1ière - Nonlinear Dynamics at the Free University...

1ière Rencontre-Workshop de Vibration et Acoustique Marrakech, 30 Juin-01 Juillet 2011

2

Mot des organisateurs

La première et la deuxième rencontre nationale de vibration et d’acoustique ont été organisées en 2009 et 2010 à Casablanca. Ces deux rencontres ont permis de rassembler des chercheurs, enseignants et doctorants marocains venus de plusieurs universités du royaume autour des thèmes de recherche en relation avec les problèmes de vibration et d’acoustique. La réussite de ces rencontres a incité les participants à élargir ces rencontres aux industriels et acteurs socio-économiques opérant dans des secteurs liés aux problèmes de vibration et de bruit. Dans ce contexte, la première rencontre-workshop de vibration et d’acoustique est organisée à Marrakech les 30 juin et 01 juillet 2011. L’objectif principal de cette rencontre-workshop est de rassembler la communauté scientifique marocaine, les doctorants et les industriels travaillant dans le domaine de vibration et d’acoustique dans le but de faire le point sur les avancées scientifiques et technologiques récentes dans ce domaine. Une place importante sera donnée à la réflexion et aux discussions sur les orientations futures de la formation et de la recherche dans le domaine de vibration et d’acoustique. Les thèmes de la rencontre traiteront, entre autres, les techniques analytiques, numériques et expérimentales pour l’analyse de la dynamique des systèmes et des structures, les méthodes de diagnostique vibratoire, le parasismique, l’acoustique, la pathologie des grandes structures, le contrôle actif des systèmes, les phénomènes de résonance, le contrôle des instabilités dans les systèmes électromécaniques, les vibrations des systèmes à multiple degrés de liberté, les interactions modales, la propagation des ondes, les fissures dans les poutres forcées, la nanotechnologie, … Cette rencontre-workshop est aussi une opportunité pour les doctorants d’assister à des conférences données par des chercheurs internationaux qui aborderont des sujets d’actualité dans le domaine de la dynamique et vibration des structures. De plus, pour enrichir les travaux de cette rencontre, des industriels et des responsables de sociétés opérant dans le domaine de vibration seront invités.

Les organisateurs Juin 2011


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COMITE D’ORGANISATION

v BELHAQ MOHAMED (Président) FS – Aïn chock v BEIDOURI ZITOUNI EST – Casablanca v BOUKSOUR OTMANE EST – Casablanca v LAKRAD FAOUZI FS – Aïn chock

COMITE SCIENTIFIQUE

v AZOUANI ABDERRAHIM FU Berlin v AZRAR LAHCEN FST – Tanger v BELHAQ MOHAMED FS – Aïn chock v BENAMAR RHALI EMI – Rabat v BEIDOURI ZITOUNI EST – Casa v BOUKSOUR OTMANE EST – Casa v BOUTYOUR E. FST – Settat v DUFOUR REGIS INSA – Lyon v FAHSI ABDELHAK FST – Mohammedia v ICHCHOU NAJIB ECL – Lyon v LAKRAD FAOUZI FS – Aïn chock v RAHMOUN M. FST – Meknès

OBJECTIFS

Cette rencontre-workshop a pour buts de :

v Communiquer les avancées récentes dans les domaines de vibration et d’acoustique.

v Permettre un échange direct des connaissances entre doctorants, chercheurs,

professionnels et industriels.

THEMES

v Analyse et diagnostic vibratoire v Contrôle des bruits et des vibrations v Contrôle de santé des structures v Acoustique pour le bâtiment v Dynamique et Vibration non linéaires v Tests expérimentaux en vibration et acoustique v Analyse modale expérimentale v Santé et sécurité


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Programme de la 1ière Rencontre-Workshop de Vibration et Acoustique

Jeudi 30 Juin 2011 09h00-09h15 : Ouverture et mot de bienvenu (M. Belhaq) 09h15-09h45: N. Ichchou. Contrôle de santé de structures complexes: methodologies et mises en oeuvre. 09h45-10h15: A. Khamlichi. Calcul sismique des bâtiments; aspects fiabilité et optimisation de la réhabilitation. Pause Café 10h45-11h15: Kjell Ahlin. Toolbox for simulation and parameter identification of nonlinear mechanical systems. Déjeuner 15h00-15h30: T. Belhoussine Drissi. Evaluation non destructive par ondes guides ultrasonore à la junction de deux plaques élastiques de nature différente en presence d’un défaut. 15h30-16h00: A. Eddanguir. Brief comments on nonlinear vibrations of continuous and discret systems : theoretical and experimental approaches to the concept of nonlinear mode shapes. Pause Café 16h30-17h00: A. Khamlichi. Validité de la méthode de Juang pour prédire le potentiel de liquéfaction des sols de Tanger. 17h00-17h30: A. Adri. Geometrically nonlinearmode shapes and resonant frequencies of multi-span beams. 17h15-18h00: O. Debbarh. Acoustique pour le bâtiment. Vendredi 01 Juillet 2011 09h00-09h30: R. Dufour. Modèle de Dahl generalise et applications industrielles. 09h30-10h00: A. Rahmouni. Geometrically nonlinear free transverse vibration of 2-dof systems involving the coupling between axial and transverse displacements, comparison between the discrete and the continuous models. Pause Café


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10h30-11h00: H. M. Abdelali. The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin fully clamped isotropic skew plate. 11h00-11h30: E. Merrimi. Non-linear free vibrations of laminated composite beams.. 11h30-12h00: T. Ecuert et I. E. Zidane. Présentation de Gunt et du système de diagnostic de machines PT500. 12h00-12h15: Clôture du workshop. Déjeuner


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Participants à la 1ière Rencontre-Workshop de Vibration et Acoustique Adri Ahmed [email protected] EST Casablanca

Alaoui Lemrani Kamal [email protected] SOGELAB

Beidouri Zeitouni [email protected] EST Casablanca

Belhaq Mohamed [email protected] FS Aïn Chock

Belhoussine Drissi Taoufiq [email protected] FS Aïn Chock

Bouksour Otmane [email protected] EST Casablanca

Debbarh Omar [email protected] FS Aïn Chock

Ecuert Tanja [email protected] GUNT

Eddanguir Ahmed [email protected] EMI Rabat El Bikri Khalid [email protected] ENSET Rabat

Fahsi Abdelhak [email protected] FST Mohammedia

Guennoun Mohamed [email protected] EST Casablanca

Hlimi Said [email protected] CI2D

Ichchou Najib [email protected] Ecole centrale de Lyon

Khamlichi Abdellatif [email protected] FS Tétouan

Kirrou Ilham [email protected] FS Aïn Chock

Kjell Ahlin [email protected] BTH, Sweden

Lakrad Faouzi [email protected] FS Aïn Chock Merrimi El Bekkaye [email protected] ENSET Rabat

Mokni Lahcen [email protected] FS Aïn Chock

Moulay Abdelali Hanane [email protected] EMI Rabat

Rahmouni Abdellatif [email protected] EST Casablanca

Régis Dufour [email protected] INSA Lyon

Rougui Mohamed [email protected] EST Salé

Souhar Khalid [email protected] FS Aïn Chock

Tri Abdeljalil [email protected] FS Aïn Chock

Yagoubi Mohamed [email protected] FS Aïn Chock

Zidane Issam Eddine [email protected] SOGELAB

Zine Abdelmalek [email protected] Ecole Centrale de Lyon

Zerbane Khalid [email protected] EST Casablanca

mailto:[email protected]































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Table des matières A. Rahmouni, Z. Beidouri & R. Benamar

Geometrically Nonlinear Free Transverse Vibration of 2-dof Systems involving the coupling between Axial and Transverse Displacements: Comparison between the discrete and the continuous models

9

K. El bikri, E. Merrimi, R. Benamar

Non linear free vibrations of laminated composite beams

12

H. M. Abdelali & R. Benamar The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin fully clamped isotropic skew plate

16

A. Eddanguir, Z. Beidouri & R. Benamar

Brief comments on nonlinear vibrations of continuous and discrete systems: Theoretical and experimental approaches to the concept of nonlinear mode shapes

20

Kjell Ahlin Toolbox for simulation and parameter identification of nonlinear mechanical systems

28

A. Adri, Z. Beidouri, M. EL kadiri & R. Benamar Geometrically non linear mode shapes and resonant frequencies of multi-span beams

32

T. Belhoussine Drissi, B. Morvan, M. Predoi, J. L. Izbicki & P. Pareige Evaluation non destructive par ondes guidées ultrasonore à la jonction de deux plaques élastiques de nature différente en présence d’un défaut

39

A. Khamlichi Seismic performance reliability assessment for irregular reinforced concrete buildings

43

N. Touil, A. Khamlichi, A. Jabbouri & M. Bezzazi Validité de la méthode de Juang pour prédire le potentiel de liquéfaction des sols de Tanger

48

R. Dufour Generalized Dahl’s model and application to nonlinear mechanical components

53

M. N. Ichchou Contrôle de santé de structures complexes : méthodologies et mises en œuvre

59

O. Debbarh Acoustique pour le bâtiment

60


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9

GEOMETRICALLY NONLINEAR FREE TRANSVERSE VIBRATION OF 2-DOF SYSTEMS INVOLVING THE COUPLING BETWEEN AXIAL AND TRANSVERSE DISPLACEMENTS: COMPARISON BETWEEN THE DISCRETE AND THE CONTINUOUS MODELS

A. Rahmouni

1, Z.Beidouri

1 and R. Benamar

2

1 Laboratoire de Mécanique Productique & Génie Industriel (LMPGI)

Ecole Supérieure de Technologie - Université Hassan II Ain Chock KM 7 Route El Jadida, Casablanca, Maroc

[email protected] [email protected]

2 Laboratoire des Etudes et Recherches en Simulation, Instrumentation et Mesures (LERSIM) Ecole Mohammadia des Ingénieurs - Université Mohammed V

Avenue Ibn Sina, Agdal, Rabat, Maroc [email protected]

Abstract

Large transverse vibration amplitudes of beams induces a significant geometrical non-linear behaviour due to the axial displacements and strains, which are usually neglected in linear theory, corresponding to small displacements. This has been shown both theoretically and experimentally in [2]. The purpose of the present work is to develop the analogy between the experimental studies of beam, and a simple 2 dof system model made of two masses and four spiral springs, similar to those recently used in [3], in addition to three axial springs representing the geometrical nonlinearity. In [1], details are given corresponding to definition of the mass, rigidity and non-linear rigidity tensors, usually encountered in previous works, dealing with non-linear vibration of structures of various types and geometry. The analogy between continuous beams and the discrete model, leading to the expressions for the equivalent spiral and axial stiffness is also presented. Some numerical results are given, showing the amplitude dependence of the frequencies with the amplitude of vibration and are compared to the backbone curves obtained previously by the experimental studies.

Keywords: Nonlinear transverse vibration, Two-degrees-of-freedom, Displacement Basis, Modal Basis, Discrete system, Hamilton’s Principle, Spectral analysis.

1. Introduction Is has been shown both theoretically and experimentally, for example in [2], that large vibration amplitudes of beams induce a significant geometrical non-linear behaviour due to the axial displacements and strains, which are usually neglected in linear theory, valid only for small displacements. The purpose of the present work is to develop the analogy between the experimental studies, and a simple 2 dof system model made of two masses and four spiral springs, in addition to three axial springs representing the same geometry as in [3]. In [1], details are given, corresponding to definition of the mass, rigidity and non-linear rigidity tensors, usually encountered in previous works, dealing with non-linear structural vibration. The analogy between continuous beams and the discrete model is developed here, leading to the expressions for the equivalent spiral and axial stiffness. Some numerical results are given, showing the amplitude dependence of the frequencies with the amplitude of vibration and are compared to the backbone curves obtained previously by the experimental studies of beam.



10

21

2/ / 1

/i i

dy dyd y dx i dx i

l ldx i

θ θ −− −−= =

11 1 2 3 22 2 3 4 12 21 2 32 2 2

1 1 2( 4 ) ( 4 ) ( )k C C C k C C C k k C C

l l l

−= + + = + + = = +

2 2 2 21 1 2 2 1 3 3 2 4 4 3

1 1 1 1( ) ( ) ( )

2 2 2 2Vs C C C Cθ θ θ θ θ θ θ= + − + − + −

Figure 1: Modelisation of a beam in flexion vibration.

The two masses m1 and m2 represent the inertia of the beam, and the two displacements y1 and y2 represent the two-degrees-of-freedom [1], the spiral springs C1 and C4 represent the limit conditions, the spiral springs of stiffness C2 and C3 represent the beam flexural rigidity, while k1, k2 and k3 represent the beam resistance to the axial load. The analogy between the experimental studies of continuous beams and the discrete model is developed here, in order to obtain the expressions for the equivalent spiral and axial stiffness.

2. Calculation of stiffness Ci for a continuous clamped-clamped beam The linear potential energy stored in the four spiral springs, subjected to the rotations shown in

Figure 1 can be written as:

(1)

On the other hand, the elementary bending potential energy in a continuous beam is given by:

22

212b

d ydV EI dx

dx

=

(2)

Which may be discretised using the finite difference technique, leading to:

(3)

After replacing in (2) the expression for the second derivative given in (3), the summation of the terms

corresponding to the four springs and the identification with equation (1) leads to:

32 3EI EI

C Cl L

= = = (4)

The values of C1 and C4 for which the non dimensional frequency curve of the model are closer to experimental one are:

25*1 4EI

C CL

= = (5)

In practice the frame whose stiffness is 25 times the stiffness of the embedded object, is considered of infinite stiffness. It gives us an idea about the robustness of the connection fitting of the experimental device. Now, in order to express the linear rigidity tensor kij in terms of the beam to characteristics, we recall the following expressions established in [1]:

(6)

y C 3 C nl 2

C2 2

C nl 1

C4 C 1 0

k1

k2

k3


11

Replacing C1, C2, and C3 and C4 by the values found, i.e. equation (4), the stiffness matrix is given by:

3

30 4[ ]

4 30EI

Kl

− = −

(7)

3. Continuous and discreet results in the non linear case The expressions for bijkl have been calculated in [1]. Experimental work was carried out on a clamped-clamped beam, of dimensions 2 x 20 x 580 mm, made of Aluminium DTD 5070 (E=67000 MPa, ρ=2700 kg/m3. ν=0.34). In [2], the dimensional frequency has been plotted experimentally. This allowed us after sketching the backbone curve for the first mode shape of a discreet system (Figure 2) and to compare the results.

0 0.5 1 1.5 2 2.5 3

x 10-3

1

1.1

1.2

1.3

1.4

1.5

1.6

Amplitude (m)

no-linear adimensional frequency

D-B

C-B

EXPERIMENTAL

Figure 2. Comparison of the non dimensional frequency of the discreet model by using the Benamar method, with experimental results and continuous clamped-clamped beam by Benamar method. The results of the discrete system are supervised by two references [2, 4].

4. Conclusion The model proposed in the reference [3] has been improved by assuming that the beam has a finite axial stiffness which has been calculated using the discreet continuous analogy. The geometrical non linear effect has been estimated by calculating the higher order components of the strains. The results of the discrete system are supervised by two references [2, 4]. The rigorous mathematical relations established for determining the parameters of the model constitute a solid basis for treating with confidence the free vibration system with N degrees of freedom [3], using the nonlinearity coefficients calculated corresponding to the characteristics of the beam.

References [1] A. Rahmouni, Z. Beidouri and R. Benamar, Geometrically nonlinear free transverse vibration of 2-dof systems involving a coupling between axial and transverse vibrations, 3th Conference on Nonlinear Science and Complexity, NSC10, 28-31 July (2010). [2] M. M. K. Bennouna. and R. G. White, The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam. Journal of Sound and Vibration (1984) 96(3) pp 309-331. [3] A. Eddanguir Z. Beidouri and R. Benamar, Geometrically Nonlinear Transverse Vibrations of Discrete Multi-Degrees of Freedom Systems with a Localised Non-Linearity. International Journal of Mathematics and Statistics, Spring, (2009). Volume 4, Number S09. [4] R. Benamar, M. M. K. Bennouna, and R. G. White, The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures, part I: Simply supported and clamped-clamped beams. Journal of Sound and Vibration (1991) 149 pp 179-195.


12

NON LINEAR FREE VIBRATIONS OF LAMINATED COMPOSITE BEAMS

K. El bikri a, E. Merrimi a, R. Benamar b

a Université Mohammed V-Souissi, Ecole Normale Supérieure de l'Enseignement Technique Rabat,

LaMIPI, B.P. 6207, Rabat Instituts, Rabat, Morocco b Université Mohammed V-Agdal, Ecole Mohammadia d’Ingénieurs, LERSIM, Av. Ibn Sina, Agdal,

Rabat, Morocco

Abstract In recent year, many papers dealing with the non linear dynamic behaviour of beams with variable cross-sectional properties such as sandwich beams, functionally graded beams and laminated composites beams have been published. The purpose of the present paper is to show that the problem of geometrically non linear free vibration of symmetrically and asymmetrically laminated composite beams with immovable ends can be reduced to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness parameters. The theoretical model of the proposed formulation is based on Hamilton’s principle and spectral analysis using the governing axial equilibrium equation of the beam in which the axial inertia and damping are ignored. Iterative solutions are presented to calculate the fundamental non linear frequency parameters and the associated non linear mode shape 1. Introduction

The study of large amplitude vibration of the composite beams which are widely used in civil and mechanical engineering applications such as nuclear reactors, aircraft, building slabs. In many cases, these beams are subjected to relatively large amplitude vibrations with respect to their thickness, which may lead to the material fatigue and structural damage. These phenomena become more significant around the natural frequencies of the structure. Therefore, the non-linear vibration analysis is essential for a reliable design. For these reasons this study has attracted much interest from researchers because such vibration must be considered in designing resonance free composite structural components. Due to the complexity of the problem, it is difficult to obtain exact analytical solutions for non-linear vibration of composite beams and plates. As far as we know, researchers have concentrated on experimental investigation, approximate analytical [1,2] and the finite element method [3]. In the present study we apply the theoretical model developed in [4] to analyze the geometrical non-linear free dynamic response of symmetrically and asymmetrically laminated composite clamped-clamped beams in order to investigate the effect of non-linearity on the non-linear resonance at large vibration amplitudes. The general formulation of the model for non-linear vibration of laminated composite beam at large vibration amplitudes is presented first, and we discuss results of the relationships between the non-linear resonance frequency ratio and the vibration amplitudes.

2. Theory and formulation

In this section a composite beam having the geometrical characteristics is considered, as shown in figure 1.

Figure 1. Laminated composite beam

x

l bh

y z


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In large amplitude vibrations of beams, the axial strain εxa and the curvature Kx are defined as:

212

ax

u wx x

ε ∂ ∂ = + ∂ ∂ ,

2

2x

wK

x∂

=∂

(1,2)

where u and w are the axial and the transverse displacements respectively. The axial resultant force Nx and the bending moment Mfy are related to εx

a and Kx respectively by:

11 11( ) ax x x x

s

N E z dS bA bB Kε ε= = +∫ (3)

11 11( ) a afy x x x

s

M zE z dS bB bD Kε ε= = +∫ (4)

E and S are Young's modulus and the area of the cross-section of the composite beam respectively. A11, B11, and D11 account for the extensional, coupling and bending parameters rigidities respectively, the values of which can be calculated using the classical laminated composite plates theory. The total elastic strain energy VT of the beam is:

0

12

la

T x x fy xV N M K dxε= +∫ (5)

Which can be written in terms of axial displacement ua and transverse displacement wa as: 22 2 22 2

11 11 112 20 0 0

1 1 1 1 12

2 2 2 2 2

l l la a a a a a

T

u w w u w wV bA dx bB dx bD dx

x x x x x x

∂ ∂ ∂ ∂ ∂ ∂ = + + − + + ∂ ∂ ∂ ∂ ∂ ∂

∫ ∫ ∫ (6)

The equilibrium axial equation for composite beam, once in plane inertia and damping are ignored, reduces to:

0xNx

∂=

∂ i.e 11 11 tana

x x xN bA bB K cons tε= + = (7,8)

Immovably axial end conditions (0) ( ) 0a au u l= = , leads to:

2 211 11

20 02 2

l la a

x

bA bBw wN dx dx

l x l x ∂ ∂

= − ∂ ∂ ∫ ∫ (9)

Then equation (6) becomes:

( ) ( )22 22

20 08 2

l la aeff eff

T

ES EIw wV dx dx

l x x

∂ ∂ = + ∂ ∂ ∫ ∫ (10)

where ( ) 11effES bA= and ( ) 11

2

1111

( )eff

BEI b D

A= + are the effective axial and bending stiffness

respectively. The expression of the total strain energy is effective for replacing the laminated composite beam problem with equivalent classical isotropic beam problem [4]. The kinetic energy is given by:

2

02

l aS wT dx

tρ ∂

= ∂ ∫ (11)

In which axial and rotary inertia are ignored, since they are expected to be small compared to the transverse inertia. For general parametric study, we use the following non dimensional formulation by putting:

* xx

l= , * w

wr

= , where I

rS

= is the radius of gyration.

Upon assuming harmonic motion and expanding the displacement w in the form of a finite series:

( )* * * * *, ( )sin( )i iw x t a w x tω= , i = 1...n (12) * *( )iw x are the linear mode shape of the beam given in [4].


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The discretised total strain energy expression VT can be written as: * 2 * * 4 *1 1

sin ( ) sin ( )2 2T i j ij i j k l ijklV a a k t a a a a b tω ω= + (13)

The discretised kinetic energy is given by:

2 * 2 *1cos ( )

2 i j ijT a a m tω ω= (14)

Where k*ij is the non dimensional classical rigidity tensor, b*

ijkl is the non dimensional non linearity tensor and m*

ij stands for the mass tensor. These non dimensional tensors are defined in [4]. The dynamic behaviour of the laminated composite beam is governed by Hamilton’s principle which leads to the following set of non linear algebraic equation:

2* * * *30

2 i j r ijkr i ir i ira a a b a k a mω+ − = , r = 1..n (15)

Where ω2 is given by:

2

* *

**

32i j ij i j k l ijkl

i j ij

a a k a a a a b

a a mω

+= (16)

3. Numerical results and discussions In present work, a graphic-epoxy laminated beam that has a length of 0.25m, a width of 0.01m and height of 0.001m has been considered. The beam has six layers of uniform thickness with the following material properties: E1= 155 GPa, E2= 12.1 GPa, ν12= 0.248, G12=4.4 GPa and ρ= 1560 Kg/m3. In figure 2 the mid span displacement of the beam is depicted as function of the fundamental non linear frequency in the case of symmetric and asymmetric cross-ply lamination scheme. It is noted that the bending-extension coupling due to the extension-bending coefficient B11, which vanishes in the case of symmetric layup orientations, affects considerably the non linear frequency of asymmetric laminated beam.

Figure 2. Back borne curve of laminated composite clamped- clamped beam.

4. Conclusion In this work the geometrically non linear free vibrations of symmetrically and asymmetrically laminated composite beams with clamped-clamped axial immovable ends have been investigated. In the theoretical formulation, based on Hamilton’s principle and multimode approach, the axial equilibrium equation of the composite beam has been used to reduce the problem under consideration to that of simple isotropic beam. it has also been shown that the extension-bending coupling affects


15

the amplitude-frequency response of asymmetric laminated beams in comparison with symmetric laminated ones. References

1. Harras B., Benamar R., Geometrically non linear free vibration of fully clamped symmetrically laminated rectangular composite plates, Journal of Sound and Vibration (2002) 251(4), 579619.

2. Harras B., Benamar R., White R.G., Experimental and theoretical investigation of the linear and the non linear dynamic behavior of a glare 3 hybrid composite panel, Journal of Sound and Vibration (2002) 252(2), 281-315.

3. Jagadish Babu Gunda, Gupta R.K., Ranga Janardhan G., Venkateswara Rao, Large amplitude vibration analysis of composite beams: Simple closed-form solutions, Composite Structures 93 (2011) 870–879.

4. Benamar R., Bennouna K. and White R. G. The e!ects of large vibration amplitudes on the fundamental mode shape of thin elastic structures. Part I: simplysupported and clamped}clamped beams. 1991 Journal of Sound and Vibration 149,179-195


16

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN FULLY CLAMPED ISOTROPIC SKEW PLATE

H.M. ABDELALI AND R.BENAMAR Ecole Mohammadia d’Ingenieurs, Rabat, Morocco

Abstract

The present work concerns the nonlinear dynamic behavior of fully clamped skew plates at large vibration amplitudes. The large amplitude free vibration problem is modeled by a set of non-linear algebraic equations using Hamilton’s principle and spectral analysis. Numerical details are presented and results are given, corresponding in the linear case to various values of θ, and in the non linear case to θ=0°, and various amplitudes of vibration. 1. Introduction In contemporary structures, skew plates are used in diverse domains, such as stiffened plate, used as floors in bridges, ship hulls, buildings, etc. Simulation of static and dynamic behaviour of skew plates is an interesting area of work for researchers. Alwar and Rao (1973) presented a non-linear analysis of orthotropic skew plates of constant thickness subjected to a uniform transverse load. Chia (1980) considered the moderately large deflection elastic behaviour of homogenous isotropic and laminated anisotropic rectangular as well as skew plates by analytical methods. Xiang et al. (1995) studied the elastic buckling behaviour of skew Mindlin plate under a shear load. In the present study, we apply the theoretical model developed in [1] to analyze the geometrically nonlinear free dynamic response of an isotropic fully clamped skew plate in order to investigate the effect of non linearity on the non linear resonance frequency at large vibration amplitudes. In the present paper, the general formulation of the model for non linear vibration of isotropic skew plates at large vibration amplitudes is presented first, and the relationships between the non linear resonance frequency ratio and the vibration amplitudes is given for values of the angle θ varying from 0° to 45°. 2. Theoretical formulation

A skew plate with skew angle θ is shown in Figure 1. For the large displacement formulation, it is assumed that the material of the plate is elastic, isotropic and homogeneous. The thickness of the plate is considered to be sufficiently small so as to avoid the effect of shear deformation. Also the stress and strain measures are based on the original dimensions of the plate.

Figure 1. Skew plate in x-y andξ-η co-ordinate system.

The theoretical formulation of the skew plate large vibration amplitude problem is presented, which leads to a numerical model. By assuming harmonic motion and expanding the transverse displacement in the form of finite series of functions, the bending strain energy Vb, the axial strain energy Va and the kinetic energy T are expressed. By discretisation of the expressions for Va, Vb and T, the tensors mij,


17

kij and bijkl are defined. The frequency equation at large vibration amplitudes is developed. Finally, the non dimensional formulation of the problem and details of the numerical solution are presented.

2.1 Expression for the bending strain, axial end kinetic energies in the skew plate Consider the transverse vibration of the skew plate shown in figure 1 having the following characteristics: a, b, S: length, width and area of the plate; x-y: plate co-ordinates in the length and the width directions; ξ-η,H : Skew plate co-ordinate and thickness of the plate; E, ν: Young’s modulus and Poisson’s ratio; D, ρ: bending stiffness of the plate and mass per unit volume of the plate. The potential energies due to bending and axial strains, and the kinetic energy are given in x-y co-ordinate by: (1) , (2, 3) The skew co-ordinates are related to the rectangular co-ordinates by: ξ=x-y tanθ ; η=y/cosθ The potential energies due to bending and axial strains and the kinetic energy are given in ξ-η co-ordinate by: (4) (5,6) , Where: dA=cosθ dξdη and W (ξ,η,t) is the transverse displacement function. The plate functions used in the discretisation process are obtained as product of beam functions. The transverse displacement can be written as:

Where:

After the discretisation, the tensors kij, bijkl and mij are obtained as:

(7)

(8)

(9)

dSSy

w

x

w

yx

w

y

w

x

wD

bV .))

2

2

2

2(

2)

2)((1(2

2)

2

2

2

2[( ]

2

1∫=

∂

∂

∂

∂−

∂∂

∂−+

∂

∂+

∂

∂νννν

dAAwwwwww

Db

V .))]2

2

2

2(

2)

2((

2cos

)1(22)

2sin2

2

2

2

2(

4cos

1[

2

1∫=

∂

∂

∂

∂−

∂∂

∂−+

∂∂

∂−

∂

∂+

∂

∂

ηξηξθ

ν

ηξθ

ηξθ

dSS y

w

x

w

H

D

aV .]

2)

2)(

2)[(

22

3∫=

∂

∂+

∂

∂

twtw ωηξηξ sin),(),,( =

),(),( ηξηξi

wi

aw =

ηξξη

θηηξξξη

θηηξξθ

ddlwkwSinlwkwlwkwjwiw

Sinjwiwjw

Aiw

H

D

ijklb ).2)(2

3cos

1

2

3(

∂

∂

∂

∂−

∂

∂

∂

∂+

∂

∂

∂

∂

∂

∂

∂

∂−

∂

∂

∂

∂+

∂

∂∫

∂

∂=

∫=A

dWiWjdHij

m ηξθρ cos

dSt

wHST .

2)(

2∫=

∂

∂ρρρρ

dAAwwww

H

DVa .]2

)sin22

)(2

)((4cos

1[

22

3∫=

∂∂

∂∂−

∂

∂+

∂

∂

ηξθ

ηξθdAA t

whT .2)(

2∫

∂∂

=ρρρρ

ηξξθ

ξηηξηηξξθ ∂∂

∂

∂

∂−

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂∫

∂

∂=

jwiwSin

jwiwjwiwjwiwjwA

iwD

ijk

2

2

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3cos

1 (

ηξηξθ

ηηξθ

ξηξθ

ηξηθ

∂∂

∂

∂∂

∂+

∂

∂

∂∂

∂−

∂

∂

∂∂

∂−

∂∂

∂

∂

∂−

jwiwSin

jwiwSin

jwiwSin

jwiwSin

222

42

222

2

222

2

2

22

ηξθνηξηξηξ

ddj

wi

wjw

iw

)))2

2

2

2

()

22

((2cos)1(2∂

∂

∂

∂−

∂∂

∂

∂∂

∂+ −


18

And use of Non dimensional parameters defined by: (10)

(11)

Where: ; ; The dynamic behavior of the isotropic skew plate is governed by Hamilton’s principle which leads to the following set of non linear algebraic equation: Where: ; (12) 3. Numerical details

In the present work, calculation has been made using 9 plate functions obtained as product the 1st three clamped-clamped beam functions, and values of θ equal to 0°, 15°, 30°and 45°. Two types of validations have been performed: 1) The linear results for various values of θ . 2) The non linear results obtained here and in previous works for θ=0°. The results, summarized in Tables 1 and 2 show a satisfactory agreement.

Table 1. Frequency parameters for a C-C-C-C Plate

For values of skew angle, θ, deg, of

0 15 30 45 Leissa [1] 36.11 36.67 38.15 40.08

Present Work 35.99 36.52 37.92 39.72

a

b=α ),(),(),( ∗∗∗=∗= ηξηξ

ηξ iHWba

iHWWi ∗= ξξa

∗=ηηb

∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂−∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂+∗∂

∗∂∗∂

∗∂∗∂

∗∂∫ ∗=∗

∗

∗

∂∂

ξηξξθα

ηηξξα

ξξξα

ξlwkwjwiw

Sinlwkwjwiwlwkwjw

Aijklbiw 3

224

3

∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂−∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂+∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂+

ξηηηθα

ηηηηξξηηα

lwkwjwiwSin

lwkwjwiwlwkwjwiw2

2

∗∗∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂+∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂−∗∂

∗∂∗∂

∗∂∗∂

∗∂∗∂

∗∂− ηξ

ξηξηα

ηηξηθα

ξξξηθα dd

lwkwjwiwSin

lwkwjwiwSin

lwkwjwiwSin

2242

32

∗∗∗∫∗

∗=∗ ηξ ddw j

Ai

wijm

n1r,0232 …==∗∗−∗+∗ir

mi

aijkr

bk

aj

ai

air

ki

a ω

∗

∗+∗

=∗

ijm

ja

ia

ijklb

la

ka

ja

ia

ijk

ja

ia

ω 24cos4

2 ∗= ωθρ

ωb

D

θρ

ω 2cos

2

Db

∗∂

∗∂

∗∂

∗∂

∗∂

−∗

∂

∗∂

∗∂

∗∂

+∗∂

∗∂

∗∂

∗∂

+∗∂

∗∂

∗∂

∗∂

+∗∂

∗∂

∫ ∗

∗∂

∗∂

=∗

ηξξ

θ

ξηηξηηξξ

αααα jwiwSin

jwiwjwiwjwiwjwA

iwijk

2

2

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

23224

∗∂

∗∂

∗∂

∗∂

∗∂

∗∂

+∗∂

∗∂

∗∂

∗∂

∗∂

−∗∂

∗∂

∗∂

∗∂

∗∂

−∗

∂∗

∂

∗∂

∗∂

∗∂

−ηξηξ

θ

ηηξθ

ξηξθ

ηξη

θ αααα jwiwSin

jwiwSin

jwiwSin

jwiwSin

222

42

222

2

222

2

2

22

23

∗∗

∗∂

∗∂

∗∂

∗∂−

∗∂∗∂

∗∂

∗∂∗∂

∗∂+ − ηξθαν

ηξηξηξdd

jwiwjwiw))

2

2

2

2()

22((

22cos)1(2

θρ

ω 2cos

2

Db


19

Table 2. Ratio frequency parameters to the first non-linear mode shape for a C-C-C-C Plate (α=1, θ=0°)

R. BENAMAR [2] Present Work w*

max ω*nl/ω*

l

w*max ω*

nl/ω*l

0.2461 1.0105 0.2000 1.0105 0.4904 1.0411 0.4005 1.0411

References [1]. A. W. LEISSA 1969 Vibrations of plates. (NASA-SP160). Washington DC: U.S Government printing office. [2]. R. BENAMAR, M.M. BENNOUNA and R.G. WHITE 1992 Journal of sound and Vibration 164(2), 295-316. The effect of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part II: fully clamped rectangular isotropic plates.


20

BRIEF COMMENTS ON NONLINEAR VIBRATIONS OF CONTINUOUS AND DISCRETE SYSTEMS THEORETICAL AND EXPERIMENTAL APPROACHES TO THE CONCEPT

OF NONLINEAR MODE SHAPES

A. Eddanguir 1, Z. Beidouri 2 and R. Benamar 1

1LERSIM, Ecole Mohammadia des Ingénieurs - Agdal, Rabat, Maroc,

Email: [email protected], and [email protected] 2LMPGI Ecole Supérieure de Technologie - Route El Jadida, Casablanca, Maroc.

Email: [email protected]

1. Introduction Real structures are complex and are made of deformable components, the mechanical description of which, in terms of displacements, strains, stresses, or load distributions, requires in general use of continuous functions of one, two or three variables, leading to models with infinite numbers of degrees of freedom. However, it is sometimes reasonable, and fully justified for given practical purposes, to describe a real structure as a discreet system with a finite number of degrees of freedom (MDOFS). This may simplify considerably both the mathematical analysis and also the physical understanding of the mechanical behavior of the system under examination [1]. The simplest case of a system with a finite number of degrees of freedom is that of a single degree of freedom (SDFS) system. In many instances, such a representation, in spite of its extreme simplicity, is sufficient to describe the structural dynamic behavior under specific conditions. Also, it permits often a good understanding of many qualitative and quantitative aspects of this behavior, such as the notions of natural frequency, frequency response function, damping, resonance, etc... However, although the single degree of freedom representation of a given structure may be often useful in describing easily its dynamic behavior in the neighbourhood of a given vibration mode, it cannot give any information about the contribution of the other modes, and about the coupling between the modes, especially in the nonlinear regime. Moreover, when the nonlinear effects, which are always present in real structures, are significant and cannot be neglected, the modeling process is more sensitive to the simplifications made, as will be shown throughout the present work, mainly concerned with nonlinear discrete models. To introduce this general presentation a short discussion is given of the types of discretisation generally made in structural dynamics, i.e. mathematical and physical, followed by a discussion of the general interest and the consequences of the process of discretisation of real structures from both the physical and mathematical points of view. Then, a brief mention is made of the attempt, started in the eighties at the ISVR1 in Southampton University, U.K. and continued during the last two decades by the team of researchers mentioned below at the EMI2 in Mohammed V University of Rabat, to build a nonlinear modal analysis theory for vibrating structures, with various types, geometries, and material characteristics. Few related topics are also briefly mentioned, such as the shift of the models from analysis to algebra when a real continuous system is discretised, the concept of nonlinear modes for both continuous and discrete systems, etc.

2. Discrete models As stated in [2] , two types of discretisations may be encountered in the modeling process, when dealing with structural dynamics, leading in both cases to the reduction of a continuous model to a model with a finite number of degrees of freedom:

1. The mathematical discretisation, resulting for example from the expansion of an unknown function, representing a dynamic parameter, such as the displacement, the strain or the stress, as a series of well chosen basic functions. Such a situation leads to a finite degree of freedom





21

problem if a truncation of the series is performed, due to practical reasons of simplicity and efficiency, followed by a validation based, on a convergence process.

2. The physical discretisation, in which the real physical system is divided into small pieces, each of which being considered as a rigid body. The adjacent components are supposed to be connected to each other appropriately in order to describe the type of liaison involved, using longitudinal springs, rotational springs, friction, viscosity, etc...

In both cases, from the mathematical point of view, the partial differential equations describing the continuous behavior in terms of one or more spatial variable are replaced by a set of differential equations involving the generalized coordinates defining at each instant the position of the (MDOFS) considered. If a harmonic motion is assumed for a linear system, the set of differential equations may be easily replaced by an algebraic eigen value problem, leading to the dynamical system mode shapes and natural frequencies. If the system is nonlinear, the assumption of a harmonic motion is known both experimentally and theoretically to be only an approximation [3-7] and approximate procedures, such as the harmonic balance method, when using the partial differential equation, or the integration of the time functions through a period of vibration, when using Hamilton's principle, are necessary in order to obtain the associated nonlinear algebraic system, often called the amplitude equation. The concept of nonlinear mode shapes will be partially discussed below, but it can be already noted that, as has been stated in [8] , surprisingly, the analytical methods, i.e. the methods of the mathematical analysis, which deal with functions and partial differential equations and lead to eigen functions and eigen parameters, and the algebraic methods, which deal with matrix algebra and lead to eigen vector and eigen value problems, show a very good agreement, with the latest converging to the formers when the number of degrees of freedom is increased. From the physical point of view, it is true that discrete systems, made of few concentrated masses connected by springs, may appear to be too simple, compared with the complexity of real systems, made of deformable bodies, subjected to continued vibrations and deformations, and requiring very sophisticated mathematical tools, in order to be described and analyzed. However, such discrete systems play an important role in the theory of vibration, for many reasons, among which:

1. They are simple, easy to deal with, and close to our usual habits of thinking. 2. They permit to simply define and illustrate the role played by the main parameters

encountered in the vibration field, such as inertia or mass, elasticity, stored energy, damping, natural frequency, modes of vibration, resonance, etc..

3. They may give a deep insight to what really happens locally in an elastic medium, with an easy transfer of concepts, from a concentrated mass to a mass density, from a force to a stress, from a displacement to a strain, and so on.

4. They are, and this may be the main practical reason, very useful for developing relatively simple models, describing complex situations, almost impossible to describe completely using continued models.

The last reason explains why the dynamic team of the EMI at Rabat, supervised by Professor R. Benamar, which has included many researchers [2], [9-16] has moved after two decades of works on nonlinear vibrations of continuous structures, such as beams, rectangular and circular homogeneous and composite plates, rings and shells, to discrete systems, started in [2] by Z.Beidouri, and followed by other researchers [17-22].The dynamics of all of the above mentioned types of structures has been modelled within the same frame work, characterised by three tensors, i.e. mij for the inertia, kij for the linear rigidity and bijkl for the nonlinear rigidity, and leading in all cases to a formally identical amplitude equation, solved by very similar methods. The main principle of the linear modal analysis, which is still widely used in the field of structural dynamics, is the assumption that the dynamics of the system under examination may be described with enough accuracy by few modes which are sufficient to represent the main aspects of its structural behavior. These modes may be exactly determined only in very rare situations with simple geometries, such as few cases of beams and plates in which an analytical solution is possible. In another class of situations, with still a quite simple geometry, they can be calculated numerically with a good precision. In all of the other cases, i.e. in most of the cases, there are only two means for estimating the modes for analyzing the system dynamics:


22

1. The finite element methods, based on various ways of physical discretisations and mathematical approaches.

2. The experimental modal analysis, in which the modes are estimated experimentally from measured data.

In the nonlinear case, which corresponds to the real case, the situation is considerably more complicated from both the theoretical and practical points of view. Theoretically, the superposition principle does not apply for nonlinear systems, the existence and uniqueness of solutions is not always guaranteed [23], and this makes the mathematical analysis much more complicated. From the practical point of view, many new physical behaviors enter into play with nonlinearity, making it almost impossible to develop a complete and simple description. These behaviors involve the mode shapes and resonance frequencies amplitude dependence, the jump phenomena, the harmonic distortion of the nonlinear response, the internal resonance, the modal coupling, the existence of bifurcation points, the occurrence of chaos, etc... [24-28] . In spite of such a complexity, or may be because of such a complexity, a considerable amount of research work has been devoted to the investigation of the nonlinear structural behavior, including:

1. Experimental investigations [29-36] . 2. The introduction of various definitions and new concepts [37-50] 3. The development of analytical approaches such as the multiple scales method, the asymptotic

numerical methods [51] , Benamar's method discused for example in [52], the hierarchical finite element methods [53-55].

4. and the investigation of solution procedures such as the perturbation procedures [56], the various forms of the linearised updated method [57-58] , kadiri and benamar's first and second formulations leading to explicit solutions, [59-60] .

Without attempting a general and complete survey, which would be very interesting but would considerably exceed the objectives of this paper this introductory chapter, the following three subsections are devoted to a quick review on two important approaches to the concept of what has been called, depending on the context, the nonlinear normal modes, or the nonlinear mode shapes, and which remains in spite of a considerable amount of work, still under examination [61,62]. Since these two approaches have been initiated mainly in a mathematical context for one of them and in an engineering context for the other, they will be designated in what followed as the mathematically based approach to the nonlinear normal modes of vibration, and the engineering based approach to the nonlinear mode shapes and resonance frequencies of thin straight structures.

3. The mathematically based approach to the nonlinear normal modes of vibration After the leading theoretical works of Poincaré in [63] and Lyapunov in [64] on nonlinear systems, Rosenberg was the first to propose an extension of the concept of normal modes of vibration, very well established in linear theory of vibration, to the nonlinear case. In his early work [37,38], he defined the nonlinear normal modes (NNM) for discrete conservative systems with a multi degree-of-freedom, as motions where all masses have periodic vibrations with the same period, reach their maximum amplitudes and pass their static equilibrium points at the same time. In the N dimensional configuration space, a nonlinear normal mode occurs on a curved line passing through the equilibrium position. It degenerates to a straight line for linear modes. The theory of NNM has been discussed by Vakakis in [65] and used to study forced resonances of nonlinear systems, and nonlinear localisation of vibrational energy in symmetric systems.. Additional applications of NNMs to modal analysis, model reduction, vibration and shock isolation designs, and the theory of nonlinear oscillators were also discussed. In [66] the study was restricted to one and two degree of freedom systems, with a unilateral constraint on one of the degrees of freedom, for which the response can be analytically determined. Generalized frequencies, modes and masses were built in the procedure. The results obtained for various sets of parameters indicated some limitations to the validity of a general modal superposition formula. In the last few years, 2-dof systems have been treated in many papers, [67-69], R. Lewandowsky has treated in ref. [70] a


23

symmetric 2-dof system using the classical Galerkin's method and compared the result with those based on Benamar's method. A further step has been accomplished in this field by S. W. Shaw, E. Pesheck and C. Pierre [43-45] and [71] , based on a new Galerkin's approach using a new definition of the NNM's based on the invariant manifolds for accurate nonlinear normal modes of discrete systems. This approach was extended to nonlinear normal modes under harmonic excitation [72] , leading to determination of the frequency response for a simple 2-dof mass-spring system with cubic nonlinearities and for a discretized beam model with 12-dof. This theoretically based approach has been recently improved by new definitions and applications in [50] and [73-78]. In [62], one can find an interesting review and comments on the subject.

4. A brief review on the engineering based approach to the nonlinear mode shapes and resonance frequencies of thin straight structures

In the seventies, R.G.White and his coworkers at the ISVR performed a considerable experimental work on various aspects of structural vibration and started the investigation of the nonlinear behaviour under both harmonic and random excitations [79-80] . In his PhD thesis, defended in 1982 [32], a systematic and carefull investigation was carried out by M.M.K. Bennouna in order to measure the amplitude dependence of the fundamental mode shape of a clamped clampd beam, with the associated nonlinear frequencies, bending and axial stress distributions [3,4]. An attempt was also made to develop simple numerical models in order to explain the experimental results. In his PhD thesis [33], R. Benamar extended the experimental work of Bennouna to plates and made systematic measurements of various homogeneous and composite plates amplitude dependent mode shapes and frequencies, under both electrodynamic and aerodynamic excitations. In order to explain the facts observed, he developed a genral model based on Hamilton's principle and spectral analysis to calculate the amplitude dependent mode shapes and resonance frequencies which is presented below, which has been called later Benamar's method [21]. The goal of this method is to contribute to develop a theory of nonlinear modal analysis, which could be applied in the same manner in the theory linear modal analysis, with unified manner, to several nonlinear problems, to have a good qualitative understanding of some fundamental aspects of nonlinear behavior, and also to allow easy quantitative calculations to be made for nonlinear dynamic characteristics such as nonlinear resonant frequencies, the nonlinear frequency response functions, etc...This method is based on a theoretical model based on Hamilton's principle and spectral analysis, using the linear mode shapes of the structure as basic functions. It reduces the large vibration amplitude problems, to a set of nonlinear algebraic equations which is solved numerically for each value of the amplitude of vibration. This method has been applied to a solve different problems of free and forced vibrations of different structures, with various geometries, materials, and boundary conditions [5-7], [9-22], [35,36], [59-60]. In all of the problems treated, the nonlinear vibration problem is presented by the mass tensor, the rigidity tensor, and the nonlinear rigidity tensor. The last tensor is most of the times a fourth order tensor. A nonlinear algebraic system is obtained, which reduces to the classical linear eigenvalue problem, well known in linear modal analysis theory. Many solution procedures have been adopted, such a the iterative method, the linearised method or the explicit procedures, leading in each case to a set of amplitude dependent mode shapes, with the associated amplitude dependent nonlinear frequencies.

5. Conclusion In spite of a considerable amount of theoretical and experimental work, a synthetic formulation of the non linear dynamic behavior, including various aspects and allowing engineering predictions and applications to be made, is still to be developed, based on a physical understanding, an experimental practice, and a deep mathematical insight in the characteristics of non linear systems. The exploration of both the continuous systems, via the non linear mathematical analysis, and discreet systems, via non linear Algebra is expected to be very usefully on the way to reach this objective.


24

Bibliography [1] R. Benamar, “The Modelling of Non-linear Structural Problems: More a Fine Art than a

Mathematical Machine", Conférence pléniére donnée à la Rencontre Nationale sur les Problémes Non Linéaires en Mécanique, Errachidia, 13-14-15, Avril 2002

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[6] R. Benamar, M. M. K. Bennouna and R. G. White “The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures, part II: Fully clamped rectangular isotropic plates", Journal of Sound and vibration, 164, 295-316, 1993 .

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[13] M. Haterbouch, Effect de la nonlinearité géometriques sur la réponse libre et forcée des plaques circulaires minces encastrées et simplement supportées", Doctorat Esciences Appliquées, Ecole Mohammadia d'Ingénieurs. Rabat, 2004.

[14] K. El Bikri, Contribution à l'étude des vibrations géométriquement non-lineares des structures minces : couplage transversal-axial dans les plaques rectangulaires & réponse dynamique des poutres fissurees", Doctorat d'état, Ecole Mohammadia d'Ingénieurs, Rabat, 2000.

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[16] M. Rougui, “Modelisation de l'effet de la nonlinearité geometrique sur la reponse libre et forcée des structures minces, cas des coques cylindriques, des anneaux circulaires et des plaques ortho tropique", Doctorat d'état, université sidi mohammed ben abdellah FST-Fes, Morroco. 2009.

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25

[19] A.Eddanguir, Z.Beidouri, and R.Benamar. “Nonlinear transverse steady-state periodic forced vibration of 2-dof systems with cubic nonlinearities” Premier Colloque International IMPACT 2010, Djerba, Tunisie, 22-24 Mars 2010 .

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[32] M. M. K. Bennouna, Non-linear dynamic behavior of a clamped-clamped beam with consideration of fatigue life", PhD Thesis, University of Southampton, England, 1982 .

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26

[41] A.F. Vakakis, C. Cetinkaya, Mode localization in a class of multidegree-of-freedom nonlinear systems with cyclic symmetry", SIAM Journal on Applied Mathematics, 53, 265-282, 1993 .

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geometrically non-linear vibrations of thin circular plates", Proceedings of the Euromech Colloquium 457 on non linear modes of vibrating systems, Frejus - France - June 7-9 2004 .

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[59] M. EL Kadiri, R. Benamar and R. G. White, Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear free response of thin straight structures. Part I: Application to C-C and SS-C beams", Journal of Sound and Vibration, 249(2), 263-305, 2002 .

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27

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28

TOOLBOX FOR SIMULATION AND PARAMETER IDENTIFICATION OF NONLINEAR

MECHANICAL SYSTEMS

Kjell Ahlin Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden

Abstract

Analysis of nonlinear mechanical systems is becoming increasingly important in many applications in automotive, aerospace and other industries. It is important to be able to accurately simulate nonlinear systems for various input signals in order to understand dynamic effects of the nonlinearities. A new digital filter based method with high performance has therefore been developed and implemented in a new toolbox for simulation and experimental analysis of nonlinear systems in MATLAB®. Nonlinear forced response from general linear systems with known nonlinearities can be computed and analyzed with fast algorithms in this toolbox. Nonlinear parameters can further be identified from experimental data. Of particular practical interest in this case are the methods proposed by Julius Bendat, which have been generalized and implemented for identification of many different nonlinear systems. Simulation results as well as experimental results from some different practical applications are presented. 1. Introduction There exist several methods to obtain the time history response of a mechanical system to an arbitrary excitation. The Duhamel integral, also known as the convolution integral, is a well-established method in the literature, its main drawback is the computational burden associated with its non-recursive nature. However, the integral can be used to derive a recursive algorithm, a digital filter, which leads to a faster solution. The method is described in detail in [1]. The mechanical system is characterized with its modal parameters, residues and poles. The parameters may come from a Finite Element Model, a lumped system description, an analytic model or from experimental modal analysis, see figure 1. When the residues and poles are known, the filter coefficients can be calculated, using different approaches. Two types of errors will occur, an aliasing error and a bias error. As we show in [1], the errors for the different methods are completely known. As is also described in [1] we can add non-linearities to the mechanical system. The non-linearities may be of simple zero memory type, but they may also have memory, like stick-slip friction performance. The simulation methods constitute the backbone of the described toolbox. There are also several functions for identification and parameter extraction for nonlinear systems. 2. Functions for simulation The toolbox contains a family of functions to calculate the forced response of linear mechanical systems. As an example, we describe the function timeresd. [y,t] = timeresd(F,fs,M,C,K,inno,outno,modes,type) The input to the function is a force time history F, given with a sampling frequency fs. The linear mechanical system is described with its mass matrix M, damping matrix C and stiffness matrix K. The force is applied to degree of freedom, DOF, inno, and the response is calculated at output DOF outno. The input parameter modes tell the function what modes to use, and type determines the approximation method used. y is the output response as displacement and we get a corresponding time vector t. The function has many options: • outno may be a vector, so many outputs can be calculated at the same time.


29

• the different types implemented are impulse invariant, step invariant, centered step invariant, ramp invariant, three point lagrange, etc. • instead of M, C,K we may give mass and stiffness matrices and modal damping • we may also describe the system with its residues and poles • we may also calculate the output(s) as velocity instead of displacement There are then many functions where non-linearities are added to the linear system. The simulation functions are optimized for fast performance. To accomplish this, the more advanced functions have the description of the non-linearities explicitly written into the code. An example of such a function is nonlindmck x = nonlindmck(F,fs,M,C,K,FDOF,nlDOFs,modes,responses) In this function a force F is applied to DOF FDOF of a linear system described with M, C and K. We may then have any number of nonlinear functions connected to the linear system. The vector nlDOFs tells which DOFs are involved. In figure 1 we give an example.

Figure 1. A four degree of freedom linear mechanical system with two nonlinear springs. A force is

input to DOF 4 and all four responses are calculated. The system in figure 1 is a four degree of freedom linear system. Between the masses m2 and m3 there is a cubic spring, and between m1 and ground there is a spring with the function “square with sign”. To use the function nonlindmck we type x = nonlindmck(F,fs,M,C,K,4,[3,2,1],[1:4],[1:4]) Inside the function, we have to give the definition of the nonlinearities, in this example f1 = 2.e16*[-(x1(1)-x1(2)).^3; (x1(1)-x1(2)).^3; -0.5e-4*x1(3).*abs(x1(3))]; To calculate all four responses to a given random force with 400 000 samples takes 65 seconds on a laptop computer. As for the other forced response functions, there are many different options: • the linear system can be described with M, C, K or with M, K and modal damping or with residues and poles. • the output(s) can be calculated as displacement or as velocity • approximation method may be selected (impulse, step, ramp,…)


30

• many examples of nonlinear functions are given, including nonlinearities with memory. In the toolbox, there are also some useful functions to create suitable input signals like swept sine and burst random. 3. Functions for analysis One method to analyze nonlinear systems builds on the work of Julius Bendat [2]. Different ways to use the basic ideas are described in [3]. The common approach is that the nonlinear problem can be reformulated to a problem of multiple input / single output, MISO. To handle that, the toolbox contains efficient functions to handle MISO and MIMO, multiple input / multiple output situations. The function xmtrx calculates the input and output spectral matrices and the cross spectral matrix. These matrices are then the input to the function mimoest that calculates the linear system Frequency Response Function, FRF, matrix and the multiple coherence function. To simplify the use of Bendat’s methods we have the function julius, which makes use of the MIMO functions above. A single degree of freedom system with a cubic spring is studied, see figure 2.

Figure 2. A single degree of freedom with a cubic spring is simulated. The system is excited with a

random force. The system is excited with a random force and the response is simulated with timeresd_cubic. The raw Frequency Response Function (best linear approximation) between force and response is calculated. When that FRF is compared with the theoretical linear part of the system, we find a great difference, figure 3. Weoverestimate the resonance frequency of the linear system. We now use julius with a guess that the nonlinearity is of cubic type. The problem we solve is a MISO problem with x and x3 as inputs and the force f as output. This approach is often called “reverse path”, as we calculate the real input f as output. As result, we get the inverse of the sought linear FRF and the unknown coefficient p. In figure 3 we find that we get an almost perfect estimate of the FRF in this way. As a quality measure, we plot the calculated multiple coherence in figure 4. It is very close to one, indicating that we have a good model. 4. Summary A MATLAB toolbox for simulation and identification of nonlinear mechanical system has been developed. The simulation functions are fast and accurate. The linear part of the system can be defined in many ways and many different nonlinearities, with or without memory, can be handled. Efficient tools to handle multiple input / multiple output (MIMO) situations are included to be used when a nonlinear identification problem is converted into an equivalent MIMO problem.


31

Figure 3. Frequency Response Functions for a single degree of freedom system with a cubic spring.

The raw FRF differs from the linear FRF, but the FRF from Julius is nearly perfect.

Figure 4. Multiple coherence function corresponding to the result in figure 3. The multiple coherence

clearly shows that the used model is good. References [1] Ahlin, Magnevall, Josefsson: Simulation of Forced Response in Linear and Nonlinear Mechanical Systems Using Digital Filters. ISMA 2006, September 2006, Leuven, Belgium. [2] Julius S. Bendat: Nonlinear Systems Techniques and Applications. John Wiley & Sons, Inc 1998 [3] Josefsson, Magnevall, Ahlin: On Nonlinear Parameter Estimation with Random Noise Signals IMAC XXV, Orlando, February 2007.


32

GEOMETRICALLY NON LINEAR MODE SHAPES AND RESONANT FREQUENCIES OF MULTI-SPAN BEAMS

A. Adri 1, Z. Beidouri1, M. EL kadiri2, R. Benamar2

1 Laboratoire de Mécanique Productique & Génie Industriel (LMPGI) Ecole Supérieure de Technologie; Université Hassan II Ain Chock KM 7 Route El Jadida, Casablanca, Maroc

2 Laboratoire des Etudes et Recherches en Simulation et Instrumentation Ecole Mohammadia des Ingénieurs; Université Mohammed V Avenue Ibnsina,Agdal, Rabat, Maroc

E-mails : [email protected] , beidouri @ est-uh2c.ac.ma , [email protected] , [email protected]

Abstract

The objective of this paper is to establish the formulation of the problem of nonlinear transverse vibration of uniform multi-span beams, with intermediate simple supports. The method used is based on the principle of Hamilton and spectral analysis for non-linear free vibration occurring at large displacement amplitudes. The problem is reduced to solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved using an approximate method developed previously (second formulation) leading to the nonlinear fundamental mode of the multi-span beam and to the corresponding backbone curve.

1. Introduction In a series of previous works, the non-linear mode shapes and resonance frequencies of beams with various boundary condition have been examined both theoretically and experimentally [1 to 4]. The theory was based on Hamilton’s principle and spectral analysis and had led to a series of amplitude dependent mode shapes and resonance frequencies. The purpose of the present work was the examination, by similar methods, of geometrically non-linear vibrations of multi-span beams.

2. Nonlinear formulation The uniform beam of fig.1 is made of a material of mass density ρ , Young’s modulus E, length L, cross-sectional area S, radius of gyration r and second moment of area of cross section I. Let x be the coordinate along the neutral axis of the beam measured from the right clamped end. Let w(x,t) be the transverse deflection of the beam, measured from its equilibrium position.

Figure 1. A beam with intermediate simply supports

If only transverse vibration is considered, the kinetic energy of the beam can be written as (neglecting the axial motion):

2

0

12

L wT S dx

tρ ∂ = ∂ ∫ (1)

The beam total strain energy can be written as the sum of the strain energy due to the bending denoted as Vlin, plus the axial strain energy due to the axial load induced by large deflection denoted as VNlin

[1].

x1 x2

xj-1 xj

xj+1

xn=L

w

x

lj lj-1 lj+1

x0=0

x1






33

22

20

12

L

Lin

wV EI dx

x

∂= ∂

∫ ; 22

0

18

L

Nlin

ES wV dx

L x

∂ = ∂ ∫ (2)

The transverse displacement function is expanded as a series of basic spatial functions (the linear modes) and the time function is supposed to be harmonic:

i i i iw( x,t ) q ( t )w ( x ) a w sin( t )= = ω (3) Where the usual summation convention for the repeated indices is used. After discretization of the

expressions (1) and (2) and applying Hamilton principle, we obtain as in [1] a set of nonlinear algebraic equations:

2 3 2 0i j ij i j k l ijkli ir i j k ijkr i ir

i j ij

a a k a a a a ba k a a a b a m

a a m

++ − = (4)

To obtain non-dimensional parameters, we put: 2

2 4

2 22

3 3

* * *i i *

ij ij ij

* * *ij ij ij

x EIw( x ) rw rw ( x )

L SLm K KEIr EIr

Sr Lm K L K L

ω = = ; = ω ρ

= ρ ; = ; = (5)

* * *ij ij ijklk m b , , are non dimensional tensors given by:

1

0

* * * *ij i jm S w w dx= ρ ∫ ;

21 2

2 20

**j* *i

ij * *

wwk EI dx

x x

∂∂=

∂ ∂∫ (6)

1 1

0 0

** * *j* * *i k l

ijkl * * * *

ww w wb dx dx

x x x x

∂∂ ∂ ∂= α

∂ ∂ ∂ ∂∫ ∫ (7)

α being the non-dimensional parameter 2

4Sr

Iα = (8)

Substituting these equations into equation (4) leads to:

2 3 2 0* *

i j ij i j k l ijkl* * *i ir i j k ijkr i ir*

i j ij

a a k a a a a ba k a a a b a m

a a m

++ − = (9)

3. Determination of the linear mode shapes of the multi span beam Before examining the nonlinear case, we start in this section by determination of the multi-span beam linear mode shapes, in order to use them as basic functions in the non linear theory. The transverse displacement function w of the beam shown in figure 1 can be defined by:

] [

] [

1 1

1

1

( ) 0,

....

( ) ( ) ,

...

( ) ,

j j j

n n n

w x x

w x w x x x

w x x x

−

−

→ = → →

(10)

A closed form solution to this eigenvalue problem can be obtained by employing transfer matrix method as in [8]. The general solution for transverse vibration in the jth span and its derivatives can be written as:

1 1

1 1

( ) cosh( ( )) sinh( ( ))

cos( ( )) cos( ( ))

ji j i j j i j

j i j j i j

w x a x x b x x

c x x d x x

β β

β β− −

− −

= − + −

+ − + − (11)

'1 1

1 1

( ) sinh( ( )) cosh( ( ))

sin( ( )) cos( ( ))

ji i j i j j i j

j i j j i j

w x a x x b x x

c x x d x x

− −

− −

= − + −

− − + −

β β β

β β (12)


34

'' 21 1

1 1

( ) cosh( ( )) sinh( ( ))

cos( ( )) sin( ( ))

ji i j i j j i j

j i j j i j

w x a x x b x x

c x x d x x

− −

− −

= − + −

− − − −

β β β

β β (13)

In which, for ,...2,1=i 2

4 ii

SEI

ρ ωβ =

(14)

are the eigenvalue parameters for the multi-span beam. The constants aj, bj, cj, dj are determined by the boundary conditions.

At jx x= and 1jx x += , the beam is simply supported and the continuity

conditions (displacement, slope and moment) are used as follows: 1( ) 0j

i jw x+ = ; 11( ) 0j

i jw x++++++++ ==== (15)

1

j j

j ji i

x x

w wx x

+∂ ∂=

∂ ∂ ;

2 1 2

2 2

j j

j ji i

x x

w wx x

+∂ ∂=

∂ ∂ (16)

The constants in the (j+1)th span (aj+1,bj+1,cj+1,and dj+1) are related to those in the jth span (aj,bj,cj, and dj) through the compatibility conditions, i:e. Eqs. (15 and 16). They can be expressed as:

1 11 12 13 14

1 ( )4 4

1

1 41 44

. . . .

. . . .

. .

j j j jj j j

j j jj

j j j

j jj j j

a a at t t tb b b

Tc c c

d d dt t

+

+×

+

+

= =

, for nj ,...,2,1= (17)

Where ( )4 4

jT × is the 4 4× transfer matrix which depends on the values of iβ , and consequently on the eigenvalue iω . The general terms of ( )

4 4jT × are:

11

1cosh( )

2j

i jt lβ= ; 12

1sinh( )

2j

i jt lβ=

13

1cos( )

2j

i jt lβ= − ; 14

1sin( )

2j

i jt lβ= − ;

1 121

1 1

cosh( ) sinh( )sin( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= − × +

1 122

1 1

sinh( ) cosh( )sin( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= − × +

1 123

1 1

cos( ) sin( )sin( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= × +

1 124

1 1

sin( ) cos( )sin( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= × −

31

1cosh( )

2j

i jt lβ= − ; 32

1sinh( )

2j

i jt lβ= −

33

1cos( )

2j

i jt lβ= ; 34

1sin( )

2j

i jt lβ=

1 141

1 1

cosh( ) sinh( )sinh( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= × +

1 142

1 1

sinh( ) cosh( )sinh( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= × +

1 143

1 1

cos( ) sin( )sinh( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= − × +

1 144

1 1

sin( ) cos( )sinh( )

2j i j i j i jj

j j

C l l lt

S S

β β β+ +

+ +

= − × −

With: ( )cosh( ) cos( )j i j i jC l lβ β= − ; ( )sinh( ) sin( )j i j i jS l lβ β= −


35

Through repeated application of Eq. (17), the four constants in the first segment (a1, b1, c1, and d1) can be mapped into those of the last segment, reducing the number of independent constants to four.

1

1( 1) (2) (1)4 4 4 4 4 4

1

1

......

n

n n

n

n

a ab b

T T Tc cd d

−× × ×

= × × ×

(18)

These four remaining constants (a1, b1, c1, and d1) can be found through the satisfaction of the boundary conditions. Case of a clamped-clamped beam

For the case of clamped-clamped beam, the corresponding boundary condition can be expressed as

1(0) 0iw = ; 11( ) 0iw l ==== ;

1

0

0iwx

∂=

∂; 0

ni

L

wx

∂=

∂ (19)

Equation (18) becomes:

1

1( 1) (2) (1)4 4 4 4 4 4 1

1

1

1

......1

n

n n

n

n

a CSb

T T T ac

d CS

−× × ×

− = × × × ×

−

(20)

Equation (19) leads to:

[ ]sinh( ),cosh( ), sin( ),cos( ) 0

n

ni n i n i n i n

n

n

a

bl l l l

c

d

− × =

β β β β (21)

Substitution of Eqs (20) into Eqs (21) leads to the frequency equation: [ ]

1

1( 1) (2) (1)4 4 4 4 4 4

1

1

sinh( ),cosh( ), sin( ),cos( )

1

...... 01.

i n i n i n i n

n

l l l l

CS

T T T

CS

−× × ×

− ×

−

× × × × × = −

β β β β

(22)

The solutions of the frequency equation (22) determine the eigenvalues nω . The coefficients of the eigenfunction, wn(x), are obtained by back substitution into Eqs.(20).

4. Determination of the non-linear of the non-linear frequencies and mode shapes of multispan beams

The values of the parameters iβ were computed by solving numerically equations (22). It is a nonlinear algebraic equation which can be solved using the standard Newton-Raphson iterations. The functions wi were normalized in such a manner that the obtained mass matrix equals

the identity matrix. Replacing the radius of gyration r by IS

in equation (8) leads to the value of the

parameter α =0,25. The parameters * *

ij ijm k , and *ijklb of equations (6-7) were computed numerically by using Simpson’s

rule in the range [0,1].


36

The non-linear algebraic system (9) has been solved using the second formulation, developed and used for the first time in [6]. It is based on an approximation which consists on writing the contribution vector as: { } [ ]1 3 11, ,...,A a ε ε= and neglecting in the expression i j k ijkra a a b of equations (24) the second terms with respect to ei, i.e., terms of the type 1 1j k jkra bε ε .

In order to illustrate the results obtained by the method used in this paper, the case of three-span

beams is presented.

Case of clamped-clamped three span beam

The three span beam is defined by: 1 2 32 2 0.2l l l l l l l= , = , = , = and is clamped at both ends. To obtain the fundamental non-linear mode shape of the beam, the first five symmetric beam functions were used. The corresponding parameters iβ for i=1,3,...9 are given in table1.

Table1. Symmetric (a) and anti-symmetric (b) eigenvalues parameters

i iβ (a) i iβ (b) 1 10,5338939979212 2 11,1550808978563 3 17,1665890954938 4 18,6739486810338 5 20,9791246712479 6 26,1783112172320 7 26,9831665579790 8 32,8690709184431 9 34,4024348142474 10 36,6503300513347

The normalized first non-linear mode shapes is plotted in figure 2. The figures show the amplitude

dependence of the mode shapes, with high increase of curvatures near to the clamps, compared with linear theory.

Figure 2. The normalized first non-linear mode shapes

Figure 3. The first non-dimensional backbone curve of the three span beams ( present results; *

results given in [7])


37

Figure 4. The first non-linear curvatures of the three span beams

5. Conclusion A theoretical model for large vibration amplitudes of a thin elastic three span beam has been developed, based on Hamilton’s principle and spectral analysis, to obtain numerical results. The theory effectively reduces a non-linear free vibration problem to a set of non-linear algebraic equations depending on the classical rigidity and mass matrix, and a fourth order tensor due to the non-linearity. If the non-linearity tensor due to finite displacements is neglected, the classical eigenvalue problem of the linear theory is obtained. By choosing the convenient basic function, results can be obtained for various boundary conditions. The theory has been applied here to a three span beam clamped at the ends with two intermediate simply supports. The fundamental backbone curves obtained is compared to the results obtained by Galerkin method [7] showing a good agreement. The values are taken from graph page 1706. The amplitude-dependent normalized mode shapes of three span beams have higher values of curvatures near to the clamp at large deflection amplitudes, which mean that the bending strain can have a higher rate of increase with increasing amplitude in such non-linear cases, compared to the bending strain in the linear state.

References [1] AZRAR R. L , BENAMAR R. , WHITE R.G.. 1999. A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at the large vibration amplitudes part1: General theory and application to the single mode approach to free and forced vibration analysis. Journal of Sound and Vibration 224 (2), 183-207. [2] Benamar,R. , Bennouna M. and White. R. G.,1991. The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures, part I: Simply supported and clamped-clamped beams. Journal of Sound and Vibration (1991) 149 pp 179-195. [3] Benamar R. 1990. Non-linear dynamic behaviour of fully clamped beams and rectangular isotropic and laminated plates Ph.D. thesis, University of Southampton. [4] Bennouna M. and White. R.G., 1984. The effects of large vibration amplitude on the fundamental mode shape of a clamped-clamped beam. Journal of sound and vibration,(3) 309-331.

[5] El Kadiri M. 2001. Contribution à l’analyse modale non linéaire des structures minces droites : réponses libres et forcées des poutres et des plaques rectangulaires aux grandes amplitudes de vibrations. Ph.D.thesis, Université Mohammed V-Agdal, Ecole Mohammadia des Ingénieurs. [6] EL Kadiri M., Benamar R. and White.R.G. 2002. Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear free response of thin straight structures. Part I: Application to C-C and SS-C beams. of Journal Sound and Vibration (2002) 249(2), pp 263-305.

[7] Lewandowsky R.,2003 “Free vibration of structures with cubic non-linerity-remarks on amplitude and Rayleigh quotient”. Computer methods in applied mechanics and engineering. 192 (2003) pp 1681-1709.


38

[8] Hai-Ping Lin, Chang S.C. Free vibration analysis of multi-span beams with intermediate flexible constraints. of Journal Sound and Vibration 281 (2005) 155-169. [9] Yozo Mikata. Orthogonality condition for a multi-span beam, and its application to transient vibration of two span beam. Of Journal Sound and Vibration 314 (2008) 851-866. [10] Lu. C.F., Lee Y.Y.,Lim C.W. ,Chen W.Q. Free vibration of long –span continuous rectangular Kirchhoff plates with internal rigid line support. Journal of Sound and Vibration 297 (2006) 351-364. [11] Hsien-Yuan Lin, Ying-Chien Tsai. Free vibration analysis of uniform multi-span beam carrying multiple spring-mass systems.. of Journal Sound and Vibration 302 (2007) 442-456.

[12] Yusuf Yesilce, Oktay Demirdag. Effet of axial force on free vibration of Timoshenko multi-span beam carring multiple spring-mass systems. International Journal of Mechanical Sciences 50 (2008) 995-1003.


39

EVALUATION NON DESTRUCTIVE PAR ONDES GUIDEES ULTRASONORE A LA JONCTION DE DEUX PLAQUES ELASTIQUES DE NATURE DIFFERENTE EN PRESENCE

D’UN DEFAUT

Taoufiq BELHOUSSINE DRISSI *, Bruno MORVAN, Mihai PREDOI, Jean-Louis IZBICKI, Pascal PAREIGE

LOMC FRE CNRS 3102 - Groupe Onde Acoustique Université du Havre, place R. Schuman - 76600 Le Havre

* [email protected]

Résumé Nous nous intéressons à une jonction droite de deux plaques de matériaux différents mises en contact bord à bord. L’objet de cette étude est l’interaction d’une onde de Lamb avec un défaut localisé à la jonction entre des plaques d’aluminium et de cuivre. Une comparaison entre des résultats théoriques, numériques et expérimentaux est effectuée. Nous nous intéressons plus particulièrement à la réflexion et la transmission du mode fondamental symétrique S0. Les coefficients de réflexion et de transmission théoriques sont obtenus par une approche multi-modale basée sur l’exploitation de la relation orthogonalité entre les différents modes. En utilisant la méthode des éléments finis, nous estimons la valeur limite du rapport entre la dimension du défaut et l’épaisseur de la structure, pour laquelle l’approche multi–modale est applicable. Dans les études expérimentale et numérique, il est également mis en évidence des effets de diffraction par le défaut. Mots clés : ondes de Lamb, défaut, jonction.

1. Introduction Les ondes guidées, et en particulier les ondes de Lamb, sont couramment utilisées pour l’inspection de structures de grandes dimensions. Les propriétés de ces ondes sont parfaitement connues lorsque celles-ci se propagent dans des structures canoniques du type plaque ou cylindre d’extension infinie. En revanche concernant des structures réelles, l’interaction des ondes de Lamb avec des soudures, des défauts ou bien des variations d’épaisseur reste encore mal comprise. Concernant une jonction droite entre deux plaques semi–infinies de même épaisseur mises en contact bord à bord mais de matériaux différents, des travaux théoriques se rapportant à ce sujet [1-3] ont permis de montrer la pertinence des ondes de Lamb pour caractériser l'interface. Ces travaux ont mis en évidence des réflexions et transmissions d’ondes à l’interface ainsi que des phénomènes de conversions de modes. Predoï et Rousseau [4], ont présenté des travaux concernant une structure de deux plaques d’acier soudées par un joint d’aluminium de façon à modéliser une soudure. L’analyse modale a été également appliquée afin d’étudier la propagation des ondes de Lamb en faisant varier la largeur de la couche d’aluminium présente entre les deux plaques en acier. Dans ce papier, nous reprenons le formalisme développé dans cette dernière étude pour l’appliquer au cas d’une jonction avec défaut. On s’intéresse à des défauts localisés dans l’épaisseur symétriques vis-à-vis du plan médian de la plaque. Le but est de présenter, pour une même configuration, une comparaison entre des résultats théoriques, numériques et expérimentaux. Nous nous intéressons plus particulièrement aux coefficients de réflexion et de transmission lorsque le mode de Lamb S0 est incident.

2. Aspects théoriques On considère une structure composée de deux plaques planes semi-infinies (Fig 1) : aluminium-cuivre, avec les notations suivantes : le solide indiqué par l’indice 1 (aluminium), est un solide élastique isotrope caractérisé par sa masse volumique ρ1, et les coefficients de Lamé λ1 et µ1. Les notations et les définitions sont identiques pour le matériau indiqué par l’indice 2 (cuivre). Les deux plaques ont la même épaisseur e et comportent un défaut (inclusion de vide) de profondeur d et de largeur h. La largeur h est considérée faible devant l’épaisseur e et on supposera dans cette partie h=0. Le défaut est localisé d’une façon symétrique par rapport au plan

z

x

I

R

T

1 1 1, ,ρ λ µ 2 2 2, ,ρ λ µ

O 2d

2d−

Largeur h

e

Figure 2. Géométrie de la jonction comportant un défaut symétrique.


40

médian. Les deux plaques sont mises en contact bord à bord en x=0, par une jonction perpendiculaire. En dehors du défaut, la jonction est supposée parfaite, de sorte que les déplacements U et les champs de contraintes σ sont continus en x=0. Le mode incident est émis depuis x<0 vers la jonction dans la plaque d’aluminium. En conséquence, les déplacements et les contraintes au niveau de la jonction de la plaque, sont ceux produits par le mode incident d’ordre q, qui est supposé d’amplitude unitaire (Aq=1). Les conditions aux limites sur l’interface (x=0) expriment la continuité des déplacements et des contraintes, elles s’appliquent dans la zone hors du défaut, sous la forme des équations (1) et (2) :

( ) ( ) ( )1 1 2 2

1 2

1 1 2' '

0 0q q m m m m

m m

A U r U t U∞ ∞

= =

+ =∑ ∑ (1)

( ) ( ) ( )1 1 2 2

1 2

1 1 2' '1 2

0 0q q m m m m

m m

A r tµ σ σ µ σ∞ ∞

= =

+ =

∑ ∑ (2)

mi (i=1,2) désigne l’ordre du mode dans le matériau. Les valeurs des coefficients rm et tm pour un mode d’ordre m représentent respectivement les amplitudes des modes réfléchis et transmis. La barre supérieure «¯» désigne le complexe conjugué. Les quantités présentant cette barre sont en fait associées à une propagation vers les x négatifs. Pour déterminer les valeurs des coefficients de réflexion rm et de transmission tm, on applique la relation d’orthogonalité [5]. On obtient alors deux équations (3) et (4) :

( ) ( ) ( ) ( )2 2 2

2

1 1 12 12' '

0

A Bq ql l ll m lm lm

m

A J r J t I I∞

=

+ = + ∑ (3)

( ) ( ) ( ) ( ) ( )1 11

1

12 1212 12 2' '

0

A BA Bm l m lq ql ql m l ll

m

A I I r I I t J∞

=

+ + + = ∑ % % (4)

Les valeurs des coefficients de réflexion rm et de transmission tm sont obtenues par un système algébrique linéaire de rang théoriquement infini (donné par l’Eqs (3) et (4)) du type : A*X=B. Les coefficients de réflexion et de transmission en énergie s’obtiennent à partir des valeurs des coefficients de réflexion rm et de transmission tm : Le coefficient de réflexion en énergie (resp coefficient de

transmission en énergie) sont défini par :

( )

( )

12

1

mm

m m

qq

JR r

J= (resp

( )

( )

22

1

mm

m m

qq

JT t

J= ). Le bilan d’énergie

à la jonction doit vérifier la relation :

1 2

1 2

1 20 0

1p p

m mm m

R T= =

+ =∑ ∑ où pi (i=1 ou 2 désigne le matériau) est le

nombre total des modes réels à une fréquence donnée. En effet, la sommation porte nécessairement sur les modes réels puisque les modes évanescents ne transportent pas d’énergie. L'épaisseur de la plaque est de 2mm. Les caractéristiques élastiques de l'aluminium et du cuivre utilisés sont données dans le Tableau 1. L’augmentation du nombre de modes n considéré dans le calcul fait converger les

coefficients de réflexion Rm et de transmission Tm en énergie vers une solution stable. Le mode S0 est le mode incident. On travaille à un Fe=1.6 MHz mm. Pour des petits défauts (d/e=0.01) et des défauts de grande taille (d/e=0.25) on trouve :

n 7 9 11 13 15 17 19 R(S0) 0.14488 0.14501 0.1449 0.14498 0.14487 0.14491 0.14481

d/e=0.01 T(S0) 0.85512 0.85499 0.8551 0.85502 0.85513 0.85509 0.85519 R(S0)+T(S0) 1 1 1 1 1 1 1 R(S0) 0.57079 0.4731 0.47246 0.56226 0.53711 0.46858 0.46717

d/e=0.25 T(S0) 0.4292 0.5269 0.52754 0.43774 0.46289 0.53142 0.53283 R(S0)+T(S0) 0.9999 1 1 1 1 1 1 Tableau 2. Coefficient de réflexion et de transmission dans le cas d’un défaut symétrique

D’après le tableau 2, on constate la conservation de l’énergie à la jonction (somme des énergies des

Aluminium Cuivre ρ(kg/m3) 2799 8705 VL(m/s) 6320 4728 VT(m/s) 3155 2360 Tableau 1. Caractéristiques des

matériaux


41

modes réfléchis et transmis). Par contre l’augmentation du nombre de modes n ne fait pas converger les coefficients de réflexion et de transmission en énergie vers une solution stable pour des grands défauts, alors que pour des petits défauts on remarque une convergence de ces coefficients. Le critère de conservation de l’énergie n’est donc pas suffisant. La théorie n’est donc pas valable pour n’importe quelle taille de défaut. Le problème qui se pose maintenant, est de savoir pour quelle taille maximum de défaut l’approche multi-modale peut être considérée comme applicable. Pour cela on réalise une étude numérique (MEF). 3. Aspects numériques L’étude numérique est réalisée en utilisant le logiciel COMSOL ®, La structure étudiée pour la simulation, est identique à la Fig 1. La largeur du défaut est h=0.001mm. Les caractéristiques des matériaux sont données dans tableau 1. Chaque plaque est de longueur 100mm et d’épaisseur e=2mm. Le mode S0 est le mode incident, il est excité en appliquant ses déplacements théoriques sur le bord libre de la plaque d’aluminium. Nous pouvons présenter l’évolution du signal spatio–temporel s(x,t) correspondant aux déplacements Uz des nœuds en surface. Sachant que la jonction avec défaut est située à x=100mm. Pour obtenir la répartition de l’énergie relative du mode de Lamb S0 incident entre les différents modes de Lamb réfléchis et transmis, on relève les amplitudes de chaque onde et on déduit les coefficients de réflexion et transmission énergétique en appliquant les relations suivantes :

( ) ( )( )

( )( )

2

0

0

refl inc refli i

i inc refl inci

A S SR S

A S Sζ φζ φ

= =

(5)

( ) ( )( )

( )( )

2

0

0

tran inc trani i

i inc tran inci

A S ST S

A S Sζ φζ φ

= =

(6)

Ainc, Arefl et Atran présentent les amplitudes des modes, où l’indice supérieur signifie respectivement incident, réflexion et transmission. Le coefficient ζ relie le module du déplacement normal à la surface

de la plaque et la racine carrée du flux de vecteur de Poynting : 2

= ± =

Φ

z

eU z

ζ

D’après la Figure 3, dans le cas d’un défaut symétrique au centre de la plaque, la comparaison entre les résultats théoriques et numériques montre qu’un erreur de moins de 3% existe pour des défauts pour lesquels d/e<0.03. Donc l’approche multi–modale sera applicable. L’intérêt de l’approche multi-modale est sa rapidité de mise en œuvre. Néanmoins son domaine de validité semble limité, mais tout dépend de l’application visée : ce qui est « petit » à l’échelle mésoscopique sera considéré comme «grand» à l’échelle microscopique.

4. Aspect expérimental Le dispositif expérimental permettant la génération des ondes et l’acquisition des signaux temporels est décrit sur la Fig 3. La structure étudiée est une structure aluminium-cuivre, où les deux plaques sont en contact bord à bord. L’épaisseur des plaques est e=2±0.1mm. La plaque de cuivre comporte un défaut interne symétrique par rapport au plan médian, telle que d/e=0.5. Il a une longueur de 2.9cm et une largeur h=0.15mm. Les caractéristiques des matériaux sont définies dans le tableau 1. Chaque

0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d/e

R(S0)

(a)

0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d/e

T(S0)

(b)

étude théoriqueétude numérique

étude théoriqueétude numérique

Figure 3. Coefficient de réflexion (a) et de transmission (b) en fonction du rapport d/e.


42

plaque a une longueur de 200mm et une largeur de 200mm. Les coefficients de réflexion R(S0) et de transmission T(S0) sont présentés Fig 4. Le bilan d’énergie, en moyenne sur l’intervalle de fréquence [1.52–1.72]MHz mm est R+T=0.89. Ce déficit dans le bilan d’énergie peut s’expliquer par la non prise en compte d’un phénomène de diffraction par le défaut. Le défaut étant de longueur finie, la largeur du faisceau incident (de l’ordre de 40mm) entraîne de la diffraction sur les bords du défaut. La Fig 4, présente les coefficients de réflexion et de transmission de l’étude numérique et de l’étude expérimentale en fonction du produit fréquence-épaisseur. On peut remarquer qu’il y a un bon accord entre les coefficients de transmission obtenus par les deux études, numérique et expérimentale. C’est moins vrai pour le coefficient de réflexion où on constate une différence entre les deux études d’environ 11% dans l’intervalle de fréquence épaisseur [1.5–1.75]MHz mm.

5. Conclusion Nous avons étudié la réflexion et la transmission à travers une jonction droite qui comporte un défaut symétrique par rapport à l’axe de propagation Ox. Théoriquement on a utilisé une approche multi–modale, cette méthode n’est valable que pour des petits défauts. Un défaut correspondant environ à une profondeur relative de 3% pourra être étudié par l’approche multi-modale qui est rapide en temps (pour une fréquence donnée un point de calcul nécessite environ 30s). Par contre pour un défaut de taille supérieure il faudra utiliser la méthode numérique (MEF) qui est beaucoup plus gourmande en temps (pour une fréquence donnée, un point de calcul, (voir Fig 4), nécessite environ 1h). Les variations expérimentales des coefficients de réflexion et de transmission présentent la même allure que dans l’étude numérique, mais les valeurs obtenues présentent une erreur maximale de 11%. Il a également été mis en évidence des effets de diffraction dus à la présence du défaut. Le bilan d’énergie en est affecté mais les phénomènes dominants restent bien la réflexion et la transmission même avec un défaut très important. Références [1] C. Scandrett, N. Vasudevan. “The propagation of time harmonic Rayleigh - Lamb waves in a bimaterial plate”. J. Acoust. Soc. Am. 89(4), 1606 – 1614, (1991). [2] M.V. Predoi, « Contribution au contrôle non destructif par ultrasons de structures planes. Aspects théoriques et expérimentaux ». Thèse Université Paris 6 (1998). [3] V. Pagneux, A. Maurel. “Lamb wave propagation in inhomogeneous elastic waveguides” Proc. R. Soc. Lond. A, 458, 1913-1930, (2002). [4] M.V. Predoi, M. Rousseau « Lamb waves propagation in elastic layers a joint strip ». Ultrasonics 43, 551-559, (2005). [5] W. B. Fraser, “Orthogonality relation for the Rayleigh – Lamb of vibration of a plate”, J. Acoust. Soc. Am., 59, 215 – 216, (1976).

BUS IEEE

Trigger

Oscilloscope Ordinateur

Aluminium Cuivre

Générateur

Signal Sortie

Vibrom

ètre

Transducteur de contact

Translation

h

Fig 3. Schéma du dispositif expérimental Fig 4. Comparaison entre numérique et expérience

1.5 1.55 1.6 1.65 1.7 1.750

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fe (MHZ mm)

Amplitude R(S0) étude expérimentale

T(S0) étude expérimentaleR(S0) étude numériqueT(S0) étude numérique


43

SEISMIC PERFORMANCE RELIABILITY ASSESSMENT FOR IRREGULAR REINFORCED CONCRETE BUILDINGS

A. Khamlichi

Analysis and Modeling of Systems Laboratory, Faculty of Sciences at Tetouan Road Sebta, M’Hannech II, Tetouan 93002, Morocco

[email protected]

1. Introduction Considering buildings at risk of earthquake events, seismic demand loads and structural capacity are random variables and vary as function of uncertainties governing the intervening basic design parameters. In particular, the ground motion acting on the building structure includes stochastic variables like the peak intensity, the frequency content as well as the seism duration. On the other hand, the nonlinear dynamic response of the structure depends on the configuration geometry of the building and the stochastic features of material properties such as stiffness, strength or damping. Taking into consideration all the uncertainties during seismic performance assessment with regards to a given limit state enables to render in a more realistic way the risk the building is undergoing. This is carried out in terms of probabilities within the framework of reliability analysis which makes it possible through uncertainty propagation to get estimation of the probability of failure [1-4]. Unlike the simple case where the building is regular, the general problem of irregular buildings at risk of earthquake events is characterized by the strong nonlinear dynamic behavior for which transient time-history analysis is needed. Since in this case explicit relationships between the input basic design variables and the structural response can not be established, it is not feasible deriving explicit forms of performance limit states. The calculation of failure probabilities relies then on either a lot of recurring simulations or on formulating a priori assumptions about the form of failure surfaces. The simulation based methodology is essentially performed by means of Monte-Carlo process [5] which does not require any assumption about the shape of the failure surface. However, the calculation of the most adverse response for each given state event necessitates the complete nonlinear dynamic history. This is a rather strict limitation as dynamic transient analysis of building structures is known to be computationally demanding. Furthermore, to get acceptable confidence in probabilities with required orders of accuracy a large number of runs are needed. To reduce the computational cost associated with Monte-Carlo simulation based methods, approaches using response surface approximation in order to represent the dynamic analysis results have been introduced. In conditions where the explicit approximation can be made to be sufficiently accurate, reliability estimation and the optimization task of performance based design can be simply conducted. However, the response surface needs to be adjusted to data, and these have to be developed by a priori runs of dynamic analysis where relevant set of combinations of the intervening variables should be used. Although this condition may be to some extent computationally challenging, the obtained response surface given in terms of the basic design variables permits absolutely quick and efficient reliability estimation. The objective of this work is to investigate feasibility of the response surface approach methodology [5-11] with the objective to assess reliability of performance based seismic design for irregular reinforced concrete buildings. The limit state regarding limitation of the maximum building roof displacement is considered and four factors are introduced in the analysis. Two of them are associated to the seismic demand while the two others are related to the building structural capacity. These factors include the seismic acceleration, the building mass, concrete members sections and steel reinforcements sections. A data set of results was obtained by using deterministic dynamic nonlinear computations which were performed according to a full factorial design of experiment table built on the four factors by choosing three levels for each factor. The results were used after that to derive a global quadratic polynomial surface interpolating the building roof displacement over the entire chosen range of design variables. This surface was used to examine the effect resulting from a particular choice of distributions of probabilities to model the random variables. A case study consisting of a 3-story reinforced concrete irregular building was considered. The building was subjected to permanent loads and to seismic excitation having intensities that are likely to occur in the northern zone of Morocco.



44

2. Nonlinear dynamic response analysis Nonlinear dynamic response analysis was performed here by means of ZeusNL software package [12-14]. ZeusNL is open source software which provides an efficient way to run structural analyses. The modelling takes into account both geometric and material nonlinear behaviour. The ZeusNL element library includes various element types that can be used to model structural elements such as beams or columns, non-structural elements like mass and damping and boundary conditions for supports and joints. Common concrete and steel material models are available, together with a wide choice of typical pre-defined steel, concrete and composite section configurations. The applied loading can include constant or variable forces, displacements or accelerations. In dynamic analysis, non-structural mass and damping elements are added to the finite element model, and the dynamic equation of motion can be solved by using Newmark time integration scheme as default option. Modelling of seismic action in ZeusNL is achieved by introducing acceleration loading at the supports. Gravitational loads are included in the analysis as static initial forces, taking into account their effect on the internal stresses in each element and their corresponding plastic behavior in critical cross-sections. In the present analysis cubic elastic plastic 3D beam-column element is used. This element gives a detailed inelastic modelling. It accounts for the spread of plasticity along the member length and across the section depth. To disretize masses, lumped mass element is used and no damping was considered. The concrete behaviour was chosen to be described by the nonlinear concrete model with constant active confinement modelling (con2) [12]. This enables accurate uniaxial concrete behaviour description where a constant confining pressure is assumed in order to take into account the maximum transverse pressure from confining steel. This is introduced on the model through a constant confinement factor denoted k and which is used to scale up the stress-strain relationship throughout the entire strain range. To enter this concrete model, four parameters are required: compressive strength cf , tensile strength ff , crushing strain 0cε and confinement factor k. The reinforcement

steel behaviour was assumed to be a bilinear elastic plastic model with kinematics strain-hardening (stl1) [12]. This model is applied for the uniaxial modelling of mild steel. To enter this model, three parameters are required: Young’s modulus E, yield strength yσ and kinematic strain-hardening µ.

There are two different convergence criteria in ZeusNL. The first is based on the norm of the out-of-balance forces. Convergence is attained when the norm is smaller than the tolerance defined in Settings. The second criterion, which is the default, is based on the maximum iterative increment of displacements. These should be lesser than the specified displacement and rotation reference values. 3. Probabilistic model for reliability performance seismic assessment The dynamic response of a structure subjected to seismic excitation depends on a large number of variables. To simplify the analysis in the context of seismic performance reliability assessment, only four factors are considered in the following. Let us denote x the vector of basic design random variables. This vector includes the peak ground motion acceleration denoted a, the mass factor denoted m, the concrete members section depth denoted c , and the steel reinforcement bars area denoted s. A single performance function g(x) is considered in this work under the following form

)()( lim xx δδ −=g (1)

where tshma ][=x is the vector of design variables with the exponent t designating the transpose,

limδ is the limit threshold displacement of the considered response being here the maximum building roof displacement )(xδ . The actual structural response )(xδ is obtained through a complete nonlinear dynamic analysis performed by ZeusNL for some given design variables vector x. The structural response will be approximately represented by an explicit polynomial function of x. The approximation is obtained through regression of the obtained results according to a full factorial design of experiment table constructed on the input variables x. A quadratic polynomial is chosen. The reliability probability for the considered limit state is obtained through evaluating the probability of the corresponding failure event g(x) < 0. This probability results from the multiple integral over the failure domain.


45

In the following, the First Order Reliability Method (FORM) is used to evaluate approximately the failure probability [2]. The main objective is to compare reliability estimates due to different distribution of probabilities describing the intervening random variable. The building dynamics is assumed to be represented accurately by the response surface approximation. Three different distributions of probabilities are investigated: Normal, Gumbel and Uniform. They are assumed to have the same means and the same standard deviations. 4. Case study The 3D model of the building as built under ZeusNL software is depicted in figure 1. The building is a three-story irregular structure with three vertical plane frames at each side. It lays on the rectangular surface9.2 m×11.7 m. The common height of stories is 3m. To enhance irregularity of the building a steel column, represented in red colour on figure 1, was introduced in the first story. Its cross section is square and have the dimensions: 25 cm× 25 cm. All the other columns have square cross sections of depth h × h. The material properties are assumed to be deterministic. The buildings columns and beams cross sections are assumed to be random and perfectly correlated among all the members. The depth of building structural members is assumed to be the same. Three levels were selected for this variable: h = 25 cm; h = 30 cm and h = 35 cm.

Figure 1: The finite element model of the considered irregular building

The steel reinforcement sections are also assumed to be random and perfectly correlated. They are proportional to the same reference section denoted s. This last was varied according to the following values: s = 250mm2 , 2 s = 300mm and 2 s = 350mm . The confined concrete properties are: cf =25MPa , tf = 2.5MPa , 0ce = 0.002 and k =1.2. The unconfined concrete properties are the same than the previous ones but with k =1.02. Steel characteristics are: 11 E = 2×10 MPa , yσ = 500MPa and µ = 0.01.

The seismic excitation is introduced in the analysis through using a scaled historical ground motion record: the Loma Prieta earthquake ground motion. This record is scaled to give the maximum seismic action expected to happen in the northern region of Morocco with a probability of exceedence higher than 50 years. Figure 2 gives the ground motion acceleration scaled to the maximal acceleration a =1.57 m/s2. During simulations intended to assess the influence of factors and to derive the response surface model, the maximum acceleration is assumed to take the following values: a =1.18 m/s2, a=1.57 m/s2 and a =1.96 m/s2. To perform reliability analysis, the peak acceleration is assumed to be a random variable having the mean value 1.57 m/s2 and the standard deviation 0.39 m/s2. The mean value of the total mass is obtained by considering the following combination G + 0.2Q , where G is the permanent load and Q the service load. The masses for all stories are assumed to be perfectly correlated. The mass scaled


46

levels that were considered are: m = 8, m =10 and m =12. The total mass is assumed to be proportional to a random variable having the mean value 10 and the standard deviation 2. The limiting value limδ is considered to be deterministic and was fixed in this work at the value

m14.0lim =δ . Other performance criteria that can be introduced to distinguish perform ance-based engineering states with regards to earthquake events such as those defined according to the Federal Emergency Management Agency [15].

Figure 2: The considered seismic accelerogram with its power spectral density Using three levels for each of the four design variable, 81 design points can be formed. Interpolating, in the mean square sense, the obtained maximum roof displacement results enables to determine the response surface which can be used to substitute the real dynamics of the building by a more simple metamodel. Accuracy of this model is tested by means of correlation coefficient, the obtained value of this coefficient in the present analysis was R2 = 0.84. It should be noticed however that the obtained response surface is valid only within the intervals used to derive the model. 5. Results Using regression of the obtained results, the building roof displacement can be interpolated as:

Giving the design point a =1.57; m =10; h = 30; s = 200 and the uncertainties means and standard deviations shown in table 1, reliability analysis can be performed by using Phimecasoft software [16].

Table 1: Characteristics of the chosen random variables

The obtained results in terms of the Hasofer-Lind reliability index and probability of failure as function of the chosen distributions of probabilities for each factor are as follows:


47

6. Conclusion Seismic performance of irregular buildings as affected by seismic action and geometry of building structural members was investigated. Reliability analysis of the performance function associated to restraining the building roof displacement was assessed. This was performed by means of surface response approach and FORM method. Four factors were considered. The obtained results in terms of probabilities of failure had shown that these last depend hugely on the chosen distributions of probabilities that describe the uncertainties. It is not sufficient to know the mean and the standard deviation, identification of the density of probabilities is also required. The Normal distributions of probabilities had provided the most severe case. References [1]Thoft Christensen P, Baker MJ. Structural reliability - theory and applications. Springer Verlag, 1982. [2]Melchers RE. Structural reliability-analysis and prediction. John Wiley and Sons, 1987. [3]Soares R., Mohamed A., Venturini W. and Lemaire M. Reliability analysis of nonlinear reinforced concrete frames using the response surface method. Reliability Engineering and System Safety, 75: 1 16, 2002. [4]Foschi R., Li H. and Zhang J. Reliability and performance-based design: a computational approach and applications. Structural safety 24: 205-218, 2002. [5]Hurtado J. Structural reliability - statistical learning perspectives. Lecture Notes in Applied and Computational Mechanics 17, Springer Verlag, 2004. [6]Wong FS. Slope reliability and response surface method. J Geotech Eng ASCE 111:32-53, 1985. [7]Faravelli L. Response surface approach for reliability analysis. J Eng Mech ASCE 115:2763–81, 1989. [8]Bucher C.G. and Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety 7:57-66, 1990. [9]Kim S.H. and Na S.W. Response surface method using vector projected sampling points. Structural Safety 19:3–19, 1997. [10]Möller O. Metodología para evaluación de la probabilidad de falla de estructuras sismorresistentes y calibración de códigos. Tesis de Doctorado en Ingeniería (PhD Dissertation), Universidad Nacional de Rosario, Argentina, 2001. [11]Möller O. and Foschi R. Reliability evaluation in seismic design. Earthquake Spectra 19:579–603, 2003. [12]Elnashai A.S., Papanikolaou V.K. and Lee D.H. Zeus NL A system for inelastic analysis: User Manual, Version 1.8.7, University of Illinois at Urbana Champaign, Mid-America Earthquake Center, 2008. [13]Haukaas T. and Scott M.H. Shape sensitivities in the reliability analysis of nonlinear frame structures. Computers and Structures 84: 964-977, 2006. [14]Haukaas T. and Der Kiureghian A. Finite Element Reliability and Sensitivity Methods for Performance- Based Earthquake Engineering. PEER Report 2003/14 Pacific Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley, 2004. [15]Federal Emergency Management Agency, FEMA-356, 2000. Prestandard and Commentary for the Seismic Rehabilitation of Buildings. ASCE, Federal Emergency Management Agency, Washington, DC. [16]Lemaire M. and Pendola M. phimeca-soft. Structural Safety 28: 130-149, 2006.


48

VALIDITE DE LA METHODE DE JUANG POUR PREDIRE LE POTENTIEL DE LIQUEFACTION DES SOLS DE TANGER

Naoufal Touil, Abdellatif Khamlichi, Abdallah Jabbouri, Mohammed Bezzazi Laboratoire AMS, Faculté des Sciences de Tétouan, BP. 2121 M’Hannech,

Tétouan 93002, Maroc

1. Introduction La liquéfaction est un phénomène dans lequel la capacité portante d'un sol est réduite à zéro à cause de l’effet dynamique lié à la secousse sismique ou pouvant résulter de tout autre chargement dynamique, intense et rapide. Elle se produit seulement dans les sols saturés. Le phénomène de liquéfaction présente un intérêt considérable à cause de l’impact destructif qu’il peut engendrer sur les infrastructures, sur l’activité économique et sur la vie de l’Homme. Parmi les travaux consacrés au problème de liquéfaction des sols, on peut citer Dobry [1] qui a introduit l’approche de type déformation, Seed et Harder [2] qui ont introduit l’approche dite de contrainte et Park et al. [3] qui ont introduit l’approche énergétique. Des corrélations entre la résistance de pointe mesurée par l’essai au pénétromètre statique et le rapport de résistance au cisaillement cyclique qui caractérise la tenue à la liquéfaction ont été introduites par Robertson et Wride [4]. Des démarches de modélisation de type milieu continu ont été introduites par Zienkiewicz et al. [5]. La résistance à la liquéfaction des sols est en général évaluée à partir de méthodes et procédures développées en se basant sur des essais in situ. L’essai au pénétromètre dynamique SPT (Standard Penetration Test) et au pénétromètre statique CPT (Cone Penetration Test) constituent les deux essais de référence les plus utilisés dans le domaine de l’analyse du potentiel de liquéfaction. Ces essais sont réputés fiables et permettent moyennant des formules de corrélation empiriques d’estimer le potentiel de liquéfaction. Ces formules dépendent du site où elles ont été établies. Les adapter à un autre site tel que le cas des sols de Tanger nécessite une étude préalable de validation. Dans un travail antérieur [6], nous avons comparé les différentes approches permettant de prédire le potentiel de liquéfaction des sols se trouvant dans la région de Tanger et nous avons mis en évidence le mérite de la méthode de Juang qui s’est avérée à la fois sécurisante et moins sensible aux incertitudes qui peuvent affecter les paramètres du sol. Le but de ce travail est de qualifier cette méthode semi-empirique en comparant les prédictions qu’elle permet d’obtenir avec les résultats d’un calcul transitoire bidimensionnel utilisant un modèle physique complet. La modélisation du phénomène de liquéfaction à partir des équations physiques permet d’introduire les paramètres réels du problème tels que l’amortissement, la durée du séisme, le spectre et le nombre de cycles du séisme afin d’estimer de manière explicite leur influence. Le logiciel DeepSoil [7] est utilisé pour développer le modèle numérique. Le choix d’un cas d’étude s’est porté sur le site qui abritera le futur complexe touristique City Center à Tanger, qui présente un risque de liquéfaction à cause de la nature particulière du sol se trouvant sous sa zone d’implantation. 2. Méthode semi-empirique de Juang Les facteurs majeurs contrôlant la liquéfaction des sols cohérents saturés sont la durée et l’intensité des mouvements sismiques, la densité du sol et la pression de confinement effective. Afin de caractériser la réponse du sol sous l’effet des sollicitations sismiques qui sont de nature cyclique, un certain nombre de méthodologies ont été développées [2]. Le but consiste à qualifier le risque de liquéfaction d’un sol donné. Actuellement, les études dans ce domaine sont basées sur trois approches : - Approche par contraintes cycliques. - Approche par déformations cycliques. - Approche par intensité d’alias. Les méthodes basées sur les contraintes cycliques et les déformations cycliques ont été développées à l’origine à partir d’essais de laboratoire. Mais étant donné que la réponse cyclique des sols est contrôlée par des facteurs tels que la nature du sol, les pré-déformations, l’histoire de chargement, et


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les effets de vieillissement qui ne peuvent pas être reproduites de manière fidèle au laboratoire, on a eu recours à des relations empiriques développées à partir des paramètres acquis in-situ. Ces paramètres sont obtenus à partir d’essais standards très répandus, à savoir l’essai au pénétromètre dynamique SPT qui permet de caractériser la résistance dynamique du sol et l’essai au pénétromètre statique CPT qui est un essai rapide et économique permettant lui de déterminer la résistance statique à l’enfonçage. Le domaine d’utilisation de ces deux essais se limite cependant à des profondeurs ne dépassant pas 30 m; au-delà, un risque de flambement de la tige apparaît. Ces essais ne s’appliquent par ailleurs qu’aux sols dont la dimension des plus gros éléments ne dépasse pas 2cm. L’essai SPT permet d’extraire un échantillon conforme au sol traversé qui peut être ainsi utilisé pour mesurer les autres caractéristiques mécaniques du sol examiné, alors que le CPT ne le permet pas. Pour représenter de manière simplifiée les mouvements du sol dus à un tremblement de terre au moyen d’un seul paramètre, une procédure a été développée par Seed et Harder [2]. Le potentiel de liquéfaction est évalué en comparant un indice normalisé lié à la résistance cyclique du sol RCR (capacité) au rapport de contraintes cycliques RCS (demande) régnant dans le sol. Ce qui permet d’évaluer le facteur de sécurité FS selon

Le rapport RCS s’exprime sous la forme

où tav est la contrainte de cisaillement moyenne due au séisme à la profondeur considérée, amax est l'accélération maximale à la surface du sol, g l'accélération de pesanteur, svo la contrainte totale verticale à la profondeur considérée, s'vo la contrainte verticale effective à la profondeur considérée et rd le facteur de réduction de contrainte. Le facteur rd est fonction de la profondeur z. Seed et Harder [2] ont donné une formule explicite qui permet de calculer la valeur moyenne de ce facteur en fonction de z exprimé en m. La magnitude du tremblement de terre influence la durée de la secousse, et peut augmenter ainsi significativement le nombre de cycles de contraintes. L’effet de l’amplitude du tremblement de terre n’est pas inclus dans l’équation (1). Pour pouvoir rendre compte de ce phénomène, un facteur d’échelle d’amplitude noté MSF (Magnitude Scaling Factor) a été introduit. L’amplitude de référence pour l’analyse basée sur l’approche par contraintes cycliques est fixée à 7.5 selon l’échelle de Richter. Différentes expressions ont été proposées pour définir le coefficient MSF. Ce facteur est calculé en fonction de la magnitude M du séisme retenu pour l’analyse du risque de liquéfaction, ce qui permet de normaliser le rapport RCS selon l’équation

La façon d’évaluer le rapport de résistance au cisaillement cyclique RCS dépend des essais réalisés. Différentes méthodes ont été ainsi proposées pour évaluer RCS. Dans le cas de méthode de Juang le coefficient RCS est exprimé de la façon suivante

Avec

où Ic est l’indice de comportement du sol calculé selon la méthode décrite dans la référence [7]. 3. Simulations numériques sous DeepSoil Le logiciel DeepSoil est utilisé dans la suite pour développer le modèle 1D. Ce logiciel qui permet de faire une analyse non linéaire ou linéaire équivalente a été développé par Hashash [7] dans l’université d’Illinois à Urbana-Champain.


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On modélise le sol comme une colonne composée de plusieurs couches dont les paramètres sont identifiés à partir d’essais de laboratoire et d’essais in-situ. On définit ensuite les conditions aux limites en précisant le niveau de la nappe phréatique, puis on applique au niveau du substratum le signal d’accélération sismique reproduisant un séisme type. On sélectionne le type d’analyse et les lois de comportement que l’on désire utiliser. L’analyse du potentiel de liquéfaction se fait sur les courbes de réponse donnant la pression interstitielle en fonction du temps. Si ces courbes présentent un point d’inflexion, il y a risque de liquéfaction, autrement le risque de liquéfaction est écarté. Nous utilisons pour décrire le comportement du sol le modèle hyperbolique dépendant de la pression. Il permet d’introduire un facteur de réduction du module de cisaillement sous la forme

où P1, P2 et P3 sont des paramètres à identifier pour le type de sol donné. Le modèle permettant de décrire la génération de la pression interstitielle est celui de Matasovic et Vucetic [8]. Il décrit l’évolution de la pression interstitielle uN par

où Nc est le nombre de cycles, tupγ le seuil de la déformation de cisaillement, ctγ la dernière

déformation connue avant changement de signe. Le coefficient tupγ est compris entre 0.01% et 0.02% pour la plupart des sables. Les paramètres p, s et

F permettent d’ajuster le modèle aux résultats expérimentaux. Des essais triaxiaux cycliques non drainés sont nécessaires pour cela. 4. Résultats Afin de comparer les prédictions de la méthode semi-empirique de Juang avec les résultats du modèle numérique, on choisit le site destiné à abriter les locaux de Tanger City Center pour lequel des essais au pénétromètre statique CPT ont été effectués. Le facteur de sécurité de Juang vis-à-vis de la liquéfaction est calculé comme mentionné précédemment. La figure 1 présente ce facteur en fonction de la profondeur. On remarque l’existence de points pour lesquels ce facteur est inférieur à 1, indiquant que le sol est susceptible de se liquéfier (sous l’effet du séisme de calcul qui est de magnitude 7.5 selon Richter). La figure 2 présente les courbes donnant la pression interstitielle en fonction du temps pour différentes profondeurs. Ces courbes ont été obtenues en utilisant le logiciel DeepSoil avec un amortissement de 4% et une fréquence de 10 Hz. La magnitude du séisme a été ajustée à la valeur 7.5 selon Richter.

Tableau 1: Comparaison des deux méthodes

Pour les profondeurs de 4.7 jusqu’à 10 m, on observe une augmentation de la pression interstitielle à partir de l’instant 3s jusqu’à 8s. La pression atteint la valeur maximale 0.4 avant de commencer à diminuer. A partir de 11.50m jusqu’à 14.50 m, on observe le même phénomène entre 3s et 6s et la pression limite est 0.33.


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Figure 1: Variation du facteur de sécurité de Juang en fonction de la profondeur

Ces résultats montrent l’apparition du phénomène de liquéfaction à 12 secondes pour les couches peu profondes, à 8 secondes pour les couches de profondeur moyenne et à 6 secondes pour les couches profondes. Ce qui montre que la liquéfaction se produit d’abord au niveau des couches superficielles avant d’atteindre les couches profondes. Le potentiel de liquéfaction tel qu’il est prédit avec la méthode de Jaung prédit une liquéfaction à partir de 2m de profondeur. En écartant la première couche pour laquelle le niveau de la nappe phréatique ne permet pas une saturation totale du sol et pour laquelle le risque de liquéfaction n’est pas concret, la méthode de Juang prédit correctement la tendance du sol à se liquéfier.

Figure 2 : Variation de la pression interstitielle en fonction du temps pour différentes

profondeurs du sol 5. Conclusion La comparaison des prédictions effectuées par la méthode semi-empirique de Juang et des résultats de simulation numérique établis avec le logiciel DeepSoil montre une bonne concordance entre les deux approches. Ceci permet de constater la capacité de la méthode semi-empirique de Juang à bien prédire le risque de liquéfaction dans le cas particulier des sols de Tanger. Ce résultat constitue une étape importante pouvant conduire plus tard à sa validation définitive. References [1] Dobry R. Soil properties and earthquake ground response, Volume 4, Proceedings of the Tenth European Conference on Soil Mechanics and Foundation Engineering, 1171-1187, 1994.


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[2] Seed H.B., and Harder L.F. Jr. SPT-based analysis of cyclic pore pressure generation and undrained residual strength. Proc., H. Bolton Seed Memorial Symposium, Vol. 2. BiTech Pub., Richmond, British Columbia, 351-376, May 1990. [3] Park K., Kim S., Lee J., and Park I. Energy-based evaluation of excess pore pressure using damage potential. International Journal of Offshore and Polar Engineering, 2007. [4] Robertson P.K and Wride C.E. Evaluating cyclic liquefaction potential using the cone penetration test Canadian geotechnical journal 35 (3): 442-459, 1998. [5] Zienkiewicz O.C., Chan A.H., Pastor M., Paul D.K. and Shiomi T. Static and dynamic behaviour of soils: a rational approach to quantitative solutions. In fully saturated problems. Proc. R. Soc. London. 429, 285-309, 1990. [6] Touil N., Khamlichi A., Jabbouri A., Bezzazi M. Adaptation d’approches en vue d’évaluer le potentiel de liquéfaction des sols de Tanger. CFM’2009, 23-28 Août 2009, Marseille, France, 2009. [7] Hashash Y.M.A, Groholski D.R., Phillips C. A., Park D. DEEPSOIL V3.7beta, User Manual and Tutorial. 88 p, 2009. [8] Matasovic N. and Vucetic M. Cyclic Characterization of Liquefiable Sands. ASCE Journal of Geotechnical and Geoenvironmental Engineering, 119(11):1805-1822, 1993.


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GENERALIZED DAHL’S MODEL AND APPLICATION TO

NONLINEAR MECHANICAL COMPONENTS

Régis Dufour Université de Lyon, INSA-Lyon, LaMCoS, UMR5259, F-69621 Villeurbanne, France

Abstract A generalized modified Dahl model and its application on several applications are presented: belttensioner and hexapod actuator. The objective is to demonstrate that this restoring force model is efficient and easy to use and can advantageously replace rheological models. It is described by a first order differential equation coupled with equations of motion. The non linear response prediction requires an integration method in the time domain. 1- Presentation of the modified Dahl model The non linear behavior of mechanical components can be modeled either by parametric models or nonparametric models, (see the overview of Vestroni and Noori, 2002, and Ibrahim 2008). The former provide stiffness and damping parameters that are introduced in the first member of the equation of motion while the latter the restoring force introduced in the second member. Mention can be made of the models developed by Masing and Dahl. Most of the time, designing mechanical components is rather complicated because they can be an assembly of several components having different shapes and materials, (Gjika et al. 1996, Gjika and Dufour, 1999). Therefore establishing a Finite Element model is not obvious for such cases. Consequently, it is logical to extract the required model parameters from measured force-deflection loops with hysteresis behavior and non linear dependences regarding temperature, forcing frequency and, above all, the deflection (Nashif et al., 1985). Such non linear behavior makes the prediction of the global assembly delicate, (Lacarbonara and Vestroni, 2003). In order to obtain the most general and easily formulated model possible, the aim is to model the force deflection loop based on the envelope curves. Existing models are limited when formulated from the envelope curves of the loop, which can be temperature, frequency and amplitude dependant. Al Majid and Dufour (2002 & 2004) proposed a modified Dahl model for force deflection loop. In Saad et al., (2006), the modified Dahl model was used and compared to an equivalent rheological model for modeling a cylinder and a plate made of carbon black filled rubber. Different behaviors such as softening, hardening or a combination of both can be modeled. A simplified formulation permits expressing it as follows:

with R the restoring force of the nonlinear component, u its deflection, β a constant. The function, given by the following equation,

depends on the sign of the velocity and could be either the upper uh or lower lh envelope curves which

can be polynomial form. Measuring the force-deflection loops permits the identification of these parameters, in particular the envelope curves. In the presence of m non linear components, m Eqs. (1) and (2) have to be available. A step-by-step time integration scheme is necessary for predicting the transient and harmonic forced response. At each time step, the coupling of the equation of the motion with the m Eq (1) permit updating the restoring force vector R. The predicted harmonic response of the beam-isolator system obeys the experimental swept sine investigation. It is performed with the step-by-step Runge-Kutta method. The forcing frequency is incremented after steady-state behavior is reached and the amplitude of displacement recorded.


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2- A mechanical belt-tensioner The tensioner used in belt drive system plays a predominant role in the dynamic behavior of the transmission belt: it maintains nominal tension in the slack span and reduces the transverse vibration level, especially in an automotive serpentine belt drive system (Beikmann et al., 1997). This type of mechanical system, known as a front-end accessory drive (FEAD), gives rise to many current technological challenges and scientific problems (Parker, 2004) and its design design leads to considerably nonlinear behavior mainly due to stick-slip motion (Leamy and Perkins, 1998). Figure 1a shows the tensioner tested. It is used in an automotive FEAD. Its stick-slip behavior is due to its specific design: it is composed of two collinear shafts on which idler, torsional spring and end-stops are connected by either oiled or dry contacts.

Figure 1. Tested mechanical belt tensioner (a). Force deflection loops given by the Dahl and Masing

models from the experimental identification (b) The tensioner is implemented on an experimental test bench, Michon et al., (2005). Its base is fixed on a very stiff frame, while its eccentric shaft is linked to a slider-crank mechanism activated by an electrical motor. The measured force-deflection loop permits providing a linear approximation of the envelope curves of the modified Dahl model. Moreover it gives the parameters of a proposed Masing model with damping presented in Bastien et al, (2007). Figure 1b compares the numerical and measured loops obtained. Now the tensioner combined with a belt and a mass constitutes a simple mechanical system for analyzing the nonlinear effect on the response to an external excitation. The experimental set-up is shown by Fig 2a. The tensioner base is fixed on a rigid frame. Its idler pulley of mass m2 = 0.15kg has a multi-ribbed belt wrapped around it. The two adjacent belt spans are joined at their other end and connected to mass m1 = 73.84kg subjected to harmonic force f generated by an electro-dynamic shaker and measured by a piezo-electric load cell. Belt tension is recorded by using a S-shape load sensor. Axial displacements u1 and u2 of masses m1 and m2 are measured with laser-optical displacement sensors.

Figure 2. Experimental set up (a) and its diagram for the axial belt-tensioner system


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The associate model is illustrated by the diagram in Fig 2b. Let g , T , 0 T be the gravity, tension and initial tension and let the belt be considered as a spring-damper of stiffness K = 0.56106N/m and equivalent viscous damping C =160Ns/m . The motion of the mass-belt-tensioner system is governed by the following set of differential equations:

which coupled with Eq. (1) permit predicting the forced response. Figure 3 shows the belt-tension measured and predicted by using either the modified Dahl model or the Masing model with viscous damping.

Figure 3. Belt tension versus forcing frequency for several increasing forcing amplitude: measured

(a), predicted with the modified Dahl model (b) and with the Masing model with damping (c) The computed frequency response represented is obtained after a series of calculations in the time domain: each point of a frequency response curve corresponds to the tension fluctuation amplitude calculated when steady state is reached for a given frequency and excitation amplitude. The solid lines correspond to the force amplitude imposed experimentally ( f∈ [13, 27, 41, 54, 67, 79, 90, 100, 110, 120 N]), while the dotted-dashed lines to higher force amplitude ( f∈[140, 160, 200, 230, 260, 300 N]). It can be observed experimentally that even if the forcing amplitude increases, the resulting belt tension variation is bounded within a frequency range. This phenomenon is predicted better with the modified Dahl model rather the Masing model with damping. This is mainly due to the lack of reliability of the numerical integration of the non-smooth differential equation in the Masing model. The stick-slip behaviour is quite well predicted by the two models: for small forcing amplitude the tensioner provides a stuck stiffness higher than the slip stiffness available for higher forcing amplitudes.

3- A passive actuator for hexapod Thales Alenia Space designed a prototype of a deployable hexapod, Fig. 4, for the future development of a deployable telescope designed to stow the secondary mirror during launch and its self deployment in orbit (Blanchard et al., 2005). The prototype has six deployable rolled tape-springs, see Fig. 5. When the stowing mechanism is released, the six passive actuators operate autonomously to bring the play load platform (diameter 0.25 m, mass 1 kg) into its final position (0.5 m from the base platform).

Figure 4. Self-deployment scenes of the tape-spring hexapod prototype

Each tape-spring is a thin curved metallic strip capable of keeping its straight configuration (Seffen and Pellegrino, 1997). Tape-spring coiling device (or passive actuator) contains a rotating roll module with spiral grooves that guide the tape-spring under a dry-friction (see Figure 5b). Coiling the strip


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induces flattens the natural curved section and generates a transition area that stores the strain energy making self deployment possible. The actuator is considered as a whole system with a restoring force model based on the modified Dahl model, see Eq. (2). The force-elongation loops are measured for different operating conditions with an experimental set-up developed specifically, see Aridon et al. 2009 and 2009. The path is anti clockwise: the self-unrolling-load is lower than the necessary rolling load. Figure 5c shows that the measured hysteresis phenomenon is captured well by the modified Dahl model. The hexapod model, designed as a Gough-Stewart parallel robot, is composed of an upper platform linked to the base by six passive actuators with prismatic joints. End-stops are modeled by adding springs and dampers. The model of the hexapod dependant on the deployment is implemented with Matlab SimMechanics. This performs the forward dynamic analysis of the manipulator by using the recursive Newton-Euler technique. A diagram of the hexapod model is presented in Fig 6a.

Figure 5. Deployed tape-spring hexapod prototype (a). Tape-spring coiling device (b). Force-

deflection loops measured (at 1 Hz) and predicted by the modified Dahl model (c)

Figure 6. A scheme of the hexapod prototype modeled with Matlab-SimMechanics©, (a). Deployment

experimental set-up, (b). Measured (Test 1, Test 2) and predicted self deployment. The experimental set-up, see Fig. 6b, takes into account gravity compensation by using a counter weight pulley equivalent to platform mass. The upper platform is kept in its theoretical stowing configuration with a tightened nylon wire clamping it against the uppermost surface of a centuring tube. The length of the latter is used to adjust the initial deployment height. Three vertical “handrail” ensure the position of the end-damper. The stowed platform is released by burning the nylon wire and its trajectory is recorded by using a high speed camera (250 frames per second) and a digital image correlation method. Figure 6c focuses on the measured and predicted vertical component of the self-deployment. Two experimental tests were carried out. The platform elevation behaviour predicted is close to that measured. A slight deviation due to the curve shape should be mentioned. This is certainly due to the characteristic dispersion. In fact all the actuator characteristics are identical, since the experimental


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identification was conducted on only one item. Nevertheless, the comparison leads to the conclusion that the modified Dahl model proposed is satisfactory. 4. Conclusions It is possible to use nonlinear system as passive actuators whose nonlinear parameters can be well adapted to device with changing characteristics. Modeling nonlinear components has become quite easy to carry out with restoring force models such as the modified Dahl model presented in the two illustrations. Now, the question is how to find the adequate nonlinearities that lead the responses expected. This could be done by using inverse problem techniques not easy to apply in nonlinear dynamics. Acknowledgments The results presented here stem from research mainly performed in the framework of different thesis PhD theses. Therefore the author would like to thank Dr A. Al Majid, Dr G. Aridon, Dr G.Michon. Thanks are also extended to Valeo SE, Thalès Alénia Space, and the Rhône-Alpes Regional Council for their support. References Al Majid, A., and Dufour, R., 2002, “Formulation of a hysteretic restoring force model. Application to vibration isolation”, Nonlinear Dynamics, Vol. 27, pp. 69-85. Al Majid, A. and Dufour, R., 2004, “Harmonic response of a structure mounted on an isolator modelled with a hysteretic operator: experiments and prediction”, J. of Sound and Vibration, Vol. 277 (1-2), pp. 391-403. Aridon G, Rémond D., Morestin F, Blanchard L., and Dufour R., 2009, “Self-deployment of a tape spring hexapod: experimental and numerical investigation”, ASME J. of Mechanical Design, 131, 021003, 8p. Aridon G., Al Majid A., Rémond D., Blanchard L. Dufour R., 2009, “A self-deployment hexapod model for a space application”, ASME Computational and Nonlinear Dynamics, 4, 011002, 7p. Bastien J., Michon G., Manin M., and Dufour R., 2007, “An analysis of the Masing and Modified Dahl models. Application to a belt tensioner”, J. of Sound and Vibration, Vol. 302, 841–864. Beikmann, R.S., Perkins, N.C. and Ulsoy, A.G., 1997, “Design and Analysis of Automotive Belt Drive Systems for Steady State Performance”, ASME J. of Mechanical Design, Vol. 119, pp. 162 168. Blanchard L., Falzon F., Dupuis J., and Merlet J.P., 2005, “Deployable hexapod using tape-springs. In Disruption in Space”, ESA/CNES Symposium, Marseille, France. Bolotin, V. v., 1964, “The Dynamic Stability of Elastic Systems”, Holden-Day, Inc. Cartmell M., 1990, Introduction to linear, parametric and nonlinear vibrations, Chapman and Hall. Craig, R.R., Bampton, M.C.C., 1968, “Coupling of substructures for dynamic analysis”, AIAA Journal, Vol. 6(7) Gjika K., Dufour R., and Ferraris G., 1996, “Transient response of structures on viscoelastic or elastoplastic mounts: prediction and experiment” J. of Sound and Vibration, Vol. 198(3), pp. 361- 378. Gjika K., and Dufour R., 1999, “Rigid Body and nonlinear mount identification: Application to on board equipment with hysteretic suspension”, J. of Vibration and Control, Vol. 5(1), pp. 75-94. Ibrahim, R.A., 2008, “Review: recent advances in nonlinear passive vibration isolators”, J. of Sound and Vibration, Vol. 314, pp. 371–452. Lacarbonara, W., Vestroni, F., 2003, “Nonclassical responses of oscillators with hysteresis”, Nonlinear Dynamics Vol.32, pp. 235–258. Leamy, M. J., Perkins, N. C., 1998, “Nonlinear periodic response of engine accessory drives with dry friction tensioners,” ASME J. of Vibration and Acoustics, Vol. 120, pp. 909–916. Michon, G., Manin, L., and Dufour, R., 2005, “Hysteretic behaviour of a belt tensioner: modelling and experimental investigation”, J. of Vibration and Control, Vol. 11 (9), pp. 1147-1158. Nashif, D., Jones, D.I.G. and Henderson, J.P., 1985, Vibration damping. New York: Wiley. Parker, R. G., 2004, “Efficient eigensolution, dynamic response, and eigensensitivity of serpentine belt drives,” J. of Sound and Vibration, Vol. 270, pp. 15–38.


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Saad, P., Al Majid, A., Thouverez F., and Dufour, R., 2006, “Equivalent rheological and restoring force models for predicting the harmonic response of elastomer specimens”, J. of Sound and Vibration. Vol. 290, 619-639. Seffen, K. A., Pellegrino, S., 1997, “Deployment of a rigid panel by tape-springs“, report CUED/DSTRUCT/TR168, Department of Engineering, University of Cambridge, UK. Vestroni, F., Noori, M., 2002, “Hysteresis in mechanical systems - modeling and dynamic response”, International J. of Non-Linear Mechanics, Vol. 37, pp. 1261–1262.


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CONTROLE DE SANTE DE STRUCTURES COMPLEXES : METHODOLOGIES ET MISES EN ŒUVRE

M. N. Ichchou

Ecole Centrale de Lyon, France Les travaux présentés dans cet exposé rentre dans le cadre de contrôle de santé structurale (SHM: Structural Health Monitoring). Les travaux consistent aux développements de méthodes de diagnostique et de reconnaissance des défauts et ceci en présence du pas d’un gradient thermique. L’analyse consiste à identifier les défauts, les localiser et en connaître la sévérité. Les techniques d’analyses vibratoires utilisées pour la détection des défauts dans les structures utilisent des méthodes basées sur les ondes guidées. Ces méthodes sont simples mais se heurtent néanmoins à plusieurs types de difficultés. Ces difficultés sont liées aussi bien à la génération de certains types d’ondes ainsi qu’au caractère hautement multimodale des structures (un grand nombre d’ondes guidées de nature différentes coexistent à la même fréquence). La génération des ondes guidées se réalise souvent en appliquant une tension aux bornes d’un transducteur piézoélectrique céramique par exemple. Ces ondes se propagent sur toute la structure et seront réfléchies au niveau des singularités. Les défauts forment des singularités particulières sur lesquelles les ondes générées se diffractent. C’est l’analyse de ce comportement qui permet de reconnaître et d’identifier les défauts. L’apport de la simulation numérique avancée de la propagation/diffusion des ondes s’avère indispensable à la mise en place et à l’optimisation des procédés de contrôle de santé. Précisément, l’objectif de ce travail consiste à étendre une formulation WFE (Wave Finite Element) développée dans pour des structures déterministe sans gradient thermique afin de tenir compte des incertitudes dans les structures (géométrique et des propriétés des matériaux,…) dans un environnement thermique. La prévision de l’effet de ces perturbations sur la propagation d’onde et leurs diffusions aux singularités sont les résultats les plus importants de ce travail. Ces résultats permettront d’imaginer des dispositifs de contrôle de santé de structures de types pipelines sous conditions réalistes. La méthode WFE (Wave Finite Element) déterministe permet de prédire les différentes propriétés des ondes propagées dans les structures (nombre d’onde, vitesses de propagation) ainsi que les différents facteurs de réflexion et de transmission de ces ondes aux interfaces. C’est un outil purement numérique qui fait appel à un modèle très réduit de type éléments finis du milieu de propagation et de la singularité à traiter. Les voies d’extension de l’approche WFE aux structures incertaines sera d’abord opéré. Le couplage de cette technique avec l’un des outils de mécanique probabiliste (technique de perturbation, éléments finis spectrales stochastiques, …) sera réalisé. Une approche au premier ordre ayant déjà été établie, le travail étendra cette analyse afin de tenir compte des ordres élevés. La sensibilité des modèles de propagation et de diffusion aux singularités aux incertitudes mécaniques et géométriques sera ainsi analysée. La prise en compte d’un environnement thermique sera ensuite opérée. Deux approches seront testées. La première consistera à tenir compte des effets thermiques par évaluation des marges d’incertitudes sur les paramètres géométriques et mécaniques. L’utilisation des résultats de la formulation stochastique permettra ainsi d’évaluer la sensibilité des paramètres de propagation et de diffusion aux gradients thermiques. La seconde approche sera plus directe. Elle consistera à réécrire la formulation WFE avec des champs mécaniques et des champs thermiques. L’approche thermomécanique permettra alors de traiter le problème couplé. Cette seconde approche devra permettra également d’évaluer les différents niveaux d’approximation que l’on peut envisager pour simplifier le traitement.


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ACOUSTIQUE POUR LE BATIMENT

Omar Debbarh Faculté des sciences Aïn Chock, Casablanca

Dans un premier temps, on définit les caractéristiques physiques d'un bruit, telles que la pression acoustique, l'intensité et la puissance acoustiques, ainsi que les niveaux acoustiques. On montre l'analyse des bruits stables (analyse d'octave et de tiers d'octave) et celle des bruits instables (notion du niveau équivalent Leq), la sensation physiologique du bruit, la pondération et la mesure des bruits. Ensuite, on étudie la propagation sonore en champ libre, puis en espace clos, avec notamment la notion de réverbération et de coefficient d'absorption de sabine. On passe ensuite au traitement acoustique des locaux sous ses trois aspects : géométrique, ondulatoire et statistique, avec la justification du choix des matériaux absorbants parmi les fibreux, les panneaux fléchissants et les résonateurs. Enfin, on étudie la transmission des bruits aériens en définissant notamment le facteur de transmission d'une paroi, l'indice d'affaiblissement, les isolements brut et normalisé, et en comparant les indices d'affaiblissement des parois simples, multiples et composites. Et on finit par l'influence des transmissions latérales.

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