PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Vibration damping capabilities of treatments with frequency andtemperature dependent viscoelastic material properties
M. GRÖHLICH1; M. BÖSWALD2; R. WINTER3
1,2,3

German Aerospace Center (DLR)

Institute of Aeroelasticity, Department for Structural Dynamics and System Identification
Bunsenstraße 10, D-37073 Göttingen, Germany

ABSTRACTDue to their low mass and stiffening components like frames, stringers and panels, aircraft fuselage structures
are prone to vibration. In order to improve acoustic comfort inside a cabin, specific vibration damping of suchcomponents would be helpful. Most common damping layouts make use of a viscoelastic material as themain source of vibration energy dissipation. Such material is placed at sensitive positions of a structure toabsorb the energy of operational deflection shapes of concern. However, the material behavior of viscoelastic
materials varies considerably with frequency and temperature and needs to be considered during the design
process. In this paper, a finite element approach for local modelling of viscoelastic damping is presented. A
viscoelastic material with frequency and temperature dependent properties is introduced as a dedicateddamping material for aeronautic applications. Using the FE method, the consequences of temperature andfrequency influence on the vibration damping capability of a Constrained Layer Damping treatment with theparticular damping material are presented.
Keywords: Viscoelasticity, Vibration Damping, Damping Layout

1. INTRODUCTION
In the construction of aeronautic structures, lightweight design is essential. Lightweight structuresoften feature discrete stiffeners as frames and stringers, typically characterized by having both, highstiffness and low mass. As a result of these characteristics, those structures are prone to vibration. In
particular, an aircraft fuselage consists of such components, as illustrated in Figure 1.

Figure 1 - Fuselage composition of Flight-LAB-demonstrator
Vibrations introduced into the structure are transmitted as structure -borne noise and lead to anincreased cabin interior noise due to sound radiation on the fuselage skin. The transmission of
structure-borne sound at steady-state excitation, e.g. from turbulent boundary layer or engine induced
vibration, can be reduced by increasing damping of the frames. Procedures for an increase of thedamping effectiveness include the deliberate generation of shear strain in viscoelastic materials. A
common application for vibration damping which utilizes this effect is the Constrained Layer
1martin.groehlich@dlr.demarc.boeswald@dlr.de3
rene.winter@dlr.de24273

Damping treatment (CLD), where a viscoelastic layer is constrained between the stiffer base structure
and a stiffer face layer. Since aircraft vibrations cover a wide range of different frequency andtemperature ranges during operation, the damping material has to cope with the corresponding
influences. Due to the dependence on frequency and temperature of viscoelastic material properties,
the damping performance will also vary during operation.
The main objective of this paper is to analyze the effect of frequency and temperature on thedamping performance of a viscoelastic material under operating condition. First, the theoreticalbackground of modelling viscoelastic material behavior with respect to frequency and temperaturedependence is exemplified. Based on the one-dimensional material model, a transformation into amaterial model for solid mechanics is presented, allowing for compilation of hysteretic damping
matrices for finite element (FE) analysis. Furthermore, a viscoelastic material is introduced and its
damping capabilities are analyzed within a CLD treatment for different frequency and temperaturelevels.

2. THEORETICAL BACKGROUND OF VISCOELASTIC MATERIAL MODELLING
In this section, the basics of idealized viscoelastic material modelling are exemplified. The
essential properties are presented and a formulation of the material behavior in frequency domain isprovided. Additionally, the temperature dependence on the material behavior is implemented in thematerial model. By using the finite element method (FEM) it is shown, how viscoelastic material canbe incorporated to physically model local viscoelastic damping and to establish a global damping
matrix for the structure.
2.1 Linear viscoelasticity in frequency domain
In general, viscoelasticity denotes time dependent material behavior. Under the assumption of aharmonic excitation, a viscoelastic material behavior can be described in terms of a complex modulus,
which is a function of the excitation frequency ȳ. If constant temperature is assumed, the complex
shear modulus ‫ כ ܩ‬can be written as (1):(1)‫ כ ܩ‬ሺȳሻ ൌ ‫ ܩ‬ᇱ ሺȳሻ ൅ ݅‫ ܩ‬ᇳ ሺȳሻThe real part of the complex notation ‫ ܩ‬ᇱ is the storage modulus and denotes the elastic behavior of
the material. In contrast to it, the imaginary part ‫ ܩ‬ᇱᇱ is the loss modulus and denotes the viscous ordissipative properties. Both, the real and imaginary part can be expressed by using the param eters of ageneralized Maxwell model, often referred to as Prony series. The composition and effects of ageneralized Maxwell model are examined in (2-4). The storage modulus can be expressed in terms of
the parameters of the generalized Maxwell element:௡‫ ܩ‬ᇱ ሺȳሻ ൌ ‫ܩ‬଴ ൦ͳ െ ෎ ቆߙ௜ െ
௜ୀଵ

ଶ
ȳଶ
ߙ௜ ߬௥௘௟ǡ௜ቇ൪ǡଶͳ ൅ ߬௥௘௟ǡ௜ȳଶ

(2)where ‫ܩ‬଴ is the instantaneous shear modulus at infinitely high frequency, ߬௥௘௟ǡ௜ the relaxation timeand ߙ௜ the so called relative modulus (5). The equation of the loss modulus is formulated as (5):௡ߙ௜ ߬௥௘௟ǡ௜ ȳ‫ ܩ‬ᇳ ሺȳሻ ൌ ‫ܩ‬଴ ෎ ቆቇଶͳ ൅ ߬௥௘௟ǡ௜
ȳଶ

(3)

௜ୀଵ

The ratio of the loss modulus and storage modulus defines the loss factor ߟ:ߟሺȳሻ ൌ‫ ܩ‬ᇳ ሺȳሻ‫ ܩ‬ᇱ ሺȳሻ

(4)Concerning damping layout, the loss factor is the most important parameter as it quantifies theenergy dissipation capability of a material. It is also known as the  ߜ resulting from dynamic
mechanical analysis (DMA).
2.2 Temperature dependence on viscoelastic material properties
The temperature dependence on material properties are taken into account by different shift factors.
Various approaches can be found in the literature regarding the shift factor calculation, e.g. in (6).

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However, it should be taken into account, that the selection of the approach for the shift factor
calculation highly depends on the given material and considered temperature region. In the scope of
this paper, the Williams-Landel-Ferry (WLF) as well as the Arrhenius approach are used and briefly
presented in the following. For analyzing the material properties' dependence o n the temperature, two
shifts have to be considered (7):xx

Horizontal shift along the frequency axis due to thermal activated rearrangements andhigher reaction rate on a molecular level
Vertical shift along the material property axis due to changes in the type and amount ofmolecular processes

The frequency shift is defined by a horizontal shift factor ߦு , resulting from the empirical WLF
equation with experimentally determined constants ‫ ͳܥ‬and ‫ ʹܥ‬relating to a reference temperature ܶ଴(7):݈‫݃݋‬ሺߦு ሺܶǡ ܶ଴ ሻሻ ൌ െ‫ ڄ ͳܥ‬ሺܶ െ ܶ଴ ሻ‫ ʹܥ‬൅ ܶ െ ܶ଴

(5)

The determination of the constants C1 and C2 is based on the evaluation of master curves. Mastercurves are curve progressions created by merging the measurement data from frequency dependent
material characterization at different temperatures.
For the description of the vertical shift, the Arrhenius approach is used. The relati on between theexperimentally determined activation energy ‫ܧ‬஺ concerning a temperature shift yields the verticalshift factor ߦ௏ (7):݈‫݃݋‬ሺߦ௏ ሺܶǡ ܶ଴ ሻሻ ൌͲǡͶ͵ ‫ܧ ڄ‬஺ ͳ ͳ൬ െ ൰ܶ ܶ଴
ܴ(6)The parameter ܴ is the universal gas constant. Considering a reference state ܵ଴ ሺܶ଴ ǡ ݂଴ ǡ ‫ܩ‬଴ᇱ ǡ ‫ܩ‬଴ᇳ ǡ ߟ଴ ሻ,
the corresponding shifted state ܵଵ ሺܶଵ ǡ ݂ଵ ǡ ‫ܩ‬ଵᇱ ǡ ‫ܩ‬ଵᇳ ǡ ߟଵ ሻ at an arbitrary temperature ܶଵ can be determined.
If the vertical shift factors for storage and loss modulus are unequal, the shift is calculated by
Equations (7) and (8):(7)݂ଵ ൌ ݂଴ ‫ߦ ڄ‬ு ‫ܩ‬ଵᇱ ሺ݂ଵ ሻ ൌ ‫ܩ‬଴ᇱ ሺ݂଴ ሻ ‫ߦ ڄ‬௏ǡீ ᇲ ǡ‫ܩ‬ଵᇳ ሺ݂ଵ ሻ ൌ ‫ܩ‬଴ᇳ ሺ݂଴ ሻ ‫ߦ ڄ‬௏ǡீ ᇴ ǡ

ߟଵ ൌ

‫ܩ‬ଵᇳ

‫ܩ‬ଵᇱ

(8)

A typical temperature shift of the storage modulus, loss modulus (a) and loss factor (b) of a virtual
viscoelastic material, modelled with a generalized Maxwell model, is presented in Figure 2. Therefore,
the vertical shift factors for storage and loss modulus are assumed to be equal.

(a)

(b)

Figure 2 – Complex modulus (a) and loss factor (b) during a temperature shift
It can be seen in both diagrams that the characteristic evolution of the respective property stays
almost the same, but it appears shifted in horizontal and in case of Figure 2(a) also in vertical direction
if the temperature changes. In Figure 2(b) the horizontal shift can be clearly observed, since themaximum of the loss factor from the reference state ܵ଴ appears on the same level at higher frequency
for a different temperature (ܵଵ). An additional vertical shift for the loss factor would have appeared, if
the vertical shift factors of storage and loss modulus had been the same.

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2.3 Implementation of viscoelasticity into the finite element method
In order to incorporate viscoelastic material properties into the finite element method, atransformation from the one-dimensional material model into a three-dimensional model is mandatory.
Assuming isotropy as an additional material property, the transformation is carried out by Hooke's law.
The relationship between stresses ࣌ and strains ࣕ is determined by an elasticity matrix ۳ (8):
‫ۍ‬
ߪ௫௫
‫ێ‬
‫ߪۍ‬௬௬ ‫ې‬‫ێ‬‫ۑ ߪێ‬‫ܧ‬‫ێ‬௭௭࣌ ൌ ࡱ ‫ ࣕ ڄ‬ൌ ‫ ۑ ߬ێ‬ൌ
‫ێ‬
௫௬
ሺͳ
൅
ߥሻሺͳ
െ
ʹߥሻ
‫ۑ ێ‬
‫ێ‬
‫߬ ێ‬௫௭ ‫ۑ‬‫ێ‬‫߬ ۏ‬௬௭ ‫ے‬
‫ێ‬
‫ۏ‬

ͳെߥ
ߥ
ߥ

ߥ
ͳെߥ
ߥ

ߥ
ߥ
ͳെߥ

Ͳ

Ͳ

Ͳ

Ͳ
Ͳ
Ͳ
ͳ െ ʹߥ
ʹ

Ͳ
Ͳ
Ͳ
Ͳ

Ͳ

Ͳ

Ͳ

Ͳ

ͳ െ ʹߥ
ʹ

Ͳ

Ͳ

Ͳ

Ͳ

Ͳ

Ͳ
Ͳ
Ͳ

‫ې‬
‫߳ ۑ‬௫௫‫߳ۍ ۑ‬௬௬ ‫ې‬Ͳ ‫߳ ێێ ۑ‬௭௭ ‫ۑۑ‬‫ߛێ ڄ ۑ‬௫௬ ‫ۑ‬Ͳ ‫ߛ ێ ۑ‬௫௭ ‫ۑ‬‫ߛ ۏ ۑ‬௬௭ ‫ے‬ͳ െ ʹߥ‫ۑ‬
ʹ ‫ے‬

(9)

The stress and strain vectors include their corresponding spatial normal parts ߪ and ߳, as well as
the spatial shear parts ߬ and ߛ. Furthermore, the Young's modulus ‫ ܧ‬and the shear modulus ‫ ܩ‬areboth related to the Poisson's ratio ߥ (8):(10)‫ ܧ‬ൌ ʹ‫ܩ‬ሺͳ ൅ ߥሻ
At this point it should be noted, that the elasticity matrix consists of complex entries, as far as
viscoelasticity is concerned. In this case, the elasticity matrix ࡱ‫ כ‬can be separated into a real andimaginary part, representing storage and loss behavior, or respectively elasticity and hysteretic
damping:(11)ࡱ‫ כ‬ሺߗǡ ܶሻ ൌ ࡱᇱ ሺߗǡ ܶሻ ൅ ݅ࡱᇳ ሺߗǡ ܶሻ
Based on the equivalence of virtual work of external loads with the virtual work of internal stressand strain, a complex element stiffness matrix ࡷ‫כ‬ா for a finite element can be defined according to thefinite element method:ࡷ‫כ‬ா ሺߗǡ ܶሻ ൌ නሺࢨ்࣐ ሻ் ࡱ‫ כ‬ሺߗǡ ܶሻࢨ்࣐ ܸ݀ா 

(12)

௏ಶ

ࢨ denotes the differential operator matrix, whereas ࣐ conforms to the matrix of element shapefunctions. Considering the complex notation of Equation (11), the real part of Equation (12) yields aelement stiffness matrix ࡷா , whereas the imaginary part yields a hysteretic element damping matrixࡰா :ࡷா ሺߗǡ ܶሻ ൌ නሺࢨ்࣐ ሻ் ࡱᇱ ሺߗǡ ܶሻࢨ்࣐ ܸ݀ா 

(13)

௏ಶ

݅ࡰா ሺߗǡ ܶሻ ൌ ݅ නሺࢨ்࣐ ሻ் ࡱᇳ ሺߗǡ ܶሻࢨ்࣐ ܸ݀ா ൌ ݅ߟሺߗǡ ܶሻࡷா ሺߗǡ ܶሻ

(14)

௏ಶ

These element matrices can be used to assemble global system matrices. For example, by using theBoolean matrices ࡮ா , elements with viscoelastic material are sorted into global stiffness andhysteretic damping matrices of the whole structure:ࡷሺߗǡ ܶሻ ൌ ෍ ࡮்ா ࡷா ሺߗǡ ܶሻ࡮ா 

(15)

ா

݅ࡰሺߗǡ ܶሻ ൌ ݅ ෍ ࡮்ா ࡰா ሺߗǡ ܶሻ࡮ா 

(16)

ா

The sum of the global stiffness and global hysteretic damping matrix forms a global complex
stiffness matrix:(17)ࡷ‫ כ‬ሺߗǡ ܶሻ ൌ ࡷሺߗǡ ܶሻ ൅ ݅ࡰሺߗǡ ܶሻ
Following this procedure, it is possible to integrate finite elements with frequency and temperaturedependent viscoelastic material to represent local dampers in arbitrary structures.

4276

3. INFLUENCE OF FREQUENCY AND TEMPERATURE ON DAMPING
In the following, the influence of frequency and temperature on the damping capability of aviscoelastically damped beam is examined by means of a numerical example.
3.1 Simulation setup
The subject of the analysis is a free-free vibrating, rectangular aluminum beam. According to a
CLD treatment a viscoelastic core layer and an aluminum face sheet are applied on the beam, in orderto damp occurring vibrations. The corresponding geometric dimensions are shown in Figure 3.

Figure 3 - Geometrical setup of the CLD treatment
The material properties of aluminum are assumed to be isotropic and constant, with a Young's
modulus of ‫ܧ‬஺௟௨ ൌ ͹ǡͳ ή ͳͲଵ଴ Pa and a Poisson's ratio of ߥ஺௟௨ ൌ Ͳǡ͵͵. Furthermore, the density isߩ஺௟௨ ൌ ʹ͹ͲͲ kg/m³ and the structural damping coefficient is chosen to be ߟ஺௟௨ ൌ ͲǡͲͲͷ. For the corelayer, the material properties of a particular viscoelastic material are applied. The material is a
dedicated mixture of bromobutyl rubber for aeronautic applications as vibration dampers, developedin cooperation with the German Institute of Rubber Technology (Deutsches Institut für
Kautschuktechnologie e.V.). The material properties are significantly frequency and temperaturedependent and can be characterized by twelve parameter sets of the generalized Maxwell model, as
well as by WLF and Arrhenius shifts. In Figure 4, the contour curves of the shear storage modulus andloss factor of bromobutyl rubber are presented.

(a)

(b)

Figure 4 – Shear storage modulus (a) and loss factor (b) of bromobutyl rubber
For the storage modulus a considerable decrease can be detected with increasing temperature - thematerial becomes more flexible. In addition, the material behaves stiffer for higher frequencies in
isothermal consideration. In comparison, the contour curves of the loss factor are completely different.
The area of maximum values extends slightly diagonally from low temperature ( -40 °C) and low
frequency (1 Hz) to higher frequencies at higher temperatures (1000 Hz at 0 °C). Outside this range,
the loss factor drops considerably.
For the simulation, commercial FEM software Ansys 18 is used. The structure is discretized by
PLANE182 elements, since this type of elements can map frequency- and temperature-dependent
behavior. Because modal analysis with frequency- and temperature-dependent material properties
cannot be carried out in the current software version, the beam is examined by means of a harmonic
response analysis. Therefore, the beam is excited by a harmonic force with constant amplitude at a free
end in vertical direction and the resulting frequency response functions (FRF's) are calculated for theexcitation point. Edge effects due to thermal expansion are not taken into account in the analysis.

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3.2 Damping performance of CLD treatment
The system described in the previous chapter is examined at five temperature levels. Using thepeak-fit method (9), the modal parameters of the first three bending modes are identified from theresulting FRF's. The identified damping ratios are shown in Figure 5. For comparison purpose, thevalues given for the eigenfrequencies ݂௠ correspond to the mean values of the respective mode fromall temperature levels. However, the influence on the eigenfrequencies is not considered in thefollowing. The dashed line in the diagram indicates the damping value of the undamped beam.

Figure 5 - Impact of frequency and temperature on damping capabilities of the CLD treatment
For the first and second mode, the order of damping ratios is identical. The highest damping ratios
occur at a temperature of 30 °C. The second highest values appear at 0 °C, while the lowest damping
ratios are recorded at lowest temperatures of -50 °C and -20 °C. However, the order of damping ratioschanges for the third mode. In opposite to the first two modes, the damping value for 0 °C is highest at
this point. In order to get a better understanding of the presented results, the contour curves of Figure
4 are plotted from top view in Figure 6. The horizontal lines indicate the particular eigenfrequencies,
the vertical ones are isothermal lines.

(a)

(b)

Figure 6 – Top view on the contour curves of the shear storage modulus (a) and loss factor (b)When considering the loss factor of the first mode in Figure 6(b), it is noticeable that the maximum
damping ratio does not occur at the same temperature as the maximum loss factor of bromobutyl
rubber. This also applies to other modes: Although the loss factor values for a temperature of -20 °C
are above the values for 30 °C for all modes, the resulting damping ratio is always lower. From this fact
it can be concluded that the loss factor is not the only decisive factor for damping, but also the stiffnessof the material, represented by the shear storage modulus, is essential. Concerning that assumption, it
is striking that the storage modulus is also lower for 30 °C than for -20 °C at all modes, as
demonstrated in Figure 6(a). From that point it could be erroneously concluded, that low stiffness isappropriated in order to obtain high damping ratios.
The reason of the above mentioned behavior is founded in the prevailing damping mechanism. For
CLD treatments, the damping mechanism is based on energy dissipation due to shear deformation. A
quantity of the real potential energy ܹ of elastic deformation is given by (10), regarding to adisplacement vector ࢛:

4278

ͳ
(18)
ܹ ൌ ்࢛ ࡷ࢛ʹIf viscoelasticity is applied, the stiffness matrix becomes complex, as demonstrated in section 2.3.
In a harmonic state of oscillation, excited by a force vector ࢌ, the displacement vector becomescomplex as well:(19)࢛‫ כ‬ൌ ሾെߗଶ ࡹ ൅ ࡷ‫ כ‬ሺߗǡ ܶሻሿିଵ ࢌ
Here, ࡹ denotes the system mass matrix of the structure. Inserting Equation (19) into Equation(18) with a complex stiffness matrix ࡷ‫ כ‬ሺߗǡ ܶሻ yields a scalar value of the complex energy ܹ ‫כ‬:ͳܹ ‫ כ‬ൌ ሾሾെߗଶ ࡹ ൅ ࡷ‫ כ‬ሺߗǡ ܶሻሿିଵࢌሿ் ࡷ‫ כ‬ሺߗǡ ܶሻሾሾെߗଶ ࡹ ൅ ࡷ‫ כ‬ሺߗǡ ܶሻሿିଵ ࢌሿʹ

(20)

The real part of the potential energy is required for the calculation of the modal loss factor ߟ௞ , anindicator of the damping capability of a vibrating system (11):σ௡௜ୀଵ ߟ௜ǡ௞ ܹ௜ǡ௞ߟ௞ ൌܹ௧௢௧௔௟ǡ௞

(21)ߟ௜ǡ௞ is the loss factor and ܹ௜ǡ௞ the modal strain energy of layer ݅ at mode ݇, while ܹ௧௢௧௔௟ǡ௞
denotes the total modal strain energy of the whole system at mode ݇. First, an essential perception canbe derived from Equation (20), in which the complex stiffness matrix occurs in both an inverse and inan original form: Low stiffness leads to high energy dissipation due to high displacement. On the other
hand, low stiffness of the viscoelastic layer causes a reduction of the modal loss factor, since the totalenergy in the denominator of Equation (21) increases. From that fact it can be derived, that an optimalcomposition of the stiffness matrix exists for each excitation frequency and temperature, which
maximizes the dissipative energy and as a consequence, maximizes the damping ratio.
Relating to the above mentioned phenomena, the loss factor of -20 °C is considerably higher thanfor 30 °C, but it does not yield any advantage due to less shear deformation as a consequence ofincreased stiffness. In the case of the third mode, the combination of loss factor and shear storage
modulus is more effective for 0 °C than for 30 °C.
3.3 Procedures for stiffness matrix modification
In order to maximize the damping performance, an optimal combination of stiffness matrix andmaterial loss factor has to be found. The evolution of the material characteristics is given for a range of
operating frequency and temperature. As shown in section 2.3, the stiffness matrix is determined fromelement shape functions and from the elasticity matrix. With given material, these cannot be used as
design variables. However, the geometry of the finite elements provides another possibility to tailor
the complex stiffness matrix for maximum damping. A geometry modification of a structure always
involves a modification of the stiffness matrix. The thickness or the width of the viscoelastic layer is apossible design variable for the adjustment of the complex stiffness matrix. Of course, it is possible touse the spatial position of a local CLD treatment as a design variable. However, in this example, the
CLD layer is modelled continuously so that this parameter cannot be used.

Figure 7 - Impact of frequency and temperature on damping capabilities of the cut CLD treatment
With the intention to create higher shear strain in the viscoelastic core layer, Lepoittevin et al.
inserted small cuts into the core and face layer and examined the effects on damping (11). By thisprocedure, the stiffness matrix of the whole structure is modified. Hence, this approach is adopted here
and applied on the beam from section 3.1 in the following. According to a rule of thumb (11), cuts at
positions of maximum bending moment are most noticeable on damping. For this reason, the core andface layers are cut at position x=500 mm for the first mode, at positions x=310 mm and x=690 mm for4279

the second mode and at positions x=220 mm, x=500 mm and x=780 mm for the third mode with a g ap
of 1 mm each. The resulting damping ratios are shown in Figure 7.
First of all it is noticeable that the eigenfrequencies are lower than in the case of the continuous
CLD treatment. This fact already indicates an overall stiffness decrement. However, the change of theeigenfrequencies is small compared to the conventional CLD treatment that it has a negligible
influence on the material parameters of bromobutyl rubber. Furthermore, the maximum damping ratios
occur in a different order compared to Figure 5. For all modes, the damping ratios for 0 °C are highest,
followed by those of -20 °C and 30 °C. For lower temperatures, the cuts seem to have a positive effect,
since an increase in the damping values can be observed. This confirms the assumption of inadequate
stiffness in the case of continuous CLD treatment. In contrast, the effect of cuts is negative at higher
temperatures. While the damping ratios are still at the same level as for the continuous CLD treatment
for 30 °C and 60 °C at the first mode, the damping ratios even decrease for higher modes.
The instance of disadvantageous stiffness modification has to be considered during the design
process as well. Otherwise, a mass increase due to damping material thickening, could even lead to the
opposite of the pursued objective - a reduction of damping due to improper stiffness.

4. CONCLUSIONSIn this paper, the modelling of viscoelastic material in frequency domain by a generalized Maxwell
model has been presented. By using Williams-Landel-Ferry and Arrhenius approaches, it was shownhow the temperature dependence on the essential damping properties, storage modulus and loss factor,
can be incorporated in the material model. Additionally, by using Hooke's law, the one -dimensional
material model can be transformed into a multi-dimensional one. Due to the complex elasticity matrix,
a complex stiffness matrix can be generated for FEM applications in which the imaginary part
corresponds to a hysteretic damping matrix. As an example, the frequency and temperature dependent
material behavior of a bromobutyl rubber mixture has been introduced and its effects on the damping
performance of a CLD treatment have been analyzed for different frequency and temperature levels.
On the one hand it has been shown that both frequency and temperature have a significant influence on
the damping performance. On the other hand it has been demonstrated that a high loss factor does notnecessarily lead to high damping ratios of a vibrating system, but especially the stiffness in terms of
the storage modulus has to be considered. Only an appropriate combination of a high loss factor and acertain stiffness leads to maximum damping ratios.

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