DUTCH-ROLL APPROXIMATION

where/(s) is the characteristic polynomial (9.7,13).

For the spiral/roll approximations, we proceed similarly with (9.7,10) to solve for the desired ratios. In the following results, the subscripts a and r are omitted from the control derivatives since the same formulas actually apply to both 6a and dT. The only difference is that &is usually zero [as in

(10.5,6) ] making the aileron transfer functions simpler than those for the rudder.

SPIRAL/ROLL APPROXIMATION

/3 _ a3s3 + a2,s2 -f ars + a0

s m

Ф_,_ b^s – f – b0

f(8)

¥ _ d2s% + drs – f – dn

$ /(*)

%=А*І

д d

whore f (s) is the characteristic polynomial (9.7,11) and

«3 = ^,; «2 = + JTr) ~ ^

A

«1 = ®+

A 2 Aju

а0=^.(^гЖд-^Жг)

2 A[i

au g> і ш

W = ; bo = 2і (J2VT, – -2VT,) + – J (J^^r –

d2 = ; d, = – ад); d0 = ^ (J2VT, – JSVQ

The accuracy of the above approximations is illustrated for the example jet transport on Figs. 10.11 and 10.12. Two general observations can be made: (1) the Dutch-roll approximation gives good results for the higher frequencies, down to a little less than that of this mode, and (2) the spiral/roll approxi­mation is correct in the low frequency limit. In this respect the situation is entirely analogous to the longitudinal case, with the spiral/roll corresponding to the phugoid and the Dutch-roll to the short-period approximation. There are ranges of frequency where neither approximation is satisfactory, as on Figs. 10.lie and 10.12e. The spiral/roll approximations for the phase angles are not shown on Fig. 10.11, since they are reasonable only at the lowest frequency. For all three variables, /?, </>, and f, they increase monotonically to about 180° at the highest frequency, whereas the exact phase angles all decrease in this range.

The reader should note that the agreement shown for the Dutch-roll approximation is not to he expected generally. We saw in Fig. 9.28 that the

damping is not given at all well by this approximation at low speed (high CLJ. Thus the approximate solution at low speed would substantially underestimate the amplitude peaks at the frequency of the lateral oscillation. We repeat that the lateral approximations must be used with caution, and that only the use of the exact equations can guarantee accurate results.