G. Leng, Flight Dynamics, Stability & Control

Lecture 3 : Static Longitudinal Stability

Or the balancing of lift forces and pitching moments

G. Leng, Flight Dynamics, Stability & Control

1.0 Trim condition

An aircraft is trimmed if there are no nett forces and momentsacting on it

E.g. The nett pitching moment coefficient is zero

M = 1/2 V2 S c Cm = 0

pitching moment coefficient

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1.1 Concept of static stability

Static stability refers to the tendency of the aircraft todevelop forces or moments to return to its trimcondition when disturbed

Question : What does that mean ?

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Figure 1.1 : Static stability

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1.2 Longitudinal static stability

Longitudinal static stability refers to the tendency of the aircraft toreturn to its trim condition after a nose up or nose down disturbance

This implies that Cm must vary with AOA in a certain manner !!

Question : What manner ?

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Figure 1.2 : Cm variation with AOA

Cm

Cm

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Figure 1.3 : F4 wind tunnel data - pitching moment

Source : NASA Technical Note D 6425

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Moral

A statically stable aircraft must have Cm / < 0

Question : Is static stability always a good thing ?

Note : Aeronautical engineers usually write Cm / as Cm

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1.3 Tail configuration - statically stable

W

c.g. wing

Lw

Ltlw

lt

tail

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1.4 Tail configuration - statically unstable

W

c.g.wing

Lw

Lt

lw lt

tail

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Moral

Aircraft stability depends critically on cg location

Note : All aircraft have forward and aft cg limits

Question : What happens if either forward or aft limits are violated ?

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2.0 Quantifying static longitudinal stability

W

c.g.

wing

Lw Lt

Xw Xt

tail

Xcg

V

em0

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The trim conditions require that

Lw + Lt = W

m = 0

Where m = m0 + Lw (Xw – Xcg) + Lt (Xt – Xcg)

= m0 + Lw lw + Lt lt

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2.1 : Lift curve slope

Lw = ½ V2 Sw CLw

The key is to express aircraft lift and pitching moment in terms ofcontributions from the wing and the tail.

= ½ V2 SW aw

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2.2 : Effect of downwash on the tail lift

Lt = ½ Vt2 St CLt

= ½ (V2) St at [ (1 - /) + e ]

Recall the tail surface is affected by downwash

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2.3 : Aircraft lift coefficient

Define the aircraft lift as L = Lw + Lt . In non-dimensional form

½ V2 Sw CL = ½ V2 Sw CLw + ½ Vt2 St CLt

CL = CLw + (Vt /V)2 (St/Sw) CLt

= CLw + (St/Sw) CLt

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The aircraft lift curve slope a = (CL)/ is given by :

a = aw + (St/Sw) at ( 1- )

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2.4 : Aircraft pitching moment coefficient

Define the aircraft pitching moment as :

m = m0 + Lw lw + Lt lt

Non-dimensionalise i.e. divide by ½ V2 Sw c

Cm = Cm0 + (lw/c) CLw + (St lt) /(Swc) CLt

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In terms of aoa and tail deflection

Cm = Cm0 + (lw/c) aw + VH at [(1-) + e]

Cm = Cm0 + [ (lw/c) aw + VH at (1-) ]

+ [ VH at ] e

How would you interpret this ?

Collecting terms…

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2.5 : Neutral point & static margin

Differentiate Cm with respect to

Cm = (lw/c) aw + VH at (1- )

Express in terms of cg location…

Cm = [(Xw - Xcg )/c] aw + (St/Sw) [(Xt - Xcg) /c] at (1- )

= [ (Xw/c) aw + (St/Sw) (Xt/c) at (1- ) ]

- [ aw + (St/Sw) at (1- ) ] (Xcg/c)

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The neutral point is the cg location where Cm = 0, i.e.

Xnp/c = [(Xw - Xt)/c] (aw/a ) + (Xt/c)

Cm = [(Xw - Xt)/c] aw + a (Xt/c) - a (Xcg/c)

Rewrite the pitching moment curve slope in terms of Xnp/c …

Cm = a (Xnp/c) - a (Xcg/c)

= - a [ ( Xcg – Xnp) /c ]