Welcome to Steve Seibel's stupendously gargantuan web posting on turns and slips in hang gliders! Changes since last (April 2000) edition: substantial changes in content in lockout and blind flying sections, substantial changes in format and presentation throughout the paper. OK, first a bit of entertainment (photo gallery), and now onto the heavy stuff...enjoy!
TURNING FLIGHT AND SIDESLIP IN HANG GLIDERS (Summary)
Steve Seibel
seibel999@hotmail.com
July 6, 2000 edition
A turning aircraft is sideslipping when the nose is angled toward the outside of the turn instead of facing directly into the direction of travel (i.e. the relative wind). This creates a sideways component in the airflow over the glider, which creates sideways aerodynamic forces that slow the turn rate. Since the aircraft is now overbanked for the turn rate, the pilot tends to fall toward the low side of the aircraft.
During experiments in my Spectrum, I found that a brief sideslip occurred whenever I rolled the glider into a turn. The major cause of this slip appeared to be rotational inertia in the yaw axis. A brief skid occurred whenever the bank angle was decreased. The amount of slip or slid depended on the roll rate, and the slip or skid largely vanished when the bank angle was stabilized. These effects can easily be seen in a yaw string (telltale) mounted in front of the pilot: whenever the bank angle steepens or shallows, the yaw string deflects away from the direction of roll.
Pitch inputs, which control the G-loading and airspeed in the turn, didn't influence the sideslip behavior (yaw coordination) of my hang glider. As I rolled the glider into a turn, I saw about the same amount of sideslip whether I coordinated the turn in the pitch axis by letting out the bar to hold the airspeed constant, or pitched the nose steeply up or down to bleed off airspeed or to put the glider into an accelerating dive. When I held the bank angle constant and pulled in the bar to pitch the nose down, I didn't see any sideslip as the glider accelerated. I also observed similar behavior in experiments in an airplane and a sailplane, but these ideas go against the conventional wisdom among hang glider pilots which holds that a turning glider will slip toward the low wing if the turn is not properly "coordinated" in the pitch axis and the G-loading (lift force) is inadequate for the bank angle. Many of the physical sensations that hang glider pilots often attribute to sideslip are really caused by dynamics in the pitch axis, involving the interplay of airspeed, angle-of-attack, G-loading, and the pitch attitude of the glider in space. When hang glider pilots talk about turn "coordination" via pitch inputs, they are actually referring mainly to these pitch dynamics rather than to yaw coordination and the prevention of sideslip. Many of our magazine articles and training handbooks show some confusion about this point.
Sideslips often coincide with changes in pitch attitude and accelerations in airspeed, because both pitch and yaw dynamics are driven by changes in bank angle.
In my Spectrum, in steady, constant-airspeed turns at a constant bank angle, including high-speed (diving) turns, a slip-skid ball or bubble indicator showed that the turn was coordinated in the yaw axis, with very little sideways airflow over the glider as a whole, and no apparent "push" upon the pilot toward either side of the aircraft. A yaw string near the front of the glider deflected slightly toward the outside of the turn, indicating a small sideways component in the airflow there. Some experiments were also made with yaw strings mounted on a "bowsprit" and on the extreme rear of the keel, to look at the curvature of the airflow (following the circumference of the turn).
On my glider a yaw string might have some value as an emergency blind flying aid; it provides more information than the float or bubble gadgets which some hang glider pilots have reported using in clouds. (This will not necessarily be true for all hang glider designs; see Appendix One for much more on this. I don't recommend intentionally entering clouds!)
Some thoughts on towing and lockout dynamics are given in Appendix Two. Appendix Six contains an outline of this paper which will also serve the reader as a Table of Contents. For a much more condensed description of these ideas on sideslip and turns, see my articles in the February and July 2000 issues of "Hang Gliding".
TURNING FLIGHT AND SIDESLIP IN HANG GLIDERS
Steve Seibel
seibel999@hotmail.com
July 6, 2000 edition
INTRODUCTION
How are the pitch and yaw axes interconnected in hang gliders? What causes a sideslip? What do we mean when we talk about turn coordination in a rudderless aircraft? I hope that this discussion of turning flight and sideslip will be illuminating to anyone who has mused upon these questions. I will provide a theoretical background, and will also present some experimental data on the sideslip characteristics of my hang glider, a sailplane, and an airplane.
My thinking on this subject has always been motivated by the assumption that all forms of fixed-wing flight are closely related in their dynamics. It's not my goal to dictate terminology or to get in the way of those who prefer to fly by pure intuition. But some pilots like to explore the physics of their flight, and when the topic of discussion is turn physics and coordination in hang gliders, there are always many conflicting points of view. This is a contribution to that exploration.
We'll start by examining the pitch-axis dynamics of turning flight in quite a bit of detail in Part One, because these dynamics are often confused with sideslip and yaw coordination in hang gliders. Then in Part Two we'll move on to talk about sideslip and yaw coordination, and in Part Three we'll look at the in-flight experiments I did in my hang glider, an airplane, and a sailplane. Part Four will revisit turn and sideslip dynamics in more detail, and the Appendices at the end will include sections on blind flying, lockout dynamics, and quantifying the aerodynamic forces produced in a slip.
Those of you with better plans for the evening than curling up with this 50-page article may want to skim directly to the sections in Part Three entitled "Suggestions for teaching methods" and "So why do we let out the bar while rolling into a turn" as they contain the main take-home lessons!
PART ONE: PRACTICAL TURN "COORDINATION" AND PITCH AXIS DYNAMICS
INTERPLAY OF ANGLE-OF-ATTACK, BANK ANGLE, AIRSPEED, AND G-LOADING
Of course the practical aspects of hang glider turn coordination are well understood. As a glider is rolled into a turn, the pilot usually lets the bar out to boost the turn rate and prevent the nose from falling. If we think of this as a pitch coordination move, it is not difficult to explain the physics involved. Letting out the bar increases the angle-of-attack and the G-load (lifting force), which maintains an adequate vertical component of lift as the bank angle increases. If we roll into a turn without changing the angle-of-attack, pulling only one "G", the turn rate will be sluggish, and the vertical component of lift (and drag) will be less than the aircraft weight. As the nose falls and the glider accelerates downward, the flight path will curve downward in the pitch dimension. The airspeed will be rising, which also increases the G-load (lift force) produced by the wing, and eventually leads to a stabilized, constant-airspeed turn at the new, increased airspeed, with the proper G-loading for the bank angle. So as the glider is rolled into a turn either the angle-of-attack must be increased, or the airspeed will increase, until the wing is again creating the appropriate G-loading for the turn and the glider is again in equilibrium.
When the angle-of-attack is held constant as the glider is rolled into a turn, the complete sequence of events in the pitch axis is actually an initial downward curvature in the flight path as the aircraft noses down and gains airspeed, and then a smaller upward curvature and slight loss of airspeed as the aircraft pulls out again into a steady glide at the new, increased airspeed. In fact the aircraft may go through several oscillations before settling into a steady glide at the new angle-of-attack. The whole maneuver is caused by the interplay between changes in the lift and drag vectors, and the subsequent gain or bleeding off of airspeed. The net effect of all these dynamics is a steepening of the glide path and an increase in speed and G-loading, all due to the increased bank angle.
A key factor in these pitch dynamics is the delay in the build-up of airspeed and G-loading (lift) as the glider noses over into a dive. If the glider accelerated instantly as the bank angle increased, then the glider would always be in an equilibrium state, with the flight path entirely determined by the L/D ratio (and the bank angle). We would still see a steepening of the glide path as the bank angle increased, but we would avoid the more dramatic dive-and-pullout dynamics that we've described above. The "falling" sensation and marked lowering of the nose occurs when the G-loading is well below the required value (for the bank angle) and the aircraft is pitching down into an accelerating dive. As the aircraft accelerates the airspeed and G-loading may then significantly overshoot the "required" values (for the bank angle at any given moment) which will then cause the nose to rise a bit and the airspeed and G-loading to begin bleeding off again. The aircraft may go through several of these pitch oscillations before settling into a steady glide, all because of the delay in the way that the airspeed and G-loading respond to changes in the bank angle and the pitch attitude.
Up to this point, we've been looking at what happens when the glider is rolled into a turn without the usual pitch "coordination" input (i.e. with no change in the angle-of-attack). However, the same dive-and-pullout dynamics occur whenever the pilot pulls in the bar to decrease the angle-of-attack and G-loading (lift force), without also decreasing the bank angle. If the pilot "unloads" the glider (reduces the G-loading) by pulling in the bar, the flight path will first curve downward as the aircraft dives and accelerates, and then curve upward again into a more moderate, steady glide as the aircraft settles into equilibrium at the new, decreased angle-of-attack and increased airspeed, with the G-loading (lift force) again properly matched to the bank angle. Again, the aircraft may actually go through several of these pitch oscillations before reaching equilibrium. Conversely if the pilot pulls "extra" G's by pushing out the bar, the flight path will first curve upward--possibly even climbing above the horizontal--as the excess airspeed bleeds off and then will curve downward again as the glider settles into a steady glide at the new, increased angle-of-attack. After the aircraft has returned to equilibrium and the airspeed has stabilized, the net effect of a pitch control input and change in angle-of-attack is a steeper glide path if the change in angle-of-attack has degraded the Lift/Drag ratio, and a flatter glide path if the change in angle-of-attack has improved the L/D ratio. In other words the net effect of the pitch input depends on where we are on the L/D vs. airspeed curve.
Again, the delay in the change in airspeed and G-loading plays a key role here. This delay is what allows the pilot to temporarily "pull" extra G's by increasing the angle-of-attack, until the airspeed bleeds off to the appropriate value for the new angle-of-attack and the G-loading is again in synch with the bank angle (i.e. 1 G in wings-level flight). Likewise this delay is what allows the pilot to temporarily "unload" the wing by decreasing the angle-of-attack, until the airspeed rises to the appropriate value for the new angle-of-attack and the G-load is again in synch with the bank angle.
Since the delay in the airspeed response plays a key role in allowing the pilot to "pull" extra G's or partially unload the wing--in relation to the "expected" G-loading for the bank angle--these extra loading or unloading effects are most visible when the control bar (or stick) is actually being moved forward or aft, or when the bank angle is changing. After the bank angle and angle-of-attack are both stabilized, the G-loading and airspeed will soon stabilize also, though in some cases this may require quite a few seconds, especially if the aircraft is going through a strong cyclical oscillation as it moves toward equilibrium.
If we strongly pull in the bar at the same time that we are rolling from wings-level into a turn, we will nose down abruptly due to the effects described above--the G-loading will be far less than "required" by the bank angle at any given instant, and initially the G-loading may even be less then 1 "G", creating a distinct "falling" sensation. Even if we want to end up in a fast, well pulled-in turn, we will accelerate much more smoothly, with a smoother flight path, if we let the bar out a bit as we are rolling into the turn. This avoids the sudden "deficit" in G-loading and spreads the acceleration in airspeed out over a slightly longer time period. We avoid the sudden nose-down motion and we also avoid overshooting our target speed and creating a series of pitch oscillations.
In summary, for every bank angle, there is one particular G-loading that will allow the glider to fly in equilibrium at a constant airspeed. The airspeed required to produce this lift force (G-loading) will depend upon the angle-of-attack, and vice versa. If the total lift force (G-loading) is excessive then this will create an upward curvature (acceleration) in the flight path in the pitch dimension, and a loss of airspeed. If the total lift force (G-loading) is inadequate then this will create a downward curvature (acceleration) in the flight path and a gain of airspeed. For every combination of bank angle and angle-of-attack (bar position or control stick position), the aircraft will eventually settle down at one particular airspeed where the vertical component of the G-load (lift force) equals the aircraft weight. (This is a slight oversimplification--see "L/D ratio and Lift Force" below.)
WHAT DETERMINES ANGLE-OF-ATTACK?
To really follow the above discussion, the reader must have a clear understanding of what determines the glider's angle-of-attack, i.e. the angle at which the wing is meeting the airflow. To a first approximation the angle-of-attack of an aircraft is determined by the position of the pilot's body on the control bar (or the position of the control stick). Pilots tend to cue onto control force rather than control position; these are related but not quite the same.
Compared to many other aircraft, a hang glider is relatively limited in control authority in the pitch axis. In particular the wing cannot be easily put at a zero or negative angle-of-attack as it can in most three-axis aircraft (which yields a 0 G ballistic trajectory or negative G's). In a hang glider the nose cannot easily be shoved steeply downwards to "unload" the wing and add a lot of airspeed, so a steep bank is often used to gain airspeed quickly, for example to begin an aerobatic maneuver. Another method used by aerobatic hang glider pilots to point the nose steeply down and build speed rapidly is to stall the glider in a very nose high condition (whipstall).
Note that the pitch attitude of the aircraft in space is really the sum of the angle-of-attack and the actual flight path through the air. When the airspeed is constant, the flight path of a glider is strictly determined by the Lift/Drag ratio (i.e. the glide ratio), and the bank angle. The Lift/Drag ratio is strictly determined by the angle-of-attack, which is determined mainly by the bar position (or control stick position) and is also somewhat influenced by the bank angle. Therefore in a steady, constant airspeed turn the bank angle and bar position (or control stick position) completely determine the pitch attitude of the glider in space, as well as the glide ratio, angle-of-attack, airspeed, and G-loading. When the airspeed is rising or falling, and the flight path is curving up or down as the glider pulls "extra" G's or is partially "unloaded", the situation is much more complex and the flight path may even temporarily arc upwards above the horizontal in a climb, wingover, loop, etc. (Of course when we refer here to the flight path we mean relative to the airmass rather than the ground; when a glider is in equilibrium with a constant airspeed it is always in a descending flight path relative to the airmass even if the glider is in "lift").
An added complexity to all this is that in a turn, the airflow curves to follow the circumference of the turn (much more on this later). At steep bank angles, this means that the curving airflow tends to "push up" on the rearmost parts of the aircraft which lowers the nose and decreases the overall angle-of-attack of the wing. The practical result of this is that as the bank angle increases, the pilot must push the control bar further out--or haul the control stick further aft--to hold a constant angle-of-attack. To see this for yourself, compare the control bar or control stick position at the stall angle-of-attack in a steep turn vs. in wings-level flight. (Note: if you are doing this in a powered aircraft pull the power to idle, as the propwash will further complicate things).
This airflow curvature effect is most noticeable at the low airspeeds and tight turn radii typical of flex-wing and rigid-wing hang gliders. However in a flex-wing the ability of the wingtips to twist (wash out) under load will significantly reduce the nose-down effect of airflow curvature in a tight turn. To a hold a given angle-of-attack as the bank angle increases, we would expect to have to push out more in a rigid-wing hang glider than in a flex-wing, at similar airspeeds and turn radii.
DO WE SEE THE SAME DYNAMICS IN POWERED AIRCRAFT ALSO?
All the dynamics involved in the interplay between G-loading, airspeed, angle-of-attack, and flight path will occur in powered aircraft as well as gliders. After all, it is the delay in the bleeding-off of airspeed which allows an aerobatic pilot to temporarily "pull" extra G's (beyond what is "required" by the bank angle) during aerobatic maneuvering such as when pitching up into a "zoom" climb, etc. However the final value of the G-loading (lift force) when the plane reaches equilibrium and the airspeed stabilizes will vary slightly according to the power setting, as noted below.
L/D RATIO AND LIFT FORCE
We've been using the terms "lift force" and "G-loading" interchangeably to mean the lift produced by the aircraft's wings, and we've been assuming that the vertical component of this force must equal the aircraft weight if the aircraft is to remain in equilibrium with a constant airspeed. In truth, if the aircraft is in a descending glide then a portion of the aircraft weight is borne by the drag vector, which slightly "unloads" the wings and decreases the lift vector (G-loading). Therefore to be technically correct we do need to recognize that the required lift vector (G-loading) will in fact vary with the angle-of-attack (L/D ratio) as well as with the bank angle. From the pilot's point of view, this effect is negligible unless the glide angle is quite steep. However if the pilot pushes the control stick far forward and leaves it there, we will see the immediate, temporary reduction in G-loading that we've been discussing, and then as the airspeed increases the G-loading will build and then stabilize at a value significantly less than 1 G (in the wings-level, unbanked case) because at this low angle-of-attack the L/D ratio is quite low, the glide angle is quite steep, the airspeed is high, and the drag vector is large enough to bear a noticeable part of the aircraft's weight. In a steeply banked, steeply diving turn the G-loading will of course stabilize at a value much greater than 1 G, but less than "required" by the bank angle if the aircraft were to maintain altitude or have a much shallower glide path. (The same would be true in a hang glider when the pilot pulls the bar in all the way, except that in a hang glider we don't have enough control authority (control bar travel) to keep the wing at the very low angle-of-attack and very steep glide angle that we're talking about here, so the reduction in the G-loading is not very noticeable). The extreme example is a sustained vertical dive an aerobatic sailplane or airplane. Here the pilot moves the control stick forward until he actually reaches the zero-lift angle-of-attack where the lift vector (G-loading) and L/D ratio are zero. This is the only case where the G-loading (lift vector) will not begin to build as the airspeed begins to increase. The aircraft is initially in a ballistic trajectory (freefall) as it arcs over into the vertical dive, and will accelerate until reaching terminal velocity, at which point the entire aircraft weight (plus thrust in a powered aircraft) is borne by the drag vector. At this point the G-loading is still zero (as we have been using the term, and as measured by the G-meter in the instrument panel which only detects "up" and "down" forces in the aircraft's reference frame) and the pilot feels none of the usual force exerted by the seat bottom against his body but he does feel a one-G force pushing on his chest as the seat belts prevent him from accelerating earthwards and falling forward into the instrument panel. (All this assuming that the wings have not departed the aircraft by this point due to flutter, excessive drag forces, or the sudden application of a heavy G-load if the pilot does allow the stick to move back and increase the angle-of-attack too quickly while the airspeed is very high).
Interestingly, the way that the drag vector begins to bear a part of the aircraft weight as the airspeed rises and the glide path steepens actually prevents the lift vector from going to infinity in a very steep turn as the bank angle approaches 90 degrees. If infinite thrust were available to sustain level flight as the bank angle approached vertical, then airspeed, lift, and drag would all go to infinity, regardless of the angle-of-attack and the L/D ratio. In the real world a very steeply or vertically banked aircraft ends up in a steeply diving corkscrew or a rolling vertical dive, with the drag vector bearing much or all of the aircraft weight, so the G-loading (lift vector) will remain finite. However we should note that this load-limiting effect does not begin to play a significant role until the bank angle is very steep and the airspeed and G-loading are already very high, especially in streamlined aircraft with high L/D ratios. In many aircraft you can still pull enough G's to remove the wings if you try to keep the aircraft in a sustained, near-vertical bank, especially if you try to "hold the nose up" by keeping the control stick moving aft to further increase the G-loading as the airspeed rises. (Of course, I'm not referring to the case where the aircraft is held indefinitely in a slip with the rudder so the fuselage, rather than the wing, supports the aircraft weight--by this method high performance aircraft can indeed sustain level flight while vertically banked).
By the way, in a powered aircraft a climb has the same "unloading" effect as a dive because part of the aircraft weight is borne by the thrust vector; here the extreme example is an accelerating vertical climb on raw thrust alone, with the wings held at zero angle-of-attack and zero G-loading (lift) to avoid arcing back into a loop. Of course this accelerating vertical climb is only possible in a few high-performance jet fighters, the space shuttle, etc but even at shallow climb angles in light planes the wings are actually generating slightly less lift in the sustained, constant-airspeed climb than in level flight. (Amaze your friends with this trivia...) The lift vector (G-loading) is exactly equal to the aircraft weight only during sustained horizontal flight with no climb or descent, with the thrust vector exactly equal and opposite to the drag vector, so none of the aircraft weight is borne by either thrust or drag. (Again we are of course talking about the flight path relative to the airmass and not considering "lift" and "sink" effects so beloved (and despised) by glider pilots which may further modify the flight path relative to the ground!)
In a powered aircraft we find the glide ratio or glide angle (in still air) by looking at the L/(D-T) ratio and the climb ratio or climb angle by looking at the L/(T-D) ratio, where T= thrust. In sustained horizontal flight thrust and drag are equal.
BUT WHAT ABOUT SIDESLIP?
Now we've considered the interplay between pitch inputs, airspeed, bank angle, G-loading, and flight path at great length without once mentioning sideslip! This should serve as a hint to the reader that I believe that the role of sideslip in hang glider turn dynamics is greatly overestimated. Skip ahead to the section entitled "So why do let out the bar while rolling into a turn?" at the end of Part 3 for the final punch line here.
PART TWO: A BRIEF OVERVIEW OF SIDESLIP DYNAMICS
In three-axis aircraft, the word "coordination" generally refers to the prevention of slips and skids, which is accomplished with the rudder. In the above discussion I've loosely used the word "coordination" to refer to the pitch inputs that a pilot makes to control airspeed and G-loading in a turn, without meaning to imply that this has anything to do with sideslip or skid. However many hang glider pilots believe that "uncoordinated", accelerating turns are in fact slipping turns. In other words, if the pilot increases the bank without letting out the control bar, or actually pulls in the bar as he rolls the glider, the glider is thought to be uncoordinated in the yaw axis as well as the pitch axis, and is thought to fall or slide sideways toward the low wingtip as it pitches down into a dive and gains airspeed. Not wanting to take this for granted, I've done some in-flight experiments to look at this further. But first let's define in more detail what we mean by a sideslip.
DEFINITION OF SIDESLIP
A slipping turn can be described in two ways. One, the nose is angled toward the outside of the curving flight path, instead of facing directly into the direction of flight (into the relative wind). In other words, the heading of the glider is lagging behind the actual direction of travel in the turn. This creates a sideways (spanwise) component in the airflow over the glider, so a yaw string (telltale) will blow toward the outside (high side) of the turn. Two, this sideways airflow over the glider creates a centrifugal aerodynamic force (toward the outside of the turn) which slows the turn rate. The bank angle is then too steep for the turn rate and the pilot tends to fall toward the inside of the turn (toward the low wing). A slip-skid ball (as used in many "conventional" aircraft) will shift toward the inside of the turn, and a slip-skid bubble (such as a carpenter's level) will shift toward the outside of the turn.
Because a sideways (sideslipping) airflow over the aircraft is really the only possible cause of a sideways or spanwise force toward the low wingtip, the yaw string and the slip-skid ball are nearly always in agreement about the whether the aircraft is slipping, skidding, or coordinated in the yaw axis (but see Appendix Three for a slight caveat here in aircraft with rudders). This is a very important point to understand. In general the reason that sailplanes use yaw strings and power planes use a slip-skid ball is that for planes with a propeller in front the propwash over the nose would render the yaw string useless; rear-engined prop planes often use yaw strings and a few jets do too (such as the U-2 spyplane with long wings and lots of adverse yaw).
Much more on the theoretical aspects of slipping turns will follow later in this discussion!
DETECTING SIDESLIP IN HANG GLIDERS
Let's consider how a hang glider pilot could be certain that he is in a sideslip. Of course there is the usual list of cues for "uncoordinated" turns in hang gliders (increasing airspeed, nose pitching down, slow turn rate, sensation of falling or inadequate G-load) but by now we've seen that most of these are caused by dynamics in the pitch axis rather than by sideslip. What specifically tells the pilot that he is actually sideslipping, i.e. that the glider is uncoordinated in the yaw axis? If there are telltales on the nose wires, they will blow toward the outside of the turn, but these are usually outside of the pilot's normal vision, and they also may be hard to read clearly if the pilot is shifted to either side of the glider centerline. The pilot might feel on his face the shift in the direction of the relative wind, but this might not be noticeable unless he is concentrating on facing straight out ahead of the glider rather than looking off to one side or the other. If the slipping airflow component is producing centrifugal forces that slow the turn rate, then the pilot will tend to fall toward the low side of the glider. This will not create the obvious, uncomfortable, sideways force that is felt by a seated, upright, pilot in a "conventional" aircraft. Instead, a freely hanging pilot will simply experience a small shift in the "neutral" hang position, meaning that if the pilot took his hands off the control bar he would tend to hang a bit to the low side of the centerline. This is only observable if the pilot is using little or no control pressure. In the real world, with turbulence, and the need for high-siding or low-siding according to the particular glider design, the pilot knows the muscular forces that he is exerting at any instant but may not be aware of the precise location of the "neutral" hang position relative to the glider centerline. In short I don't think it's always that obvious to the pilot what his glider is doing in regard to sideslip. To carefully investigate the sideslip behavior of a glider, a pilot should use a slip-skid ball or bubble, or simply a yaw string that is mounted where it is easy to read in flight.
A hang-glider sideslip is often described as a marked sensation of "falling toward the low wing". Actually, a pilot would have to firmly pull in the bar if he wished to temporarily decrease the G-load to noticeably less than one. Perhaps many pilots also describe a sense of "falling" whenever the G-load is less than normal for a given bank angle, or whenever the G-load decreases regardless of the actual magnitude of the G-load. In any case, it seems to me that anytime a turning glider pitches downward, dropping the nose, with a high sink rate and an accelerating airspeed, the pilot will describe a falling sensation regardless of the actual magnitude of the G-load. And he will likely perceive that he is falling toward the ground, i.e. towards the low wing, regardless of whether or not he actually tends to fall toward the low side of the control frame, or is able to sense the airflow coming up from the low side of the glider. In short he may greatly overestimate the amount of sideslip toward the low wing as the glider pitches down into a dive.
SOME COMMON IDEAS ABOUT SIDESLIP IN HANG GLIDERS
Many hang glider pilots use the word "slip" to describe the falling sensation as the nose drops and the airspeed rises when the G-loading is inadequate in a turn. As we've already discussed in detail, these effects can be produced either by pulling in the bar while banked or by failing to let the bar out adequately as the glider is rolling into a turn. As noted above, many hang glider pilots believe that these pitch-axis dynamics also involve a true sideslip in the usual aviation sense of the word, with a sideways or spanwise component in the airflow as the glider falls toward the low wing, and an imbalance of forces (in the pilot's reference frame) that allows the pilot to fall toward the low side of the control bar. Proper pitch coordination, to maintain the appropriate G-loading and a constant airspeed as the glider rolls into a turn, is thought to also ensure yaw coordination and prevent this sideslip. If the pilot actually pulls in the bar while rolling into a turn the glider is thought to enter a very marked sideslip, which later disappears as the airspeed and G-loading build and then stabilize at the appropriate values for the bank angle and angle-of-attack. In all cases the slip is said to be a temporary event because of the inherent yaw and pitch stability of the glider. Clearly this view of sideslip assumes a close interconnection between pitch and yaw axis dynamics. This viewpoint is well represented in Dennis Pagen's Hang Gliding Training Manual (pp. 128-129, 141, 149-150 (relates to spanwise airflow), and 358) and Performance Flying (pp. 34, 45, 52-54).
An important detail of this model as outlined by Pagen is that a coordinated turn may be initiated at any airspeed, as long as the airspeed is kept steady in the turn entry. Pagen is careful to separate the issue of turn coordination from the issue of the optimum speed to fly for minimum sink rate (HGTM p.129).
An alternative description of slips in hang gliders holds that all turns at high airspeed are always slipping turns, and by keeping the airspeed high, the glider can be kept in the slip indefinitely. This view was communicated to me by a very experienced instructor/ aerobatic pilot.
A third view held by some pilots is that the pitch and yaw axis dynamics are largely unrelated. By this viewpoint the nose-down motions that most pilots call "slips" really involve dynamics in the pitch axis only, with little or no actual sideslip, except for effects such as adverse yaw which are not influenced by the pilot's pitch "coordination" inputs.
SOME IDEAS ABOUT SIDESLIP IN 3-AXIS AIRCRAFT
Until I started my hang glider training, it never occurred to me that pitch control inputs might have an effect on sideslip. While flying sailplanes and airplanes, I've always assumed that yaw coordination was controlled entirely by the rudder, while pitch inputs affected only airspeed and G-loading. I've never noticed that pulling extra "G's" in a turn had any tendency to create a skid, or that allowing the nose to fall due to inadequate back pressure as I rolled into a turn had any tendency to create a slip. Nor have I come across such suggestions in the current flight training literature for 3-axis aircraft. However about the same time that I started digging into Dennis Pagen's Hang Gliding Training Manual and Performance Flying, I also ran across the idea that that pitch inputs will have an effect on yaw coordination in a couple of older books for 3-axis aircraft (Wolfgang Langewiesche's 1944 physics-for-pilots classic Stick and Rudder pp. 203-206 and 225-226, and p.308 in the 1966 3rd edition of Modern Airmanship (Van Sickle, ed.)). I didn't find either of these sources to be very convincing on this particular subject, though Langewiesche's book is a real gem in most respects and contains a full chapter on the physics of turns.
PART THREE: THE HEART OF THE MATTER: MAKING THE IN-FLIGHT EXPERIMENTS, INTERPRETING THE RESULTS, AND SUGGESTIONS FOR TEACHING
TOOLS FOR INVESTIGATION
With all of these different ideas floating around about sideslip in hang gliders, I decided to take some data of my own in my Spectrum. To see what the glider was really doing in turns, I rigged three small bubble levels to simulate a single, larger tube with the desired amount of curvature. This gadget measured up to four degrees tilt to either side. This seemed to be adequate as the bubbles rarely reached full deflection in flight, and other arrangements with more curvature yielded little movement of the bubbles in flight. Deflection of the bubble to the high side of the turn indicates a slip--a standard aircraft slip-skid ball would move the opposite direction. I also put a yaw string (telltale) on a dowel projecting forward 60 cm from the center of the base tube, for easy viewing in flight. The dowel provided a centerline reference to that helped the pilot judge the deflection angle. There was also a long yaw string attached to the rear of the keel, and on some flights, a yaw string on a long "bowsprit" projecting almost 2 meters forward from the apex. A small crossbar "index" mounted near the free end of each of the two forward yaw strings helped the pilot to compare the angular deflections of these strings, which were at different viewing angles and distances from the pilot's eyes. The "bowsprit" and rear keel yaw strings were intended to explore the curvature of the airflow in the yaw axis, following the circumference of the turn.
MAKING THE TEST FLIGHTS
I accumulated the data slowly over several flights in ridge lift. Obviously there must be no nearby traffic, and conditions in general must be mellow enough to allow some diversion of the pilot's attention. Glass-off conditions are ideal.
I watched the yaw strings and slip-skid bubbles as I rolled the glider from wings-level into a turn while letting out the bar to "coordinate" the turn in the pitch axis, and while pulling in the bar, and while making no pitch input. To isolate the effect of pitch inputs on yaw coordination from the effect of roll inputs on yaw coordination, I also watched the yaw string and bubbles as I pulled in or let out the bar while holding the glider in a steady, constant-banked turn. Some of these pitch inputs were quite marked and produced strong changes in the glider's airspeed, pitch attitude, and flight path.
Because a hang glider has little structure out in front of the pilot, I found that I was usually much more aware of the yaw rotation rate than of the bank angle itself. I instinctively tended to make the required roll inputs to keep the glider tracking around the horizon at a steady rate. When I wanted to hold a constant bank, it took some extra effort to overcome these habits. I used various bank angles up to about 45 degrees in the experiments involving the various pitch inputs while turning. I didn't use bank angles steeper than 45 degrees because of the difficulty in flying smoothly and precisely at very steep bank angles. In maneuvers involving simultaneous and dramatic changes in bank angle, pitch attitude, and G-loading it was challenging to carefully observe the precise timing of all the dynamics involved. In some experiments in an airplane which I'll describe later, the additional visual references provided by the cowling, windscreen, etc., and also the gyro instruments, made it much easier to make precise observations of the bank angle and to carefully relate changes in the bank angle to changes in the airspeed and pitch attitude.
I also made observations of the yaw strings and slip-skid bubbles in stabilized, constant-banked, constant-airspeed turns. In these observations in particular I was looking at small displacements of the yaw strings so it was essential that the bank angle, turn rate, and airspeed all be held steady. I made these observations at bank angles of 20-30 degrees. In 180 degree turns or single 360's, there often wasn't time to settle into a steady turn rate and it was easy to draw hasty conclusions. Flying multiple revolutions produced better results. Viewing the rearmost yaw string at the extreme rear of the keel was an extra challenge. When making quick glances to the rear of the glider, I inevitably would see a skid because I had allowed the bank angle to shallow. Longer observations tended to have the opposite effect as I inadvertently tightened the bank. I tried flying with a convex mirror, but the image was too small. Finally I found that by facing steadily rearward through multiple circles, I could keep the turn rate steady for short intervals. By controlling the glider with a slight rotation of the body (moving primarily the feet and legs), I could keep a clear sightline to the rear of the keel, while adjusting the turn rate via my view of the rear horizon, and holding a constant pitch pressure on the bar. It took some practice, and lots of clear air! Viewing the "bowsprit" yaw string involved less contortion but it was above the pilot's line of normal vision so some care was needed to keep the turn rate steady as I raised my head above the normal flying posture. These observations of the yaw string deflections in steady, constant-speed, constant-bank turns were made with a bank angle of 20 to 30 degrees.
When viewing the yaw string in front of the base bar, near the pilot's eyes, allowance had to be made for a parallax effect when the pilot was shifted away from the glider centerline.
ACTUAL DATA, AND INTERPRETATION OF RESULTS: SLIP-SKID BEHAVIOR OF MY GLIDER (Spectrum 144)
Many of the ideas introduced here will be explored in much more detail in later sections of this paper.
I found that the slip-skid bubble was centered most of the time. The slip-skid bubble was centered in all steady, constant-bank, constant-airspeed turns. However, whenever the bank angle was increasing, the yaw strings and slip-skid bubble showed a slip, and they showed a skid when the bank angle was decreasing. So a turn entry always created a brief sideslip, and a roll-out to wings level created a brief skid. The magnitude of these slips and skids depended strongly on the roll rate. All three yaw strings (on the "bowsprit" 1.9 meters forward of the apex, and on the probe projecting 60 cm forward from the base bar, and on the tail end of the keel) were basically synchronized in their motions while the glider was rolling in or out of a turn. This point, plus the time lag described below, showed that none of the yaw strings were unduly influenced by changes in the local airflow around my body as I made roll inputs.
The slip caused by a turn entry lagged behind the initial pilot roll input. As the glider began to bank, the amount of slip gradually increased along with the bank angle, and then faded away soon after I stabilized the bank angle. This was most visible when I used a large roll input.
Much can be learned from this time lag between the pilot roll input and the resulting slip. This reveals that the major cause of the slip was probably not adverse yaw from differential airfoil shapes, which should correlate closely with the position of the pilot's body on the control bar. Instead, the major cause of the sideslip in my Spectrum appeared to be rotational inertia in the yaw axis, which will be most pronounced once the glider has reached a high roll rate and a significant change in bank angle has occurred. Other hang glider designs with more span and less sweep may well show both more adverse yaw and more yaw rotational inertia than did my Spectrum.
Pitch inputs while holding the bank constant did not cause a sideslip or skid, either in the bubble or the yaw strings. Pitch inputs while the glider was rolling into a turn did not seem to either decrease or augment the slip caused by the roll input. For some of these trials, I pulled in or pushed out the bar quite strongly which resulted in large changes in airspeed and pitch attitude, but did not appear to affect sideslip and yaw coordination. All observed deflections of the slip-skid bubble and yaw string appeared to correspond entirely to roll inputs by the pilot and seemed to be unaffected by changes of the bar position in the pitch axis, whether the glider was rolling into a turn or was established in a steady turn. I saw no evidence that the usual hang glider "turn coordination" inputs had any effect on yaw coordination and the prevention of sideslip.
When I pulled in and shifted fully to the side to enter a steeply banked diving turn, there was a marked increase in airspeed and bank angle, and a lowering of the nose. There was also a sensation of falling which was due in part to the obvious visual effect of the nose dropping. The bubble showed a slip, and then became centered within one half of a revolution from the initial input. This roughly coincided with the end of my roll input, though the whole maneuver was so dynamic that it was hard to observe such details with certainty. The G-forces seemed to increase steadily along with the airspeed as the glider slipped and rolled, until after about half a revolution the G-forces were as high as I wished to experience, so I relaxed the roll input and allowed the bank angle to stabilize, and the G-forces also stabilized at a sustained, high value. The maneuver was very dramatic and it was difficult to carefully note the precise timing of all the dynamics; as we've already discussed while looking at pitch dynamics in Part Two, and will see in more detail in the airplane experiment described below, I suspect that the airspeed and G-loading actually continued to increase for several seconds after the bank angle was stabilized. A video camera to carefully record the bank angle and pitch attitude, and a tape recorder to record the pilot's perception of the airspeed and G-loading, would allow a more detailed investigation of these points.
When I performed the same full shift to one side but without pulling in the bar, the results were much the same. In this case, it was particularly noticeable that after I shifted my weight, there was a time lag before the yaw string and bubbles showed a slip and also before the airspeed started to increase. The ultimate airspeed increase was dramatic and seemed nearly as great as when I pulled in while shifting to the side. (No airspeed data were taken). The amount of sideslip seen as the glider rolled into the turn was about the same whether or not I pulled in the bar during the maneuver.
Here is my overall view of the behavior of my hang glider during a turn entry: if the glider is rolled into a turn without the usual pitch "coordination" input, this creates both a sideslip due to rotational inertia in the yaw axis and (to a lesser extent in my glider) adverse yaw, and a marked dive and an acceleration in airspeed due to an inadequate G-loading for the bank angle at any given instant. The amount of slip and the rate of airspeed acceleration both depend on the roll rate. Once the bank angle is stabilized and the roll rate is zero, then there is little or no sideslip. The pitch dynamics will occur on a slightly longer timescale than the yaw dynamics: the airspeed and G-loading will continue to build for several seconds after the bank angle is stabilized. As the airspeed and G-loading build to the appropriate values for the bank angle and angle-of-attack or bar position, the glider will pull out of the initial dive into a more moderate glide path. As noted in our discussion of pitch-axis dynamics in Part One, the slight delay in the build-up of airspeed and G-loading are the reason that the glider goes through these dive-and-pullout dynamics rather than settling immediately into a stabilized glide that is exactly matched to the angle-of-attack and bank angle (at any given instant). During these oscillations the airspeed and G-loading are continually "out of phase" with the glide path and pitch attitude, and this is why the glider noses sharply down and accelerates, then slightly overshoots the "target" airspeed and G-loading, then pulls up into a more moderate glide, and then may actually go through several more pitch oscillation cycles before settling into the final, stabilized glide path. As I've noted, in my Spectrum it was challenging to observe the precise timing of all these events but we'll see them more clearly in some experiments I did in an airplane (see below). Of course the severity of these oscillations depends upon the abruptness of the changes in bank angle (and angle-of-attack): in many cases they will be almost undetectable in ordinary flight especially as we gain experience and learn to subconsciously correct for them.
The events in the pitch and yaw axes are both driven by the change in the bank angle, but are independent. If the bar is let out to increase the angle-of-attack as the glider is rolled into the turn, the airspeed can be held constant (if the stall angle-of-attack is not reached) but the sideslip will still occur. If the glider had a rudder the sideslip could be completely eliminated, but the glider would still pitch down into an accelerating dive if the control bar were not let out as the glider rolled into the turn. In fact with gliders with a lot of adverse yaw the initial sideslip toward the high wing will actually yaw the nose up above the horizon; correcting this with a rudder will yaw the nose down toward the low wing and move the nose further "down" relative to the horizon.
If a glider suddenly dips a wing due to turbulence, the glider will be in a slipping turn until the glider's yaw rotational inertia is overcome, at which time the turn will become coordinated in the yaw axis. (Alternatively the glider's inherent roll stability may return the glider to wing's level before the turn becomes coordinated; we will talk about this more in a later section entitled "Balancing yaw and roll stability".) The initial slip due to yaw rotational inertia is what causes the pilot to swing slightly toward the low side of the control bar as the glider is tipped to one side in turbulent air.
Getting back to my experimental observations in my Spectrum: at no time did my roll inputs create a strong sensation of being forced to one side of the control bar. On the whole I would say that sideslip had very little to do with the physical sensations that I experienced during these trials. On the other hand changes in G-loading due to pitch inputs were very noticeable. For example, "unloading" the wing by pulling in the bar during a constant-banked turn produced a general falling sensation due to the visual effect of the nose dropping and also due to a noticeable reduction in G-loading. The maneuvers involving full roll inputs, when I shifted fully to one side of the control frame, were the only instances where I saw full deflection of the slip-skid bubble (about four degrees from "level"). In these maneuvers, the yaw string in front of the base tube showed a maximum deflection of about 25 degrees. Based on previous experiments with other slip-skid bubble arrangements with more curvature, I don't think that the actual amount of slip (as "felt" by the slip-skid bubble) was ever much greater than the full deflection of four degrees from "level" described above. A bit of geometry based on the dimensions of my control frame shows that four degrees of bubble deflection corresponds to a shift in the "neutral" (hands-off) pilot hang position of only about 4 inches to the low side of the turn. While the pilot is making a large roll input and is fully shifted to the low side of the control bar, he will feel this slight change in the "neutral" hang position as a slight decrease in the muscular force that he must exert to hold himself against the low side of the control frame. (We can "simulate" this on the ground by hanging in a control frame that is tilted four degrees from "level"). Amidst all the other sensations of the steep turn entry, I had no awareness of this tendency to hang slightly low on the bar.
In all turns in my glider I had to remain slightly on the low side of the control bar to prevent the bank from decreasing (the maximum bank angle tested was about 40 degrees).
The main purpose of these experiments was to look at the slip and skid behavior of my glider while the bank angle or G-load was changing. I also had some interest in looking at the behavior of the glider in a steady, constant banked, constant speed turn: would the slip-skid bubble be centered? Would a yaw string be centered? Would yaw strings at various points on the glider show the effects of "airflow curvature", i.e. the way that the airflow follows the circumference of the turn? These questions were explored by flying with three yaw strings: one on a long "bowsprit" projecting almost 2 meters forward from the apex, one on the "probe" extending 60 cm from the center of the base tube, and one attached to the rear of the keel.
All the yaw strings generally agreed with the slip-skid bubble, deflecting the toward the high side of the glider in a slip when the bank angle was increasing, and toward the low side in a skid when the wings were rolling towards level. In addition the subtle deflections of the yaw strings in a steady, constant-bank, constant-airspeed turn did show some indication of airflow curvature, but my data weren't precise enough to look at this in great detail. About 8 degrees of airflow curvature (in the yaw axis) would be expected over the length of the keel, at 30 degrees bank and 24 mph. (See the table in Part Four for more on these calculations). The yaw string mounted 60 cm in front of the base tube showed a slight deflection toward the high side of the glider in a stabilized turn, indicating a slight sideslipping component in the airflow at this location. This deflection was roughly six degrees in a turn of 20 to 30 degrees bank. The yaw string at the tail end of the keel appeared to stream straight back in steady turns. The indication of the "bowsprit" yaw string wasn't noticeably different from the yaw string in front of the base tube. As noted earlier, it was challenging to make an accurate estimate of these deflections in steady turns because of the viewing angles involved, and because the indications were greatly affected by any accidental roll inputs. At steeper bank angles it was hard to hold the turn steady, so no comparison was made of the deflections at various bank angles or airspeeds.
So I couldn't see any difference in airflow direction over the 1.5 meters between the yaw string on the "bowsprit" and the middle yaw string (60 cm in front of the base bar), but I did see a noticeable difference over the 2.9 meters between the middle yaw string and the rear yaw string (at the tail end of the keel). The rear yaw string seemed to mark the point where the keel was tangent to the curving airflow. However, the fact that the slip-skid bubble appeared to be centered in steady turns shows that little or no slipping (centrifugal) aerodynamic forces were being generated, suggesting that the "average" airflow over the glider as a whole was well aligned with the keel. These results seem to be somewhat in conflict, since the center of surface area of the wing is about half a meter aft of the control frame, not back by the rear of the keel. In any case, regardless of the exact location of the point where the keel was tangent to the airflow, it is not surprising that some slip was indicated in the forward yaw strings (much more on this in Part Four). To gather more accurate data on these airflow angles, and to better explore various bank angles and airspeeds, cameras should be used to view the yaw strings and record bank angle and turn rate information, leaving the pilot free to concentrate on flying smooth circles!
DO THESE RESULTS APPLY TO OTHER HANG GLIDERS?
Since I did all these experiments in my Spectrum, I've left myself open to the criticism that my results are biased because I did my experiments in such a docile, user-friendly glider, and that higher-performance gliders will for some reason show more of a linkage between pitch inputs and sideslip. On this point I think its worth bearing in mind that the idea that pitch inputs control sideslip has deep roots in the early beginnings of hang gliding and did not arise with the advent of high-performance "blade wings". Nonetheless I certainly can't assume that other hang gliders will show exactly the same characteristics as my Spectrum. All rudderless aircraft will show some slip in turn entries and some skid while rolling out of a turn, due to rotational inertia in the yaw axis and adverse yaw. Gliders with more span, more mass, more rotational inertia, and less sweep and yaw stability will likely show more slip in a turn entry (and skid in a roll-out) than did my Spectrum. In general I would expect that G-loading changes (pitch inputs) would not affect yaw coordination and sideslip in other gliders any more than they do in mine, particularly if we are looking at the dynamics at a constant bank angle or at a particular rate of roll. However, Part Four we will look at some dynamics involving anhedral which may create some interaction between pitch changes, roll rate, and sideslip.
The subtle slip or skid characteristics in stabilized, constant-speed, constant-bank turns will be certain to vary markedly from one glider to another, depending on an interrelated web of factors including differential airspeed across the span, airflow curvature along the length of the keel, flex wing airfoil changes which depend on airspeed and airframe flexibility, adverse yaw created by flex wing effects and by pilot roll inputs, and the way that slip or skid interacts with sweep or dihedral to create a roll torque. Gliders that require high-siding may show a skid rather than a slip in a steady, constant-bank turn--more on all this in the second half of Part Four.
SUGGESTIONS FOR TEACHING METHODS
My findings suggest that yaw coordination is generally not affected by pitch inputs. These findings are at odds with the general understanding of sideslip among hang glider pilots. As we've already mentioned, we seem to believe that aircraft show a general tendency to slip toward the low wing whenever a turn is not correctly "coordinated" in the pitch axis, i.e. whenever the G-loading is less than "required" by the bank angle, and the flight path is arcing downward, and the airspeed is rising. My experiments in my Spectrum argue against this idea; as do some experiments I performed in an airplane and a sailplane which I'll describe in the next sections. As hang pilots we often fail to distinguish between a general sensation of falling or diving due to inadequate G-loading, and a definite swing toward the low side of the bar which is the mark of a sideslip. Looking at our training manuals, I believe that these points of confusion stem in part from some confusion about the basic physics of turning flight. The key points of interest are the net force on an aircraft in a turn at various G-loadings, and whether the pilot will "feel" a tendency to swing toward either side of the control bar, and whether the net force on the aircraft will drive a change in the yaw rotation rate and so create a temporary slip or skid. In this discussion (we'll get deeper into the physics in Part Four) I believe I've accurately explained why sideslip is driven primarily by the rate of change in bank angle and is generally not affected by changes in G-loading due to pitch inputs.
I think that our current lack of clarity about these issues originally began in the early days of hang gliding when the pioneers of our sport adopted aviation terms like "sideslip" and "coordination" without thinking carefully about how these concepts related to a 2-axis control system.
In fact I believe that our training materials and USHGA exams could be made both simpler and more accurate if many of the current references to sideslip were simply omitted. In many cases we should stick to terms like "min. sink speed", "high speed", "constant speed", and "accelerating" or "diving" to describe various types of turns. These words are clear and accurate and apply across the whole spectrum of hang glider designs. When the word "coordination" is used, we should be very clear whether we mean roll control to hold a constant bank angle (this is a loose but common usage), pitch coordination which controls our G-loading in the short run and our airspeed in the long run (this is usually what we mean in the hang gliding context), or yaw coordination which is the prevention of sideslip and is largely beyond our control in a rudderless aircraft. Clearly we need to teach students about the relationship between angle-of-attack (bar position) and turn performance, and also about all the nuances of the sensations that they will feel in flight, including the way that changes in G-loading relate to airspeed control. All these ideas can be communicated effectively and accurately without any reference to sideslip. By the way, I'm not the first to have these thoughts--one hang gliding instructor has recently told me that he has been using a similar approach in his own teaching for many years. When we do discuss sideslip we need to try for greater accuracy, regardless of how deeply we choose to delve into the underlying physics.
What would be the practical impact of adopting a more accurate point of view? I can't speak for everyone but I know that some students, especially those with some 3-axis flight time, find our current ideas overly complex and somewhat baffling. (Don't tell me, I know some readers would say the same about my own ideas too!). In my own case, coming to hang gliding with some 3-axis experience, I remember that when I completed my first altitude flights and first started ridge-soaring it took me several flights before I felt that I was starting to dial into smooth, safe turns without excessive diving or mushing. The basic advice in the training manuals about letting the bar out as I rolled into a turn was helpful, but the emphasis on yaw coordination and the prevention of sideslip led me to overdo the pitch inputs in a somewhat mechanical matter because I wasn't completely clear as to what the pitch inputs were supposed to be accomplishing. If I had been advised to simply "pull in for a bit of speed before turning, and then control the turn rate with the bank angle and control the airspeed with your pitch inputs", I'm sure that I would have dialed into the turns and gained a true feel for the glider a few flights sooner.
I also believe that a more accurate viewpoint would be helpful to any hang pilots who are transitioning to any kind of 3-axis aircraft, including "rigid-wings" like the Millennium. In many 3-axis aircraft yaw coordination with the rudder is quite important both for overall efficiency and also for spin avoidance, and it will be very dangerous if a pilot believes that holding the nose up with a pitch input is in some way equivalent to centering the yaw string or slip-skid ball with the rudder. I also think that a more accurate viewpoint would help us to analyze complex dynamics like lockouts on tow (see Appendix Two). Finally, regardless of the level of detail that we choose to include in the training manuals, I believe that increased accuracy will allow a simpler and more direct presentation of our dynamics.
I'm not arguing against the possibility of a coupling between pitch inputs and sideslip in specific maneuvers in specific gliders, as long this can be demonstrated through observations of a yaw string and slip-skid bubble. Later in this discussion I'll explore how anhedral may create a feedback between roll and sideslip. If the anhedral effect is strongest at low angles-of-attack, then we might simultaneously drive both slip and roll by pulling in the bar while rolling. The real point that I'm trying to drive home is that we can't even begin a thoughtful conversation about such interactions until we set aside our prevailing idea that all gliders sideslip whenever a turn is "uncoordinated" in pitch, i.e. whenever the G-loading is not properly matched to the bank angle. Nor can we make accurate observations in flight unless we are clear on the difference between the sensations caused by G-loading changes and the sensations caused by sideslip.
SO WHY DO WE LET OUT THE BAR WHILE ROLLING INTO A TURN?
What are the reasons behind our usual pitch "coordination" inputs? I'll list several here, most of which stem directly from our discussion of pitch dynamics in Part One, and all of which are unrelated to sideslip and yaw coordination. One, pulling in before for extra airspeed before rolling gives more control authority and a better roll rate. Then as we approach the desired bank angle, we often wish to go back to a higher angle-of-attack to minimize our sink rate. Two, a little extra G-loading as we are rolling into the turn may augment our weight-shift control input and the related flex-wing effects, and thus help the glider roll faster. Three, airflow curvature effects increase the angle-of-attack of the "tail" (i.e. the wingtips), which tends to pitch down the nose and decrease the overall angle-of-attack of the wing. This requires that we let the bar further out as the bank angle increases if we wish to maintain a constant overall angle-of-attack. (As previously noted, this effect is less pronounced in a flex-wing hang glider than in a rigid-wing, because the wing tips can flex (wash out) and shed some of their load). Four, imagine that we are rolling into a turn and want to end up at the same angle-of-attack (or the same bar position) that we had in wings-level flight. If we try to maintain a constant angle-of-attack (or bar position) as the bank angle begins to increase, we will eventually see the airspeed and G-load increase as needed but there will be some time lag in this process and the nose will initially drop sharply as the glider "falls" into an accelerating dive because the G-loading is inadequate for the bank angle at any given instant. The glider may then go through several pitch oscillations before settling into a steady glide. If we let the bar out a bit to increase the angle-of-attack as we roll into the turn, we can avoid these abrupt changes in pitch attitude, and our turn entry will be much smoother. Even if we want to end up in a fast, well pulled-in turn, we may chose to let the bar out a bit as are rolling into the turn, and then pull back in after the bank angle is stabilized. This will spread our acceleration out over a slightly longer time interval and take us more smoothly to our desired flight path, as we can avoid the marked changes and oscillations in pitch attitude, flight path, airspeed, and G-loading which would be caused by abrupt changes in bank angle with no pitch "coordination" input.
ACTUAL DATA: SLIP-SKID BEHAVIOR OF SAILPLANES AND AIRPLANES
(I wanted to get the "punch line" sections above before relating some further experiments in 3-axis aircraft, because I know that some pilots will be skeptical that we can draw parallels between weight-shift and 3-axis dynamics. Nonetheless the 3-axis aircraft afforded much better control and measurement of the bank angle and I was able to learn some things that were not obvious during the experiments in my Spectrum.)
When I performed the same experiments in a sailplane (Slingsby Swallow) and a light airplane (Cessna 152), flying with my feet off the rudder pedals, I saw dynamics similar to what I saw in my Spectrum: slip occurred mainly while rolling into a turn, and skid occurred while mainly rolling out of a turn. In turning flight, pitch inputs didn't seem have an effect on sideslip or skid, even when the aircraft were dramatically pitched up under a high G-load or were "unloaded" all the way to weightlessness (zero G's). The only significant difference between these aircraft and my hang glider was that in the three-axis aircraft there was no delay between the roll input and the slip: the slip began as soon as the ailerons were moved, and in fact the nose initially swung away from the direction of roll. (Many higher performance hang gliders will behave the same way). These points indicate that adverse yaw was a major cause of the sideslip in the airplane and the sailplane, in contrast to my Spectrum where adverse yaw seemed minimal.
I didn't look closely at the behavior of the airplane and sailplane in a steady, constant-bank, constant-speed turn; in general sailplanes tend to slip in a steady, constant-bank turn (if the rudder is not used) for reasons I'll explain in more detail in Part Four; in airplanes the engine torque (and p-factor, etc.) is a complicating factor so a steady turn may tend to slip or skid depending on the turn direction and airspeed.
See Appendix 4 for a rough comparison of the spanwise force created by my Spectrum 144 and by a Schweizer 2-22 sailplane, flying at similar sideslip angles.
I've done just a bit of aerobatic flying in 3-axis aircraft (wingovers, aileron rolls, and spins in airplanes, and one spectacular, fully aerobatic lesson in a sailplane with loops, rolls, a cloverleaf, a hammerhead turn, etc). These experiences were not controlled experiments but it never seemed that pitch inputs were affecting sideslip and yaw coordination, even during dramatic maneuvers with marked changes in the angle-of-attack and G-loading.
ACTUAL DATA: TIMING OF PITCH AND YAW DYNAMICS IN AN AIRPLANE
As noted above, in my experiments with my feet off the rudders, the airplane slipped mainly while rolling. The pitch dynamics occurred on a much longer time scale. I did some experiments where I fixed the control yoke so that it was free to move in roll but not in pitch--so no pitch "coordination" inputs were possible and the angle-of-attack was nearly constant--and looked closely at the timing of the changes in pitch attitude, airspeed, and G-loading as I rolled from wings-level into a steep (60 degree) banked turn. (I used a low power setting so the plane was normally in a descending glide). Sideslip was seen mainly while the bank angle was changing but the G-loading and airspeed continued to rise for about 10 seconds after the steep bank was established. This delay in the build-up of airspeed and G-loading created an initial "deficit" in G-loading (in relation to the bank angle) which caused the nose to pitch down quite steeply. The G-loading and airspeed then remained somewhat out of phase with the pitch attitude; for example the nose then began to rise toward a more moderate glide path and was actually approaching the horizon by the time that the airspeed and G-loading peaked put at their maximum values (after significantly overshooting the equilibrium values seen when the pitch attitude and airspeed finally stabilized) and began to decrease. The nose actually rose well above the horizon into a climbing attitude and flight path before beginning to drop again. Altogether the pitch attitude and airspeed went through at least two complete oscillation cycles before stabilizing into a steady descending spiral; this whole process took about 20 seconds after the steep bank was established.
One more detail: the nose actually rose briefly during the initial roll into the steep turn, because adverse yaw from the ailerons was yawing the nose toward the high wing. (The roll from wings-level to 60 degrees bank only required about 2 seconds). When I used the rudder to prevent this initial slip and keep the slip-skid ball centered throughout the experiment, this prevented the initial yaw toward the high side of the turn but had little effect on the subsequent pitch-axis dynamics described above.
In a hang glider the time scales and the magnitudes of the oscillations will be different but we will see the same basic dive-and-pullout dynamics when we roll quickly into a turn without making the usual pitch "coordination" input. I'm sure that the airspeed and G-loading will require several seconds to build and that the glider will go through several small oscillations in pitch attitude, airspeed, and G-loading before settling into a steady glide. Of course in a hang glider it's not so easy to perform experiments where the control bar position is completely constant in the pitch axis. The airplane experiment with the yoke fixed in the pitch axis vividly pointed out that as we enter a turn or make other changes in the bank angle, many of our small, almost unconscious pitch inputs are aimed at preventing large oscillations in pitch attitude and airspeed, whether we wish to hold a steady speed or to smoothly accelerate or decelerate to a new target speed.
Interestingly, in these experiments the turn rate seemed to correspond mainly to the bank angle. For example, immediately upon establishing the steep bank, the G-loading (total lift force) and the turning force (i.e. the horizontal or centripetal component of the total lift force) were still low but this was largely compensated by the low airspeed, so the turn rate was only a little less than after the G-load and airspeed built to their normal values
ACTUAL DATA: STEEP, REVERSING TURNS IN AN AIRPLANE
In these experiments I looked at several techniques involving a series of reversing turns with the goal of maximizing the sink rate. Dennis Pagen has suggested that to maximize the sink rate of a hang glider, for example to escape strong lift near cloud base, the pilot should fly as series of steep reversing turns, reversing the turn direction each time the airspeed and G-loading build to their peak values, and pulling in the bar each time the glider rolls into a new steep turn. The theory given behind this recommendation is that the glider is thought to be slipping whenever the airspeed and G-loading are building. (See Pagen's Hang Gliding Training Manual p.344 and Performance Flying pp.34 and 53). Dennis notes that the airspeed and G-loading may continue to build for several turns after the steep bank is established. Note that I've described a similar lag in the build-up of airspeed and G-loading, but based on my experiments in hang gliders and 3-axis aircraft I would expect the glider to be actually sideslipping only while the bank angle is increasing. Regardless of the role that sideslip plays in increasing the total drag, we would certainly expect the initial sink rate to be will quite high as the glider arcs over into a steepening dive until the airspeed and G-loading build up to their appropriate values. However I'm skeptical of the overall benefit of a series of changes in the bank angle and the angle-of-attack. For example, in low-speed wings-level flight we could create a high initial sink rate by firmly pulling in the bar, because the glider would dive steeply until the airspeed built up to match the new angle-of-attack and the G-loading returned to the normal value of (nearly) one "G", at which point the glider would round out into a more moderate glide path. Yet we would not use a series of these pulling-in pitch motions to sustain a high sink rate, because the nose would rise dramatically each time we let the bar back out, until we bled off some of the airspeed from the previous dive. We would expect to achieve the highest average sink rate over time by holding the bar fully pulled in, to maximize our average airspeed. Likewise the glider will pitch down steeply when we roll into a steep turn without the usual pitch "coordination" input, but it will also pitch up steeply as it bleeds off excess airspeed if we roll quickly from a steep, diving turn to wings-level without also pulling in the bar. We would expect to achieve the highest sink rate over time by holding the glider in a steeply banked turn rather than by using a series of turn reversals, unless perhaps sideslip due to adverse yaw and yaw rotational inertia was creating a great deal of drag during the turn reversals. By the same logic it's not at all clear why a repeated series of combined pitch and roll inputs would be the most effective way to sustain a high average sink rate, particularly if we don't expect our pitch inputs to contribute to sideslip during the turn reversals. Nonetheless in a recent telephone conversation Dennis told me that his reversing-turn method did yield a higher average sink rate than a sustained, pulled-in, steeply banked turn in experiments he performed in a Sensor and a Classic.
I performed my reversing-turns experiments in an airplane because this allowed for repeated climbs for altitude (altogether I burned off about 50,000 feet of altitude in all the repeated trials to look at all the nuances of the dynamics!) and also allowed much more precise measurement and control of the bank angle and better conditions for the recording of data (via a tape recorder). I used a low power setting so the plane was normally in a descending glide. Since I had already observed that pitch inputs had no apparent effect on sideslip in this aircraft, I fixed the control yoke so that it was free to move in roll but not in pitch, so that I could isolate the effect of changes in the bank angle upon the sink rate. No pitch inputs were possible; all turns were "uncoordinated" in pitch. This loosely simulated a hang glider with the bar "stuffed" or at some other constant bar position and nearly constant angle-of-attack. Therefore I did not exactly replicate Dennis's experiments but did gain some valuable insights into pitch and yaw dynamics which I believe do bear the sink rate question. In particular I saw very clearly that the pitch and yaw dynamics operated on different timescales (as described in the previous section), and that sideslips produced by adverse yaw and yaw rotational inertia as the bank angle was changing had a minimal effect on the overall drag and sink rate during these radical turning maneuvers.
For "method 1" I held the plane in a 60 degree bank. This produced the highest average sink rate. For "method 2" I flew a series of reversing 60 degree banked turns, reversing the turn direction every time the airspeed and G-loading reached their peak values. This method produced a slightly lower average sink rate than "method 1", presumably because the average bank angle and average airspeed were slightly less. Sideslip and skid were seen mainly when the bank angle was actually changing, just as in our hang gliders. The non-equilibrium pitch axis dynamics at play while the G-load was changing were complex, as I've described in more detail in the preceding section. Each time the steep bank was established, the airspeed and G-loading continued to rise for about 10 more seconds until they reached, and then significantly overshot, the steady-state values seen in "method 1". During part of this interval the sink rate was quite high as the glide path became very steep as the nose pitched steeply down, and then the nose began to rise again before the G-loading and airspeed reached their peak values. Each time the aircraft was rolled through wings-level as the turn direction was reversed, the excess airspeed and G-loading retained from the steep turn caused the nose to rise dramatically. One time I stopped the roll in the wings-level position, and a truly dramatic series of pitch oscillations resulted as the excess airspeed was bled off; in general it was obvious throughout "method 2" that the dynamics were far from the steady state and involved a lot of oscillations. For "method 3" I kept the aircraft constantly rolling between left and right 60 degree banks without waiting for the airspeed and G-load to peak out, to take advantage of the slips and skids that occurred mainly while the bank angle was actually changing. This method produced the lowest average sink rate, again presumably because of the lower average bank angle and airspeed. Clearly the slips and skids produced by adverse yaw and rotational inertia as the bank angle changed did not contribute a great deal to the overall drag of the maneuver in this aircraft, even with a fuselage and vertical fin to "feel" the full force of the sideways airflow.
Despite the fact that the control yoke was fixed in the pitch axis, I saw some variation of the angle-of-attack during some of the oscillations, and in particular the angle-of-attack was slightly lower at steep bank angles and high airspeeds. (Changes in angle-of-attack were detected by comparing the predicted, and observed, change in airspeed between the wings-level case and the stabilized 60 degree bank turn). This was appeared to be due mainly to propwash effects, but airflow curvature effects would certainly also contribute to this angle-of-attack change as described earlier in Part One. I also saw that the pitch control force (as distinct from the pitch control position) increased dramatically at high speeds, which of course is a familiar effect to both hang pilots and three-axis pilots.
How do these observations relate to flex-wing dynamics? Are there flex-wing effects that are somehow used to best advantage in the reversing-turns method in some gliders? In Part One we've already described how the curvature of the airflow (following the circumference of a turn) tends to pitch the nose down and reduce the angle-of-attack for a given bar position (or control stick position). We've noted that these effects are most pronounced when the turn radius is small (i.e. at hang-glider airspeeds) though they are somewhat alleviated by flex-wing effects. Also, the Lift / Drag ratio will tend to decrease as the whole wing flexes and "sheds G's" under load. All these factors would weigh in favor of the sustained, steeply banked, well pulled-in turn as the best method to sustain a high average sink rate in a flex-wing hang glider. We do also need to consider the linkage between pitch inputs, roll rate, and sideslip in any gliders where such a linkage can in fact be shown to occur. As the reader can no doubt tell I find myself a bit skeptical of the reversing-turns idea; at the very least it seems to me an open and interesting question whether Dennis's recommendations on this subject will apply to all flex-wing gliders. At some point when I have lots of altitude to burn I'd like to further explore this question both on my Spectrum and on a higher performance wing.
By the way, I'm presenting all this primarily as a point of aerodynamic interest. Dennis Pagen's comments on the dangers of vertigo in sustained steep turns are well taken, and at any rate I'm well aware that in very strong lift a glider must be flown out of the airmass before it can descend by any method. I should also emphasize that I'm not recommending any radical turning maneuvers for a glider that actually enters cloud.
PART FOUR: EXPANDED THEORY OF TURNS AND SIDESLIP IN HANG GLIDERS
We began this discussion with a detailed look at pitch-axis dynamics, and then we went through an overview of sideslip and turn dynamics before describing the in-flight experiments. Now I'm going to cover in much more detail the physics of turning flight and sideslip, considering more carefully the various effects that create the overall flight characteristics of the glider. This will fill out the theory behind the conclusions given above. We will focus first on the dynamics while the bank angle is changing, and then on the dynamics in a steady, constant-banked, constant airspeed turn. I'll also briefly describe an experiment I did in my Spectrum to look at the roll rate with and without a vertical fin. We'll end with some interesting Appendices covering topics like blind flying, and lockout dynamics, and also detailing one more experiment that attempted to quantify the sideways aerodynamic force produced in a slip, both in my Spectrum hang glider and in a sailplane.
FRAME OF REFERENCE IN TURNING FLIGHT
Except where otherwise stated, my reference frame is the outside world, not the accelerated reference frame of the pilot. I always use "centripetal" and "centrifugal" to mean real aerodynamic forces acting horizontally toward the inside or outside of the turn, not apparent forces as perceived by the pilot. Occasionally I will make reference to apparent side forces upon the pilot, meaning the tendency of the pilot and the slip-skid ball to "fall" toward the low or high side of the aircraft during a slip or a skid. I've chosen to use the term "G-loading" to refer to the lift vector produced by the wings, which always acts perpendicular to the wingspan; this is also the G-loading measured by the G-meter in the instrument panel of an aerobatic plane, which only detects forces acting "up" or "down" in the aircraft's reference frame. I've generally chosen not to use the term "G-loading" to refer to the total aerodynamic force at play in any given instant, which would include the drag (and thrust) vector and also any spanwise aerodynamic force vector created by sideslip.
When I use the term "airflow" I mean the "relative wind" created by the motion of the glider through the airmass. This "relative wind" or "airflow" is assumed to be aligned with the glider's flight path; I'm not taking into consideration any of the countless changes in the direction of the airflow as it encounters the wing and all the other parts of the glider's structure.
However we have already mentioned "airflow curvature" and will discuss this in much more detail later. By "airflow curvature" I mean the curvature in the airflow or relative wind that is caused by the fact that different points on the surface of the glider are actually moving through space in slightly different directions at any given instant during turning flight. Yes, this may seem strange at first--see the section entitled "Airflow curvature in turning flight" for much more!
WHAT MAKES AN AIRCRAFT TURN?
A turn is a curve in the flight path. The sole cause of a turn is a net force that is perpendicular to the flight path. This is called a centripetal force because it points toward the center of the circular flight path. The most efficient way of creating this centripetal force is by tilting the lift vector of the wings (banking). To keep this scenario going, the aircraft must rotate in the yaw axis to keep the heading in synch with the changing direction of the flight path (relative wind). Since aircraft are directionally stable relative to the airflow, this is easily accomplished, but an initial torque is needed to overcome rotational inertia. This is provided by the rudder, or by the force of the airflow against the vertical tail, as the direction of the relative wind changes at the start of the turn. In other words the turn will begin with a bit of sideslip if the rudder is not used.
In hang gliders, the wingtips provide the same function as would a vertical tail. Because the wing is swept, the surfaces which have the greatest moment arm (i.e. the greatest distance from the Center of Mass) are well aft of the Center of Mass, and generate drag in such a way that the glider always tends to weathervane into the relative wind so that the nose of the glider is aligned with the flight path through the airmass. (Later we'll see that airflow curvature effects will slightly complicate this picture.)
Once we establish a steady rotation in the yaw axis, the net torque in the yaw axis must be zero. In a steeply banked aircraft there is also a significant rotation in the pitch axis. In this case the aircraft's inherent pitch stability, which governs the angle-of-attack, must overcome rotational inertia to begin this rotation.
It is very common to see incorrect or incomplete descriptions of turning flight in the hang gliding training literature. For example, Dennis Pagen's bobsled analogy (p.128 Hang Gliding Training Manual and p.45 Performance Flying) runs into problems translating from a 2-dimensional flat surface to 3-dimensional space, and completely misses the fundamental connection between a sideways or centripetal force vector (acting perpendicular to the flight path) and the resulting curvature of the flight path which will produce a circular path through the sky. Pagen states as a general principle of flight that a banked wing will simply produce a sideways slipping force, and that a curvature of the flight path will occur only the pilot pitches up the nose to increase the angle-of-attack. Certainly the nose will drop and the airspeed will increase if the pilot enters the turn without the usual pitch "coordination" input, but I strongly disagree with the widespread idea that a glider will not begin to turn until the pilot makes a pitch-up "coordination" input.
If the pilot fails to make the usual pitch "coordination" input, then the G-load and turn rate will initially be somewhat below normal for the bank angle until the airspeed builds (although the low airspeed will partly compensate for the low G-loading so the initial turn rate will not be as low as we might expect). If the pilot temporarily loads the aircraft with "extra" G's while rolling into the turn then the G-load and turn rate will initially be above normal (for the bank angle) until some of the excess speed and G-loading bleed off. So it's very clear that the angle-of-attack is playing an important role in the turn dynamics; yet we cannot say that the aircraft will not turn if the angle-of-attack is not increased during the turn entry. The only instances where rolling into a bank will not produce a turn at all are if the pilot is holding the wing at the zero-lift, zero-G angle-of-attack (this is only possible in 3-axis aircraft; hang gliders don't have enough pitch control authority to keep the wing at the zero-lift angle-of-attack) or if the pilot is holding enough "top" rudder to create a strong sideslip which generates enough side force to completely cancel the centripetal (turning) force from the banked wing (this is obviously not possible in a rudderless aircraft).
On a related note, as long as the wing's lift force is the only aerodynamic force at play (besides drag), and there are no sideways aerodynamic forces such as may be created if there is a slipping airflow due to adverse yaw or yaw rotational inertia, then the net aerodynamic force is acting squarely "up" in the reference frame of the aircraft and pilot, and the pilot will feel no tendency to fall toward either side of the aircraft. This is true even if the aircraft is steeply banked and the nose is falling and the aircraft is accelerating into a steeper dive because the lift force (G-loading) is inadequate for the bank angle. This is also true even during radical aerobatic maneuvers such as a roll or loop. (Much more on this in later sections).
It is also not at all uncommon to see flawed descriptions of turning flight in the training literature for "general" aviation. I've seen one soaring manual that presented a coordinated turn as some kind of a balance between a loop (due to elevator action only) and a flat skidding turn (due to rudder action only). And then there is the ubiquitous table of G-loads versus bank angles, which states that the loads go infinite at vertical bank but rarely mentions that this analysis also assumes that infinite thrust and airspeed are available to maintain level flight. One excellent resource for anyone trying to work through the physics of turning flight is Wolfgang Langewiesche's 1942 physics-for-pilots classic Stick and Rudder (but see my comment at the end of Part Two regarding some details of his treatment of pitch inputs in turns!).
EFFECT OF SIDESLIP ON TURN RATE
Any centrifugal aerodynamic forces (toward the outside of the turn) will decrease the net centripetal force and slow the turn rate. A sideslip, where the airflow strikes one side of the aircraft (and also flows crosswise over wings and other surfaces) will create such a centrifugal force in most aircraft. The bank angle is then too steep for the net centripetal force and acceleration, and the pilot tends to "fall" toward the low side of the aircraft. A slipping turn describes a circle in space, just as any other turn, but the nose of the aircraft is always pointing toward the outside of the circular flight path, and the airflow against the "slewed" fuselage and other components of the aircraft generates forces that slow the turn rate.
During a slip at a steep bank angle, the side force created by the spanwise (slipping) airflow includes a significant vertical component that will bear part of the aircraft weight, and this will reduce the lift vector generated by the wing. For this reason, a slip actually reduces the net aerodynamic load on the aircraft and the net force or net G-loading experienced by the pilot at a particular bank angle. Again, the turn rate will be reduced, even though the centrifugal component in the aerodynamic side force from the slip starts to decrease as the bank angle increases past 45 degrees. The extreme case is sustained, high-speed, vertically banked, knife-edge flight by an aerobatic airplane. Here the wing is "unloaded" to zero G's, and the entire aircraft weight is borne by fuselage and vertical fin, which are flying at an "angle-of-attack" equal to the sideslip angle. Although the side force created by the slip no longer includes a centrifugal (horizontal) component, the turn rate is zero because the wing's lift vector has disappeared. The total aerodynamic load is one "G".
In aircraft with rudders, a sideslip can easily be made to completely cancel the turn at moderate bank angles--see Appendix 3 for more.
MORE ABOUT TORQUE
Note that unbalanced torques persist only briefly in aircraft. The aircraft will find its own equilibrium state. For example, if tip drag is pulling the right wing backwards during a turn to the left, that will yaw the vertical tail to the left, so that the airflow against it provides a counterbalancing torque. In hang gliders, the wingtips provide the equivalent effect of a vertical tail because they are well aft of the center of mass. The turn will continue at a steady rate in this slipping condition, with the nose yawed a bit to the outside of the turn.
REFERENCE FRAME IN A SIDESLIP, AND GENERATION OF SIDEWAYS FORCES AND DRAG
It is important to notice that aerodynamic terms like "centripetal", "centrifugal", "lift", and "drag", are all defined relative to the flight path (relative wind), not the aircraft heading. Therefore a sideslip is best described by saying that the nose is yawed to the outside of the flight path, so that the heading and the flight path are not the same. This could happen either by changing the flight path changing without changing the heading, or by yawing the glider to change the heading without changing the direction of the flight path. When I refer to the aerodynamic side force created by a slip, I mean the force component which is aligned with the glider's wingspan in the roll axis--acting toward the high wing--and is at the same time perpendicular to the flight path. At shallow bank angles this is mainly a horizontal force acting toward the outside of the turn (i.e. it is a centrifugal force) but as the bank angle steepens this side force remains aligned with the wingspan in the roll axis and so gains an increasing vertical component, which at very steep bank angles may bear a significant portion of the aircraft weight. Also, if the bank angle is very steep and if the flight path is steeply descending rather than level, then as this sideways aerodynamic force vector remains perpendicular to the flight path, it will be tilted significantly forward of the true vertical and so will contain a horizontal component which will point "straight ahead" along the aircraft's direction of travel. (It's not easy to pin this side force vector down with terms like "vertical" and "horizontal"; its primary reference frame relates to the direction of the flight path and airflow.)
This aerodynamic side force is actually a form of lift generated as the sideways (spanwise) airflow component impinges against the flat side areas on the fuselage, vertical tail, winglets, and other suitably shaped components of the aircraft. The sideslip angle can be viewed as the "angle-of-attack" of the fuselage and other aircraft components in the yaw axis. The sideways "lift" forces will be associated with induced drag loads which will steepen the glide path; other sources of drag will also increase as the aircraft meets the airflow in a less-than-optimal manner; these drag loads will act parallel to the airflow and so are not included in the aerodynamic side force vector.
Note that the sideways aerodynamic force vector created by the sideslip is not quite parallel to the wingspan because it is not strictly aligned with the wingspan in the yaw axis. As the aircraft's nose is yawed further away from the flight path, the direction of the sideways force vector does not change relative to the flight path (though its magnitude will increase). For example, in the case of the aerobatic airplane flying at a 90 degree bank angle and held in an approximation of horizontal flight by the application of lots of top rudder to hold the nose up and to generate a "lift" force from the slipping airflow over the fuselage, vertical tail, etc, the side force vector acts basically upwards and its direction (relative to the flight path) does not change as the pilot works the rudders to yaw the nose further up or down and change the "angle of attack" of the fuselage in the yaw axis.
HOW DOES A HANG GLIDER PRODUCE A SIDEWAYS AERODYNAMIC FORCE IN A SLIP?
How can a slipping hang glider, which has very little vertical surface area, produce a sideways aerodynamic force (and the associated increase in drag)? Think carefully about how the different components of a hang glider will react to a sideways (spanwise) airflow component. Vertical, round tubing and wires will simply change the direction of their drag vectors and therefore will generate some spanwise force components but will not generate any true spanwise "lift" forces acting perpendicular to the airflow. Lengthwise units such as the keel and the pilot's body, and streamlined downtubes, will be flying at an "angle-of-attack" (in the yaw axis) equal to the sideslip angle and so will generate some sideways "lift" components. Stepping and looking at the three-dimensional shape of the glider as a whole it is really quite hard to imagine how a hang glider in a sideslip could generate strong sideways "lift" components acting perpendicular to the airflow. With no fuselage side area and no vertical tail, we might expect the hang glider to be relatively "transparent" to a sideways airflow, so that the aerodynamic side force generated by a given sideslip angle might be minimal in a hang glider compared to other aircraft. This means that we might see a great deal of slip in the yaw string, but only a little displacement of the slip-skid bubble, and only a little apparent side force upon the pilot. If the total aerodynamic side force is small than the centrifugal (horizontal) component will also be small and will not slow the turn rate very much. This kind of slip would likely generate only a small amount of additional drag and so glide angle might not steepen very much. (See Appendix Four for a rough comparison between the forces generated in a sideslip by my Spectrum 144 and by a Schweizer 2-22 sailplane).
A slip generates drag in several ways: first, by the sideways motion of various aircraft surfaces through the air as described above. Second, if a given turn rate is to be maintained in spite of the centrifugal force from the slip (or if we desire to cancel the turn entirely and hold a constant heading in a slip in a 3-axis aircraft), the bank angle must be increased, leading to an increase in the wing's lift and drag vectors. Third, regardless of whether significant centrifugal forces are being generated, the wing is simply less efficient in the slipping airflow so the wing's induced drag increases. This may be the most important source of drag in hang glider slips.
DYNAMICS WHILE THE BANK ANGLE AND AIRSPEED ARE CHANGING:
SIDESLIP DUE TO YAW ROTATIONAL INERTIA
In a rudderless aircraft, the first thing that happens when the wings are banked is that the aircraft accelerates sideways toward the low wing. This is a curvature in the flight path: we are starting to turn. Since the actual heading of the glider has not changed yet, the airflow over the glider gains a spanwise component and centrifugal aerodynamic forces build in opposition to our sideways (centripetal) acceleration. These centrifugal aerodynamic forces keep the turn rate low. Meanwhile the slipping airflow interacts with the yaw stability of our aircraft, generating the torque to overcome inertia and start a rotation in the yaw axis. Once we are rotating in synch with the turn rate, then the sideways component in the airflow disappears, and we settle into a steady turn driven by the centripetal force from the banked wings. The yaw rotation rate is now steady, and there is no more need for a sideslip (at least so far as rotational inertia is concerned--later we will discuss reasons that we might see sideslip in a steady, constant-bank turn).
By the same reasoning we expect a bit of skid (airflow from the outside of the turn) when the bank angle is decreased and rotational inertia continues to yaw the nose around at the original yaw rotation rate.
This initial sideslip is not unique to hang gliders. As we will see in the section entitled "Balancing yaw and roll stability", any aircraft with sweep or dihedral depends upon this principle to keep the wings level in straight-ahead flight.
In higher performance hang gliders with more mass and wingspan and less wing sweep, the sideslip due to rotational inertia will be much more pronounced, especially at low airspeeds where the turn rate is high and the profile drag forces on the airframe are low.
One important caveat: as we roll past 45 degrees, the yaw rotation rate actually begins to decrease (while the pitch rotation rate continues to increase). Therefore above 45 degrees of bank, yaw rotational inertia reverses its effects, tending to promote a skid as the bank angle steepens and a slip as the bank angle decreases.
ADVERSE YAW
Another cause of slip in turn entries in hang gliders is the adverse yaw produced by the way the wing twists as the pilot weight-shifts. Just as in an aircraft with ailerons, the left and right airfoil shapes change, and there is a lift and drag increase on the wing that is being raised. This adverse yaw torque acts as an "anti-rudder", and initially can swing the nose in the opposite direction of the turn. This inhibition or reversal of the yaw rotation is a form of sideslip. The sideslip then interacts with the yaw stability of the aircraft to create its own torque, into the direction of turn. At some angle of sideslip all torques are balanced: we would expect to see this sideslip angle in a stabilized, constant-bank turn if the pilot had to low-side the bar. Conversely if the pilot had to high-side the bar in a constant-bank turn, this would create an anti-skid or pro-skid influence. In a hang glider adverse yaw is produced whenever the pilot is exerting a muscle force on the control frame, regardless of whether or not the bank angle is changing, and regardless of whether his body is on the glider centerline or is shifted to one side. (See the lockout discussion in Appendix Two for much more on this). Adverse yaw is greater at high angles of attack (i.e. at low airspeeds, or when the glider is "pulling" extra G's in an accelerated maneuver), and is greater in higher performance gliders with less wing sweep and more span.
As adverse yaw tends to push the nose the wrong way when a turn is initiated, it will certainly appear to the pilot that the turn rate is being greatly slowed. However, the actual change in the turn rate is not the same as the yaw rotation rate, and depends on how much centrifugal force the glider produces as it flies sideways through the air.
I can't come up with any way that our usual pitch "coordination" inputs would act to decrease adverse yaw, though flex-wing aerodynamics are undoubtedly so complex that almost anything as possible in some specific maneuver in some particular glider!
As Dennis Pagen has frequently pointed out, a weight-shift roll input would create adverse yaw even if there were no physical change at all in the airfoil shape, because the more heavily loaded inner wing would tend to fly faster. The way I like to think about this is that relative to the center of gravity of the whole system, the outboard wing has become slightly longer and has gained in surface area, while the inboard wing has become slightly smaller.
CONSIDERING ADVERSE YAW AND YAW ROTATIONAL INERTIA TOGETHER
The effects of adverse yaw and yaw rotational inertia must be considered in combination, not separately. For example, when a pilot is rolling a rudderless aircraft into a steeper turn, and the bank angle is already above 45 degrees, then yaw rotational inertia promotes a skid while adverse yaw promotes a slip. What is really happening here? Let's give a hypothetical example with some (completely hypothetical) numbers thrown in. Imagine that because of adverse yaw, when some particular rudderless aircraft is given a strong roll input by the pilot, the aircraft "wants" to fly with the nose yawed 20 degrees toward the high wing, relative to the airflow and flight path: this is the attitude where are all the yaw torques are in balance as the aircraft rolls. In this attitude, the increased drag of the outboard wing is balanced by the slipping airflow over the aircraft, so the net yaw torque is zero. As the aircraft rolls from 0 to 45 degrees bank, we know that the yaw rotation rate is increasing so we might find that the nose is actually yawed 30 degrees toward the high wing, relative to the airflow. This additional slip is caused by yaw rotational inertia as the yaw rotation rate lags behind the actual turn rate, and this additional slip is what provides the yaw torque that drives the increase in yaw rotation rate. As the aircraft continues to roll from 45 to 90 degrees bank, we know that the yaw rotation rate is now decreasing, so we might find that the nose is actually yawed now only 10 degrees toward the high wing, relative to the airflow. The decrease in the slip angle is caused by yaw rotational inertia as the nose tends to keep swinging toward the low wingtip at a high rate. Now the nose is actually aimed to the inside of the "balanced" position of 20 degrees, so the aircraft is now experiencing a net torque toward the low wing, which acts to drive a decrease in the yaw rotation rate.
(I haven't analyzed this closely but these rotational inertia effects may help to explain why an aileron roll involves some rather complex changes in rudder input to keep the slip-skid ball centered, even though the aileron deflection is constant and the airspeed is smoothly decreasing so adverse yaw should generally be smoothly increasing. The changes in G-loading through the roll will have some effect on the amount of adverse yaw present, too. See fig. 22-2 in William Kershner's The Flight Instructor's Handbook 3rd ed. (1993).)
In a rigid-wing glider like the ATOS or Exxtacy where roll control is achieved with spoilers and adverse yaw is minimal, then we would see only the effects of yaw rotational inertia, and so we might actually see a skidding airflow as we roll past 45 degrees.
Clearly, adding a vertical tail or winglets to a hang glider will increase the amount of torque generated by a given degree of sideslip, so rotational inertia and adverse yaw will be overcome more easily and less sideslip will be seen as the glider rolls into a turn.
EFFECT OF SIDESLIP ON ROLL RESPONSE
We've already described how a sideslip interacts with a glider's inherent yaw stability to create a yaw torque. Since roll creates sideslip through yaw rotational inertia and adverse yaw, this leads to a coupling between roll and yaw. A sideslip also interacts with sweep or dihedral to create a roll torque away from the side component in the airflow. Therefore the sideslip created by yaw rotational inertia and by adverse yaw will inhibit the roll response of the glider, not just the yaw rotation rate. Glider designs that show minimal sideslip while rolling into turns will also have good roll rates.
BALANCING YAW AND ROLL STABILITY
When an aircraft dips a wing in turbulence, it begins to turn. The rotational inertia of the aircraft creates a slip as the yaw rotation rate lags behind the turn rate. While the aircraft is slipping, the airflow interacts with sweep or dihedral and tends to roll the aircraft back to wings-level. However if the aircraft has excessive yaw stability (perhaps due to a large vertical fin or excessive sweep), it will quickly "weathervane" into a steady or tightening turn before it can roll back to level flight. On the other hand, with too little yaw stability the aircraft will slip excessively when the pilot begins an deliberate turn. As the glider slips it resists the pilot's intentions in both the yaw and roll axes.
ANHEDRAL EFFECTS AND POSSIBLE LINKS TO PITCH INPUTS
The above comments refer to an aircraft that is stable in roll. The same principles, but acting in the reverse sense, will apply to a glider that is unstable in roll. Consider a hypothetical high performance "blade wing" glider with little sweep, little yaw stability, and much rotational inertia and adverse yaw. This glider will be designed with a lot of anhedral to increase roll response. With enough anhedral, the glider will actually be unstable in roll, by which I mean that it will tend to roll toward rather than away from a slipping airflow and will therefore show a feedback effect between sideslip and roll when the pilot makes a roll input or a wing is lifted in turbulence. A careful look at the geometry suggests that the anhedral effect might be significantly stronger at low angles-of-attack, so we might simultaneously drive both slip and roll by pulling in the bar while rolling. This might explain the strong connection between pulling in the bar while rolling into a turn, and sideslipping, that pilots report for some high-performance gliders. (See for example p.21 of Dennis Pagen's article "Hang Glider Turn Perspectives" in the April issue of Hang Gliding; in a recent telephone conversation Dennis mentioned that he noticed this on the Moyes CSX glider in particular). I need to strongly emphasize that I've reached the hypothetical stage here: I don't have experience with high performance "blade wings" and don't know whether are fundamentally unstable in roll in this way or show this linkage between pitch inputs and the effectiveness of the anhedral. I also want to emphasize that this effect would not create a coupling between pitch inputs and slip in cases where the pilot was holding a steady bank angle as he pulled in the bar.
EFFECT OF A FIN ON ROLL RATE
I was thinking about the effect of a vertical fin on roll rate during a recent visit to Wallaby Ranch. It was clear that many pilots preferred to fly without the fins that new students use to learn aerotowing, saying that many gliders roll poorly and don't thermal well with fins. For gliders that are stable in roll, a fin should actually increase the roll rate by damping sideslip as described just above in "Balancing yaw and roll stability". In the case of the hypothetical blade wing with extreme anhedral and an unstable coupling between slip and roll, a fin would decrease the roll rate (but I'm a bit skeptical that any actual hang gliders behave this way!) Notice that a rigid-wing glider with positive dihedral as well as some sweep will likely benefit quite a bit from a vertical fin, in terms of roll rate. I took a couple of tows in the calm morning air in my Spectrum to look at roll rate with and without a fin, and found no measurable change. More on how a fin might affect handling in steady, constant-bank thermal turns later.
WHY DO AIRSPEED CHANGES AND SIDESLIPS OCCUR TOGETHER? A COMPLETE DESCRIPTION OF THE DYNAMICS IN THE PITCH AND YAW AXES AS THE GLIDER IS ROLLED INTO A TURN
This section is really a repetition of information contained in the earlier sections on the pitch-axis dynamics and the analysis of the in-flight experiments; I give it again here for the sake of completeness.
Let's review once again the sequence of events in the pitch axis as the glider is rolled to a steeper bank angle, in an artificially "telescoped" breakdown of events: first, the pilot shifts his weight to the side. Any adverse yaw due to differential airfoil shapes will come into play at this time. Second, a significant bank angle is developed, and the flight path begins to curve due to the centripetal force from the wing (the turn has started). The heading of the glider has not yet changed, so there is a sideways component in the airflow which creates centrifugal aerodynamic forces that keep the turn rate low. Third, the aircraft begins to rotate in the yaw axis due to its inherent yaw stability (the "weathervane" effect), which creates an aerodynamic torque to overcome yaw rotational inertia and swing the nose into the relative wind. This yaw rotation removes the sideslip and allows the turn rate to increase. As long as the roll rate is substantial, a significant sideslip angle can be maintained, because the bank angle and centripetal force can keep "ahead of" the yaw rotation rate of the glider. When the bank angle is stabilized, the slip will soon end except for effects such as differential wingtip drag, and adverse yaw if the pilot is low-siding the bar.
If the angle-of-attack is not increased by letting out the bar as the glider rolls, then the vertical component of lift is being diminished as the bank angle increases, and so the airspeed must also increase as we've already noted. The airspeed takes some time to build, so the airspeed and G-loading remain lower than their equilibrium values (for the bank angle and angle-of-attack) until the glider stops rolling and the airspeed can "catch up". This makes the flight path curve downwards as the glider "falls" and the airspeed builds. While the airspeed is in transition, the airspeed, lift and drag vectors, G-loading, turn rate, and glide ratio are all slightly less then their equilibrium values for the angle-of-attack and bank angle at any given instant, and there is a downward curvature in the flight path. The airspeed and G-loading will then build and then slightly overshoot their steady-state values, leading to a slight upward curvature of the flight path that brings the glider toward its final, equilibrium glide path. Over several seconds these oscillations will damp out and the glider will settle into its stabilized, constant-airspeed flight path. The slight delay in the build-up (and loss) of airspeed and G-loading (lift) is the reason that the aircraft goes through these oscillations instead of immediately settling into the appropriate glide path for the bank angle and angle-of-attack at any given instant. The airspeed will eventually stabilize at some value dependent only upon the bank angle and the position of the pilot's body on the control bar (angle-of-attack), and the wing's lift vector (G-loading) will then be stabilized at a value depending only mainly the bank angle (and to a small extent upon the L/D ratio). The glide path or glide ratio is now stabilized at some angle determined only by the bank angle and the L/D ratio (which is governed by the angle-of-attack; the angle-of-attack in turn is governed mainly by the bar position and to a small extent by the bank angle as described earlier in Part One).
The airspeed may be held constant by letting out the bar and increasing the angle-of-attack as the glider is rolled. (However, before the glider is rolled very far, the stall angle-of-attack would be reached, so for very large changes in bank angle an increase in airspeed cannot be avoided.)
All of these pitch-axis dynamics are independent of yaw coordination and sideslip. The sideslip due to rotational inertia is roughly synchronized in time with the airspeed increase (if any) created by the pitch dynamics, but the slip is not caused by the pitch dynamics or the airspeed increase. If the airspeed is held constant by letting out the bar to "coordinate" the turn, the sideslip will still occur. The degree of sideslip depends on the roll rate rather than on the pilot's pitch inputs. In an aircraft with rudders where the sideslip can be prevented altogether, the airspeed increase will still occur unless the angle-of-attack is increased. But the fact that the roll rate simultaneously drives these dynamics in the pitch and yaw axes is no doubt one reason that many hang glider pilots believe that pitch inputs must have an effect on sideslip, or that the sideslip itself is the main cause of the airspeed increase as the glider seems to "fall" off toward the low wing.
Interestingly the component of sideslip that is due to adverse yaw (and not rotational inertia) will initially actually raise the nose relative to the horizon, as the glider yaws toward the high wing. On the long run a sideslip usually does cause the nose to fall and the airspeed to increase, because the wing is less efficient in the slipping airflow. On the other hand the fact that the sideways aerodynamic force vector bears a significant part of the aircraft weight during a sideslip at a very steep bank angle means that a sideslip will actually relieve some of the pitch-down motion and acceleration in airspeed that we would otherwise see as a glider rolls to an extreme bank angle. But all these are one-way relationships: I'm maintaining that pitch dynamics and pilot pitch inputs have little effect on the sideslip angle.
If a pilot pulls in the bar in while shifting to the side, this will exaggerate the effects that we see when we roll into a turn without letting out the bar. The initial G-loading and turn rate will be decreased and more airspeed will be gained. As long as the control bar is kept moving aft, it contributes to the same "unloading" effects as are caused by the steepening bank angle: because the airspeed is in transition, the airspeed, G-loading, turn rate, and glide ratio are less than the values predicted by the bank angle and angle-of-attack at any given moment, and the flight path curves downward. Again, these effects are independent of the sideslip condition of the glider which has its own effect upon the turn rate.
These pitch-axis effects look much like the effects of an increased sideslip angle: the nose pitches down as the glide path steepens. The crucial difference is that dynamics in the pitch axis convert altitude into kinetic energy as well as into drag, while a sideslip produces an increase in the drag coefficient that need not coincide with an increase in airspeed. This is why a sideslip is such a valuable tool for controlling the final approach path in a 3-axis aircraft, where it may be sustained indefinitely (and with no change in airspeed) even in straight-ahead flight. The large sideslip angles available to 3-axis pilots allow this maneuver to be effective even at low airspeeds, whereas a hang glider sideslip might not produce a lot of drag unless performed at a high airspeed.
If a pilot pushes out excessively to strongly increase the angle-of-attack and really "load up" the glider and "carve out the turn", the G-loading and turn rate will immediately increase, but then will settle back to their equilibrium values (for the new angle-of-attack) as the excess airspeed is bled off. The final value of the G-loading will depend only upon the bank angle (with small variations according to the L/D ratio), while there will be some net increase in the turn rate due to the lower airspeed. All these dynamics will occur regardless of the sideslip condition of the glider before and after the pitch input.
In a steady turn with a stabilized G-loading, the turn rate is inversely proportional to the airspeed, regardless of the sideslip condition of the glider. The turn radius is inversely proportional to the airspeed squared. As the airspeed is decreased, the tighter turn might seem more "coordinated" to the pilot, but this really has nothing to do with yaw coordination and sideslip.
Since a sideslip in a hang glider occurs primarily while the bank angle is changing, the only way to keep the glider slipping is to fly the glider in a series of reversing turns; see the section in Part Three entitled "Actual data: steep reversing turns..." for some thoughts on the effectiveness of this technique.
FUNDAMENTAL RELATIONSHIPS: DOES AN "IMBALANCED" G-LOADING CREATE A SIDEWAYS FORCE ON THE PILOT?
Many pilots believe that if the magnitude of the G-load produced by the wings (lift force) is not properly matched to the bank angle, then there will be some kind of force imbalance that will push the pilot (and the slip-skid ball) toward the low side of the aircraft. (For example see figure 3-1 in Pagen's Performance Flying). I've been arguing that this isn't the case as long as there is no sideways component in the airflow over the glider. The G-load produced by the wing always acts upward (or downward when pulling negative G's) in the reference frame of the pilot, and does not exert any side force on the pilot. We can easily see this by flying aerobatic maneuvers in an airplane or sailplane. As long as we use the rudder to offset effects like adverse yaw and keep the yaw string or slip-skid bubble centered, we won't experience any tendency to fall toward the low side of the aircraft even as we are passing through a vertical bank angle during an aileron roll or wingover. This is true regardless of whether we are sinking into the seat-bottom cushion under a heavy positive G-loading or have "unloaded" the wing to zero G's and are floating weightless in the cockpit as the aircraft follows a ballistic trajectory.
However, if we used the rudder to make the aircraft fly in a sideslip (or if we allow yaw rotational inertia or adverse yaw to create a sideslip), then the fuselage and other components would produce sideways aerodynamic forces that would push us toward the low side of the cockpit. The extreme example of this is sustained, knife-edge flight in an aerobatic aircraft at a 90 degree bank angle, where top rudder is applied to hold the fuselage at a high "angle-of-attack" to the airflow to keep the flight path horizontal. In this case the wing is unloaded to zero G's, and the aerodynamic side force produced by the slipping airflow over the fuselage, vertical fin, etc. is entirely vertical and supports the entire weight of the aircraft. The pilot "feels" a 1 G force toward the low side of the fuselage as the aircraft pushes up on him (through the seat belts or the cockpit sidewall) in the opposite direction (away from the earth, and towards the high side of the cockpit) with a force equal to his body weight.
A fascinating point here is that the pilot feels only the forces produced by the aircraft. These forces are transmitted to him through the seat and seatbelts (or in a hang glider, through the hang strap and through the pilot's arms on the control bar). The force of gravity itself is not apparent in the pilot's reference frame, because it accelerates the aircraft and the pilot together. (Also the force of gravity is not felt by the pilot's muscles and nervous system because it accelerates every molecule of the pilot's body equally). If the total aerodynamic (and propulsive) forces equal zero G's, then the pilot will be weightless in ballistic flight--astronauts experience weightlessness for precisely this reason since drag, thrust, and lift are all absent in space once the engines are switched off. Gravity is present but not felt. And anytime the lift vector (plus other aerodynamic and thrust vectors) happens to equal +1G in the "upward" direction, relative to the pilot and aircraft, the pilot will feel the "normal" forces of level flight, even while banked 90 degrees, or inverted at the top of a loop or aileron roll.
Clearly, any vector diagram illustrating why the pilot "falls" toward the low side of aircraft during a sideslip must include the sideways aerodynamic force (aligned with the wingspan in the roll axis) that is created by the sideways or spanwise airflow, as this is the sole reason for an apparent sideways force in the pilot's reference frame. Yet nearly all vector diagrams that attempt to illustrate slips in hang gliders omit this sideways aerodynamic force vector, and "put the cart before the horse" by implying that an imbalance between the lift vector (G-loading) and the weight or gravity vector is the direct, immediate cause of an apparent sideways force on the pilot which allows him to fall toward the low side of the control bar. This erroneous conclusion invariably stems from an inaccurate treatment of the weight (gravity) vector and also of "centrifugal force". Much more on this later when we will learn how to draw our own, accurate vector diagrams for turning and slipping flight.
FUNDAMENTAL RELATIONSHIPS: WHY DOESN'T THE AIRCRAFT SLIP TOWARD THE LOW WING WHEN THE G-LOADING IS INADEQUATE IN A TURN?
In the previous section we saw that the aircraft must slip sideways through the air to create an apparent sideways force upon the pilot. If the G-loading