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National Advisory Committee for Aeronautics, Report - 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 chordwise 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 indioial-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 4.

One of the basic problems arising in the analysis of wing
flutter boundaries is the calculation of the aerodynamic forces
on wings undergoing small but arbitrary spanwise and chord—
wise distortions. When the wing aspect ratio islarge (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 chord-
wise 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 rectan—
gular wings (limited, however, to the range where effective
aspect ratio (m A) is 21) 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. However, 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 general enough to represent
the first few mode shapes of rectangular plates.