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Aircraft impact on a spherical shell
Nuclear Engineering and Design 37 (1976) 205-223 © North-Holland Publishing Company
AIRCRAFT IMPACT ON A SPHERICAL SHELL* J. H A M M E L
Institut j~ir Mechanik, Technische Itochschule Darmstadt, D-6100 Darmstadt, Germany Received 2 December 1975 For nuclear power plants located in the immediate vicinity of cities and airports safeguarding against an accidental aircraft strike is important. Because of the complexity of such an aircraft crash the building is ordinarily designed for loading by an idealized dynamical load F(t), which follows from measurements (aircraft striking a rigid wall). The extent to which the elastic displacements of a structure influence the impact load F(t) is investigated in this paper. The aircraft is idealized by a linear mass-spring-dashpot combination which can easily be treated in computations and which can suffer elastic as well as plastic deformations. This 'aircraft' normally strikes a spherical shell at the apex. The time-dependent reactions of the shell as a function of the unknown impact load F(t) are expanded in terms of the normal modes, which are Legendre functions. The continuity condition at the impact point leads to an integral equation for F(t) which may be solved by Laplace transformation. F(t) is computed for hemispheres with several ratios of thickness to radius, several edge conditions and several 'aircraft' parameters. In all cases F(t) differs very little from that function obtained for the case of the aircraft striking a rigid wall. The calculation of the normal displacements w(t) at various points of the shell shows that the influence of the impact is bounded on a small region around the impact point. Therefore boundary conditions are not of interest. When calculating the normal displacement w(t) of the shell in consequence of the impact load F(t), we see a vibration of large amplitudes and of low frequency with a superposed oscillation of small amplitudes and a higher frequency. The fundamental frequency corresponds very well to that of the idealized aircraft, while the higher frequency belongs to a natural frequency of the hemispherical shell. It is of interest that not the fundamental natural frequency of the shell becomes visible but a higher one. If the 'aircraft' strikes a thin plate of similar parameters, the impact load is influenced more by the elastic deformation of the plate. Contrary to the hemisphere in this case only tl3e two lowest eigenmodes are of interest. This different behaviour of shells and plates is investigated in the present paper. When calculating the amounts of energy of plates and these of shells with similar parameters we obtain very different values which show the different response of plates and of shells to impact load.
1. Introduction The safety o f m o d e r n reactor buildings requires that' dynamical loads, such as earthquakes, explosions, aircraft crashes, etc. m u s t be taken into consideration. In the analysis o f the p r o b l e m of an aircraft crash on a building, the building structure is usually considered as loaded by a dynamical load F ( t ) which follows f r o m m e a s u r e m e n t s (aircraft strikes a rigid wall). H o w far the elastic displacements o f a structure influence the impact load F(t) is investigated in this paper.
2. The aircraftmodel For c o m p u t a t i o n the aircraft m u s t be very simple. The aircraft is considered as a mass p o i n t and the c o n t a c t between the mass p o i n t and the shell is established by a linear s p r i n g - d a s h p o t - c o m b i n a t i o n . We can therefore describe elastic as well as plastic d e f o r m a t i o n s of the aircraft and we obtain a linear relation b e t w e e n the 'aircraft de* Paper S1/3 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguards and Containment Structures (ELCALAP), Berlin, Germany, 8-11 September 1975.
205
J. Hammel / Aircraft impact on a spherical shell
206
formation' u(t) and F(t) the unknown contact force between aircraft and shell. For our computation we take the simplest case: a Maxwell model, a linear elastic spring (constant c), and a linear dashpot (constant rT) (fig. 1). We then get
u(t)
t 1 . F ( t ) + l fo F(z)dr = c
(1)
The displacement of the mass point is Wm(t ) and by Newton's law we get
m(32Wm/Ot2 ) = - F ( t ) .
(2)
The weight of the aircraft is omitted in eq. (2). At the beginning of impact the aircraft model just touches tile shell and has the velocity %:
Wm(t = O) = O,
(OWm/3t) (t = 0) = o0 .
(3)
Integrating eq. (2) and utilizing the initial conditions, eqs. (3), gives the displacement Wm(t ) as a function of F(t): l _ _
Wm(t)=v 0 •
t
t - m f F(r)(t
~-)dr.
(4)
0 We assume that the aircraft is attached to the shell for t > 0 (a negative force F(t) is possible). So the deformation u(t) can be related to the linear elastic deflection Ws(t) of the shell at the impact point and to the displacement Wm(t):
u(t) = Wm(t ) - Ws(t ) .
(5)
If it is possible to describe Ws(t) as a function o f F ( t ) likewise, we get from eq. (5) together with eqs. (1) and (4), a linear integral equation for F(t): t t 1. F(t) + 1 fo F('r)d'r = Vo " t _ 1 f FQ') (t - r) d~- - ws(F(t)) . c
(6)
o
3. Shell under dynamical load The equations of motion for a normally loaded undamped shell of thickness h, reference length a, and density p may be written in the form [3,5]:
~l(U,V,w)=_(1/co2).[;,
~2(u,v,w)=_(1/co2).~)" '
223(u,v,w)=_(1/co2).~-(a2/D)q(t).
(7)
Rotatory inertia of the shell is neglected; u, v, w are the displacements of the middle surface in the directions of the shell coordinates (fig. 2); Z?1, Z?2, ~3 [2] are differential operators and the dot means differentiation with respect to time. D = [Eh/(1 - v2)] is the extensional rigidity, and coO = {El [pa2(1 - v2)]) 1/2 is a reference frequency. The solutions of the homogeneous problem (q(t) = 0) are the eigenfunctions Un(~O, 0), Un(~0,0), Wn(~O,O) [2] with eigenvalues con , the natural frequencies. The orthogonality condition of the eigenfunctions can be written in the form
* ,m + un, ,um + WnW~z) sin ~0 d~o dO = ¢/o, •f f (UnU o ~ Bn'
for n :/: m for n = m
(8)
207
J. Hammel ] Aircraft impact on a spherical shell
IV
Fig. 2. Sphericalshell-coordinates.
Fig. 1. Idealizedmodel of aircraft.
Expanding the displacements of the inhomogeneous problem in terms of the normal modes yields displacement equations in the form u(~o, O, t)= ' ¢bn(t)Un(~O,O ), n=l
v(~o, O, t) =
n=l
¢bn(t)On(~O,O ) ,
w(~o,O,t) = ~¢bn(t)Wn(~O,O), n=l
(9)
where ~bn(t ) denotes the generalized coordinates and represents the time-dependent aspect of the solution. Substituting the assumed solution eq. (9) into eq. (7) yields a set of equations for ~n(t):
¢Pn(t)"
~l (Un,On, * * Wn) + ~-~0 " ~ n U
n =0
I
n=l Cbn(t ) " ~3(u~, on, , w*) +1_1_. 6o02 ~ n W ~
I
~ n ( t ) " ~2(Un, On, Wn)*-~O2"'~nOn
=0,
= --~q(t)
(10)
Since Un, On, w n are the eigenfunctions of the homogeneous problem, ., l(Un
[°~n~2 .
)
*
On ,
I¢°nX2
"
Un ,
)
On ,
.
O.
,
16°n\2
Wn
(11)
the differential-operators 2?1, ~2, Z?3 in eqs. (10) can be replaced by eqs. (11). Together with the orthogonality condition, eq. (8), we obtain an uncoupled equation for ~n (t):
~ n + 602 ~ n =
ayfq(t) ~ o
w n sin ~0d~odO hpaBn
= _ fin(t) a • ffl n "
(12)
The initial displacements and velocities of the shell are also evaluated in terms of the normal modes:
o0 w0
=
~,.(o)
v;, "w~,"
,
bo ~,o
=
'i'n~O)
v~,
.
w~,
With the orthogonality condition we obtain the following pair of initial conditions [4] :
(13a),(13b)
J. Hammel / Aircraft impact on a spherical shell
208
qbn(O)=kf f(uOu*+vOVj+WOWn)Sin+d+dO,
(14a)
SoO
~n (0) : Bnnlf f(~oUn + ~OVn+ flOWn) sin ~pdCdO
(14b)
So 0
If we assume, that the shell is at rest at the beginning, we can see from eqs. (14) that all qSn(0) and all ~n(0) are zero. So we get the solution ~bn(t ) by integrating eq. (12): t
d)n(t)_
1 _= a • con • m~
f Fn(r)sincon(t- r)dr. 0
(15)
Now ~n(t) and with it all shell variables (the displacement Ws(t) at the impact point tO, too) are known as functions of the load q(t).
w(t0, 0 = ~ n=l
w2,(t0) a • conmn
f t Fn (r) sin con (t
7) d r .
06)
0
Assuming that the impact area is small in relation to the shell surface the load q(t) is approximated by a concentrated force (17)
q(t) = F(t)8 (t - ~0) ,
where 6(t - tO) is the Dirac function in shell coordinates. In the special,case of a spherical shell with coordinates ~oand 0 we obtain 1 - a2sin~p a(~o- % ) 6 ( 0 - 0 0 ) .
6(t-t0)~8(t.o -~00;0-00)
(18)
Then the generalized force Fn(t) is given by
F n (t) = a f f F ( t )
6 (t - t O) Wn (t) sin ~pd~0d O - F(t) Wna (t 0 )
(19 )
Thus, the shell displacements can be expressed as functions of the impact force F(t):
"w(t,
O"
"Wn(t)"
and especially the normal displacement at the impact point is given by o~
W(to, t) = ~ n=l
t
[Wn(tO)]2 fF(r)sincon(t
- r)dr.
(21)
a2conffln ~0
Replacing the displacement at tile impact point in eq. (6) by eq. (21) and observing the different directions of w(t0, t) and Ws(t), we obtain
J. Hammel / Aircraft impact on a spherical shell
1 t
_1j F ( r ) d r + ~F(t) = V o . t - - ~ f F(r)(t - z ) d r - £ [Wn(~O)]2 t; F(r) sfn con (t - T) dr. r/0 0 n=l a2confftn ~0
209
(22)
This linear integral equation, eq. (22), can be solved for example by Laplace transformation:
1_~ f(s) +f(s) _ VO r/• s c s2
f(s) m s2
~ [Wn(~0)] 2 n=l a 2 • t~ n
f(s) s 2 + 092 '
(23)
where f(s) is the Laplace transform of F(t):
mo o
(24)
f(s) =
m 2 +m 2.J 1 +ms +--s rl c n=l
[w~(~0)] 2
a2fft n
s2 " - s 2 + c62
If we end the infinite sum of the denominator after n = N terms, the function f(s) can be written as a quotient of two polynomials
f(s) = m v 0 [Z(s)/R(s)] ,
(25)
which can be decomposed to partial fractions
f(s)=mvo
(26)
= s - sn
The constants C n are determined by
Z(sn)
Z(sn)
(27)
Cn - I-I (sn - s]l-R'(sn 1' ]¢n where the sn are the roots of the denominator polynomialR(s), and a prime means differentiation with respect to s. The inverse transformation gives a sum of exponential functions: N
(2s)
F(t) = m v 0 n~t Cn exp(s nt) . If the impact force is not a concentrated but a distributed load, the generalized force must be replaced by
Fn(t) = [(fq(~o) Wn(~O) sin ~0d~o)/(afq(~o)sin ~od~o)]F(t). ~o
(29)
~o
Then the solution F(t) is obtained by replacing one Wn(~0) in eq. (21) by the expression
jq(~o) w~ (~o)sin ~o ~o
sin ~ ~o
Eliminating F(t) in eq. (15) by use of eq. (28) yields the solution of the generalized coordinates
(30)
210
J.
d#n (t) -
m OoWn(~O) pha2B n
Hammel / Aircraft impact on a spherical shell
]=1 co2 + $2
exp(sit )-coscont-.
sJ sinwnt COn
,
(31)
and the displacements of the shell can be written as double sums:
(u(~o, O,t)]
N ]v(~°' O't) l=mvo,~l:
Vn(~'O) t" Wn(~O)pha2Bn ,=l ~
"w(~, o, t)"
wT,(~, o) j
CO2CJ+,2
s~ . (c°Scont +--slncont" c o , ,exp(sjt))
(32)
The constants C/. and sj depend on all considered eigenfunctions, as can be seen from eq. (27), and so Cbn(t) is not only a function of the nth mode of vibration. It is not necessary that the n u m b e r s N and N of the double sum in eq. (32) are equal, but for computation it is suitable to have N = N. How many terms N must be considered is a question of shape and of duration of the impact. If the impulse has a very short duration and an abrupt shape (unit step function), many terms must be considered. Because in reality such impulses are not observed, we can expect that the function F(t) can be described relatively accurately by a small number N. In refs. [1] and [7] estimates of N for several impulse shapes are given. In our case we also have an expansion o f F ( t ) over the shell area. Therefore a rule-of-thumb cannot be given for our expansion ofF(t). We consider the amplitude of deviation when taking into account the next eigenfunction as a measurement of convergence. Considering the impact on plates, a few terms (N = 2) are sufficient to describe the influence of the elastic layer. Here the lowest eigenfrequencies have the leading part. When we consider the impact on shells we find that the eigenfunctions of shells contain extensional energy as well as bending energy terms. Therefore, in this case the higher frequencies are of greater interest than the lower ones. In our example it was necessary to take into account N = 10 until 20 terms to obtain a fairly good approximation. 4. The pushing deformable body The aircraft - idealized by a mass with an elastic spring and a viscous dashpot - pushes the shell with velocity vO. If the shell is rigid (Ws(t) = 0), the impact force F(t) can be calculated exactly:
F(t) = F(t) =
[c/m -
VO
exp
(-ct/2r?)
sin
[c/m - (c/2rl)2]l/2t,
exp
(-et/2~2)
sin [(c/2r/) 2 -
for
c/m
> (c/2r/) 2 ,
(33a)
for
c/m
< (C/2~'/) 2 .
(33b)
(c/2r/)211/2
v0 • c c/m]1/2
c/m]
1/2 t ,
[(c/2~7) 2 -
We assume that the aircraft is connected with the shell after its impact. Then the impulse I 0 on the shell is given by
I 0 = j F(t) dt = m v o .
(134)
0 The impulse I 0 does not depend on the functions e and r/; but the shape, the duration of impact, and the size of the impact load depend on c and 7). If the pushing mass is m = 10 000 kg, ~ = 20 kp sec/m and the velocity v0 = 1 m/sec (fig. 3), we have a duration of impact of about 0.17 sec and the peak of the impact load Fma x ~ 11.6 Mp. A negative impact load does not yet mean a lifting of the mass, because in the case of a vertical impact we had to superpose its weight. If the damping constant rt is increasing, the duration of collision is decreasing. The argument of the e function in 'eqs. (33) becomes smaller with increasing 7, i.e. the influence of r/is decreasing. In the same way the peak of
J. Hammel /Aircraft impact on a spherical shell
211
F I MpI 80
15
c =500 M p l m ~Z >. . . . . . . .
b4p.seklm
,..
1o
•
0
~
; t Isekl
Fig. 3. Transient force F(t) of an impact against a rigid wall for different m
m:pl
? :~jsoo
"°I :A': oo I/I.
m!l
\\ ,oo
I~
:o
m o
,'~
t Isekl
Fig. 4. F(t) of an impact against a rigid wall for different c.
the impact load is increasing because the plot of the force becomes more and more similar to that of an undamped vibration (r/= ~). N o w we take ~7 fixed and vary the spring constant c (fig. 4). the peak load is increasing with increasing c. The duration of collision tl, the positive range of impact force, given by the term sin [c/m - (c[2rl)2]l/2t in eq. (33a), is t 1 = n/[c/m - (c/2rl) 2] 1/2. If the parameters fulfill the condition c/m < 2n2/m 2, the duration t 1 is decreasing with increasing c. A large spring constant causes a short and hard impact. For our computation we estimate c, 7/and m so that the duration of impact and the peak load agree as closely as possible with experience.
212
.L Itammel / Aircra/t impact on a spherical shell
5. Load of an impact against an elastic structure In the following the structural reactions, when a mass strikes an elastic plate or an elastic spherical shell, are discussed. The method to calculate the impact force is explained in section 3. 5.1. Impact on a circular plate
We take the differential equation of the vibrating plate A A w = - (ph/K) w ,
(35)
where K is the bending rigidity, h the thickness, p the density, w the displacement of the plate and A the Laplace operator. The analysis of a circular plate yields solutions of eq. (35) in terms of Bessel functions. The boundary conditions of the plate lead to the eigenvalues and to tile eigenmodes of the vibrating plate. The impact force is expanded in series of these eigenfunctions. If the mass strikes the circular plate at its center, only the axial-symmetrical eigenmodes have to be considered. For a plate clamped at its edge (fig. 5) the lowest frequencies of the symmetrical vibration (m = 0) are calculated as: w 1 =41.3 1/sec; o)2= 160.8; o)3=320.3; co4 = 639.6; co5 = 998.7; and co6 = 1437.7. The frequencies of the circular plate are very clearly separated and we approximately have con ~ n2co I . Now we compute the impact force considering these eigenvalues and compare the result with the impact on a rigid wall (fig. 5). The velocity of the crashing aircraft is v0 = 1 m/sec; v0 is a multiplier when calculating the impact force (see eq. (24)). The first curve of fig. 5 shows the force F(t) of an impact against a rigid wall (n = 0), the lower curve is the force F(t) of an impact on the given elastic plate under consideration of n = 2, 4, 6 eigenfunctions. We see that the amplitude of deviation is so small in the last three cases that the calculation leads to the same curve F(t). In the case of an impact on an elastic plate it is sufficient to consider the two lowest eigenfunctions. The maximum force o f F ( t ) of the 'elastic' impact is about 7% lower than that of the 'rigid' impact. The later shape of the curve is entirely different when taking into account the elastic displacements of the plate. For later comparison with impact on a shell of h/a = 1/100 we compute F(t) of impact on a plate of h/a = 1/100 (fig. 6). We only consider the two lowest eigenfrequencies: co I = 20.7 1/sec, and co 2 = 80.4 1/sec, and plot the impact force F(t) against time t. Again, the upper curve indicates F(t) of impact on a rigid wall (n = 0). The crashing 'aircraft' in figs. 5 and 6 is the same, so the upper curves of both figures are the same. The maximum force F(t) (n = 2) has decreased again. The drop is 20% now, and the shape o f F ( t ) is entirely different from that of n = 0. 5. 2. Impact on an elastic hemispherical shell
First we determine the eigenfrequencies and eigenfunctions of a freely vibrating hemispherical shell. If the shell is pushed at its pole, we must only consider the axially-symmetrical eigenfunctions (m = 0). For a clamped hemisphere of v = 0.2 and h/a = 1/100 the eigenffequencies ~ have been calculated. The eigenfunctions u*, w* are plotted against meridian (fig. 7). The displacement o* is zero (m = 0). The displacement w* is symmetrical, u* is asymmetrical to the pole (tp = 0). All eigenfunctions are standardized to the normal displacement w* at the pole. The amplitude of the u displacement of the first eigenvalue is rather large but it decreases for higher frequencies except for the eigenfunctions at frequency ~ = 2.6381. At that frequency u* is again rather large and w* does not oscillate around the position of rest. This is due to the influence of extensional vibration. If we calculate the frequencies of a hemispherical membrane shell with a roller hinged edge, we get ~1 = 1.549 and ~ 2 = 2.656. So these curious eigenfunctions of fig. 7 in the region of ~ = 1.5 and co = 2.6 are nothing else but membrane vibrations with superposed bending vibrations. Since the impact area is small in relation to the shell surface, we first consider the impact load as a concentrated
J. Hammel / Aircraft impact on a sphericalshell F[Mpl
213
.Frtfp]
o .'n =O
• n = 2,4,6 n=2
c
~
ioookg
5ookp/cm
~1 t
4 Mp.seklm
h l a .'11,I00 ~,0.£
E
0.5
h/a = l /50 ~'=0.2 E =300000 kp/cm "~ ~ = 2500 kg/m ~
,
~';-'.P kp/cm 'f O0 /¢~/m .~
l 3 ~ 0
=
0.5
o
Fig. 5. F(t) of impact on an elastic plate (h/a = 1/50).
: SOOkp/cm
N/o.,
Fig. 6. F(t) of impact on an elastic plate (h/a = 1/100).
force; furthermore the weight of the crashing 'aircraft' is omitted. The aircraft impinges with velocity u0 = 1 m/sec upon the apex of the shell of fig. 8. The shell is considered to be of homogeneous isotropic material with specified material constants. The maximum force of n = 0 (impact upon rigid wall) is 15.47 Mp. If we consider n = 3 eigenfunctions the maximum force is 15.37 Mp. With n = 9 the force only decreases to 15.06 Mp, that is by 2.6%. The shape of the impact force F(t) is nearly unchanged. Now we vary the parameters of the 'aircraft'. The spring constant c remains fixed and the dashpot constant is halved (fig. 9). The shape of F(t) changes as discussed in section 4: the maximum force becomes smaller and the force is absorbed after a short time. But if we compare the 'elastic' and the 'rigid' impact we have the same tendency as seen in fig. 8. The decrease of the maximum force (n = 9) is only 1.8%. If we vary the spring constant c (figs. 10(a) and (b)), there is nearly no difference between an impact on an elastic and a rigid shell. If we have a smooth spring (fig. 10(a)) the difference is not visible. In the case of a very hard spring (fig. 10(b)) the decrease of the maximum force (n = 9) is only 0.5%. But it is of interest that now not the whole force F(t) gets absorbed, there remains an oscillation ofF(t). The hard spring excites the shell to very strong vibrations and, because we assume that the aircraft remains connected with the shell surface after impinging upon it, these strong shell vibrations entail an obvious oscillating force F(t). If appears (fig. 10(a)) that there is one distinguished frequency in which the excited shell vibrates. The period of the remaining oscillation is about T = 0.016 sec, to which a frequency co = 392 1/sec belongs. The sixth mode of vibration of the considered hemisphere has the frequency co = 390 1/sec, indicating that the excited shell mainly vibrates in its sixth mode. Now we take the 'aircraft of fig. 8 and vary the shell parameters. We consider shells of radius a = 10 m and 20 m but of the same h/a = 1/100. The impact force (figs. 8, 11 and 12) equates more and more to that of an impact on a rigid wall, i.e. the influence of the elastic shell becbmes smaller, although the shell frequencies decrease with increasing radius a. The frequencies alone do not describe the behaviour of an excited continuum: frequencies of the vibrating masses of the eigenfunctions, which increase with increasing
214
J. H a m m e l / A i r c r a f t i m p a c t o n a spherical shell
~
.
~
y
~ = 0.7624
• • 0
/w
u"
/-~
"---/
~ -09257
/~ ~
,-/", ~
~
~= 7.4754
--
--
J = [5503
A /
d, W
('~/'-----".~
~ :0.9666 0
q/ / ~
/-x
/~
:~
~
1.7041 y
uV~wA :'(~" :~vf'-.vfm = 1.00003 O• V k,_/ ~
-
: y. ~
~ =19226
),.~_
~ =2.1738
w
o ~
/'x
/
~
v
v
v
..j
L[,W
~
/.,, A / . . . . ~J,,,,v
j.,~j" •
w
/-", _/C'-~..._..: ~
G=!.2108
~ : 2.6381
u,~
'I.o ~/ /"(w
~
"~ ~,'U"/ "
~
"~"
/~
~'=½
~ =2.7791
Fig. 7. Eigenmodes of a clamped hemisphere (v = 0•2, h/a = 1/100).
shell-radius, are also of interest• Poisson's ratio v (figs. 8, 13 and 14) does not influence the impact force in any way. Edge conditions (fig. 15) also are not of interest• The impact is bounded to a small region at the apex. Only in the case of very thin shells (h/a = 1/200, fig. 16) can the influence of shell elasticity be of interest• The maximum force decreases from 15.47 Mp (n --- 0) to 14.75 Mp (n = 12). It is interesting that the deviation of the maximum force is greater from n = 6 to n = 9 than from n = 0 to n = 3, i.e. the higher eigenfunctions have a greater influence upon the maximum force than the lower ones. In the foregoing the impact force was considered as concentrated at the apex. In the following we consider the load as evenly distributed over a small area. Thereby in the calculation o f F ( t ) the function w(~ 0, t) - the normal displacement at impact point - is replaced by eq. (30), i.e. the eigenfunctions are evaluated in relation to their w displacements in the impact area. Thereby the lower eigenvalues are emphasized, so that the impact force hardly differs from the force of impact upon a rigid wall (fig. 17), We can summarize that the force F ( t ) essentially is influenced by the parameters of the considered 'aircraft'. The shell parameters - besides the case of very thin shells - influence the shape o f F ( t ) on!y very little. The shell - different from a p l a t e - behaves like a rigid wall when we consider the crash of an aircraft on it, especially in the case of distributed loads•
6. Displacements of the hemisphere If the impact force is known, the displacements of the shell can be calculated from eq. (32). In fig. 18 we have plotted the normal displacement w at the apex (full line) and at ~ = 10 ° (dashed line) against time t under concentrated impact force at the apex. n = 14 eigenfunctions have been considered• We notice a fundamental oscillation of
215
J. Hammel / Aircraft impact on a spherical shell F/Mpl v.ff
I II
,-I
m=lOOOOkg
E~
~
Ill
,olj I /I
I
II
I
..,ooo ~,o~
..o;,.,oo .--0.2
FtM~
E=300000 ,p/cm"L
' I
v.=lm/,W
j...-.n.O rl
,~.
m=lOOOOkg c.5000kp/cm
~n=9
h/a= I I100 ~'=0.2 E • 300000 kp/crnA tfsekl
~ = 2500 kglm '~
o~ o - T
Fig. 8. F(t) of impact on a hemisphere (.1 = 40000 kp sec/m).
o~
,, ,tt,,*l
Fig. 9. F(t) of impact on a hemisphere (.1 = 20000 kp sec/m).
F/Mpl v~z,,,I ~
h•=2O
mf,OOOOkg
OOOl~.sddm
F IMp)
'°l
h / o f 1/100
~
~=0.2 E . 300000 kp/cm=
t
:_~
~.
/--
h/o= III00
I!
,,.o.2
2500 kg/~'
t/51k/
O
~
tlse/d
Fig. 10. F(t) of impact on a hemisphere (c = 1000 kp/cm, c = 15000 kp/cm).
216
J. H a m m e l / A i r c r a f t i m p a c t o n a spherical shell FtMp]
,./.] /
~
F/Mp) m . ,o ooo kg
.. ~
n=O n=3
r~-~,
I
~Z=40WOkp.seklm
10
h/a=I II00
h/cl= 11700
-p=0.2
"~'= 0.2
E = 300 (220 kp/¢rn L
E = 300 000 kp/cm "L
= 2500 kg/rn J
9 -- 2500 k.q/rn~
'l 0
Fig. 1 i. F ( t ) of impact on a hemisphere (a = 10 m).
F [Nip}
. . . . i n =O ~n=3 ~n=6 "rt =9
t [sek]
Fig. 12. F ( t ) of impact on a hemisphere (a = 20 m).
F[MpI
V,,=l'/s#x[ ~ ~>
rn=lOOOOkg
Xn=9
c 5000 kplcm
~ I . / , ~ L[ ~
=
m=toooo,u
~.
c = 5000 kp/cm
/~...x,
10
~ =40 000~.sok/m
I0
h/o= I / l O0
h/a= III00
~:0.0
")'=0.4
E= 300000 kplc rn s"
E = 300000 kplcm "L
= 2500 kg/m 3
9 = 2500 kg/rn 3 5
0
t [sek/
Fig. 13. F ( t ) of impact on a hemisphere (u = 0.0).
6
~ : t(sek]
Fig. 14. F ( t ) of impact on a hemisphere (v = 0.4).
217
J. Hammel / Aircraft impact on a spherical shell F I MI~
F[Mp]
FIMpl
/4.o ~n =9
v,=t'/..,][~ m=tOOOO kg J/ ~ . tG
c ,$O00 kp/cm
h ~ = 4
0 000kp.sek/m
| hlo = I II, O0
[
"2=0.2 E = 300 000 kp/cm ~ = 2500 kg/rn ~
= t [sek]
5
0
Fig. 15.
h/o= 1/100
hie = 11200
i
,P=0.2 E = 300 0t20 kp/crn~ ~ = 2500 kg/m J
^,~ |(l # I/ 0u ..~ 8 " ~ 2
~J
Fig. 16.
__ t[sekl
5:
"P=0.2 E= 300000 kp/cm J" 9 = 2500 kglm "1
0
0.4 ~ 1 6
"¢"-~.:
; t IsekJ
I Fig.17
Fig. 15. F(t) of impact on a hemisphere (hinged supported). Fig. 16. F(t) of impact on a hemisphere (h/a = 1/200). IFig. 17. F(t) of impact on a hemisphere (distributed load).
about 3.5 Hz with a distinct superposed oscillation of about 60 Hz and small amplitudes. The frequency of the fundamental oscillaiton agrees very well with the frequency of our model for the crashing aircraft: co = (c/m) 1/2 = 22.1 I/s;- 3.52 Hz.
The superposed oscillation has a frequency, which agrees very well with that of the fifth or sixth eigenmode of the shell, i.e. the crashing 'aircraft' excites the fifth or sixth eigenmode first of all. If we consider w(t) at ¢ = 30 ° (fig. 19) the fundamental oscillation has disappeared. At this point the shell is at rest for a short time (about 0.007 sec) and then a steady oscillation with small amplitudes starts. In fig. 20 the displacement w is plotted against the meridian at different instants. The first curve is w(¢) under a force from an impact on a rigid wall, the second is w(~) under the force F(t) from an impact on the elastic shell. Again we can state that the difference is hardly visible. The graphs of fig. 20 show how the displacement at the impact point increases and how waves are propagating over the shell. After 0.064 see (fig. 20(c)) the maximum displacement at the apex occurs, while the force F(O has its maximum after 0.061 sec. Because the frequencies of our shell-aircraft model system are decoupled (section 7), we have this good agreement o f F ( t ) and of w(t). Large displacements are confined to the immediate vicinity of the impact area, which is why edge conditions cannot be of any interest; it is probably sufficient to consider only a small region of the excited shell. To obtain information about the convergence of our computation we consider fig. 21. The load is uniformly distributed over ¢ = 10°, n is the number of the considered eigenfunctions. We see that the problem cannot be represented by two eigenfunctions, and the increment of the third and fourth eigenmode is greater than that of the first and second; beyond ten eigenmodes the increments have different signs. If we consider 16 eigenmodes we get the maximum displacement of 0.63 mm after 0.067 see. The statical solution (the shell statically loaded by the maximum load of F(t)) is 0.61 mm. There is only a small increase in the normal deflection if we make a dynamical computation. This again is a consequence of the uncoupled frequencies of our considered shell-aircraft system.
218
Jr. Hammel /Aircraft impact on a spherical shell wlmml 1.0
{~
,.
~1
~:
.'~
~ ' ~ "-
j/
. . . . . . . .
o.2
V "V" ~
; t/sekl
" o.~
:.., /
u/I
,,,
\~! , ,,, -~.0
. . .
,~, i',d /
\
'
~
/
,
~,.,J rq
.
,,,,,oooo,,
%
.z°,o=,~,.,,,,.
"--",.,J
h/a= I / I 0 0
~=0.2
E = 300000 kp/cm*" "9 "
Fig. 18. D i s p l a c e m e n t
w(t)
under impact force at apex;
w(t)
2 5 0 0 k.qlm "1
at apex, - ....
w {ram } O.O5
o
.A~
VV
A A A,.,AAAAA b,{',.
-
_,,,,,
°'""1q °"°~" v
-o.o5
~.
~
' ~6m.~ h/a= 1/100
-~-= 0.2 E = J00000 kplcm J" =
2500 k g / m J
Fig. 19. Displacement w ( t ) at ~o= 30 ° under impact at apex.
w(t)
a t ~o = 10 °
J. Hammel / Aircraft impact on a spherical shell
219
• O. OOS L ~
~I~
r~
~.~
= o.o3 =,I~
(d)
0.5
(b)
0.5 ,~[ , . . , }
t • 0.t5 8o~ I
(e)
v~ J£.,n] f%'--
, v
(f) v, ~j,J t',,.,,/
I
(9) Fig. 20. Displacement x(~o) over meridian at different time t. w Imm]
1.0
-
"#
~
~=~O~s~lm
1T~0
h/a= 11100 "P =0.2 E = 300000 kplcm ~ 9 =
2500 k g l m ~
Fig. 21. Displacement w ( t ) at apex under distributed impact load at apex considering n = 2, 4, 6, 8, 10, 12, 14, 16 eigenmodes.
220
J. Hammel / Aircraft impact on a spherical shell
7. Participation of normal modes in the forced vibration solution We have seen that plates and shells have a quite different response to impact forces. To clarify this, we consider a clamped hemispherical shell connected to a undamped mass-spring system at its apex (fig. 22). But now we assume that the shell has only one degree of freedom, i.e. the vibration of the shell can be described by one eigenfunction, for example the ith. Utilizing the method of Rayleigh-Ritz we calculate the frequencies X of the coupled system. We take the functions w = Wi • w~(~o,0 ) ,
u = Wi • ui*(¢, O),
v = O,
(36)
where Wi is a constant and u~, w~ are eigenmodes of the ith mode of vibration of the shell (u~, w~ satisfy the boundary conditions). Substituting u, v, w into the expressions of the potential and kinetic energy of the shell, we get U = ~ pa2032ih W2B i,
T = ~1 pa2hB.W2 , , X2 ,
(37)
where w i is the eigenfrequency of the ith mode of vibration of the shell and B i stands for eq. (8). Replacing M = 2Trpa2h (the whole mass of the shell) we obtain the energy of the coupled system BiM Uges = ~co2 ~ W~ + ~c(Y - w(O)) 2 ,
T ges =m---X2y2+~x2BiM 2 ~ W2,
(38)
where Y is the amplitude of the mass m, and w(O) is the,displacement of the shell at the apex. w(O) equals Wi, if the eigenfunction w~ is a unit at the apex. The variation of the energy leads to a set of algebraic equations for Wi and Y: (
+2~rm ) 2zrm y = 0 ~22 BiM - ' Y 2 W i _ Bi___ ~ ,
_Wi+(l_72)y=o,
(39)
where ~2i = 03i(m/c) 1/2, 7 = X(m/c) 1/2. From eq. (39) we get the frequency equation 74_ 72 ( ~ 2 + 1 + ~ ) + ~ 2 2 = 0 .
(40)
If the term 2 lrm/BiM < ~22 i , we obtain the two uncoupled solutions 72 ~- 122 and 722~ 1. If we take the aircraft and shell as defined by fig. 11 and consider the first mode of vibration we get the couple-term 21rm/B1M = 0.275, a very small value in relation to $22. We get the frequency equation 74 - 72(145.2 + 1 + 0.275) + 145.2 = 0
(41)
with the solutions 712 = 145.5 and 72 = 0.998. The framed couple-term influences the fre~luencies of the coupled system very little. If we double the shell radius keeping the other parameters constant, I2~ decreases in the second power but the couple-term in the third power. Therefore, the frequencies 03i of the shell and to = (c/m) 112 of the spring-mass system get closer together; however, the frequencies of the coupled system agree better with those of the uncoupled system (~oi, 03). If we change the parameter h/a but keep the radius a constant, the lower frequencies ~oi of the shell change very little. In contrast, the mass of the shell and so the couple-term decreases nearly linearly with h/a. For a shell as given by fig. 16 we get I212 = 142.5 and 27rm/B1M 0.518. The couple-term is still very small in relation to g22 but it is twice the value of a shell with h/a = 1/100. Considering in the same way a clamped circular plate connected with a string-mass system at the center (fig. 23) we obtain a similar frequency equation: %4 - [~27 + 1 + (Trm/BiMp)]72 + ~2 = 0,
(40a)
£ Hammel/Aircraft impacton a sphericalshell
"
I
C. /
ill
Fig. 22. Clampedhemispherewith a mass-spring system at apex.
221
Fig. 23. Clamped circularplate with a mass--~ring systemat its middle.
Fig. 24. Two mass system.
where Mp = plr a 2 is the mass of the plate. The frequency equation (40a) is identical to eq. (40) (34 = 2Mp). But the frequencies [2/2 and the couple terms, respectively, are of other orders of magnitude. For a plate ofa = 10 m and h/a = 1/100 connected with the same spring-mass system, we get [212 = 0.22 and ;rm/B1Mp = 0.70. The frequency of the shell is 660 times greater and simultaneously the couple term is 2.5 times smaller than the values of the plate, respectively. So we can understand that the vibrations of plates are strongly influenced by the joined spring-mass system, i.e. the response of an elastic plate has a great influence upon the impact force. If we replace the shell by a spring-mass system (the frequency Q1 and the static displacement are the same) we must take a spring constant ~ = Eh/45 (1 - v2) and a mass ~ = 0.0061 M (M = mass of shell). This system (fig. 24) leads to a similar frequency equation: 74 - 72([Z12+ 1 + m/ffz) + [22 = O.
(40b)
The term ~21 in eq. (40b) is identical to that of eq. (40) but now the couple-term looksquiet different. Instead of 27.1 m/M (40) now we have m/~ = 164 m/M, i.e. the system of fig. 24 simulates a strong coupling of the two systems which does not exist in reality. It is problematic to replace a vibrating continuum by a single spring-mass system: besides the frequencies I22 the coupling strongly depends on the couple-terms, respectively. The ratio/g = fi12/(2 nm/B 1M) is about 500 for the considered shell and about 0.3 for the circular plate. Large values of/a (shell) mean that the influence of the elastic structure upon the impact force is negligible. To solve the question why higher modes of the shell are excited, we use again the Raleigh-Ritz-method. Because we want to compare the eigenmodes of the shell, we now assume that the shell shall only have two degrees of freedom, i.e. the vibration of the shell can be described by two eigenfunctions, e.g. the ith and ]th mode:
w(~o,O, t) = (Wi w~(~o,O) + Wi . w; (~o,O)sin ?~t , u(~o,O,t)=(Wiu~.(~o,a)+Wl~*(~o,d)sinXt,
v@,O,t)=O,
(42)
where u~ and w~ are the displacement functions of the ith mode with eigenfrequency coi and orthogonality integral
B i. Wi, W/are free constants. Together with assumption eq. (42) we get the kinetic and potential energy of the coupled system:
Uf~pha2(6°2BtW2+6o~B/W~)+~c(Y- w(O))2 ,
T=½pha2(BiW2+B/W2)X2+½my2)~2.
(43)
We differentiate the total energy (U - T) with respect to Wi, W/and Y, the amplitude of the displacement of the mass m and get a homogeneous algebraic system for Wi, I¢/and Y:
J. Hammel / Aircraft impact on a spherical shell
222 (f22+21rm
Bi-'---M-')'2
) W +27rm
21rm
i ~iM W J - ~ i M Y = O '
2 rrm ( BjM wi +
+ 2 rrm ) 2 rrm BjM - 32 wj r-- 0,
(44)
- W i - Wj + ( 1 - 3'2 ) Y = 0 . If we consider the first and second eigenmode of the shell (fig. 11), we obtain 2 ~rm ~22 = 145.3 , B - - ~ = 0.275;
~2
27rm = 214.2, B~-2-2-2 ~ = 0.698,
(45)
72 = 0.995.
(46)
with solutions 72=145.6,
3,2=214.9,
We receive the normal displacement at the apex together with the initial conditions (all initial displacements and velocities are zero besides the velocity v0 of mass m):
w(t) = 10-4vo(m/c)l/~(1.5 sin 267 t + 2.2 sin 325 t - 51.5 sin 22.1 t ) .
(47)
A fundamental oscillation of frequency co3 = 22.1 sec- 1 is superposed by two oscillations of high frequencies and small amplitudes. Moreover, the amplitude belonging to the second mode of vibration is 1.5 times greater than that of the first mode. Colasidering the first and the third mode of vibration we have ~2
21rm = 145.3, B ~ = 0.275;
27rm f22 = 233.6, B - ~ -= 1.087 ,
(48)
and together with the same initial conditions we get the displacement at apex:
w(t) = 10-400(m/c)1/2i(1.5 sin 267t + 3.05 sin 339t - 61 sin 22.1t).
(49)
The amplitudes of the fundamental as well as that of the superposed oscillation of the third mode have been increased, i.e. the interaction between mass-spring and shell has become greater. As we have seen, the coupling of the two systems is given by the ratio//i = ~22i/(21rm/BiM). Therefore, we calculate//i for each eigenmode of the shell and we obtain //1 = 528.7,
//2 = 307.1,
//3 = 214.9,
//7 = 179.1,
//8 = 214.9,
//9 = 265.0.
//4 = 188.1,
//5 = 178.2,
//6 = 177.5, (50)
The ratios//i first decrease to the minimum of the sixth mode and then increase again. Considering the fifth and sixth eigenmodes of the shell and doing the same calculation as above we get the displacement at the apex:
w(t) = 10-4o 0 m(3.05 sin 367.5 t + 3.4 sin 391.6 t - 112 sin 2 2 t ) .
(51)
Indeed, combining the fifth and sixth mode, eq. (51), gives the largest amplitudes of the fundamental as well as of the superposed oscillations. The ratios//i indicate how strong the eigenmodes are influenced by the spring-mass system; small ratios//i indicate a large and large ratios a small coupling of the two shell-spring-mass systems. That is the reason why the superposed oscillation in fig. 18 coincides with the fifth or sixth eigenfrequency and not with the first. If we calculate the ratios//i for a very thin shell, e.g. h/a = 1/200 (fig. 16), we obtain //1 = 276,
//2 = 168,
//3 = 113,
//4 = 88,
//5 = 77,
//8 = 6 3 ,
//9 = 6 2 ,
/~10 = 6 5 ,
//11 = 6 6 ,
//12 = 7 6 .
//6 = 70,
//7 = 65, (52)
J. Hammel / Aircraft impact on a spherical shell
223
The ratios g/have considerably decreased and the minimum value of/ai occurs at the ninth mode. The influence of the elastic shell upon the impact force has increased (fig. 16) and the higher modes have become still more important. For a plate (fig. 5) we get #1 = 12.6,
#2 = 107.6,
P3 = 358,
P4 = 819.
(53)
The ratios Pi of a plate have another order of magnitude; they increase from the beginning and are distinctly separated. As we have seen from fig. 5 the impact force is rather strongly influenced by the elastic plate and it is sufficient to consider the first and the second eigenmode. What is the mechanical meaning of/ai? If we consider the same shell and the same spring-mass system, the ratio p/is proportional to Bico 2: gi "" ~°2iBi , where wi, the eigenfrequency, is the velocity of the eigenmode; and B i indicates the degree of participation of the mass of the shell in that mode of vibration. Thus,/~. is nothing more than the volume of kinetic energy of the ith mode of vibration (mass times velocity squared). A high ratio of Pi means that a high energy is necessary to produce this mode of vibration. In a continuum those modes of vibration are excited which have the lowest volume of energy. 8. Conclusion
The force F(t) of an impact on plates is influenced by the elasticity of the plate whereas the F(t) of impact on shells is unchanged by the elasticity of shells. The volumes of the energy of modes of vibration of a shell are so high that the shell almost behaves like a rigid wall for impact loads. As we have seen from the plots o f F ( t ) (figs. 8 - 1 7 ) and w(t) (fig. 18) higher modes and frequencies are especially emphasized for impinged shells (for plates the first eigenmode is predominant). The first eigenmode of the shells may not have the lowest volume of energy. Similar effects are known for cylindrical shells [6], where the lowest frequency occurs at higher wavenumber m, because the shells contain bending energy as well as extensional energy. The sum of both energies has its minimum at higher wave numbers. Plates only contain bending energy which, of course, has its minimum for the lowest modes. Shell parameters hardly influence the 'elastic' impact force. The reactions upon impact are confined to a small area around the impact point which is why boundary conditions a great distance from the impact point are of no interest. If we replace the concentrated force by a distributed load the lower modes of the shell obtain more importance, i.e. the force F(t) approaches that of an impact against a rigid wall. The result is that the transient force F(t) of impact of a deformable 'aircraft' upon an elastic shell is much more influenced by the considered aircraft model than by the elastic displacements of the shell. If we discuss the response of the shell we must consider many eigenmodes (many more than in the case of impact on plates); moreover, higher eigenmodes of shells are excited much more than lower ones. The ratios #i give information about the participation of an eigenmode in the final solution. References
[1] R. Burton, Vibration and Impact, Dover Publications, Inc. New York (1965). [2] J. Hammel, Stosa eines FlugkSrpers auf eine Kugelschale,Diss. TH Darmstadt, D 17, 1974. [3] A. Kalnins, Effect of bending on vibrations of spherical shells, J. Aoaust. Soc. Amer. 36 (1964) 74-81. [4] H. Kraus and A. Kainins, Transient vibration of thin elastic shells, J. Appl. Mech. 31 (1964) 994-1002. [5] H. Ktaus, Thin elastic shells, Wiley, New York/London/Sydney (1967). [6] W. Schneli, Schwingungs-und Stabilit~tsverhaltend0nnwandiger Schalen, 3. Lehrgangfur Raumfahrttechnik der DGF, Aachen (1964) 308, 1-56. [7] E. Skurdzyk, Die Grundlagender Akustik, Springer Verlag (1954).
AIRCRAFT IMPACT ON A SPHERICAL SHELL* J. H A M M E L
Institut j~ir Mechanik, Technische Itochschule Darmstadt, D-6100 Darmstadt, Germany Received 2 December 1975 For nuclear power plants located in the immediate vicinity of cities and airports safeguarding against an accidental aircraft strike is important. Because of the complexity of such an aircraft crash the building is ordinarily designed for loading by an idealized dynamical load F(t), which follows from measurements (aircraft striking a rigid wall). The extent to which the elastic displacements of a structure influence the impact load F(t) is investigated in this paper. The aircraft is idealized by a linear mass-spring-dashpot combination which can easily be treated in computations and which can suffer elastic as well as plastic deformations. This 'aircraft' normally strikes a spherical shell at the apex. The time-dependent reactions of the shell as a function of the unknown impact load F(t) are expanded in terms of the normal modes, which are Legendre functions. The continuity condition at the impact point leads to an integral equation for F(t) which may be solved by Laplace transformation. F(t) is computed for hemispheres with several ratios of thickness to radius, several edge conditions and several 'aircraft' parameters. In all cases F(t) differs very little from that function obtained for the case of the aircraft striking a rigid wall. The calculation of the normal displacements w(t) at various points of the shell shows that the influence of the impact is bounded on a small region around the impact point. Therefore boundary conditions are not of interest. When calculating the normal displacement w(t) of the shell in consequence of the impact load F(t), we see a vibration of large amplitudes and of low frequency with a superposed oscillation of small amplitudes and a higher frequency. The fundamental frequency corresponds very well to that of the idealized aircraft, while the higher frequency belongs to a natural frequency of the hemispherical shell. It is of interest that not the fundamental natural frequency of the shell becomes visible but a higher one. If the 'aircraft' strikes a thin plate of similar parameters, the impact load is influenced more by the elastic deformation of the plate. Contrary to the hemisphere in this case only tl3e two lowest eigenmodes are of interest. This different behaviour of shells and plates is investigated in the present paper. When calculating the amounts of energy of plates and these of shells with similar parameters we obtain very different values which show the different response of plates and of shells to impact load.
1. Introduction The safety o f m o d e r n reactor buildings requires that' dynamical loads, such as earthquakes, explosions, aircraft crashes, etc. m u s t be taken into consideration. In the analysis o f the p r o b l e m of an aircraft crash on a building, the building structure is usually considered as loaded by a dynamical load F ( t ) which follows f r o m m e a s u r e m e n t s (aircraft strikes a rigid wall). H o w far the elastic displacements o f a structure influence the impact load F(t) is investigated in this paper.
2. The aircraftmodel For c o m p u t a t i o n the aircraft m u s t be very simple. The aircraft is considered as a mass p o i n t and the c o n t a c t between the mass p o i n t and the shell is established by a linear s p r i n g - d a s h p o t - c o m b i n a t i o n . We can therefore describe elastic as well as plastic d e f o r m a t i o n s of the aircraft and we obtain a linear relation b e t w e e n the 'aircraft de* Paper S1/3 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguards and Containment Structures (ELCALAP), Berlin, Germany, 8-11 September 1975.
205
J. Hammel / Aircraft impact on a spherical shell
206
formation' u(t) and F(t) the unknown contact force between aircraft and shell. For our computation we take the simplest case: a Maxwell model, a linear elastic spring (constant c), and a linear dashpot (constant rT) (fig. 1). We then get
u(t)
t 1 . F ( t ) + l fo F(z)dr = c
(1)
The displacement of the mass point is Wm(t ) and by Newton's law we get
m(32Wm/Ot2 ) = - F ( t ) .
(2)
The weight of the aircraft is omitted in eq. (2). At the beginning of impact the aircraft model just touches tile shell and has the velocity %:
Wm(t = O) = O,
(OWm/3t) (t = 0) = o0 .
(3)
Integrating eq. (2) and utilizing the initial conditions, eqs. (3), gives the displacement Wm(t ) as a function of F(t): l _ _
Wm(t)=v 0 •
t
t - m f F(r)(t
~-)dr.
(4)
0 We assume that the aircraft is attached to the shell for t > 0 (a negative force F(t) is possible). So the deformation u(t) can be related to the linear elastic deflection Ws(t) of the shell at the impact point and to the displacement Wm(t):
u(t) = Wm(t ) - Ws(t ) .
(5)
If it is possible to describe Ws(t) as a function o f F ( t ) likewise, we get from eq. (5) together with eqs. (1) and (4), a linear integral equation for F(t): t t 1. F(t) + 1 fo F('r)d'r = Vo " t _ 1 f FQ') (t - r) d~- - ws(F(t)) . c
(6)
o
3. Shell under dynamical load The equations of motion for a normally loaded undamped shell of thickness h, reference length a, and density p may be written in the form [3,5]:
~l(U,V,w)=_(1/co2).[;,
~2(u,v,w)=_(1/co2).~)" '
223(u,v,w)=_(1/co2).~-(a2/D)q(t).
(7)
Rotatory inertia of the shell is neglected; u, v, w are the displacements of the middle surface in the directions of the shell coordinates (fig. 2); Z?1, Z?2, ~3 [2] are differential operators and the dot means differentiation with respect to time. D = [Eh/(1 - v2)] is the extensional rigidity, and coO = {El [pa2(1 - v2)]) 1/2 is a reference frequency. The solutions of the homogeneous problem (q(t) = 0) are the eigenfunctions Un(~O, 0), Un(~0,0), Wn(~O,O) [2] with eigenvalues con , the natural frequencies. The orthogonality condition of the eigenfunctions can be written in the form
* ,m + un, ,um + WnW~z) sin ~0 d~o dO = ¢/o, •f f (UnU o ~ Bn'
for n :/: m for n = m
(8)
207
J. Hammel ] Aircraft impact on a spherical shell
IV
Fig. 2. Sphericalshell-coordinates.
Fig. 1. Idealizedmodel of aircraft.
Expanding the displacements of the inhomogeneous problem in terms of the normal modes yields displacement equations in the form u(~o, O, t)= ' ¢bn(t)Un(~O,O ), n=l
v(~o, O, t) =
n=l
¢bn(t)On(~O,O ) ,
w(~o,O,t) = ~¢bn(t)Wn(~O,O), n=l
(9)
where ~bn(t ) denotes the generalized coordinates and represents the time-dependent aspect of the solution. Substituting the assumed solution eq. (9) into eq. (7) yields a set of equations for ~n(t):
¢Pn(t)"
~l (Un,On, * * Wn) + ~-~0 " ~ n U
n =0
I
n=l Cbn(t ) " ~3(u~, on, , w*) +1_1_. 6o02 ~ n W ~
I
~ n ( t ) " ~2(Un, On, Wn)*-~O2"'~nOn
=0,
= --~q(t)
(10)
Since Un, On, w n are the eigenfunctions of the homogeneous problem, ., l(Un
[°~n~2 .
)
*
On ,
I¢°nX2
"
Un ,
)
On ,
.
O.
,
16°n\2
Wn
(11)
the differential-operators 2?1, ~2, Z?3 in eqs. (10) can be replaced by eqs. (11). Together with the orthogonality condition, eq. (8), we obtain an uncoupled equation for ~n (t):
~ n + 602 ~ n =
ayfq(t) ~ o
w n sin ~0d~odO hpaBn
= _ fin(t) a • ffl n "
(12)
The initial displacements and velocities of the shell are also evaluated in terms of the normal modes:
o0 w0
=
~,.(o)
v;, "w~,"
,
bo ~,o
=
'i'n~O)
v~,
.
w~,
With the orthogonality condition we obtain the following pair of initial conditions [4] :
(13a),(13b)
J. Hammel / Aircraft impact on a spherical shell
208
qbn(O)=kf f(uOu*+vOVj+WOWn)Sin+d+dO,
(14a)
SoO
~n (0) : Bnnlf f(~oUn + ~OVn+ flOWn) sin ~pdCdO
(14b)
So 0
If we assume, that the shell is at rest at the beginning, we can see from eqs. (14) that all qSn(0) and all ~n(0) are zero. So we get the solution ~bn(t ) by integrating eq. (12): t
d)n(t)_
1 _= a • con • m~
f Fn(r)sincon(t- r)dr. 0
(15)
Now ~n(t) and with it all shell variables (the displacement Ws(t) at the impact point tO, too) are known as functions of the load q(t).
w(t0, 0 = ~ n=l
w2,(t0) a • conmn
f t Fn (r) sin con (t
7) d r .
06)
0
Assuming that the impact area is small in relation to the shell surface the load q(t) is approximated by a concentrated force (17)
q(t) = F(t)8 (t - ~0) ,
where 6(t - tO) is the Dirac function in shell coordinates. In the special,case of a spherical shell with coordinates ~oand 0 we obtain 1 - a2sin~p a(~o- % ) 6 ( 0 - 0 0 ) .
6(t-t0)~8(t.o -~00;0-00)
(18)
Then the generalized force Fn(t) is given by
F n (t) = a f f F ( t )
6 (t - t O) Wn (t) sin ~pd~0d O - F(t) Wna (t 0 )
(19 )
Thus, the shell displacements can be expressed as functions of the impact force F(t):
"w(t,
O"
"Wn(t)"
and especially the normal displacement at the impact point is given by o~
W(to, t) = ~ n=l
t
[Wn(tO)]2 fF(r)sincon(t
- r)dr.
(21)
a2conffln ~0
Replacing the displacement at tile impact point in eq. (6) by eq. (21) and observing the different directions of w(t0, t) and Ws(t), we obtain
J. Hammel / Aircraft impact on a spherical shell
1 t
_1j F ( r ) d r + ~F(t) = V o . t - - ~ f F(r)(t - z ) d r - £ [Wn(~O)]2 t; F(r) sfn con (t - T) dr. r/0 0 n=l a2confftn ~0
209
(22)
This linear integral equation, eq. (22), can be solved for example by Laplace transformation:
1_~ f(s) +f(s) _ VO r/• s c s2
f(s) m s2
~ [Wn(~0)] 2 n=l a 2 • t~ n
f(s) s 2 + 092 '
(23)
where f(s) is the Laplace transform of F(t):
mo o
(24)
f(s) =
m 2 +m 2.J 1 +ms +--s rl c n=l
[w~(~0)] 2
a2fft n
s2 " - s 2 + c62
If we end the infinite sum of the denominator after n = N terms, the function f(s) can be written as a quotient of two polynomials
f(s) = m v 0 [Z(s)/R(s)] ,
(25)
which can be decomposed to partial fractions
f(s)=mvo
(26)
= s - sn
The constants C n are determined by
Z(sn)
Z(sn)
(27)
Cn - I-I (sn - s]l-R'(sn 1' ]¢n where the sn are the roots of the denominator polynomialR(s), and a prime means differentiation with respect to s. The inverse transformation gives a sum of exponential functions: N
(2s)
F(t) = m v 0 n~t Cn exp(s nt) . If the impact force is not a concentrated but a distributed load, the generalized force must be replaced by
Fn(t) = [(fq(~o) Wn(~O) sin ~0d~o)/(afq(~o)sin ~od~o)]F(t). ~o
(29)
~o
Then the solution F(t) is obtained by replacing one Wn(~0) in eq. (21) by the expression
jq(~o) w~ (~o)sin ~o ~o
sin ~ ~o
Eliminating F(t) in eq. (15) by use of eq. (28) yields the solution of the generalized coordinates
(30)
210
J.
d#n (t) -
m OoWn(~O) pha2B n
Hammel / Aircraft impact on a spherical shell
]=1 co2 + $2
exp(sit )-coscont-.
sJ sinwnt COn
,
(31)
and the displacements of the shell can be written as double sums:
(u(~o, O,t)]
N ]v(~°' O't) l=mvo,~l:
Vn(~'O) t" Wn(~O)pha2Bn ,=l ~
"w(~, o, t)"
wT,(~, o) j
CO2CJ+,2
s~ . (c°Scont +--slncont" c o , ,exp(sjt))
(32)
The constants C/. and sj depend on all considered eigenfunctions, as can be seen from eq. (27), and so Cbn(t) is not only a function of the nth mode of vibration. It is not necessary that the n u m b e r s N and N of the double sum in eq. (32) are equal, but for computation it is suitable to have N = N. How many terms N must be considered is a question of shape and of duration of the impact. If the impulse has a very short duration and an abrupt shape (unit step function), many terms must be considered. Because in reality such impulses are not observed, we can expect that the function F(t) can be described relatively accurately by a small number N. In refs. [1] and [7] estimates of N for several impulse shapes are given. In our case we also have an expansion o f F ( t ) over the shell area. Therefore a rule-of-thumb cannot be given for our expansion ofF(t). We consider the amplitude of deviation when taking into account the next eigenfunction as a measurement of convergence. Considering the impact on plates, a few terms (N = 2) are sufficient to describe the influence of the elastic layer. Here the lowest eigenfrequencies have the leading part. When we consider the impact on shells we find that the eigenfunctions of shells contain extensional energy as well as bending energy terms. Therefore, in this case the higher frequencies are of greater interest than the lower ones. In our example it was necessary to take into account N = 10 until 20 terms to obtain a fairly good approximation. 4. The pushing deformable body The aircraft - idealized by a mass with an elastic spring and a viscous dashpot - pushes the shell with velocity vO. If the shell is rigid (Ws(t) = 0), the impact force F(t) can be calculated exactly:
F(t) = F(t) =
[c/m -
VO
exp
(-ct/2r?)
sin
[c/m - (c/2rl)2]l/2t,
exp
(-et/2~2)
sin [(c/2r/) 2 -
for
c/m
> (c/2r/) 2 ,
(33a)
for
c/m
< (C/2~'/) 2 .
(33b)
(c/2r/)211/2
v0 • c c/m]1/2
c/m]
1/2 t ,
[(c/2~7) 2 -
We assume that the aircraft is connected with the shell after its impact. Then the impulse I 0 on the shell is given by
I 0 = j F(t) dt = m v o .
(134)
0 The impulse I 0 does not depend on the functions e and r/; but the shape, the duration of impact, and the size of the impact load depend on c and 7). If the pushing mass is m = 10 000 kg, ~ = 20 kp sec/m and the velocity v0 = 1 m/sec (fig. 3), we have a duration of impact of about 0.17 sec and the peak of the impact load Fma x ~ 11.6 Mp. A negative impact load does not yet mean a lifting of the mass, because in the case of a vertical impact we had to superpose its weight. If the damping constant rt is increasing, the duration of collision is decreasing. The argument of the e function in 'eqs. (33) becomes smaller with increasing 7, i.e. the influence of r/is decreasing. In the same way the peak of
J. Hammel /Aircraft impact on a spherical shell
211
F I MpI 80
15
c =500 M p l m ~Z >. . . . . . . .
b4p.seklm
,..
1o
•
0
~
; t Isekl
Fig. 3. Transient force F(t) of an impact against a rigid wall for different m
m:pl
? :~jsoo
"°I :A': oo I/I.
m!l
\\ ,oo
I~
:o
m o
,'~
t Isekl
Fig. 4. F(t) of an impact against a rigid wall for different c.
the impact load is increasing because the plot of the force becomes more and more similar to that of an undamped vibration (r/= ~). N o w we take ~7 fixed and vary the spring constant c (fig. 4). the peak load is increasing with increasing c. The duration of collision tl, the positive range of impact force, given by the term sin [c/m - (c[2rl)2]l/2t in eq. (33a), is t 1 = n/[c/m - (c/2rl) 2] 1/2. If the parameters fulfill the condition c/m < 2n2/m 2, the duration t 1 is decreasing with increasing c. A large spring constant causes a short and hard impact. For our computation we estimate c, 7/and m so that the duration of impact and the peak load agree as closely as possible with experience.
212
.L Itammel / Aircra/t impact on a spherical shell
5. Load of an impact against an elastic structure In the following the structural reactions, when a mass strikes an elastic plate or an elastic spherical shell, are discussed. The method to calculate the impact force is explained in section 3. 5.1. Impact on a circular plate
We take the differential equation of the vibrating plate A A w = - (ph/K) w ,
(35)
where K is the bending rigidity, h the thickness, p the density, w the displacement of the plate and A the Laplace operator. The analysis of a circular plate yields solutions of eq. (35) in terms of Bessel functions. The boundary conditions of the plate lead to the eigenvalues and to tile eigenmodes of the vibrating plate. The impact force is expanded in series of these eigenfunctions. If the mass strikes the circular plate at its center, only the axial-symmetrical eigenmodes have to be considered. For a plate clamped at its edge (fig. 5) the lowest frequencies of the symmetrical vibration (m = 0) are calculated as: w 1 =41.3 1/sec; o)2= 160.8; o)3=320.3; co4 = 639.6; co5 = 998.7; and co6 = 1437.7. The frequencies of the circular plate are very clearly separated and we approximately have con ~ n2co I . Now we compute the impact force considering these eigenvalues and compare the result with the impact on a rigid wall (fig. 5). The velocity of the crashing aircraft is v0 = 1 m/sec; v0 is a multiplier when calculating the impact force (see eq. (24)). The first curve of fig. 5 shows the force F(t) of an impact against a rigid wall (n = 0), the lower curve is the force F(t) of an impact on the given elastic plate under consideration of n = 2, 4, 6 eigenfunctions. We see that the amplitude of deviation is so small in the last three cases that the calculation leads to the same curve F(t). In the case of an impact on an elastic plate it is sufficient to consider the two lowest eigenfunctions. The maximum force o f F ( t ) of the 'elastic' impact is about 7% lower than that of the 'rigid' impact. The later shape of the curve is entirely different when taking into account the elastic displacements of the plate. For later comparison with impact on a shell of h/a = 1/100 we compute F(t) of impact on a plate of h/a = 1/100 (fig. 6). We only consider the two lowest eigenfrequencies: co I = 20.7 1/sec, and co 2 = 80.4 1/sec, and plot the impact force F(t) against time t. Again, the upper curve indicates F(t) of impact on a rigid wall (n = 0). The crashing 'aircraft' in figs. 5 and 6 is the same, so the upper curves of both figures are the same. The maximum force F(t) (n = 2) has decreased again. The drop is 20% now, and the shape o f F ( t ) is entirely different from that of n = 0. 5. 2. Impact on an elastic hemispherical shell
First we determine the eigenfrequencies and eigenfunctions of a freely vibrating hemispherical shell. If the shell is pushed at its pole, we must only consider the axially-symmetrical eigenfunctions (m = 0). For a clamped hemisphere of v = 0.2 and h/a = 1/100 the eigenffequencies ~ have been calculated. The eigenfunctions u*, w* are plotted against meridian (fig. 7). The displacement o* is zero (m = 0). The displacement w* is symmetrical, u* is asymmetrical to the pole (tp = 0). All eigenfunctions are standardized to the normal displacement w* at the pole. The amplitude of the u displacement of the first eigenvalue is rather large but it decreases for higher frequencies except for the eigenfunctions at frequency ~ = 2.6381. At that frequency u* is again rather large and w* does not oscillate around the position of rest. This is due to the influence of extensional vibration. If we calculate the frequencies of a hemispherical membrane shell with a roller hinged edge, we get ~1 = 1.549 and ~ 2 = 2.656. So these curious eigenfunctions of fig. 7 in the region of ~ = 1.5 and co = 2.6 are nothing else but membrane vibrations with superposed bending vibrations. Since the impact area is small in relation to the shell surface, we first consider the impact load as a concentrated
J. Hammel / Aircraft impact on a sphericalshell F[Mpl
213
.Frtfp]
o .'n =O
• n = 2,4,6 n=2
c
~
ioookg
5ookp/cm
~1 t
4 Mp.seklm
h l a .'11,I00 ~,0.£
E
0.5
h/a = l /50 ~'=0.2 E =300000 kp/cm "~ ~ = 2500 kg/m ~
,
~';-'.P kp/cm 'f O0 /¢~/m .~
l 3 ~ 0
=
0.5
o
Fig. 5. F(t) of impact on an elastic plate (h/a = 1/50).
: SOOkp/cm
N/o.,
Fig. 6. F(t) of impact on an elastic plate (h/a = 1/100).
force; furthermore the weight of the crashing 'aircraft' is omitted. The aircraft impinges with velocity u0 = 1 m/sec upon the apex of the shell of fig. 8. The shell is considered to be of homogeneous isotropic material with specified material constants. The maximum force of n = 0 (impact upon rigid wall) is 15.47 Mp. If we consider n = 3 eigenfunctions the maximum force is 15.37 Mp. With n = 9 the force only decreases to 15.06 Mp, that is by 2.6%. The shape of the impact force F(t) is nearly unchanged. Now we vary the parameters of the 'aircraft'. The spring constant c remains fixed and the dashpot constant is halved (fig. 9). The shape of F(t) changes as discussed in section 4: the maximum force becomes smaller and the force is absorbed after a short time. But if we compare the 'elastic' and the 'rigid' impact we have the same tendency as seen in fig. 8. The decrease of the maximum force (n = 9) is only 1.8%. If we vary the spring constant c (figs. 10(a) and (b)), there is nearly no difference between an impact on an elastic and a rigid shell. If we have a smooth spring (fig. 10(a)) the difference is not visible. In the case of a very hard spring (fig. 10(b)) the decrease of the maximum force (n = 9) is only 0.5%. But it is of interest that now not the whole force F(t) gets absorbed, there remains an oscillation ofF(t). The hard spring excites the shell to very strong vibrations and, because we assume that the aircraft remains connected with the shell surface after impinging upon it, these strong shell vibrations entail an obvious oscillating force F(t). If appears (fig. 10(a)) that there is one distinguished frequency in which the excited shell vibrates. The period of the remaining oscillation is about T = 0.016 sec, to which a frequency co = 392 1/sec belongs. The sixth mode of vibration of the considered hemisphere has the frequency co = 390 1/sec, indicating that the excited shell mainly vibrates in its sixth mode. Now we take the 'aircraft of fig. 8 and vary the shell parameters. We consider shells of radius a = 10 m and 20 m but of the same h/a = 1/100. The impact force (figs. 8, 11 and 12) equates more and more to that of an impact on a rigid wall, i.e. the influence of the elastic shell becbmes smaller, although the shell frequencies decrease with increasing radius a. The frequencies alone do not describe the behaviour of an excited continuum: frequencies of the vibrating masses of the eigenfunctions, which increase with increasing
214
J. H a m m e l / A i r c r a f t i m p a c t o n a spherical shell
~
.
~
y
~ = 0.7624
• • 0
/w
u"
/-~
"---/
~ -09257
/~ ~
,-/", ~
~
~= 7.4754
--
--
J = [5503
A /
d, W
('~/'-----".~
~ :0.9666 0
q/ / ~
/-x
/~
:~
~
1.7041 y
uV~wA :'(~" :~vf'-.vfm = 1.00003 O• V k,_/ ~
-
: y. ~
~ =19226
),.~_
~ =2.1738
w
o ~
/'x
/
~
v
v
v
..j
L[,W
~
/.,, A / . . . . ~J,,,,v
j.,~j" •
w
/-", _/C'-~..._..: ~
G=!.2108
~ : 2.6381
u,~
'I.o ~/ /"(w
~
"~ ~,'U"/ "
~
"~"
/~
~'=½
~ =2.7791
Fig. 7. Eigenmodes of a clamped hemisphere (v = 0•2, h/a = 1/100).
shell-radius, are also of interest• Poisson's ratio v (figs. 8, 13 and 14) does not influence the impact force in any way. Edge conditions (fig. 15) also are not of interest• The impact is bounded to a small region at the apex. Only in the case of very thin shells (h/a = 1/200, fig. 16) can the influence of shell elasticity be of interest• The maximum force decreases from 15.47 Mp (n --- 0) to 14.75 Mp (n = 12). It is interesting that the deviation of the maximum force is greater from n = 6 to n = 9 than from n = 0 to n = 3, i.e. the higher eigenfunctions have a greater influence upon the maximum force than the lower ones. In the foregoing the impact force was considered as concentrated at the apex. In the following we consider the load as evenly distributed over a small area. Thereby in the calculation o f F ( t ) the function w(~ 0, t) - the normal displacement at impact point - is replaced by eq. (30), i.e. the eigenfunctions are evaluated in relation to their w displacements in the impact area. Thereby the lower eigenvalues are emphasized, so that the impact force hardly differs from the force of impact upon a rigid wall (fig. 17), We can summarize that the force F ( t ) essentially is influenced by the parameters of the considered 'aircraft'. The shell parameters - besides the case of very thin shells - influence the shape o f F ( t ) on!y very little. The shell - different from a p l a t e - behaves like a rigid wall when we consider the crash of an aircraft on it, especially in the case of distributed loads•
6. Displacements of the hemisphere If the impact force is known, the displacements of the shell can be calculated from eq. (32). In fig. 18 we have plotted the normal displacement w at the apex (full line) and at ~ = 10 ° (dashed line) against time t under concentrated impact force at the apex. n = 14 eigenfunctions have been considered• We notice a fundamental oscillation of
215
J. Hammel / Aircraft impact on a spherical shell F/Mpl v.ff
I II
,-I
m=lOOOOkg
E~
~
Ill
,olj I /I
I
II
I
..,ooo ~,o~
..o;,.,oo .--0.2
FtM~
E=300000 ,p/cm"L
' I
v.=lm/,W
j...-.n.O rl
,~.
m=lOOOOkg c.5000kp/cm
~n=9
h/a= I I100 ~'=0.2 E • 300000 kp/crnA tfsekl
~ = 2500 kglm '~
o~ o - T
Fig. 8. F(t) of impact on a hemisphere (.1 = 40000 kp sec/m).
o~
,, ,tt,,*l
Fig. 9. F(t) of impact on a hemisphere (.1 = 20000 kp sec/m).
F/Mpl v~z,,,I ~
h•=2O
mf,OOOOkg
OOOl~.sddm
F IMp)
'°l
h / o f 1/100
~
~=0.2 E . 300000 kp/cm=
t
:_~
~.
/--
h/o= III00
I!
,,.o.2
2500 kg/~'
t/51k/
O
~
tlse/d
Fig. 10. F(t) of impact on a hemisphere (c = 1000 kp/cm, c = 15000 kp/cm).
216
J. H a m m e l / A i r c r a f t i m p a c t o n a spherical shell FtMp]
,./.] /
~
F/Mp) m . ,o ooo kg
.. ~
n=O n=3
r~-~,
I
~Z=40WOkp.seklm
10
h/a=I II00
h/cl= 11700
-p=0.2
"~'= 0.2
E = 300 (220 kp/¢rn L
E = 300 000 kp/cm "L
= 2500 kg/rn J
9 -- 2500 k.q/rn~
'l 0
Fig. 1 i. F ( t ) of impact on a hemisphere (a = 10 m).
F [Nip}
. . . . i n =O ~n=3 ~n=6 "rt =9
t [sek]
Fig. 12. F ( t ) of impact on a hemisphere (a = 20 m).
F[MpI
V,,=l'/s#x[ ~ ~>
rn=lOOOOkg
Xn=9
c 5000 kplcm
~ I . / , ~ L[ ~
=
m=toooo,u
~.
c = 5000 kp/cm
/~...x,
10
~ =40 000~.sok/m
I0
h/o= I / l O0
h/a= III00
~:0.0
")'=0.4
E= 300000 kplc rn s"
E = 300000 kplcm "L
= 2500 kg/m 3
9 = 2500 kg/rn 3 5
0
t [sek/
Fig. 13. F ( t ) of impact on a hemisphere (u = 0.0).
6
~ : t(sek]
Fig. 14. F ( t ) of impact on a hemisphere (v = 0.4).
217
J. Hammel / Aircraft impact on a spherical shell F I MI~
F[Mp]
FIMpl
/4.o ~n =9
v,=t'/..,][~ m=tOOOO kg J/ ~ . tG
c ,$O00 kp/cm
h ~ = 4
0 000kp.sek/m
| hlo = I II, O0
[
"2=0.2 E = 300 000 kp/cm ~ = 2500 kg/rn ~
= t [sek]
5
0
Fig. 15.
h/o= 1/100
hie = 11200
i
,P=0.2 E = 300 0t20 kp/crn~ ~ = 2500 kg/m J
^,~ |(l # I/ 0u ..~ 8 " ~ 2
~J
Fig. 16.
__ t[sekl
5:
"P=0.2 E= 300000 kp/cm J" 9 = 2500 kglm "1
0
0.4 ~ 1 6
"¢"-~.:
; t IsekJ
I Fig.17
Fig. 15. F(t) of impact on a hemisphere (hinged supported). Fig. 16. F(t) of impact on a hemisphere (h/a = 1/200). IFig. 17. F(t) of impact on a hemisphere (distributed load).
about 3.5 Hz with a distinct superposed oscillation of about 60 Hz and small amplitudes. The frequency of the fundamental oscillaiton agrees very well with the frequency of our model for the crashing aircraft: co = (c/m) 1/2 = 22.1 I/s;- 3.52 Hz.
The superposed oscillation has a frequency, which agrees very well with that of the fifth or sixth eigenmode of the shell, i.e. the crashing 'aircraft' excites the fifth or sixth eigenmode first of all. If we consider w(t) at ¢ = 30 ° (fig. 19) the fundamental oscillation has disappeared. At this point the shell is at rest for a short time (about 0.007 sec) and then a steady oscillation with small amplitudes starts. In fig. 20 the displacement w is plotted against the meridian at different instants. The first curve is w(¢) under a force from an impact on a rigid wall, the second is w(~) under the force F(t) from an impact on the elastic shell. Again we can state that the difference is hardly visible. The graphs of fig. 20 show how the displacement at the impact point increases and how waves are propagating over the shell. After 0.064 see (fig. 20(c)) the maximum displacement at the apex occurs, while the force F(O has its maximum after 0.061 sec. Because the frequencies of our shell-aircraft model system are decoupled (section 7), we have this good agreement o f F ( t ) and of w(t). Large displacements are confined to the immediate vicinity of the impact area, which is why edge conditions cannot be of any interest; it is probably sufficient to consider only a small region of the excited shell. To obtain information about the convergence of our computation we consider fig. 21. The load is uniformly distributed over ¢ = 10°, n is the number of the considered eigenfunctions. We see that the problem cannot be represented by two eigenfunctions, and the increment of the third and fourth eigenmode is greater than that of the first and second; beyond ten eigenmodes the increments have different signs. If we consider 16 eigenmodes we get the maximum displacement of 0.63 mm after 0.067 see. The statical solution (the shell statically loaded by the maximum load of F(t)) is 0.61 mm. There is only a small increase in the normal deflection if we make a dynamical computation. This again is a consequence of the uncoupled frequencies of our considered shell-aircraft system.
218
Jr. Hammel /Aircraft impact on a spherical shell wlmml 1.0
{~
,.
~1
~:
.'~
~ ' ~ "-
j/
. . . . . . . .
o.2
V "V" ~
; t/sekl
" o.~
:.., /
u/I
,,,
\~! , ,,, -~.0
. . .
,~, i',d /
\
'
~
/
,
~,.,J rq
.
,,,,,oooo,,
%
.z°,o=,~,.,,,,.
"--",.,J
h/a= I / I 0 0
~=0.2
E = 300000 kp/cm*" "9 "
Fig. 18. D i s p l a c e m e n t
w(t)
under impact force at apex;
w(t)
2 5 0 0 k.qlm "1
at apex, - ....
w {ram } O.O5
o
.A~
VV
A A A,.,AAAAA b,{',.
-
_,,,,,
°'""1q °"°~" v
-o.o5
~.
~
' ~6m.~ h/a= 1/100
-~-= 0.2 E = J00000 kplcm J" =
2500 k g / m J
Fig. 19. Displacement w ( t ) at ~o= 30 ° under impact at apex.
w(t)
a t ~o = 10 °
J. Hammel / Aircraft impact on a spherical shell
219
• O. OOS L ~
~I~
r~
~.~
= o.o3 =,I~
(d)
0.5
(b)
0.5 ,~[ , . . , }
t • 0.t5 8o~ I
(e)
v~ J£.,n] f%'--
, v
(f) v, ~j,J t',,.,,/
I
(9) Fig. 20. Displacement x(~o) over meridian at different time t. w Imm]
1.0
-
"#
~
~=~O~s~lm
1T~0
h/a= 11100 "P =0.2 E = 300000 kplcm ~ 9 =
2500 k g l m ~
Fig. 21. Displacement w ( t ) at apex under distributed impact load at apex considering n = 2, 4, 6, 8, 10, 12, 14, 16 eigenmodes.
220
J. Hammel / Aircraft impact on a spherical shell
7. Participation of normal modes in the forced vibration solution We have seen that plates and shells have a quite different response to impact forces. To clarify this, we consider a clamped hemispherical shell connected to a undamped mass-spring system at its apex (fig. 22). But now we assume that the shell has only one degree of freedom, i.e. the vibration of the shell can be described by one eigenfunction, for example the ith. Utilizing the method of Rayleigh-Ritz we calculate the frequencies X of the coupled system. We take the functions w = Wi • w~(~o,0 ) ,
u = Wi • ui*(¢, O),
v = O,
(36)
where Wi is a constant and u~, w~ are eigenmodes of the ith mode of vibration of the shell (u~, w~ satisfy the boundary conditions). Substituting u, v, w into the expressions of the potential and kinetic energy of the shell, we get U = ~ pa2032ih W2B i,
T = ~1 pa2hB.W2 , , X2 ,
(37)
where w i is the eigenfrequency of the ith mode of vibration of the shell and B i stands for eq. (8). Replacing M = 2Trpa2h (the whole mass of the shell) we obtain the energy of the coupled system BiM Uges = ~co2 ~ W~ + ~c(Y - w(O)) 2 ,
T ges =m---X2y2+~x2BiM 2 ~ W2,
(38)
where Y is the amplitude of the mass m, and w(O) is the,displacement of the shell at the apex. w(O) equals Wi, if the eigenfunction w~ is a unit at the apex. The variation of the energy leads to a set of algebraic equations for Wi and Y: (
+2~rm ) 2zrm y = 0 ~22 BiM - ' Y 2 W i _ Bi___ ~ ,
_Wi+(l_72)y=o,
(39)
where ~2i = 03i(m/c) 1/2, 7 = X(m/c) 1/2. From eq. (39) we get the frequency equation 74_ 72 ( ~ 2 + 1 + ~ ) + ~ 2 2 = 0 .
(40)
If the term 2 lrm/BiM < ~22 i , we obtain the two uncoupled solutions 72 ~- 122 and 722~ 1. If we take the aircraft and shell as defined by fig. 11 and consider the first mode of vibration we get the couple-term 21rm/B1M = 0.275, a very small value in relation to $22. We get the frequency equation 74 - 72(145.2 + 1 + 0.275) + 145.2 = 0
(41)
with the solutions 712 = 145.5 and 72 = 0.998. The framed couple-term influences the fre~luencies of the coupled system very little. If we double the shell radius keeping the other parameters constant, I2~ decreases in the second power but the couple-term in the third power. Therefore, the frequencies 03i of the shell and to = (c/m) 112 of the spring-mass system get closer together; however, the frequencies of the coupled system agree better with those of the uncoupled system (~oi, 03). If we change the parameter h/a but keep the radius a constant, the lower frequencies ~oi of the shell change very little. In contrast, the mass of the shell and so the couple-term decreases nearly linearly with h/a. For a shell as given by fig. 16 we get I212 = 142.5 and 27rm/B1M 0.518. The couple-term is still very small in relation to g22 but it is twice the value of a shell with h/a = 1/100. Considering in the same way a clamped circular plate connected with a string-mass system at the center (fig. 23) we obtain a similar frequency equation: %4 - [~27 + 1 + (Trm/BiMp)]72 + ~2 = 0,
(40a)
£ Hammel/Aircraft impacton a sphericalshell
"
I
C. /
ill
Fig. 22. Clampedhemispherewith a mass-spring system at apex.
221
Fig. 23. Clamped circularplate with a mass--~ring systemat its middle.
Fig. 24. Two mass system.
where Mp = plr a 2 is the mass of the plate. The frequency equation (40a) is identical to eq. (40) (34 = 2Mp). But the frequencies [2/2 and the couple terms, respectively, are of other orders of magnitude. For a plate ofa = 10 m and h/a = 1/100 connected with the same spring-mass system, we get [212 = 0.22 and ;rm/B1Mp = 0.70. The frequency of the shell is 660 times greater and simultaneously the couple term is 2.5 times smaller than the values of the plate, respectively. So we can understand that the vibrations of plates are strongly influenced by the joined spring-mass system, i.e. the response of an elastic plate has a great influence upon the impact force. If we replace the shell by a spring-mass system (the frequency Q1 and the static displacement are the same) we must take a spring constant ~ = Eh/45 (1 - v2) and a mass ~ = 0.0061 M (M = mass of shell). This system (fig. 24) leads to a similar frequency equation: 74 - 72([Z12+ 1 + m/ffz) + [22 = O.
(40b)
The term ~21 in eq. (40b) is identical to that of eq. (40) but now the couple-term looksquiet different. Instead of 27.1 m/M (40) now we have m/~ = 164 m/M, i.e. the system of fig. 24 simulates a strong coupling of the two systems which does not exist in reality. It is problematic to replace a vibrating continuum by a single spring-mass system: besides the frequencies I22 the coupling strongly depends on the couple-terms, respectively. The ratio/g = fi12/(2 nm/B 1M) is about 500 for the considered shell and about 0.3 for the circular plate. Large values of/a (shell) mean that the influence of the elastic structure upon the impact force is negligible. To solve the question why higher modes of the shell are excited, we use again the Raleigh-Ritz-method. Because we want to compare the eigenmodes of the shell, we now assume that the shell shall only have two degrees of freedom, i.e. the vibration of the shell can be described by two eigenfunctions, e.g. the ith and ]th mode:
w(~o,O, t) = (Wi w~(~o,O) + Wi . w; (~o,O)sin ?~t , u(~o,O,t)=(Wiu~.(~o,a)+Wl~*(~o,d)sinXt,
v@,O,t)=O,
(42)
where u~ and w~ are the displacement functions of the ith mode with eigenfrequency coi and orthogonality integral
B i. Wi, W/are free constants. Together with assumption eq. (42) we get the kinetic and potential energy of the coupled system:
Uf~pha2(6°2BtW2+6o~B/W~)+~c(Y- w(O))2 ,
T=½pha2(BiW2+B/W2)X2+½my2)~2.
(43)
We differentiate the total energy (U - T) with respect to Wi, W/and Y, the amplitude of the displacement of the mass m and get a homogeneous algebraic system for Wi, I¢/and Y:
J. Hammel / Aircraft impact on a spherical shell
222 (f22+21rm
Bi-'---M-')'2
) W +27rm
21rm
i ~iM W J - ~ i M Y = O '
2 rrm ( BjM wi +
+ 2 rrm ) 2 rrm BjM - 32 wj r-- 0,
(44)
- W i - Wj + ( 1 - 3'2 ) Y = 0 . If we consider the first and second eigenmode of the shell (fig. 11), we obtain 2 ~rm ~22 = 145.3 , B - - ~ = 0.275;
~2
27rm = 214.2, B~-2-2-2 ~ = 0.698,
(45)
72 = 0.995.
(46)
with solutions 72=145.6,
3,2=214.9,
We receive the normal displacement at the apex together with the initial conditions (all initial displacements and velocities are zero besides the velocity v0 of mass m):
w(t) = 10-4vo(m/c)l/~(1.5 sin 267 t + 2.2 sin 325 t - 51.5 sin 22.1 t ) .
(47)
A fundamental oscillation of frequency co3 = 22.1 sec- 1 is superposed by two oscillations of high frequencies and small amplitudes. Moreover, the amplitude belonging to the second mode of vibration is 1.5 times greater than that of the first mode. Colasidering the first and the third mode of vibration we have ~2
21rm = 145.3, B ~ = 0.275;
27rm f22 = 233.6, B - ~ -= 1.087 ,
(48)
and together with the same initial conditions we get the displacement at apex:
w(t) = 10-400(m/c)1/2i(1.5 sin 267t + 3.05 sin 339t - 61 sin 22.1t).
(49)
The amplitudes of the fundamental as well as that of the superposed oscillation of the third mode have been increased, i.e. the interaction between mass-spring and shell has become greater. As we have seen, the coupling of the two systems is given by the ratio//i = ~22i/(21rm/BiM). Therefore, we calculate//i for each eigenmode of the shell and we obtain //1 = 528.7,
//2 = 307.1,
//3 = 214.9,
//7 = 179.1,
//8 = 214.9,
//9 = 265.0.
//4 = 188.1,
//5 = 178.2,
//6 = 177.5, (50)
The ratios//i first decrease to the minimum of the sixth mode and then increase again. Considering the fifth and sixth eigenmodes of the shell and doing the same calculation as above we get the displacement at the apex:
w(t) = 10-4o 0 m(3.05 sin 367.5 t + 3.4 sin 391.6 t - 112 sin 2 2 t ) .
(51)
Indeed, combining the fifth and sixth mode, eq. (51), gives the largest amplitudes of the fundamental as well as of the superposed oscillations. The ratios//i indicate how strong the eigenmodes are influenced by the spring-mass system; small ratios//i indicate a large and large ratios a small coupling of the two shell-spring-mass systems. That is the reason why the superposed oscillation in fig. 18 coincides with the fifth or sixth eigenfrequency and not with the first. If we calculate the ratios//i for a very thin shell, e.g. h/a = 1/200 (fig. 16), we obtain //1 = 276,
//2 = 168,
//3 = 113,
//4 = 88,
//5 = 77,
//8 = 6 3 ,
//9 = 6 2 ,
/~10 = 6 5 ,
//11 = 6 6 ,
//12 = 7 6 .
//6 = 70,
//7 = 65, (52)
J. Hammel / Aircraft impact on a spherical shell
223
The ratios g/have considerably decreased and the minimum value of/ai occurs at the ninth mode. The influence of the elastic shell upon the impact force has increased (fig. 16) and the higher modes have become still more important. For a plate (fig. 5) we get #1 = 12.6,
#2 = 107.6,
P3 = 358,
P4 = 819.
(53)
The ratios Pi of a plate have another order of magnitude; they increase from the beginning and are distinctly separated. As we have seen from fig. 5 the impact force is rather strongly influenced by the elastic plate and it is sufficient to consider the first and the second eigenmode. What is the mechanical meaning of/ai? If we consider the same shell and the same spring-mass system, the ratio p/is proportional to Bico 2: gi "" ~°2iBi , where wi, the eigenfrequency, is the velocity of the eigenmode; and B i indicates the degree of participation of the mass of the shell in that mode of vibration. Thus,/~. is nothing more than the volume of kinetic energy of the ith mode of vibration (mass times velocity squared). A high ratio of Pi means that a high energy is necessary to produce this mode of vibration. In a continuum those modes of vibration are excited which have the lowest volume of energy. 8. Conclusion
The force F(t) of an impact on plates is influenced by the elasticity of the plate whereas the F(t) of impact on shells is unchanged by the elasticity of shells. The volumes of the energy of modes of vibration of a shell are so high that the shell almost behaves like a rigid wall for impact loads. As we have seen from the plots o f F ( t ) (figs. 8 - 1 7 ) and w(t) (fig. 18) higher modes and frequencies are especially emphasized for impinged shells (for plates the first eigenmode is predominant). The first eigenmode of the shells may not have the lowest volume of energy. Similar effects are known for cylindrical shells [6], where the lowest frequency occurs at higher wavenumber m, because the shells contain bending energy as well as extensional energy. The sum of both energies has its minimum at higher wave numbers. Plates only contain bending energy which, of course, has its minimum for the lowest modes. Shell parameters hardly influence the 'elastic' impact force. The reactions upon impact are confined to a small area around the impact point which is why boundary conditions a great distance from the impact point are of no interest. If we replace the concentrated force by a distributed load the lower modes of the shell obtain more importance, i.e. the force F(t) approaches that of an impact against a rigid wall. The result is that the transient force F(t) of impact of a deformable 'aircraft' upon an elastic shell is much more influenced by the considered aircraft model than by the elastic displacements of the shell. If we discuss the response of the shell we must consider many eigenmodes (many more than in the case of impact on plates); moreover, higher eigenmodes of shells are excited much more than lower ones. The ratios #i give information about the participation of an eigenmode in the final solution. References
[1] R. Burton, Vibration and Impact, Dover Publications, Inc. New York (1965). [2] J. Hammel, Stosa eines FlugkSrpers auf eine Kugelschale,Diss. TH Darmstadt, D 17, 1974. [3] A. Kalnins, Effect of bending on vibrations of spherical shells, J. Aoaust. Soc. Amer. 36 (1964) 74-81. [4] H. Kraus and A. Kainins, Transient vibration of thin elastic shells, J. Appl. Mech. 31 (1964) 994-1002. [5] H. Ktaus, Thin elastic shells, Wiley, New York/London/Sydney (1967). [6] W. Schneli, Schwingungs-und Stabilit~tsverhaltend0nnwandiger Schalen, 3. Lehrgangfur Raumfahrttechnik der DGF, Aachen (1964) 308, 1-56. [7] E. Skurdzyk, Die Grundlagender Akustik, Springer Verlag (1954).