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Continuous and Finite Element Methods for theVibrations of Inflatable Beams

Jean-Christophe Thomas, Zhihong Jiang, Christian Wielgosz

To cite this version:Jean-Christophe Thomas, Zhihong Jiang, Christian Wielgosz. Continuous and Finite Element Meth-ods for the Vibrations of Inflatable Beams. International Journal of Space Structures, Multi-SciencePublishing, 2006, 21 (4), pp.197-222. 10.1260/026635106780866033. hal-01385737

Continuous and Finite ElementMethods for the Vibrations of

Inflatable Beams J.C. Thomas, Z. Jiang and C. Wielgosz

GeM, Institute of Research in Civil Engineering and Mechanics, UMR-CNRS 6183Université de Nantes, 2 rue de la Houssinière, 44322 Nantes, France

ABSTRACT: Inflatable structures are under increasing development invarious domains. Their study is often carried out by using 3D membrane finiteelements and for static loads. There is a lack of knowledge in dynamicconditions, especially for simple and accurate solutions for inflatable beams.This paper deals with the research on the natural frequencies of inflatableTimoshenko beams by an exact method: the continuous element method(CEM), and by the classical finite element method (FEM). The dynamicstiffness matrix D(ω) is here established for an inflatable beam; it depends onthe natural frequency and also on the inflation pressure. The stiffness and massmatrixes used in the FEM are deduced from D(ω). Natural frequencies andnatural modes of a simply supported beam are computed, and the accuracy ofthe CEM is checked by comparisons with the finite element method and alsowith experimental results.

Key words: Continuous element, mass matrix, stiffness matrix, followerforces, inflatable tubes, natural modes

1. INTRODUCTIONInflatable structures are increasingly used nowadays.They find applications in well-known structures (tires,boats) and in more recent applications, mainly in thespatial industry (inflatable antennas, shields, houses)or in civil engineering (buildings, temporary hospitals,inflatable scales). Other industrial sectors aredeveloping inflatable structures. Inflatable wings areaeronautical applications. The naval industry isinterested in many different structures: boats,inflatable laths to rigidify the sails and inflatablelanding stages. Environmental protection is concernedwith inflatable dams. Medical inflatable prosthesis arebeing increasingly used.

The growing interest for inflatable structures comesfrom the large amount of advantages that they offer:

they are light, easy to fold, quite easy to built, easy todeploy. In the spatial industry, the focus is put on theirability to attain large sizes for a small storage volumeand to present low specific mass and an interestingreliability. For these applications, the inflationpressure is quite low. Earthly structures are mainlyused for temporary applications, but durable ones arealso built nowadays: autonomous or in associationwith metallic reinforcements (tensairity systems [1]).Generally, they need greater pressure than for spatialapplications and may become very strong structures ifthe pressure is large enough. All these developmentsrequire engineers to be able to calculate theirmechanical behaviour.

The type of material used, the geometricalcharacteristics of the beams, and the internal pressurelead to the strength and stiffness of the inflatable

1

structure. Several types of fabrics are used. They areoften manufactured from high modulus fibres such asKevlar, Vectran. For instance, Vectran was used for theMars Pathfinder [2]. Polyester fabrics with a PVCcoating are more often used for inflatable structures incivil engineering: Ferrari pre-stressed fabric is one ofthem. Materials are robust and can be packed anddeployed many times with minimal degradation inperformance.

Inflatable structures are built from elementary partslike tubes, panels, cones or tori. Due to the geometryof these elements, the strength of materials is welladapted to the study of these structures in order topredict their behaviour. A beam approach gives usefultools for easy model assembly of longitudinalelements. Several authors have worked on staticanalysis. Comer and Levy [3] have proposed an EulerBernoulli formulation for an inflatable cantileverbeam. Main and al. [4] have realized experiments andcompared their results to Comer and Levy’s theory. Infact, the effect of the pressure isn’t obtained explicitlyby using the usual beam theory. Clearly, thedeflections of fabric tubes are dependent on theinternal pressure, on the fabric’s constitutive law, onthe initial geometry and on the external loads. Fichter[5] used the minimization of the global potentialenergy with Timoshenko’s beam assumptions and wasthe first to put forward a theory for the deflectionsincluding the pressure effects. Thomas [6] hasconstructed a model for the tubes, which lead the wayto quite complicated formulas and to a non-symmetrical stiffness matrix. Even if the comparisonbetween theoretical and experimental results havegiven evidence of a good accuracy of these formulas,the choice has been made here to use the recent LeVan’s approach [7] who obtained simpler static beamformulations, which leads to bending formulas wherethe inflation pressure appears also explicitly in therigidities of the beam. In fact, Le Van’s study is animprovement on Fichter’s. Equations are written bythe use of the virtual work principle in lagrangianvariables with finite rotations. This approach has beenextended in this paper by introducing the dynamicterms in order to predict the influence of the pressureeffects on the natural frequencies.

There are only a few papers which deal withtheoretical studies on the dynamics of inflatablebeams with a strength of materials approach.Membrane or shell analysis is generally used. Forinstance, the vibrations of inflatable dams (rubberdams) have lead to several studies where the authors

mainly use displacement finite element computations[8]. Recently, Jha and al. [9] have suggested a studyfocused on the vibrations of an inflatable torus madeof kapton dedicated to the spatial industry, and themodel used was based on a shell theory. Concerningthe beam approach, Main and al [10] put forward workdealing principally with the damping mechanisms; anEuler-Bernoulli theory was used again. In fact theexperimental results clearly show the dependency ofthe frequencies on the pressure. The beam dynamictheory that will be established in this paper follows LeVan’s work on static analysis. We have chosen to usethe continuous element method to analytically obtainthe dynamic stiffness matrix because it allows an exactvibration analysis of the problem. The most popularcomputational method used in structural dynamic isthe finite element method. In this work, the stiffnessmatrix and the mass matrix used in the FEM have alsobeen calculated.

The physical intuition is that increasing the pressurein a structure leads to an increase of the materialproperties. In order to take into account the effects ofthe inflation pressure and the shear deformation, aTimoshenko beam theory has been used in this paper.The internal pressure is the pre-stress effect: it allowspre-stressing the walls and inducts the initial geometryand stiffness. The second special feature comes fromthe fact that the force inducted locally follows thewalls and will be denoted by the follower force effect.The equations of motion therefore have to be writtenon the current configuration, which explains thechoice of the lagrangian approach. This means that thefollower load effect due to the pressure can be takeninto account properly, which is of importance for thesekinds of structures.

The framework of this paper is the following: in thefirst part, the differential equations of motion areobtained by using the virtual work principle written inlagrangian variables following [7]. In the second part,the dynamic stiffness matrix of an inflatable beam isbuilt by connecting the values of the loads and thedisplacements at the two ends of the beam. FEMstiffness and mass matrixes are deduced from thedynamic stiffness matrix. In the fourth part, thestiffness and mass matrix are directly derived from theFEM and the coherence with the continuous elementanalysis is verified. Then the natural frequencies arecalculated for an inflatable simply, supported beam byusing the CEM. Comparisons are finally made withexperimental results and show the accuracy of thetheory.

2

2. THE DIFFERENTIAL EQUATIONS OFMOTIONInflatable structures are made of pressurized fabrics.The inflation pressure leads to a pre-stress state in thewalls of the structure, which ensures its stiffness. Theshear deformations of these thin-walled structuresmust not be overlooked and one must useTimoshenko’s beam kinematics because the cross-section doesn’t remain orthogonal to the neutral fibre(see Fig 2 in [6]). Fig.1 shows a straight uniformTimoshenko beam model.

The use of the virtual work principle written inlagrangian variables leads to differential equations ofmotion where the effect of the inflation pressureappears. This principle is written as follows:

(1)

U denotes the displacement field and U* the virtualdisplacement field. Π is the first Piola-Kirchhoff stresstensor. ρ0 is the density in the reference configuration,Ω0 is the volume of the element in the referenceconfiguration and δΩ0 its boundary. N is the outwardnormal unit in the reference configuration, and f0 thebody force per unit reference volume. In thefollowing, details are expressed for the virtual work ofthe inertia terms. The principle of calculus and thecorresponding results for the other virtual works(internal virtual stress work, external virtual deadloads work, external virtual work due to the pressure)have just been condensed. Details can be found in [7].

2.1 The virtual work of the inertia termsThe cross-section is assumed to have a rigid bodydisplacement. In a right-handed rectangular Cartesiancoordinate system, Fig. 1 shows an inflatable tube oflength 0 in the reference configuration and a currentcross section on the reference and in its currentconfiguration. The displacement of any point P0 of thecross section comes from:

x

y

z G0

P0

Y

Z..

.P

U(X,t)

V(X,t)

θ(x,t)

G

X

0

curent configuration

referenceconfiguration

Figure 1. reference and current configuration

0*

00*

0*

0*T3*

ddS

ddgrad:)(,R

00

00

Ωρ

ΩΩ

ΩΩδ

ΩΩ

UUNΠU

UfUΠU 0

⋅=⋅⋅+

⋅+−∈∀

∫∫

∫∫

3

(2)

Here, U(G0, t) = (U(X, t), V(X, t), 0) is the displacement of the centre of the current cross-section, and Rcorresponds to the rotation of the section.

The virtual displacement field is chosen as follows:

(3)

where U*(G0) = (U*(X), V*(X), 0), Ω* = (0, 0, θ*(X)) and where the particularity comes from GP whichcorresponds to the final configuration. This gives in lagrangian variables:

(4)

The virtual work of the inertia terms is:

(5)

where (6)

This leads to:

(7)

By introducing the second moment of area of the beam cross section about the z axis I, and considering thatthe first moment of area is nil, the virtual work of the inertia terms is:

(8)∫ ∂∂

+∂∂

+∂∂

=0

0 2

2*

2

2*

2

2*

00*acc dX

tI

t

VV

t

UUSW

θθρ

∫ ∫ ∂∂+

∂∂−

∂∂−

∂∂−

∂∂+

∂∂−

∂∂−

+∂∂+

∂∂=

0

0

0S

*2

22

*2

*2

2*

2

2

*2

*2

2*

2

2

2

2*

2

2*

0*acc dXdS

tY

Vcost

Vsint

sint

V

Usint

Ucost

cost

U

Yt

VV

t

UUW θθ

θθθθθθ

θθθθθθρ

0

sinYV

cosYU

.

0

cost

Ysint

Yt

V

sint

Ycost

Yt

U

)P(t

)t,P( **

**2

2

2

2

2

2

2

2

2

2

0*

20

2θθθθ

θθθθ

θθθθ

−−

∂∂−

∂

∂−∂

∂∂∂+

∂

∂−∂

∂

=∂

∂U

U

0*

2

2

0*ac d

tW

0

ΩρΩ

UU ⋅

∂

∂= ∫

Z

cosY

sinY

Z

Y

0

100

0cossin

0sincos

: θθ

θθθθ −

=−

== 00 PGRGP

GPΩUU ∧+= *0

*0

* )G()P(

( )

0

)cos1(YV

sinYU

Z

Y

0

000

01cossin

0sin1cos

0

V

U

.)t,G()t,P( 0.

0

θθ

θθθθ

−−−

=−−−

+=

−+=++== 00PGR

0000 PGIRUGPGGGPPPU00

4

2.2 Strain tensor, stress tensor, constitutive lawThe Green strain tensor is given by E = 1_2(H + HT + HT · H), where H = gradU is the displacement gradienttensor. One obtains:

(9)

(10)

The matrix of the second Piola-Kirchhoff stress tensor Σ is assumed to be a “beam” matrix:

(11)

The following generalized stresses can therefore be introduced:

(12)

which are respectively considered as the normal force, the bending momentum and the shear force acting on thecross-section in the reference configuration.

Different kinds of materials are commonly used to build inflatable structures. Thin isotropic membrane maybe used (example of the kapton in the spatial industry). Composites made of fabric embedded in a matrix materialare also used (temporary buildings are often made of polyester fibres and rubber coating). Even if this kind ofmembrane follows an orthotropic behaviour, the material is considered here as an hyperelastic isotropic Saint-Venant Kirchhoff material and the constitutive laws are:

(13)

∑0XX and ∑0

XY are the initial stresses. They are induced by the initial inflation with inducts pre-stress effects in thewalls of the structures. E and G are respectively the Young modulus and the shear modulus.

2.3 The virtual work of the internal forcesThe virtual work of the internal forces is:

(14)

Π is calculated by using the deformation gradient F: Π = FΣOn the assumption that the material is linear and elastic and by taking into account the existence of pre-

stressing, we have:

(15)

[ dXdX

d

XK

XKE

2

1

S

INsin

X

Vcos

X

U1M

sinX

Vcos

X

U1T

Xcos

X

V

Xsin

X

U1M

dX

dVcosT

XsinM

X

VN

dX

dUsinT

XcosM

X

U1NW

*0

2

0

0

*

*

0

**int

0

∂∂+

∂∂++

∂∂+

∂∂++

∂∂+

∂∂+−

∂∂

∂∂+

∂∂

∂∂+−+

+∂∂+

∂∂+−

∂∂+

∂∂+=− ∫

θθγθθθ

θθθθθθθ

θθθθθθ

0*T*

int d:)(W

0

ΩΩ

gradUΠ∫ −=

XY0XYXYXX

0XXXX E.G2E.E +=+= ΣΣΣΣ

∫ =

0S

0XX dSN Σ , ∫ −=

0S

0XX dSyM Σ and ∫ =

0S

0XY dST Σ

[ ]=000

00

0

XY

XYXX

ΣΣΣ

∑

∂∂+−

∂∂= θθ sin

X

U1cos

X

V

2

1EXY and 0EYY =

∂∂

∂∂−

∂∂

∂∂−

∂∂+

∂∂+

∂∂+

∂∂−

∂∂=

Xsin

X

VY2

Xcos

X

UY2

XY

X

V

X

U

2

1

XcosY

X

UE

22

22

XXθθθθθθθ

5

Here, and γ0 comes from the development of the initial stresses. The strain expression

allows writing(16)

2.4 The virtual work of the external loads Two kinds of load can be considered for these structures: dead loads (loads which always go in the samedirection) and follower loads due to the inflation pressure (which inducts local forces acting normally to thesurface, and following this surface from the reference configuration to the current configuration). The inflationpressure is considered here as an external force even if it is acting inside the structure.

The virtual work of the external loads (except the pressure) is given by:

(17)

Where T = Π.N and px, py, µ are respectively the in-plane components of the dead load and momentum perunit length; X(.), Y(.), Γ(.) are the components of the resultant force and the resultant momentum, which will leadto the boundary conditions.

Concerning the external loads due to the pressure, it is supposed that the beam is subjected to a uniformpressure p on the entire internal surface. The surface is broken down into two different parts: the lateral wallsand the surfaces at the end sections. In the following, the force due to the pressure acting on the end section inthe initial configuration will be noted: P = pS0.

Figure 2. following forces

Under the assumption that reference volume is a cylinder, the virtual work of the surface density pn iscalculated.

(18)=

PS

*pre dS.W n

* pU∫

)()()(V)(Y

)(U)(X)0()0()0(V)0(Y)0(U)0(XdX)VpUp(

dS))G()(t,P(d))G().(t,P(

dS)P().t,P(d)P().t,P(W

0*

00*

0

0*

0***

0

**Y

*X

S

000000

000000*ext

00

0

00

θΓ

θΓµθ

Ω

Ω

Ω

ΩΩ

++

++++++=

∧++∧+=

+=∂

GPUTGPU W Wf

UTUf

****0

**0∫ ∫

∫ ∫

∫

2XX YY γβαΣ ++= and 20000

XX YY γβαΣ ++= .

−=

0S0

20

04

S

IdSyK ∫

1e

3e

2e

Ω0

pn

lateral walls

end section

end section

Initial configuration

6

Sp corresponds to the lateral external surfaces and the two bases of the tube. After development (see [7]), thework can be written as follows:

(19)

2.5 Equations of motion and boundary conditions:The use of the virtual work principle written in lagrangian variables then leads to the following equations wherethe dynamic terms appear:

(20)

After part integration and considering the terms U*, V*, θ*, we can obtain the equations of motion:

(21)

(22)

(23)

With the boundary conditions:

(24)

µθθθγθ

θθθθθρ

=∂∂+−

∂∂−

∂∂+

∂∂+−

∂∂+

∂∂++

∂∂

∂∂−

∂∂+

∂∂−

∂

∂

))X

U1(sin

X

V(cosP

XK

XEK

2

1

S

IN

T)sinX

Vcos)

X

U1((sin)

X

VM(

Xcos))

X

U1(M(

XtI

X,

02

0

2

2

0

Y2

2

00 pX

cosP)cosT(X

)X

sinM(X

)X

VN(

Xt

VS =

∂∂+

∂∂−

∂∂

∂∂−

∂∂

∂∂−

∂

∂ θθθθθρ

X2

2

00 pX

sinP)sinT(X

)X

cosM(X

))X

U1(N(

Xt

US =

∂∂−

∂∂+

∂∂

∂∂−

∂∂+

∂∂−

∂

∂ θθθθθρ

*int

*pre

*ext

*acc WWWW ++=

00

0***

0

***pre sinVcosUPdX))

X

U1(sin

X

V(cos

XcosV

XsinUPW θθθθθθθθθ ++

∂∂+−

∂∂+

∂∂−

∂∂= ∫

)0(X)0(cosP)0(sin)0(T)0(X

)0(cos)0(M))0(X

U1)(0(N −=−−

∂∂+

∂∂+ θθθθ

)(X)(cosP)(sin)(T)(X

)(cos)(M))(X

U1)((N 000000000 =−+

∂∂+

∂∂+ θθθ

θ

θ

)0(Y)0(sinP)0(cos)0(T)0(X

)0(sin)0(M)0(X

V)0(N −=−+

∂∂+

∂∂ θθθ

)(Y)(sinP)(cos)(T)(X

)(sin)(M)(X

V)(N 000000000 =−+

∂∂+

∂∂ θθθθ

)0()0(X

K)0(X

EK2

1

S

I)0(N)0(

X

V)0(sin)0(M)0(cos))0(

X

U1)(0(M 0

2

0

0 Γθγθθθ −=∂∂+

∂∂++

∂∂+

∂∂+

() )(X

K)(X

EK2

1

S

I)(N)(

X

V)(sin)(M)(cos))(

X

U1)((M 00

00

2

0

00000000 Γθγθθθ =

∂∂+

∂∂++

∂∂+

∂∂+

7

In order to calculate analytical solutions, the same assumptions than those made in [7] are made. Afterlinearization, the reference and spatial configurations can be considered as the same approximately. Thecomputational viewpoint remains Lagrangian; however, use will be made of small characters (i.e. Euleriannotations) in order to designate the variable. The constitutive laws become:

(25)

here, k is the correction shear coefficient. For circular thin tubes, k =0.5.The equations of motion can therefore be deduced:

(26)

(27)

(28)

The boundary conditions are written for the generalized stresses:

(29)

(30)

(31)

(32)

(33)

(34)

3. CONSTRUCTION OF THE ELEMENTARY DYNAMIC STIFFNESS MATRIX3.1 Solutions for the differential equation of motionIn the following, we will consider an inflated tube submitted to bending local external loads and torques at itstwo ends, and to its internal pressure. Hence, we consider that px, py, µ are nil. Eqns (25), (29) and (30) give:

(35)P)(N)0(N 000 ==

P)(N)0(N 000 ==

)0()0(x

)S

I)0(NEI(

0

000 Γθ −=

∂∂+

)(Y)()kGSP()(x

V)kGS)(N( 000000

0 =+−∂∂+ θ

)0(Y)0()kGSP()0(x

V)kGS)0(N( 00

0 −=+−∂∂+ θ

)(XP)(N 000 =−

)0(XP)0(N 0 −=−

µθθθρ =−∂∂+−

∂

∂+−∂

∂)

x

V)(kGSP(

x)

S

INEI(

tI 02

2

0

0002

2

00

y02

2

00

2

2

00 px

)kGSP(x

V)kGSN(

t

VS =

∂∂++

∂

∂+−∂

∂ θρ

xx,0

2

2

00 pNt

US =−

∂

∂ρ

0NN = )x

V(kGST 0 θ−

∂∂=

xEIM 0 ∂

∂= θ

8

Figure 3. nodal forces and torques

By neglecting the effect of rotation inertia, the equation of motion for free vibrations can be obtained:

(36)

(37)

which allows to write, after some manipulations:

(38)

One assumes that the solution takes the shape:

(39)

The differential equation can be written as follows:

(40)

and leads to the usual result:

(41)2

0

00

002

2

0

00

4

4

2

2

v

S

PIEI

S

dx

vd

kGSP

S

dx

vd

gdt

gd

ωρρ

−=

+−⋅

+

=

0dx

vdg

dt

gd

dx

vd

kGSP

S

dt

gdv

S

PIEI

S4

4

2

2

2

2

0

002

2

0

00

00 =⋅+⋅⋅+

−⋅⋅+

ρρ

)t(g)x(v)t,x(V ⋅=

0x

V

tx

V

kGSP

S

t

V

S

PIEI

S4

4

22

4

0

002

2

0

00

00 =∂

∂+∂∂

∂⋅+

−∂

∂⋅+

ρρ

0)x

V)(kGSP(

x)

S

IPEI( 02

2

0

00 =−

∂∂+−

∂

∂+− θθ

0xx

V)kGSP(

t

VS

2

2

02

2

00 =∂∂−

∂

∂+−∂

∂ θρ

F1

F2Γ1Γ2

v(0)θ(0

( )0v

( )0A

B

x

y

θ

9

Harmonic variation of g with circular frequency ω is assumed then g(t) = cos(ωt – ϕ); then:

(42)

Let us consider a solution of eqn (42) in the exponential form:

(43)

Introducing solution eqn (43) into eqn (42) yields to the characteristic equation:

(44)

The roots can now be obtained:

(45)

or

(46)

Finally we can obtain the solutions:

(47)

(48)

(49)

λ1 and λ2 are functions of the circular frequency ω and of the internal pressure.

S

PIEI

S4)

kGSP

S(

kGSP

S

0

00

002

2

0

002

0

002

2+

++

++

−=ρωρωρω

λ2

1

S

PI2

1

EI

S4)

kGSP

S(

kGSP

S

0

00

002

2

0

002

0

002

1+

++

++=

ρωρωρωλ

11 is λ±= ; 22s λ±=

2

S

PIEI

S4)

kGSP

S(

kGSP

Ss

0

00

002

2

0

002

0

002

2

++

++

+−=

ρωρωρω

2

S

PIEI

S4)

kGSP

S(

kGSP

Ss

0

00

002

2

0

002

0

002

2

++

+−

+−=

ρωρωρω

0

SPI

EI

Ss

kGSP

Ss

0

00

002

2

0

002

4 =+

−⋅+

+ ρωρω

sxae)x(v =

0v

S

PIEI

S

dx

vd

kGSP

S

dx

vd

0

00

002

2

2

0

002

4

4=⋅

+−⋅

++ ρωρω

10

3.2 Expression of displacements in the matrix formThe expression of the transversal displacement and the cross-section rotation can be written as:

(50)

In eqn (50), A, B, C and D are undetermined constants. After reduction, one can obtain:

(51)

with

(52)

The values of the displacements and rotations at the ends of the beams are:- At the left-end x = 0:

(53)

(54)

- At the right-end x = 0:

(55)

(56)

We can now write the displacements in a matrix form:

(57)

where

(58)

and with UT = v1, θ1, v2, θ2

[ ]

−

=

)(ch)(sh)cos()sin(

(sh) )(ch)sin()cos(

00

0101

)(

022

21

022

21

011

22

011

22

02020101

2

21

1

22

λλλλ

λλλ

λλλ

λλ

λλλλλλ

λλ

ωW

[ ] )tcos(

D

C

B

A

)( ϕωω −= WU

)tcos(D)(chC)(shB)cos(A)sin( 020201012 ϕωλΘλΘλΛλΛθ −⋅⋅+⋅+⋅−⋅=

)tcos()(Dsh)(Cch)sin(B)cos(AV 020201012 ϕωλλλλ −⋅+++=

)tcos()DB(1 ϕωΘΛθ −⋅⋅+⋅−=

)tcos()CA(V1 ϕω −⋅+=

221 λλΛ −=⋅ ; 2

12 λλΘ =⋅

)tcos(D)x(chC)x(shB)xcos(A)xsin( 2211 ϕωλΘλΘλΛλΛθ −⋅⋅+⋅+⋅−⋅=

[ ]

[ ]

[ ][ ]

)tcos()x(Dsh)x(Cch)xsin(B)xcos(A)t,x(V 2211 ϕωλλλλ −·+++=

)()x(Dch)x(Csh)xcos(B)xsin(A)x(~

)x(Dch)x(Csh)xcos(B)xsin(A)x(~

)x(Dch)x(Csh)xcos(B)xsin(A)x(~

)tcos()x(~

kGSP

S)x(

~kGSP

S

PIEI

)x(~

)t,x(

2222211113

23

223

213

113

12

222211111

30

002

0

0

00

1

ωλλλλλλλλθ

λλλλλλλλθ

λλλλλλλλθ

ϕωθρθθθ

−·+++−=

++−=

+++−=

−·+

−+

++=

11

3.3 Expression of the external loads in the matrix formIn the same way, we will obtain the expression of the external loads at the ends of the beam by using the boundaryconditions (31-34):

- For the transversal forces F1 and F2:

(59)

- For the torques Γ1 and Γ2:

(60)

Substituting into ∂V∂x,

∂θ∂x

, θ and simplifying gives:

(61)

With

(62)

(63)

The values of the shear loads and bending momenta at the ends of beam are finally:At the left-end x = 0:

(64)

(65)

At the right-end x = 0:

(66)

(67)

The external loads are also written in matrix form:

(68) [ ] )tcos(

D

C

B

A

)( ϕωω −= ZF

)tcos(D)(shC)(chB)sin(A)cos()EI( 022

1022

1012

2012

2p2 ϕωλλλλλλλλΓ −⋅⋅+⋅+⋅−⋅−=

[ ] )tcos(D)(ch)(C)(sh)(B)cos(A)sin()(F 0210210120122 ϕωλλλλλλλλϑ −⋅⋅−+⋅−+⋅+⋅−=

)tcos()CA()EI( 21

22p1 ϕωλλΓ −⋅⋅+⋅−=−

)tcos()DB(F 121 ϕωλλϑ −⋅⋅−⋅=−

0

00P

S

PIEI)EI( += and )kGSP()kGS( 0p +=

p002 )EI(Sρωϑ =

10

00 )0(

x)

S

IPEI( Γθ −=

∂∂+ 20

0

00 )(

x)

S

IPEI( Γθ =

∂∂+

10 F)0()0(x

V)kGSP( −=−

∂∂+ θ 2000 F)()(

x

V)kGSP( =−

∂∂+ θ

[ ] )tcos(D)x(ch)(C)x(sh)(B)xcos(A)xsin()()x

V()kGS( 21211212p ϕωλλλλλλλλϑθ −⋅⋅−+⋅−+⋅+⋅−=−

∂∂

)tcos(D)x(shC)x(chB)xsin(A)xcos()EI(x

)EI( 22

122

112

212

2pp ϕωλλλλλλλλθ −⋅⋅+⋅+⋅−⋅−=∂∂

12

where

(69)

and with FT = F1, Γ1, F2, Γ2

3.4 The dynamic stiffness matrixThe dynamic stiffness matrix [D(ω)] is given by:

(70)

and [D(ω)] = [Z(ω)][W(ω)]–1

(71)

The dynamic stiffness matrix is symmetrical, and its components are:

(72)

[ ]=

44

3433

242322

14131211

D

DDsym

DDD

DDDD

)(ωD

( )[ ]=

2

2

1

1

2

2

1

1

V

V

wF

F

θ

θΓ

ΓD

[ ]

−−

−−−−

−

=

)(sh)EI((ch)EI() )sin()EI()cos()EI(

)(ch)(sh)cos()sin(

0)EI(0)EI(

00

)(

022

1p022

1p012

2p012

2p

021021012012

21p

22p

12

λλλλλλλλ

λϑλλϑλλϑλλϑλλλ

ϑλϑλ

ωZ

[ ])(sh)sin()(d

DD 023

1013

22

22

121

3113 λλλλλλλϑλ ++−==

[ ])(ch)cos()(d

DD 02012

22

1

22

21

4114 λλλλλϑλ −+−==

[ ])(sh)cos()(ch)sin()(d

)EI(D 0201

320201

31

22

21

21p22 λλλλλλλλ

λλ−+=

[ ])(ch)cos()(d

DD 02012

22

1

22

21

3223 λλλλλϑλ −+==

)(ch)sin()(sh)cos()(d

D 02013

202013

12

22

121

11 λλλλλλλλλϑλ ++=

)1)(ch))(cos(()(sh)sin(d

DD 02013

2123

102014

24

121

2112 −−++== λλλλλλλλλλλϑλ

13

where

The inflation pressure appears in the equations and each term of the matrix D is a transcendental function of ω. Simple changes of coordinates give the matrixes needed in order to compute the bending natural frequencies

and mode shapes of inflatable tubes. This leads to the non-linear eigenvalue problem:

(73)

Natural frequencies can be calculated by the use of the well-known algorithm defined by Wittrick andWilliams [11], which enables the calculation of the number of natural frequencies of the structure that exist belowan arbitrary chosen trial frequency. It allows us to calculate the natural frequencies with a chosen accuracy.Numerous papers deal with the determination of natural frequencies and natural modes of beams with the CEMwith accuracy ([12], [13]). In this paper, we have preferred to focus our attention on two points: the first is thecoherence of the results with the conventional finite element method; the second point is concerned with theinfluence of the inflation pressure in the results which is of great importance for inflatable structures.

4. F.E. STIFFNESS MATRIX AND MASS MATRIXThe aim of this section is to deduce FEM stiffness and mass matrixes dedicated to inflatable tubes from thedynamic stiffness matrix.

4.1 From CEM to FEM:Free vibrations may be calculated by the use of the dynamic stiffness matrix, which leads to the non-lineareigenvalue problem:

(74)

This problem may be rewritten as follows:

(75)

The mass and stiffness matrixes are obtained with the help of Leung’s theory [14]:

(76)

then

(77)( ) ( ) ( )ωωωω MDK 2+=

( ) ( )2ωωω

∂∂−= D

M

( ) ( )( ) 0=UM-K 2 ωωω

( ) 0UD =ω

( ) FUD =ω

)1)(ch)(cos(2)(sh)sin()(d 02013

23

102016

26

1 −−−= λλλλλλλλ

[ ])(sh)sin()(d

)EI(DD 02

3201

31

22

21

21p4224 λλλλλλ

λλ−+−==

[ ])(ch)sin()(sh)cos()(d

D 02013

202013

12

22

121

33 λλλλλλλλλϑλ ++=

( )[ ])1)(ch))(cos(()(sh)sin(d

DD 02013

2123

102014

24

121

4334 −−++−== λλλλλλλλλλλϑλ

[ ])(sh)cos()(ch)sin()(d

)EI(D 0201

320201

31

22

21

21p44 λλλλλλλλ

λλ−+=

14

Free vibrations may also be studied by the use of the finite element method. The displacement field isdiscretized:

(78)

where N is the shape function matrix, and V is the vector of generalized co-ordinates. The FEM leads to the lineareigenvalues problem:

(79)

The FEM stiffness and mass matrixes will then be calculated with:

(80)

(81)

4.2 The stiffness and mass matrix obtained with the dynamic stiffness matrixBy calculating the limit of [D(ω)] when ω→0, one obtains [K].

(82)

where ΦP has been introduced for convenience:

(83)

The pressure P appears explicitly in the stiffness matrix. If the pressure tends to 0, one can recognize theTimoshenko beam stiffness matrix. This stiffness matrix can be deduced from the results derived by Wielgosz[15], by replacing E*I* by EIo+PIo/So and pbh by P+kGS0.

The limit of M(ω) with ω→0, gives the mass matrix M:

(84)

200

0

00

P)kGSP(

)S

PIEI(12

+

+=φ

[ ]

+

−

−−+

−

+

+=

)4(sym

612

)2(6)4(

612612

)1(

)S

PIEI(

p20

0

p200p

20

00

p30

0

00

φ

φφ

φK

( ) ( )200 limlim

ωωω ωω

∂∂−== →→D

MM

( )ωω DK 0lim →=

( ) 0=UM-K 2ω

[ ] )t()x()t,x(v VN=

[ ]

++

++−++

++−++++

++−++++++

+

=

20

2PP

02PP

2PP

20

2PP0

2PP

20

2PP

02PP

2PP0

2PP

2PP

2P

00 0

120

1

60

1

105

1sym

24

1

120

11

210

11

3

1

10

7

35

13120

1601

1401

241

403

42013

1201

601

1051

24

1

40

3

420

13

6

1

10

3

70

9

24

1

120

11

210

11

3

1

10

7

35

13

)1(

S

φφ

φφφφ

φφφφφφ

φφφφφφφφ

φρ

M

15

The inflation pressure also appears in the mass matrix. If P tends to 0, the mass matrix also becomes theTimoshenko beam mass matrix.

4.3 The stiffness matrix by the left quotient operation:The foundations of this algebraic operation can be found in [16]. The diagram given figure 4 can draw the leftquotient operation.

Figure 4. the left quotient operation

The continuous element is described by a “mechanical element”: the two vector spaces V (displacement space)and Φ (loads space) are dual for a bilinear form <.;.>1, which represents the work of a load ϕ ∈ Φ in adisplacement v ∈ V. The load law L (application from V into Φ) summarizes the equilibrium equations andthe boundary conditions. This has already been written for an inflatable panel [17] element submitted to nodalconcentrated loads. The vector spaces are:

(85)

The static load law and the load vector are:

(86)

with ϕT = F1, Γ1, F2, Γ2The principle of the left quotient operation is to describe the equilibrium element by another “mechanical

element”. Φ1 is the discreet subspace of the loads space vector and contains the nodal loads (here Φ1 = R4). If jis the canonical injection from Φ1 into Φ, then V/Φ1

0 is the quotient space (i.e. the subspace of the nodaldisplacements, V/Φ1

0 = R4). ϕ~ ∈ Φ1 represents the nodal loads (F1, Γ1, F2, Γ2) and v~ ∈ V/Φ10 represents the

nodal displacements (v1, θ1, v2, θ2). These two vector spaces are dual for the bilinear form <.;.>2, work of thenodal loads in the nodal displacements:

(87)

The principle of the left quotient operation is to solve the following problem: if the nodal loads ϕ~ ∈ Φ1 aregiven, find the nodal displacements v~ ∈ V/Φ1

0.The displacement v~ ∈ V is obtained by integration of the equilibrium equations. The nodal displacements

are therefore given by v~ = jt(v). The result gives the compliance of the element v~ = (jt o L–1 o j)(ϕ~). Ifthe nodal displacements v~ ∈ V/Φ1

0 are given, find the nodal loads ϕ~ ∈ Φ1. The result now gives the stiffnessmatrix of the element: ϕ~ = (jt o L–1 o j)–1(ϕ~)

We have to inverse the compliance given by the first problem, but it always leads to a singular system (because

)v~(jv)~(j,v~,v~),/V()~,v~(1t

12101

−∈∀>=<><→∈ ϕϕΦΦϕ

( ) ( ) ,)0()0(dx

dvPSkGS,

dx

d

dx

vd,

dx

dv)kGSP(

dx

dEI),v(L 002

2

02

2

0 −+−−−++= θθθθθ

( ) ( ) ( )−+− )(dx

dEI,)()(

dx

dvpSkGS),0(

dx

dEI 0000000

θθθ

[ ]( )R;,0CV 04= [ ]( ) 4

0 RR;,0C ×=Φ

V

V/

j

<.,.>1 Φ

T↓ ↑ j

Φ1o <.,.>2 Φ1

16

of the rigid body displacements) and the building of the stiffness matrix needs other equations, which come fromthe equilibrium equations.

Following the process detailed in [17], one obtains:

(88)

Concerning the inflatable tubes, the load law is obtained with the static equilibrium equations obtained byelimination of the dynamic terms in eqns (36-37-59-60):

(89)

Hence, an obvious analogy allows us to obtain the exact stiffness matrix of the inflatable tube. Only the

bending rigidity changes (just replace EI0 by ). This proves that the stiffness matrix obtained by

calculating the limit of [D(ω)] is the exact stiffness matrix.

4.4 The FEM stiffness and mass matrix The left quotient operation doesn’t allow the calculation of the mass matrix. Once the coherence between FEMand CEM is verified for the stiffness matrix, the usual method is used to propose shape functions and calculatethe stiffness and mass matrix by the use of the virtual work principle.

After linearization, by neglecting the distributed loads from eqn (17), the following expression for the flexionof an inflatable beam in dynamics is obtained:

(90)

The use of the constitutive equations allows us to write the virtual work principle in terms of displacements,and after some manipulation, the equation may be written under a simple and practical form:

(91)

+−−∂∂+=+++ ∫ )

dx

dV)(

x

V)(kGSP()()(VF)0()0(VF *

0

*

00*

20*

2*

1*

10

θθθΓθΓ

dxt

IVt

VSdx

dx

d

xS

PIEI

0

0

*2

2

00*

2

2

00

*

0

00 ∫ ∂

∂+∂∂+

∂∂+ θθρρθθ

=+−∂∂+

∂∂−++++ ∫ 00

0

0**

0

*0

*20

*2

*1

*1 VPdx)

x

V(

xVP)()(VF)0()0(VF θθθθθΓθΓ

dxdx

d

xS

IPMT

dx

dV)T

x

VP(dx

tI

t

VVS

*

0

0*

0

*

0

*2

2

002

2*

000 θθθθθρρ

∂∂++−++

∂∂+

∂∂+

∂∂ ∫ ∫

0

00

S

PIEI +

( ) ,)0()0(dx

dvpSkGS,

dx

d

dx

vd,

dx

dv)kGSpS(

dx

d

S

PIEI),v(L 002

2

002

2

0

00 −+−−−+++= θθθθθ

( ) +−++− )(dx

d

S

PIEI,)()(

dx

dvpSkGS),0(

dx

d

S

PIEI 0

0

000000

0

00

θθθ

[ ]

+

−

−−+

−

+=

)4(sym

612

)2(6)4(

612612

)1(

)EI(

p20

0

p200p

20

00

p30

0

φ

φφ

φK with

200

0P

)kGSP(

)EI(12

+=φ

17

The virtual work principle for static analysis can then be written, which is used here to obtain the stiffnessmatrix:

(92)

The equations of motion are easily obtained by elimination of the dynamic terms in eqns (36-37).For an Euler Bernoulli beam, the displacement is written as follows:

(93)

where the shape functions are the usual cubic expansions:

(94)

The relation is valuable for an Euler Bernoulli beam. It allows us to obtain the shape functions for the section’srotation. For inflatable beams, which are Timoshenko’s beams, the cross-section does not remain orthogonal tothe neutral fibre. Hence dv/dx ≠ θ. The preceding functions interpolate displacements and the derivative ofdisplacements, which is now written as:

(95)

where the shape functions Nv′1(x) and Nv′

2(x) are:

(96)

In this case, the following boundary conditions are, of course, checked:

(97)'2020

'11 v)(

dx

dVv)(Vv)0(

dx

dVv)0(V ====

−=00

2

vx

1x

)x(N '1

−−=00

2

vx

1x

)x(N '2

[ ][ ] VN v~

)x(

v

v

v

v

)x(N)x(N)x(N)x(N)x(V

'2

2

'1

1

vvvv '22

'11

==

+−=0

2

01v

x21

x1)x(N −=

0

2

02v

x23

x)x(N

−=00

2

1x

1x

)x(Nθ −−=00

2

2x

1x

)x(Nθ

[ ] [ ] VN==

)t(

)t(v

)t(

)t(v

)x(N)x(N)x(N)x(N)x(V

2

2

1

1

vv 2211

θ

θθθ

dxdx

d

dx

d

S

PIEI)

dx

dV)(

dx

dV)(kGSP()()(VF)0()0(VF

*

0

00

*

0

*

00*

20*

2*

1*

10 θθθθθΓθΓ ++−−+=+++ ∫

18

By changing the unknowns V~ = v1, v ′1, v2, v ′2 to the unknowns V = v1, θ1, v2, θ2, the equilibriumequations lead to:

(98)

where

(99)

The boundary conditions give the linear system:

(100)

The following relation is then easily obtained:

(101)

Thus:

(102)

And the shape functions can be calculated by:

(103)

The shape functions depends on the inflation pressure, the Young modulus and the shear modulus:

(104)

( )( )( )

( )( )

( )( )( )

( )( )( )P

20

0P02

P20

0P001

P30

20P0

2

2vP

30

20

20P0

20

1v

1

x2xx

2

1)x(N

1

2x2xx

2

1)x(N

1

x3x2x)x(N

1

xx2x)x(N

φφ

φφ

φ

φ

φ

φ

θθ+

+−=+

−−−=

+

−−−=

+

−−−−=

[ ][ ] VAN=)x(V

[ ] VA=

+−−

−+−

+=

2

2

1

1

P

0

PP

0

P

P

0

PP

0

P

P'2

2

'1

1

v

v

21

2

010022

1

0001

1

1

v

v

v

v

θ

θ

φφφφ

φφφφ

φ

+−−

−+−

+=

2

2

1

1

P

0

PP

0

P

P

0

PP

0

P

P'2

'1

v

v

21

2

221

1

1

v

v

θ

θφφφφ

φφφφ

φ

[ ]=

'2

2

'1

1

422

1

v

v

v

v

Mθθ

200

0

00

P)kGSP(

)S

PIEI(12

+

+=φ

)x(dx

Vd

12)x(

dx

dV)x(

dx

Vd

)kGSP(

)S

IPEI(

)x(dx

dV)x(

3

3P

2

3

3

0

0 0

00 φθ +=+

++=

19

The boundary conditions allow the calculation:

(105)

The virtual work of the internal loads can then be written as:

(106)

And by use of the shape functions:

(107)

The stiffness matrix of the inflatable tube is then calculated, which is nothing else than the stiffness matrixcalculated from the dynamic stiffness matrix:

The use of the shape functions now allows the calculation of the mass matrix:

(108)

(109)

The term gives this mass matrix. The calculation of the different terms of the

matrix is simplified by the use of formal calculus software. Finally, the mass matrix and the stiffness matrixobtained here are exactly the same as those given by eqns (82-84), which shows the coherence of thesedevelopments.

5. SOLUTIONS FOR A SIMPLY SUPPORTED INFLATABLE BEAMThe theory is now applied to a simply supported inflatable beam. The boundary conditions are:

(110)0)t,0(V = 0)t,(V 0 = 0)t,0(x

=∂∂θ 0)t,(

x0 =

∂∂θ

[ ] [ ]dx)x(N)x(NS0

0

T00 ∫ ρ

[ ] [ ] [ ] [ ]=∂∂ ∫ ∫ 2

2

0

TTT*00

0 2

2*

dt

VdAdx)x(N)x(NAVS00 S dx

t

VV

00

ρρ

[ ][ ] )t(VA)x(N)t,x(V =

[ ]

+

−

−−+

−

+

+=

)4(sym

612

)2(6)4(

612612

)1(

)S

PIEI(

p20

0

p200p

20

00

p30

0

00

φ

φφ

φK

[ ] Vdxdx

dN

dx

dNN

dx

dNN

dx

dN)

12(AV

S

PIEIW

Tv

0

Tv

P20

*

0

00

* 0

+−−+= ∫ θθθθ

φδ

dxdxd

dxd

)dxdV

)(dxdV

)(12

(SPI

EIS

dxdx

d

dx

d

S

PSEIS)

dx

dV)(

dx

dV)(kGSP(

**

0

*

P200

00

*

0

00

*

0

*

0

0

0

θθθθφ

θθθθ

+−−+=

++−−+

∫

∫

+++−

+−=−

2

2

1

1

p

p0

p

p

p

p0

p

p

v

v

121121dx

dV

θ

θφ

φφ

φφ

φφ

φθ

20

The natural frequencies of vibration of an inflatable simply supported beam are solutions of:

(111)

One gets:

(112)

Then

(113)

Hence, with d ≠ 0, one can obtain:

(114)

We find:

(115)

The natural frequency can finally be calculated by the use of eqn (48):

(116)

The inflation pressure appears explicitly in the formulation of the frequency, which increases with thepressure.

Figure 5. natural frequencies of a simply supported inflatable tube: influence of the internal pressure

Fig 5 shows the evolutions of the first three natural frequencies of an inflatable tube versus pressure. Here, 0

= 1,858m, R0 = 0,0831m, ρ0e = 0,3759kg.m–2. Two Young moduli and three shear moduli are used here to show

E=150000 Pa.m

G=20000 Pa.m

G=20000 Pa.m

G=20000 Pam

G=50000 Pam

G=50000 Pa.m

G=50000 Pam

G=80000 Pa.m

G=80000 Pa.m

G=80000 Pam

0

20

40

60

80

100

120

140

50 70 90 110 130 150

Pressure (kPa)

Fre

qu

ency

(H

z)

E=200000 Pa.m

G=20000 Pam

G=20000 Pam

G=20000 Pam

G=50000 Pam

G=50000 Pam

G=50000 Pa.m

G=80 00 Pa.m

G=80 00 Pa.m

G=80000 Pa.m

0

20

40

60

80

100

120

140

50 70 90 110 130 150

Pressure (kPa)

Fre

qu

ency

(H

z)

n=1

n=2

n=3

n=1

n=2

n=3

0

20

2200

0

00

4000

2

n

kGSP

nS

S

PIEI

S

1

2

nf

++

+

=πρρ

π

01

nπλ ±= or 02 =λ

0)sin( 01 =λ or 0)(sh 02 =λ

0)1)(ch)(cos(2)(sh)sin()()(sh)sin( 02013

23

102016

26

10201 =−−− λλλλλλλλλλ

0)(sh)sin(()(d

EI

)(sh)cos()(ch)sin()(d

EI

2

023

2013

12

22

1210

2

02013

202013

12

22

1210

=−+−

−−+

λλλλλλ

λ λλλ λ

λ λ λ

λλ

λλ

0DD

DD

4442

2422 =

21

their influence on the solution. Increasing the pressure inducts an increase of the natural frequencies, but thesegrowths are quite small and depend on the natural frequency. Multiplying by three the pressure leads in this caseto an increase of the frequencies by 2.7%, 6% and 9% respectively for n=1, 2, 3 in the most varying cases. Infact, the first frequency is mainly dependent on the Young modulus. For a given Young’s modulus, the shearcoefficient G has a great influence on the interval between the natural frequencies.

6. COMPARISONS WITH THE FINITE ELEMENT METHOD RESULTSThe first three natural frequencies are calculated with the two methods. The CEM values are simply obtainedwith eqn (116). The three first frequencies are calculated for the simply supported inflatable tube, which is usedfor the experiments (see section 7). Geometry and material are such that:

0 = 1,858m, R0 = 0,0831m, Ee = 179000 Pa.m, Ge = 20000 Pa.m, ρ0e = 0,3759kg.m–2.

Figure 6. Evolution of frequency according to the number of element reference

Figure (6) shows that the results of FEM converge with the results of CEM with the increase of the numberof elements. If the CEM gives the exact frequency with a chosen accuracy, it is not the case of the FEM,However, the results of FEM are always greater than the ones of the CEM, as expected.

7. COMPARISONS WITH THE EXPERIMENTAL RESULTSExperiments have been carried out to measure the natural frequencies of an simply inflatable supported beam.Sewing two layers of fabric manufactures the beam used for these experiments. This fabric is a woven network,for which the fibbers are initially orthogonal and imply an orthotropic behaviour. Because we focus the studieson beam models, it will be considered here as an hyperelastic isotropic Saint-Venant Kirchhoff material. A thinPVC membrane incorporated inside the beam ensures the air tightness. The vibration of the beam comes froman electromagnet. The induction coil has been separated form the electromagnet. It is very light and just stickedon the beam.

17,56

17,57

17,58

17,59

17,6

17,61

17,62

17,63

17,64

17,65

17,66

0 20 40 60 80 100number of elements

firs

t fr

equ

ency

(H

z)

f1_FEM

f1_CEM

60

60,5

61

61,5

62

62,5

63

63,5

64

0 20 40 60 80 100number of elements

seco

nd

fre

qu

ency

(H

z)

f2_FEM

f2_CEM

100

110

120

130

140

150

160

170

180

190

200

0 20 40 60 80 100number of elements

thir

d f

req

uen

cy (

Hz)

f3_FEM

f3_CEM

22

Figure 7. Simply supported beam experiment

Figure 8 presents the results for the first three natural frequencies and the evolution of the length and radiusof the inflated beam versus the pressure. The notable point is that increasing the pressure leads to an increase inthe rigidity of the beam, which inducts the natural frequency to grow.

a) natural frequencies versus pressure b) length and radius versus pressureFigure 8. Experimental results

In order to compare experiments and theoretical results, the Young modulus has to be identified. This is donewith the variation of the initial geometry, which clearly shows the importance of the pres-tress state. It is easy tocalculate the circumferential and longitudinal stresses in the membrane due to the internal pressure of the beamin the initial configuration:

(117)

e corresponds to the thickness of the fabric. Since we consider an elastic isotropic material, the length and theradius may be modelled with:

(118)

Here, φ and Rφ are the length and the radius for a very low internal pressure, and can be obtained with thecurves Fig. 8. b with the linear interpolations of the two curves length versus pressure and radius versus pressure.The Young modulus is also determined with these linear interpolations.

+=Ee2

pR10

φφ and +=

Ee

pR1RR0

φφ

e

pR0c =σ and

e2

pR0=σ

f1

f2

f3

0

20

40

60

80

100

120

140

40 90 140 p (kPa)

freq

uen

cy (

Hz)

lenght (m)

1,81

1,82

1,83

1,84

1,85

1,86

1,87

1,88

1,89

40 90 140 P (kPa)

0,08

0,081

0,082

0,083

0,084

0,085

0,086

0,087

radius (m)

23

Figure 9. pre-stress effect

The average value of the product of the Young modulus with the thickness e is estimated to Ee=178973 Pam.The Young modulus is the main influent parameter for the first natural frequency. The product Ge is thenestimated in order to ensure the interval between the other natural frequencies (here Ge=20000 Pam). Thecomparison between theoretical and experimental results is shown fig 10 a. The analytical results are close to theexperimental results for the first pressure (50 kPa) but the slopes of the theoretical curves are smaller than theexperimental ones. This can be corrected by the use of the influence of the pressure on the radius and on thelength on the natural frequencies. The initial state is then correctly defined. Fig 10. b presents the comparisonbetween the experimental frequencies and the theoretical frequencies taking into account the variation of thelength and radius versus pressure, and the results show a correct accuracy between theory and experiment.

a) without pre-stress effect b) with pre-stress effectFigure 10. Comparison between theoretical and experimental results

8. CONCLUSIONSInflatable structures are increasingly used nowadays. They are often made of beam elements. In this paper, twomethods to calculate the eigenvalues and eigenmodes of structures made of inflatable beams have beenpropounded. The equations of motion have been written in lagrangian variables for a Timoshenko beam in orderto take into account the following forces due to the internal pressure and the shear behaviour of this kind of thin-walled fabric structure. We first have dealt with the continuous element method to establish the dynamic stiffnessmatrix, which allows us to get exact results. The finite element method stiffness and mass matrixes have beenderived from the CEM and also directly from the virtual work principle. In each case, the inflation pressureappears in the matrixes. The natural frequencies have been calculated for a simply supported inflatable beam withthe dynamic stiffness matrix, and these results have been compared to FEM and to experimental ones. Theinfluence of the material parameters has been pointed out. The first frequency is mainly dependent on the Youngmodulus and for a given Young’s modulus, the shear coefficient has a great influence on the interval between thenatural frequencies. The comparison between the experimental results and the theoretical results has shown theimportance of the pre-stress effect, which increases the radius and the length of the initial configuration.

f1_exp.

f2_exp.

f3_exp

f1_theo.

f2_theo.

f3 theo.

0

20

40

60

80

100

120

140

40 90 140 p (kPa)

freq

uen

cy (

Hz)

f1_exp.

f2_exp.

f3 exp.

f1_theo.

f2_theo.

f3_theo.

0

20

40

60

80

100

120

140

40 90 140 p (kPa)

freq

uenc

y (H

z)

φ

0

φR 0R

p 0

Initial configuration pression p

24

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Computationnal Methods in Applied Sciences andEngineering, ECCOMAS 2004

2. Cadogan,D., Sandy, C. and Grahne, M., Developmentand evaluation of the mars pathfinder inflatable airbaglanding system, Acta Astronautica, 2002, 50, 633-640.

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11. Wittrick, W.H. and Williams, F.W., A general algorithmfor computing natural frequencies of elastic structures,Quarterly Journal of Mechanics And AppliedMathematics, 1971, 24, 263-284

12. Banerjee, J.R. and Williams, F.W., Exact dynamicstiffness matrix for composite Timoshenko beams withapplications, Journal of sound and vibration, 1996, 194,573-585

13. Banerjee. J. R. and Su. J., Development of dynamicstiffness matrix for free vibration analysis of spinningbeams, Computers and structures, 2004, 82, 2189-2197.

14. Richards, T.H. and Leung, Y.T., An accurate method instructural vibration analysis, Journal of sound andvibration, 1997, 55, 363-376

15. Wielgosz, C. and Thomas, J.C., An inflatable fabricbeam finite element, Communications in NumericalMethods in Engineering, 2003, 19, 307-312.

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17. Wielgosz, C.,Thomas, J.C. and Le Van, A., Equilibriumfinite elements for inflatable beams, 6th WorldConference on Computational Mechanics, Bejing, 2004

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