A Survey of Unsteady Hypersonic Flow Problems

-4z-
where C . is the rate of change of pitching moment coefficient Cm with tune
rate of c"e: ange of angle-of-attack parameter, ah/U.
Comparison of this equation with that for the short period motion of
an aircraft in level flight (Ref. 42: Section 6.7) shows that the equations are
the same, apart from the time dependence of U and n and the factor
in the stiffness term fa(t). The fac;or i(Cb T) can be
shown to be negligible in comparison with the other terms tiers. 47-k 49) so
that It is to be expected that the frequency at a given altitude will be close
to that for level flight. This is confirmed by Kistler and Capalongan in
Ref. 51 where they give the results of analogue studies of the motion of
hypervelocity vehicles.
Fram the solution of equation (3.11), it can be shown that the
requirement for convergence of the oscillations is that
where q is the free-stream dynamic pressure &p,V’, . . . (3.12)
s is distance along the flight path,
B is density parameter in p = Poe-Bh,
y flight path angle to local horizontal,
ca, = aha)
as .
The parameter K occurs also in the equation for the short period
motion of an aircraft in level flight (Ref. J+2: Section 6.7). The convergence
criterion is then
K < 0. . . . (3.13)
It can be shown from equation (3.12) that the conditions of re-entry
flight introduce a destabilising influence from decelerating effect of the drag
of the vehicle, and a stabilizing effect from the rate of increase of air density.
After they have established the equations of motion of a vehicle and
the convergence criterion, equation (3.12)) Sommer and Tobak examine the
oscillation histories of a range of lifting and non-lifting vehicles for a
range of entry conditions to give examples of the significance of the damping
criterion./