In this article we discuss off-axis responses to pilot control inputs. Off-axis responses refer to helicopter attitude (pitch, roll or yaw) changes after a control input aimed at a different axis. Prominent examples are roll changes due to pitch (longitudinal cyclic) inputs and pitch changes due to roll (lateral cyclic) inputs, sometimes called pitch-roll coupling. Sections below are allocated to each of several different off-axis responses.
There are two primary sources of off-axis pitch/roll responses: control coupling and rate coupling. Control coupling applies an off axis moment roughly proportional to the stick movement. E.g. if the longitudinal cyclic is moved an amount \(\delta_x\), it results in a change of roll moment \(L_x\delta_x\). Rate coupling is a roll moment proportional to pitch rate \(q\), i.e. \(L_q q\). Hence we have a total off axis roll moment \(L_x\delta_x + L_q q\) due to a longitudinal cyclic input \(\delta_x\).
Control coupling is associated with a combination of control rigging (e.g. where longitudinal cyclic peak pitch is applied to the swashplate / rotor) and rotor dynamics (e.g. the rotor flapping response to this cyclic input). Rate coupling is primarily due to changes in aerodynamics at the rotor due to the on-axis response. For example, when the helicopter pitches nose down the front of the rotor experiences an upward air velocity due to the pitch motion but an increase in induced velocity (downward) as it sinks into vortices emitted by the rotor.
For a long time these off-axis effects were poorly predicted by computer models, often even missing the direction of the movement. For example, many models would predict a right wing down roll motion after a longitudinal cyclic input when the real helicopter rolled left wing down. Models have improved in this regard, primarily due to better estimates of wake distribution over the main rotor disk.
In hover and low speed flight, the primary off-axis response to collective is yaw. Assuming the main rotor turns counter clockwise (CCW) when viewed from above, an increase in collective will cause a clockwise (CW) yaw response when viewed from above. This is because the engine provides a CCW torque to the main rotor (hence a CW torque to the fuselage). The tail rotor counters this torque to maintain fuselage heading, but increasing collective causes the engine to produce more torque (to maintain rotor speed), which overpowers the tail rotor and yaws the fuselage CW. Normally a pilot will add left pedal - adding tail rotor thrust - when increasing collective to counter this yaw effect and maintain heading.
At higher speeds the picture is different. Modern helicopters have large vertical fins that reduce the aforementioned yaw responses at high speed. However, there’s another effect that creeps in. At high speed, collective input adds more lift to the advancing side (right side for CCW) of the main rotor than to the retreating side. (This follows from the fact that the advancing side operates at a larger airspeed / dynamic pressure.) This lift imbalance changes rotor flapping, increasing it roughly 90 degrees later over the nose. This causes the helicopter to pitch up when the collective is increased.
Left pedal increases tail rotor thrust (CCW) which immediately adds a “right wing down” roll moment for a typical tail rotor located above the helicopter CG. However, with forward speed, this effect is quickly overcompensated by another effect. The left pedal input pushes the nose left and increases sideslip (the angle of the relative wind to the fuselage). This changes the main rotor flapping to be lower over the nose and higher over the pilots right side. This results in a left wing down roll and a nose down pitch.
The effects described above are the "raw" effects associated with the control movement. Helicopters with augmentation system activated will partially or completely counter many of these effects. For example, most augmentation systems, upon detecting a pure longitudinal cyclic input, will adjust lateral cyclic "underneath the covers" to minimize roll.
A paper about modeling off-axis responses may be found here.