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National Advisory Committee for Aeronautics, Technical Notes - Generalized Indicial Forces on Deforming Rectangular Wings in Supersonic Flight

A method is presented for determining the time-dependent flow over
a rectangular wing moving with a supersonic forward speed and undergoing
small vertical distortions expressible as polynomials involving spanwise
and chordwisé distances. The solution for the velocity potential is
presented in a form analogous to that for steady supersonic flow having
the familiar "reflected area" concept discovered by Evvard. Particular
attention is paid to indicial-type motions and results are expressed in
terms of generalized indicial forces. numerical results for Mach numbers
equal to 1.1 and 1.2 are given for polynomials of the first and fifth
degree in the chordwise and spanwise directions, respectively, on a wing
having an aspect ratio of h.

One of the basic prdblems arising in the analysis of wing flutter
boundaries is the calculation of the aerodynamic forces on wings under-
going small but arbitrary spanwise and chordwise distortions. When the
wing aspect ratio is large (actually, when the distance between spanwise
nodal lines is large), these forces are usually estimated by some strip
theory in which the loading on each spanwise section is approximated
from that on a two—dimensional wing having the same chordwise distortion.
This report is concerned with low—aspect-ratio rectangular wings for
which tip effects are important and the full three-dimensional theory
must be used.

The exact linearized solution for the forces on thin rectangular
wi s limited, however, to the range where effective aspect ratio
( M2-l A) is 2 l) traveling at supersonic speeds has been presented by
both Gardner (ref. 1) and Miles (refs. 2 and 3) in terms of multiple
integrals involving arbitrary surface undulations. waever, the use of
such solutions in evaluating, numerically say, the forces induced'by
specific wing distortions still presents some difficulties. It is the
purpose of this report to discuss certain techniques that can simplify
the labor involved in these calculations and to present numerical tables
for the forces induced by a class of surface deformations, a class gen-
eral enough to represent the first few mode shapes of rectangular
plates.