# The derivatives

There are 36 stability derivatives and 24 control derivatives in the standard 6 DoF set. In this section we shall discuss a limited number of the more important derivatives and their variation with configuration and flight condition parameters. The complete set of numerical derivatives for all three reference aircraft are contained as charts in the Appendix, Section 4B.2, and the reader may find it useful to refer to these as the discussion unfolds. It should be noted that the derivatives plotted in Appendix 4B include the inertial and gravitational effects from eqn 4.44. For example, the elements Zq and Yr tend to be dominated by the forward velocity term Ue. Each derivative is made up of a contribution from the different aircraft components – the main rotor, fuselage, etc. In view of the dominant nature of the rotor in helicopter flight dynamics, we shall give particular, but certainly not exclusive, attention to main rotor derivatives in the following discussion. The three most significant rotor disc variables are the rotor thrust T and the two multi-blade coordinate disc tilts fiic and fBs. During disturbed motion these rotor states will vary according to the algebraic relationships derived in Chapter 3 (eqns 3.90, 3.65). Considering the simple approximation that the rotor thrust is normal to the disc, for small flapping angles, the rotor X and Y forces take the form

Xr = TPic, Yr = – TPis (4.52)

The derivatives with respect to any motion or control variable can then be written as, for example, d Xr d T BBc

—R = —p1c + T — d u d u d u

Rotor force and moment derivatives are therefore closely related to individual thrust and flapping derivatives. Many of the derivatives are strongly nonlinear functions of velocity, particularly the velocity derivatives themselves. The derivatives are also nonlinear functions of the changes in downwash during perturbed motion, and can be written as a linear combination of the individual effects, as in the thrust coefficient change with advance ratio, given by

dCt / dCt dCt dko ЭCt 3kis dCt 3kic

Bp, Bp. Jx=const dko dp. 3kis Bp Bkc Bp

where Ct is the thrust coefficient and д the advance ratio defined by

T V

CT =———- »—г, д =—— (4.55)

T p(V R)2nR2 V R  and the Xs are the components of the rotor induced inflow in the harmonic, trapezoidal form     The thrust coefficient partial derivative with respect to д can be written as

The rotor force, moment and flapping equations as derived in Chapter 3 are expressed in terms of the advance ratio in hub/wind axes. The relationships between the velocity components at the aircraft centre of mass and the rotor in-plane and out-of-plane velocities are given in Chapter 3, Section 3A.4. It is not the intention here to derive general analytic expressions for the derivatives; hence, we shall not be concerned with the full details of the transformation from rotor to fuselage axes except where this is important for enhancing our understanding.

The translational velocity derivatives

Velocity perturbations give rise to rotor flapping, changes in rotor lift and drag and the incidence and sideslip angles of the flow around the fuselage and empennage. Although we can see from the equations in the 3.70 series of Chapter 3 that the flapping appears to be a strongly nonlinear function of forward velocity, the longitudinal cyclic required to trim, as shown in Fig. 4.10(a), is actually fairly linear up to moderate forward speeds. This gives evidence that the moment required to trim the flapping at various speeds is fairly constant and hence the primary longitudinal flapping derivative with forward speed is also relatively constant. The orientation between the fuselage axes and rotor hub/wind axes depends on the shaft tilt, rotor flapping and sideslip angle; hence a u velocity perturbation in the fuselage system, say, will transform to give дх, Ду and дг disturbances in the rotor axes. This complicates interpretation. For example, the rotor force response to дг perturbations is much stronger than the response to the in-plane velocities, and the resolution of this force through only small angles can be the same order of magnitude as the in-plane loads. This is demonstrated in the derivatives Xu and Zu at low speed where the initial tendency is to vary in the opposite direction to the general trend in forward flight.