Table 10.1 presents the maximum roll mode time constants that should not be exceeded in order to achieve a given flying quality level. In order to interpret this table, one needs to understand the meaning of the term “time constant.”
A first-order, linear system x(t) obeys a differential equation, which can be written as
x + — x=f(t) (10.36)
where r is defined as the time constant. For the homogeneous solution, x will be of the form
x = ХоЄ^’
so Equation 10.36 becomes
(<r + i)x = °
v = – (10.37)
The homogeneous solution for x can therefore be written as
X = x0e~tlr (10.38)
Thus, the time constant, r, is a measure of the damping in a first-order system.
The more heavily damped a system, the smaller will be its time constant. Given an initial displacement and released, the system will damp to Це or 0.368 of its initial displacement in a time equal to the time constant. The time to halve amplitude and the time constant, r, are related by
Tm = 0.693 t
For the Cherokee 180 example, for the roll mode, cr = —2.79, so
t = 0.350 air sec
or, in real time,
t = 0.033 sec
This value is well within the level 1 criteria for the class I airplane for all flight phases.
Dutch Roll Mode
Table 10.2 presents criteria for the frequency and damping ratio for the Dutch roll mode. Note that minimum values are specified for £, ш„, and for the product £шп. The minimum value for is determined from the column labeled o)„. However, the governing damping requirement equals the largest value of £ obtained from either of the two columns labeled £ and £<a„.
For the Cherokee 180 example, in real time,
tr = -0.601 ± 3.03/
Thus, from Equation 9.51,
£ = 0.194 a>„ = 3.09 rad/sec
With reference to Table 10.2, the Cherokee’s damping in this mode is governed by the column labeled £ and is seen to be nearly equal to the minimum value prescribed for a flying quality level of 1 in the flight phase category A. Thus one would not expect to encounter any problems from the Dutch roll mode with this airplane.