Design of viscoelastic damping for

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Structural and Material Computational Mechanics department

Thèse originale présentée en vue de l’obtention du grade de Docteur en Sciences Appliquées

Année académique 2006-2007

Promoteur: Prof. Philippe BOUILLARD Co-promoteur: Prof. Jean-Louis MIGEOT

Laurent Hazard

Design of viscoelastic damping for

vibration and noise control: modelling, experiments and optimisation

- Almost the last version ...

December 9, 2006

Universit´e Libre de Bruxelles

Structural and Material Computational Mechanics department

Design of viscoelastic damping for vibration and noise control:

modelling, experiments and optimisation

Laurent Hazard

Committee in charge:

XXX Chairman

XXX Reporters

XXX

XXX Examiners

XXX XXX

XXX Advisor

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Laurent Hazard

Design of viscoelastic damping for

vibration and noise control: modelling, experiments and optimisation

December 21, 2006

Universit´e Libre de Bruxelles

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Acknowledgements

A PhD thesis is a very personal work but, at the same time, involves so many people that I ﬁnd of capital importance to acknowledge their contributions.

First of all, this research work would not have been possible without the support, motivation and strong faith in the project of a few people: Dr.

Jean-Yves Sener (ex. Arcelor), Prof. Jean-Louis Migeot (ULB, FFT) and Prof. Philippe Bouillard (ULB). Thank you all for making it happen...

I am grateful to both the Arcelor group and the R´egion Wallonne for sharing the ﬁnancial support of this thesis. I would particularly like to acknowledge Mr Raymond Montfort (DGTRE), for showing a real interest in the project and always asking pertinent questions during meetings.

I would like to thank all the people at Arcelor Li`ege who where involved in this research: Dr. Jean-Yves Sener, Emmanuel Bortolloti, Muriel Chaidron and Jacques Mignon.

I am deeply grateful to the Structural and Material Computational Me- chanics department of the Universit´e Libre de Bruxelles, for allowing me to make my comeback to the academic world after many years of industrial wan- derings. I especially thank Prof. Guy Warz´ee, head of the department, for his kindness and patience. I would like to thank Prof. Philippe Bouillard for his interest in the project from the start and for his personal involvement as thesis advisor. I would also like to express my gratitude to Prof. Jean-Louis Migeot for his support and for accepting the role of co-advisor.

I thank Guy-Michel Hustinx and Philippe Lemaire from OPTRION SA, for the help they gave me during the setup of the experimental measurements.

I am also grateful to Dr. Ezio Gandin from the Solvay Central Laboratory, for his valuable recommendations concerning the characterisation of viscoelastic materials.

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Acknowledgements

I am, of course, indebted to all my collegues and friends, past and present, at the ULB for making this short moment in my life a great and unforget- table journey. I would like to thank Tanguy, Genevi`eve, Katy, Yannick, Kﬁr, Benoit, Louise, Berta, Adama, Peter, Vincent, Erik, Nathalie, Guy and others that I apologise for not naming.

A special “Thank you !” to Guy Paulus for his great support in bothhard skills (such as L^ATEX or UNIX) and soft skills (such as english grammar).

He contributed immensely to the readability of this dissertation and it was always a pleasure to share long discussions with him about physics, politics and other “-ics” .

I would also like to thank all my friends for their support and encourage- ments. Laurent, Carine, Philippe, C´ecile, Eric, Vincent: I am lucky to know you all and that you still like to spend time with me, even if I am obsessed with physics, mathematics and punk-rock music...

Finally, I would like to express a very warm thanks to my family. My parents have always supported me in all my adventures, even when I decided to start this PhD research. I know that I can always rely on them and I am proud of the education and values they gave me.

Last but not least, I want to thank Julie for her constant support and Nathan¨ael, simply for being the most wonderful little boy I know. We shared together good and hard times during these last years. Too often, I had so many things to do and so little time that it all perturbed our life as a family.

Julie handled these moments with care and patience and I am immensely grateful to her for all she has done and continues to do...

Laurent Hazard, Brussels, December 2006.

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1 INTRODUCTION

Nowadays, the acoustic and vibrational properties are increasingly considered as key constraints during the design process of new products. Many factors explain this evolution:

• public realisation of the importance of noise as a major pollution and annoyance source;

• the publication of extremely rigorous noise emission norms, speciﬁcally at the european level;

• the emergence of the sound quality notion among the consumer’s choice criteria.

Consequently, market pressure and regulations dictate quiet products. Steel- makers, especially, are experiencing hard times: the competition with other materials, like aluminium and polymers, is tough and they must continuously struggle to innovate. While steel has many intrinsic qualities that make it the most widely used material for many industrial applications, ranging from transportation to civil engineering or home appliances, it also has a major drawback in an NVH (noise, vibration and harshness) context: ﬂat steel products are naturally noisy and this is essentially due to the very low natural damping of thin steel sheets.

Many steelmakers are now proposing damped sandwich sheets, composed of two steel layers separated by a dissipative material. These products are based on the concept of passive damping (as opposed to active damping) where, usually viscoelastic, dissipative material is added to the steel to complete the properties of the designed product with noise and vibration performance. These sandwich products have already found applications in the transportation industries, such as railway or automotive transport, but are

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1 INTRODUCTION

actually too expensive to really be competitive in the building or civil engineering markets.

However, an alternative design is possible: recent developments in man- ufacturing tools allow the economic production of ﬂat steel products with localised bonded reinforcements. Initially developed by the steel-makers for the production of light car body members (B-pillar, body sides, etc.) with variable thicknesses (the patchwork technique), this methodology has already been adopted by many automotive manufacturers for recent car models [Arc05].

The adaptation of the concept to noise and vibration constraints leads to a variant of the viscoelastic sandwich that we call viscoelastic patches (see ﬁgure 1).

Fig. 1.1. Full and partial sandwich plate. The left image illustrates the classical constrained layer sandwich concept where the whole structure contains a dissipative core. The right image illustrates the concept of patch constrained layer damping where only a small portion of the structure is locally covered with a dissipative material and a constraining plate.

The physical damping principle behind the patches is the so-called Con- strained Layer Damping (CLD) technique [Ker59]. The energy loss comes from the shear deformation energy of the viscoelastic material layer which is partially dissipated in the form of heat.

Since it is generally impossible to foresee the eﬀects of alterations of designs by experiments, designers require predictive numerical simulation tools. Such tools should help to position the damping patches and optimise its use.

The first aim of the thesis is to develop original and efficient modelling techniques to simulate the dynamic and acoustic behaviour of structures with viscoelastic damping devices such as patches. The finite element method of-

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1 INTRODUCTION

to simulate the behaviour of each material layers. It is possible to describe accurately the shear deformation and the damping of the viscoelastic layer by using tridimensional elements [BB02] for the core. One drawback of this technique is that any modifications of the thickness of the polymer layer involves a re-meshing step, which is sometimes expensive. Another drawback that affects all FEM models is the following: as the frequency increases, the use of low-order finite elements leads to prohibitive mesh refinement, very large matrices and huge amounts of computational resources, limiting their potential application to low frequencies. The main issue with low-order Galerkin discretizations is the dispersion error [IB95] [BI99]. For high wave numbers, Ainsworth [Ain03] proved improved efficiency by using higher polynomial degrees in the approximation. This idea is revisited in this thesis in a partition of unity framework [BM97]. Other alternatives include the use of wave-based methods (WBM [Des98], developed by Desmet and his team), the variational theory of complex rays (VTCR, [LARB01]) or the developments of the discontinuous Galerkin method (DGM, [FHF01][FHH03]).

The manuscript is organised as follows: the ﬁrst chapter introduces the use of viscoelastic materials. We work under the assumptions of linear viscoelasticity, applied to homogeneous and isotropic materials, and introduce the concept of complex modulus. We present both tabulated and parametric models and discuss their use. The processing of experimental data is also illustrated on the 3M ISD112 material, that we tested in laboratory.

In the second chapter, we introduce the partition of unity ﬁnite element method as a generalisation of the ﬁnite element method and discuss advan- tages and drawbacks on uni- and bidimensional problems.

Chapter 3 focuses on the formulation of an original PUFEM Mindlin plate element that answers our needs for an eﬃcient element for vibration analysis.

This element benefits from polynomial enrichment and is more efficient than classical, low-order finite elements. Our first contribution consists in the application of the partition of unity finite element method (PUFEM), based on the work of Babuˇska et al [BM97], to the development of efficient Mindlin plate elements. These elements perform significantly better than the elements available in the commercial software ACTRAN [Fre05]. Convergence studies on both static and dynamic tests are performed in Chapter 4. This part also demonstrates the performance and cost of the polynomial PUFEM approach, as opposed to standard elements available in a commercial software. This work has been already presented in [HB06].

Chapter 5 introduces an interface element technique for the modelling of the viscoelastic core layer, in full and partial sandwich conﬁgurations. This

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1 INTRODUCTION

interface technique was developed initially for the modelling of bond assem- blies by FEM, but its application to PUFEM elements and to viscoelastic layers is the second original contribution of this thesis.

Chapter 6 oﬀers a review of classical analysis schemes for the dynam- ics of structures, such as modal extraction and direct frequency response.

Some peculiarities linked to the analysis of structures containing viscoelastic materials are developed and implementation tricks are also discussed. Most authors addressing the design of passive damping devices in the literature only focus on the dynamic behaviour of structures. We present also a simple acoustic propagation model, based on the Rayleigh integral, that allows the acoustic design of such devices. This approach, thethird contribution of this thesis, is lacking in most passive damping works though common in publications on active control of structures.

Chapter 7 covers a handful of applications and focus on the validation of our approach (PUFEM + interface element + modal or direct frequency response). Our results are compared to published numerical or experimental data. The methodology and some applications were already presented in [HBS06].

We also developed our own experimental test bench for the medium frequency validation, in collaboration with OPTRION SA. This validation test is based on the measurement of frequency response curves for diﬀerent con- ﬁgurations of sample plates (naked or patched). This work is an important part of the thesis and is covered in Chapter 8. The quality of the measured data and the broad range of frequencies covered make it a major achievement and the fourth contribution of this dissertation.

Finally, we tackle the design optimisation of viscoelastic patches. We give some simple design rules based on bidimensional models. This work is presented in chapter 9 and was already the subject of conference articles [HDBB⁺04],[HBS05] and [BHB06]. An original optimisation technique is proposed for the optimal positioning of patches on structures in chapter 10. This strategy takes advantage of the ﬂexibility brought by the polynomial enrichment of the Mindlin element. A variable ﬁdelity optimisation framework is monitored by a hybrid optimisation sequence. This strategy does perform well on our application and succesfully reaches optimal design points, while coping with the multimodal character of the response functions. In this chapter, we also compare optimisation based on damping response functions with

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2 VISCOELASTIC MATERIALS

Vibrations represent a major engineering issue in many areas of civil, mechanical, or aeronautic design: wind excitation of long bridges and skyscraper buildings, earthquake-proof foundations, sea waves actions on offshore plat- forms, takeoff efforts on satellites transported rocket launchers, turbofan- induced noise in airplanes - to name a few hot topics of active research.

Quite often, vibrations are undesirable and engineers struggle to reduce them or to damp the structure. The term damping refers to the removal of some part of the vibration energy from a vibrating structure. This suppression may result from transferring energy to other structural components or ﬂuids which are not of concern or from converting this energy into other forms like heat or radiation, for instance.

Since a long time, characterisation of damping forces of vibrating structures has been an area of active research in the broad ﬁeld of structural dy- namics. Since the publication of Lord Rayleigh’s book “Theory of Sound” at the end of the 19th century [Ray45], a vast body of literature can be found on this subject. Nevertheless, even if this topic is an old one, the demands of modern engineering have led to a continuous increase of research in the recent years.

At this point of the discussion, the term damping is still purely theoret- ical; in fact, it covers a large number of physical mechanisms involving the dissipation of vibration energy in a system. Some of these processes can be linked to the intrinsic properties of the structural material, like the friction of macromolecular chains in polymers, for instance. Others can be generated by various types of boundary effects: friction at the contact of two bodies, fluid-structure interaction - a moving body in a fluid or a moving fluid in a hollow body -, by radiative energy loss, etc. In all these cases, a fraction of the vibration energy is transformed into another, generally unrecoverable,

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2 VISCOELASTIC MATERIALS

form of energy (often, heat) or transported to another part of the system where it is dissipated.

In this work, we are interested in the damping brought about by the use of viscoelastic materials (VEM) in structures. The concept seems to date back to a few years after the end of the First World War (WWI), when the Lord Corporation company was founded by H.C. Lord in 1919 with the objective of “exploring the potential of bonding rubber to metal to isolate and control shock, noise and vibration.”¹

The ﬁrst publications appeared in the 1950’s, by Li´enard in France and Oberst in Germany (see [Aus98]). The work of Oberst was focused on the application of rubber layers on automotive panels to reduce acoustic radiation.

Most of his articles dealt with experimental methods of testing the performance of various combinations of materials. At that time, the viscoelastic layer was considered alone, with free-layer damping as the only loss mechanism.

Later, Kerwin ([Ker59], 1959), introduced the concept of constrained layer damping (CLD), in which the VEM is sandwiched between two layers of solid material. In that case, the loss mechanism is primarily due to shear in the VEM core. Kerwin was also the first to introduce mathematical modelling of this dissipative sandwich configuration. Together with Ross and Ungar, Kerwin compared the effectiveness of free- and constrained-layer damping on large plates and concluded that the constrained-layer treatments were the most weight efficient in most cases. Since the 1950’s, hundreds of papers have been published on the theory and application of constrained layer damping.

The relationship between damping and energy had already been noted by Ross, Ungar and Kerwin in 1962 (see [BV92], [Cro98]). In 1981, Johnson, Kienholz and Rogers presented an important work, introducing the Modal Strain Energy (MSE) technique for the specific design of viscoelastic damping structures by the finite element method [JK82]. With this technique, the analysis of damping treatments by numerical simulation became much easier than before, and analytic formulations lost some appeal, essentially because of their limitation to very simple geometries. The rise of the finite element method, together with the MSE technique, opened the way for the industrial damping applications, primarily in aerospace engineering.

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The design of a viscoelastic damping treatment involves ﬁve main design options (see [Aus98]):

1. the thickness of the VEM layer,

2. the modulus of the VEM (which is both temperature- and frequency- dependent),

3. the location of the VEM,

4. the thickness of the constraining layer 5. and the modulus of the constraining layer.

The design process requires ﬁnding the right combination of all these options which will result in the maximum damping of the vibration modes of interest.

An eﬃcient damping treatment conﬁguration is the one that focus the strain energy into the VEM, and the ideal VEM is the one that dissipates this energy the most.

We treat this matter extensively in the chapter devoted to the optimisation of damping patches (see chapter 11). Speciﬁcally, we study the optimal thickness ratio of damping material and constraining material for viscoelastic sandwiches and partial damping treatments. We also address the problem of optimal placement of damping patches on product surface.

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2.1 Viscoelastic materials

Materials for which the relationship between stress and strain depends on time are called viscoelastic.

Some phenomena are typical of such behaviour: creep or relaxation, strain-rate dependent apparent stiﬀness, hysteresis in the stress-strain curves (dissipation of mechanical energy), etc.

Most polymers are viscoelastic materials made up of long chains of mole- cules or macromolecules. When submitted to an applied stress, polymers can deform by either one or both two fundamentally diﬀerent atomistic mechanisms [Roy01]:

1. The lengths and angles of the chemical bonds connecting the atoms may distort, moving atoms to new positions of greater internal energy; this small motion can occur quickly.

2. If the polymer has sufficient molecular mobility, larger-scale rearrange- ments of atoms are possible. This mobility is influenced by various physical and chemical factors such as molecular architecture, temperature or presence of fluid phases in the polymer.

The degree of mobility is determined by the rate of conformational change, which is often described by an Arrhenius expression [MBB97] of the form

rate ∝e^−Eact^RT (2.1) where Eact is the activation energy of the process, R = 8.314J/molK is the Gas constant and T is the temperature, expressed in Kelvin.

The glass transition temperature T_g is an important property of polymer materials. At temperatures much below T_g, the rates are so slow as to be negligible. The mobility is, sort of,frozen and the polymer can only adapt to applied stress by bond stretching. It responds in a glassy manner, incapable of being strained beyond some few percent before brittle fracture. The material exhibit a high modulus and low loss factor (“glassy” region (a) in ﬁgure 2.1).

In the neighbourhood of T_g, the material is in a transition phase. Its response is a combination of viscous ﬂuidity and elastic solidity. This change of phase corresponds to a fast variation of the modulus and to the high- est value of the loss factor, obtained at the glass transition temperature T_g

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2.1 Viscoelastic materials

At temperatures much above Tg, the rates are so fast that the changes are almost instantaneous, and the polymer acts in a rubbery manner in which it exhibits large, instantaneous and fully reversible strains in response to applied stress. It is characterised by low storage modulus and loss factor, both varying weakly with temperature (“rubbery” region (c) in ﬁgure 2.1).

At even higher temperatures, far above the T_g, the material exhibits the properties of a fluid and is therefore instable (“flow” region (d) in figure 2.1).

Fig. 2.1. Variation of storage modulus (continuous line) and loss factor (dashed line) of a viscoelastic polymer with temperature. We can distinguish four regions of diﬀerent viscoelastic behaviour, namely: (a) the glassy region, (b) the transition region, (c) the rubbery region and (d) the ﬂow region. Existing materials can be found in each regions at ambient temperature: grease, rubbers, elastomers, epoxies or others polymers are all viscoelastic materials. [Roy01]

The effect of frequency on the dynamic properties of viscoelastic materials is similar (but inverse) to the effect of temperature. The increase of excitation frequency has the same effect on materials than a decreasing temperature.

This is called the temperature-frequency equivalence and this property is used in most experimental measurement techniques for the characterisation of the dynamic properties of materials.

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2.2 Linear viscoelasticity and complex modulus

In linear viscoelasticity (see [Oya04]), the stress σ is supposed to be a linear function of the strain history. For a onedimensional tensile test, there exists a relaxation functionh such that the responseσ(t) to a strainε(t) is obtained by

σ(t) = t

−∞

h(t−τ)dε

dτ (τ)dτ. (2.2)

This assumption is strictly equivalent to the existence of a complex modulus, notedΛ. First let us operate a change of variable in relation 2.2:s=t−τ, then

σ(t) =− _+∞

0

h(s)dε

dτ (t−s)ds. (2.3)

In a second step, we suppose an harmonic excitation (ie. strain) of the form² ε=ε₀e^jωt; the response (stress) is then σ =σ^∗₀e^jωt, for which the amplitude is a complex variable (the stress is out-of-phase with the strain).

Substituting these expressions into 2.3 leads to the next equation σ₀^∗ =jωε₀

_+∞

0

h(s)e⁻^jωsds. (2.4)

Previous expression can be rewritten in the form : σ₀^∗

ε₀ =Λ(ω) +jΛ(ω) (2.5)

where ³

Λ(ω) = ωε₀ _+∞

0

h(s) sin (ωs)ds Λ(ω) = ωε₀

_+∞

0

h(s) cos (ωs)ds. (2.6) The terms Λ and Λ are the real and imaginary parts of the complex modulus, respectively. The complex modulus approach treats the viscoelastic materials like frequency dependent elastic materials with complex properties.

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2.2 Linear viscoelasticity and complex modulus

2.2.1 Dynamic loading

For isotropic and homogeneous materials, the complex modulus is described by a complex Young modulus E^∗ and a complex Poisson ratio ν^∗ (see [Ker04]). The experimental measurement of these two quantities is however very tricky, such that the Poisson ratio is usually considered real and constant (not frequency-dependent) and that the experiment aims at the determination of the complex Young modulus E^∗ or complex shear modulusG^∗ alone.

In the frequency domain, if we consider a simple dynamic, onedimensional, tensile test, the following relations hold

σ(ω) =E^∗(ω)ε(ω)

= [E(ω) +jE(ω)]ε(ω)

=E(ω) [1 +jη(ω)]ε(ω) (2.7) where we introduce the storage modulusE(ω), the loss modulusE(ω) and the loss factor η(ω) =E(ω)/E(ω).

Fig. 2.2.Stress-strain cycle for a linear viscoelastic material [Ker04].

For each frequency, the complex modulus describes an elliptical trajectory in the stress/strain plane (see ﬁgure 2.2):

σ = Re

E^∗ε₀e^iωt

= Re

E(1 +iη)ε₀e^iωt

=Eε₀(cosωt−ηsinωt). (2.8)

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The shape of the ellipse varies with the loss factorη; it describes an hysteresis trajectory and the surface of this hysteresis is proportional to the dissipated energy.

2.3 Selection of the material

The choice of a viscoelastic material will be based on its properties (loss factor and Young modulus), in the frequency and temperature range of interest for the application. But these properties are also aﬀected by the stress state of the material and environmental factors such as humidity.

The temperature dependance of viscoelastic materials has already been covered in a previous section, and is intrinsequely linked to the nature of the polymeric material. Temperature is the factor which has the most inﬂuence on the VEM properties [Fer80]. Referring to ﬁgure 2.1, we can say that the material of choice for damping should have its transition temperature right in the range of functionning temperature of the application of concern.

Speciﬁcally, the loss factor peek should ideally corresponds to the standard temperature for the system in function. The ideal material should also has a broad loss factor peek, to exhibit good damping properties for a wide range of temperatures.

The frequency dependence of the material is taken into account by the complex modulus approach, with tabulated frequency properties. A methodology, calledreduced variables methods (RVM [Fer80]) can be used to modify the parameters of this frequency-dependent model to reﬂect the temperature dependence. The use of RVM leads to a unique set of material parameters that can be handled at any predeﬁned temperature, with the help of a single scalar variable.

2.3.1 The method of reduced variables (RVM)

The eﬀect of temperature and frequency on viscoelastic materials can be combined into a single parameter dependence. This can be achieved through a temperature function variable, called theshift factor, which means that the viscoelastic behaviour at diﬀerent temperatures can be related to each other by a change (or shifting) in the time-scale only.

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2.3 Selection of the material

Fig. 2.3. Representation of the time-temperature superposition principle (from [dS03])

an arbitrary viscoelastic material, at three diﬀerent temperatures T1, T2 and T3. When the method of reduced variables is applied to the curves, one of the curve is chosen as reference and the others are shifted along the frequency axis to overlap the reference curve. This horizontal shift, extrapolated from the raw material data, islog(a_T), where a_T is the shift factor. This characteristic behaviour is referred to as the time- or frequency-temperature superposition principle, which implies the existence of a reduced frequency, related to the actual one through the shift factor:

ω_r =a_Tω. (2.9)

The single reference curve that comes up from the method is called master curve.

The materials for which this superposition principle applies well are numerous and are generally called thermorheologically simple (TS) materials.

Some complex materials do not obey the rule but they will not be considered in the scope of this research: they are essentially non-homogeneous materials, like copolymers (obtained by the chemical assembly of two diﬀerent

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polymers) or polymers including added inorganic or organic material in the melted phase (like ﬁber-reinforced or chalk-reinforced polymers).

The reduced variable method is often combined with an analytical expression for the shift factor temperature dependence. A simple data ﬁt of the experimental shift values is suﬃcient to obtain a convenient analytical represention of the shift factor temperature dependence. In 1955, Williams, Landel and Ferry [WLF55] proposed an empirical relation, referred since as the WLF equation.

To quote the original article: “In an amorphous polymer above its glass transition temperature, a single empirical function can describe the temper- ature dependence of all mechanical and electrical relaxation processes. The ratio A(T) of any mechanical relaxation time at temperature T to its value at a reference temperature, T(0), derived from transient or dynamic viscoelas- tic measurements or from steady flow viscosity, and the corresponding ratio b(T) of the values of any electrical relaxation time, appear to be identical over wide ranges of time scale.”

The WLF equation is written :

log₁₀(a_T) = −C₁(T −T₀)

C₂+ (T −T₀), (2.10) with T₀, the reference temperature.

For a given material, the constants C₁ and C₂ are obtained from a plot of (T −T₀)/log₁₀(a_T) versus (T −T₀).

2.3.2 Experimental determination of material data: the ISD112 case

Our objective is not to detail every steps of the procedure: the technique is developed in many documents, refer for instance to [dS03] for a comprehen- sive overview.

To obtain the shift factor and master curve for a given material, we need to carry on a few sets of experiments at diﬀerent temperatures. For the ISD112 material, manufactured by 3M Company in the form of tapes, the tests were performed at the Central Laboratory of Solvay S.A., under supervision of E.

Gandin (see report [Gan04]). Dynamic shear tests on samples were made at

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To validate the superposition principle for this material, we can manually translate the curves for the modulus, in order to ensure the continuity of the storage module. The resulting modulus curve is the master curve of the ISD112 material (see ﬁgure 2.4). The reference temperature was chosen equal to 20^oC. One can see that, by doing so, the loss factor segments also form a continuous curve (2.5). The behaviour of the ISD112 perfectly matches the superposition assumption. The shift factor, necessary to align each modulus segment, is also plotted against the temperature (expressed in Kelvin) (2.6).

Using the material shift factor curve and the two master curves, we can generate modulus and loss factor frequency dependent data at all temperatures covered by the shift factor plot.

Fig. 2.4.Master curve of the shear modulus of the ISD112 material, as a function of the reduced frequency. This curve was build from experimental measurements carried out at Solvay S.A.

Central Laboratory.

2.3.3 Representation of the Complex Modulus

In this section, we brieﬂy present diﬀerent numerical representations of the complex modulus of a material. We present the tabulated data model and parametric models.

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Fig. 2.5. Master curve of the loss factor of the ISD112 material, as a function of the reduced frequency.

-1 0 1 2 3 4 5

240 250 260 270 280 290 300 310

Temperature [Kelvin]

Shift factor log(aT)

Fig. 2.6.Shift factor, for the ISD112 material, as a function of temperature.

Tabulated data

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reduced frequency and the shift factor temperature dependence. For each temperature, we can generate tabulated data containing, for example, the storage and loss modulus as functions of the frequency, using simple interpo- lation procedures.

The advantage of this tabulated approach is that the material behaviour law is very general and that complex frequency and temperature dependence can be captured, even for wide ranges of frequencies and/or temperatures.

The direct use of data also avoid the process of model parameters ﬁtting that can be a source of discrepancy between the virtual and the real behaviour.

To conclude, the tabulated model is the most general form of complex modulus representation and is also convenient for most numerical simulations.

Parametric models

With a parametric model, we attempt to approach the material behaviour with analytical expressions. The ﬁrst type of models are the rheological models, which are build by combining springs and dashpots: such simple models like Maxwell, Kelvin-Voigt or Zener models are present in the literature for the representation of linear viscoelastic material behaviour. These simple models are however limited, for real materials, to narrow range of frequencies.

More complex frequency dependence can be taken into account by fractional derivatives models, such as those presented in [Ker04].

To be complete, recent work involving damping materials often men- tion the Golla-Hughes-McTavish parametric model developed by Golla et al. (GHM, see [GH85], [MH93]) and the Anelastic Displacement Field model (ADF, see [LB95]) proposed by Lesieutre and Bianchini. Both models are compatible with frequency domain and transient simulations.

For the ISD112, da Silva present the following analytical expressions, involving fractional derivatives, for the complex (shear) modulus and the shift factor:

G=G_real+iG_imag =B₁+ B₂ 1 +B₅

ifr

B₃

₋B₆

+ ifr

B₃

₋B₄, (2.11)

wheref_ris the reduced frequency (in Hz). The material loss factor is obtained as η= ^G_G^imag

real.

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The shift factor can be found from the expression:

log (aT) =a 1

T − 1 T₀

+ 2.303 2a

T₀ −b

log T

T₀

+ b

T₀ − a

T₀² −S_AZ

(T −T₀). (2.12)

The parametersB_i, i= 1, ...,6 anda, b, S_AZ have no physical meaning and are just chosen for the best data ﬁt. Their values, for the ISD112 material, are given in table A.1, in appendix. Figures 2.3.3 and 2.3.3 show respectively the master curves and the shift factor curve corresponding to this parametric model.

10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

Reduced frequency f r [Hz]

Storage modulus [MPa]

Loss modulus [MPa]

Loss factor

Fig. 2.7.ISD112 material : Master curves for the storage modulus, the loss modulus and the loss factor (reference temperature = 290 K)

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200 220 240 260 280 300 320 340 360

−4

−2 0 2 4 6 8 10

T [K]

log10(aT)

Fig. 2.8.ISD112 material : Shift factor curve.

2.3.4 Manufacturer’s data: what are nomograms ?

Often, performing experimental measurements for the determination of material properties is just too expensive or too time-consuming to be aﬀorded in a short project, and there are no analytical models for the chosen material in the literature. If that is the case, the only source of information is the material’s brochure delivered by the manufacturer.

Manufacturer data are almost always found in the form of nomograms, giving the essential material properties as function of both the temperature and the frequency. A typical example is presented in ﬁgure 2.3.4, extracted from the ISD112 material brochure. To each temperature corresponds a shift factor a_T that deﬁnes an isothermal line in a plate (a_Tf, f), where f is the frequency. The frequency values are then read on the vertical right axis.

For a given frequency f_i, and a temperature T_j, the nomogram can be read as follows:

1. The intersection point (P) between the horizontal at frequencyfi and the isothermal (oblique) line T_j is found;

2. drawing a vertical line through point P, the intersection with the modulus or loss factor curves give the corresponding values, respectively.

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Fig. 2.9.Nomogram for 3M’s ISD112 material.

2.4 Summary

In this chapter, we gave a short history of the use of viscoelastic materials for the damping of structural vibrations. The characteristics of viscoelastic materials was developed and their temperature dependence explained.

In the context of the linear viscoelasticity theory, we introduced the notion of Complex Modulus. The dependence with frequency was explained and the superposition principle was deﬁned. Application of this frequency- temperature superposition, through the reduced variables methodology, allows to express simply the temperature and frequency dependence of materials, through the introduction of the shift factor. This assumption leads to nomograms, giving modulus and loss properties as function of the reduced frequency. We illustrated the technique on experimental data that we col- lected for the ISD112 material.

To take into account the complex modulus data of viscoelastic materials in computer simulations, diﬀerents representations are possible. We presented both tabulated and parametric models principles. In our own calculations, we choose to use tabulated data, deﬁned from experimental measurements.

The implementation of such tables of frequency dependence in our numerical code is straightforward and does not need to be detailed here.

The general formalism used for the numerical calculation of damped struc-

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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD

The fundamental concepts of the partition of unity (PU) approximation were established initially by Babuˇska and Melenk, with the partition of unity method [BM97] and the partition of unity ﬁnite element method (or PUFEM) [MB96]. Early work was also found in the doctoral thesis of J.M. Melenk, concerning generalized ﬁnite element methods [Mel95], in 1995, or in the thesis of C.A. Duarte, which concerned the hp-cloud method [Dua96], in 1996.

To summarise, the partition of unity approximation was applied to develop different kinds of generalised finite element methods (GFEM, or PU- based GFEM). The PU-based GFEM family includes the hp−cloud method of Duarte and Oden [ODZ98], the generalised finite element method of Strouboulis et al. [SCB00] [SBC00], the generalised finite element method of Duarte et al. [DBO00], and the partition of unity-based hierarchical finite element method of Taylor et al. [TZO98].

A well-known problem of the PU-based GFEM is the arising of linear dependencies in the system matrices. Even after correct specification of the Dirichlet boundary conditions, the number of unknowns is generally larger than the number of generalised shape functions generated by the PU approximation. These unknowns must therefore be linearly dependent which leads to the rank deficiency of the stiffness matrix.

In this chapter, we further develop the partition of unity finite element method (PUFEM) on a onedimensional model problem and illustrate the linear dependencies phenomenon. We start by presenting the model problem and the corresponding differential equation. We then develop its variational form and introduce the Galerkin method to solve the problem. The classical finite element method is covered and the PUFEM technique is presented as a generalisation of the finite element concept.

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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD

3.1 Model problem

Consider the problem in onedimensional linear elasticity of a rod of lengthL, subjected to a distributed traction loadf(x) along its length, and a localised tractionT at positionx=L. The rod is constrained at the left end (x= 0).

The location of each point is given by a coordinatexand the displacement of each point from its original position is denotedu(x). To simplify the developments in this section, we assume a constant section area and Young modulus along the rod (no material changes, no section changes).

Fig. 3.1.Onedimensional model problem: a rod of lengthLis subjected to a distributed traction loadf(x) and a traction forceT at locationx=L.

Under these assumptions, the boundary value problem (BVP) which describes the displacement u(x) as a function of the positionx along the rod is given by the diﬀerential equation (strong form):

AEd²u

dx² =f(x), ∀x∈ Ω= [0, L]

u(0) = 0 AEdu

dx(L) =T.

(3.1)

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3.2 Galerkin method

3.1.1 Variational formulation

LetV be a linear space. The variational boundary value problem equivalent to 3.1 is stated as follows:

Find the function u∈ V, that satisﬁes

a(u, v) = b(v) ∀v ∈ V, (3.2)

where a:V × V →Ris the bilinear form

a(u, v) = _L

0

AEdu dx

dv

dxdx (3.3)

and b :V →R is the linear functional

b(v) = L

0

f(x)vdx+T v(L). (3.4)

3.2 Galerkin method

The Galerkin method builds an approximate solution to the variational boundary value problem from afinite dimensional subspaceV^h ofV spanned by N linear independent functions in V:

V^h ⊂ V, span{Φ_i}^N_i₌₁=V^h (3.5) where the Φi are the basis functions (or approximation functions). The parameter hcharacterises the dimension of the subspace and decreases with an increasing number of basis functions N. We pose the variational form of the BVP in V^h as follows:

Find the function u^h ∈ V^h, that satisﬁes a

u^h, v^h

=b v^h

∀v^h ∈ V^h (3.6)

and

b v^h

= _L

0

f(x)v^hdx+T v^h(L). (3.7)

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To solve for the approximation u^h, we write :

u^h(x) = N

i=1

Φ_i(x)u_i, v^h(x) = N

i=1

Φ_i(x)v_i (3.8) where u_i and v_i are the unknown coeﬃcients of the approximation.

We now substitute the approximations of equations 3.8 in relations 3.6 and 3.7, and use the arbitrariness of the values v_i to obtain the following linear system of equation

Ku=F (3.9)

where

K_ij =a(Φ_i, Φ_j) (3.10)

F_j =b(Φ_j). (3.11)

3.3 Finite element method

The ﬁnite element method (FEM) is based on two principles: ﬁrst, the original problem is approximated following the application of the Galerkin method;

secondly, the constuction of an approximation space of finite dimension based on finite partitioning of the domain and the definition of approximation function on each subdomains. The Galerkin method has been intoduced previ- ously. We focus here on the definition of the approximation functions Φi in the case of the FEM.

We start by partitioning the domain Ω into a set of M subdomains (or elements). Nodes are located at the vertices of each elements. A total of N nodes is distributed through the domain. The coordinates of the nodes are labelled x₁, x₂, ..., x_N and the elements are labelled Ω₁, Ω₂, ..., Ω_M.

We associate to each nodex_i a shape functionΦ_i with a compact support ωi. The support of the nodal shape functions is deﬁned as the union of the element subdomains sharing the node x_i.

The ﬁnite element basis functions are the nodal shape functions spanning a space of piecewise polynomials of at least order one. Figure 3.2 shows a

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3.3 Finite element method

Fig. 3.2. Mesh for the one-dimensional problem. The linear shape functionΦi used at nodexi, deﬁned on the supportωi is illustrated.

The ﬁnite element approximation is written as follows

u^h(x) = N

i=1

Φ_i(x)u_i. (3.12)

The nodal coeﬃcients u_i have a physical meaning: they are the values of the approximated ﬁeld (lateral displacements, for our 1D model problem) at the nodes x_i. The linear precision property follows from the fact that the shape functions satisfy

i

Φ_i(x) = 1,

i

Φ_i(x)x_i =x. (3.13) Hence, if the nodal values are prescribed according to an arbitrary linear field, the FE approximation will reproduce this field exactly. The equations 3.13 are therefore often calledreproducing conditions. The first expression implies that the shape functions form a partition of unity [BM97]. This property relates to the ability of the FE model to represent rigid body modes and is closely linked to the convergence properties of the approximation.

For the construction of the stiﬀness matrix Kand force vectorF, the integrals overΩare replaced by a sum of elemental integrals of the subdomains ω_i. One important thing to note is that the the shape functions restricted to the subdomains are polynomial functions. These functions are often numeri- cally integrated with a Gauss quadrature, since the order of the scheme can be chosen such as to integrate the bilinear form 3.10 exactly.

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3.4 Partition of unity ﬁnite element method

The Partition of unity finite element method (PUFEM) was first proposed by Melenk and Babuˇska [BM97] and is also known today as the Generalized Finite Element Method (GFEM) [Mel95] [SCB00]. It is a particular case of the general class formed by the Partition of Unity Methods, that covers numerous different meshless approaches such as the Diffuse element method (DEM)[NTV92], the Element Free Galerkin method (EFGM)[BLG94] or the Reproducing Kernel Particle method (RKPM)[LJZ95].

The basic idea consists in the use of partition of unity functions, a set of functions whose sum equals the unity on the whole domain. Let the functions ϕ_α,α = 1, ..., N, denote a partition of unity subordinate to the open covering domain TN ={ω_α}^N_α₌₁ of the domainΩ , such that

αϕ_α(x) = 1, ∀x∈Ω. (3.14)

This set of functions ϕ_α composes the partition of unity attached to the support ωα. In the case of the partition of unity ﬁnite element method (or generalized FEM), ω_α is the union of the ﬁnite element sharing the vertex nodex_α.

We now introduce the local spaces χ_α deﬁned on ω_α, α= 1, ..., N

χ_α(ω_α) = span{L_iα}_i_∈I₍_α₎, (3.15) whereI(α) are index sets andLiα denotes the local approximation functions or local enrichment functions.

The family of generalized ﬁnite element shape functions of order p, FN^p, is constructed by multiplying each partition of unity functionϕ_α by the local approximation functions

F_N^p ={Φ^α_i =ϕ_αL_iα, α = 1, ..., N, i∈ I(α)}. (3.16) An obvious choice to form a basis forχ_α are polynomial functions, which can approximate well smooth functions. We use the following notation:

L_iα(x) = ˆL_i◦F⁻¹(x) (3.17)

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3.4 Partition of unity ﬁnite element method

The operator F⁻¹_α operates a change of coordinate system such that F⁻¹_α :ω_α →ω,ˆ F⁻¹_α (x) := x−x_α

h_α (3.18)

where x_α is the coordinate of the node α, h_α is the diameter of the largest element sharing the node α.

The PUFEM approximation foru(x) can be written in the following form:

u^h_{GF EM}(x) = N α=1

i∈I

ϕ_α(x)L_iα(x)a^α_i =Φ {a}. (3.19) Figures 3.3 and 3.4 illustrate the principle of the PUFEM, in the case of the one dimension model problem presented earlier (for nonshifted and shifted nodal enrichment functions). The problem is discretised by a two elements mesh. The central node as coordinate x = 5, for a rod of length L= 10. For this central node, the support is the union of both elements and the PU functions are simply the linear hat functions. The nodal enrichment functions, deﬁned on this support, take diﬀerent forms in the non-shifted case and the shifted case.

The PUFEM technique is also illustrated on a two-dimensional support mesh in ﬁgure 3.5.

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0 2 4 6 8

0 0.5 1

x PU function

0 5 10

0 0.5 1 1.5 2

x

Nodal enrichment functions

0 5 10

x

0 5 10

0 50 100

x

0 2 4 6 8

0 0.5 1

x

Generalized shape functions

0 2 4 6 8

0 1 2 3 4 5

x

0 2 4 6 8

0 10 20

x {1}

{x²} {x}

PU X {1}

PU X {x}

PU X {x²} PU

Fig. 3.3. Illustration of the principle of the PUFEM, for a 2 elements mesh. The generalized shape functions are obtained as the product of the PU functions, deﬁned over the support around node 2, with the unshifted nodal enrichment functions (monomials ofx). The last column shows each generalized shape function generated in the process (up to cubic functions).

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3.4 Partition of unity ﬁnite element method

0 2 4 6 8

0 0.5 1

x PU function

0 2 4 6 8

0 0.5 1 1.5 2

x

Nodal enrichment functions

0 2 4 6 8

−4

−2 0 2 4

x

0 2 4 6 8

0 5 10 15 20 25

x

0 2 4 6 8

0 0.5 1

x

Generalized shape functions

0 2 4 6 8

−1

−0.5 0 0.5 1

x

0 2 4 6 8

0 1 2 3

x

PU {1}

{x−x_α}

{(x−x_α)²}

PU X {1}

PU X {x−x_α}

PU X {(x−x_α)²}

Fig. 3.4. Illustration of the principle of the PUFEM, for a 2 elements mesh. The generalized shape functions are obtained as the product of the PU functions, deﬁned over the support around node 2, with the shifted nodal enrichment functions (monomials of (x−xα)). The last column shows each generalized shape function generated in the process (up to cubic functions).

0 0.5

1

−0.5 0 0.5

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

Partition of unity function

0 0.5

1

−0.5 0 0.5

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

Enrichment function

0 0.5

1

−0.5 0 0.5

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04

Shape function

Fig. 3.5.Partition of unity ﬁnite element principle, for a problem in two dimensions. The support mesh is made of four elements. The PU functions at the central node are plotted is the ﬁrst window, the enrichment function in the second window and the corresponding generalised shape function in the last window.

Design of viscoelastic damping for

Laurent Hazard

Design of viscoelastic damping for

vibration and noise control: modelling, experiments and optimisation

- Almost the last version ...

December 9, 2006

Universit´e Libre de Bruxelles

Structural and Material Computational Mechanics department

Design of viscoelastic damping for vibration and noise control:

modelling, experiments and optimisation

Laurent Hazard

Committee in charge:

XXX Chairman

XXX Reporters

XXX

XXX Examiners

XXX XXX

XXX Advisor

Laurent Hazard

Design of viscoelastic damping for

vibration and noise control: modelling, experiments and optimisation

December 21, 2006

Universit´e Libre de Bruxelles

Acknowledgements

Contents

1

INTRODUCTION

2

VISCOELASTIC MATERIALS

2.1 Viscoelastic materials

2.2 Linear viscoelasticity and complex modulus

2.3 Selection of the material

2.4 Summary

3

INTRODUCTION TO THE PARTITION OF UNITY METHOD

3.1 Model problem

3.2 Galerkin method

3.3 Finite element method

3.4 Partition of unity ﬁnite element method