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There seems to be some confusion about where the 'zone of reversed commands' really is, i.e. where a decrease of airspeed results in an increase of drag.

Some references place that zone to the left of the minimum of the power required curve, while others place it (correctly, in my opinion) to the left of the minimum of the thrust required curve... Those minima are separated by a 30% difference in airspeed, hence the distinction isn't academic...

Two examples of these contradictory references, one taken from the internet, and the other from a book on gyros ('Flugphysik der Tragschrauber'):

enter image description here

enter image description here

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    $\begingroup$ notice both references put it at before minimum sink speed, or highest L/D, or "best endurance". Here, it is not contradictory, and applies to all aircraft. In a glider, if you trimmed back to a lower airspeed, your sink rate increases, burning more "fuel" (altitude). $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 11:04
  • $\begingroup$ @RobertDiGiovanni. No... You are very wrong... The best endurance speed is NOT the speed for best L/D... The minimum of the drag curve is indeed the airspeed for highest L/D, but the minimum of the power required curve marks a different airspeed, the 'best endurance speed', that is about 30% less than the airspeed for best L/D. $\endgroup$ – xxavier Oct 8 '19 at 12:27
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    $\begingroup$ You wrote minimum sink speed, or highest L/D, or "best endurance" For propeller propulsion, that is wrong. Minimum sink speed is indeed the best endurance speed, but it's not the highest L/D speed... $\endgroup$ – xxavier Oct 8 '19 at 13:22
  • $\begingroup$ we were both looking at the wrong curve. In level flight, "best endurance" is lowest fuel consumption, not maximum power available, nor even most efficient generation of thrust per gallon (optimal PROP AOA). No, it's just thrust, which is (perhaps not linearly, but always proportional) to RPM, which is proportional to fuel burn. So the curve to look at in this case is prop RPM. A minimum will be found at minimum sink rate. Think if it this way, glide first, find min sink speed, add any engine you want. In ALL cases (In level flight) you will burn more fuel going faster or slower. $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 16:06
  • $\begingroup$ Vorderseite = front side, Ruckseite = back side $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 23:54
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This is commonly known as the Speed Stability, not to be confused with speed stability in the sense of static longitudinal stability. I think there's some common confusion with this phenomenon. The first part of this answer addresses the OP's question; the second part clarifies a common confusion.

1. Main Answer

In level flight (i.e. zero vertical rate), the longitudinal equation of motion can be succinctly written as:

$$m\dot{V}=T-D$$

$m$ is airplane mass, $V$ is airspeed, $T$ is thrust and $D$ is drag. Now if we express thrust and drag as first order approximation as a function of change in airspeed ($\Delta V$) from the trimmed condition, we have:

$$T=T_0+\frac{dT}{dV}\Delta V=T_0+T_V\Delta V$$ and $$D=D_0+\frac{dD}{dV}\Delta V=D_0+D_V\Delta V$$

At trim condition, we necessarily have $T_0=D_0$. So now we have a new equation of motion:

$$m\dot{\Delta V}=(T_V-D_V)\Delta V$$

This equation is a first order ordinary differential equation, and is stable if $T_V-D_V<0$ and unstable otherwise.

  • For a jet plane, thrust is fairly constant in flat rated conditions and $T_V$ is approximately zero. Thus, $D_V=0$ corresponds exactly to minimum drag or minimum thrust required (where $C_{D_0}=C_{D_i}$ for high aspect ratio, low Mach airplanes).

  • For a propeller plane, power is constant, but now the stability criterion is $-\frac{P}{V^2}-D_V<0$. This corresponds neither to minimum power required nor minimum thrust required.

2. Addendum

What does this result mean, exactly? If an airplane is trimmed in the speed unstable regime, will it decay toward stall if it experiences a speed perturbation with pilot hands-off, even if it's statically longitudinally stable?

Remember, the equation we began with only holds in level condition where the airplane is neither climbing nor descending. Thus, the pilot must be holding altitude with elevator while the speed is changing. The conclusion for speed instability is:

  • If the altitude deviates below the trim altitude, pulling up will decrease the airspeed, resulting in further energy deficiency. Speed will continuously decay in this manner.
  • If the altitude deviates above the trim altitude, pushing down will increase the airspeed, resulting in further energy excess. Speed will continuously accelerate until reaching a stable point on the other side of the polar.

Since the pilot is in the loop, it has nothing to do with the basic aircraft eigenmodes.

The same result can be obtained if we aim for constant flight path angle. In approach configuration, this is also called flight path stability, which I think is a better name than speed stability.

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  • $\begingroup$ May the downvoter please add a comment on what is perceived to be inaccurate? $\endgroup$ – JZYL Oct 7 '19 at 9:24
  • $\begingroup$ hands off recovery from a pitch perturbation is a hallmark of static stability, which is a created with aerodynamic tail downforce (pitches up when plane too fast) and forward set CG (pitches plane down when plane too slow). While this esoteric "speed unstable" "zone of reversed command" lingo makes for interesting theoretical discussion, there is no way I would even mention the terminology to a student pilot. Flying at 35 knots indicated with full throttle kind of got the point across for me. $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 22:54
  • $\begingroup$ now on to your (good) work. Power supplied is not constant, it is controlled by the throttle. Efficiencies aside (for now), more thrust requires more RPM, more RPM requires more fuel. The curve we should be looking at (for endurance) is again minimum thrust. Do we have a case where lower prop RPM produces more thrust? No. So maximum endurance will be at the lowest RPM setting possible. Honestly, the force x distance definition pains me. Power as force over time (fuel burn per hour) makes more sense. $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 23:04
  • $\begingroup$ @RobertDiGiovanni I don't understand what you're trying to get across here... $\endgroup$ – JZYL Oct 8 '19 at 23:24
  • $\begingroup$ 1. Your work is interesting. 2. Minimum thrust is proportional to throttle setting. Maximum power available curve doesn't seem to have anything to do with this. $\endgroup$ – Robert DiGiovanni Oct 8 '19 at 23:30
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Both are right in their own ways.

First the basics: Power is thrust times speed and is constant over speed for variable-pitch propeller-powered piston aircraft. Therefore, propeller thrust is proportional to speed inversed. The minimum power required coincides with the point of maximum excess power since it needs the lowest power setting for trimmed flight. The minimum thrust required is when the absolute drag is at its minimum. Since variable-pitch propeller thrust is inverse to speed, flight at minimum drag needs more power and happens at a higher speed than trimmed flight at the maximum excess power speed in propeller-powered piston aircraft.

Thrust, drag and excess power over speed

Thrust (green), drag (red) and excess power (blue) over speed for a propeller aircraft. The broken line is for trimmed flight at the lowest power setting. Numbers for a light GA aircraft with 106 kW and 1300 kg mass.

Now to the instability: Normally, if you increase speed you need more power to trim the aircraft at this higher speed. Without pilot intervention, the aircraft will slow down to the trimmed speed. This is a stable process. Works in reverse, too. In the plot above in the stable range, you can see that if you move away from the trim point, drag will drop less than thrust, so any speed increase will need a higher power setting and the aircraft falls back to the trim point. This works for every power setting to the right of the maximum excess power speed.

Below that point this condition inverses: Now drag will increase more with a speed decrease than thrust will increase. To the left of the maximum excess power speed, a deviation from the initial speed will either speed the aircraft up or slow it down with no hope of ever returning to the initial state without pilot intervention.

Obviously, the maximum excess power speed (which is the minimum power required speed and the best endurance speed for propeller aircraft) is the boundary between stable and unstable behavior. For propeller aircraft.

With turbojets, thrust is roughly constant over most of the subsonic speed range and now the maximum excess power speed coincides with the minimum thrust required speed. Your book deals with turbojets while those web pages explain speed stability for variable-pitch propeller aircraft.

Note that fixed-pitch propellers show a linear increase of efficiency over speed below their optimum advance ratio, so here again the thrust is approximately constant over speed. But that holds only over the slow speed range when the aircraft flies slower that what the propeller advance ratio would like.

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    $\begingroup$ I disagree with your characterization in the third paragraph. Speed instability (imo a terrible misnomer) has nothing to do with speed stability in the sense of static longitudinal stability. $\endgroup$ – JZYL Oct 6 '19 at 10:01
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    $\begingroup$ @Jimmy: Look at the equations of motion and solve them for different speeds. The eigenvalues will produce an unstable condition for speeds below that instability boundary. This is the same kind of instability as the roll instability. And then change cg location and see what that does to the eigenvalues. After having done this, would you still think that both have noting in common? $\endgroup$ – Peter Kämpf Oct 6 '19 at 10:33
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    $\begingroup$ I have done just that and I can confirm this is not true. Drag only improves phugoid stability, and short-period cannot go unstable without static long stab going unstable. $\endgroup$ – JZYL Oct 6 '19 at 10:54
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    $\begingroup$ @Jimmy: When the cg is shifted aft to an unstable position, the complex eigenvalue pair of the phygoid splits into two real eigenvalues, one of which crosses into the positive side, signaling instability. Just what the speed stability eigenvalue does when speed drops below the stability limit. If this did not happen, you did something wrong. Drag is not an independent variable in this. The short period mode will split into two real pairs later, when cg is shifted behind the maneuver neutrality point. $\endgroup$ – Peter Kämpf Oct 6 '19 at 12:58
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    $\begingroup$ I think may be erroneous to link the concept of the "region of reversed command" to pitch stability. See my alternative answer. $\endgroup$ – quiet flyer Oct 6 '19 at 13:50
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Well, this answer is not from me, but I found it, minutes ago, in Richard von Mises' 'Theory of Flight' Dover Books, ISBN 978-0-486-60541-8.

'The abscissa where the two power curves have parallel tangents...'

enter image description here

enter image description here

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One definition of the "region of reversed command" is the part of the flight envelope where a small aftwards movement of the stick or yoke, causing a small decrease in airspeed, with no change in the position of the throttle or thrust lever, will eventually lead to a net increase in sink rate, not a net decrease in sink rate.

In the "region of reversed command", you must advance the throttle or thrust lever, not retard it, to maintain altitude (or to maintain a constant climb rate or sink rate) as you slowly move the control stick or yoke aft to decrease airspeed.

Note that this does NOT mean that in the "region of reversed command", if you want to maintain a constant altitude as you move the throttle or thrust lever forward, you are forced to move the stick or yoke aft and let the airspeed decrease. You could instead put the stick forward to allow the aircraft to accelerate out of the "region of reversed command".

If for a given position of the throttle or thrust lever, your aircraft's engine is putting out constant horsepower regardless of airspeed, which is characteristic of a piston or turboprop engine, then the "region of reversed command" will be the part of the flight envelope where the airspeed is lower than the airspeed for minimum power-required. I.e. the part of the power-required graph that lies to the left of the airspeed where the minimum power is required.

On the other hand, if your aircraft is putting out constant thrust regardless of airspeed, which is characteristic of a jet engine with no propeller, then the "region of reversed command" will be the part of the flight envelope where the airspeed is lower than the airspeed for minimum thrust-required. I.e. the part of the power-required graph that lies to the left of the airspeed where the minimum thrust is required.

A more nuanced approach would consider the shape of the power-required curve and the power-delivered curve for any given position of the throttle or thrust lever. On such a graph, the "region of reversed command" is any region where, as we decrease airspeed, we make the value of (power delivered minus power required) get less positive or more negative. This is the part of the flight envelope where a decrease in airspeed will lead to a decrease in climb rate or an increase in sink rate, with no change in the position of the throttle or thrust lever.

This answer could be expanded to consider the effects of constant-speed propeller.

All the above content could also be restated and simplified to eliminate any reference to the position of the control stick or yoke, and focus solely on airspeed. Also if we wished, we could eliminate any reference to the position of the throttle or thrust lever, and focus solely on thrust-required or power-required. It's just a matter of definitions. If by "region of reversed command", we simply mean that we need more thrust to maintain altitude while flying a little slower than while flying a little faster, than obviously the "region of reversed command" is the part of the thrust-required graph that lies to the left of the minimum thrust-required point. Similarly, if by "region of reversed command", we simply mean that we need more power to maintain altitude while flying a little slower than while flying a little faster, than obviously the "region of reversed command" is the part of the power-required graph that lies to the left of the minimum power-required point. It's just a matter of defining our terms.

We could also define the "region of reversed command" slightly differently-- as the part of the flight envelope where a small aftwards change in the position of stick or yoke, leading to a small decrease in airspeed, causes the glide angle (relative to the airmass) to get steeper rather than shallower. In the power-off case, this region would include any airspeed lower than the best L/D airspeed.

With any of these various definitions, it would seem to be an error to suggest that being in the "region of reversed command" profoundly changes an aircraft's basic pitch stability dynamics and/or pitch control response dynamics and/or speed stability dynamics, unless we've introduced an autopilot (or human pilot) into the loop that is trying to use pitch control inputs to maintain altitude or maintain a set climb or descent rate or stay on a fixed glide slope, etc.

Another answer has inspired the following thoughts:

In the "region of normal command", we may control the aircraft in any of the following ways:

1) Increase airspeed by moving the control stick or yoke forward, and decrease airspeed by moving the control stick or yoke aft, with no change to the position of the thrust lever or power lever (throttle). The climb rate or sink rate will not remain exactly constant.

2) Increase airspeed by moving the control stick or yoke forward, and decrease airspeed by moving the control stick or yoke aft, while simultaneously adjusting the thrust or power level (throttle) as needed to hold altitude or climb rate or sink rate constant. (Move the lever forward to increase climb rate or decrease sink rate, and aft to decrease climb rate or increase sink rate.)

3) Increase sink rate (or decrease climb rate) by moving the control stick or yoke forward, and decrease sink rate (or increase climb rate) by moving the control stick or yoke aft, with no change to the position of the thrust or power lever (throttle). The airspeed will not remain exactly constant.

4) Increase sink rate (or decrease climb rate) by moving the control stick or yoke forward, and decrease sink rate (or increase climb rate) by moving the control stick or yoke aft, while simultaneously adjusting the thrust or power level (throttle) as needed to hold airspeed constant. (Move the thrust or power lever forward to increase airspeed, and aft to decrease airspeed.)

In the "region of reversed command", only methods 1 and 2 will work. Methods 3 and 4 will not work.

Other methods of controlling the aircraft that will work in the "region of reversed command"--

5) Move the control stick or yoke aft to increase the sink rate, and move the control stick or yoke forward to decrease the sink rate, while leaving the thrust or power level in a fixed position. The airspeed will not stay exactly constant. Do not try this near the stall angle-of-attack!

6) Move the control stick or yoke aft to increase the sink rate, and move the control stick or yoke forward to decrease the sink rate, while simultaneously adjusting the thrust or power level (throttle) as needed to hold airspeed constant. (Move the thrust or power lever aft to increase airspeed, and forward to decrease airspeed-- very counterintuitive-- just as with method 4, this method only works because it forces the pilot to modulate his pitch control inputs in a way that leads to the desired change in airspeed.) Again, it is unwise to try this near the stall angle-of-attack.

However, methods 5 and 6 will ONLY work if the pilot waits a good while after any pitch input to see the ULTIMATE, not IMMEDIATE, result in the sink rate or climb rate before making an additional follow-up pitch input. Thus they are really not very practical in most cases.

To help illustrate the difference between the immediate and ultimate results of a pitch input in the "region of reversed command", consider this-- it is perfectly possible to execute a landing flare in the "region of reversed command", keeping the stick or yoke moving aft to generate an abnormally low sink rate (essentially zero or nearly zero) for the power setting (which may be near zero, especially in a light plane or glider.) Here the stick is or yoke is being moved AFT, not forward, to arrest the sink rate as the airspeed decreases. However, this maneuver is not sustainable-- if the pilot keeps moving the stick or yoke aft, the plane will eventually stall, while if he stops the aft movement of the stick or yoke just short of the stall position, the plane will start sinking at a high rate.

In the "region of reverse command", for maneuvers other than the landing flare, it is usually best to keep things simple and use pitch inputs ONLY for airspeed control, not control of sink rate or climb rate. In the "region of reverse command", it is most practical to accomplish changes in sink rate or climb rate by moving the thrust or power lever.

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  • $\begingroup$ I've now reworked my answer considerably. $\endgroup$ – quiet flyer Oct 6 '19 at 16:54
  • $\begingroup$ I still don't agree that a glider, or any other aircraft, is going to experience "speed instability" while flying on the backside of the sink rate polar, unless it is caused by the way an autothrottle and/or autopilot is connected into the control loop. $\endgroup$ – quiet flyer Oct 6 '19 at 17:16
  • $\begingroup$ In additions to variations 1 through 6 listed above, the answer could also consider variations based on holding a constant pitch ATTITUDE while manipulating the throttles or power levels to control sink rate or stay on a glide path. $\endgroup$ – quiet flyer Oct 21 '19 at 6:49
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According to Bold Method, and importantly pertaining to your checkride, the region of "reversed command" is essentially slow flight technique, where pitch controls speed and power controls altitude, as compared with cruising flight, where one "levels off" with pitch and "throttles back" to cruising speed. This, crucially, is technique used for approach and landing, where a safe speed can be trimmed in.

This is not a definition to be remembered by rote, as it only serves to confuse common sense: that speed control is critical for slow flight and you can't "count" on your engine to save you if you "screw it up" (or down). Glider pilots know this well. Managing speed with pitch, and altitude with power, sounds very "normal" to me, and yes, looking at it properly, works the same way at cruise.

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  • $\begingroup$ Folks, for any trim condition (staticly stable), a plane will climb if power is added and sink when power is taken out (from it's flight path). Set that in stone. In the region of "reversed command" moving the elevator trim for a lower airspeed (higher AOA for same lift) requires MORE throttle due to increased induced drag. In the "normal" region moving elevator trim for lower airspeed requires LESS throttle due to less parasitic drag. One can see how the least amount of throttle is required at minimum sink rate speed, aka "best endurance speed", aka highest L/D AOA. $\endgroup$ – Robert DiGiovanni Oct 7 '19 at 18:13

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