Rotary Derivatives

Rotary stability derivatives are the variations in force and moment coefficients with airplane angular velocity. Angular velocity is almost always made dimensionless in rotary derivatives by multiplication by a factor 1 /(2 V), where 1 stands for either the wing chord c or span b and V is the velocity. A typical rotary derivative is the pitch damping derivative Cmq, defined as dCm/(dqc/2V).

Rotary derivatives were neglected in the original Bryan and Williams equations of motion (Bryan and Williams, 1903), since there was then no way to measure them. However, Bryan was later able to describe two techniques for rotary derivative measurement: putting a model on a whirling table or at the end of a whirling arm, and oscillating a model in an otherwise conventional wind tunnel (Bryan, 1911).

The oscillation technique survived right up to modern times. It is used in supersonic as well as in low-speed wind tunnels. An ingenious forced oscillation technique for measuring rotary derivatives uses feedback control to stabilize the amplitude and frequency of a forced oscillation regardless of the model’s level of stable or unstable derivatives (Beam, 1956).

An additional feature of Beam’s forced oscillation method is the separation of pitch and yaw damping derivatives from the cross-rotary derivatives, such as the rolling moment due to yawing, by oscillating the model around different axes. In – and out-of-phase torque measurements are solved simultaneously for the answers. The drawback in Beam’s work is that the damping derivatives such as Cmq and Cnr are inseparable from angle of attack and side-slip rate derivatives, such as Cma and Cn^. This separation is possible in specialized forced-motion wind-tunnel tests.

One of the few wind tunnels that produced pure damping derivatives was the NACA Langley Stability Wind Tunnel. The past tense is used because the Stability Wind Tunnel was

dismantled some years ago and shipped to the Virginia Polytechnic Institute. The Stability Wind Tunnel had curved test sections in which the forces and moments on an ordinary model were the result of rotary flows. This yielded the rotary derivatives uncombined with attitude rate derivatives. The same effect was produced with curved airship models tested in ordinary wind tunnels back in the 1920s.

The Stability Wind Tunnel also used radial turning vanes ahead of the test section to produce rolling flow. Flow angularity with respect to the wind-tunnel centerline was a linear function of distance from the centerline to the tunnel walls. The aerodynamic forces on a model held rigidly at the tunnel center would be identical to those of a rolling model in an ordinary wind tunnel, except for some transverse boundary layer motion caused by radial pressure gradient. The DVL in Germany experimented with rolling flow in wind tunnels in the 1930s.

The whirling arm as a device for measuring rotary derivatives had a rebirth of sorts at the Cranfield College of Aeronautics in the early 1960s (Mulkens and Ormerod, 1993). The motivation is support of a Royal Aircraft Establishment flight research program called HIRM, for High-Incidence Research Model. Carbon-fiber-reinforced plastic, foam, and fiberglass models are whirled on an 8.3-meter arm inside a toroidal test channel. Moving the models at constant angle of attack along circular paths provides pure rotary derivative data, equivalent to that gotten from curved flow wind tunnels.