Are turn rate indicators calibrated to a certain speed? If they are, what speed are they calibrated to, and why aren't all the turn rate indicators calibrated for the same speed?
They are not calibrated for a certain speed.
If by "calibrated", you mean they deflect together / to the same side, then you're using the instruments wrong. The two instruments have are not linked to each other, they operate independently and they operate on different principles. They are located next to each other in the cockpit out of convenience.
The "slip indicator" is a ball in a U-shape tube. It tells you whether gravity (or experienced acceleration) is pointed down, or to the side. Period. It is as simple as that.
The roll is restricted by a spring, which is calibrated to indicate standard rate turn when the yaw rate is 3 degrees per second. The turn coordinator works at any airspeed (part of the preflight check is to turn the plane while taxing and verify the turn coordinator works).
As per the clear intent of the original question, this answer deals with the deflection of the needle in a turn rate indicator (or the deflection of the symbolic airplane in a turn "coordinator"), not the deflection of the slip-skid ball. "Turn and slip indicator" or "turn and bank indicator" are names that are commonly applied to the whole package (at least in the case of the turn rate indicator with the needle rather than the turn "coordinator" with the little symbolic airplane), but it's clear that the original question is not asking about the movement of the slip-skid ball (inclinometer).
Also, this answer only deals with actual gyroscopic instruments, not their more modern counterparts.
Also, we'll assume throughout this answer that the slip-skid ball (inclinometer) is kept centered via appropriate rudder inputs by the pilot, or by the aircraft's yaw damper or other automated systems, if needed.
Also, please don't confuse the ideas explored in this answer, with the fact that various turn rate indicators designed for use in various types of aircraft with varying performance use different values for the turn rate that is achieved when the needle is at the primary index marking, and also use different values for the turn rate achieved when the needle approaches full deflection. For example, airliners often perform half-standard-rate turns (four minutes per turn), so many turn rate indicators intended for use in such aircraft have the primary index markings set to correspond to a half-standard-rate turn. Such instruments will be marked with the label "4 minute turn" rather than the more customary label "2 minute turn". But that's not what this answer is about.
Are turn and slip indicators calibrated to a certain speed?
Essentially, yes. On any given turn rate indicator, each turn rate index marking will only be completely accurate at one specific True airspeed. On an instrument with multiple turn rate index markings, they wouldn't necessarily all need to all be calibrated to be accurate at the exact same True airspeed, though it seems logical that they might be.
Most illustrations of turn rate indicators with multiple turn rate indices appear to depict older instruments, often of British or other non-U.S. manufacture. Many turn rate indicators bear only one index for turn rate in each direction, corresponding to either a standard-rate (two minute) or half-standard-rate (four minute) turn. In this case the answer to the question would be "yes, on any given turn rate indicator, the turn rate index is only completely accurate at one specific True airspeed."
However, the variation in the accuracy of a turn rate indicator as True airspeed varies is small enough that it is not generally something that a pilot needs to take into account in actual practice.
But it does exist.
For example, in one source we read that:
The indicators are calibrated for rate 1, 2 and 3 turns at specified angles of bank and TAS, and looping error is taken into account in this calibration
Original source: "a rather elderly version of RAF publication AP1234"
Quoted in: this post on the PPRuNe aviation forum
In another source we read:
Turn rate indicators are calibrated to Rate 1, 2 and 3 turns at a specific TAS. A rate 1 turn is 180° a minute or 3° a second. A rate 2 turn is twice that, 360° a minute or 6° a second. A rate 3 turn is three times as much, 540° a minute or 9° a second. The angle of bank required to achieve a given rate of turn increases with the TAS.
It is inherent in the design of the instrument that in any yaw condition the gyro axis will tilt. and the gyro will become sensitive to pitch rate.
If the aircraft is then rapidly pitched nose up (in a loop or recovery from a spiral dive) this pitch input can deflect the gyro to read maximum turn rate. This is called looping error. This pitch rate error also affects the instrument readings in normal turns.
To compensate for this and for the difference between yaw and turn rates the indicators are calibrated to show rates of turn correctly in balanced turns for Rate 1, 2 and 3 turns at specific angles of bank and TAS.
Although the indicated rate of turn will be incorrect at speeds away from these datums the errors are not significant in normal operation.
The direction of rotation of the gyro is chosen so that when an aircraft banks into a balanced turn the gyro precesses in the opposite roll sense to keep its axis more or less horizontal and therefore keep it more sensitive to turn rate. If the gyro rotated in the opposite direction it would only work satisfactorily at very low yaw rates and small angles of bank.
Source: Bristol Groundschool ATPL Training Material sample, v6.1.5, page 3.13, published November 13, 2013, accessed via this on-line link, italicization and bolding added.
In other words, the calibration of the indicators to show specific turn rates is carried out at specific bank angles and specific True airspeeds.
Actually, both of these quoted statements are a little less than crystal clear, because if the turn rate is specified, then we only need to specify one additional parameter, either bank angle or True airspeed, in order to fully constrain the other parameter as well. They can't vary independently of each other, assuming that the slip-skid ball is kept centered. Still, the point is made that the calibration of the instrument for turn rate is not independent of True airspeed.
Here's a quote from an on-line discussion forum that actually cites the specific airspeed used to calibrate the turn rate indicator in one specific aircraft:
Also, if you consider that in a balanced turn you have yaw roll and pitch, the pitch element makes the instrument over-read as the aircraft will be at more than 1g. For this reason, the turn indicator is calibrated to read correctly with this increased g by adjusting the tension on the springs, but at one speed only . The instrument fitted to the Chipmunk in RAF service was actually calibrated to 180 knots, so I don't think this correction is too significant.
Source: this post on the PPRuNe aviation forum
and why aren't all the turn rate indicators calibrated for the same speed?
It would make no sense that turn rate indicators intended for use in aircraft with vastly different performance would all be calibrated to have maximum accuracy at exactly the same airspeed. Especially given the fact that the critical parameter is actually True airspeed, which varies even more widely between different aircraft types than does Indicated airspeed.
Why should the calibration of a turn rate indicator be airspeed-dependent?
The basic reason that the calibration of a turn rate indicator is airspeed-dependent is that for a given turn rate, the spin axis of the gyro wheel cannot be horizontal relative to the earth at all possible bank angles.
We can start by thinking of a turn rate indicator as a yaw rate sensor. A turn is comprised of a pitch rotation as well as a yaw rotation. For a given turn rate, the yaw rate is dependent on the bank angle. For a given turn rate, the steeper the bank angle, the lower the yaw rate. And for a given turn rate, the higher the True airspeed, the steeper bank angle. So for a given turn rate, the yaw rate is dependent on the True airspeed.
But the story is more complicated than that, because a turn rate indicator is actually more than just a yaw rate sensor.
Let's explore this in more detail. (This will get a bit complicated-- the reader is encouraged to make some sketches, or better yet, to make a simple model of a gyro wheel such as a pencil skewered through a round disk that he or she can hold in different attitudes in space.)
In this You Tube video entitled How Gyroscopic Turn & Slip Indicator Works, we can see what a turn rate indicator looks like inside. At rest, the gyro's spin axis is oriented laterally (parallel to the aircraft's wingspan). The direction of spin is clockwise as viewed from the right wingtip, so that the top of the gyro wheel is spinning away from the pilot. The instrument's pivot axis or "gimbal axis" is oriented longitudinally-- fore and aft.
In the video, we can see the gyro's spin axis roll (bank) to a new orientation in response to yaw. The degree of movement is rather substantial. The rotation of the gyro wheel converts the aircraft's yaw rate to a roll torque on the gyro, forcing the gyro's spin axis to roll (bank) against the direction of the yaw, i.e. opposite to direction that the aircraft is banking in a coordinated turn. Gears convert this movement to a motion of the pointer in the opposite direction, i.e. in the same direction that the aircraft is banking in a coordinated turn.
Note that the gyro is not acting like a "stable platform" like the artificial horizon-- purely rolling the instrument has no effect on the pointer, as it would if the gyro's spin axis were tending to stay fixed in space. Apparently the centering springs are strong enough to overcome any such tendency. Instead, as the aircraft yaws, the roll torque created by gyroscopic precession is acting against the centering springs to cause the gyro's spin axis to come to rest in an orientation that is "banked" relative to the instrument case. Note also that the direction that the gyro's spin axis "banks" relative to the instrument case is also the direction that would tend to keep the gyro's spin axis more or less level (horizontal) with respect to the earth. However, since the gyro is not acting like a stable platform, there's no reason to think that the gyro's spin axis will stay exactly horizontal with respect to the earth. In fact, for a steady-state turn at a given turn rate, the gyro's spin axis can only be horizontal with respect to the earth at one particular bank angle, i.e. at one particular True airspeed.
Whenever the gyro's spin axis is banked relative to the instrument case and the aircraft-- i.e. whenever the aircraft is turning-- the gyro responds to pitch rotation as well as yaw rotation. For simplicity, consider a case where the gyro's spin axis is banked almost ninety degrees relative to the aircraft, which would never actually happen. Now yaw rotation will barely be sensed at all, but any pitch rotation by the aircraft would be converted by gyroscopic precession to a roll torque on the gyro, tending to roll the gyro's spin axis to a steeper bank angle.
This effect is called "looping error". One consequence of "looping error" is that any time the flight path is curving upward-- i.e. any time the aircraft is experiencing more than the normal G-loading for the bank angle-- the Turn Rate Indicator will read higher than the actual turn rate.
It appears that whenever the gyro's spin axis happens to be exactly horizontal with respect to the earth and the aircraft's flight path is not curving upward or downward, the deflection of the indicator needle will be exactly proportional to the turn rate. (Citation needed.) In essence, "looping error" is exactly compensating for the fact that the turn comprises of both pitch and yaw, rather than pure yaw. However, as noted above, a turn rate indicator cannot be designed so that this happy situation occurs at all possible bank angles for any given turn rate. Rather, in a steady-state turn at a given turn rate, this situation can only occur at one possible bank angle. In many instances the gyro's spin axis will not be exactly level with respect to the earth. So for any given actual turn rate, the indicated turn rate ends up being somewhat dependent on bank angle, which means that the indicated turn rate cannot be independent of True Airspeed.
It also appears that the stiffer the centering springs of the instrument, the less sensitive the instrument will be to transient "looping error" caused by pulling excess "G" in relation to the bank angle, but the less accurate the instrument can be over a wide range of True airspeeds in a steady-state turn, because the spin axis of the gyro will tend to stay more aligned with lateral axis of the aircraft (i.e. the wingspan) and less horizontal relative to the earth.
In a turn coordinator, with a symbolic moving airplane on the front of the instrument rather than a needle, the situation is even more complicated. The orientation of the gimbal axis is different. Rather than being oriented purely fore-and-aft, i.e. parallel to the aircraft's longitudinal axis, it is tilted (with the forward end raised and the aft end lowered) by about 30 degrees. The purpose of this is to make the instrument sensitive to roll as well as yaw, to better signal the earliest stage of a developing turn or change in bank angle.
Apparently not all turn coordinator gyros spin the same direction. If the gyro spins clockwise as seen from the right, with the top of the gyro moving away from the pilot, "looping error" is similar to what we see in the turn rate indicator, and if the gyro spins counter-clockwise as seen from the right, with the top of the gyro moving toward the pilot, "looping error" is reversed, so that the instrument under-reads the turn rate whenever the G-loading is excessively high for the bank angle. Note that in the latter case, with the gyro spinning counter-clockwise as seen from the right, the symbolic airplane must be geared to tilt in the same direction that the gyro is tilted, not the opposite direction. This means that in a steady-state coordinated turn the spin axis of the gyro actually is tilted away from level with the earth's surface, rather than toward level with the earth's surface. In a steady-state turn, this would seem to make the displayed turn rate more dependent on bank angle, and therefore more dependent on True airspeed, than would be the case if the gyro spun the opposite direction. For more on the direction of spin of the gyros in turn coordinators, and the resulting effect on the direction and magnitude of "looping error", see this related answer.
Any slight error in the turn rate indicated by a turn rate indicator or turn coordinator in a stabilized turn due to the calibration being only valid for one True airspeed is insignificant enough that it is rarely taken into account in actual practice. This error is the clearly the least of a pilot's worries when flying in cloud. But the answer to the question is that the calibration of any given turn rate index on any given turn rate indicator or turn coordinator can only be exactly accurate at one specific True airspeed.
Some related links--
How Gyroscopic Turn & Slip Indicator Works (You Tube video) -- in the cutaway, note how far the gyro's spin axis tilts to the side (rolls) in response to yaw rotation-- the tilt at full deflection appears to be about 40 degrees. We can see that the tilt is in the direction that would bring the gyro's spin axis closer to vertical with respect to the earth during a normal coordinated turn. By the fact that the gyro's spin axis rolls to the left when the instrument is yawed to the right, we can see that the gyro must be spinning clockwise as viewed from the right, so that the top of the gyro is spinning away from the pilot.
"How a Turn Coordinator Works-- Inner Workings" -- Again we can see that gyro's spin axis rolls in response to yaw, and in this particular turn coordinator, the roll is in the direction that would bring the gyro's axis of rotation closer to vertical with respect to the earth during a normal coordinated turn. Again, by the fact that the gyro's spin axis rolls to the left when the instrument is yawed to the right, we can see that the gyro on this particular turn coordinator must be spinning clockwise as viewed from the right, so that the top of the gyro is spinning away from the pilot.
Bristol Groundschool ATPL Training Material sample v6.1.5, published November 13, 2013 -- see page 3.13 regarding the dependency of the calibration of the turn rate indicator upon True airspeed. See figure 3.16 on page 3.12 for an illustration of a turn rate indicator with multiple rate markings-- the typical British practice?
A link to an aviation website giving a rule of thumb for bank angle for a standard rate turn-- the rule of thumb is that the bank angle in degrees should be roughly .15 * (True airspeed in knots). Reaffirming the idea that if the calibration of a Turn Rate Indicator or Turn Coordinator depends on bank angle, then it cannot be independent of True Airspeed.
https://www.pprune.org/tech-log/82721-turn-gyro-axis.html -- from an airline pilot's discussion forum-- not all the ideas here appear to be correct, especially the comment about the rotation about the gimbal axis being limited to 6 degrees of motion.
https://www.pprune.org/tech-log/528255-turn-rate-indicator-turn-coordinator-looping-error.html#post8164645 -- from the same forum as above-- a comment on the problems that can sometimes be posed by a Turn Rate Indicator's tendency to over-read under excess G-loading, and the greater problems that can be posed by a Turn Coordinator's tendency to under-read under excess G-loading.
Related ASE answers --
May turn rate indicators (and turn coordinators) be calibrated to be accurate at all airspeeds, or only for one? -- a good answer to what is essentially a duplicate question
In some "turn coordinators", does the gyro spin opposite to the direction it spins in a "turn rate indicator"? -- more on how the tilted gimbal axis of a turn coordinator allows roll sensing as well as yaw sensing, and also affects "looping error". More on why if the gyro spins counterclockwise rather than clockwise as seen from the right, the instrument will under-indicate rather than over-indicate the turn rate whenever the G-load is unusually high for the bank angle. Includes several links to videos and diagrams showing the internal workings of turn rate indicators and turn coordinators.
What is the formula for the bank angle required for a turn in line-abreast formation? -- calculating the required bank angle as a function of turn rate and True airspeed