The air-driven gyro-horizon suffers from acceleration errors. These are due to two design features:
1. The whole gyro case is slightly bottom-heavy, to reduce the time for erection*; and
2. The pendulous vanes, alluded to above, are also, as their name suggests, pendulous and free to swing on their pivots.
The acceleration can be fore and aft, as in an increase or decrease in speed, or sideways, as in the acceleration towards the centre of a circular flight-path (turning flight).
During a longitudinal acceleration, the whole unit will experience a forward-tilting force, due to the inertial lag of the bottom-heavy gyro case. This force is precessed and becomes a sideways-titling force, resulting in an apparent angle of bank.
At the same time, the transverse-mounted vanes will lag rearwards, covering one port and uncovering its opposite counterpart. This will result in a jet of air acting to tilt the bottom of the case sideways. This force is again precessed, resulting in an actual fore-aft tilt, indicating an apparent angle of pitch.
A similar effect occurs due to the centripetal acceleration experienced during a steady turn, the combination of precessed forces acting to show small cyclical, out-of-phase bank and pitch errors, reaching their maxima and minima at 90, 180, 270 and 360 degrees of heading change.
An attempt to minimise these errors is made by slightly off-setting the axis of the gyro.
The classic example of this was during the introduction of the early jet fighters, where the rapid acceleration of a night or IMC take-off could result in an apparent climb to starboard. The pilot might over-correct and then induce a descending turn to port, crashing to the left of the extended centreline.
Most light aircraft are unable to produce longitudinal accelerations sufficient to cause a dangerous effect. The newer electrically-driven gyros and our modern laser-gyros have reduced the errors.
*note to self; must investigate further....