A Survey of Unsteady Hypersonic Flow Problems

-%-
wiierc h and au are the flutter displacements in plunge and pitch, and the
? a
primes denote the operator - -. The coeffxients Z& involve the characteristics
w at
of the section, the mean incl 8 ence, Mach number, and frequenoy parameter: there
arc two sets of coeffuSxdx,one for the plunge equation, the other for the
pitch equation. These non-linear equations are first simplified under the
assumption that, if the non-linearities are small ! information on the orders of
magnitude of terms can be obtained from the solution of the linearized problem,
and certain terms in the non-linear equations can then be neglected because of
their smallness. It is also assumed that the mean incdence is zero. An
approximate analysis of the simplified equations is then carried out. It is
assumed, first, that if the section is flying at a speed close to the flutter
speed predicted by a linearized analysis, and is subjected to a disturbance,
It will stabilize to a finite periodic motion and that this motion can be
represented. by
m
h, = T hneinwt b = 0; hBn = h;
*=-ccl
au = r a* eirwt a, = 0; a-* = a;
*=-co
. . . (4.15)
where 19 and afi are the complex conjugates of hn and an. h and a
allowd eo
be complex so that it is possible to diow phase anglesnbetween the are
aegrecs of freedom, but h, and au fan be shown to be real. For simple
harmonic motion (single frequency) Ihn/ and IanI are equal to one half of
the corresponding amplztudes.
It is then assumed that the fundamental harmonic of the two component
motions dominant in the flutter motion, and. the equations are found which ensure
that these components are balanced. Finally, if it is further assumed, on the
basis of a linearized analysis, that the phase angle between the hu and au
motions is very small, the equations for the motion become
where the coefficients Zn are given below.
h,, motion/