A Survey of Unsteady Hypersonic Flow Problems

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vehicle. (The difference between the damping curves for the phugoid motion for
the vehicle with air-breathing engine, and for the vehicle with rocket engine,
is a result of the assumed variation of thrust with altitude for the sirbreathing
vehicle.)
The values of frequency parameter given by Etkln's analyst can be used
to estimate the order of magnitude of the maximum frequericy parameters that are
likely to be met in practice, except for vehicles with very low wing loadings.
This result arises from the fact that It can be shown, by a dimensional argument,
that the frequency parameter of the pitching osclllatlons of a vehicle is
independent of vehicle size and speed, and 1s dependent only on the aerodynamic
characteristics, the altitude, the weight loading, and the inertia properties.
The hypothetical vehicle 1s sufficiently representstlve in these respects.
In general terms, the frequency of the pitching oscillations of a
vehicle is given by:
where M,
of inertia.
where
!Y
. . . (3.8)
1s the aerodynamic stiffness in pitch and IB is the vehxle moment
Then, if the aerodynamic coefficients are independent of speed,
pU2Sb
fa -
I---
5
is the radius of gyration of the vehicle, and the frequency parameter
2db PSb
k=-oc -.
U rmK=
Y
. . . i3.9)
From equation (3.9) it can be concluded that the frequency parameter is
independent of the velocity of the vehicle, and depends only on the vehicle
characteristics and the altitude. It decreases with altitude.
For geometrically and inertially similar bodies, since
and
5
it follows from equation (3.9) that
S cc b2,
m oc aba where cr is the vehicle density,
~1 b,
k cc ,'Tc. . . . (3.10)
From equation (3.10),it can be concluded that the frequency parameter
is independent of the vehicle size, and inversely proportional to the square root
of/