them go forward is the fact that they are going forward, and the law says that they will continue to do so unless there is something to alter that state of affairs. If the forces are balanced then there is nothing to alter that state of equilibrium, and the aircraft and glider will keep moving at a constant speed.

The mass of an object can be loosely described as the quantity of matter in it. The greater the mass of an object, the greater will be the force required to start it moving from rest or to change its speed if it is already moving.

Mass is measured in units of kilograms (kg) in the SI metric system or pounds (lb) in the Imperial and Federal systems. Unfortunately, the same names are commonly used for the units of weight (which is a force), and this causes a great deal of confusion, as will be explained a little later under the heading Units. In this book, we will always use kilograms for mass, and newtons for weight.

Momentum

The quantity that decides the difficulty in stopping a body is its momentum, which is the product of its mass and the velocity of movement.

A body having a 20 kg mass moving at 2 m/s has a momentum of 40 kg m/s, and so does a body having a 10 kg mass moving at 4 m/s. The first has the greater mass, the second the greater velocity, but both are equally difficult to stop. A car has a larger mass than a bullet, but a relatively low velocity. A bullet has a much lower mass, but a relatively high velocity. Both are difficult to stop, and both can do considerable damage to anything that tries to stop them quickly.

To change the momentum of a body or even a mass of air, it is necessary to apply a force. Force = Rate of change of momentum.

Flying and mechanics

The flight and manoeuvres of an aeroplane provide glorious examples of the principles of mechanics. However, this is not a book on mechanics. It is about flying, and is an attempt to explain the flight of an aeroplane in a simple and interesting way; the mechanics are only brought in as an aid to understanding. In the opening chapter I shall try to sum up some of the principles with which we are most concerned in flying.

Force, and the first law of motion

An important principle of mechanics is that any object that is at rest will stay at rest unless acted upon by some force, and any object that is moving will continue moving at a steady speed unless acted upon by a force. This state­ment is in effect a simple statement of what is known as Newton’s First Taw of Motion.

There are two types of forces that can act on a body. They are:

(1) externally applied mechanical forces such as a simple push or pull

(2) the so-called body forces such as those caused by the attraction of gravity and electromagnetic and electrostatic fields.

External forces relevant to the mechanics of flight include the thrust produced by a jet engine or a propeller, and the drag resistance produced by movement through the air. A less obvious external force is that of reaction. A simple example of a reactive force is that which occurs when an object is placed on a fixed surface. The table produces an upward reactive force that exactly balances

the weight. The only body force that is of interest in the mechanics of flight is the force due to the attraction of gravity, which we know simply as the weight of the object.

Forces (of whatever type) are measured in the units of newtons (N) in the metric SI system or pounds force (lbf) in the Imperial or Federal systems. In this book, both sets of units are used in the examples and questions.

We are grateful to the following for permission to reproduce copyright material:

Figures IB, 2G, 2E, 8D courtesy of the Lockheed Aircraft Corporation, USA; Figures 2B, 2C, ЗА, 3B, 6B, 11B, 12D courtesy of the former British Aircraft Corporation; Figures 3C, 6E, 9F, 13B courtesy of General Dynamics Corporation, USA; Figure 3D courtesy of Paul MacCready; Figures ЗЕ, 5H courtesy of the Grumman Corporation, USA; Figure 3F courtesy of Fiat Aviazione, Torino, Italy; Figures 4D, 13D, courtesy of the Bell Aerospace Division of Textron Inc., USA; Figure 4G courtesy of Beech Aircraft Corporation, USA; Figures 4H, 5B courtesy of Cessna Aircraft Company, USA; Figure 41 courtesy of the former Fairey Aviation Co. Ltd; Figures 5C, 8C courtesy of Flight; Figure 6A courtesy of Slingsby Sailplanes Ltd; Figure 6C (bottom) courtesy of Terry Shwetz, de Havilland, Canada; Figure 6F courtesy of Bell Helicopter Textron; Figure 6G courtesy of Nigel Cogger; Figures 7C, 13C courtesy of the Boeing Company; Figure 9H courtesy of SAAB, Sweden; Figure 91 courtesy of Piaggio, Genoa, Italy; Figure 11A courtesy of the Shell Petroleum Co. Ltd; Figure 11E courtesy of McDonnell Douglas Corporation, USA; Figure 12A courtesy of the Lockheed-California Company, USA; Figure 12B courtesy of British Aerospace Defence Ltd, Military Aircraft Division; Figure 12C courtesy of Avions Marcel Dassault, France; Figures 13A, 13E courtesy of NASA.

Quotation from The Stars in their Courses on p.391 (Sir James Jeans) reprinted courtesy of Cambridge University Press.