Dynamic loads on airplane structures during landing

62 Pages·1998·4.08 MB·English

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NACA ARR No. 4HlO RESTRICTED
NATI Ol!TALA DVISORY COMHITTEE FOR AERO1JAUTICS
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ADVANCE RESTRICTED REPORT
DYNAMIC LOADS OX AIRPLAlJE STRUCTURES DURING LANDING
By MI A. Biot and R, L. Bisplinghoff
1. SUMMARY
The application of transient theory to the rational
determination of dynamic loads on airplane structures during
initial landing impact is discussed. Simplified procedures
are described by which the distribution of the maximum at-
tainable value of the dynamic stresses in the structure may
be obtained. Illustrations of the procedure are given by
numerical examples for tbe case of airplane wings. This
indicates approximate orders of magnitude to be expected in
a typical problem. The v.alidity of the underlying simplifi-
cations and assumptions is discussed. A brief outline of
the general mathematical theory of transients in undamped
elastic system is presented.
.2 INTRODUCTION
During landing; the airplane structure undergoes
transient oscillations which are excited by the inftial land-
ing impact. Secent exyorience has shown that in the case of
large airc raft these oscillations may produce critical design
conditions for the structure; whereas heretofore design loads
for the landing condition have been based upon calculations
which assume the structure to be rigid. Since tho advent of
larger aircra,ft has resulted in heavier and more fiexible
structures, these calculations are considerably in error.
This together with the fact that flight 1oe.C factors are re-
duced in the case of large aircraft makes it necessary that
methods be developed for predicting dynamic landing loads.
The present theoretical. investigation of this problem is of a
preliminary nature. It has the twofold purpose first of pre-
senting methods by which the designer may predict the dynamic
loads and second of serving as a guide in the experimental
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NACA A?3 Bo. 4HlO 2
investigation by defining the significant factors involved
and determining the an _ n .t. roximate magnitude of the quantities
to be measured.
It appears that the problem may be approached in two
different ways:
In the case of the landplane to consider the airplane
structure and its landing gear as a whole, and to introduce
the actual force displacement characteristics of the landing
into the theory. In this procedure the dynamic stresses re-
sult from the sudden application of moving constraints im-
posed on the airplane during landing. Similarly for the
seaplane the elastic structure and the water surrounding the
hull may be considered as interacting bodies. While this
method is not precluded. in tha investigation of specific
cases or for research purposes, it involves inherent complex-
ities, such as those resulting from the nonlinear properties
of the landing gear and the variable-mass effects of the
water surrounding the hull of a seaplane, which tends to make
this ty?e of approach less adequate for design purpose,
In the other procedure, in which stresses en tho struc-
ture are considered to be caused by a landing impact force
applied directly to the structure, it is assumed that the
time history of the impact force may be investigated inde-
pendently of the elastic properties of the structure. In
this way the investigation involves two separate phasos - a
study of the landing forces and a study of the dynamic be-
havi or of the structure under such forces. This procecure
involves the assumption that a landing impact force may be
defined in such a way that its time history is for all prac-
t.ical purposes independent of the elastic properties of the
structure. ecz3iS ti si believed that the relativs simplicity
of the latter approach overshadows tho approximations in-
volved, It has been adopted as the basis for the present work.
This procedure was described previously in reference 1 in
connocti on with tho problem of dynamic stresses in buildings
during an earthquake and the present work is essentially an
ada+ptation to aircra,ft structures of the methods described
in this reference.
It is assumed that in first approximation the damping
and the aerodynamic forces may be neglected. The landing
im-pact force is applied directly to the elastic structure as
an external force of given time history. The response of
the structure Is represented as a supsrposltion of natural
modes excZted by the landing impact. A first step in the
NACA RXA .oN 01EG 7.
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ehT method. has its limitations and is not necessarily
applicable to all types af transient problems. Some of
these limitations are pointed out in t>e discussion,but the
extont to which the methods aro valid for some particular
class of problems will have to be determined experimentally.
mk
mass
k spring constant
Q(t) generalized ecrof
w circular frequency
NACA AR3 Ho. 4ElO 4
tI period
T,t time variable
Y dynamic response factor
9 generalized. coordinate
z deformation displacement of any point on the wing
,d
normal function describing tho wing mode shape
cc normal function doscribing tho modo shapa of twisting
about the olsstic axis
h normal function describing tho mode shape of bending
of the elastic axis
I momsnt of inortia
S static mass moment
u4 subscript denoting jth mode
k subscript denoting kth spanwise wing station
P mass per unit volane of prismatic beam
A cross- sectional area of prismatec beam
E modulus of elasticity
n vertical landing load fact or
Iqtj landing reaction
rj(tj shock strut axial force - time relation
D(t) drag force - time relation cau.sed by wheel spin-up
a(t) observed acceleration time history in a drop or
landing test
W gross weight of airplane
WI gross weight of air;3lane less landing gear weight
.
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k = 'WN
_*
_ _ .-.. q
I-------
-L __.*_.-.P&*,-,A,/../ ...-- .-..I hi +____ .._.... ml
,I ’ I, ~.,-~----yy-- -k------~ , , ii , , -
/ I ,
KACA ARR No. 4BlO 6
Denoting by M the mass and w the natural frequency of
0sci11sti0n (W is the circular frequency expressed in
rsdians/scc), the spring constant is k = M @, The expres-
sion giving the displacement q of this mass under a force
FW of arbitrary time history is well known (refcrenccs 2,
3, and 4).. It may be written as:
r t
F(T) sin w(t - T)~T (1)
%O
where T is a variable of integration. It is usually desig-
nated as Duhamel’s integral. According to this formula the
computation of the displacement q at the instant t re-
quires the evaluation of a definite integral between the
limits of integration 0 and t, and the tfme history of q *
is obtained by repeating this process for every value of t.
It is noted that even in the simple case of a single-mass
system the process of computing the transient response is
quite elaborate. Fortunately, this difficulty may be avoided
in adant$.nr the theory to practical problems of design by
considering only the maximum value of q instead of its com-
plete time history. Bow this is achieved will be explained
later (sec. 7).
Assume for the present tha t the complete time history
of the deforma.tion in the airplane structure is desired.
Such a structure differs from the simple system of figure 1
by two features:
(a) It is a free body.
(b) It is an elastic body -crrith many degrees of freedom.
In order to show how the ‘t\revious discussion may be ex-
tended to include these fcaturks, consider a free system of
two mass06 K and m connoctcd by a spring of constant X
as illustrated by figure 2.
.&_. ___ ..., x
1
-__.- -
- -.--.--x 2 k s --. I“ 1
Figure 2
ACAN 9RA .oN OlH4 7
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Nl
NACA ARR Ko. 4HlO 8
elastically rentrained and under the action of a’ force Q&j
proportional to F(t) (fig. 4b).
.__-+. _ -s&h-- q1
90
1 ,& - ~-!_.-_ .-F(t)
y----T-r
, .’
’)ai ” ’
er.l_ggiF 4
The natural frequency of this equivalent system is the same
w1 as the natural frequency of the mode which it represents
in the actual physical system. Such coordinates as aq and
q.1 are called ge_n_eIr_a..l-i zed .-._ co_o.-r,d-.i-n ates. The mass Ml of the
eo.uivalent system is referred to as the generalized mass of
the corresponding mode and Q,(t) is referred to as the
generalized force for this same mode. It is also shown (see
-1 -._
a-onendfx II) that the value of this generalized mass is de-
rived quite simply by expressing that the kinetic energy T
in the equivalent system is the same as in the corresponding
.
mode
T =- lM l 2 1 *2+lM@2(;12
1 q1 =yq, (3)
2 5
where @ = i is the ratio of the amplitudes of the ma,sses m
and M in the ql mode.
Hence JJ!l = rn+N@a (4)
Similarly the generalized force Q,(t) is derived by
expressin.g that the work done by E’(t) on the actual system
is eoual to the work done by El(t) on the equivalent sys-
tern i that is,
@ q1 F(t) = Qdt) 91
or (5)
Q&j = cb a(t)
The problem of finding $ile transient motion of two masses
connected elastically has thus been reduced to that of two
independent single masses for which the motion may be ex-
pressed by Duhamel Is integral (equation (1)).
NACA ARR Ro. 4910 9
Consider now a simplified airplane in which the wing is
represented by two masses elastically connected to a rigid
fuselage as illuetrated in figure 5a.
___-_--~iiiip! -__-- j’, _...-____-J._~~- qf;pI
9(
Zigure 5
Obviously, for symmetric deformatfons this simplified
structure is equivalent to the two-mass free system of fig-
ure 2. The motion under the landing impact force may be
described as the superposition of a rigid translation (fig.
5b) which represents the motion of the center of gravity and
a natural mode (fig. 5c) which represents the motion about
the center of gravity. The equivalent single mass systems
are the same as in figure 4.
This procedure may be immediately generalized to a com-
plex airplane structure. Instead of a single deformation
mode as in the case of the simplified airplane discussed
above, there are actually an infinite number. It can be
shown that the deformation of the structure may be repre-
sented by a superposition of these modes. If damping is
neglected as is the case here, these modes are uncoupled.
The behavior of each one under the landing impact force is
independent of the other. The motion of each mode is repre-
sented by that of an equivalent single mass oscillator of
the same natural frequency and excited by a generalized
force. As an example, take the case of a wing in Ture bend-
ing. There are an infinite number of bending modes, a few
of which are represented in figure 6. The motion of the
center of gravity represented by the rigid airplane with the
generalized coordinate q. is referred to in the present
text as the “zero mode.n