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Groundschool — Theory of Flight
Aircraft manoeuvring forces
Revision 72 — page content was last changed 15 August 2012
|The performance of an aircraft in the hands of a competent pilot — at a given altitude — results from the sum of power, angle of attack, attitude and configuration. Power provides thrust and consequently contributes to acceleration, airspeed, lift, drag and radius of turn. The angle of attack dictates the dimensions of the lift force and the induced drag and contributes to airspeed; also the angle of attack is a significant contributor to the aircraft attitude. Attitude is the angle the aircraft longitudinal axis subtends above or below the horizon (usually called the 'pitch' which has another meaning associated with propellers) plus the angle of bank and the degree of slip. Attitude dictates the direction of the lift, thrust and drag vectors and, consequently, converts power into velocities and accelerations in the three planes. Configuration relates to the deployment of lift/drag changing devices.
Airspeed is dependent on power, angle of attack, configuration and attitude — under any given set of conditions — and attitude in flight is readily checked by reference to the horizon. The lift, thrust and drag forces produce manoeuvring loads on the aircraft structure, generally expressed in terms of 'g', that must be kept within defined limits. There is a fourth performance factor — energy management — which is an art that supplements attitude plus power plus height to produce maximum aircraft performance. The epitome of such an art is demonstrated by air-show pilots who produce extraordinary performances from otherwise relatively mundane aircraft.
(Equation #1.3) Power required for level flight [watts] = CD × ½rV³ × S (note V cubed).
The total drag curve can be converted into a 'power required' diagram — usually called the power curve — if you know the total drag at each airspeed between the minimum controllable speed and the maximum level flight speed. It is a different curve from that for total drag, because the power required is proportional to speed cubed rather than speed squared. This means that (ignoring the related CD change) if speed is doubled, drag is increased four-fold but power must be increased eight times — which indicates why increasing power output from, say, 75% power to full rated power, while holding level flight, doesn't provide a corresponding increase in airspeed.
The diagram above is a typical level-flight power curve for a light aircraft. The part of the curve to the left of the minimum power airspeed is known as the back of the power curve — where the slower you want to fly, the more power is needed, because of induced drag at a high angle of attack. The lowest possible speed for controlled flight is the stall speed, which is discussed in the 'Airspeed and properties of air' module. Two aerodynamic cruise speeds are indicated — the speed associated with minimum drag (the point on the curve where the drag force factor has the lowest value) and the speed associated with minimum power (the point on the curve where drag force × speed has the lowest value). To maintain level flight at speeds less than or greater than the minimum power airspeed, power must be increased. propeller. The propellers used in most light aircraft have a maximum efficiency factor, in the conversion of engine power to thrust power, of no more than 80%. (Thrust power = thrust × forward speed.) The pitch of the blades, the speed of rotation of the propeller and the forward speed of the aircraft all establish the angle of attack of the blades and the thrust delivered. The in-flight pitch of ultralight and light aircraft propeller blades is usually fixed (though many such types are adjustable on the ground) so that the maximum efficiency will occur at one combination of rpm and forward speed — this is usually in the mid-range between best rate of climb and the performance cruise airspeeds. Propeller blades are sometimes pitched to give the best efficiency near the best rate of climb speed (climb prop), or pitched for best efficiency at the performance cruise airspeed (cruise prop). The efficiency of all types of propellers falls off either side of the optimum; one with a too high pitch angle may have a very poor take-off performance, while one with a too low pitch may allow the engine to overspeed at any time.
With the advent of higher-powered four-stroke light engines, such as the Jabiru 3300, there has been a corresponding increase in the availability of more advanced light-weight propeller systems, providing maximum effective power utilisation during all stages of flight. For more information refer to the 'Engine and propeller performance' module.
engine performance, select a cruise altitude where the throttle is fully open and the engine is delivering 65% to 75% power.
A couple of points to note from the speed-power diagram above:
The region between the maximum thrust power curve and the power required (to maintain level flight) curve indicates the excess power available at various cruise speeds — this excess power is available for various manoeuvres if the throttle is fully opened. The simplest use would be a straight unaccelerated climb, in which case the maximum rate of climb would be achieved at the airspeed where the two curves are furthest apart. It can be seen that the best rate of climb speed is around the same airspeed as the minimum drag airspeed shown in the earlier powered required diagram.
The rate of climb will decrease at any speed either side of the best rate of climb speed because the power available for climb decreases. The rate of climb (metres/second) = excess power available (watts)/aircraft weight (N).
One thing to bear in mind is that we have assumed the aircraft's aerodynamic shape — its configuration is constant. However if the aircraft is fitted with flaps, high lift devices or spoilers the pilot is able to change its configuration and consequently its performance. Thus performance is dependent on power, plus attitude (pitch, bank, sideslip and aoa) plus configuration.
If an aircraft is maintained in a continuous full-throttle climb, at the best rate of climb airspeed, the rate of climb will be highest at sea-level; it will decrease with altitude, as engine power decreases. The aircraft will eventually arrive at an altitude where there is no excess power available for climb, then all the available power is needed to balance the drag in level flight and there will be only one airspeed at which level flight can be maintained. Below this airspeed the aircraft will stall. This altitude is the aircraft's absolute ceiling. However, unless trying for an altitude record, there is no point in attempting to climb to the absolute ceiling so the aircraft's service ceiling should appear in the aircraft's performance specification. The service ceiling is the altitude at which the rate of climb falls below 100 feet per minute; this is generally considered the minimum useful rate of climb.
This diagram of forces in a climb and the subsequent mathematical expressions, have been simplified, aligning the angle of climb with the line of thrust. In fact the line of thrust will usually be 4 to 10° greater than the climb angle. The climb angle (c) is the angle between the flight path and the horizontal plane.
The relationships in the triangle of forces shown are:
Lift = weight × cosine c
Thrust = drag + (weight × sine c)
In a constant climb the forces are again in equilibrium, but now thrust + lift = drag + weight.
Probably the most surprising thing about the triangle of forces in a straight climb is that lift is less than weight. For example, let's put the Jabiru into a 10° climb with weight = 4000 N. (There is an abridged trig. table at the end of this page.)
Then, Lift = W cos c = 4000 × 0.985 = 3940 N
It is power that provides a continuous rate of climb, but momentum may also be used to temporarily provide energy for climbing; see 'Conserving aircraft energy' below. It is evident from the above that in a steady climb, the rate of climb (and descent) is controlled with power, and the airspeed and angle of climb is controlled with the attitude and particularly the included angle of attack. This is somewhat of a simplification, as the pilot employs both power and attitude in unison to achieve a particular angle and rate of climb or descent.
The angle of attack in a climb is the pitch attitude minus the angle of climb being achieved plus the wing incidence.
A very important consideration, particularly when manoeuvring at low level at normal speeds, is that the steeper the climb angle the more thrust is required to counter weight. For example, if you pulled the Jabiru up into a 30° 'zoom' climb the thrust required = drag + weight × sine 30° (= 0.5) so the engine has to provide sufficient thrust to pull up half the weight plus overcome the increased drag due to the increased aoa in the climb. Clearly, this is not possible, so the airspeed will fall off very rapidly and will lead to a dangerous situation if the pilot is slow in getting the nose down to an achievable attitude. Never be tempted to indulge in zoom climbs — they are killers at low levels.
If the pilot pushed forward on the control column to a much steeper angle of descent, while maintaining the same throttle opening, the thrust plus weight resultant vector becomes greater, the aircraft accelerates with consequent increase in thrust power and the acceleration continues until the forces are again in equilibrium. Actually, it is difficult to hold a stable aircraft in such a fixed angle 'power dive' as the aircraft will want to climb — but an unstable aircraft might want to 'tuck under'; i.e. increase the angle of dive, even past the vertical. We discuss the need for stability in the 'Stability' module.
When the pilot closes the throttle completely, there is no thrust, the aircraft enters a gliding descent and the forces are then as shown in the diagram on the left. In the case of descent at a constant rate, the weight is exactly balanced by the resultant force of lift and drag.
From the dashed parallelogram of forces shown, it can be seen that the tangent of the angle of glide equals drag/lift. For example, assuming a glide angle of 10° (from the abridged trigonometrical table below, the tangent of 10° is 0.176), the ratio of drag/lift in this case is then 1:5.7 (1/0.176 =5.7).
Conversely, we can say that the angle of glide depends on the ratio of lift/drag [L/D]. The higher that ratio is, then the smaller the glide angle and consequently the further the aircraft will glide from a given height.
For example, to calculate the optimum glide angle for an aircraft with a L/D of 12:1.
Drag/lift equals 1/12, thus tangent = 0.08 and, from the trigonometrical table, the glide angle = 5°.
Although there is no thrust associated with the power-off glide, the power required curve is still relevant. The minimum drag airspeed shown in that diagram is roughly the airspeed for best glide angle and the speed for minimum power is roughly the airspeed for minimum rate of sink in a glide. This is examined further in the 'Airspeed and the properties of air' module.
It may be useful to know that in a glide, lift = weight × cosine glide angle and drag = weight × sine glide angle. There is further information on glide angles and airspeeds in the lift/drag ratio section of module 4.
The acceleration, as you know from driving a car through an S curve, depends on the speed at which the vehicle is moving around the arc and the radius of the turn. Slow speed and a sweeping turn involves very little acceleration. But high speed and holding a small radius involves high acceleration, with consequent high radial g or centripetal force and difficulty in holding the turn. Even when an aircraft enters a straight climb from cruising flight, there is a short transition period between the straight and level path and the straight and climbing path, during which the aircraft must follow a curved path — a partial turn in the vertical plane.
An aircraft turning at a constant rate turn is continuously accelerating towards the centre of the turn. The acceleration towards the centre of the turn is V²/r m/s². The centripetal force required to produce the turn is m × V²/r newtons, where m is the aircraft mass in kilograms and r is the turn radius in metres. Note this is aircraft mass, not weight.
In a level turn, the vertical component of the lift (Lvc) balances the aircraft weight and the horizontal component of lift (Lhc) provides the centripetal force.
(Note: in a right-angle triangle the tangent of an angle is the ratio of the side opposite the angle to that adjacent to the angle. Thus, the tangent of the bank angle is equal to the centripetal force [cf] divided by the weight — or tan ø = cf/W. Or, it can be expressed as tan ø = V²/gr . In the diagram, I have created a parallelogram of forces so that all horizontal lines represent the centripetal force or Lhc and all vertical lines represent the weight or Lvc.)
The centripetal force of 4000 N is provided by the horizontal component of the lift force produced by the wings when banked at an angle from the horizontal. The correct bank angle depends on the airspeed and radius; think about a motorbike taking a curve in the road. During the level turn, the lift force must also have a vertical component to balance the aircraft's weight, in this case it is also 4000 N. But the total required force is not the sum of 4000 N + 4000 N = 8000 N; it is less and we have to find the one — and only one — bank angle where Lvc is equal to the weight and Lhc is equal to the required centripetal force.
What then will be the correct bank angle (ø) for a balanced turn? Well, we can calculate it easily if you have access to trigonometrical tables. If you haven't then refer to the abridged version below.
acceleration caused by the force of gravity. When an aircraft is airborne maintaining a constant velocity and altitude — the total lift produced equals the aircraft's weight and that lift force is expressed as being equivalent to a '1g' load. Similarly, when the aircraft is parked on the ground, the load on the aircraft wheels (its weight) is a 1g load.
Any time an aircraft's velocity is changed, there are positive or negative acceleration forces applied to the aircraft and felt by its occupants. The resultant manoeuvring load is normally expressed in terms of g load, which is the ratio of all the aerodynamic forces experienced during the acceleration to the aerodynamic forces existing at the normal 1g level flight state.
You will come across terms such as '2g turn' or 'pulling 2g'. What is being implied is that during a particular manoeuvre the lift force is doubled and a radial acceleration is applied to the airframe — for the Jabiru a 2g load = 400 kg × 20 m/s² = 8000 N. The occupants will also feel they weigh twice as much. This is centripetal force and 'radial g'; it applies whether the aircraft is changing direction in the horizontal plane, the vertical plane or anything between.
You may also come across mention of 'negative g'. It is conventional to describe g as positive when the lift produced is in the normal direction relative to the aircraft. When the lift direction is reversed, it is described as negative g. Reduced g and negative g can occur momentarily in turbulence. An aircraft experiencing a sustained 1g negative loading is flying in equilibrium, but upside down. It is also possible for some high-powered aerobatic aircraft to fly an 'outside' loop; i.e. the pilot's head is on the outside of the loop rather than the inside, and the aircraft (and its very uncomfortable occupants), will be experiencing various negative g values all the way around the manoeuvre.
It can be a little misleading when using terms such as 2g. For instance, let's say that a lightly loaded Jabiru has a mass of 340 kg, and if you again do the preceding centripetal force calculation in a 45° banked turn using 340 kg mass you will find that the centripetal acceleration is 10 m/s², centripetal force is 3400 N, weight is 3400 N and total lift = 4800 N. The total lifting force is 15% less than in the 400 kg mass calculation but it is still a 1.41g turn; i.e. the ratio 4800/3400 = 1.41.
Rather than thinking in terms of ratios, it may be appropriate to consider the actual loads being applied to the aircraft structures. The norm is to use the lift load produced by the wing as a primary structural load reference. In the 400 kg mass calculation the load produced is 5660/8 = 707 N/m², compared to the 500 N/m² load in normal cruise. However, even if the total weight of the aircraft changes, the forces experienced individual structural items — the engine mountings for example — will vary according to the g force produced by the wings.
The radius of turn = V²/g tan ø metres. For a level turn, the slowest possible speed and the steepest possible bank angle will provide both the smallest radius and the fastest rate of turn. However there are several limitations:
(For more information on turn physics see 'Turning back — procedure and dynamics'.)
If you consider an aerobatic aircraft weighing 10 000 N and making a turn in the vertical plane —such as a loop — and imagine that the centripetal acceleration is 2g; what will be the load factor at various points of the turn? Actually, the centripetal acceleration varies all the way around because the airspeed and radius must vary. For simplicity we will ignore this and say that it is 2g all around. If the acceleration is 2g then the centripetal force must be 20 000 N all the way around.
A turn in the vertical plane differs from a horizontal turn in that, at both sides of the loop, the wings do not have to provide any lift component to counter weight, only lift for the centripetal force — so the total load at those points is 20 000 N or 2g. At the top, with the aircraft inverted, the weight is directed towards the centre of the turn and provides 10 000 N of the centripetal force while the wings need to provide only 10 000 N. Thus, the total load is only 10 000 N or 1g, whereas at the bottom of a continuing turn the wings provide all the centripetal force plus counter the weight — so the load there is 30 000 N or 3g.
This highlights an important point: when acceleration loads are reinforced by the acceleration of gravity, the total load can be very high.
If you have difficulty in conceiving the centripetal force loading on the wings, think about it in terms of the reaction momentum, centrifugal force which, from within the aircraft, is seen as a force pushing the vehicle and its occupants to the outside of the turn and the lift (centripetal force) is counteracting it. Centrifugal force is always expressed as g multiples. gross wing area [S]. (There are some complications when national regulations specify a maximum allowable weight for an aircraft category that is lower than the design weight of a particular aircraft type; see the 'Weight and balance' module.) Aircraft with low W/S have lower stall speeds than aircraft with higher W/S — so consequently have shorter take-off and landing distances. High W/S aircraft are less affected by atmospheric turbulence. W/S is expressed in pounds per square foot [psf] or kilograms per square metre [kg/m²].
The second wing loading connotation is as the operating wing loading; if the aircraft takes off at a gross weight lower than the designer's maximum, then the operating wing loading — in level unaccelerated flight — will also be lower than the design W/S, as will its stall speed.
The third is the load applied by the pilot in manoeuvring flight. As we saw above, pulling 2g in a steep turn will produce a manoeuvring wing loading that is double the operating wing loading. So, if a pilot takes off in an overloaded aircraft (i.e. the aircraft's weight exceeds the design MTOW) and conducts a 2g steep turn, then that manoeuvring wing loading will be greater than the designer's expectations.
type certification the design of a general or recreational aviation factory-built aircraft must conform to certain airworthiness standards among which the in-flight manoeuvring loads and the loads induced by atmospheric turbulence, that the structure must be able to withstand, are specified. The turbulence loads are called the gust-induced loads. The U.S. Federal airworthiness standards FAR Part 23 are the recognised world standards for light aircraft certification and the following are extracts [emphasis added]:
"... limit loads ... [are] the maximum loads to be expected in service (i.e. the highest load expected in normal operations) and ultimate loads ... [are] limit loads multiplied by [a safety factor of 1.5 — 1.75 for sailplanes]."
"The structure must be able to support limit loads without detrimental, permanent deformation. At any load up to limit loads, the deformation may not interfere with safe operation ... The structure must be able to support ultimate loads without failure for at least three seconds ..."
Three seconds is not much time, so any inflight excursion above the ultimate load will probably result in structural failure.
The 'utility' category (which includes training aircraft with spin certification) limit loads are +4.4g and –2.2g while the 'acrobatic' category (i.e. aircraft designed to perform aerobatics) limit loads are +6g and –3g. Sailplanes and powered sailplanes are generally certificated in the utility or acrobatic categories of the European Joint Airworthiness Requirements JAR-22, which is the world standard for sailplanes; aerobatic sailplanes have limit loads of +7g and -5g.
The 'flight load factor' calculation is defined as the component of the aerodynamic force acting normal (i.e. at right angles) to the aircraft's longitudinal axis, divided by the aircraft weight. A positive load factor is one in which the force acts upward, with respect to the aircraft; a negative load factor acts downward. The inflight load factor is a function of wing loading, dynamic pressure and the aoa, i.e. lift coefficient, but see the flight envelope.
It should not be thought that aircraft structures are significantly weaker in the negative g direction. The normal level flight load is +1g so with a +3.8g limit then an additional positive 2.8g acceleration can be applied while with a –1.5g limit an additional negative 2.5g acceleration can be applied.
The manufacturer of a particular aircraft type may opt to have the aircraft certificated within more than one category, in which case there will be different maximum take-off weights and centre of gravity limitations for each operational category. See weight/cg position limitations.
The sustainable load factors only relate to a new factory-built aircraft. The repairs, ageing and poor maintenance that any aircraft has been exposed to since leaving the factory may decrease the strength of individual structural members considerably. Read the current airworthiness notices issued by the RA-Aus technical manager.
Because it is the accumulation of the work done to raise the aircraft 6500 feet, the potential energy is 16 times the kinetic energy, and is obviously an asset that you don't want to dissipate. It is equivalent to 2% of your fuel.
It is always wise to balance a shortage of potential energy with an excess of kinetic energy, and vice versa. For example, if you don't have much height then have some extra speed up your sleeve for manoeuvring or to provide extra time for action in case of engine or wind shear problems. Or if kinetic energy is low (because of flying at lower speeds than normal) make sure you have ample height or, if approaching to land, hold height for as long as possible. The only time to be 'low and slow' is when you are about to touch down.
However, during take-off it is not possible to have an excess of either potential or kinetic energy; thus, take-off is the most critical phase of flight, closely followed by the go-around following an aborted landing approach. Ensure that a safe climb speed is achieved as quickly as possible after becoming airborne — or commencing a go-around — and before the climb-out is actually commenced; see take-off procedure. Ground speed represents the horizontal component of that velocity, and rate of climb/sink represents the vertical component.
Kinetic energy, gravitational potential energy and energy conservation are complex subjects. If you wish to go further, google the search terms 'kinetic energy' and 'reference frame'.
If we recalculate the preceding figures — doubling the initial (100 m/s) and final velocities (90 m/s) — the height gained will increase fourfold to 96 metres, or about 18 feet per knot. Conversely, if we halve the initial velocity to about 50 knots, the height gained per knot converted is halved, to about 4 feet. Note that as mass appears in both the kinetic energy and the weight expressions, it can be ignored; thus the figures are the same for any mass. Sometimes momentum (mass × velocity) is confused with inertia (a particular quality of mass).
You will come across the expression 'low inertia / high drag' applied to some recreational light aircraft. This means that although all recreational light aircraft are low-inertia aircraft, compared to other recreational light aircraft this minimum aircraft has a relatively low inertial mass combined with a relatively high parasite drag profile; thus if the thrust is reduced or fails, the drag reduces the airspeed very rapidly. This is exacerbated if the aircraft is climbing. An aluminium tube and sailcloth aircraft at one end of the spectrum may be termed 'low momentum' or 'draggy', while an epoxy composite aircraft at the other end may be termed 'slippery'; some are very slippery indeed. The standing world speed record for an aircraft under 300 kg is 213 miles per hour; that amateur-designed and amateur-built aircraft was powered by only a 65 hp two-stroke Rotax. The handling characteristics for a low inertia/low drag aircraft differ considerably from those of a low inertia/high drag (low momentum) aircraft.
The next module in this Flight Theory Guide examines aspects of airspeed and air properties, but you may first wish to read the notes below.
|Abridged trigonometrical table
Relationship between an angle within a right angle triangle and the sides:
Tangent of angle=opposite side/adjacent
Sine of angle=opposite/hypotenuse
Cosine of angle=adjacent/hypotenuse
Things that are handy to know
Stuff you don't need to know
Groundschool — Flight Theory Guide modules
| Flight theory contents | 1. Basic forces | (1b. Manoeuvring forces) |
| 2. Airspeed & air properties | 3. Altitude & altimeters | 4. Aerofoils & wings |
| 5. Engine & propeller performance | 6. Tailplane surfaces | 7. Stability | 8. Control |
| 9. Weight & balance | 10. Weight shift control | 11. Take-off considerations |
| 12. Circuit & landing | 13. Flight at excessive speed | 14. Safety: control loss in turns |
| Operations at non-controlled airfields | Safety during take-off & landing |
Copyright © 2000—2012 John Brandon (contact information)
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