Static stability

Longitudinal static margin

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The third of Eqns (4.7) is a simplified statement of the pitching moments on the aircraft.

The last term on the left-hand side is the inertial moment required to bring the aircraft to a quasi-static balance, that is, it is equal in magnitude but of opposite sign to the algebraic sum of the aerodynamic and effects. Thus, replacing the inertial moment by M, the equation may be rewritten in the

loading a n d structural layout

Expressed in coefficient this becomes:

+

+

or if the dynamic at the tail is assumed to be the same as that at the wing:

The horizontal stabilizer volume coefficient is defined as:

For steady conditions and with the dynamic pressure at the tail equal to that at the wing, from Eqn. (4.19):

-

+ +

This may be substituted Eqn. (4.25) to give:

Equation (4.26) can be rearranged to state the elevator angle required to trim the aircraft, that is to achieve = 0:

The condition for longitudinal static stability is that the rate of change of the pitching moment with should be negative. Noting that and are constant and that = differentiation of Eqns (4.26) with respect to gives.

= - - - (4.28)

Rigid airframe dynamics

Different stability conditions when the elevator angle is fixed, known as the controls-fixed case, as opposed to when it is free to move, the controls-free case. The controls-free condition is dependent upon the elevator hinge moment characteristics hut the controls-fixed condition follows directly from Eqn. (4.28) since is zero:

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Many references cover the controls-free condition. for example Let the fixed static margin be defined as:

(h,

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where is the position on he aerodynamic mean chord of the controls-fixed neutral point. From Eqn. (4.29):

Equation (4.30) shows that controls-fixed static stability is present when is positive and that stability increases with increase of the distance of the centre of gravity ahead of the neutral point. Thus the controls-fixed neutral point defines the aft-most centre of gravity position consistent with natural, controls-fixed static stability. For aircraft with conventional control systems it is usually considered that the value of should not be less than about The forward centre of gravity limit is determined by the need to ensure that is not so large as to create a situation where the control force required to the aircraft is unacceptably high. The actual control force required is a function of the characteristics of the control system.

Aircraft equipped with advanced control systems may be designed to allow for the possibility that the static margin becomes negative, artificial stability being achieved means of automatic control deflection. Reference to Eqns (4.28) and (4.30) shows that

may be written in the form:

Thus a negative numerical value of the sum of the first two terms on the right-hand side of the equation may be compensated for by an appropriate movement of pitch control, Since such an automatic control system is invariably designed to be

irreversible the still applies.

M. V. Principles. 1997.

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Longitudinal manoeuvre margin

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When the aircraft is in a steady pitching manoeuvre the body angle, and the rate of change of pitching moment, are both constant so that

The increment in the normal acceleration is given by Eqn. as:

Also the increment in the wing-body lift may be written as:

The increment in the angle of attack of the horizontal stabilizer is due both to the increase in the body angle consistent with Eqn. and the steady rotational effect given the second term in the brackets on the right-hand side of Eqn. (4.19). When

is equal to the latter becomes:

In the steady level flight trimmed condition:

and

where

is the longitudinal relative density, see Section

The pitching moment equation, Eqn. (4.25). applies to this condition with modified values of the variable parameters and, for completeness, the inclusion of the

Write:

Rigid airframe dynamics

where

From Eqns and

Now from Eqns (4.19) and (4.26):

where use has been made of Eqn. Hence:

and noting that collecting like terms:

The first of the on the right-hand side of the equation is the steady flight pitching moment and, as initial trim is assumed. it is zero. Therefore:

For stability and for a steady pitching manoeuvre = 0.

loading and structural layout

Write Eqn. (4.36) in the form:

When the controls are fixed:

and then. since and are constant:

The controls-fixed manoeuvre margin, is defined as:

= h,,, h where, using Eqn. (4.31):

Hence from Eqn. (4.30):

It follows from Eqn. that the controls-fixedmanoeuvre margin is greater than the controls-fixed static margin providing that is positive. but see Section

4.3.3

The tendency of an aircraft to return to a wings level condition after a disturbance about the Ox, rolling. axis is defined as lateral static For an aircraft with the usual vertical plane of along the Ox axis ideally there are no forces or moments in the lateral trim condition.

Lateral static stability requires that:

where is the rolling moment coefficient and p i s the hank angle. Although a number of effects contribute to the value of including the vertical position of the wing on the body, the most significant ones are the dihedral and sweep of the wing. Positive dihedral, that is when the wing tips are above the root. and sweepback both give negative

Rigid airframe dynamics

contributions to the rolling moment coefficient, that is confer lateral static stahility. Thus less dihedral. or even is associated with swept hack wings. See also Section

4.3.4 Directional static stability

The tendency of an aircraft to align itself, or weathercock. into the wind direction is defined as directional static stability. The required condition is:

where is the yawing moment coefficient and the yaw angle. When the wind and earth axes coincide the sideslip angle. is equal to the required stability condition may be written as:

The main positive contribution to the static directional stability arises from the vertical stabilizing surface, or fin. The inclusion of a dorsal extension to the fin is advantageous in delaying the fin stall and thus ensures directional stability up to higher yaw angles.

See also Section

4.4 General equations of motion