In this pap er we investigate the inuence of p erio dicwater waves on the deection of a
thin exibleshell,with large radius, mounted at the sea b ottom at depth h and piercing
through the free surface. The uid is assumed to b e inviscidand rotationless, hence the
velo city p otential ob eys the Laplace equation. The motion of the shell is describ ed by
the classicaltheory of thins shells. The equationsare then reducedto simplied"two di-
mensional" equationsinvolvingthe normal,tangential andlongitudinal motionu;v;wof
the middlesurface. This theory requiresthat the shellthicknessdis smallincomparison
withshellradiusR,andthewaterdepthh. Theotherclassicalassumptionscanb efound
in Markus [1]. We will fo cus on the p erio dic deections only and we pay no attention
to thestaticdeectionduetohydrostaticpressure. Bylinearity,thiseectscanb eadded.
The harmonic water waves diracted by a rigid cylinder will b e describ ed by means of
the wellknowneigenmo deexpansion. These mo desare solutions ofthe Laplace equation
and t the b oundary conditions at the free surface and the sea b ottom. To solve the
coupledequationsforthe refractedp otentialand the motionofthe shellweuse the same
set of orthogonal functions. Due to the b oundary conditions of the variables of motion
of the shell this can not lead to a converging series. To assure convergence we expand
these highest derivatives, as they o ccur in the equations of motions, in series of these
eigenmo des. The lower order derivativesand the functions themselvesfollowfrom these
expansions. Thisgivesrisetoasetofp olynomialswith unknownco eÆcients. Makinguse
of the orthogonality relationsof the eigenmo des andthe expansionof the p olynomials in
these mo des leads to a set of equations with a sparse blo ck, structured by the orthogo-
nality relation, combined with full rows due to the expansions of the p olynomials. The
b oundary conditions of the variables describing the motion of the shellgive an extra set
of equations. The nal 'square' matrixequationcan b e solvedby astandard metho d.
In section 6, an application of this mathematical approach is treated for the case of a
exibleb eamloadedbyadistributiondescrib edbytherst eigenmo deofthe water-wave
problem. Convergenceof the deection and itsderivativesis shown. In the next section,
the mathematicalmo delfor the shellis givenand the metho dis tested forseveral values
of the exural rigidity of the shell.
2 Derivation of the main equations
Westudy the b ehaviorof the thin cylindricalshellin p erio dicwaves in asea of constant
depth h. The shell,emergingfromthe sea surface isxed on the sea b ottom.
We assume the uid to b e p otential and intro duce the velo city p otentialV = r(x;t)
whereVistheuidvelo cityvector. Wegetforthep otential( x;t)theLaplaceequation
=0 inthe uid domain.
The linearized freesurface z =0, the linearized condition g
z +
tt
=0 , at the b ottom
we have
@
@z
= 0 and for the linearizedb o dy b oundary condition
@
@n
= u where u is the
normaldeection ofthe shell.
W
U V
Following Love'stheory, we derivethe equations of motion of the shell.
R
@u
@z +
R
2
(1+)
@ 2
v
@z@
+R 2
@ 2
w
@z 2
+ 1
2
(1 )
@ 2
w
@ 2
0
E
(1
2
)R 2
@ 2
w
@t 2
=0 (1)
@u
@ +
@ 2
v
@ 2
+ R
2
2
(1 )
@ 2
v
@z 2
0
E
(1
2
)R 2
@ 2
v
@t 2
+ R
2
(1+)
@ 2
w
@z@
=0 (2)
R 4
@ 4
u
@z 4
+2R 2
@ 4
u
@z 2
@ 2
+
@ 4
u
@ 4
u
0
E
(1
2
)R 2
@ 2
u
@t 2
+
@v
@
(3)
+R
@w
@z
=
(1
2
)R 2
Ed
P(r;z; ;t)
with = 1
12 (
d
R )
2
and P the dynamicpressuredue to water waves.
P(r;z; ;t)=
@
@t
(r; ;z;t) (4)
By dierentiatingequation(1), one can obtain.
R
@ 2
u
@z 2
+ R
2
(1+)
@ 3
v
@z 2
@ +R
2
@ 3
w
@z 3
+ 1
2
(1 )
@ 3
w
@z@
2
0
E
(1
2
)R 2
@ 3
w
@z@t 2
=0 (5)
The dierentiationwillallowusto expand later this equationin the sameeigenfunctions
than the two others shell equations. If we cho ose equation [1] to b e zero at z = h it is
automatically fullled everywhere.
Wesplit the p otentialand the pressure into twoparts.
The rst ones ( d
and p d
) are due to the diraction when the shell is rigid. The second
ones ( and p) are due to the exural deections (radiation).
with
P(r; ;z;t)=(p d
(r; ;z)+p(r; ;z))e i! t
(6)
(r; ;z;t)=( d
+)e i! t
(7)
p d
(r=R)= g
cosh(kh)
h+ 1
sinh 2
(kh)
2
1=2
f
0 (z)
X
n=0 2
n (i)
n+1
kRH 0
n (kR)
cos (n )
= 1
X
n=0 A
n
cos(n ) f
0 (z)
3 Expansion of the solutions in eigenfunctions
Werst expandthe p otential in the usual eigenfunctionsf
m .
(r; ;z;t)= 1
X
n=0 1
X
m=1 X
mn K
1
n (k
m
r )cos (n )f
m (z)e
i! t
(8)
+ 1
X
n=0 X
0n H
1
n
(kr )cos (n )f
0 (z)e
i! t
(9)
Instead of following the standard approachfor Sturm-Liouvilleb oundary value problems
where expansions in the eigen-function space of the op erator is sought, we assume that
the fourth derivative of u, the second derivative of v and third derivative of w can b e
expanded in the sameclass of eigenfunctions. Then, if we integrate those functions and
nd for u;v and w
u(z; ;t)= 1
X
n=0
1
X
m=0 u
mn
k 4
m f
m (z)+
1
6 C
1n
(z+h) 3
+C
2n 1
2
(z+h) 2
+C
3n
(z+h)+C
4n
cos(n )e i! t
(10)
v(z; ;t)= 1
X
n=1
v
0n
k 2
f
0 (z)
1
X
m=1 v
mn
k 2
m f
i
(z)+C
8n
(z+h)+C
9n
sin(n )e i! t
(11)
w (z; ;t)= 1
X
n=0
w
0n
k 3
g
0 (z)
1
X
m=1 w
mn
k 3
m g
m (z)+
1
2 C
5n
(z+h) 2
+C
6n
(z+h)
+C
7n
cos (n )e i! t
(12)
with
f
0 (z)=
p
2 coshk(z+h)
h+ 1
sinh 2
(kh)
1=2 g
0 (z)=
p
2sinhk(z+h)
h+ 1
sinh 2
(kh)
1=2
(13)
with k givenbythe usual disp ersionrelation =ktanh (kh)
f
n (z)=
2 cosk
n
(z+h)
h
1
sin 2
(k
n h)
1=2 g
n (z)=
2 sink
n
(z+h)
h
1
sin 2
(k
n h)
1=2
(14)
k
n
givenby = k
n tan (k
n h)
The f
n
functions are orthonormaland formacompleteset ofeigenfunctions forthe solu-
tion .
In order to guarantee the convergence of the termwize integrated series to the integral
of the expanded function, we haveto assumethat the series of the successive derivatives
converge uniformly. This requirement can b e to o strong and is not necessary as can b e
shown by a slightly dierent approach. However, numerical results conrm the conver-
gence of the fourth derivative of u although slowly. The convergence of the series for u
is then guaranteed and fast. The eigenfunctions f
n
for are not eigenfunctions of the
mechanicalequations. It turns out that the p olynomials are convenientto b e added, the
found solution liesin theprop er function space. It fullsthe b oundary conditions and in
a weak sense the equations. It is reasonable to exp ect somelo cal convergence problems
for the series of the highest derivatives.
4 Boundary conditions
The b oundary condition on the cylinder for the p otential in frequency domain is as
follow:
@
@n
= i!u (15)
Theb oundaryconditionattheb ottomofthe uiddomainalsorepresentsthattheb ound-
ary isa rigid wall,therefore
@
@z
=0 at z = h (16)
The mechanicalb oundary conditions at the ends of the shell are then derivedbyconsid-
ering that the shell is xed at z = h but free at z = 0. Following Markus, the eight
extraequations read:
u=v =w=
@u
@z
=0 forz = h (17)
@w
@z +
R
@v
@ u
=0 (18)
@ 2
u
@z 2
+ 1
R 2
@ 2
u
@ 2
=0
1
R
@w
@ +
@v
@z
=0
@ h
@w 1 @v u
i
(1 ) @ h
1@w @v @ 2
u i
to zequals zero. This givesan extracondition for the co eÆcientsC
8n
and C
5n :
nR
2
(1+)C
8n +R
2
C
5n
=0 (19)
5 Numerical method
For the numericalresolution of the problem,we truncate the sumsto N for n,and to M
form. aftermultiplyingeachequationbyf
m
(z),weintegratetheequations[5],[2],[3]and
[15]fromz = hto z =0usingtheorthonormalityofthesefunctions. For eachequation,
we also expand the p olynomials inthe f
m
series.
Æ
0i
= Z
0
h f
i
(z)dz Æ
1i
= Z
0
h
(z+h)f
i
(z)dz (20)
Æ
2i
= Z
0
h
(z+h) 2
f
i
(z)dz Æ
3i
= Z
0
h
(z+h) 3
f
i
(z)dz (21)
This yields, for each n 2 [0N], to a linear system of 4M +13 equations, with un-
known values U
mn
;V
mn
;W
mn
;X
mn
and the C
i
co eÆcients. The force termof the system
is logicallyfound to b e the pressure due to the diractedproblem fora rigid shell.
6 Numerical results and test of convergence
Tomakeourmetho d clearlyunderstandable,letusrst solve,for illustrationasimplied
case. Let describ e the axisymmetric, radial vibration of a cylindrical shell under the
symmetricload q(z). The former system of equations can b e reducedto the dierential
equation
D
@ 4
u
@z 4
+ Eh
R 2
u=q(z) with q(z)=kf
0
(z) (22)
Weassume that we havethe following b oundary conditions
u =0 u 0
=0 at z=-h (23)
u 00
=0 u
000
=0 at z=0 (24)
the function uwrites
u(z)= M
X
m=0 u
m
k 4
m f
m (z)+
1
6 C
1
(z+h) 3
+C
2 1
2
(z+h) 2
+C
3
(z+h)+C
4
(25)
Cho osing k = 1, D = 1, h = 50 and Eh
R 2
= 10 2
, we give the numerical solutions for u,
u (1)
, u (2)
,u (3)
and u (4)
obtained for dierentvalues of M(5;10;20;50;150 ).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
-25 -20 -15 -10 -5 0
m=5 m=10 m=20 m=50 m=150
0 0.002 0.004 0.006 0.008 0.01 0.012
-25 -20 -15 -10 -5 0
m=5 m=10 m=20 m=50 m=150
-0.001 -0.0005 0 0.0005 0.001 0.0015 0.002
-25 -20 -15 -10 -5 0
m=5 m=10 m=20 m=50 m=150
-0.0003 -0.00025 -0.0002 -0.00015 -0.0001 -5e-05 0 5e-05 0.0001 0.00015 0.0002 0.00025
-25 -20 -15 -10 -5 0
m=5 m=10 m=20 m=50 m=150
-0.00015 -0.0001 -5e-05 0 5e-05 0.0001
-25 -20 -15 -10 -5 0
m=5 m=10 m=20 m=50 m=150
Figure1: convergence
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-25 -20 -15 -10 -5 0
Uabs
’Uabs’
0 0.02 0.04 0.06 0.08 0.1 0.12
-25 -20 -15 -10 -5 0
Wabs
’Wabs’
Figure 2: shelldeection
The convergence of u app ears very fast and we only need a couple of terms to have a
very go o d approximation of our solution. The series of the fourth derivative seems to
have a slow converge. Esp eciallyto obtain a right value at the p oint z = 0. This is due
to the fact that the f
m
functionsdo not ob ey the b oundary equations of the mechanical
problem.
For the complete mo delit turns out that the convergenceof the series for the deection
shows similar bahaviour. Dep ending on the physical parameters we cho ose the upp er
b ounds of the series. In general the choice M = 50 is reasonable. We present here some
numerical resultsfor a cylindricalshellwith a diameterof 6m, in a25m depth sea. The
thickness of the shell is 1cmand itsdensity is 2500Kg=m 3
. The rigidityequals E = 10 9
and the incoming wave frequency ! = 0:3r ad=s. The chosen value for the rigidity is
rather low,so the thin structure is very exible
Examples of the computation of the amplitude of the deection for u and w against z,
for =0 are shown in gure [2]As we suggested we have chosen 50 termsfor the series
although, moretermsareneededto obtaina go o d convergenceofthe highestderivatives.
In gure[3],weshowthe cylinderdeplacementat twodierent depths(z=-15mand z=-
5m) and in ve dierents time moments. The main contribution to the deplacement is
due to the b ending of the cylinder.
7 Conclusion
It willb e shown that for certainvalues of the wave frequencythe cylinderb ecomes reso-
nant. This means that our homogeneousb oundary value problem has no solution at all.
This phenomenon is well known for forced Sturm-Liouville systems with homogeneous
b oundary conditions. For the structural parameters chosen inour examplethis happ ens
intheregion ofthefrequecy bandofarealisticsea-state. Ifwecho osethe thicknesstob e
larger,forinstance5cm,the resonancephenomenono ccursat afrequencylargerthanthe
frequenciesof a sea-sp ectrum. We also willshow results for a largervalue of the rigidity
parameter. Ifwe takea value for asteelstructure we can use athicknessof 1cmleads to
no problem ina sea-state. It turns out that this approach can b e used for a wide range
of parameters
4 2 0 2 4
-15m -5m
4 2 0 2 4
-15m -5m
4 2 0 2 4
-15m -5m
4 2 0 2 4
-15m -5m
4 2 0 2 4
-15m -5m
Figure 3: shelldeection
References
[1]
Stefan
Markus . The Mechanics of Vibrationsof CylindricalShells (1988).
Studies in Applied Mechanics.
