# Rolling Moment with Rudder

If the center of pressure of the fin-rudder combination lies a distance of Zv above the longitudinal axis passing through the center of gravity, an increment, ALV, in the lift (side force) will produce an increment in the rolling moment, given by

AL = Z„ ALV

In coefficient form this becomes

Zv A Lv b qS

= тгСг. в,

The side force derivative CYs is obtained from

Rolling Moment with Yaw Rate

Figure 8.37 illustrates a wing having a tapered planform that is yawing at the rate of R rad/sec. The wing is operating at a lift coefficient of CL. It will be assumed that the section lift coefficients are constant and equal to CL. Because – of the rotational velocity R, a section on the right side located at у experiences a local velocity equal to V – Ry while the corresponding section on the left wing has a velocity of V + Ry. Thus a differential rolling moment is produced equal to

dL = уpcCL[{V + Ryf-{V-Ryf] dy

Expanding and integrating from 0 to Ы2 results in

, pcCJtVb3 12

=qSb? CL

As an exercise, show that, for a linearly tapered wing,

^ Cl/1 + 3A

C, f 6 u + A j

The vertical tail can also contribute to Ctf Because of the yaw rate, its angle of sideslip is decreased by RIJV. Thus, an incremental side force on the tail is generated, given by

A V c ^

Д Y = – y-

Flgure 8.37 A wing yawing and translating.

In coefficient form,

CY = 2r),av Vvr (8.104)

This incremental side force acting above the center of gravity produces an increment in the rolling moment, given by

Д L = r)tqSvaK Zv

or, in coefficient form,

Clf = Cyrf (8.105)

= 2т

The total C(, equals

<8I06)