High Frequency Oscillations of First Eigenmodes in...

Operator Theory:Advances and Applications, Vol. 258, 89–110c! 2017 Springer International Publishing

High Frequency Oscillations of FirstEigenmodes in Axisymmetric Shellsas the Thickness Tends to Zero

Marie Chaussade-Beaudouin, Monique Dauge,Erwan Faou and Zohar Yosibash

Abstract. The lowest eigenmode of thin axisymmetric shells is investigatedfor two physical models (acoustics and elasticity) as the shell thickness (2!)tends to zero. Using a novel asymptotic expansion we determine the behaviorof the eigenvalue "(!) and the eigenvector angular frequency k(!) for shellswith Dirichlet boundary conditions along the lateral boundary, and naturalboundary conditions on the other parts.

First, the scalar Laplace operator for acoustics is addressed, for whichk(!) is always zero. In contrast to it, for the Lame system of linear elasticityseveral di!erent types of shells are defined, characterized by their geometry,for which k(!) tends to infinity as ! tends to zero. For two families of shells:cylinders and elliptical barrels we explicitly provide "(!) and k(!) and demon-strate by numerical examples the di!erent behavior as ! tends to zero.

Mathematics Subject Classification (2010). 74K25, 74H45, 74G10, 35Q74,35C20, 74S05.

Keywords. Lame, Koiter, axisymmetric shell, sensitive shell, developable shell.

1. Introduction

The lowest natural frequency of shell-like structures is of major importance in engi-neering because it is one of the driving considerations in designing thin structures(for example containers). It is associated with linear isotropic elasticity, governedby the Lame system. The elastic lowest eigenmode in axisymmetric homogeneousisotropic shells was address by W. Soedel [14] in the Encyclopedia of Vibration:

[We observe] a phenomenon which is particular to many deep shells,namely that the lowest natural frequency does not correspond to the sim-plest natural mode, as is typically the case for rods, beams, and plates.

90 M. Chaussade-Beaudouin, M. Dauge, E. Faou and Z. Yosibash

This citation emphasizes that for shells these lowest natural frequencies may hidesome interesting “strange” behavior. The expression “deep shell” contrasts with“shallow shells” for which the main curvatures are of same order as the thickness.Typical examples of deep shells are cylindrical shells, spherical caps, or barrels(curved cylinders).

In acoustics, driven by the scalar Laplace operator, it is well known that,when Dirichlet conditions are applied on the whole boundary, the first eigenmodeis simple in both senses that it is not multiple and that it is invariant by rotation.We will revisit this result, in order to extend it to mixed Dirichlet–Neumannconditions. In contrast to the scalar Laplace operator, the simple behavior of thefirst eigenmode does not carry over to the vector elliptic system – linear elasticity.Relying on asymptotic formulas exhibited in our previous work [5], we analyzetwo families of shells already investigated in [2]. Doing that, we can comparenumerical results provided by several di!erent models: The exact Lame model,surfacic models (Love and Naghdi), and our 1D scalar reduction.

The first of these families are cylindrical shells. We show that the lowest eigen-value1 decays proportionally to the thickness 2! and that the angular frequency kof its mode tends to infinity like !!1/4.

The second family is a family of elliptic barrels which we call “Airy barrels”.Elliptic means that the two main curvatures (meridian and azimuthal) of themidsurface S are non-zero and of the same sign. Airy barrels are characterized bythe following relations:

• The meridian curvature is smaller (in modulus) than the azimuthal curvatureat any point of the midsurface S,

• The meridian curvature attains its minimum (in modulus) on the boundaryof S.

For general elliptic barrels, the first eigenvalue tends to a positive limit a0 as thethickness tends to 0. This quantity is proportional to the minimum of the squaredmeridian curvature. More specifically, for Airy barrels, the azimuthal frequency kis asymptotically proportional to !!3/7 as !! 0. Besides, for the particular familyof Airy barrels that we study here, a very interesting (and somewhat non-intuitive)phenomenon occurs: For moderately thick barrels k stay equal to 0 and there is athreshold for ! below which k has a jump and start growing to infinity.

We start by presenting the angular Fourier transformation in a coordinate-independent setting in Section 2 followed by the introduction of the domains andproblems of interest in Section 3. Section 4 is devoted to cases when the angularfrequency k of the first mode is zero or converges to a finite value as !! 0. Cylin-drical shells are investigated in Section 5 and barrels in Section 6. We conclude inSection 7.

1With the natural frequency f , the pulsation ! and the eigenvalue ", we have the relations

" = !2 = (2#f)2.

High Frequency Oscillations of First Eigenmodes 91

2. Axisymmetric problems

Before particularizing every object with the help of cylindrical coordinates, wepresent axisymmetric problems in an abstract setting that exhibits the intrinsicnature of these objects, in particular the angular Fourier coe"cients. This intrin-sic definition allows one to prove that the Fourier coe"cients of eigenvectors areeigenvectors if the operator under examination has certain commutation proper-ties.

2.1. An abstract setting

Let us consider an axisymmetric domain in R3. This means that # is invariantby all rotations around a chosen axis A: For all " " T = R/2#Z, let R! be therotation of angle " around A. So we assume

#" " T, R!# = #.

Let t = (t1, t2, t3) be Cartesian coordinates in R3. The Laplace operator $ =$2t1 + $2

t2 + $2t3 is invariant by rotation. This means that for any function u

#" " T, #t " R3, $!u(R!t)

"= ($u)(R!t).

The Lame system L of homogeneous isotropic elasticity acts on 3D displace-ments u = u(t) that are functions with values in R3. The definition of rotationinvariance involves not only rotation of coordinates, but also rotations of displace-ment vectors. Let us introduce the transformation G! : u $! v between the twodisplacement vectors u and v

G!(u) = v %& #t, v(t) = R!!

!u(R!t)

".

Then the Lame system L commutes with G!: For any displacement u

#" " T, L(G!u) = G!(Lu). (2.1)

Now, in the scalar case, if we define the transformation G! by (G!u)(t) =u(R!t), we also have the commutation relation for the Laplacian

#" " T, $(G!u) = G!($u). (2.2)

The set of transformations!G!

"!"T has a group structure, isomorphic to that of

the torus T:G! ' G!! = G!+!! , ", "# " T.

The rotation invariance relations (2.1) and (2.2) motivate the following angularFourier transformation T ( " $! k " Z (here u is a scalar or vector function u or u)

#uk(t) =1

2#

$

T(G"u)(t) e

!ik" d%, t " #, k " Z. (2.3)

Then each Fourier coe"cient #uk enjoys the property:

(G!#uk)(t) = eik! #uk(t), t " #, " " T , (2.4)


and each pair of Fourier coe"cients #uk and #uk!with k )= k# satisfies

$

T#uk(R!t) · #uk!

(R!t) d" = 0, t " #,

which implies that #uk and #uk!are orthogonal along each orbit contained in # and

hence in L2(#).The inverse Fourier transform is given by

u(t) =%

k"Z#uk(t), t " #. (2.5)

The function u is said unimodal if there exists k0 " Z such that for all k )= k0,#uk = 0, and #uk0 )= 0. Such a function satisfies

(G!u)(t) = eik0! u(t), t " #, " " T.

For an operator A that commutes with the transformations G!, i.e., G!A =AG!, as in (2.1) and (2.2), there holds for any k " Z

&Auk =1

2#

$

T(G"Au) e

!ik" d% =1

2#

$

T(AG"u) e

!ik" d% = A#uk.

In particular, if Au = &u, then &Auk = &#uk. We deduce from the latter equalitythat

&#uk = A#uk.

Therefore any nonzero Fourier coe"cient of an eigenvector is itself an eigenvector.We have proved

Lemma 2.1. Let & be an eigenvalue of an operator A that commutes with the groupof transformations {G!}!"T. Then the associate eigenspace has a basis of unimodalvectors.

Remark 2.2. If moreover the operator A is self-adjoint with real coe"cients, theeigenspaces are real. Since for any real function and k )= 0, the Fourier coe"cient#u!k is the conjugate of #uk, the previous lemma yields that if there is an eigenvectorof angular eigenfrequency k, there is another one of angular eigenfrequency *kassociated with the same eigenvalue.

2.2. Cylindrical coordinates

Let us choose cylindrical coordinates (r,%, ') " R+ + T + R associated with theaxis A. This means that r is the distance to A, ' an abscissa along A, and % arotation angle around A. We write the change of variables as

t = T (r,%, ') with t1 = r cos%, t2 = r sin%, t3 = ' .

The cylindrical coordinates of the rotated point R!t are (r,% + ", ').


2.2.1. Scalar case. The Laplace operator in cylindrical coordinates writes

$ = $2r +

1

r$r +

1

r2$2" + $2

# .

The classical angular Fourier transform for scalar functions is now

uk(r, ') =1

2#

$

Tu(T (r,%, ')

"e!ik" d%, (r, ') " (, k " Z, (2.6)

where ( is the meridian domain of #. We have the relations

uk(r, ') eik" = #uk!T (r,%, ')

", (r, ') " (, % " T, k " Z (2.7)

and the classical inverse Fourier formula (compare with (2.5))

u(T (r,%, ')"=

%

k"Zuk(r, ') eik" .

The Laplace operator at frequency k is

$(k) = $2r +

1

r$r *

k2

r2+ $2

# ,

and we have the following diagonalization of $

($u)k = $(k)uk, k " Z.

2.2.2. Vector case. Now, an option to find a coordinate basis for the representationof displacements is to consider the partial derivatives of the change of variables T

Er = $rT , E" = $"T , and E# = $#T .

If we denote by Et1 , Et2 , and Et3 the orthonormal basis associated with Cartesiancoordinates t, we have

Er = Et1 cos%+ Et2 sin%, E" = *rEt1 sin%+ rEt2 cos%, and E# = Et3 .

We note the e!ect of the rotations R! on these vectors (we omit the axial coordi-nate ')

Er(r,%+ ") = (R!Er)(r,%) and E"(r,%+ ") = (R!E")(r,%) (2.8)

and E# is constant and invariant.The contravariant components of a displacement u in the Cartesian and cylin-

drical bases are defined such that

u = ut1Et1 + ut2Et2 + ut3Et3 = urEr + u"E" + u#E# .

Covariant components uj are the components of u in dual bases. Here we have

uti = uti , i = 1, 2, 3, and ur = ur, u" = r2u", u# = u# .

Using relations (2.8), we find the representation of transformations G!

G!u(t) = ur(r,%+ ", ')Er(r,%) + u"(r,%+ ", ')E"(r,%) + u# (r,%+ ", ')E# ,

with t = T (r,%, '). Then the classical Fourier coe"cient of a displacement u is:

uk(r,%, ') = (ur)k(r, ')Er(r,%) + (u")k(r, ')E"(r,%) + (u# )k(r, ')E# ,


where (ua)k is the Fourier coe"cient given by the classical formula (2.6) for u = ua

with a = r,%, ' . We have a relation similar to (2.7), valid for displacements:

uk(r, ') eik" = #uk!T (r,%, ')

", (r, ') " (, % " T, k " Z. (2.9)

Let L be the Lame system. When written in cylindrical coordinates in the basis(Er,E",E# ), L has its coe"cients independent of the angle %. Replacing the de-

rivative with respect to % by ik we obtain the parameter-dependent system L(k)

that determines the diagonalization of L

(Lu)k = L(k)uk, k " Z. (2.10)

3. Axisymmetric shells

We are interested in 3D axisymmetric domains # = #$ that are thin in one direc-tion, indexed by their thickness parameter !. Such #$ is defined by its midsurfaceS: We assume that the surface S is a smooth bounded connected manifold withboundary in R3 and that it is orientable, so that there exists a smooth unit normalfield P $! N(P) on S. For ! > 0 small enough the following map is one to one andsmooth

% : S + (*!, !) ! #$

(P, x3) $! t = P+ x3 N(P).(3.1)

The actual thickness h of # is 2! (we keep this thickness h = 2! in mind to bridgingwith some results of the literature). Such bodies represent (thin) shells in elasticity,whereas they can be called layer domains or thin domains in other contexts.

The boundary of #$ has two parts:

1. Its lateral boundary $0#$ := %!$S + (*!, !)

",

2. The rest of its boundary (natural boundary) $1#$ := $#$ \ $0#$.

The boundary conditions that will be imposed are Dirichlet on $0#$ and Neu-mann on $1#$. We consider the two following eigenvalue problems on #$, posedin variational form: Let

V!(#$) := {u " H1(#$) , u = 0 on $0#$},

and

VL(#$) := {u = (ut1 , ut2 , ut3) " H1(#$)3 , u = 0 on $0#

$}.(i) For the Laplace operator: Find & " R and u " V!(#$), u )= 0 such that

#u$ " V!(#$),

$

"!

,u ·,u$ dt = &

$

"!

u u$ dt. (3.2)

(ii) For the Lame operator: Find & " R and u " VL(#$), u )= 0 such that

#u$ " VL(#$),

$

"!

Aijlmeij(u) elm(u$) d#$ = &

$

"!

utiu$ti d#$. (3.3)


Here we have used the convention of repeated indices, Aijlm is the material tensorassociated with the Young modulus E and the Poisson coe"cient )

Aijlm =E)

(1 + ))(1* 2))*ij*lm +

E

2(1 + ))(*il*jm + *im*jl), (3.4)

and the covariant components of the strain tensor are given by

eij(u) =1

2($tiutj + $tjuti).

The associated 3+ 3 system reads

L = * E

2(1 + ))(1 * 2))

'(1 * 2))$+, div

(.

Both problems (3.2) and (3.3) have discrete spectra and their first eigenval-ues are positive. We denote by &!(!) and &L(!) the smallest eigenvalues of (3.2)and (3.3), respectively. By Lemma 2.1, the associate eigenspaces have a basis ofunimodal vectors. By Remark 2.2, in each eigenspace some eigenvectors have anonnegative angular frequency k. We denote by k!(!) and kL(!) the smallest non-negative angular frequencies of eigenvectors associated with the first eigenvalues&!(!) and &L(!), respectively.

In the next sections, we exhibit cases where the angular frequencies k(!)converge to a finite limit as ! ! 0 (the quiet cases) and other cases where k(!)tends to infinity as !! 0 (the sensitive or excited cases).

4. Quiet cases

We know (or reasonably expect) convergence of k(!) for the Laplace operator andfor plane shells.

4.1. Laplace operator

Let us start with an obvious case. Suppose that the shells are plane, i.e., S is anopen set in R2. Then #$ is a plate. The axisymmetry then implies that S is a discor a ring. Let (x1, x2) be the coordinates in S and x3 be the normal coordinate.In this system of coordinates

#$ = S + (*!, !) (4.1)

and the Laplace operator separates variables. One can write

$"! = $S - I(!$,$) + IS -$(!$,$) . (4.2)

Here $"! is the 3D Laplacian on #$ with Dirichlet conditions on $0#$ and Neu-mann conditions on the rest of the boundary,$S is the 2D Laplacian with Dirichletconditions on $S, and $(!$,$) is the 1D Laplacian on (*!, !) with Neumann con-ditions in ±!. Then the eigenvalues of $"! are all the sums of an eigenvalue of$S and of an eigenvalue of $(!$,$). The first eigenvalue &!(!) of problem(3.2) is


of course the first eigenvalue of *$"! . Since the first eigenvalue of *$(!$,$) is 0,we have

&!(!) = &S

and the corresponding eigenvector is u(x1, x2, x3) = v(x1, x2) where (&S , v) is thefirst eigenpair of*$S . Thus k!(!) is independent of !, and is the angular frequencyof v.

In the case of a shell that is not a plate, the equality (4.2) is no more true.However, if $S denotes now the Laplace–Beltrami on the surface S with Dirichletboundary condition, an extension of the result2 of [13] yields that the smallesteigenvalue of the right-hand side of (4.2) should converge to the smallest eigenvalueof $"! . An extension of [13, Th. 4] gives, more precisely, that

&!(!) = &S + a1!+O(!2), as !! 0, (4.3)

for some coe"cient a1 independent of !. Concerning the angular frequency k!(!),a direct argument leads to the following conclusion.

Lemma 4.1. Let #$ be an axisymmetric shell. The first eigenvalue (3.2) of theLaplace operator is simple and k!(!) = 0.

Proof. The simplicity of the first eigenvalue of the Laplace operator with Dirichletboundary conditions is a well-known result. Here we reproduce the main steps ofthe arguments (see, e.g., [9, Sect. 7.2]) to check that this result extends to moregeneral boundary conditions.

Let u be an eigenvector associated with the first eigenvalue &. A consequenceof the Kato equality

,|u| = sgn(u),u almost everywhere3

is that |u| satisfies the same essential boundary conditions as u and has the sameminimum Rayleigh quotient as u. Therefore |u| is an eigenvector too and it satisfiesthe same eigenequation *$|u| = &|u| as u. The latter equation implies that *$|u|is nonnegative, and hence |u| satisfies the mean value property

|u(t0)| .1

meas(B(t0, +))

$

B(t0,%)|u(t)| dt

for all t0 " #$ and all + > 0 such that the ball B(t0, +) is contained in #$. Hence|u| is positive everywhere in #$. Therefore u = ±|u| and we deduce that the firsteigenvalue is simple.

By Lemma 2.1, this eigenvector is unimodal. Let k be its angular frequency.If k )= 0, by Remark 2.2 there would exist an independent eigenvector of angularfrequency *k for the same eigenvalue. Therefore k = 0. !

2In [13], the manifold S (denoted there by M) is without boundary. We are convinced thatall proofs can be extended to the Dirichlet lateral boundary conditions when S has a smoothboundary.3With the convention that sgn(u) = 0 when u = 0.


4.2. Lame system on plates

The domain #$ is the product (4.1) of S by (*!, !). For the smallest eigenvalues&L(!) of the Lame problem (3.3), we have the convergence result of [6, Th. 8.1]

&L(!) = &B !2 +O(!3), as !! 0. (4.4)

Here, &B is the first Dirichlet eigenvalue of the scalar bending operator B that, inthe case of plates, is simply a multiple of the bilaplacian (Kirchho! model)

B =1

3

E

1* )2$2 on H2

0 (S). (4.5)

The reference [6, Th. 8.2] proves convergence also for eigenvectors. In particular thenormal component u3 of a suitably normalized eigenvector converges to a Dirichleteigenvector of B. This implies that the angular frequency kL(!) converges to theangular frequency kB of the first eigenvector of the bending operator B.

4.3. Lame system on a spherical cap

A spherical cap #$ can be easily defined in spherical coordinates (+, ",%) " [0,/)+[*&

2 ,&2 ]+ T (radius, meridian angle, azimuthal angle) as

#$ =)t " R3, + " (R* !, R+ !), % " T, " " (&,

#

2]*.

Here R > 0 is the radius of the midsurface S and & " (*&2 ,

&2 ) is a given meridian

angle. Numerical experiments conducted in [7, Sect. 6.4.2] exhibited convergencefor the first eigenpair as !! 0 (when & = &

4 ), see Fig. 10 loc. cit.. We do not have(yet) any theoretical proof for this.

5. Sensitive cases: Developable shells

Developable shells have one main curvature equal to 0. Excluding plates that areconsidered above, we see that we are left with cylinders and cones4.

The case of cylinders was addressed in the literature with di!erent levels ofprecision. In cylindrical coordinates (r,%, ') " [0,/) + T + R (radius, azimuthalangle, axial abscissa) a thin cylindrical shell is defined as

#$ =)t " R3, r " (R* !, R+ !), % " T, ' " (*L

2 ,L2 )*.

Here R > 0 is the radius of the midsurface S and L its length. The lateral boundaryof #$ is

$0#$ =

)t " R3, r " (R * !, R+ !), % " T, ' = ±L

2

*.

One may find in [15, 14] an example of analytic calculation for a simply sup-ported cylinder using a simplified shell model (called Donnel–Mushtari–Vlasov).We note that simply supported conditions on the lateral boundary of a cylinder

4Since we consider here shells with a smooth midsurface, cones should be trimmed so that theydo not touch the rotation axis.


allow reflection across this lateral boundary, so that separation of the three vari-ables using trigonometric Ansatz functions is possible. This example shows thatfor R = 1, L = 2 and h = 0.02 (i.e., ! = 0.01) the smallest eigenfrequency doesnot correspond to a simple eigenmode, i.e., a mode for which k = 0, but to a modewith k = 4.

In Figure 1 we plot numerical dispersion curves of the exact Lame modelL for several values of the thickness h = 2! (0.1, 0.01, and 0.001). This meansthat we discretise the exact 2D Lame model L(k) obtained after angular Fouriertransformation, see (2.10), on the meridian domain

($ =)(r, ') " R+ + R, r " (R * !, R+ !), ' " (*L

2 ,L2 )*.

We compute by a finite element method for a collection of values of k "{0, 1, . . . , kmax} so that for each !, we have reached the minimum in k for the firsteigenvalue.

! " #! #"$%"

"

"%"

&

&%"

'

'%"

(

(%"

)

*

+,- #!.

/

0

0

1020#3 #1020#3 41020#3 5

Figure 1. Cylinders with R = 1 and L = 2: log10 of first eigenvalue of L(k)

depending on k for several values of the thickness h = 2!. Material constantsE = 2.069 · 1011, # = 0.3, and $ = 7868 as in [2]

So we see that the minimum is attained for k = 3, k = 6, and k = 11 whenh = 0.1, 0.01, and 0.001, respectively. We have also performed direct 3D finiteelement computations for the same values of the thickness and obtained coherentresults. In Figures 2–4 we represent the shell without deformation and the radialcomponent of the first eigenvector for the three values of the thickness.

In fact the first 3D eigenvalue &L(!) and its associated angular frequency kL(!)follow precise power laws that can be determined. A first step in that direction isthe series of papers by Artioli, Beirao Da Veiga, Hakula and Lovadina [4, 1, 2]. In


Figure 2. Cylinder with R = 1, L = 2 and h = 10!1: First eigenmode(radial component).

Figure 3. Cylinder with R = 1, L = 2 and h = 10!2: First eigenmode(radial component).

Figure 4. Cylinders with R = 1, L = 2 and h = 10!3: First eigenmode(radial component).


these papers the authors investigate the first eigenvalue of classical surface modelsposed on the midsurface S. Such models have the form

K(!) = M+ !2B. (5.1)

The simplest models are 3 + 3 systems. The operator M is the membraneoperator and B the bending operator. These models are obtained using the as-sumption that normals to the surface in #$ are transformed in normals to thedeformed surfaces. In the mathematical literature the Koiter model [10, 11] seemsto be the most widely used, while in the mechanical engineering literature so-called Love-type equations will be found [14]. These models di!er from each otherby lower-order terms in the bending operator B. As we will specify later on, thisdi!erence has no influence in our results.

Defining the order , of a positive function ! $! &(!), continuous on (0, !0],by the conditions

#- > 0, lim$%0+

&(!) !!'+( = 0 and lim$%0+

&(!) !!'!( =/, (5.2)

[4, 1, 2] proved that , = 1 for the first eigenvalue of K(!) in clamped cylindricalshells. They also investigated by numerical simulations the azimuthal frequencyk(!) of the first eigenvector of K(!) and identified power laws of type !!) for k(!).They found . = 1

4 for cylinders (see also [3] for some theoretical arguments basedon special Ansatz functions in the axial direction).

In [5], we constructed analytic formulas that are able to provide an asymptoticexpansion for k(!) and &(!), and consequently for kL(!) and &L(!):

k(!) 0 /!!1/4 and &(!) 0 a1! , (5.3)

with explicit expressions of / and a1 using the material parameters E, + and ),the sizes R and L of the cylinder, and the first eigenvalue µbilap 0 500.564 of thebilaplacian - $! $4

z- on the unit interval (0, 1) with Dirichlet boundary conditions-(0) = -#(0) = -(1) = -#(1) = 0, cf [5, Sect. 5.2.2]:

/ =

+R6

L43(1* )2)µbilap

,1/8

and a1 =2E

+RL2

-µbilap

3(1* )2). (5.4)

We compare the asymptotics (5.3)–(5.4) with the computed values of kL(!) by2D and 3D FEM discretisations, see Figure 5. The values of kL(!) are determinedfor each value of the thickness:

• In 2D, by the abscissa of the minimum of the dispersion curve (see Figure 1)• In 3D, by the number of angular oscillations of the first mode (see Figures2–4)

Finally we compare the asymptotics (5.3)–(5.4) with the computed eigenval-ues &L(!) by four di!erent methods, see Figure 6.

Problems considered in Figure 6:

a) Lame system L(k) on the meridian domain ($, computed for a collection ofvalues of k by 2D finite element method.


! " # $ %&

!

%&

%!

$&

$!

'()%&*+,

-

.

.$/.0(1234546(78#/.0(1234546(7898:124(4608

Figure 5. Cylinders (R = 1, L = 2): Computed values of kL(!) versus thethickness h = 2!. The asymptotics is h !" 9.2417 · !!1/4 # 11 · h!1/4 (with

# = 0.3).

! "#$ " %#$ % &#$ &!

!#$

$

$#$

'

'#$

(

)*+&,-./

)*+ &,-

/

0

0

123456*6782%90:;<&90=>+.?7&90@*AB"90:;<

Figure 6. Cylinders (R = 1, L = 2): Computed values of "L(!) versusthe thickness h = 2!. Material constants E = 2.069 · 1011, # = 0.3, and$ = 7868. 1D Naghdi and Love models are computed in [2]. The asymptoticsis h !" 6.770 · !E/$ = 3.385 · hE/$.


b) Naghdi model [12] on the meridian set C = (*1, 1) of the midsurface, com-puted for a collection of values of k by 1D finite element method in [2].

c) Love-type model [14] on C, computed for a collection of values of k by collo-cation in [2].

d) 3D finite element method on the full domain #$.

In methods a), b) and c),

&(!) = min0&k&kmax

&(k)(!)

where &(k)(!) is the first eigenvalue of the problem with angular Fourier parameterk (remind that &(k)(!) = &(!k)(!)). In method d), &(!) is the first eigenvalue.

We can observe that these four methods yield very similar results and that theagreement with the asymptotics is quite good. In [5, Sect. 5] the case of trimmedcones is handled in a similar way and yields goods results, too.

6. A sensitive family of elliptic shells, the Airy barrels

In this section we consider a family of shells defined by a parametrization withrespect to the axial coordinate, which is denoted by z when it plays the role ofa parametric variable: The meridian curve C of the surface S is defined in thehalf-plane R+ + R by

C =)(r, z) " R+ + R, z " I, r = f(z)

*

where I is a chosen bounded interval and f is a smooth function on the closureof I. We assume that f is positive on I. Then the midsurface is parametrized as(with values in Cartesian variables)

I + T *! S(z,%) $*! (t1, t2, t3) = (f(z) cos%, f(z) sin%, z).

(6.1)

Finally, the transformation F : (z,%, x3) $! (t1, t2, t3) sends the product I + T+(*!, !) onto the shell #$ and is explicitly given by, see also [5, Sect. 1.3.1]

t1 =!f(z)+x3

1s(z)

"cos%, t2 =

!f(z)+x3

1s(z)

"sin%, t3 = z*x3

f !(z)s(z) , (6.2)

where s is the curvilinear abscissa

s(z) =.1 + f #2(z).

With shells parameterized in such a way, we are in the elliptic case (thatmeans a positive Gaussian curvature) if and only if f ## is negative on I. In thissame situation the references [4, 1, 2] proved that the order (5.2) of the firsteigenvalue of K(!) is , = 0. In [2], numerical simulations are presented for thecase

f(z) = 1* z2

2on I = (*0.892668, 0.892668), (6.3)


Figure 7. Shell (6.2)–(6.3) with h = 10!1.

by solving the Naghdi and the Love models. A power law k(!) 1 !!2/5 is suggestedfor the angular frequency of the first mode. The shells defined by (6.2)–(6.3) havethe shape of barrels, Figure 7.

Before presenting the analytical formulas of the asymptotics [5], let us showresults of our 2D and 3D FEM computations. In Figure 8 we plot numerical dis-persion curves of the exact Lame model L for several values of the thickness h = 2!(0.01, 0.004, 0.002 and 0.001).

We can see that, in contrast with the cylinders, k = 0 is a local minimumof all dispersion curves. This minimum is global when h . 0.005. A second local

! " #! #" $! $"%&'

(

(&$

(&)

(&%

(&'

*

+,-#!.

/

0

01020#3 $1020)3 41020$3 41020#3 4

Figure 8. Shell (6.2)–(6.3): log10 of first eigenvalue of L(k) depending on kfor several values of the thickness h = 2!. Material constants as in [2]


Figure 9. Shell (6.2)–(6.3) and h = 0.01, 0.004, 0.002, 0.001: First eigen-mode (radial component).

minimum shows up, which becomes the global minimum when h 2 0.004 (k = 9,12 and 16 for h = 0.004, 0.002, and 0.001, respectively. In Figure 9 we representthe radial component of the first eigenvector for these four values of the thicknessobtained by direct 3D FEM.

Comparing with the cylindrical case, we observe a new phenomenon: theeigenmodes also concentrate in the meridian direction, close to the ends of thebarrel, displaying a boundary layer structure as ! ! 0. In [5] we have classifiedelliptic shells according to behavior of the function (proportional to the square ofthe meridian curvature bzz)

H0 =E

+

f ##2

(1 + f #2)3. (6.4)

If H0 is not constant, the classification depends on the localization of the minimumof H0. If the minimum is attained at a point z0 that is at one end of I, we are inwhat we called the Airy case. We observe that for the function f = 1* z2

2

H0 =E

+

1

(1 + z2)3.


Its minimum is clearly attained at the two ends ±z0 of the symmetric interval I.In the Airy case our asymptotic formulas take the form, see [5, Sect. 6.4],

k(!) 0 /!!3/7 and &(!) 0 a0 + a1!2/7 with a0 = H0(±z0) . (6.5)

To give the values of / and a1 we need to introduce the functions

g(z) = *2E

+

'ff ##

s6+

f2f ##2

s8

((z) and B0(z) =

E

+

1

3(1* )2)

1

f(z)4. (6.6)

Withb = B0(z0) and c = z(1)Airy

!g(z0)

"1/3 !$zH0(z0)

"2/3, (6.7)

(here z(1)Airy 0 2.33810741 is the first zero of the reverse Airy function) we have

/ =' c

6b

(3/14and a1 = (6bc6)1/7

+1 +

1

6

,. (6.8)

We compare the asymptotics (6.5)–(6.8) with the computed values of kL(!) by2D and 3D FEM discretisations, see Figure 10. The values of kL(!) are determinedfor each value of the thickness by the same numerical methods as in the cylindriccase.

Finally we compare the asymptotics (6.5)–(6.8) with the computed eigenval-ues &L(!) by the same four di!erent methods as in the cylinder case, see Figure 11.

Here we present 2D computations with two di!erent meshes. The uniformmesh has 2+8 curved elements of geometrical degree 3 (2 in the thickness direction,8 in the meridian direction) and the interpolation degree is equal to 6. In the

! " # $ %&

$&

"&

'&

(&

%&&

%$&

)*+%&,-.

/

01234*546*78)49

4

4$:4;<6#:4;<6=>?2@A*ABC>

Figure 10. Shell (6.2)–(6.3): Computed values of kL(!) versus the thicknessh = 2!. The asymptotics is h !" 0.51738·!!3/7 # 0.6963·h!3/7 (with # = 0.3).


! " # $ %&

&'$

&'"

&'&

&'(

)

*+,%-./0

*+,%-.

112-0

1

1

$314561789/1:;<8=;>1?;@/$314561789/1A=8<+:?1?;@/%31B2,/>8%31C+D;#314562%1

Figure 11. Shell (6.2)–(6.3): Computed values of "L(!)$a0 versus thicknessh = 2!. 1D Naghdi and Love models in [2]. Asymptotics h !" a0 + a1 ! #(0.1724 + 1.403 · !)E/$.

refined mesh, we add 8 points in the meridian direction, at distance O(!), O(!3/4),O(!1/2), and O(!1/4) from each lateral boundary, see Figure 12. So the mesh hasthe size 2 + 16. The geometrical degree is still 3 and the interpolation degree, 6.In this way we are able to capture those eigenmodes that concentrate at the scaled/!2/7, where d is the distance to the lateral boundaries. In fact eigenmodes alsocontain terms at higher scales, namely d/!3/7 (membrane boundary layers), d/!1/2

(Koiter boundary layers), and d/! (3D plate boundary layers).A further, more precise comparison of the five families of computations with

the asymptotics is shown in Figure 13 where the ordinates represent now log10(&*a0 * a1!2/7). These numerical results suggest that there is a further term in theasymptotics of the form a2!4/7. We observe a perfect match between the 1D Naghdimodel and the 2D Lame model using refined mesh. The Love-type model seemsto be closer to the asymptotics. A reason could be the very construction of theasymptotics: They are built from a Koiter model from which we keep

• the membrane operator M,• the only term in $4

" in the bending operator. Note that this term is commonto the Love and Koiter models. After angular Fourier transformation, thecorresponding operator becomes B0(z) k4, with B0 introduced in (6.6).

The exponent * 37 in (6.5) is an exact fraction arising from an asymptotic

analysis where the Airy equation *$2ZU + ZU = 'U on R+ with U(0) = 0 shows

up. Thus, the exponent * 25 in [2] that is only an educated guess is probably


!"# ! !"# !"$ !"% !"& ' '"# '"$ '"% '"&

!"&

!"%

!"$

!"#

!

!"#

!"$

!"%

!"&

!"#$ !"% !"%$ !"$ !"$$ !"& !"&$ !"' !"'$ !"( !"($

!"($

!"(

!"'$

!"'

!"&$

!"&

!"$$

!"$

!"%$

! !"# $ $"#

!"%

!"&

!"'

!"(

!

!"(

!"'

!"&

!"%

!"# !"$ !"% !"& !"'

!"'$

!"'

!"&$

!"&

!"%$

!"%

!"$$

!"$

!"#$

Figure 12. Meshes for shell (6.2)–(6.3) and ! = 0.01:Uniform (top row), refined (bottom row).

! " # $ %$&"

#

#&"

"

"&"

!

!&"

'()*+,-.

'()*+,

//0+/

/0*%12 .

/

/

%3/456/789-/:;<8=;>/?;@-%3/456/789-/A=8<(:?/?;@-*3/B0)->8*3/C(D;$3/456-#12

Figure 13. Shell (6.2)–(6.3): Computed values of "L(!)$ a0$ a1!2/7 versusthickness h = 2!. 1D Naghdi and Love models in [2]. The reference line ish !" !4/7 E/$.


! " #! #"$

"

%

&

'

(

)

*+,#!-

.

/

/

0/1/#2 #0/1/#2 30/1/#2 4526789:2

! " #! #" $! $"%&'

(

(&$

(&)

(&%

(&'

*

+,-#!.

/

0

0

1020#3 $1020)3 41020$3 41020#3 4536789:3

Figure 14. Cylinders (top) and Airy barrels (bottom): First eigenvaluesof L(k) depending on k for several values of thickness compared with firsteigenvalues of M(k) (membrane).

incorrect. We found in [5, Sect. 6.3] this * 25 exponent for another class of elliptic

shells that we called Gaussian barrels, for which the function H0 (6.4) attains itsminimum inside the interval I (instead of on the boundary for Airy barrels).


7. Conclusion: The leading role of the membrane operatorfor the Lame system

We presented two families of shells for which the first eigenmode has progressivelymore oscillations as the thickness tends to 0. The question is “Can we predict sucha behavior for other families of shells? What are the determining properties?”

In [5] we presented several more families of shells with same characteristicsof the first mode. The common feature that controls such a behavior seems to bestrongly associated to the membrane operatorM. If we superpose to our dispersioncurves k $! &(k)(!) of the Lame system the dispersion curves k $! µ(k) of themembrane operator, we observe convergence to the membrane eigenvalues as !! 0for each chosen value of k, see Figure 14. We also observe that for each chosen !,the sequence &(k)(!) tends to / as k !/. The appearance of a global minimumof &(k)(!) for k that tends to/ as !! 0 occurs if the sequence µ(k) has no globalminimum: its infimum is attained “at infinity”.

For cylinders and cones, the sequence µ(k) tends to 0 as k ! /. Hence thesensitivity. For elliptic shells, the sequence µ(k) tends to a limit that coincideswith the minimum of the function H0. Sensitivity depends on whether µ(k) hasa minimum lower than this value. From our previous study it appears that anyconfiguration is possible. For hyperbolic shells, µ(k) tends to 0 so sensitivity occurs,cf. [4, 1, 2] but the analysis of the coe"cients in asymptotics cannot be performedby the method of [5].

A natural question that comes to mind is: Are there other types of axisym-metric structures that behave similarly? Rings (curved beams) are conceivable –the recent work [8] suggests that sensitivity does not occur for thin rings withcircular or square sections.

References

[1] E. Artioli, L. Beirao da Veiga, H. Hakula, and C. Lovadina, Free vibrationsfor some Koiter shells of revolution, Appl. Math. Lett., 21 (2008), pp. 1245–1248.

[2] , On the asymptotic behaviour of shells of revolution in free vibration, Compu-tational Mechanics, 44 (2009), pp. 45–60.

[3] L. Beirao Da Veiga, H. Hakula, and J. Pitkaranta, Asymptotic and numericalanalysis of the eigenvalue problem for a clamped cylindrical shell., Math. ModelsMethods Appl. Sci., 18 (11) (2008), pp. 1983–2002.

[4] L. Beirao Da Veiga and C. Lovadina, An interpolation theory approach to shelleigenvalue problems., Math. Models Methods Appl. Sci., 18 (12) (2008), pp. 2003–2018.

[5] M. Chaussade-Beaudouin, M. Dauge, E. Faou, and Z. Yosibash, Free vibra-tions of axisymmetric shells: parabolic and elliptic cases, arXiv,http://fr.arxiv.org/abs/1602.00850 (2016).

[6] M. Dauge, I. Djurdjevic, E. Faou, and A. Rossle, Eigenmode asymptotics inthin elastic plates, J. Maths. Pures Appl., 78 (1999), pp. 925–964.


[7] M. Dauge, E. Faou, and Z. Yosibash, Plates and shells : Asymptotic expan-sions and hierarchical models, Encyclopedia of Computational Mechanics, 1, Chap.8 (2004), pp. 199–236.

[8] C. Forgit, B. Lemoine, L. Le Marrec, and L. Rakotomanana, A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of generalcross-section, to appear, (2016).

[9] B. Helffer, Spectral theory and its applications, vol. 139 of Cambridge Studies inAdvanced Mathematics, Cambridge University Press, Cambridge, 2013.

[10] W.T. Koiter, A consistent first approximation in the general theory of thin elasticshells, Proc. IUTAM Symposium on the Theory on Thin Elastic Shells, August 1959,(1960), pp. 12–32.

[11] , On the foundations of the linear theory of thin elastic shells: I, Proc. Kon.Ned. Akad. Wetensch., Ser.B, 73 (1970), pp. 169–182.

[12] P.M. Naghdi, Foundations of elastic shell theory, in Progress in Solid Mechanics,vol. 4, North-Holland, Amsterdam, 1963, pp. 1–90.

[13] M. Schatzman, On the eigenvalues of the Laplace operator on a thin set with Neu-mann boundary conditions, Appl. Anal., 61 (1996), pp. 293–306.

[14] W. Soedel, SHELLS, in Encyclopedia of Vibration, S. Braun, ed., Elsevier, Oxford,2001, pp. 1155–1167.

[15] , Vibrations of Shells and Plates, Marcel Dekker, New York, 2004.

Marie Chaussade-BeaudouinMonique Dauge and Erwan FaouIrmar, (Cnrs, Inria)Universite de Rennes 1Campus de BeaulieuF-35042 Rennes Cedex, France

e-mail: [email protected]@[email protected]

Zohar YosibashBen-Gurion University of the NegevDept. of Mechanical EngineeringPOBox 653Beer-Sheva 84105, Israel

e-mail: [email protected]

High Frequency Oscillations of First Eigenmodes in...

Documents

Transcript of High Frequency Oscillations of First Eigenmodes in...