PIO Theory

[Al_Hammer]

The phugoid mode , or so-called long period is not the basis for PIO in my opinion. Hollmann has a website with mathematical treatment, but he ,too, does not adequately address PIO. He does attempt to model the coupling between rotor and fuselage, but this alone may not tell the whole story, if pilot inputs are left out.

Ray Prouty explains, "His (Hollmanns) explanation of "porpoising" does not make sense to me. Both airplanes and helicopters can have an inherent "phugoid" mode in which the aircraft goes up and down as it interchanges the kinetic energy of forward speed with the potential energy of altitude. This mode usually has a period of 10 to 20 seconds and can be either stable or unstable. Even when unstable, the pilot will easily overcome it with subconscious control inputs since the period is so long."
http://www.gyroplanestability.com/article_1.html

The Phugoid , or long period longitudinal oscillation, spoken of by Prouty, would be measured , I assume, with stick fixed. The gyro will show variations in airspeed/altitude that are periodic. As I said, I don't think this is related to the oscillation associated with PIO.

In PIO, the pilot's control inputs are out of phase with the pitching movements of the aircraft. The RTV(rotor thrust vector) must be moved away from, or towards the cg to affect a rotation.

With a HTL (high thrust line) machine and no stab, the rotor thrust will pass in front of the cg such that the net moment about the cg is zero.
To make the nose go up, the stick is, of course, pulled back, unbalancing the moments and allowing the aircraft to rotate nose up until the pilot stops the motion by once agin shifting the rotor thrust line.

Let's assume that PIO has developed and the pilot's inputs are out of phase with the pitch diplacement of the aircraft, thus driving the amplitude higher and higher.
A "restoring force" arises as the pilot moves the RTV vector away from, or towards the CG, and it is a function of the sine of the angle of deflection just as the restoring force of gravity is a function of the sine of the angle of deflection in a pendulum.

The pilot who is unfortunate enough to phase his inputs 180 degrees away from the sinusoidal pitching of the gyro , will excite an oscillation who's frequency can be estimated by using the tosional pendulum formula given below.

Oscillating system - Any system that always experiences a force acting against the displacement of the system (restoring force).
With PIO, the displacement is the angular pitching of the gyro as it rotates about the CG, measured in radians or degrees. The pilot's actions are "late" and phased such that the restoring force applied by the rotor is proportional to the angular displacement.
In other words, just as the gyro pitches up to its maximum angle in the cycle, the pilot applies maximum control force in the opposite direction, which drives the oscillation.

Simple Harmonic Motion - Any motion that experiences a restoring force proportional to the displacement of the system.
We are assuming that the restoring force is proportional to the angular displacement, at least to a first approximation.
This force acts as a torsional spring.

Resonance - The phenomena in which a driving force causes a rapid increase in the amplitude of oscillation of a system.
The Pilot's inputs are driving the system to higher and higher amplitudes, so this is a resonance condition.

Equation for the angular frequency of a torsional oscillator (depends on moment of inertia, I and torsional spring constant, k)
w=square root of [(k / I)]
where
w = angular frequency in radians/sec
k= torsional spring consant, in ft-lbs/radian
I = moment of inertia of gyro in slug-ft2

to find k, we can plug in some numbers for a typical 2 place gyro:
weight = 1200 lbs, rotorhead is 4 ft from CG
Using these numbers, assume the rotor thrust line has moved 20 degrees off the CG.
From this, a little trig allows us to calculate the moment being exerted. It works out to be 1634 ft-lbs of torque trying to rotate the gyro at that instant. I have neglected the fact that rotor thrust will vary as the rotor pitches up and down, but assuming it to be a constant 1200 lbs makes things a bit simpler. (See diagrams)
Vector diag_1 (http://theolddub.com/hammer/vectors_1.jpg)
Vector diag_2 (http://theolddub.com/hammer/vectors_2.jpg)



K , the spring constant, is in units of ft-lbs/radian. 20 degrees is 0.35 rad. So, k = 1634 ft-lb/0.35= 4668 ft-lb/rad
I = moment of inertia, and I'll use a figure of 105 slug-ft2 as estimated by graphical analysis of a 1200 lb AUW gyro and all its major parts.

Plugging these numbers into the equation, we get
w= square root of [(4668/105)] =6.5 rad/sec
(6.5 rad/sec)/(6.28 rad/cycle) = 1.038 cycles/sec
Period = 1/freq= 0.96 sec.

This particular gyro will PIO at about 1 cycle per second, consistent , with observed PIO rates.


The PIO oscillations can stopped by either removing the driving force( the pilot stops chasing the movement and holds the stick still), retiming the inputs to be out of phase rather then in phase, and/or by providing a pitch damping mechanism, such as a horizontal stab.

Note that a RAF type stabilator on the rotor will only work in stick-free mode(Pilot not forcing stick) and is ineffective at low airspeeds.
Of course, PPO (power push over) can occur in a HTL machine with or without PIO and the only way to eliminate the at all airspeeds and power settings is to design for CTL (center line thrust) or near CTL + effective horizontal stab with appropriate downthrust.


It's worth noting that a stab provides two functions. It adds 1) static stability and 2) damping.
Normally, the role of static stability is emphasized, because the stab must be sized and angled such that it will balance the nose down moment from a high thrust line.

Note that, if a PIO gets started, that the stab provides a restoring "spring force" similar to the rotor spring force described above.
Any angular rotation away from the relative wind is opposed by the stab. It seems to me that this does nothing to decrease PIO, in fact, by increasing the spring constant in the equation, it increases the frequency of the PIO.
The stab does its job of balancing the moments and preventing a sudden nose down pitching moment if the rotor becomes unloaded. However, if the pilot, for some reason puts in a large stick moevement, (or rapidly pumps the stick at the PIO frequency*) then the stab only contributes to PIO in terms of its static stability qualities.
What is important in this case, is its damping qualities, defined as the force that develops in proportion to velocity of rotation. This damping force increases as the square of the distance of the stab 1/4 chord line from the CG.

It would be interesting to calculate whether a stab that has sufficient control power to balance a HTL machine, automatically has sufficient damping to prevent a PIO for a given gyro with a given moment of Inertia. Standard formulas for finding critical damping coefficient could be used. See references below.

-Al Hammer

=======
web sites of interest:

http://scienceworld.wolfram.com/physics/TorsionalPendulum.html
frequency of torsional pendulum

http://www.engin.brown.edu/courses/en4/notes/Dampedvibes/Dampedvibes.html
critically damped, underdamped, overdamped

=========

*
Inputs don't have to be large to cause a PIO, just frequent. In fact,
we fixed the problem[on the Space Shuttle] with the FCS by adding a bandaid PIO suppressor
that reduced the gain on pilot inputs when the frequency of those
inputs increased. Move the stick too fast and the PIOS filter cuts
input authority to match.

-Mary Shafer
Senior Handling Qualities Research Engineer
NASA Dryden Flight Research Center, Edwards, CA

[Doug Riley]

Al, I agree that long-period (so-called phugoid) pitch oscillation probably doesn't play much of a role in PIO. It's just too slow to get in time with. My little, light, low moment-of-inertia Air Command had a lazy oscillation of this type. The period was 10-15 sec., as well as I can recall. Time a 15-second interval on your watch and you'll find that it seems like an eternity. After a day's flying, I could still feel it as I was driving home, like the left-over feel of the waves if you've been on a boat all day.

You'd expect that long-term pitch oscillations in a gyro would result from variations in airspeed and RRPM. OTOH, the nasty short-term ones would result from the nearly instantaneous rotor thrust variations caused by changes in rotor AOA. If these quick AOA changes result in lagged responses from the airframe, or (even worse) are amplified by the effects of a high prop thrustline, you're set up for PIO.

I guess that the dominant effect of HS's on seriously HTL machines (such as stock RAF, early full-dress Air Commands and so on) is damping. I think this because few people bother to put enough incidence into these (usually smallish) HS's to force the rotor thrust line back where it belongs. The pilot can usually handle the remaining static instability manually -- and probably without even realizing he's doing it.

I'd quibble with the idea that, by making the system's "spring" stiffer, a HS "does nothing" from the static-stability viewpoint to cut off the development of PIO. Lag is an important factor in the human part of operator-induced oscillations. The human is apt to over-control precisely because of lag, and the longer the lag the more excessive the human's input. If the machine's visible response (frame rotation) follows the initial disturbance more quickly, then, yes, the system frequency is higher. However, the human is less likely to over-control and therefore won't supply the control pulses needed to build oscillations.

You can't clap to music that's playing at 600 beats per minute, even if you badly want to. Goose up the system frequency and PIO becomes impossible. Mathematically, to capture this idea, it seems you'd have to make the amplitude of pilot-supplied exciting forces some inverse function of the system frequency.

[Udi]

The stab has another important function not mentioned by Al and Doug. The stab is providing control feedback to the pilot.

The airframe of a stab-less gyro does no respond to changes in the flight path. When the pilot pulls back on the stick, causing the rotor disk to immediately to tilt back, the flight path of the gyro is also changing pretty much immediately. The airframe, however, does not follow the flight path and so the pilot does not have an immediate feedback for his action.

A stab, even a small one, is making the gyro react to changes to the flight path like a weathercock, thus providing the pilot with an immediate feedback. I believe that this attribute of the stab may be more important than adding dynamic stability. I don't think short coupled gyros with stabs located 4-5 ft from the GC have a great deal of pitch damping.

Al - any otherwise stable control loop can become unstable due to delay in feedback.

Udi

[Al_Hammer]

Thanks, Udi and Doug. I was afraid you guys wouldn't come out to play.
Nobody, so far has questioned my torsion pendulum idea of PIO, so I can't complain too much.

A stab, even a small one, is making the gyro react to changes to the flight path like a weathercock

Udi, let me play devil's advocate:

The only time the nose will tend to stay completely aligned with the flight path is in a gentle climb or descent, or other coordinated turn type of flying. In any kind of rapid pitch manuever, the stab will slow down the response and add to the delay, won't it?

In a landing flare, a stabbed gyro will not go as nose high. We know that.
Similarly, at higher airspeed, if the stick is pulled back crisply, the fuselage will experience an immediate moment and pitch up quickly, the exact rate depending on the moment of inertia in the pitch axis. The stab will limit the initial pitch up away from the existing flight path, and this is probably a good thing in terms of PIO.
The flight path will change, because the rotor thrust increases, but since the lift slope of the rotor is something like 1/10 that of the stab, the total lift will not change as rapidly as the pitching moment will.
Further, due to the large mass of the gyro, the velocity upwards will not immediately be very great and there will always be a delay before the stab swings the nose into the flight path.There will be overshoot if the swing is too fast, which also would tend to add to PIO tendencies. Ever see an arrow hunting back and forth in flight?
It is only the damping property of the stab that acts to suppress PIO, by critically damping the amplitude of the oscillations. Otherwise, the stab is mainly acting to put the rotor thrust behind the cg and create a stable static condition, from which PIO is less likely to be triggered by a sudden change in rotor thrust.

Doug, I'm still thinking about the PIO freq vs the stiffness of the spring and how it might affect stability. I agree that beyond some critical frequency range, PIO does not exist. I saw it listed as topping out around 8 radians/sec(1.27 cycles/sec) in one study.

[C. Beaty]

Al, when dealing with a flying machine controlled either by an autopilot or a by a real pilot, like any closed loop control system, it will oscillate at a frequency determined by the open loop gain/phase shift and the amount of feedback.

I think Udi was hinting about a Nyquist plot but he didn’t quite get there.

A Nyquist plot can be done is a couple of ways; the most common being a root/locus plot but a magnitude/phase plot may be more understandable to people not familiar with closed loop systems.

The phase shift of our gyro example is mostly a function of control response lag. Its frequency response is determined mostly by the ratio of inertia to control power.

Control system theory is a complex subject touched only lightly at the undergraduate level.

[KenSandyEggo]

Huh? I'm glad you guys have it nailed. Doug, I did understand one thing you said though. I got out my RCA Victrola and played a 33 1/3 record of "Hand Jive" at the 78 setting and couldn't clap fast enough to keep up with the beat. I think I'm beginning to understand this stuff!

[Doug Riley]

Ken: The music analogy came to mind because I'd been to a couple holiday parties this weekend. I would never be a success as a Baptist; can't clap in time to save my life.

Udi and Al -- we may be talking past each other because of inconsistent definitions here. The HS does or does not change the time lag in the system, depending on what you define to be the output. If you define the output (as Al has) as a change in the flight path, then, no, the HS doesn't affect the lag in the aircraft's rate of initial reaction to a control input. If OTOH, you define the output as a pitching rotation of the airframe, then the HS does shorten the reaction time and therefore the resonant frequency.

[Al_Hammer]

Actually, Doug, 600 beats per minute is being conservative. :D

PIOs are caused by the pilot being _faster_ (higher bandwidth)
than the aircraft. Pilots are much faster than four or five hertz.
We used to use ten hertz because the hydraulic system filtered out
anything higher, but now we go up to at least twenty.
(20 hz= 1200 beats per minute)
-Mary Shafer NASA Dryden Flight Research Center, Edwards, CA


Time lag is one cause of PIO, but not the only one.
Oscillatory behavior can be caused by inexperience or incompetence,
and is usually known as “overcontrol”. A typical example occurs when learners
or otherwise inexperienced pilots attempt to achieve a precise altitude, and
oscillate about it, a phenomenon known as “chasing the altitude”. It is common
in pilots beginning their instrument-flight training. Such oscillatory phenomena
have a typical frequency of a few seconds, and are put down to lack of judgement
or skill – indeed, one has to maintain proficiency to keep out of such habits.

Another characteristic of divergent PIO is that the pilot’s control inputs are
high-gain: that is, his/her control inputs are large in comparison with that control
input which would stably achieve the desired state. High-gain behavior can be
exacerbated by a rate-limiting design, in which the rate of change of control lags
behind (hysteresis) the control input.

In order to model PIO mathematically, as in the case with the Chuck's Nyquist plots, etc, it is necessary to model the pilot's behavior.
One way is to model the pilot as a "PD" controller ("Proportionl-Derivative") from classic control theory.
A PD control scheme emulates the effect of a linear spring and damper acting between the current state and the desired state.
This is the idea I was getting at when I modelled PIO as a torsion pendulum.

The pilot's response is in proportion to the pitch excursions.
I used only the proportional term, but PD adds in a damping term to account for the fact that the pilot is sensitive to rate of change as well as absolute deviation of pitch angle.
Derivative Gain (Kd)—Derivative gain is the damping effects on the system. It determines the contribution of restoring force proportional to the rate of change (derivative) of position error. This force is much like viscous damping in a damped spring and mass mechanical system—a shock absorber, for example.

[mceagle]

PIOs are caused by the pilot being _faster_ (higher bandwidth)
than the aircraft. Huh ??? more information required.

[Al_Hammer]

Huh ??? more information required.

Sure, Tim, here is more from where that came from..


>For cars, it's a matter of the untrained reactions being
> slow; for air- and spacecraft, it's a matter of oscillations at
> (but near) the fastest speed the pilot _can_ respond, no matter how
> good his training is. I think that's around four or five hertz.
======
No, PIOs are caused by the pilot being _faster_ (higher bandwidth)
than the aircraft. Pilots are much faster than four or five hertz.
We used to use ten hertz because the hydraulic system filtered out
anything higher, but now we go up to at least twenty.
The PIOs came when the pilots put in
inputs faster than the aircraft responded.
==========

>If the pilot's reaction rate was much_ faster, I'd expect that
>the pilot could easily learn to avoid PIOs.
=======
We call this pilot compensation. It's an integral part of the
Cooper-Harper Pilot Opinion Rating scale, which was first published in
the '50s.

It's a form of overtraining, however, and pilots have a nasty habit of
abandoning the special technique when their gain gets high enough.

In addition, pilot compensation _always_ involves the pilot doing more
work.
=======

>But there are two issues I'm still not sure
> about. One, a bad pilot could create PIOs any time his reaction
> rate was greater than that of the aircraft.
>Two, why is the reaction time for pilots so much less than than the reaction time assumed by
> successful, police courses, etc...
======
No, not reaction time, but the ability to generate an input of that
high a frequency.

Take, for example, a standard bank-to-bank roll. The normal technique
is to put in a command to start the roll and, based on the established
roll rate, lead the plane slightly in putting in an opposite command
to stop the roll. However, a plane with a lot of time delay, for
example, might require that the pilot compensate by making a lot of
little, open-loop step inputs, pausing between inputs to see what the
plane does. Those little step inputs can be pretty high frequency
taps, but they're open loop, not linked directly to the aircraft
response.
There's a vast literature on this stuff (I've written my share), but I
really can't think of a good introductory work to recommend. In fact,
I can't even think of a bad introductory work.

=======

> I think of a PIO as a pilot sustained oscillation. when the
> frequency content of the pilot's input is high, he gets close
> to the gain or phase margins.

Yes, it's really trying to over-drive the system, asking the system to
respond faster than it can.

-- Add a little time delay, decrease the
control rates, and tighten the task--it works almost every time.

Mary Shafer NASA Dryden Flight Research Center, Edwards, CA
SR-71 Flying Qualities Lead Engineer

====

[drifter57]

What? I refuse to believe a human being can generate a physical oscillation at 20 cycles per SECOND! Unless it's the lips of a trumpet player.

I want some proof.

Am I reading this right? Basically, the way to arrest a PIO is to relax and "let go", letting the aircraft fix itself?

[Al_Hammer]

What? I refuse to believe a human being can generate a physical oscillation at 20 cycles per SECOND! Unless it's the lips of a trumpet player.
Okay, maybe a little off topic, but I can't resist a guitar reference..
The tremolo piece, Receurdos de la Alhambra was and is one of my favorites.
Here's an mp3 of it being done nicely by an artist named Elixir.
http://www.free-albums.net/download-Elixir-Recuerdos_De_La_Alhambra-8775.mp3

if played fast, it can exceed 20 notes/sec sustained for several minutes.. This is not reaction time, but rather preprogrammed muscle patterns.

I 've seen a helicopter pilot move the stick maybe 1/2 this this fast as he made tiny corrections in a hover. It all happens at the subconscious level. After years of practice, the reaction requires no thought.


Am I reading this right? Basically, the way to arrest a PIO is to relax and "let go", letting the aircraft fix itself?

Yes, if you stop making inputs, you remove the driving force behind the oscillations and the pitching will damp itself out, or should. Best to hold the stick still, though ,rather than let go, which achieves the same thing.

A stability test that is sometimes done in sailplanes is called "Mr Toad's Wild Ride." It consists of pulling up quickly to about a 30 degree nose high attitude and then releasing the stick. The aircraft should pitch 2 or 3 times times and then stop.

[Udi]

What? I refuse to believe a human being can generate a physical oscillation at 20 cycles per SECOND! Unless it's the lips of a trumpet player.
I guess you haven’t met a 22 y/o fighter pilot saturated with caffeine and adrenaline...:eek:

Udi

[C. Beaty]

If a gyro is statically stable but dynamically unstable, holding the stick will cause the oscillations to diverge. If such a gyro is equipped with a Bensen rotorhead, it might be dynamically stable stick free. If equipped with swashplate cyclic control, releasing the stick would not cause the oscillations to stop.

A gyro with CLT or HLT and without horizontal stabilizer can be statically stable but most likely dynamically unstable.

The only documented exploration of stick free vs. stick fixed stability that I’m aware of was performed by Roger Wood a number of years ago with his Bensen.

[Al_Hammer]

Yes, thanks Chuck. PIO will only stop if you have dynamic stabilty. :eek:

Static stability means that the gyro will initially tend to recover from a pitch disturbance, but it will continue to overshoot without dynamic stability, which is another word for damping.

The bad thing about static stability in a stabless gyro, is it isn't a fixed quantity. The rotor is providing the balancing moment to the high prop thrust line, so static balance depends on stick position.
If the gyro has a high thrust line and the stick is held fixed, it may , in fact push over without continuing to oscillate at all, depending on which half of the cycle you are in when you stop moving the stick.
If the rotor thrust is angled so that it happens to oppose the current direction of pitching, it will slow down the pitching for 1/2 cycle, (depending on throttle setting)and then it will get worse in the next 1/2 cycle, when the gyro starts to pitch the other way.

Chuck, a swashplate alone won't help stability, but it's been shown that RC pusher gyros can be stabilized when equipped with a swashplate controlled head and Bell-Hiller flybar. Actually, just Hiller paddles alone will do. The Hiller paddles slow down the rotor following rate and provide correcting inputs against any pitch disturbance. No stab is needed. I posted more on this in another thread.
http://www.rotaryforum.com/forum/showthread.php?p=89765#post89765

[C. Beaty]

The Bell gyroscopic stabilizer bar, Al, uncouples the rotor from the airframe in the short term via a differential linkage mechanism in the cyclic control path. The fuselage can flop any which way without affecting the rotor.

In the long term, the stabilizer bar is slowly precessed into alignment with the airframe by dampers between bar and mast.

The Hiller bar is similar except it is aerodynamically precessed by the paddles which also act as a servo. It’s a battle between responsiveness and stability.

There are pictures somewhere of a Hiller prototype hovering with sandbags instead of a pilot but that’s a somewhat misleading PR stunt. In order to do that, the paddles would have contained so much lead that it wouldn’t do anything but hover.

When I first became interested in rotorcraft, I built a number of electric motor powered models for hands on experience. Usually 13A vacuum cleaner motors controlled by a Variac and rotorblades made from spruce yardsticks. That was in an age when building supply stores gave away yardsticks as promotional items.

[karlbamforth]

Has anyone ever built a full size gyro with stabaliser/fly bar ?
Would it help ? Would it make a better gyro If a little more complex ?

[Doug Riley]

Karl, I can't answer your question directly as I don't know, but let me answer a related one.

The pitch stability of the airframe is a somewhat different issue in a gyro than it is in a standard helo. This is because the propulsive thrust comes from a separate device -- the propeller -- in a gyro, while it comes from the rotor itself in the helo. The propulsive thrust coming from a separate place than the lift and control forces introduces a possible complication. If the frame is laid out so that the propulsive thrust causes pitching reactions in the frame, then the rotor forces and prop forces have to be played off against each other to maintain trim.

Whatever the virtues of a stabilizer bar (mostly in hovering, I would think) , it misses the issue of frame layout as it applies to the gyro. It may in fact distract the designer away from laying out the frame in a way that eliminates this "playing off" scenario.

Sorry for the slight digression.

[Al_Hammer]

Chuck, it's my understanding that Arthur Young used an electric drill motor to power his prototype model helicopters, while developing the stabilizer bar.
He could only fly as far as the power cord reached, which was about as far as the barn door.

karlbamforth:
I don't know of any full size gyro with bell, or bell-hiller flybars.
On RC models, it has been shown that they help dramatically with stability. However, the advantage is primarily due to the slowing down of the rotor following rate. On a model, the rotor responds far too quickly for a human to manage with direct control. The Hiller paddles slow the following rate down and make the rotor flyable.
On a full size machine, the following rate is already slow enough, although making it slower could be an advantage depending on how much responsiveness you were willing to trade away.

The complexity is so much greater than a gimbal head that it really isn't worth it, unless you are building a jump takeoff machine and happen to need collective pitch control.
As Doug, says, it still wouldn't solve the problem with a HTL machine.

Also, the RC model has the teetering locked out. This improves the control at zero g, at least until the rotor stops, which is not a problem with stiff blades. On a full size rotor, stopping the blades would be a one time occurence, before they snapped off. On the model, if you let the rotor rpm get too slow at the top of a loop- no problemmo, just wait for them to spin back up.

The photo shows the Hiller paddles on a pusher RC gyro designed by Mickey Nowell.
Compare that to the more complex Bell-Hiller paddles on a RC helo which allows mechanical mixing of the pilot and flybar control inputs.
The last photo shows an even simpler way of doing it, with slip rings and motor controlled paddles.

[drifter57]

OK. Yes, I play the guitar, and if I use MULTIPLE fingers I can possibly generate 20 notes a second. But you only have ONE arm (or maybe two) on the cyclic.

Hadn't considered the caffeine and adrenaline scenario.

I still maintain it is impossible to get better than maybe 5 to 10 Hz max.

Try it! Smack your closed fist rapidly between two closely spaced objects. Have someone time you for say 5 seconds counting the times you hit ONE side (one complete oscillation), divide by 5. I bet you can't get a number above 10.

[C. Beaty]

It is quite simple to incorporate a Hiller style rotor on a tilt head gyro.

The only person I know with certainty to have done so was Harris Woods who served a short time as Bensen’s chief engineer.

He had previously designed a series of bolt together FW aircraft; one of which, a biplane, achieved some popularity but I can’t remember the name of it. Another was the Woody pusher.

Harris recounted his experiences with a Hiller rotor in a long ago issue of the PRA magazine. My recollection was that control was difficult during rotor startup but I don’t imagine he had a decent prerotator.


But Doug is correct; there are better ways of addressing gyro stability issues than by bandaids applied to the rotor.

[Al_Hammer]

Drifter, Django Reinhardt only used two fingers...but I see your point.

I agree that 10 hz is pushing the envelope for back and forth motions, but the NASA engineer was talking about light taps in one direction.

The effect of providing force feedback can be likened to that of a drum-roll. The human bandwidth for limb motion is about 10 Hz, but it has been observed that
humans can produce actions such as drum-rolls at over 40 Hz by allowing the
drumstick to bounce through suitable control of the passive impedance of the hand
joints. Thus, by allowing the end-effector to "bounce", faster
movements can be performed.

So, if the stick has some stiffness, or springiness in it....

[Al_Hammer]

The only person I know with certainty to have done so was Harris Woods
Interesting, Chuck. That's a name I hadn't heard mentioned.


But Doug is correct; there are better ways of addressing gyro stability issues than by bandaids applied to the rotor.

Agreed, although in the case of the RC gyros, of course, the Hiller rotor turns out to be more than a bandaid, due to the "following rate" problem.

Following rate is increased when the blades get lighter or move faster, so at model sizes, following rate with a gimbal head is too fast for human control to deal with.
Solutions usually involve large stab, delta 3 heads, gyro rate control, massive blade weights, limiting control travel, etc. The hiller paddles are a great solution for the model. All this has nothing to do with full size gyros, but is fascinating...
the equation for the following rate of the flybar as

G= (w.a.p.S.(R)^3)/4.I Radians/sec/Radian lag

where w is the angular velocity of the rotor in radians/sec
a is a constant derived from the paddle size and angle of attack
and is about 3.75
p is the density of air in Slugs/cu.ft (0.0024)
S is the area of the paddle in sq.ft
R is the radius of the flybar
and I is the Moment of Inertia of one paddle in slugs/ft^2
The typical following rate of a Hiller flybar is about ten times slower than that of the rotor alone.
-from a book on helicopter design by John A. Drake.

[raghu]

The lagged response to cyclic displacement of rotary wing aircraft along with the consequent implications on PIO is well known. However what exactly lags and causes pilot handling difficulties is sometime misunderstood.

When the cyclic is displaced the rotor takes a short time to respond. While there certainly is a lag, as the rotor has to precess to its new orientation, the rotor responds very quickly ( typically 3 - 5 revs) and as such poses no control difficulty to the pilot.

Another misconception is that the lag is due to the delay in aligning with the flight path or relative wind. While this delay certainly has some effect on handling, it is more related to the long term response ( phugoid response) and plays no role in the short term immediately after a cyclic displacement.

When a pilot displaces the cyclic during a maneuver there are two interrelated cues that he/she uses to monitor the progress of the maneuver- pitch rate and normal acceleration ( or G loading). When the cyclic is pulled, the initial response is a small increase in G load due to the increase in rotor AOA and secondly the displacement of the RTV starts a pitch angular acceleration of the fuselage.

It turns out the G loading experienced in a pitch up maneuver is a function of the initial increase (due to change in AOA) and a further increase due the pitch rate . The pitch rate of the fuselage starts at zero and begins to increase due to the displaced RTV and stabilizes some time after the cyclic has been pulled. Hence the G loading builds slowely initially and then starts building more rapidly until the pitch rate stabilizes to a steady state. If the gyro is lightly damped in pitch (as most are) then the final pitch rate takes a while to develop and consequently there is a delay in the maximum G load occurring. This lag in reaching a stabilized pitch rate and hence maximum G load is what causes handling difficulties in both helicopters and gyros.

All this has been well researched and there are prescriptions in terms of 'time to reach maximum G's' for good handling (if you can say that for any rotorcraft :-) )

In theory a horizontal stab improves handling in two ways
1. improves damping so that the pitch rate achieves a steady state value more rapidly after cyclic displacement. Hence less lag in the development of G load

2. It improves the AOA stability of the gyro which also reduces the time for the pitch rate to reach a steady state value. Hence less lag.

Mathematically, the lag time is a function of

( pitch damping derivative )*( vertical damping ) - mu * (AOA stability)

As mu is small at low speeds AOA stability has less impact on the lag than pitch damping at slow speed.

[mceagle]

Sure, Tim, here is more from where that came from..

"PIOs are caused by the pilot being _faster_ (higher bandwidth)
than the aircraft"....and..... "a bad pilot could create PIOs any time his reaction rate was greater than that of the aircraft".You blokes are way ahead of me on this stuff but these statements seem a bit hard to swallow.
In my experience it has always been that the pilot is slow to react to the disturbance, and attempts a correction, the mean reaction being greater than 180 degrees after the event, thus perpetuating or magnifying the disturbance.
A pilot cannot cause a PIO if he is too fast simply because he doesn't know that the disturbance is coming until after it comes.
Are we talking about the same thing here?, or are you referring to the "movement / unit time" of the pilots input rather than the timing of the input?

[C. Beaty]

The lady from NASA, Tim, was talking about classic closed loop stability. Rather than saying the pilot was too fast, it might have been more appropriate to say the aircraft was too slow. But it’s the same thing. There will be some frequency where pilot and machine are out of phase when PIO exists.

To compare FWs to rotorcraft in relation to what Raghu just posted; a fixed wing aircraft reaches its maximum rate of pitching or rolling in a very short time in comparison with stabless rotorcraft.

The stabilization of pitch/roll rate in FWs is fast enough that stick displacement is very nearly a pitch or roll rate control; whereas the initial response of a stabless rotorcraft to a stick displacement is very much an acceleration control. Rate control is much easier than acceleration control.

Without a stab, a rotorcraft begins accelerating about the pitch axis until the rate becomes stabilized by rotor damping. Rotor damping is a consequence of rotor flapping; the rotor lags behind until its moment equals the control moment. But that takes a long time and the pitch rate might have reached several hundred degrees per second.

There simply is no practical substitute for a horizontal stabilizer.

[Al_Hammer]

Tim, I think it's a case in some aircraft,( like the shuttle) that the controls reach a programmed rate limit and it is possible for the pilot to move the controls faster than the aircraft can respond. This tends to lead to PIO because the pilot drives the inputs harder and harder, waiting for a response.
The pilot inputs would need to be slowed down or filtered.

Gyro controls are not rate limited, so time lag in the response to an input is more likely to be the cause of PIO. A pilot can lead with the controls(feed-forward) to anticipate lag, or use the stick to damp unwanted motions, which in effect, counteracts 1st order delay. Damping from a stab does the same thing.

[drifter57]

Al, I will concede that if you add the cyclic rebound (bounce) into the equation, it would be possible to achieve 20 Hz. The inclusion of the rebound in the initial description was not clear to me. That would equate to extremely small linear movements however. More like vibration.

As to the drum-rolls, again, two sticks and multiple rebounds per human induced motion. If I could actually move at 40 Hz, the Flash would have some competition.

So, if just holding the stick immobile will not necessarily stop PIO in an unstable gyro, how do you then correct the situation - besides trading it in for a stable bird?

[Al_Hammer]

I found these diagrams for a spring mass system interesting. They show how phase changes rapidly near resonance in the absence of damping. In fact the phase reverses if damping is near zero.(Pilot's inputs out of phase with aircraft)

Also, it can be seen that amplitude increases rapidly at resonance without damping. (Pitching gets out of control rapidly)

Damping can be calculated from the graph of amplitude vs driving frequency. This might suggest a way to flight test a gyro in such a way as to determine damping, assuming the pilot could control the frequency of his/her inputs. (Not very likely, I know.)

[Udi]

Thanks Raguh, Chuck and Al.

Question - wouldn't one consequence of this delay, which I don't think anyone has mentioned in this thread, be that there will be an almost equal delay in stopping the pitching motion, once it has been established (considering the lack of damping in such rotorcraft)? Wouldn't that be a large part of PIO?

Udi

[C. Beaty]

Udi, the cruise control of a small car could be made unstable by packing in 18 college kids.

The governor of a rotary lawnmower could be made unstable by increasing the moment of inertia of the flywheel.

So it is with flying machines. Increasing the pitch axis moment of inertia increases both undershoot (lag) and overshoot. Damping addresses both.

For the same degree of stability, a tandem gyro requires more damping from the horizontal stabilizer than does a SBS machine.

In a stabless gyro, the pilot supplies the damping if he survives.

Bensen wrote an article called the second jab. Establish an angular rate and then stabilize it with a reverse jab of the stick. This is the damping which must supplied by the pilot of a stabless gyro to snub overshoot.

[Doug Riley]

A safer way to explore this overshoot/reverse jab issue than playing with pitch oscillations is... to play with roll oscillations. Nobody bothers with wings on gyros anymore, so the only roll damping we have is rotor damping. Rotor damping isn't much.

I've gotten a gyro into roll PIO a couple times. The first time it happened was on my first-ever solo flight in a gyroglider. I got a mile behind the machine and was using (overhead) stick excursions of at least a foot, overcontrolling entirely out of phase with the machine. I weaved back and forth behind the car like a drunk. A collision with a snowdrift on the side of the runway kindly "damped" the whole process short of a crash.

It happened again with my Gyrobee, nearly 30 years later. In both cases I was using an overhead stick. These sticks have a very slow leverage ratio (especially the one on the 'Bee) and extremely light control forces. The ratio on the 'Bee is over 2" stick travel per degree of head travel; it takes a measurable amount of time just to move the stick that far. Compare the mere flick of a wrist needed with a joystick.

Precise control requires "second jabs" after a bank angle is established, and again when rolling out. Students get this pretty quickly if it's explained to them before the flight starts.

[raghu]

Thanks Raguh, Chuck and Al.

Question - wouldn't one consequence of this delay, which I don't think anyone has mentioned in this thread, be that there will be an almost equal delay in stopping the pitching motion, once it has been established (considering the lack of damping in such rotorcraft)? Wouldn't that be a large part of PIO?

Udi

Absolutely Udi! The lag essentially is between moving of the stick and getting the result you want. The result you want could be starting a pitching motion or for that matter stopping it.

[raghu]

Al,
I recall a day or so ago seeing a post by you asking for the units of the stability derivatives. I didn't get around to posting a reply then but in case you are still curious here it is.

All the force derivatives have the unit 'per second' or 1/sec. This is because they are normalised by dividing by aircraft mass. So, vertical damping would have the units 1/sec

The moment derivatives have the units per ft. sec or 1/(ft. sec.) as they are normalised by dividing by MOI. The exception is moment derivatives based on angular velocity (pitch damping), which have the units 1/sec. You can see why by doing a dimensional analysis ( I can show it to you if you find it difficult to work out).

I recall you also asked about ballpark numbers...here are some for the Magni vpm 16 based on measurements made at Univ. Glasgow by S. Houston:

pitch damping Mq(1/sec) = between -0.8 and -1.2
vertical damping Zw(1/sec) =between -0.3 and -0.6
pitch stability Mw(1/(meter-sec) ) = between -0.02- -0.07

the AOA stability (M-theta) can be got from the pitch stability by multiplying the pitch stability (Mw) by the velocity of the aircraft. Hence M-theta for the Magni would be (approximately) in the range -.6 to -2.1 for a typical gyro speed range of 30 to 70 mph.

What does all this mean ...Given control response is a function of:
(pitch damping)* (vertical damping) - mu*(AOA stability)

Hence at low speed satisfactory cyclic response can be got by increasing pitch damping even if the gyro is unstable wrt AOA. However at high speed AOA stability is almost mandatory to get satisfactory control response.

[raghu]

Nobody, so far has questioned my torsion pendulum idea of PIO, so I can't complain too much.


It seems from your posts Al, you are looking for a model of the short term response of a gyro. That is not too hard to accomplish. As an approximation you can assume that the short mode is essentially a motion about the pitch axis and you can ignore any change in rotor rpm or forward speed. Now what you have is essentially a damped harmonic oscillator.

You can create a model of the harmonic oscillator and study frequency, damping and play with forceing function that cause PIO. To create the model use:

spring constant: Mq*Zw - (mu)* M-theta

Damping coefficient: Mq+Zw (ignores change in rotor downwash)

You can get ball park values for Mq, Zw, M-theta for a stable gyro from my previous post. Play around with these values and increase the value of M-theta (+ve M-theta is unstable) and see the effect on damping and frequency.

[Ga6riel]

excuse my total inexperience but
is anyone considering the effects of gyroscopic pression in this ?
e.g. take the front wheel off a bicycle, get it to spin at speed, input some movement, and see if u can counter the forces in your hands, note too the lag. Surely a 20+ ft rotor contains much more force than this, and more lag ?

[C. Beaty]

You would be quite right, Ga6riel, if the rotor was bolted solidly to the rotorhead. There would be a powerful 2/rev shake as well as a precessional torque of several hundred ft-pounds.

Fortunately for all rotorcraft, Cierva invented flap hinges in the 1920s to address that problem.

Tilting the rotorhead of a gyro does not directly tilt the rotor on account of the teeter bolt (all the flap hinge needed with a 2-blade rotor), it can only rotate the blades about their feathering axes, producing a cyclic pitch variation until the rotor disc flies itself into alignment with the new rotorhead position.

Aerodynamic power steering.