The derivatives Nu, Nw, Lu, Lw
These four derivatives play an important role in coupling the low-frequency longitudinal and lateral motions of the helicopter. The yawing moment derivatives stem largely from the changes in main rotor torque with velocity perturbations, although there is also an effect from the fin (Nu), similar to the contribution of the horizontal stabilizer to Mu. The Nw effect can be quite significant since torque changes to vertical velocity are similar to the direct control coupling-torque changes from collective inputs. The Lu effect reflects the changes in lateral cyclic to trim with forward speed, being dominated by the main rotor effect at low speed. Forward velocity perturbations increase the incidence on the forward part of the (coned) disc and reduce the incidence at the rear. The disc will therefore tilt to starboard for anticlockwise rotors (port for clockwise rotors). As forward speed increases, the four derivatives show similar trends and the dominating main rotor components are closely related through the shaft tilt and product of inertia.
The angular velocity derivatives
Our discussion on derivatives with respect to roll, pitch and yaw rate covers three distinct groups – the force derivatives, the roll/pitch moment derivatives due to roll and pitch and the roll/yaw derivatives due to yaw and pitch. Derivatives in the first group largely share their positions in the system matrix (eqn 4.44) with the trim inertial velocity components. In some cases the inertial velocities are so dominant that the aerodynamic effects are negligible (e. g., Zq, Yr). In other cases the aerodynamic effects are important to primary response characteristics. Two such examples are Xq and Yp.