157_168

Development of a numerical modelling methodology forthe NVH behaviour of elastomeric line connections

A. Stenti, D. Moens, P. Sas, W. DesmetK.U.Leuven, Department of Mechanical Engineering,Celestijnenlaan 300 B, B-3001, Leuven, Belgiume-mail: [email protected]

AbstractThe modelling of the NVH behaviour of elastomeric line connections, such as car door weather strip sealsand car windshield connections, involve the computational burden of including a detailed model of the jointinto the system model and of including variability and uncertainty, typical for elastomeric joints, in a full-scale system model.In the general framework of developing engineering tools for virtual prototyping and product refinement,a methodology is being developed which allows simplified equivalent joint models to be readily includedin full-scale NVH models at a reasonable computational cost. The methodology is based on a three-levelmodelling approach involving a material, a component and a system level. This study focuses on the dynamicmodelling of car door weather strip seals and illustrates the methodology through a numerical case study.

1 Introduction

The research on weather strip seals is mainly focusing on the sound transmission characteristics of the joint( [1], [2], [3], [4]), while the structure-borne vibration transmission characteristics of the component hasreceived little attention. This study starts with the numerical modelling of the structure-borne transmissioncharacteristic of the weather strip seals, extending the work presented by Wagner et al. [5] to the dynamicdomain [6], and aims at developing a more general methodology for the numerical modelling of the NVHbehaviour of elastomeric joints including viscoelasticity and amplitude dependent effects, typical for elas-tomeric connections.

Automotive weather strip seals are typically dual extrusion bulbs of sponge and dense rubber that are attachedto either the car door or the car body in order to seal the passenger compartment. The seal strip runs typicallyall around the perimeter of the car door. When closing the car door onto the car body, the door remainsin contact with the body through the hinges at the front side, through the lock mechanism at the rear sideand through the seal strip all around the car door perimeter. The seal strip induces some residual stiffnessand viscoelastic contribution to the car door support conditions and plays a role in the way exterior noiseand vibration can be transmitted to the interior of the car. Initial use of seal in automotive applications wasaimed at accommodating for manufacturing variations. Lately, the isolation issue became important, and thedesign driver was to better isolate the passenger compartment from dust, water and air leakage. Nowadays,the general trend in seal design mainly focuses on the isolation of the passenger compartment from noise andvibration.

Elastomers are viscoelastic materials. Their mechanical properties are strain, frequency and temperaturedependent. In an early design phase, not knowing a priori the working condition of the joints, it is difficult tochoose the right material properties in a modelling context. This reasoning suggests the definition of materialproperties in terms of intervals rather than in terms of crisp deterministic values.

In addition, the geometry of the component is often rather complex which generally implies modelling with

157

a high level of detail, including the modelling of the friction behaviour of the seal with its mechanicalsurroundings.

Including such a detailed component model in a system level model increases the computational burden ofthe solution process and becomes impractical when the simulation aims, for instance, at the investigation ofthe effects of material uncertainties on the dynamic behaviour of the full system.

The numerical methodology presented in this study allows the development of an equivalent linear model ofthe weather strip seal, and builds up a strategy to include uncertainty and variability typical for elastomericjoints in full-scale NVH models at a reasonable computational cost.

2 Three-level modelling approach

The numerical methodology developed in this study splits the modelling problem into three different levels:the material, the component and the system level:

• material level: in this level, a suitable material model is selected and the material parameters areidentified. Intervals on the material parameters due to variability and uncertainty are identified.

• component level: this level consist of two steps. The first step involves the definition of a non-linearfinite element model of the component using the previously defined material model. It includes the in-vestigation on the propagation of the variability at material level to the variability at component level,and the identification of intervals on the component properties.The second step involves a sensitivity based analysis on the detailed finite element model of the com-ponent. An equivalent linear parametric component model is developed and intervals on the equivalentmodel parameters are identified.

• system level: prediction of the scatter on the system dynamic response using the equivalent linearisedcomponent model.

This paper illustrates this methodology for a numerical case study, which involves the static non-linearbehaviour of the seal component along with its viscoelastic behaviour. The connection type consideredin the numerical case study is a typical car door weather strip seal, as shown in figure (1). The fastener inthe lower part of the seal is used to attach the seal to the car body, while the ring shape in the upper part isdirectly in contact with the car door. The weather strip seal has a total height of 30mm, with the ring shapein the upper part having a diameter of 15mm.

Figure 1: Typicalcar door weatherstrip seal profile.

Figure 2: FE detailed model of the set-up used inthe full system numerical simulation.

158 PROCEEDINGS OF ISMA2006

Figure (2) shows the numerical case considered for the validation of the developed approach. It consist of analuminium plate of dimensions: L = 140mm, W = 140mm and t = 1mm. The plate is clamped at one endand connected along one side to the ground through the weather strip seal. In this study, the aim is to evaluatethe effects of the weather strip seal on the dynamic behaviour of the plate. Therefore, only the seal upperring part is included in the numerical model. The dynamic strain dependency of material properties, knownas the Payne effect [7], is considered in the simulation, whereas temperature effects are not considered andthe Mullin’s effect [7] is considered negligible.

In the remainder of the paper, two numerical solution methods are compared. The first solution methoddescribed in section 3 consists of solving an entire system model that comprises all components. Section 4describes the developed three-level approach. Finally, both methods are being compared in section 5 in termsof accuracy and computational load.

3 Full system solution

In this section, the effects of the weather strip seal on the dynamic behaviour of the plate are evaluated con-sidering a fully detailed model of the elastomeric component coupled with the plate, as shown in figure (2).Non-linear material behaviour is identified using a test set-up.

3.1 Model choices

The non-linearities involved in the simulation are due to material, geometry and contact behaviour.

The non-linear material model used to describe the behaviour of the elastomeric component is a hyper-elasticmaterial model [8]. Hyper-elastic material models are characterised by their strain energy density functionW. The material is implicitly assumed isotropic and elastic. Stresses in the material are obtained by derivingthe strain energy density function with respect to strain:

Sij = 2∂W

∂Cij(1)

Cij = FkiFkj (2)

Fij =∂xi

∂Xj(3)

xi = ui + Xj (4)

where Sij is the 2ndP-K stress, Cij is the right Cauchy deformation tensor and Fij is the deformation gra-dient tensor, which maps the deformation of a material vector from the undeformed configuration X to thedeformed configuration x [8].

The hyper-elastic material model chosen for the case study is a Neo-Hookean material model [8], whosestrain energy density function W takes the form:

W = C10(I1 − 3) (5)

I1 = trCij (6)

where C10 is a material constant obtained from measurements and I1 is the first strain invariant.A finite element pre-analysis was performed to evaluate the amount of equivalent strain in the componentdue to the pre-load. The equivalent strain never reached values above 30% except in some small regions ofthe component and at maximum pre-load, justifying the choice of the material model.For a Neo-Hookean material model C10 = G∞/2, where G∞ is the equilibrium shear modulus evaluatedfrom a static shear test.

AMS3 - APPLICATIONS 159

The material frequency behaviour is modelled via the theory of small amplitude vibrations superimposed ondeformed viscoelastic solids [9], where it is shown that the expression of the 2ndP-K stress takes the form:

Sij = S0ij + L∗ijkl∆ε∗klexp(iωt) (7)

where S0ij denotes the static equilibrium stress tensor defined in equation (1), ∆ε∗kl is the strain increment

due to the superimposed small amplitude vibration, and L∗ijkl are the components of a fourth-order tensordefined by:

L∗ijkl = Dijkl + 2iωΦ∗ijkl (8)

In (8), Dijkl is the static equilibrium nonlinear elastic material constitutive tensor defined by

Dijkl = 4∂2W

∂C0ij∂C0

kl

(9)

and

Φ∗ijkl = φ∗0(ω)[(C0ik)−1(C0

jl)−1 + (C0

il)−1(C0

jk)−1] +

+ φ∗1(ω)[δik(C0jl)−1 + (C0

il)δjk] +

+ φ∗2(ω)[(C0ik)(C

0jl)−1 + (C0

il)−1(C0

jk)] +

+ φ∗10(ω)[δij(C0kl)−1] +

+ φ∗11(ω)[δijδkl] ++ φ∗12(ω)[δijC

0kl] +

+ φ∗20(ω)[C0ij(C

0kl)−1] +

+ φ∗21(ω)[C0ijδkl] +

+ φ∗22(ω)[C0ijC

0kl]

where δij is the Kronecker delta, C0ij is the static equilibrium right Cauchy deformation tensor and the phi-

functions are defined as:

φ∗0(ω) = 2g∗(ω)I2W2

φ∗1(ω) = 2g∗(ω)(W1 − I1W2)φ∗2(ω) = 2g∗(ω)W2

φ∗10(ω) = 4g∗(ω)I1W2

φ∗11(ω) = 4g∗(ω)(W11 + 2I1W12 + I21W22)

φ∗12(ω) = φ∗21(ω) = −4g∗(ω)(W12 + I1W22)φ∗20(ω) = −4g∗(ω)W2

φ∗22(ω) = 4g∗(ω)W22

where I1 is the first strain invariant defined previously, I2 = 12(CiiCjj−CijCji) is the second strain invariant,

Wij is the second derivative of the strain energy density function W with respect to the strain invariant Ii andto the strain invariant Ij , and g∗(ω) is the material complex shear modulus identified from measurements.

The solution procedure presented in this study considers only linear viscoelasticity. As a consequence thePayne effect, which is the dependency of the material properties on the amplitude of the input vibration,cannot be modelled directly. In the numerical simulation, the material measurements done at two inputvibration levels of interest are applied.

To simulate contact behaviour in the contact area between the plate and the seal, a Coulomb friction model isused [10]. A pre-analysis showed that the accuracy of the results is rather insensitive to the particular value


of the friction coefficient, while high values result in a fast convergence of the solution procedure. A functionvalue of unity was chosen for the numerical simulation.It is important to note that the case study shows a flat contact region between the weather strip seal andthe plate, whereas in reality this is often not the case. When the geometry of the contact area is complex,a sensitivity analysis of the assembly to the friction model and the friction coefficient can be of crucialimportance to correctly identify the final deformed shape of the component, and as a consequence, thecorrect stiffness and damping contribution of the component to the assembly.

Measured Material Properties

The plate is assumed to be made of aluminium, with Young’s modulus E = 72GPa, Poisson coefficientν = 0.33, and density ρ = 2800kg/m3.

In the Neo-Hookean hyper-elastic material model the only constant which needs to be determined is G∞. Inthe visco-elastic model, the only unknown is the complex function g∗(ω), which can be written in the form:

g∗(ω) = gstorage(ω) + jgloss(ω) (10)

where the real and imaginary part of the complex shear modulus are known, respectively, as the storageand loss shear modulus. The set-up shown in figure (3) was built with the aim of obtaining, with a single

Figure 3: Sketch of the set-up. Figure 4: Measured material properties.

Amplitude gstorage(2Hz) gstorage(500Hz) gloss(0Hz) gloss(500Hz)A1 0.1N/mm2 0.45N/mm2 ≈ 0 0.09N/mm2

A2 0.1N/mm2 0.4N/mm2 ≈ 0 0.07N/mm2

Table 1: Bounds on the measured material properties.

measurement, the quantities needed by the material models used in the simulation. The set-up consist of aBruel & Kjaer 4809 shaker and a PCB 288D01 mechanical impedance head on which the elastomer sampleis mounted. The other end of the elastomer sample is forcing a rigid wall. The material sample is obtainedby cutting a 5mm section from the weather strip seal component. It is placed as a vertical (hollow) cylinderbetween the two place holders. Ten different samples were tested and averaged. Assuming a one degree offreedom behaviour, where the known mass is the dynamic mass of the impedance head plus the mass of thecylinder bottom holder, the complex shear modulus of the elastomer is evaluated as a function of frequencyand for two particular values of the vibration amplitude (A2 > A1). Results are shown in figure (4), wherethe green and blue curves represent the modulus values, respectively, for amplitude A1 and A2, in agreementwith the fact that an increase in the amplitudes of vibration corresponds to a decrease in the storage and lossshear modulus. The measurements were done at an ambient temperature of 17oC.


To obtain the single parameter needed to define the Neo-Hookean hyper-elastic material model, it is assumedthat G∞ ∼= gstorage measured at very low frequencies. To define the viscoelastic model, the function g∗(ω)is required in an analytical form. The measured values of gstorage(ω) and gloss(ω) are curve-fitted withpolynomials, and the resulting g∗(ω) is used in the numerical simulation. To simulate the Payne effect, twosets of polynomial curve fits are obtained, one for each input amplitude of vibration (A1 and A2).

3.2 Numerical implementation

The MSC.Marc non-linear finite element solver is used to model the case study. However, the choice of thesolver type does not limit the generality of the developed methodology.

The plate is modelled via 784 isoparametric 4-noded thin shell element with six degrees of freedom per node.The material is assumed isotropic.

The weather strip seal is modelled via 5558 isoparametric 9-node brick elements. Those elements are specialbrick elements with an extra pressure node (Herrmann formulation) taking into account the incompress-ibility characteristic of the material [10]. The Neo-Hookean hyper-elastic material model is equivalent inMSC.Marc to a Mooney hyper-elastic material model with one constant. The elastomer is assumed masslessand incompressible.The theory of small amplitude vibrations in deformed viscoelastic solids [9] is implemented in MSC.Marc.In addition MSC.Marc allows to pass directly to the solver the analytical g∗(ω) function via an external userdefined UPHI.f subroutine.

3.3 Solution sequence

A first analysis is performed to take into account the weather strip seal pre-load effects on the plate. Thisanalysis is a quasi-static non-linear analysis which takes into account the non-linearities involved in thesimulation due to material, geometry and contact behaviour. In this analysis inertial effects are ignored.

A second analysis is performed to calculate frequency response functions of the plate at different weatherstrip seal pre-loads. For a given pre-load, the problem is linearised around the equilibrium state and considersall the effects of the non-linear deformation on the dynamic solution. In this analysis inertial effects areconsidered. MSC.Marc evaluates the frequency response functions with the Direct Stiffness Method [10]. Byincluding the aforementioned UPHI.f subroutine, the visco-elastic behaviour of the material is considered inthe dynamic solution process.

To simulate the Payne effect, two UPHI.f subroutines are considered in the analysis, respectively for theparameters identified with the amplitude of vibration A2 and the amplitude of vibration A1 (A2 > A1).

4 Application of the three-level modelling approach

The developed methodology splits the problem into three different levels of analysis: the material, the com-ponent and the system level:

4.1 Material level

The material complex shear modulus is measured, with the set-up shown in figure (3), as a function of fre-quency and as a function of the amplitude of the input vibration. Results of the measurements are shown infigure (4).At this level, intervals on the material properties are identified. In the following component level, these


Figure 5: Interval on the real part of the complexg∗(ω) modulus.

Figure 6: Interval on the imaginary part of thecomplex g∗(ω) modulus.

intervals are used to identify intervals in the component properties which are then used to identify the inter-vals on the parameters describing the linear equivalent component model. To pass to the component level afull description of the material behaviour within the frequency and amplitude bounds of interest, advancednon-deterministic numerical tools would be required. As the interest in this study is to explain the developedmethodology, a linear behaviour within these bounds is assumed. The bounds on the measured materialproperties used in the numerical simulation are shown in table (1). From these bounds, intervals in the realand imaginary part of the complex shear modulus are obtained, as shown in figure (5) and (6). The caseof higher-order (polynomial) curve fits of the measured properties in figure (4) will be analysed in the nearfuture.

4.2 Component level: step 1

Figure 7: Finite element model ofthe component section.

Figure 8: Static pre-load:1. Figure 9: Static pre-load:2.

A quasi-static non-linear analysis is performed on a two-dimensional section of the component shown infigure (7) (thickness of 1mm in the third dimension). In figure (10) the compression load deflection shape(CLD) of the component is shown, which results from applying a static displacement pre-load on the com-ponent and evaluating the needed force. Due to the non-linear behaviour of the curve, the tangent stiffnessrather than the linear stiffness needs to be considered in the equivalent model. In figure (11), the tangentstiffness is obtained from the numerical differentiation of the CLD with respect to the static pre-load. Twoparticular static pre-load cases are highlighted in figure (11). The geometric deformations and strain distrib-utions for those two cases are shown in figures (8) and (9). By assuming that the actual pre-load conditionsof the considered test case is situated between the two bounds shown in figures (8) and (9), an interval on thecomponent tangent stiffness can be identified.

On the same component section, a dynamic analysis is performed considering viscoelasticity via the MSC.Marc


UPHI.f subroutine adopting a linear curve fit. Two sources of uncertainty affecting the analysis results arebeing investigated, i.e. the amount of pre-load and the vibration level. A vertex analysis is performed tak-ing into account each combination of assumed extreme values of the numerical properties representing theuncertainties under investigation. In total, four analyses are performed, combining both values of selectedpre-loads with both UPHI.f subroutines taking into account the Payne effect at the analysed amplitudes A1and A2. Bounds on the real and imaginary part of the component complex tangent stiffness:

K∗t (ω) = Kstorage

t (ω) + jK losst (ω) (11)

are shown in figure (12) and (13), where the blue and red areas are related, respectively, to the blue andred areas of figure (5) and (6). Note that the interval on the component stiffness due only to the pre-load isrepresented by the vertical line at 0Hz.

Figure 10: Component CLD as a function of pre-load.

Figure 11: Component tangent stiffness Kt as afunction of pre-load.

Figure 12: Frequency-dependent interval on thereal part of the complex K∗

t (ω) modulus (uncer-tainty on pre-load and vibration level).

Figure 13: Frequency-dependent interval on theimaginary part of the complex K∗

t (ω) modulus(uncertainty on pre-load and vibration level).

4.3 Component level: step 2

The equivalent linear parametric interval model, shown in figure (14), consists of replacing the actual weatherstrip by a single linear complex spring whose values of stiffness and damping agree with the intervals evalu-ated in the previous step. In MSC.Marc the complex spring element represents a parallel linear spring-dashpot


Figure 14: The equivalent interval parametricmodel.

Figure 15: FE equivalent model of the set-up.

connection [10]. Such a system is known in linear viscoelasticity as the Kelvin-Voigt model [7]. The dynamicstiffness of the Kelvin-Voigt model shows a constant behaviour with frequency for the real part and a linearlyincreasing behaviour with frequency for the imaginary part.

The real part of the equivalent linear parametric interval model (i.e. the linear spring in the Kelvin-Voigtmodel) will be defined such that it covers the frequency dependency identified for the stiffness on the materiallevel. It is derived from the shaded area shown in figure (12). For pre-load:1 and pre-load:2 the identifiedintervals will be, respectively, defined by the bounds (A,B) and (C,D).

For the imaginary part of the equivalent linear parametric interval model the reasoning is the same, except forthe fact that the imaginary part of the dynamic stiffness of the Kelvin-Voigt model increases with frequency.To have an average distribution of damping within the frequency band of interest, only one interval forpre-load is identified. According to the behaviour of the gloss(ω) shown in figure (4), it is assumed thatthe bounds of this interval are reached at the middle of the frequency band of interest. For pre-load 1 andpre-load 2 the identified interval will be defined by the bounds (AB,CD), as shown in figure (13).

4.4 System level

The equivalent linearised model is then distributed according to its dimensions, along the contact area be-tween the plate and the weather strip seal. The equivalent model is connected to the mesh of the plate throughrigid connections (RBE2 elements) [10], in such a way that the spring end points can be centred with respectto the contact area independently from the plate mesh size. The assembly of the plate and the equivalentmodel is shown in figure (15).

5 Result comparison

Figures (16) to (18) show the results of the simulation in terms of the frequency response function betweenthe normal displacements in one point of the plate and the applied dynamic force at another point of the plate.Figure (16) shows the results of the full system solution approach, with viscoelasticity and Payne effect takeninto account. The black curve represents the baseline undamped dynamic response of the plate when theweather strip seal is not considered. The blue curve is related to pre-load:1 and amplitude of vibration A2,while the red curve is related to pre-load:2 and amplitude of vibration A1, such that the maximum scatter onthe system response is represented.The effect of the weather strip seal on the dynamic behaviour of the plate is clearly visible in the entirefrequency range of interest. In line with the component stiffness behaviour shown in figure (11), the responseof the plate at 0Hz is decreased as the pre-load increases. As a result of the combined effect of pre-loadand viscoelastic behaviour, the plate resonances are shifted to higher frequencies. The weather strip seal is


Figure 16: Frequency response function between two points on the plate (full system approach).

introducing damping in the system, which is clearly increasing with pre-load as the compression increasesthe contact area between the seal and the plate. It can be noted that the plate third resonance is only slightlyinfluenced by the weather strip seal. This resonance corresponds to the plate mode shape (0/1) [11], asshown in figure (19), for which the seal is located near the mode shape nodal line.

Figure 17: Frequency response function betweentwo points on the plate.

Figure 18: Frequency response function betweentwo points on the plate.

The dashed curves in figures (17) and (18) were obtained from the three-level methodology. For pre-load:1,the K∗

t − lower and K∗t − upper curves represent the response of the equivalent system respectively for the

bounds (A,AB) and (B,AB), in figures (12) and (13). For pre-load:2, the K∗t − lower and K∗

t − uppercurves represent the response of the equivalent system respectively for the bounds (C,CD) and (D,CD),in figures (12) and (13).

The response of the full system at 0Hz lies within the boundaries, while the K∗t − lower curve clearly

underestimates the system resonances. This is because of the amount of conservatism introduced by theselection of the bounds on K∗

t . According to the measurements shown in figure (4), the transition region onthe complex shear modulus is around 20Hz. This means that, except at frequencies below 20Hz, the system


meets the conditions defined by the bounds [B,AB] for pre-load:1 and [D,CD] for pre-load:2 in almost allthe frequency range of interest. The K∗

t − upper curve, while overestimating the response of the system at0Hz, clearly represents the full system resonances in the frequency band of interest within an accuracy of5%.It can be seen from figures (17) and (18) that the amplitude of the full system resonances lay within thebounds in the frequency band of interest. For frequencies above 250Hz, as the imaginary part of the dynamicstiffness of the Kelvin-Voigt model increases with frequency, the amplitude of the full system resonances tendto be underestimated.

The total CPU time needed for finding the solution with the developed three-level methodology was approx-imately 10% of the CPU time for the full system solution.

A comparison of plate mode shapes is shown in figures (19) and (20).

Figure 19: Full system - plate mode shapes: (0/0), (1/0), (0/1), (2/0), and (1/1).

Figure 20: Equivalent system - plate mode shapes: (0/0), (1/0), (0/1), (2/0), and (1/1).

6 Conclusions and future work

A methodology based on a three-level modelling approach involving material, component and system level,has been developed to model the NVH behaviour of elastomeric line connections. The focus was on the dy-namic modelling of car door weather strip seals and a numerical case study was built-up to test the method-ology. Two solution methods are compared. The first solution method consist of solving the entire systemmodel with the component modelled in full detail. In the second solution method the developed methodol-ogy is applied and the difference between the two methods in terms of accuracy and CPU solution time areidentified.

It is shown that an equivalent system can represent the behaviour of the full system with reasonable accuracyat a reasonable computational cost. The accuracy of the methodology depends mainly on the choice of theequivalent linear model, and the limitations introduced by the use of the Kelvin-Voigt model were shown.Other mechanical analog of springs and dashpots, which better simulates the viscoelastic behaviour of thematerial, will be used in a near future.

A consistent follow up to this study is the experimental validation at the component and system level. Themeasurement set-up shown in section (3.1) can be adapted to validate the weather strip seal componentmodel. An additional set-up has been built to validate the numerical approach at system level, and correlationof the results with the numerical approach will be presented in a near future.

Temperature effects have not been considered in this study, despite their importance for the NVH behaviourof elastomeric line connections. Identifying intervals in the complex shear modulus also as a function of


temperature perfectly fit in the developed methodology solution sequence and will be the subject of futureresearch.

A last comment concerns the choice of the intervals used in the simulation. The scatter of the system responsewas evaluated by simply considering the crisp values of the parameters at the vertex bounds of the intervalsand assuming a linear behaviour between those extremes. Advanced non-deterministic numerical techniquesare in general needed to describe the material behaviour within the intervals. In this respect an approachbased on the Fuzzy Finite Element [12] method has been presented by the authors in a previous work [13]and will be adopted for the present case study in a near future.

Acknowledgements

This research is part of the Belgian Program on Interuniversity Poles of Attraction initiated by the BelgianFederal Science Policy Office (AMS IAP V/06).

References

[1] J. Park, T. Siegmund, and L. Mongeau. Sound transmission through elastomeric bulb seals. Journal of Sound andVibration, 259(2):299–322, 2003.

[2] J. Park, L. Mongeau, and T. Siegmund. Effects of geometric parameters on the sound transmission characteristicof bulb seals. In SAE Noise anf Vibration Conference and Exhibition, Grand Traverse, MI, USA., 2003.

[3] Y. Gur and K. N. Morman. Sound transmission analysis of vehicle door sealing system. In Proceedings of theSAE Noise and Vibration Conference, Traverse City, MI, USA, number 1804 in 01, pages 1187–1196, 1999.

[4] Ruben van Schalkwijk. Simulation of noise penetration through car weather seals. In Proceedings of the ABAQUSUser Conference, 1997.

[5] D. A. Wagner, K. N. Morman Jr., Y. Gur, and M. R. Koka. Nonlinear analysis of automotive door weatherstripseals. Finite Elements in Analysis and Design, 28:33–50, 1997.

[6] A. Stenti, D. Moens, and W. Desmet. Dynamic modeling of car door weather seals: A first outline. In Proc. ofISMA 2004, International Conference on Noise and Vibration Engineering, pages 1249–1262, 2004.

[7] Aklonis J.J and W.J. MacKnight. Introduction to Polymer Viscoelasticity. John Wiley & Sons, 2nd edition, 1983.

[8] Bonet J. and Wood R.D. Nonlinear continuum mechanics for finite element analysis. Cambridge University Press,1st edition, 1997.

[9] Morman K.N.Jr. and J.C. Nagtegaal. Finite element analysis of small amplitude vibrations in pre-stressed non-linear visco-elastic solids. Int. J. Num. Meth. Engng., 19:1079–1103, 1983.

[10] MSC/MARC. Theory and User Information, 2003.

[11] Arthur W. Leissa. Vibration of Plates. NASA-SP160, 1969.

[12] D. Moens and D. Vandepitte. Non-probabilistic approaches for non-deterministic dynamic fe analysis of impre-cisely defined structures. In Proc. of ISMA 2004, International Conference on Noise and Vibration Engineering,pages 3095–3119, 2004.

[13] A. Stenti, D. Moens, M. De Munck, W. Desmet, and D.Vandepitte. Application of the fuzzy finite element methodto the dynamic modelling of car door weather seals. In Proc. of ICSV12 2005, International Conference on Soundand Vibration, 2005.


157_168

Documents

Transcript of 157_168