In normal flight, a light aircraft derives its forward motion from the thrust provided by the engine-driven propeller. If the aircraft is maintaining a constant height, direction and speed then the thrust force will balance the air’s resistance to the aircraft’s motion through it. The forward motion creates airflow past the wings and the dynamic pressure changes within this airflow create an upward-acting force or lift, which will balance the force due to gravity — weight — acting downward. Thus, in normal unaccelerated flight, the four basic forces acting on the aircraft are approximately in equilibrium. The pilot is able to change the direction and magnitude of these forces and thereby control the speed, flight path and performance of the aircraft.
- 1.1 Introduction
- 1.2 Vector quantities
- 1.3 Weight
- 1.4 Lift
- 1.5 Thrust
- 1.6 Air resistance to motion
- Things that are handy to know
When a well-trimmed aircraft is cruising (i.e. flying at a constant speed, and maintaining a constant heading and a constant altitude) in non-turbulent air, there are two sets, or couples, of basic forces acting on it. The two forces in each couple are equal and approximately opposite to each other otherwise the aircraft would not continue to fly straight and level at a constant speed; i.e. the aircraft is in a state of equilibrium where all forces balance each other out so there is no change in its motion.
The couple that acts vertically is the lift, generated by the energy of the airflow past the wings and acting upward, and the weight acting downward. So, being equal and approximately opposite, the lifting force being generated must exactly match the total weight of the aircraft.
The couple that acts horizontally is the thrust, generated by the engine-driven propeller, and the air resistance, caused by the friction and pressure of the airflow, or drag, trying to slow the moving aircraft. The thrust, acting forward along the flight path, exactly equals the drag. The thrust provides energy to the aircraft and the drag dissipates that same energy into the atmosphere. The forces are not all equal to each other. In fact, an aircraft in cruising flight might generate ten times more lift than thrust.
When all forces are in equilibrium a moving aircraft will tend to keep moving along the same flight path at the same speed — whether it is flying straight and level, descending or climbing — until an applied force or a displacement force changes that state of motion. For instance, if the pilot opens the engine throttle fully, and maintains level flight, the thrust force is initially greater than drag and the aircraft accelerates. However, as the speed of airflow over the aircraft increases, the air resistance also increases and the aircraft will soon reach the speed — its maximum — where the forces are again balanced.
This property of resisting any change in motion, or continuing in the same state of motion or state of rest, is inertia. The mass of a body is a measure of its inertia; i.e. its resistance to being accelerated by an applied force increases with mass. The unit of mass we will be using is the kilogram [kg]. Some older texts refer to the ‘slug’, which is the unit of mass in the old British gravitational system of measurement units and equals 32.174 lb. A slug accelerates by one foot per second per second when a force of one pound-force is applied to it.
Air also has mass and thus inertia, and will resist being pushed aside by the passage of an aircraft. That resistance will be felt both as drag and as pressure changes on the aircraft surfaces.
A moving aircraft has momentum, which is mass × velocity and is a measure of the effort needed to stop it moving. (Momentum and inertia are not synonymous). The same aircraft also has the energy of motion — kinetic energy — which is related to mass × velocity squared. Also, because it has climbed above the Earth’s surface, it has acquired additional gravitational potential energy which, in this case, is weight × height gained. Energy is discussed further in the section on conserving energy in the next module.
An aircraft in flight is ‘airborne’ and its velocity is relative to the surrounding air, not the Earth’s surface. (A ground-based observer sees the aircraft movement resulting from the sum of aircraft velocity and the ambient air velocity — horizontal motion [the wind] plus vertical motion [updrafts, downdrafts and wave action].) However, when the aircraft encounters a sudden change in the ambient air velocity — a gust — inertia comes into play and momentarily maintains the aircraft velocity relative to the Earth or, more correctly, relative to space. This momentarily changes airspeedand imparts other forces to the aircraft. (The fact that inertia over-rides the physics of aerodynamics is sometimes a cause of confusion.) A more massive (heavier) aircraft has more inertia than a less massive (lighter) one, so is more resistant to random displacement forces — wind shear and turbulence. Ultralights — whose mass is less than 750 kg — are all regarded as ‘low inertia’ aircraft and particularly affected by acceleration loads produced by turbulence.
If the aircraft’s control system allows, it can rotate about each of three axes — longitudinal, lateraland normal (or vertical) — and move forward along the longitudinal axis. An aerobatic aircraft can also move backward along the longitudinal axis — in a tail slide. Rotation about the longitudinal axis isroll, about the lateral axis is pitch — as in a ship pitching in a heavy sea — and about the normal axis is yaw — again, like much in aviation, a nautical term.
Other movements can include a bodily movement along the lateral axis (sideslipping, slipping or skidding) or the normal axis (rising or sinking). Thus, an aircraft has six degrees of freedom of movement — three rotational and three translational. The three axes are relevant to the aircraft and each other (not to the horizon) so that when an aircraft is steeply banked its normal axis is closer to horizontal, rather than vertical to the Earth’s surface. The axes are orthogonal (at right angles to each other) and, by convention, all are represented as passing through the aircraft’s centre of gravity.
When manoeuvring, an aircraft may experience any combination of the rotational and translational movements; for example, it may be rolling, pitching, yawing, slipping and sinking all at the same time.
When an aircraft is in straight and level flight lift acts vertically upward with thrust and drag acting horizontally. In fact, lift acts perpendicular to both the flight path and the lateral axis of the aircraft, drag acts parallel to the flight path, and thrust usually acts parallel to the longitudinal axis of the aircraft.
So if you imagine an aircraft doing a loop, as in the diagram below, you can see that at one point, when it is going up, thrust will be acting vertically upward, drag vertically downward and lift acting horizontally towards the centre of the loop. At a point on the other side of the loop the thrust acts downward, drag upward and lift again horizontally. Weight, of course, always acts from the centre of mass of the aircraft towards the centre of mass of the Earth, so on the downside of the loop, weight and thrust are acting together and the aircraft will accelerate rapidly unless thrust is reduced.
You might ask yourself this: if the aircraft is using its maximum thrust when it starts the loop, how can it climb vertically when lift no longer counters weight, and there is no extra thrust available to also counter the drag plus the weight, which are both now acting downwards? The answer is extra momentum which enables the aircraft to accomplish a fast pull-up, and usually provided by the pilot, of a lower-powered aircraft, accelerating the aircraft in a shallow dive before beginning the manoeuvre.
The lift only matches the weight when the aircraft is flying straight and level. When the aircraft is in a steady descent or in a steady climb the lift is a bit less than the weight. We will explore this in the climb/descent modules but just be aware that when the line of thrust is inclined above the horizon the thrust will have a vertical component; i.e. it will provide a lifting force. When the aircraft is turning in the horizontal plane or in the vertical plane as in the loop, or anywhere in between, the lift is greater than the weight. In high-performance military aircraft it can be seven or eight times greater, because the lift provides the centripetal force to make the turn.
Note: it’s not always true that lift and drag act relative to the flight path. Imagine an aircraft flying straight and level, which encounters a substantial atmospheric updraught. Due to inertia the aircraft will, for the first milliseconds anyway, maintain its flight path relative to the Earth. During that time the ‘effective airflow’ passing by the wings will no longer be directly aligned with the flight path but will have acquired a vertical component. The lift will now act at 90° to this new ‘effective airflow’ rather than the actual flight path, and have a significant effect. Also, the wing itself modifies the effective airflow so just for now, until we look at aerofoils and wings, it is simpler to ignore the ‘effective airflow’ and other concepts and stay with the flight path.
Vector quantities have both magnitude and direction. Velocity is a vector quantity having both a magnitude (the airspeed) and a 3-dimensional direction. A force has both a direction in which it is pushing or pulling and a magnitude (in newtons [N]), thus it is a vector quantity. Momentum, having mass and velocity, is also a vector quantity, but inertia is not.
Ignoring weight and friction for now; when only one force is applied to a stationary object, the object will accelerate in the same direction as the force applied. Acceleration is the rate of change of velocity, the change being either in speed or three-dimensional direction, or both.
If an aircraft accelerates in a straight line from an airspeed of 25 metres/second [m/s] to 75 m/s in 10 seconds then the average change in airspeed per second is 75 –25 / 10 = 5 m/s, thus the acceleration is 5 metres per second per second [5 m/s²].
The common usage term ‘deceleration’, referring to a reduction in linear speed only, is generally not used in physics as, in that science, ‘acceleration’ has both positive and negative connotations.
When more than one force is being applied to an object there will be a resultant force, probably imparting an initial acceleration until all forces are again balanced at a new velocity. It is common practice to estimate resultant forces non-mathematically by drawing scaled, arrowed lines to represent each vector quantity, producing the resultant of two vector quantities in a vector parallelogram. The lengths of the lines represent the magnitude of each force and the placements indicate the application points and directions. The diagram is an exaggerated representation of an unpowered aircraft in a constant rate descent, showing that the lift/drag resultant is equal and opposite to the weight vector. In the diagram the resultant shown is the net aggregation of the aerodynamic forces generated by the wings, and it is conventional to then resolve that into its lift and drag components. We will be looking at aerodynamic forces in later modules.
There is a common, if not universal, tendency to equate the mass of a body with its weight. This is not surprising, as both are usually expressed using the same unit — kilograms [kg]. You need to appreciate that weight is a ‘body force‘, the product of the body mass and the acceleration due to gravity.
The force due to gravity — or weight — of an aircraft on the ground or in flight is expressed as W = m × g, where — somewhat confusingly — m is the symbol for mass (rather than metres) and g is a gravity constant applied to objects on, or near, the Earth’s surface. That constant is not a force but an acceleration of 9.806 m/s² — also known as the acceleration of free fall.
|In coming calculations we will use the performance of an early version of the Australian designed Jabiru aircraft as representative of the general aviation/ultralight four cylinder, two-seat, fixed-pitch and fixed-undercarriage recreational aircraft. All forces, including weight, are measured in newtons. The maximum allowed take-off weight [MTOW], of the Jabiru of mass 430 kg sitting on the runway is m = 430 × g= 9.806 = 4216 N. The Jabiru with an 80 kg pilot on board and with half the maximum fuel load would have a mass of only 340 kg which makes some difference to performance. We shall explore this in other modules. To simplify calculations, we will load full fuel, plus a lightweight passenger, into the Jabiru giving it a loaded mass of 400 kg and use g=10 m/s², thus weight = 4000 N.|
The position of the centre of mass or centre of gravity [cg] within the aircraft will vary according to the seating of passengers and stowage of luggage. The knowledge of the total mass of the loaded aircraft and the cg position — the weight and balance — is very important, as we will see in the ‘Weight and balance‘ module. Weight is always presumed to act from the cg position to the centre of the Earth. We will also see, in the ‘Altitude and altimeters‘ module, that atmospheric conditions affect aircraft performance, and subsequently the appropriate MTOW.
Vector quantities are sometimes very easy to calculate; for example if a Jabiru, weighing 4000 N, is cruising straight and level, then the lift force must be 4000 N pushing vertically upward.
When an aircraft is cruising in straight and level flight, at low altitudes, the wings are set at a small angle — 3 to 5° — to the ‘flight path’ (or the ‘line of flight’ or the ‘effective airflow’ or the ‘free stream airflow’ or the ‘relative wind’, all of which mean much the same thing in cruising flight in a non-turbulent atmosphere). The net sum of the aerodynamic reaction on the wing is a resultant force directed upwards and backwards. Aerodynamicists have found it convenient to resolve that resultant force into just two components: that part acting backward along the flight path is the wing drag, and that acting perpendicular to the flight path is the lift. The amount of lift, and drag, generated by the wings depends chiefly on:
(a) the angle at which the wing meet the airflow or flight path — the angle of attack
(b) the shape of the wing, particularly in cross-section — the aerofoil
(c) the density (i.e. mass per unit volume) of the air
(d) the speed of the free stream airflow; i.e. the airspeed
(e) and the wing plan-form surface area.
There is a standard equation to calculate lift from the wings, which will be often referred to in these notes:
The expression ½rV² (pronounced half roe vee squared) represents the dynamic pressure of the airflow in newtons per square metre [N/m²].(Please note — if a ‘Symbolic’ font is not available, your browser will not display the Greek letter rho, the accepted symbol for air density, and may display r or ? instead.) The dynamic pressure expression, ½rV², is very similar to the kinetic energy expression ½mv², where m = mass. Air density is mass per unit volume; i.e. kg/m³, so the dynamic pressure of the airflow is the kinetic energy per unit volume.
The values in the expression are:
- r (the Greek letter rho) is the density of the air, item (c), in kg/m³
- V² is the airspeed, item (d), in m/s
- S is the wing area, item (e), in m²
- CL (C sub L) is a dimensionless quantity — the lift coefficient — which relates mostly to item (a), but also to item (b).
In normal operations for very light aircraft, and when there are no high lift devices incorporated in the wing structure, CL usually has a value between 0.1 and 1.5. It can be regarded as the ratio of the conversion of dynamic pressure into lift, by the wing, at varying angles of attack.
Item (a) above, the angle at which the wings meet the flight path — more properly termed the geometric angle of attack — is near 16° at minimum controllable airspeed and around 2 to 5° when cruising at low altitudes; less at higher speeds, greater at higher altitudes. We will cover the close relationship between CL, angle of attack (aoa or alpha) and airspeed in the aerofoils and wings module.
The diagram shows a typical CL vs angle of attack curve for a light aircraft not equipped with flaps or high-lift devices. From it you can read the CL value for each aoa, for example at 10° the ratio for conversion of dynamic pressure to lift is about 1.0.
Note that CL still has a positive value (about 0.1) even when the aoa is –1°. This is because of the higher camber in the upper half of the wing; some highly cambered wings may still have a positive CLvalue when the aoa is as low as –4°. A light non-aerobatic aircraft pilot would not normally utilise negative aoa because it involves operating the aircraft in a high-speed descent, but we will discuss this further in the ‘Flight at excessive speed‘ module.
Also note that the lift coefficient increases in direct relationship to the increase in angle of attack, until near 16° aoa where CL reaches its maximum and then decreases rapidly as aoa passes that critical angle. A rule of thumb for light aircraft with simple wings is that each 1° aoa change — starting from –2° and continuing to about 14° — equates to a 0.1 CL change.
Also, it is not just the wings that produce lift. Parts of a well-designed fuselage — the aircraft body — can also produce lift and the vertical component of the thrust vector can supplement lift when that vector is angled upwards.
We can calculate CL for the Jabiru cruising at an altitude of 6500 feet and an airspeed of 97 knots (50 m/s). The wing area is very close to 8 m²:
• lift = weight = 4000 N
• r = 1.0 kg/m³ (the approximate density of air at 6500 feet altitude)
• V² = 50 × 50= 2500 m/s
• S = 8 m²
Lift = CL × ½rV² × S
Dynamic pressure = ½ ×1.0 × 2500 = 1250 N/m²
So, 4000 = CL × 1250 × 8, — thus — CL = 0.4.
Now what happens if the pilot decides to decrease airspeed to 88 knots (45 m/s) while maintaining the same altitude? First, the pilot decreases power to give 88 knots then adjusts control pressure to maintain the same altitude — but look at the changes in the lift equation.
• lift still equals weight = 4000 N
• air density still = 1.0 kg/m³
• V² changes and is now = 45 × 45 = 2025 m/s
• S can’t change = 8 m²
Dynamic pressure = ½×1.0 × 2025 = 1012.5 N/m²
So, 4000 = CL × 1012.5 × 8, — and — CL = 0.5 approximately.
So, the result of decreasing airspeed, while maintaining straight and level flight, is an increase in the lift coefficient from 0.4 to 0.5. That has two possible contributors — the shape of the aerofoil and the angle of attack; items (a) and (b) above. Because the pilot can’t change the aerofoil shape (unless flaps are extended, which we discuss in the ‘Aerofoils and wings‘ module) the angle of attack must have changed. How? By the pilot adjusting control pressure to apply an aerodynamic force to the aircraft’s tailplane (or some other control surface), which has the effect of rotating the aircraft just a degree or so about its lateral axis. Once the pilot has achieved the desired aoa, as indicated by the new airspeed (which will be explained in the ‘Airspeed‘ module), the tailplane trim control is adjusted and the aircraft will then maintain that aoa.
It may be appropriate to slip in another slight complication at this point. Lift, like weight, may be taken as acting through a central point — the centre of pressure [cp]. The position of the cp changes with aoa and this movement has a significant effect — it causes the nose of the aircraft to pitch up or down. So, the lift and weight are usually not in equilibrium and the rotational moment must be counteracted by aerodynamic forces produced by the horizontal stabiliser. Other tailplane surfaces also produce aerodynamic forces for trim and control, so to maintain an aircraft in straight and level flight — apart from the four forces mentioned — there will always be another force, or forces, generated by the fixed tailplane of most aeroplanes or its movable surfaces. We will look at this in the ‘Stability‘ module.
In the Jabiru the engine supplies torque directly to the propeller shaft and the propeller converts the torque to thrust; we will amplify how this is accomplished in the ‘Engine and propeller performance‘ module. The propeller pushes backwards a tube of air with the same diameter as itself; i.e. it adds momentum to the tube of air where momentum = mass × velocity, and is also a vector quantity. Increasing the speed also imparts kinetic energy to the air. This tube of accelerated, energised air is the slipstream.
Considering Isaac Newton’s third law you expect an equal and opposite reaction to the action of adding momentum. This reaction is the application of forward momentum to the propeller, which pulls the rest of the aircraft along behind if the engine/propeller installation is a ‘tractor’ type, or pushes it if the engine/propeller installation is a ‘pusher’ type.
We need to clarify the line of thrust. This line is extended forward through the propeller shaft, which is usually aligned with the longitudinal axis of the aircraft, but not always. For instance, the engine and propeller installation in the carrier-borne Hellcat, of Second World War fame, was vertically offset so that the thrust vector was 3° down; consequently, the aircraft flew with a rather jaunty tail-down attitude. I think the reason was that the thrust line then extended back over the cg, making the aircraft more stable at the very low speeds required for deck landing. (You can read a little about the deck landing techniques of those days — my youth — in this magazine article).
The relationship of the longitudinal axis with the horizontal flight path — the aircraft’s attitude — varies with the speed of the aircraft. At maximum allowable airspeed, or Vne, the longitudinal axis might coincide exactly with the flight path but as speed decreases, the axis starts to angle up and could be inclined 15° to the flight path at minimum controllable airspeed. Because the line of drag is always aligned with the flight path, then the thrust vector does not directly oppose the drag vector.
The diagram, slightly exaggerated for clarity, shows the relationship between angle of attack and line of thrust to the flight path, for an aircraft maintaining level flight at a very slow speed. The flight path is horizontal so the drag vector will also be horizontal; i.e. aligned with the relative airflow. The line of thrust is aligned with the longitudinal axis, so the angle between the thrust line and the horizontal flight path is the aircraft attitude — in this case, its attitude in pitch. The wing chord line is extended so that the geometric angle of attack can be seen — the angle between the chord line and the flight path. The lift and weight vectors would both be at right angles to the flight path.
You might notice from the diagram that the thrust vector will have quite a substantial vertical component, so that part of the thrust is supplementing lift. Thus we have just destroyed our previous assertion that if an aircraft is flying straight and level, lift must always equal weight. In this instance, the lift is less than weight and the (very small) shortfall is provided by the vertical component of thrust. So it is more correct to say that, if an aircraft is flying straight and level, lift plus the vertical component of thrust must equal weight.
|We can estimate the thrust delivered by the Jabiru’s propeller, cruising at a speed of 97 knots at 6500 feet:
From the pilot’s operating handbook we find that the engine is rated at 80 hp × 746 = 60 kilowatts [kW], and cruise power for 97 knots is 65%, or 39 kW. The fixed pitch propeller is about 70% efficient at cruising speed so the effective power from the propeller is 27 kW.
One watt is the work accomplished by a force of one newton moving an object one metre in one second. The aircraft is moving at 97 knots, or 50 m/s (knots × 0.514 = m/s), so, 27 000 W divided by 50 [m/s] equals 540 N and that is the thrust being provided, which also means the drag is 540 N. Compare that to the weight and lift of 4000 N, stated in section 1.3, and you see that the lift force — silently and efficiently generated simply by the angle of the wings and the velocity of the airflow — is 7.4 times the thrust force — noisily and inefficiently generated by the engine burning expensive fuel. That is being a bit unfair because the wings are really converting much of the thrust into a lifting force.
Put another way the lift to drag ratio when cruising at 97 knots is 7.4:1. We will examine lift/drag ratios in the ‘Aerofoils and wings’ module.
The slipstream speed of an aircraft at cruising speed might be 20% greater than the aircraft speed, during a climb it could be 50% faster, and when the aircraft is maintaining height near its minimum controllable speed, slipstream velocity might be 100% greater. Some aircraft are designed so that the slipstream over the centre section of the wings increases V and thus lift, and the combination of the vertical component of thrust plus the slipstream effect means that possibly 25% of the thrust output is contributing lift when flying in a tail-down attitude.
Drag is the resistance of the air to an aircraft pushing through it. The resistance depends on:
(a) the streamlining of the aircraft body
(b) i. the excrescences attached to the airframe
ii. turbulence at the junctions of structural components
iii. the cooling airflow around the engine
(c) the roughness of the surface skin
(d) the ‘wetted’ area; i.e. the amount of surface exposed to the airflow
(e) the density of the air
(f) the speed of the airflow
(g) the angle of attack.
These components of drag are classified in several ways and we will look at them in the ‘Aerofoils and wings‘ module. Part of the air resistance, theinduced drag, is a consequence of item (g) the angle of attack. Induced drag is very high, maybe 70% of the total, at the high aoa of the minimum controllable airspeed, but induced drag decreases as speed increases, being possibly less than 10% of the total at full throttle speed.
However the balance of the air resistance, known as parasite drag, increases as speed increases until the total air resistance equals the maximum thrust that can be produced.
You can see from the diagram that parasite drag is directly proportional to dynamic pressure [½rV²] while induced drag is inversely proportional to it.
Thus in normal straight and level flight, air resistance is high at both minimum and maximum airspeeds and lowest at some mid-range speed where — as resistance is at a minimum — the thrust required to maintain constant height will also be at a minimum; consequently, that is the speed — Vbr — which provides maximum range. If drag is at a minimum, then the lift/drag ratio will be at a maximum; consequently, this is very close to the best engine-off glide speed —Vbg.
Air density (and thus air resistance) decreases with increasing altitude. So, the parasite drag component for a given airspeed decreases with increasing altitude while the induced drag component increases, because the wing has to fly at a greater aoa to produce the lift required.
The standard expression for total aircraft drag is very similar to the lift equation:
where CD is the total drag coefficient and the ratio of total aircraft drag to dynamic pressure. CD increases as aoa increases.
The next module in this Flight Theory Guide discusses the forces involved in manoeuvring an aircraft. But first, read the notes below.
- The generic term aircraft covers a wide range of airborne vehicles: lighter-than-air (aerostats); e.g. airships and hot-air balloons, and heavier-than-air (aerodynes). The latter includes gliders, powered parachutes, weight-shift controlled trikes or microlights, rotorcraft, helicopters, unmanned aerial vehicles and aeroplanes of all types from high performance supersonic fighters, jump jets or jumbo jets, to those like our friendly single-engine Jabiru.
- However, for the purpose of these notes we use the term aircraft to refer only to a class of ‘general aviation’ and ‘sport and recreational aviation’ aeroplanes with piston-engines driving propellers; relatively simple wing configurations including parawings; minimum power-off controllable airspeeds possibly as low as 20 knots; and possibly a maximum cruise airspeed, for a really high-performance aeroplane in that class, of 250 knots. These aircraft generally have one or two engines rated at 20 hp to 400 hp. The general aviation aircraft have normal wing loadings (i.e. weight/wing area) between 7 and 24 pounds per square foot (35–120 kg/m²) and maximum lift/drag ratios of 9:1 to 12:1, while ultralights have wing loadings between 4 and 12 pounds per square foot (20–60 kg/m²) and aircraft lift/drag ratios of 5:1 to 12:1.
- A newton [N] is the force required to give a mass of one kilogram an acceleration of one m/s².
- Isaac Newton’s third law: If one body exerts a force on another there is an equal and opposite force — a reaction — exerted on the first body by the second.
- A knot is a speed of one nautical mile per hour, 0.5144 m/s or roughly 100 feet per minute. One nautical mile is the length, at the Earth’s surface, of one minute of arc of a great circle and currently accepted to equal 1852 metres or 6076.115 feet.
• Objects do not fall freely in the Earth’s atmosphere. The air resistance (drag) increases as both fall velocity and air density increase until a terminal velocity is reached — where the drag force and the weight (the force due to gravity) are balanced — and the object stops accelerating. If the fall continues the object will start to slow slightly because of increasing air density at lower altitude, which increases drag. A streamlined body will have a higher terminal velocity than a non-streamlined body, of the same mass, because of the lower drag.