The significance of these roots and the instability exhibited by the positive real root can be examined by looking at the shapes of each mode in the same manner that was followed for the longitudinal motion.
For the transient solution, 8„ and 8r are zero. Using Equation 10.27a and 10.27b, we can eliminate r to solve for the ratio of ф to /3. This result, which holds for any a, is
ф _ a + 0.513
J~~ 0.766o-2 + 2.150- – 0.0180
rlj3 can then be obtained from any one of the equations. Using Equation 10.27a
^ = -(1.007a – + 0.0131) – & (0.00129a – – 0.0180) (10.30)
For cr = -2.79, фіі8 and rip become
Since r — ф, we can replace r by стф, so that
For this mode, it is seen that both /3 and ф are small compared to ф. Thus, this mode is predominantly a damped rolling motion. Indeed, if one neglects all but the ф terms in Equation 10.27b, a value for <r of —2.80 that is very close to the exact value is obtained immediately.
The time for the roll rate to damp to half of its initial value can be found from Equation 9.72 to be
For this mode, the motion is seen to be predominantly a heading change with a small roll angle and sideslip angle. With o – being positive, these angles increase with time, so the mode is actually unstable! With ф increasing exponentially with time, the flight path of the airplane describes a spiral. Thus, this mode is referred to as the spiral mode. If it is unstable, as in this case, the motion is referred to as spiral divergence; otherwise, it is referred to as spiral convergence.
The time to double amplitude is found from
= 330 air sec
Tdbi = 30.2 sec
This time is characteristic of many aircraft and is sufficiently long so that the pilot compensates for the divergence without realizing it. Although spiral divergence cannot be described as unsafe, it can result in extreme attitudes if the pilot should be studying a chart and forgets to fly the airplane for a few moments. It can prove catastrophic for the noninstrument-rated pilot who finds herself or himself in instrument conditions.
The root for the spiral mode is normally small, so it can be closely approximated by the constant term in the characteristic equation divided by the coefficient of a to the first power. From Equation 10.17, the determinant defining the characteristic equation is:
When this determinant is expanded for typical values of the stability derivatives, one obtains approximately
2n(Cl0CN. – СщСч)<т = C^Cn’Q, – ClfCNf) or
Equation 10.31 neglects terms in a of order higher than the first. It also neglects some first-order terms in a that are typically small. For the Cherokee, this approximation for the root of the spiral mode gives a value of 0.00227 that is 8% higher than the exact value.
The denominator of Equation 10.31 is usually positive, so the combination of terms in the numerator governs whether or not the spiral mode
is stable. For spiral convergence,
Сц0Сір < C^Cn. (10.32)
Since most of the contribution to Ctf results from the wing, this derivative is not too easily adjusted. Varying the vertical tail size will change and CNf approximately in the same proportion. Also, the vertical tail size is normally fixed by other considerations. Hence the primary control on the spiral mode is exercised through Ch, the dihedral. Increasing the dihedral effect will tend to make the spiral mode more stable. However, as stated previously, too much dihedral leads to an unpleasant feel to the airplane.