Approximate Equations for the Lateral Modes

As with the longitudinal modes we should like if possible to have useful analytical approximations to the lateral characteristics. We find that there are reasonable ap­proximations to all three modes, but the application of all such approximations must be made with caution. Their accuracy can really be verified only a posteriori, by comparison with exact solutions. They can only be used with confidence in situations similar to those in which they have previously been found to work well.

SPIRAL MODE

Comparison of the eigenvalues in Sec. 6.7 shows that Л for the spiral mode is two or­ders of magnitude smaller than the next larger one. This suggests that a good approx­imation to this root may be obtained by keeping only the two lowest-order terms in the characteristic equation, that is,

£>A + £ = 0 (6.8,1)

or Av = —E/D

where As denotes the real root for the spiral mode. Before deriving expressions for D and E, we rewrite the matrix of (4.9,19) in a more compact notation for convenience, including the approximation Yp = 0.

%

0

%

g cos 0O

A =

£v

£r

0

К

к

К

0

_ 0

1

tan 0O

0

The meanings of the symbols in (6.8,2) are obtained by comparison with (4.9,19), for example

Approximate Equations for the Lateral Modes Подпись: К h

and in the special case when the stability axes are also principal axes, I„ = 0 and

With the notation of (6.8,2), expanding det (A — ЛІ) yields

E = g[(£vNr – £rKv) cos 0O + (£рХи – £vMp) sin 0O] (a)

D = ~g(£v cos 0O + Jfv sin 0O) + °Hv(£rMp – £pKr) (6.8,3)

+ Щ<£^0 – £VMP) (b)

When the orders of the various terms in D are compared, it is found that the second term can be neglected entirely and YT can be neglected in rj>£. The approximation that then results is

D = ~g(£„ cos 0O + Mv sin 0O) + u0(£vXp – £pMv) (6.8,4)

The result obtained from (6.8,1), (6.8,3a), and (6.8,4) for the jet transport example of Sec. 6.7 is As. = —0.00725, less than 1% different from the correct value. Equation

(6.8,1) is seen to give a good approximation in this case.

It will be recalled that the coefficient E has special significance with respect to static stability (see Sec. 6.1). We note here that in consequence of (6.8,1) the spiral mode may exhibit exponential growth, and that the criterion for static lateral stability is

(£vMr – £rXv) cos 0O + (£PMV – £v. Kp) sin 0O > 0 (6.8,5)

On substituting the expanded expressions for £v and so forth, (6.8,5) reduces to

{Cifnr – C, Cn/1) cos 00 + (ClpCnfi – chcn) sin 00 > 0 (6.8,6)

Since some of the derivatives in (6.8,6) depend on CLo, the static stability will vary with flight speed. It is not at all unusual for the spiral mode to be unstable over some portion of the flight envelope (see Table 6.10 and Exercise 6.3).

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