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RESPONSE TO DISCRETE AND CONTINUOUS GUSTS OF AN AIRPLANE HAVING WIND-BENDING FLEXIBILITY
ANALYSIS OF RESPONSE TO ARBITRARY GUSTS
EOVA_ONSroa ST_UCTUaALamPONSE
The following analysis treats the problem of determining
the stresses that develop in an airplane flying through vertical
gusts on the assumption that the airplane is free to respond
only in vertical motion and wing bending. The case of the
transient response to arbitrary gusts is considered first.
A subsequent section is then devoted to the case of random
disturbances in which explicit consideration is given the
continuous nature of atmospheric turbulence.
Equations of motion.--It is convenient to treat the problem
simply as one of determining tile elastic and translational
response of a free-free elastic beam subject to arbitrary dy-
namic forces. For dynamic forces of intensity F per unit.
length, the differential equation for wing bending is, if struc-
tural damping is neglected,
_2 _. _ ....
_=-mwt_ O)
where w is the deflection of the elastic axis referred to a fixed
reference plane. The task of detelmMning the deflection that
results from the applied forces F may be handled conveniently
by expressing the deflection in terms of the natural free-free
vibrational modes of the wing.
The wing deflection is thus assumed to be given by the
equation
w----acwo+atwl +oaw2+ . . • (2)
where the a_'s are functions of time alone, and the w,'s
represent the deflections of the various modes along the
elastic axis of the wing, each being given in terms of a unit
tip deflection. In equation (2), w0 represents the rigid-body
mode and has a constant unit displacement over the span;
the other w's are elastic-body modes which satisfy the differ-
ential equation
and the orthogonality condition
n mw,,,w, dy=O (re#n) (4)
t'/2
=M, (m----n) (5)
In accordance with the Galerkin procedure for solving
differential equations, equation (2) is first snbstituted into
equation (l) to give, after use is made of equation (3),
al_, 2mwt + a2ca22mw2+ ..... m (_owo+ fi,wl + • • .) + F
(6)
Now if this equation is multiplied through by w,, then is
integrated over the wing span, and u_ is made of equations
(4) and (5), the following basic equation results:
/-b/2
M.a. +o_.2M.a.= J__,, Fw.@ (7)
which allows for the solution of the coefficients a, if the
applied forces F are known. This equation applies for the
translational mode n=O, for which ease oJ0=0, as well as for
all the elastic-body modes. The quantity M, appearing in
the equation is commonly ealled the generalized mass of
the nth mode.
For the present ease of the airplane flying through a gust,
the force F is composed of two parts: a part designated by
L, due to the vertical motion of the airplane (including both
rigid-body and bending displacements) and a part L, resulting
directly from the gust (this latter part is the gust force which
would develop on the wing considered rigid and restrained
against vertical motion). On the basis of a strip type of
analysis, these two parts are defined as follows:
F=L.+ L,--'----2 ocV fo' _b [l--, (t--r)]dr+ a ocl" f'
where t----O is taken at the beginning of gust penetration.
l--¢(t) is a function (commonly referred to as the Wagner
function) which denotes the growth of lift on a wing following
a sudden change in angle of attack and for two-dimensional
incompressible flow is given by the approximation
[1--_b (t)la - ®= 1--0.165e-°'°_t--o.335e -°6_'t (9)
and _(t) is a function (commonly referred to as the Ktissner
function) which denotes the growth of lift on a rigid wing
penetrating a sharp-edge gust anti for two-dimensional
incompressible flow is given by the approximation
[_ (t)]a.,, = 1-- 0.Se-°_v'-- 0.Se -2v' it0)
Figure 1 is a plot of equations (9) anti (10).
An additional term which involves the apparent air mass
should be included in equation (8) ; this mass term is inertial
in character and may be included with the structural mass
(see ref. 16) althol,gh it is usually small in comparison. The
lift-curve slope a may be chosen so as to include approximate
overall corrections for aspect ratio and compressibility effo_'ts.
The remainder of the analysis is restricted to uniform
spanwise gusts and the assumption is nmde that the response
will be given with sufficient accuracy by considering only
two degrees of freedom: vertical motion anti fundamt, ntal
wing bending. On this basis, if w as given by the first two
terms in equation (2) is substituted into equation (S) anti
the resulting equation for F is substituted into equation (7 }.
the following two resl)onsc equations result when rt is set