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Sailplane Moments

Introduction

These notes consider the moment forces around the pitch axis of a typical aircraft. The analysis results in the CG location based on normalized dimensions, wing loading, tail volume coefficient, and flight speed.

Analysis

A sailplane is affected by lift, weight, drag, tail-force, and the pitching moment of a cambered airfoil. Unlike a powered aircraft, a sailplane is not affected by the thrust of an engine. Each force generates a torque, or moment, around some reference point. The moment is simply the force multiplied by the distance between the force and reference. In quilibrium, or straight and level flight, the sum of these moments must be zero. Their sum is:
W Xc = Mo + L Xa + T Xb - D Xz
where
W weight
Xc distance from leading edge to center of gravity
Mo wing moment around the leading edge
L lift generated by wing
Xa distance from leading edge to center of lift
T lift generated by tail
Xb distance from leading edge to center of tail lift
D drag generated by wing
Xz verticle distance between center of gravity and center of drag

Eliminating Lift

Consider that the lift generated by the wing and tail equal the weight:

L = W - T

and substituting

W Xc = Mo + (W - T) Xa + T Xb - D Xz
W (Xc - Xa) = Mo + T (Xb - Xa) - D Xz

The result above shows that the lift can be dropped from the equation, and that the tail and CG moment arms can be taken, more conveniently around the center of lift. However, the pitching moment need also be taken around the center of lift. The center of lift is typically located at 1/4 of the mean aerodynamic chord, MAC, and it is standard practice to measre the pitching moment at this point. Mo is therefore replaced with Mg.

W Xg = Mg + T Xt - D Xz
where
Xg distance from center of gravity to 1/4 MAC location
Mg wing moment around the 1/4 MAC location
Xt distance from center of tail lift to 1/4 MAC location

Affect of Airspeed

The moment and force terms can be expanded to consider airspeed:

W Lg = 1/2 r V2 S c Cm + 1/2 r V2 St Clt Lt - 1/2 r V2 S Cd Lz
where
rho air density divided by viscosity (e.g. 0.002378 slugs / ft^4)
V air speed (e.g. ft per second)
S wing area (e.g. ft^2)
c mean aerodynamic chord of wing
Cm wing moment coefficient
St tail area (e.g. ft^2)
Clt lift coefficient of tail
Cd drag coefficient of wing
and simplifying
W Lg = 1/2 rho V2 { S c Cm + St Clt Lt - S Cd Lz }

Normalizing Dimensions

By dividing each term by S c, the dimensions in the above equations can be made relative to the MAC of the wing. In addition, the weight can effectively be replaced by wing loading. :
(W/S) Lg / c = 1/2 rho V2 [ Cm + Clt (St Lt) / (S c) - Cd Lz / c ]
recognizing that:
W / S wing loading
(St Lt) / (S c) tail volume coefficient
Also notice that all distances are divided by c and can be represented by normalized values:
(W/S) xg = 1/2 rho V2 [ Cm + Clt (St xt) / S - Cd xz ]
where
xg normalized distance from center of gravity to 1/4 MAC location
xt normalized distance from center of tail lift to 1/4 MAC location
xz normalized verticle distance between center of gravity and center of drag

CG Location

Finally, dividing both sides by the wing loading isolates the CG location, referenced to the wing's 1/4 MAC location:
xg = 1/2 rho V2 / (W/S) [ Cm + Clt St xt / S - Cd xz ]

Observations

Some observations:

Some questions:


Dynamic Behavior

How does the plane react to a pertubation of its angle of attack?

Moment vs Center-of-Pressure

The following show the relationship between the moment coefficient and center-of-pressure
1/2 rho V2 S c Cm = 1/2 rho V2 S Cl x
where
x distance from center of pressure to the 1/4 MAC location
Simplifying
c Cm = Cl x
and rearranging
x / c = Cm / Cl
The reference can be shifted from the 1/4 MAC location to the leading edge by adding a constant
x / c = 0.25 + Cm / Cl
It's easy to see that at zero lift, the center-of-pressure is an infinite distance behind the wing. Similarly, at infinite lift, the center-of-pressure can be no further than 1/4 MAC.
Last modified Friday, 06-Sep-2002 06:49:54 EDT