Let take a look at all the forces and torques that act in a kite and how they are acting:
- Wind Generated Forces
- Gravity Force
- Line Tension
- Balance of forces and torques
- Forces and Torques on a Kite – Summary
- Lift: This is the vertical force upward perpendicular to the wind that provide lift to the kite. The Lift of the kite is proportional to the Lift Coefficient (Cl) of the airfoil which varies dependent on the angle of attack (AoA) of the kite. From 0 to around 20 degrees (most dominant AoA range for kiting), Cl increases as AoA increases. After the peak in around 15 to 20 degrees, Cl will decrease. More info about Lift can be found at http://www.grc.nasa.gov/WWW/Wright/airplane/lifteq.html.
- Drag: (or profile Drag) is the horizontal force in the same direction as the wind that drags the kite rearward. Similar to Lift, the Drag force is proportional to the Drag Coefficient (Cd) which varies dependent on the AoA. Cd normally increases as AoA increases. More info about drag can be found athttp://www.grc.nasa.gov/WWW/Wright/airplane/drageq.html.
- Lift/Drag (L/D) ratio: The L/D ratio shows how Lift changes as compare to Drag. The faster Lift increases compare to Drag or the slower Lift decreases compare to Drag, the higher the L/D ratio. From 0 to around 20 degrees (the most dominant AoA range for kiting), the L/D ratio is inverse proportional to AoA: low AoA means high L/D and high AoA means low L/D.
- Induced Drag: Induced Drag is the drag occurred when a physical wing or kite is flying. The total Drag of the kite is sum of the profile Drag and the induced Drag. Induced Drag is proportional to square of Cl (Lift Coefficient), inverse proportional to the Aspect Ratio (AR) of the kite and also inverse proportional to the shape of the kite (Induced Drag is minimum for an elliptical planform).
- Moment (or Torque): this is the rotational force that either flips the kite over its nose (for traditional airfoil, negative moment) or over its tail (for reflex airfoil, positive moment). The point along the chord where Moment force is constant for all AoAs is called the Aerodynamic Center. History and experiments have shown that for most subsonic airfoils, the “quarter-chord” point at 25% of chord from the leading edge has a fairly constant Moment (for AoAs from -5 to 20 degrees, the range of AoA most important for airplanes and kites) and most if not ALL modern data are measured using the quarter-chord point as the AC. Moment is proportional to Moment Coefficient (Cm) and the chord of the airfoil. More info about AC and Moment can be found at http://www.grc.nasa.gov/WWW/Wright/airplane/ac.htmlandhttp://www.grc.nasa.gov/WWW/Wright/airplane/cp.html.
- There are TWO MATHEMATICAL MODELS used by designers to consider how the Lift, Drag forces and the Moment act on the airplane or kite:
- The AC and Moment model: In this model, the Lift, Drag forces and the Moment are considered acting at the AC. The Moment, if negative will flip the kite over the nose around the AC (traditional airfoil) and if positive will flip the kite over the tail around the AC (reflex airfoil).
- The Center of Pressure (CoP) model: In this model, there is only Lift and Drag forces acting at a single point on the chord line called the Center of Pressure (CoP). Since the CoP is not necessary at the same location as the AC, the Lift and Drag forces (the sum of Lift and Drag forces component that are perpendicular to the chord line; let’s call it Fcop) will generate a torque around the AC. Since the torque around the AC is constant, the CoP is close to the AC when the force Fcop is high (large AoA), and the CoP is far from the AC when the force is small (small AoA).
For traditional airfoil, the Moment is negative, therefore, the CoP moves along the rearward side of AC (from 25% of chord to the rear).
For reflex airfoil, the Moment is positive, therefore the CoP moves along the frontward side of AC (from 25% of chord to front).
Since the Moment is constant, it moves either along the front side or rear side of the AC (dependent on the airfoil type) and will NEVER cross the AC.
Some CoP chart can be found at http://naca.larc.nasa.gov/reports/1921/naca-report-93/naca-report-93.pdf.
Both the AC/Moment and the CoP mathematical models are useful and can be used in different situations in the wing and kite design process. For some reasons, airplane designers are more comfortable with the AC/Moment model (this is one of the main reasons why airplanes have tail-wing to counter balance the Moment of the main wing) and kite designers are more comfortable with the CoP model (it is easier to deal with 1 force than with a force and a moment). In any event, from these 2 models, one can approximately determine the position of the CoP mathematically as:
CoP position = AC – Cm/Cl
The exact equation is:
CoP position = AC – Cm/( Cl(Cos(AoA) + Cd(sin(AoA) )
So at any AoA, once Cl, Cd and Cm are known (normally measured in the wind tunnel at quarter-chord point), we can determine the position of CoP at 0.25 – Cm/Cl of chord from the leading edge. Please note the minus sign, if Cm is negative, the CoP is on the rear side of the quarter-chord point (traditional airfoils normally used in LEI) and if Cm is positive, the CoP is on the front side of the quarter-chord point (reflex airfoil normally used in Arc).
Why is CoP so important to a kite designer?
CoP is important to the kite designer because it is the location of the kite that the effective tow point has to be at or the bridle system has to support during the flight of the kite. Since CoP of the kite is changing during flight, the effective tow point of the kite has to change accordingly (either support statically via bridle line tensions, change automatically via a spherical shape, or change manually via the back and front lines of a spherical shape).
The kite weight is centered at its Center of Gravity. The Lift of the kite must be larger than the weight of the kite for it to fly. A kite is an unbalanced device and won’t be able to fly by itself unless:
- It has a “thrusters system” that pushes it forward to counter balance the Drag force and some mechanism to counter balance the Moment (e.g., tail plane. For airplane without tail plane, it needs to use reflex profile and place its CoG in front of the main wing). This is the simple model for an air plane.
- Has a tether line placed at an appropriate location to counter-react the Drag force and the Moment. This is the model for a kite.
The line tension is the main force component of the kite that act similar to the thrust force of an airplane. While the thrust force is an active force, Line tension is a static force. The effective tow point is a location along the chord line that the line tension acts on. A tow point can be a single fixed tow point, a bridled tow point, a dynamic sled tow point or a full control sled tow point.
- Single fixed tow point: the tether line is connected directly to the kite. This is almost never used in a traction kite (there is however a kite patent on fixed tow point at http://v3.espacenet.com/textdoc?DB=EPODOC&IDX=WO0158755&F=0&RPN=WO9005663)
- Bridled tow point: the effective tow point is determine by a bridle system consisting of multiple connections to the kite. While the built-in effective tow point is the most optimum location for a bridle system to counter-react the lift from the kite, the bridle system can accommodate some variation around the built-in effective tow point during flight by automatically adjusting the tension on various parts of the bridle lines.
- Dynamic sled tow point: Using a spherical form, a sled intrinsically has a dynamic tow point configuration for the kite (more for the central part, less for the wingtip) where the effective tow point can varies quite a range dynamically while the kite is flying (this is the case of an original 2 line LEI as described in Bruno’s LEI patent).
- Full control sled tow point: Using a spherical form and a system of front lines and back lines, a 4 line sled has a dynamic fully controllable configuration where the effective tow point are dynamically readjusted during the fly and also be fully manipulated by the kiter (this is the case of a 4 line sled, LEI or Arc).
When a kite is balanced on the sky, all the forces and torques acting on it must be equal. This means:
- The Lift of the kite has to be larger than the weight of the kite; the left over Lift will create line tension to generate pull and also “thrust” ( this thrust force T is equal to(L-W)*Tan(AoA) when the kite is fully balance) due to the inclination of the line to move the kite forward (only if this “thrust” is larger than the sum of Profile Drag and Induced Drag).
- Using the AC and Moment model, the line tension and the weight of the kite has to balance the Moment of the kite about its AC or quarter-chord point. Normally when a kite is flying (especially a traction kite), lift is much higher than weight and the Moment about the AC is higher than the torque created by the weight of the kite. The effective tow point should be rearward of the AC (25% of chord) for traditional airfoil (which has negative Moment) and frontward of the AC for reflex airfoil (which has positive Moment).
- Using the CoP model, the difference of the Lift force at CoP and the kite weight at CoG is a force L1 slightly less than Lift and very close to CoP (since Lift is normally much larger than Weight; otherwise, we won’t feel any force on the line). Let’s call this spot CoPg. If CoP is frontward of CoG, CoPg is slightly rearward of CoP and if CoP is rearward of CoG, CoPg is frontward of CoP.The kite is balance longitudinally when the effectivetow point is around the CoPg. For a bridle system, the CoPg should be within the range supported by the bridle system. For a dynamic sled tow point system (2 line LEI), the kite will rotate and change the effective tow point to match that of the CoPg (the kite line will point straight to the CoPg). For a full control sled, the kite can readjust the tow point automatically or the kiter can do it manually. While the kite is flying the CoP is changing and the kite balances its tow point correspondingly to keep longitudinal balance.
Balancing the tow point will change the kite AoA to the wind (more drastically with sled) depending on the shape of the kite. For example, for a typical AR 5 sled (with a kite angle of 50 degrees – read the Sphere theory for kite angle), decrease the tow point 1% of chord is equal to increase the AoA 0.5 degree and vice and versa). So for a sled using a traditional airfoil (LEI), during the flight path, the wind direction changes and the AoA increases, the CoP will decrease and the the kite will rotate itself to increase the AoA further. In this case, the kite amplifies the AoA increase so the kite designer has to make sure that at all places along its path, the kite will not luff nor stall due to this additional automatic adjustment from the kite.
This phenomenon from sled kite using traditional airfoil (for LEI only as Arc uses mostly reflex airfoil due to concern about wingtip or shoulder collapsing) makes it an excellent performer as it accelerates the Lift during the growth phase (AoA from 0 to around 20) and decelerates the Lift loss during the decline phase (AoA from 20 and above)
For AoA from minus 5 to 20:
AoA increases during flight path -> CoP moves frontward -> Tow point follows CoP frontward -> AoA further increases -> Even more Lift
For AoA from 20 and above:
AoA increases during flight path -> CoP moves rearward -> Tow point follows CoP rearward -> AoA decreases -> Hang on to the Lift as long as possible.
This effect is called “Sled Boosting” effect and is the reason why many kiters feel that they can jump easier and higher with LEI and LEI won the battle over foil in the early days of kitesurfing.
For sled using reflex airfoil (such as Arc or LEI using reflex airfoil), the “Sled Boosting” effect is reverse, which means that it would protect the kite from exposing itself to very large or very small AoA. This mean that for Arc, the kite will try to retain within a range AoA with excellent L/D ratio. So this type of kite will be fast and leverage power from speed instead of lift like the case of LEI.
Furthermore, with a 4 line sled, the kiter can adjust the effective tow point (adjusting the length of the front line and back line) when the AoA become too little or too much. This is what depower really mean for a 4 line sled. At any CoP position, a kiter can adjust the front line and back line such that the kite will fly a certain degrees of AoA smaller or larger than the case of a 2 line sled kite.
The kite will fly properly once it reaches longitudinal balance and will continue to adjust its longitudinal balance automatically (or the kiter can help manually) during the flight.
- The kite will stop flying when its “thrust” force is equal the sum of all the drag forces (Profile Drag and Induced Drag). So the wind-window and the AoA of the kite at the wind window is determined by the kite itself (the airfoil characteristics). The built-in effective tow point should be selected accordingly to be as close as the wind window CoP as possible (via bridle setting for bridle kite and via Profile Attachment Points for sled kite).
- A kite has Lift and Drag similar to an airplane.
- CoP of a kite varies dependent on AoA. CoP is closer to AC (25% of chord from leading edge) for large AoA and farther from AC for small AoA. For traditional foils, CoP is normally around 27% (AoA around 20 degrees) to 55% (AoA around 0) of chord from leading edge. For reflex airfoil (Arc), the CoP is normally frontward from 0 to 25% of chord.
- The tow point of the kite should either statically (bridled kite) or dynamically (sled kites) support the range of variation of the CoP when the kite is flying across the wind window. In the case of sled kite it is automatic. By following the CoP automatically, a sled kite using a traditional airfoil (such as LEI) “amplifies” the acceleration of Lift and sustain the peak. This “Sled Boosting” effect is one of the main reason why a LEI kite jumps higher and easier to jump.
- The Lift, Weight and Drag of the kite determine the wind window and the AoA of the kite at the wind window.