Vibrations, damping and power dissipation in Car Tyres

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Vibrations, damping and power dissipation in Car Tyres

Martin Fraggstedt

Stockholm 2008

Doctoral Thesis TRITA-AVE 2008:24

ISSN 1651-7660 ISBN 978-91-7178-996-9

Royal Institute of Technology School of Engineering Sciences

Department of Aeronautical and Vehicle Engineering

The Marcus Wallenberg Laboratory for Sound and Vibration Research

Postal address Visiting address Contact

Royal Institute of Technology Teknikringen 8 Tel: +46 8 790 9202

MWL / AVE Stockholm Fax: +46 8 790 6122

SE-100 44 Stockholm Email:mfragg@kth.se

Sweden

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Abstract

Traffic is a major source of green house gases. The transport field stands for 32 % of the energy consumption and 28 % of the total CO2 emissions, where road transports alone causes 84 % of these figures. The energy consumed by a car travelling at constant speed, is due to engine inefficiency, internal friction, and the energy needed to overcome resisting forces such as aerodynamic drag and rolling resistance. Rolling resistance plays a rather large role when it comes to fuel economy. An improvement in rolling resistance of 10 % can yield fuel consumption improvements ranging from 0.5 to 1.5 % for passenger cars and light trucks and 1.5 to 3 % for heavy trucks.

The objective of this thesis is to estimate the power consumption in the tyres. To do this a car tyre is modelled with waveguide finite elements. A non-linear contact model is used to calculate the contact forces as the tyre is rolling on a rough road. The contact forces com- bined with the response of the tyre is used to estimate the input power to the tyre structure, which determines a significant part of the rolling resistance. This is the first rolling resistance model based on physical principles and design data.

The elements used in the waveguide finite elements tyre model are derived and validated. The motion of the tyre belt and side wall is described with quadratic anisotropic curved deep shell elements that includes pre-stress and the motion of the tread on top of the belt by curved quadratic, Lagrange type, homogenous, isotropic two dimensional solid elements. The tyre model accounts for: the curvature, the geometry of the cross-section, the pre-stress due to inflation pressure, the anisotropic material properties and the rigid body properties of the rim and is based on data provided by Goodyear.

To validate the tyre model, mobility measurements and an experimental modal analysis have been made. The model agrees very well with point mobility measurements up to roughly 250 Hz. The eigen- frequency prediction is within five percent for most of the identified modes. The estimated damping is a bit too low especially for the anti- symmetric modes. The non-proportional damping used in the model is based on an ad hoc curve fitting procedure against measured mobilities.

The non-linear contact force prediction, made by the division of applied acoustics, Chalmers University of Technology takes the tyre, the road texture and the tread pattern into account.

The dissipated power is calculated through the injected power and the power dissipated within each element. It is shown that a rough road leads to more dissipation than a smooth road. A demonstration on real existing motor ways, for which rolling resistance measurements also have been made, show the potential of the method.

The damping is very important for the rolling resistance prediction. The damping properties of the tyremodel are therefore updated based on measurement, equivalent structure modelling and viscoelastic material models. This updated model is slightly better at the point mobility prediction and is far better at predicting the damping level of the identified modes from the experimental modal analysis.

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Doctoral Thesis

The thesis consists of an introduction and the following four papers:

Paper A

S. Finnveden, M. Fraggstedt, Waveguide finite elements for curved structures, Journal of Sound and Vibration 312 (2008) 644-671.

Paper B

M. Fraggstedt, S. Finnveden, A Waveguide Finite Element Model Of A Pneumatic Tyre, Submitted August 2007 to Journal of Sound and Vibra- tion.

Paper C

M. Fraggstedt, S. Finnveden, Power dissipation in car tyres, 2008. To be submitted to the Journal of Sound and Vibration .

Paper D

M. Fraggstedt, S. Finnveden, The influence of the road on rolling resistance, 2008.

Paper E

M. Fraggstedt, S. Finnveden, Estimates of the visco-elastic properties of car tyres, 2008. To be submitted to the Journal of Sound and Vibration.

Contribution from the author of this thesis Paper A

Developed the straight elements. Performed the validation study. Cowrote the paper.

Paper B

Experimental modal analysis and the mobility measurements. Performed simulations. Fine tuning of the model developed by the supervisor. Wrote the paper.

Paper C

Performed the power calculations. Literature study on rolling resistance.

Wrote the paper.

Paper D

Performed the power calculations. Wrote the report.

Paper E

Performed the measurements, simulation and the optimisation. Wrote the paper.

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Material from this thesis has been presented at eight workshops in the ITARI project plus at three conferences:

SVIB, Nordic Vibration Research Conference, Stockholm Sweden, M. Fraggstedt , Estimation of Damping in Car Tyres, 2004.

Novem conference 2005, S. Finnveden, C.-M. Nilsson and M. Fraggstedt, Waveguide FEA of the Vibration of Rolling Car Tyres.

Euronoise 2006, Tampere, Finland, M. Fraggstedt, Rolling Resistance Of Car Tyres, 2006.

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1 Introduction

1.1 Background

For over fifty years traffic has been an irritating noise polluter. For higher speeds tyres have been found to be the major contributor for traffic noise.

Also the interior noise in the vehicle due to the tyres are becoming more important as other noise sources such as engines, exhaust systems and gear boxes are better managed.

The negative effect on the environment has been highlighted for a number of years, given that traffic is a major source of green house gases. The transport field is representing 32% of the energy consumption and 28% of the total CO2 emissions, where road transports alone stands for 84 % of these figures [1].

When it comes to the dynamics of the car the tyres are crucial, as they provide the grip required for cornering, braking and acceleration. In addi- tion, tyres are also highly involved in the cars handling abilities. As a final point it is the tyres and the suspension system that assures a comfortable ride.

The energy consumed by a car travelling at constant speed, is due to engine inefficiency, internal friction, and the energy needed to overcome resisting forces such as aerodynamic drag and rolling resistance, which is one of the main topics of this thesis.

The rolling resistance Fr is defined as the energy consumed per unit of distance travelled [2]. The unit is N m/m = N which is equivalent to a drag force in Newtons. Tyres are made of reinforced rubber, which is a viscoelastic material. As it deforms a part of the energy is stored elastically but the remainder is dissipated as heat. These hysteretic losses, as well as aerodynamic drag and friction in the contact patch and with the rim are losses that contribute to the total drag force on a moving vehicle. Rolling resistance has a rather large impact when it comes to fuel economy. A 10

% improvement in rolling resistance can give fuel consumption reductions ranging from 0.5 to 1.5 % for passenger cars and light trucks and 1.5 to 3

% for heavy trucks [3].

In undergraduate texts, the rolling resistance is given as a dimensionless constant times the gravity force,

Fr = Crm g, (1)

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where m is the mass, g is the constant of gravity and Cr is the rolling resistance coefficient. Cr is normally in the range 0.01-0.02 with a typical value of 0.012 for a passenger car tyre on dry asphalt [4]. The power consumed by this force is

P = V Fr= V Crm g (2)

where V is the speed of the vehicle. In equation (1) the only explicit parameter is the load. The variation with other parameters are concealed in Cr. Studies has shown that the rolling resistance coefficient is influenced by a number of parameters such as speed, driving torque, acceleration, rubber compound, internal and ambient temperature, road texture, road roughness, and wear. The model is however usually sufficient for some applications. In this thesis the influence of the road texture and tyre characteristics is studied.

The aim of this thesis is to model a radial car tyre with waveguide finite elements and to use this model to estimate the power dissipation as the tyre is rolling on a rough road. These losses determine a significant part of the rolling resistance. As a demonstration the procedure is used on real existing motor ways, for which rolling resistance measurement data are available, making a comparison possible. This is the first model based on physical principles to estimate the rolling resistance.

1.2 Car tyres

Car tyres are made of several different materials including steel, fabric and of course numerous rubber compounds, see Figure 1. To get different dynamic properties in the tyre sub regions the materials are used in many ways. The three major sub regions of the tyre are the upper side wall, the lower side wall and the central area. The ply is a layer of embedded fabric in the rubber.

At the lower side walls the ply encloses a volume filled with both steel wires and hard rubber materials, this makes the lower side walls relatively stiff.

The upper side walls are on the other hand quite flexible, since the ply layer there is simple and there is less steel in there. The central area consists of the belt and the tread. The belt consists of a rubber embedded steel lining in the circumferential direction to give support and rigidity. The tread is an about 13 mm thick rubber layer which is there to provide the grip. This makes the central area rigid with respect to bending waves in the circumferential direction but fairly flexible when it comes to motion within the cross-section. The high loss factor of the tread rubber makes the latter motion highly damped.

The tyre studied here is a Goodyear, radial, passenger car tyre, with the dimensions 205/55ZR16, mounted on an Argos rim. The tyre is ’slick’, i.e. it does not have a tread pattern or groves, but in all other aspects has properties typical of a production tyre.

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Figure 1: The tyre consists of three major sub regions. Upper side wall, lower side wall and the central area.

To make use of the rotational symmetry of the tyre a waveguide finite element approach is employed, where only the cross-section is discretised, and hence the calculation time is reduced.

1.3 Waveguide finite elements

A waveguide is a wide-ranging term for a device, which constrains or guides the propagation of mechanical waves along the waveguide. Here it is also assumed that a waveguide has constant geometrical and material properties along one direction.

Waveguide Finite Elements (FE) yield equations of motion for systems with wave-propagation along a single direction, in which the structure is uniform. It is then possible to separate the solution to the wave equation into one part depending on the cross-section, one part depending on the coordinate along the waveguide and one part depending on time.

As an example of a waveguide, a generalised beam, in which longitudinal, torsional, shearing and flexural waves can travel, can be considered. The main idea with a waveguide approach is to study waves propagating in the structure.

The most important benefit with waveguide FE, is that it decreases the calculation time compared to ordinary finite elements since only the cross-section has to be discretised and the number of degrees of freedom is reduced. Another advantage compared to conventional FE methods, is that it is straight forward to identify and analyse different wave types, which allows a physical understanding of the structure under investigation. The

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ability to handle infinite waveguides, is an additional good feature of this method.

Forced response solutions for waveguide FE models can be handled in several different ways. Four of these methods for forced responses will be briefly explained.

For infinite waveguides an approach based on Fourier transforms may be used. The equations of motion are transformed to the wave number domain through a spatial Fourier transform. The solution in the wave number domain then has to be transformed back to the spatial domain through an inverse Fourier transform which generally involves residue calculus [[5], [6]].

’Super Spectral Elements’, (SSE), are derived by using wave solutions, given from a generalised eigenvalue problem, as test and shape functions in the variational form of the wave equation [7]. At the ends, the spectral elements can be coupled to other spectral elements or to regular finite elements.

Certain convenient boundary conditions are fullled by a wave solution together with a companion wave travelling in the opposite direction. For such boundary conditions, the eigenmodes of a finite length structure are directly identified by a two-dimensional waveguide FE analysis, which has a much lower computational cost than a conventional three-dimensional FE analysis. Once the eigenmodes are identified, a modal analysis gives the forced response of the structure. In reference [6] the convenient boundary condition for a straight waveguide was the shear diaphragm condition, in which all motion within the cross section is blocked while all motion along the waveguide is free. Another convenient boundary condition is that for a circular structure, such as a car tyre. In this case the response is a periodic function of the angular coordinate. The non-proportional damping used in the present analysis, however, leads to non-orthogonal eigenmodes and therefore this method is not used.

In an assumed modes procedure the response is assumed to be an expo- nential Fourier series in the spatial domain. This approach is suitable, since the tyre is a circular structure and the solutions to the wave equation will be periodic with respect to the circumferential angle. The sum is inserted into the variational statement, and upon variation follows the equations of motion. One advantage with this direct methodology in the frequency domain is that it is uncomplicated to handle fluid-structure interactions [8]. The car tyre including the air cavity has been modeled successfully by Nilsson [5] with a waveguide FE approach similar to the one presented here. Also, frequency dependant materials are easily included. This is an especially good quality when considering a structure such as a car tyre, which is built from rubber, whose material properties show a strong frequency dependency.

This is the procedure used in the present analysis.

Straight waveguide finite elements were first formulated by Alaami [9]

and Lagasse [10] in 1973. Curved waveguides are used by Hladky-Hennion 4

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[11] and Nilsson [5]. In reference [5] there is a comprehensive review of the applications of waveguide FE for vibro-acoustic problems.

The waveguide finite elements used in the tyre model are developed and validated in Paper A.

1.4 Tyre model

1.4.1 Existing tyre models.

There are several earlier attempts to describe the dynamic behaviour of the car tyre. There are many different approaches based on equivalent structure modelling. Andersson [12] and Larsson et al. [13] used simple orthotropic plate strips and Muggleton et al. [14] used an assembly of ortotropic plate strips. B¨ohm [15], Kropp [16], Kung et al. [17] and Dohrmann [18] used a circular ring model to capture the dynamic behaviour whearas Molisani et al. [19] and Kim et al. [20] used circular thin shells. Pinnington and Briscoe [21], chose to use straight beams, Pinnington [22] curved beams and Larsson et al. [23] coupled elastic layers. A finite element model was used by Kung et al. [17] , Richards [24] and Pietrzyk [25]. Nilsson [5] used a waveguide finite element model built up from pre-stressed conical thin shell elements. This approach is similar to the one presented here. Kim et al. [26]

divided the tyre response into different frequency regimes in which diffrent wave types are dominating. They found that below 500 Hz most waves are inefficient sound radiators due to the cancelation effect. In references [19], [24], [5], [27] and [28] the air cavity is also taken into account which is of main importance for structure borne noise into the passenger compartment, but is disregarded here.

More recent attempts include the work by Lopez et al. [29] on the vibrations in a deformed rolling tyre using a FE model in a fixed (eulerian) frame, and the work by Brinkmeier et al. [30], where FE methods are used to simulate tyre noise.

1.4.2 Waveguide FE tyre model.

The Waveguide FE tyre model is based on an Abaqus finite element model given by Goodyear [25], specifying the geometry and most of the elastic data. The big disadvantage with this procedure is the need to rely on a tyre manufacturer for input data. The advantage is that the model is based on design data, therefore communication with the design engineers is possible.

It is of very good quality. The model was originally designed for tyre road noise predictions.

The motion of the tyre belt and side wall is described with quadratic anisotropic, deep shell elements that include pre-stress and the motion of the tread on top of the tyre by quadratic, Lagrange type, isotropic two dimensional elements.

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Two different tyre cross-section meshes have been used. The medium mesh with 42 elements, 113 nodes and 516 degrees of freedom (DOF) is seen in Fig. 2 and an enriched Abaqus full mesh with 214 elements, 569 nodes and 2562 DOF in Fig. 3. Assuming that the same nodal density is used in the circumferential direction, the full mesh is equivalent to a conventional FE model with a total of 2562 × 1300 ≈ 3.3 10⁶ DOF. Based on numerical experiments involving forced response and dispersion curves calculations, it was concluded that the medium mesh is sufficient up to at least 1000 Hz.

The Abaqus model does not include any damping information and a rather simplistic damping definition is used in most of the work presented here. It is based on an ad hoc curve fitting procedure against measured mobilities.

The model accounts for: the curvature, the geometry of the cross-section, the pre-stress due to inflation pressure, the anisotropic material properties and the rigid body properties of the rim.

The tyre model is described and validated in Paper B.

2 Analytical modelling of rolling resistance

The analytical rolling resistance investigations available in the literature are all based on rather simple equivalent structures. Stutts and Soedel [31] used a tension band on a viscoelastic foundation. Kim and Savkoor [32] used an elastic ring supported on a viscoelastic foundation. Yam et al [33] based their calculation on experimentally deduced modal parameters. Popov et al [34] modeled a truck tyre, based on the model developed by Kim and Savkoor [32]. The stiffness and damping parameters needed, came from an experimental modal analysis.

The model used in this study, has the correct tyre geometry and stiffness, and it includes the road texture effect. None of the models above are treating a rough road even though the road texture and roughness have a significant effect on the rolling resistance [35].

2.1 Rolling resistance estimation based on dissipated power The approach used throughout this work is to estimate the rolling resistance from the power dissipation in the tyre due to internal viscoelastic forces.

The contact forces are calculated by Chalmers University of Technology (CTH), based on a non-linear contact model in which the tyre structure is described by its flexibility matrix [36]. Topographies of the surface are scanned, the tread pattern is accounted for, and then the tyre is ’rolled’

over it. The calculations are made in the time domain and is based on a Lagrange multipliers approach. The non-linear conditions used are: i) the tyre cannot indent into the road, ii) if a point is not in contact the force

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is zero and iii) that the force cannot be negative (road pulling tyre down) ([37] chapter 6). Only forces acting normal to the road are considered.

Newton’s law applies for a fixed piece of matter: a particle. The equations of motion are therefore solved in a Lagrangian coordinate system fixed to the rotating tyre. In doing so, the Corioli forces are neglected. The cen- trifugal force is also neglected and it is assumed that the increased static tension and radial expansion that it induces are already included in the definition of the tyre’s steady state. Upon this basis, the tyre vibration is predicted.

Energy is always conserved, it is just transformed from one form to another. The energy going into the system must therefore equal the dissipated energy in the system. The same argument is valid for the power, conse- quently, the injected power must equal the dissipated power.

Since only normal forces are considered, the dissipated power stems from damping alone.

In Paper C the dissipated power procedure is developed and the results are shown to be in the right range compared to measurements. It is also shown that a rough road dissipated more power than a smooth road. Paper D contains a demonstration of the procedure on real existing motor ways for which measured rolling resistance data are available. The results are promising and if the roads are different enough a ranking is possible.

To clarify the rolling resistance - power dissipation equivalence a simple example can be considered. To keep a car running at a constant speed, in the presence of rolling resistance forces, requires that energy is added. This added energy must be equal or greater than the energy dissipated in the tyres. The equal sign applies if there are no other dissipation mechanisms, such as friction in the tyre rim interface or losses in the suspension system, than the internal dissipation in the tyre.

2.2 Rolling resistance - dynamic method

One could argue that once the contact forces are known the rolling resistance can be estimated from those alone. The power needed to keep the tyre rolling at a given angular velocity Ω subject to a resisting moment M is given by

P = MΩ = Fd Ω (3)

, where F is the contact force resultant and d is the centre of contact forces, see Figure 4. To produce the typical power 800 W, at a speed of 80 km/h and a vertical load corresponding to 300 kg, d should be around 3 mm, but the spatial force resolution is only 4 mm meaning that the dynamic method is unreliable, and numerical experiments shows that it often fails. The energy approach used in this work is more robust.

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3 Tyre model updating

The tyre model from Paper B, is in Paper E updated to include viscoelastic properties making it more realistic. This is made in a modular way with inverse methods. The reason that this work is undertaken is because the rolling resistance is depending solely on the damping which therefore has to be modelled as good as possible. The necessary building blocks in the updating procedure are: i) Measurements on tread samples, a beam sample and the whole tyre, ii) equivalent structure modelling, iii) the initial tyre model and iv) a viscoelastic material model. The tyre is divided into three substructures, tread, belt and sidewall. Each measurement result provides input for each substructure.

The updating procedure concerns the stiffness and damping of the whole tyre. A good article on model updating is reference [44]. Relevant articles regarding inverse methods includes McIntyre and Woodhouse’s [45] work on the experimental determination of material properties of orthotropic sheet material and the study by Geng et al. [46], on damping in truck tyres, where a generally distributed viscous damping model is fitted to observations from an experimental modal analysis. The modal techniques, used in this work, can only provide estimations of the loss factor at resonant frequencies. This fact can, at least for simple structures, be circumvented by instead making, a more challenging, spatially distributed measurement to estimate the loss factor from the complex wave order [47], while this is not pursued here.

3.1 Damping - Viscoelastic materials.

Since the rubber in the tyre is highly damped, the elastic properties alone, are not sufficient to accurately model tyre dynamics. The viscoelastic properties are needed too. A good damping description is also crucial for rolling resistance estimations. The tyre model used in Paper B, C and D has a very simplistic damping model which is updated in Paper E.

Damping can be introduced in many ways. For structure borne sound applications the linear structural damping, with a frequency independent lossfactor, is widely used. This model is, however, physically unrealistic since it leads to non-casual behavior [38]. The standard linear model, where the response depends on a convolution between the response history and relaxation functions, first introduced by Boltzman [39], does not have this drawback. However, it turns out that many relaxation functions, and therefor many unknown parameters, are needed to accurately describe the frequency dependence of rubber materials over a wide frequency range [40].

Another approach is the fractional derivative model. A five parameter fractional derivative model in the frequency domain is suggested by [40]. It is argued that the reason that only five parameters are needed is that these parameters are more consistent with the physical principles involved [40].

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Rubber is a visco-elastic material exhibiting frequency dependant material characteristics. A rather simple mathematical model describing the visco-elastic behaviour of rubber, valid for a harmonic time dependence e⁻^iωt, is the fractional Kelvin-Voigt model described in for example [41].

This model uses three fractional derivative material model (FDMM) parameters to describe the frequency dependence.

D(ω) = Dˆ 0(1 + (−iω ω0

)^α). (4)

,where ω is the angular frequency, D0 is a characteristic static stiffness (stiffness at zero frequency), and ω0 and α describes the frequency dependance.

ω⁰ has dimension [rad/s] and α is dimensionless. Note that the Fourier transform of the fractional derivative of order α of x(t) is (−iω)^α times the Fourier transform of x(t) [40], which is the reason for using the term ’frac- tal’. When ω⁰ tends to infinity the model becomes purely elastic and when α equals unity the model reduces to the classic Kelvin-Voigt material model.

The lossfactor is defined by:

η = −Im( ˆD)

Re( ˆD). (5)

The lossfactor at zero frequency is zero as it should be, but this simple model does not lead to a vanishing loss factor when the frequency goes to infinity. This is, however, not a problem since the focus of this investigation is up to 1000 Hz where this simple model is deemed sufficient.

The fractional Kelvin - Voigt model is a simplification compared to the five parameter model. The reason that the three parameter model is used, is exclusively to reduce the number of parameters and thereby simplifying the analysis and making the damping estimation more robust. It should be emphasized that any fitting function is possible, as long as it depicts the observed material behaviour sufficiently accurately within the considered frequency domain, [42].

The tread of the tyre is made of homogenous, isotropic, almost incompressible rubber and its stiffness properties are thus defined by the shear modulus alone. Results from a dynamic shear modulus measurement on tread samples is in Paper E used to determine the FDMM parameters of the tread.

3.2 Equivalent structure modelling

The elastic properties of the deep shell elements used in the waveguide FE model are given by Goodyear. There are in total 29 shell elements in the tyre model (see Figure 2) of which 15 are unique The local stiffness matrix of each shell element is an 8 by 8 matrix , D, with in total 21 independent

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elements (23 if the the transverse shear stiffnesses are different to the inplane shear stiffness).

Damping is added by making the stiffness matrices complex . A complete description of the damping thus requires 21 · 15 = 315 lossfactors, which, furthermore, are frequency dependent. This is not feasible and therefor some kind of parametrisation has to be made.

The idea is to model each shell element by an equivalent structure made up of rubber embedded steel wires. It is inspired by reference [43], where Pinnington studies wave transmission in flexible tubes reinforced by steel wires.

The equivalent structure depends on a number of geometrical parameters such as layer thicknesses, angle of wires, steel to rubber ratios. It also depends on two elastic parameters: the shear modulus of rubber and the Young’s modulus of steel, which facilitates the modelling.

The equivalent structure is first determined for the static case. The shell elements can be divided into two groups based on the angle of the wires:

belt and sidewall elements.

Viscoelastic properties can be included by instead of using static stiffness values making use of Equation (4), where the static stiffness is known and α and ω0 are to be determined.

Measurements of the eigenfrequencies and corresponding loss factors of a beam sample cut out from the the tyre belt with tread on top of it are used to determine the FDMM parameters of the belt. Modal parameters from the experimental modal analysis on the complete tyre made earlier, are used to determine the FDMM parameters of the side wall.

This work was very successful and the updated tyre model has a more realistic damping behavior, which also agrees better with measurements.

4 Summary of the papers

4.1 Paper A. Waveguide finite elements for curved structures

A Waveguide finite elements formulation for curved structures is derived.

The formulation is valid for structures having constant properties along one axis, such as infinite straight beams, cylinders or car tyres.

The formulation is based on a modified Hamilton’s principle valid for general viscoelastic motion. This is proved rigorously mathematically. This means that frequency dependent losses in the structure can be easily taken into account also at the theoretical level.

Curved and straight isoparametric solid and deep shell elements are developed.

To validate the developed elements a finite length cylinder with shear diaphragm end conditions is considered. This example structure is chosen

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because all four elements can be used to model the vibration field. Another nice property of this structure is that the exact solution is known from literature. The calculated results compares well with those from literature.

For very thin shells, the low order modes can have eigenfrequencies below higher order modes. This means that to accurately resolve them, a substantially higher nodal density is needed than one would initially expect.

4.2 Paper B. A waveguide finite element model of a pneumatic tyre

A waveguide finite elements model based on design data is used to describe the dynamic properties of a passenger car tyre. The response of the tyre belt and side wall is described with quadratic anisotropic, deep shell elements that include pre-stress and the motion of the tread on top of the tyre by quadratic, Lagrange type, isotropic two dimensional elements. These are the curved elements derived in paper A.

To validate the tyre model, mobility measurements and an experimental modal analysis have been made. The point mobility prediction agrees very well with measurements up to roughly 250 Hz for the radial point mobilities, see Figures 5 and 6 for excitation in the middle of the tread. The eigenfre- quency prediction are within five percent for the identified modes, except for the axial semi rigid body mode (error 12 %), the anti-symmetric mode of order two (error 10 %) and the anti-symmetric mode of order seven (error 7

%). The predicted damping is overestimated for the symmetric mode family and underestimated for the anti-symmetric mode family.

The ’cut-on’ frequency, of the belt bending modes, is the lowest frequency at which the corresponding waves are propagated. It appears at lower frequencies in the prediction than in the measurement. This is perhaps due to aging of the tyre since comparable measurements performed in the spring of 2001 is in agreement with the calculation [12]. In the range 500 - 1000 Hz there is an error ranging from 1.5 dB up to 3.5 dB for the squared amplitude of the point mobility. For the transfer mobilities, the error is larger since they are more sensitive to the exact position of the accelerometer, particularly so for the anti-resonances, see Figure 7.

The non-proportional damping is found with an ad hoc curve fitting procedure based on the measured mobilities. The applied damping is structural and frequency independent in the modal frequency region (below 250 Hz).

For higher frequencies losses proportional to the mass matrix, increasing linearly with frequency, are added to the mix, since the apparent damping level is higher in this frequency region.

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4.3 Paper C. Power dissipation in car tyres

The tyre model described in Paper B is used to estimate the power consumed by viscoelastic losses. External forces resulting from a non-linear contact model, for two different roads are inserted and the responses are calculated.

The dissipated power is then equated to the injected power. The dissipated power can also be studied element wise, making it possible to pinpoint where the dissipation occurs.

The contact force predictions are made by the division of applied acoustics, Chalmers University of Technology (CTH) as described in reference [36]. It is based on a non-linear contact model in which the response of the tyre is described with its flexibility matrix. Topographies of the surface are scanned, the tread pattern is accounted for, and then the tyre is ’rolled’ over it in the time domain. The nonlinear conditions used are: i) the tyre cannot indent into the road, ii) if a point is not in contact the force is zero and iii) the force cannot be negative (road pulling tyre down). Only forces acting normal to the road is considered.

The contact forces are used to calculate the response of the tyre. A power spectral density comparison of the measured and predicted acceleration can be seen in Figure 8. The measured acceleration signal comes from an accelerometer placed in a groove of a tyre rotating on a tyre test drum.

The low frequency peak is underestimated by around 3 dB and at higher frequencies the curves diverge even more. The curves do bear similarities and the result is promising. One possible explanations for the discrepancies is that the measurements are taken for a ’warm’ tyre since the tyre heats up during the measurements and the calculation is made for a ’cold’ tyre model. This temperature increase could change the viscoelastic properties of the rubber compounds.

When the force and motion are known the injected power can be calculated. The predicted power dissipation is underestimated but has the same order of magnitude compared to literature [4] and measurements. The power dissipation is larger on the rough road than on the smooth road, this showing the influence of the road on the rolling resistance. To the best of the author’s knowledge, this influence is neglected in all previous calculations of rolling resistance.

The dissipated power for a slick tyre running on a smooth road (ISO road), as a function of frequency and wave order can be seen in Figure 9 and 10 respectively. The reason that the frequency spectrum looks so rough is that two revolutions have been used for the calculation. If the contact forces were truly periodic every other frequency component would cancel out. A significant part of the dissipation occurs below 100 Hz and at a wave order around 3. The fact that the dissipation is low frequency and that the response is underestimated for those frequencies, is probably the reason why the power dissipation is underestimated. Another explanation could be that

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friction forces are omitted in the contact force calculation.

By studying the power dissipated within the elements it can be concluded that most of the power dissipation occurs in the tread and in the belt elements below the tread, see Figures 11 and 12. Roughly. 15 % of the losses occur in the side wall which is in conflict with reference [3], who says that roughly 30 % of the total dissipated power appears in the upper and lower side wall. The overall damping level in the model is estimated quite accurately (see Paper B), but the distribution of the damping, in the different parts of the tyre, is probably wrong. Since the viscoelastic data are very important for rolling resistance predictions, the damping should be established in a more scientific way. Furthermore, rolling resistance is a low frequency phenomena meaning that the use of a constant structural loss definition cannot be justified.

4.4 Paper D. The influence of the road on rolling resistance The calculation procedure developed in Paper C is used to predict the rolling resistance of real existing roads. The investigated roads were used in a round robin test for rolling resistance measurements in the autumn of 2004.

Six of the roads were re-visited in 2007 when texture measurements were made. The measured rolling resistance results spreads a lot, depending on measurements system and the tyres used, but the ranking of the roads is consistent. The predicted values by the procedure developed in Paper C are low when compared to the measured values. Also the predicted differences between the roads are smaller in the calculation. Nevertheless, if the roads are different enough, a ranking based on calculations is possible.

4.5 Paper E. Estimates of the viscoelastic properties of car tyres.

The tyre model introduced in Paper B is updated. As mentioned in Paper C, the viscoelastic properties of the tyre are very important when rolling resistance is of interest. In this work the fractional Kelvin - Voigt viscoelastic material model is put to use. The updating procedure is made in a modular way.

The tread of the tyre, for which solid elements are used, is considered isotropic meaning that only two stiffness parameters are needed. Poisson’s ratio and the dynamic shear modulus. Rubber is near incompressible, meaning that Poisson’s ratio is close to 0.5. A measurements of the dynamic shear modulus have been made in the interesting frequency region. The parameters in the fractional derivative model are then fitted to the measurements in a least square sense. Comparisons of the measured and simulated dynamic shear modulus and lossfactor are in Figures 13 and 14, showing a very good agreement.

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The belt and sidewall, for which shell elements are used, are more complex and needs many stiffness parameters to be described. Each shell element is replaced by an equivalent structure defined by two stiffness parameters along with a number of geometry parameters. The reduction of stiffness parameters by a factor 10 is the motivation for this work.

The equivalent structure consists of two rubber layers sandwiching a layer with rubber embedded steel wires. The geometry parameters are; the angle of the steel wires, the thickness of the layer with steel wires, the total thickness of the element, the distance from the origin to the edge of the element and the position of the layer with wires within the element.

As a first step the equivalent structure is fitted for the static case. It turns out that the shell elements can be arranged in two subgroups based on the angle of the steel wire: belt elements and sidewall elements. For the belt elements the angle is around 21 degrees and for the sidewall elements the angle is 90 degrees.

Each shell element is now described with two stiffness parameters, corresponding to steel and rubber respectively. The stiffness parameters are made frequency dependent according to the fractional Kelvin - Voigt material model. The static stiffness’ are considered known. The FDMM parameters of the steel are set to give a lossfactor of around 0.1 %, while only marginally effecting the real part and are therafter considered to be known.

The rubber FDMM parameters are determined based on results from measurements. Two sets of FDMM parameters for the belt and sidewall are used leading to a total of four unknown parameters.

To determine the FDMM parameters of the belt, measurements on a beam specimen cut out from the tyre were made. The beam consisted of the tyre belt with the tread on top of it. Modal parameters for the three first bending modes were extracted. The tread properties are known from the tread measurements. The beam is modelled with regular 3D finite elements.

The geometry is specified by a submesh of the tyremodel mesh. In an optimisation routine the modal damping of the measured and calculated modes are compared in a least square sense.

Result from the experimental modal analysis described in Paper B provided information to find the FDMM parameters for the sidewall. An opimi- sation routine compares the calculated and measured eigenfrequencies and lossfactors.

The updated tyre model remedies some of the problem with the origi- nal tyremodel. The damping estimation is better, see Figures 15 and 16.

Furthermore, the damping at low frequencies is modelled in a more realistic way.

The measured and simulated point mobility show excellent agreement in the modal frequency region, see Figures 17 and 18.

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5 Future Work

The most obvious next step is to use the updated tyre model for rolling resistance predictions. Unfortunately there was no time to finalise this study.

An investigation of the influence of certain tyre parameters would be interesting. It would be possible to change the speed, the load on the tyre and perhaps also to model wear of the tyre.

The contact modelling can be improved to include friction forces and with faster and more memory empowered computers the spatial resolution can be increased. Material non-linearities effects may also be included.

A statistical study, similar to the demonstration in Paper D, on the correlation between road texture and rolling resistance is possible and would be very informative.

6 Conclusion

A car tyre is modelled with waveguide finite elements. The model is employed to calculate the power dissipation as the tyre is rolling on a rough road showing promising agreement with measurements. The road roughness is seen to have a significant effect on the dissipated power, which is neglected in all previous works. It is concluded that rolling resistance is a low frequency phenomena and the damping properties of the tyre model have therefore been updated. This is, to the best of the authors knowledge, the first rolling resistance model, based on physical principles and design data.

7 Acknowledgements

The early development of the tyre model was funded by the Swedish Re- search Council (621-2002-5661) and the European Commission (G3RD-CT- 2000-00097). Many thanks to the members of the Ratin consortium and in particular to Roger Pinnington, ISVR, for helpful discussion, to Wolfgang Gnörich and Andrzej Pietrzyk, Goodyear, for advice and for sharing data for tyres and to Wolfgang Kropp, Patrik Andersson and Frédéric Wullens, Applied Acoustics, Chalmers, for advise and calculation of contact forces.

The final tyre model and the work presented in this thesis were funded by the European Commission, ITARI, FP6-PL-0506437. The measurement of rolling resistance was made by Gdansk University of Technology.

I am very thankful to my supervisor Svante Finnveden, for his construc- tive ideas, support and encouragement. He also has a decent taste in music.

Many thanks to the EDSVS and especially Professor Bj¨orn Petersson for

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making my stay at Technische Universit¨at Berlin possible. I would also like to thank to Ulf Carlsson, Kent Lindgren and Danilo Prelevic for assisting me with measurements. Special thanks to Carl-Magnus Nilsson.

Finally I would like to thank my family, my friend and the people at MWL.

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−0.1 −0.05 0 0.05 0.1 0.18

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

x [m]

r [m]

Figure 2: Medium mesh.

−0.1 −0.05 0 0.05 0.1

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

x [m]

r [m]

Figure 3: Enriched mesh.

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Figure 4: The dissipated power can also be estimated from the forces directly. This approach is not as stable as the energy approach and requires a much finer spatial resolution.

10¹ 10² 10³

−70

−65

−60

−55

−50

−45

−40

−35

Frequency [Hz]

Magnitude of point mobility dB rel 1 (m/Ns)2

Figure 5: Magnitude of point mobility for excitation in the middle position.

Measured (solid) and calculated (dashed).

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10¹ 10² 10³

−2

−1.5

−1

−0.5 0 0.5

Frequency (Hz)

Phase of point mobility (rad)

Figure 6: Phase of point mobility for excitation in the middle position.

Measured (solid) and calculated (dashed).

10¹ 10² 10³

−100

−90

−80

−70

−60

−50

−40

Frequency [Hz]

Magnitude of transfer mobility dB rel 1 (m/Ns)2

Figure 7: Magnitude of transfer mobility for excitation in the middle position. The response is measured 23.5 cm avay in the circumferewntial direction and 4.3 cm above the geometric centre. Measured (solid) and calculated (dashed).

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0 200 400 600 800 1000 1200 1400 1600 1800 2000

−5 0 5 10 15 20 25 30 35 40

Frequency [Hz]

Power spectral density of acceleration dB [rel 1 (m/s2)2/Hz]

Figure 8: Power spectral density of the Acceleration signal for the rotating accelerometer, measured; solid, simulated; dashed. The measured curve is an average over 320 revolutions and the calculated curve is a spatial average around the tyre.

0 50 100 150 200 250 300 350 400 450 500

0 5 10 15 20 25 30 35 40

Frequency [Hz]

Power [W]

Figure 9: Dissipated power as a function of frequency. The bandwidth is 5.6 Hz. Most of the dissipation occurs below 100 Hz.

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0 5 10 15 20 25 30 0

10 20 30 40 50 60

Waveorder [−]

Power [W]

Figure 10: Dissipated power as a function of wave order. A substantial part of the dissipated power occur at a wave order of around 3.

0 5 10 15 20 25 30

Element number

Power [W]

Figure 11: Power dissipation in the different elements. Belt elements (solid), Tread elements (dashed). Most of the dissipation occurs in the tread.

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−0.1 −0.05 0 0.05 0.1 0.2

0.22 0.24 0.26 0.28 0.3 0.32

X (m)

Y (m)

Figure 12: The ellipses and circles indicates elements where a lot of power is consumed.

0 200 400 600 800 1000 1200

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5

12x 10⁶

Frequency [Hz]

Re(G) [Pa]

Figure 13: Real part of dynamic shear modulus. Measured (solid) and calculated with equation (4) (dashed)

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0 200 400 600 800 1000 1200 0.18

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38

Frequency [Hz]

Lossfactor

Figure 14: Measured (solid) and calculated with equation (4) (dashed) lossfactor.

3 4 5

0 0.5 1 1.5 2 2.5 3 3.5 4

Wave order [−]

Damping [Percent of critical damping]

Figure 15: Measured (X), calculated (Paper B) (Rings) and calculated (updated model, Paper E) (Squares) lossfactors for the symmetric mode family.

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3 4 5 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Wave order [−]

Damping [Percent of critical damping]

Figure 16: Measured (X), calculated (Paper B) (Rings) and calculated (updated model, Paper E) (Squares) lossfactors for the anti symmetric mode family.

10² 10³

−65

−60

−55

−50

−45

−40

−35

−30

Frequency [Hz]

Magnitude of point mobility dB rel 1 (m/Ns)2

Figure 17: Magnitude of point mobility for excitation in the middle position.

Measured (solid), model in Paper B (dashed) and updated model in Paper E (dash-dotted).

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10¹ 10² 10³

−2

−1.5

−1

−0.5 0 0.5

Frequency (Hz)

Phase of point mobility (rad)

Figure 18: Phase of point mobility for excitation in the middle position.

Measured (solid), model in Paper B (dashed) and updated model in Paper E (dash-dotted).

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Vibrations, damping and power dissipation in Car Tyres