Steady Rolling Motion
A measure of aileron control power is afforded by the steady roll rate, P, produced by the aileron. This rate is such that the rolling moment from the ailerons equals the damping moment resulting from P. The origin of this damping moment can be seen from Figure 8.35.
A section of the right wing located a distance у from the centerline will experience an increment in its angle of attack because of P, given by
Neglecting induced effects, Да results in a differential moment given by
dL = – yqcao-ydy
The total rolling moment is obtained by integrating this equation from 0 to b/2 and multiplying by two to account for the left wing. In coefficient form this becomes
c–¥dv)/.’ (§)*’* <8-98)
The quantity (Pb/2V) will be denoted by p and is comparable to the dimensionless pitch rate, q. p can be interpreted geometrically as the tangent of the helix angle prescribed by the wing tip as the rolling airplane advances through the air.
For a linearly tapered wing, Equation 8.98 integrates to
_ a0p 1 + 3A
n — ао 1 + ЗА 12 1 + A
The derivation of Equation 8.99 provides some insight into the origin of the roll damping but, otherwise, is of little value because of induced effects that were neglected. Correcting a0 for aspect ratio improves the accuracy somewhat.
Calculated values of Clf based on lifting surface theory are presented in Figure 8.36 (taken from Ref. 8.11). The original source of these calculations is Reference 5.5. These graphs are all for zero sweep angle of the midchord line. Results for other sweep angles can be found in the references. Generally, C((S is insensitive to А ід up to approximately ±20°. This range increases as the aspect ratio decreases.
Calculations of Ct. for a complete airplane configuration must also include contributions from the horizontal and vertical tails. The horizontal tail’s contribution can be determined on the basis of Figure 8.36 multiplied by Stb2/Sb2. The vertical tail’s contribution can also be determined on the same basis by visualizing the tail to be one-half of a wing. Thus the geometric aspect ratio of the vertical tail is doubled to enter Figure 8.36. The resulting value of Ci – is multiplied by Svb2ISb2 and then halved. If the horizontal stabilizer is mounted on top of the vertical tail, the value of Av must be increased, possibly by as much as 20%, to account for the end-plate effect of the horizontal tail on the vertical tail.
In a steady roll, the aileron-produced rolling moment and the damping moment will be equal in magnitude but of opposite sign. To find the steady
0 2 4 6 8 10
Figure 8.36 Roll damping coefficient derivative. Note that Cif should be corrected by ratio of section lift curve slope to 2тг.
rolling velocity as a function of aileron angle, we set the sum of the aileron and damping moments to zero.
Ch 5a + C,/ = 0 or
Thus, the dimensionless roll rate is directly proportional to the aileron deflection angle. PbtlV will typically range from approximately 0.05 to 0.10, depending on the maneuverability requirements imposed on the airplane.
Past practice was to specify values of p but both civil and military requirements (Refs. 8.1 and 8.12) now state specific requirements on P or, to be more specific, on a displacement of ф in a given time. For example, FAR Part 23 requires that an airplane weighing less than 6000 lb on the approach must have sufficient lateral control to roll it from 30° in one direction to 30° in the other in 4 sec. In landing at 20% above the stalling speed, this number is increased to 5 sec. Mil-F-8785 В requires an air-to-air fighter to be rolled 360° in 2.8 sec, an interceptor aircraft 90° in 1.3 sec, a transport or heavy bomber 30° in 1.5 sec, and a light utility aircraft 60° in 1.4 sec.