Influence of material damping on the prediction of road texture
and tread pattern related rolling resistance
Citation for published version (APA):
Lopez Arteaga, I. (2010). Influence of material damping on the prediction of road texture and tread pattern related rolling resistance. In P. Sas (Ed.), ISMA 2010 International Conference on Noise and Vibration Engineering, 20-22 September 2010, Leuven Katholieke Universiteit Leuven.
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Influence of material damping on the prediction of road
texture and tread pattern related rolling resistance
I.Lopez
Eindhoven University of Technology, Department of Mechanical Engineering, Den Dolech 2, 5600MB, Eindhoven, The Netherlands
e-mail: i.lopez@tue.nl
Abstract
A comprehensive analysis of the influence of material damping properties on the prediction of the road texture and tread pattern contribution to rolling resistance is presented. It is well known that rolling resistance is directly related to the tyre material damping and increases as the internal damping in the tyre increases. However, it is unclear what the accuracy of the material damping should be when the goal is to predict the (relative) effect of road texture and tread pattern on rolling resistance. Recently a FEM-based modeling approach to predict the road texture influence on the rolling resistance of pneumatic tyres has been proposed. In this approach the total rolling resistance is approximated as the sum of two components: smooth road rolling resistance and road texture induced rolling resistance. In the present paper the influence of material damping on both components of the predicted rolling resistance will be studied. The steady-state rolling resistance is obtained using the ’Steady State Transport’ algorithm in the software package Abaqus. The viscoelastic rubber properties are modeled using a second order Prony series and the coefficients are varied to represent a realistic range for tyres. The road texture induced rolling resistance depends on both the belt and tread damping. The belt damping is modeled as Rayleigh damping and the tread damping as frequency independent (viscous) damping. Measured road texture profiles are used as an input to study the combined influence of road texture and tread pattern on rolling resistance.
1
Introduction
In Europe, 18% of the total CO2emission originates from road transport [1]. Aerodynamic resistance,
iner-tial forces, climbing forces and rolling resistance contribute to the total force a vehicle has to overcome to maintain constant speed. Rolling resistance is an important factor in this respect since it accounts for approx-imately 20-30 % to the energy consumption of a typical passenger car [2, 3]. Lowering rolling resistance in pneumatic tyres can therefore greatly contribute to the reduction of greenhouse gas emissions.
This work focuses on the influence of road parameters on rolling resistance and, more specifically, on the influence of road texture. Existing literature on the influence of road texture on rolling resistance mainly uses experimental data to relate various texture metrics to the rolling resistance level. The SILVIA project [4] provides an interesting overview of the empirical work done up to 2004. Significant increases in rolling resistance and fuel consumption are found for increasing texture severity [5, 6]. No general consensus can be found on the relative importance of different texture wavelength bands. Moreover, rolling resistance often shows a poor measurement reproducibility. Developing models of tyre/road interaction to study the effects of road texture on rolling resistance would allow to assess the relative importance of different texture wavelength bands more systematically.
Only a few models to predict the influence of road texture on rolling resistance have been proposed in the literature. In [7] a waveguide finite element model is presented which predicts rolling resistance as amount of power dissipated in the tyre. The contact problem is solved in the tyre-fixed (Lagrangian) reference
system and the stiffening due to rotation and Coriolis forces are not taken into account. The predicted rolling resistance for a few different road surfaces is compared to measured data and a good qualitative agreement is found. An alternative modeling approach is presented in [9, 10] where the Modal Arbitrary Lagrangian-Eulerian (M-ALE) approach described in [11] is applied to the prediction of road texture induced rolling resistance. In [9, 10] the total rolling resistance is approximated as the sum of two components: smooth-road rolling resistance and smooth-road texture-induced rolling resistance. Measured texture profiles of 30 different roads are used as input to the model and the calculated rolling resistance coefficient shows a good qualitative agreement with experimentally determined rolling resistance coefficients.
In the present paper a comprehensive analysis of the influence of material damping properties on the pre-diction of the road texture and tread pattern contribution to rolling resistance is presented. The influence of material damping on both the smooth-road and road texture-induced rolling resistance is studied. The smooth-road rolling resistance is obtained using the ’Steady State Transport’ algorithm in the software pack-age Abaqus. The viscoelastic rubber properties are modeled using a second order Prony series and the co-efficients are varied to represent a realistic range for tyres. It is shown that relatively small variations of the Prony series coefficients lead to large variations in the predicted rolling resistance. The road texture-induced rolling resistance depends on both the belt and tread damping. The belt damping is modeled as Rayleigh damping and the tread damping as frequency independent (viscous) damping. The rolling resistance coeffi-cient is calculated for 30 different road surfaces with varying texture depth. The simulation results indicate that the belt damping has a negligible influence on the predicted increase of rolling resistance coefficient with texture depth. Regarding tread damping, it is shown that the influence of texture depth on the rolling resistance coefficient seems to be larger for very low and very high damping values. In all cases the model overpredicts the increase of rolling resistance with texture depth.
2
Background
2.1 Definition of rolling resistance
There are a number of definitions of rolling resistance: it can be expressed as either a resisting force at the wheel axle, a resisting moment around the wheel axle or a power dissipation. To avoid confusion, an accurate definition of rolling resistance is important. Rolling resistance can be defined as the power dissipation Pdis
at a certain axle load Naxle. The power dissipation depends on the resistant force Fres acting on the wheel
axle and the vehicle velocity v. In this study, the rolling resistance coefficient, RRC, is used to quantify rolling resistance: RRC = Fres Naxle = Pdis Naxlev . (1)
It has been shown that the resistant force Fres is linearly dependent on the applied axle load Naxle in the
practical range of axle loads [3]. Therefore, the rolling resistance coefficient does not change with changing axle loads. Furthermore, the rolling resistance coefficient has been shown to be fairly constant with in-creasing velocity [6]. This simplifies the comparison between results found in literature. Rolling resistance coefficients generally range from 0.006 to 0.013 for tyres on modern passenger cars [3].
2.2 Two step approximation to tyre response
In [9, 10] a new computationally efficient modeling approach for the prediction of the road texture con-tribution to rolling resistance is proposed. The large steady state tyre deformations and the small texture induced tyre vibrations are studied separately and the total rolling resistance is approximated as the sum of the smooth road rolling resistance and the road texture rolling resistance (as illustrated in Fig. 1). The
smooth road rolling resistanceis the energy dissipation due to the large cyclic deformation of the cross sec-tion of the tyre. A nonlinear steady-state rolling analysis on a FEM tyre model is used to determine this energy dissipation. The road texture rolling resistance is the additional energy dissipation resulting from road texture induced tyre vibrations. A reduced modal representation is extracted from the FEM tyre model and the M-ALE approach from [11] is used to include the tyre belt as a boundary condition in a tyre/road interaction model.
(b) c)
(a) (
Figure 1: Schematic representation of the proposed modeling approach. (a) Base state (b) The smooth road rolling resistance resulting from steady state tyre deformations. (c) The road texture rolling resistance due to texture induced tyre vibrations.
The main assumption in this approach is that the principle of superposition holds; the total deformation can be expressed as the sum of a large deformation due to the nominal load and (small) deformations due to the road texture and the total force as the sum of nominal load and force variations due to the road texture. In this paper the term road texture includes all wavelengths from micro texture (< 0.5 mm) to unevenness (> 0.5 m). This implies that the rolling resistance due to contact area variations caused by the road unevenness is captured in the road texture rolling resistance. Although this approximation would be too coarse if the goal is to improve the tyre design for reduced rolling resistance, it might be sufficiently accurate for the design of low rolling resistance pavements. This expectation is supported by experimental results that show the same dependence of the rolling resistance coefficient for different tyres [12].
2.3 Smooth road rolling resistance
The smooth-road rolling resistance is defined as the part of the rolling resistance which is due to the energy losses induced by the large cyclic deformation of the cross-section of the tyre as it enters, travels along and exits the contact patch. This is obviously the main contribution to the total rolling resistance. The energy dissipation in a tyre which is rolling on a smooth road surface can be modeled using a steady state rolling analysis. This analysis uses an Arbitrary Langrangian Eulerian transformation which removes the explicit time dependency from the problem so that a purely spatially dependent analysis can be performed. The (Eulerian) reference frame moves at the vehicle speed but does not spin along with the tyre. This choice of reference frame allows the finite element mesh to remain stationary so that only the part of the body in the contact zone requires fine meshing [8]. Figure 2 displays the finite element discretization of the tyre (175 SR14). The simplified model consists of a tyre belt with internal reinforcement bars. The rubber material in the tyre is described by a second order Prony series to model the visco-elastic modulus E(t). The material parameters are condensed from data used by Fraggstedt [7] as explained in [9].
The tyre is loaded with an axle load of Naxle = 4100N and pressed against a rigid smooth road surface.
Subsequently, a steady state rolling analysis is performed at vehicle speeds, v, ranging from 20 to 100 km/h. At 80 km/h the RRC equals approximately 0.0135, which exceeds the expected values reported in literature [3]. This result is discussed in 3 and the influence of the material parameters and modeling simplifications is illustrated.
Figure 2: Tyre discretization
2.4 Road texture rolling resistance
The road texture rolling resistance is defined as the additional energy dissipation resulting from road texture induced contact force variations and tyre vibrations. Based on the M-ALE approach from [11], a tyre/road interaction model is developed to determine these contact force variations. Figure 3 presents a schematic overview of the tyre/road interaction model.
Figure 3: Schematic overview of the tyre/road model The model consists of three layers:
• Dynamic response of the deformed rotating tyre: Green’s functions are used to represent the deformed rotating tyre [14]. These Green’s functions serve as a boundary impedance condition.
• Tread dynamics: The tread dynamics are modeled using a linear spring damper system.
• Contact mechanics: The contact mechanics between the tread blocks and the road surface are modeled using a nonlinear stiffness function which accounts for the indentation of the tread block by the road asperities [15].
The dynamic equations of the deformed rotating tyre in a fixed (Eulerian) reference frame is derived applying the M-ALE approach proposed in [11]. The M-ALE equation of motion reads
¨
η(t) + ˜D(Ω) ˙η(t) + ˜K(Ω)η(t) = ΦTf (t) (2) where η represent the modal coordinates and ΦΦΦ represent the eigenmodes of the tyre which are calculated with the FE tyre model. Therefore the displacement vector in cartesian coordinates is given by x = Φη. The
vector f (t) contains the applied forces in the reference frame and Ω is the rotating velocity of the tyre. The modified damping and stiffness matrices ˜D and ˜K are defined as
˜
D = 2P(Ω, M, ΦΦΦ) + Dmod (3)
˜
K = S(Ω, M, ΦΦΦ) + DmodP(Ω, M, ΦΦΦ) + Kmod (4)
The mass matrix, M, is extracted from the FE tyre model. The reduced stiffness matrix Kmodis a diagonal
matrix with elements kii= ωi2, where ωiare the eigenfrequencies of the tyre. The reduced damping matrix
Dmodis the projection of the system damping matrix on the retained modes. In general Dmod can be a full
matrix, since there is no fundamental assumption in the formulation that requires this matrix to be diagonal. However, as a first approximation, Rayleigh damping is assumed in [9]. Definitions of the additional stiffness and damping matrices caused by tyre rotation, S and P, can be found in [13].
In the time domain, the displacement at position i on the tyre as a result of a arbitrary forces at positions j can be determined by the convolution of the forces fj and the Green’s functions gij(t, Ω).
xi(t, Ω) =
X
j
gij(t, Ω) ⊗ fj(t, Ω) (5)
Note that in the above expression the Green’s functions, tyre displacement and contact forces are a explicit function of the rotational velocity Ω, since (5) is expressed in the vehicle-fixed (Eulerian) reference system. These Green’s functions can be determined by solving
¨
η(t) + ˜D(Ω) ˙η(t) + ˜K(Ω)η(t) = ΦTjδ(t) (6) where ΦTj is the jth row of Φ and δ(t) represents the Dirac delta function. If this set of equations is solved a semi-analytical expression for the Green’s functions in the Eulerian reference frame can be obtained [14]. These Green’s functions form the boundary impedance condition for the tread layer in the tyre/road interaction model (figure 3).
The tread layer is modeled as regularly spaced tread blocks represented by an array of spring-damper systems (figure 3). In a first approximation frequency independent viscous damping is used to model the tread losses, but more complex models will be developed in the future. The details of the implementation and parameter data used can be found in [9].
The contact stiffness between the tread blocks and the road surface is modeled by a nonlinear spring to account for small wavelength texture components following the approach presented in [15]. This approach uses a scan of the geometry of the road surface, the elastic properties of the tread compound and a model of a flat circular punch indenting an elastic layer [16]. This results in an approximate stiffness function that is unique for every pair of contact elements. Although in [9] only 2D texture profiles are considered, the model can easily be extended to be used with full 3D texture data [10].
The power dissipated in the tyre due to the interaction between tyre and road texture can be determined with the following equation
Pdis= 1 T Cn X i=1 Z T 0 Fi(t) ˙xi(t)dt (7)
where the time interval T represents the time of one tyre revolution in which the averaged total contact force has converged to the desired load. Furthermore, Cn represent the total number of contact nodes in the contact patch, Fi(t) the contact force at contact point i and ˙xi(t) the velocity at the same position. Finally,
2.5 Preliminary results on the influence of road texture on rolling resistance
The rolling resistance coefficient for 30 road surfaces with different texture properties from the test site at Kloosterzande (The Netherlands) is calculated with the model described above and compared to measure-ments performed using a trailer from the University of Gdansk, Poland. The measuremeasure-ments are taken at a velocity of 80 km/h and an axle load of 4100 kg. Further details on these measurements and the road texture measurements can be found in [12]. The 30 road surfaces can be divided into six road structure categories shown table 1.
Table 1: Road structure categories
Pavement type Designation
International standardized road surface ISO
Stone Mastic Asphalt SMA
Dense Asphalt Concrete DAC
Thin Layered Asphalt TLA
Single layer Porous Asphalt Concrete PAC Double layer Porous Asphalt Concrete DPAC
In the following the root mean square (RMS) texture depth is used to quantify road texture in a single value (see [12] for a discussion on possible indicators). Figure 4 shows the rolling resistance coefficient (RRC) as a function of the texture depth RM Stexfor a traveling velocity of 80 km/h and a nominal load of 4100 kg.
The dashed line corresponds to a linear regression fit of the calculated RRC and the solid line is the linear regression fit of the measured RRC.
0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMS tex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC
Figure 4: Comparison of predicted (dashed) and measured (solid) rolling resistance coefficient vs. RMS texture depth on 30 different test tracks at 80 km/h
There is a clear correlation between road texture and rolling resistance. Rolling resistance increases linearly with texture depth and a decrease in RM Stexof 1mm results in a decrease in rolling resistance of
approx-imately 8%. The trends in the numerical and experimental results match despite the rough estimation of the model parameters. Due to the excessively high value of the smooth-road RRC obtained in 2.3 the pre-dicted RRC vs. RMS texture (dashed) line is shifted upwards with respect to the experimental (solid) line. Nevertheless, the numerical results are promising as they illustrate the potential of the proposed modeling concept.
2.6 Discussion
Although promising, the results presented above give rise to a number of question regarding the modeling choices and the validity of the approach. The first obvious question is what is the reason for the high value of the smooth-road RRC found. Is it a shortcoming of the FE model used or of the modeling approach or of both?
Furthermore, a close look at the trend lines of the numerical and experimental data in 4 reveals that the slope of the numerical data (1.18) is higher than the slope of the experimental data (0.81). In other words if the RMS texture depth increases by 1mm the experiments show an increase in rolling resistance of 8% while the model predicts an increase of 12%. This overestimation of the influence of texture depth might be due to inadequate values for the damping of belt and tread blocks, to the simplicity of the damping models or to the fact that in the current implementation the tyre/road interaction is modeled in 2-dimensions (contact along a line extruded to the full width of the contact patch).
In the present paper a close look at the damping models and corresponding parameter values is taken for both the FE model used for the smooth-road rolling resistance calculation and the tyre/road interaction model developed for the road texture-induced rolling resistance prediction. For each of these models the damping model is described and an assessment of the influence of the parameter values used is performed.
3
Influence of material damping on the smooth-road rolling
resis-tance
3.1 Material model
Rubber is a visco-elastic material with frequency dependent characteristics which can be modeled by a Prony series as follows, E(t) = E0 1 − n X i=1 pi(1 − e −t τi) ! (8) where E0 is the instantaneous elastic modulus, τiare relaxation times and pi are the modulus ratios defined
as E∞= E0(1 −Pni=1pi) with E∞the long-term elastic modulus.
The structure of a tyre is a complex combination of reinforcement chords and rubber, where de properties of the rubber compounds are different for different parts of the tyre cross-section. In this work a single rubber compound is used for the whole tyre. The material data is obtained from experiments performed in [7] to characterize the viscoelastic properties of a tread compound. The data from [7] is scaled to ensure that the eigenfrequencies predicted by the FE model match the experimentally determined values as shown in [13]. The parameters of the second order Prony series obtained are summarized in table 2 and the resulting real and imaginary parts of the elastic modulus are plotted in figures 5(a) and 5(b) together with the scaled experimental data from [7] (thick solid line).
In order to study the influence of the Prony series parameters on the smooth-road rolling resistance, the relaxation times τi are lowered to 75% and 50% of the originally fitted values. The resulting real and
imaginary of the elastic modulus are given in figures 5(c) and 5(d) for 0.75τi and figures 5(e) and 5(f) for
10−2 10−1 100 101 102 103 104 105 106 1 1.5 2 2.5 3 3.5x 10 6 Frequency [Hz] Re(E)[Pa] (a) 10−2 10−1 100 101 102 103 104 105 106 0 1 2 3 4 5 6 7 8 9x 10 5 Frequency [Hz] Im(E)[Pa] (b) 10−2 10−1 100 101 102 103 104 105 106 1 1.5 2 2.5 3 3.5x 10 6 Frequency [Hz] Re(E)[Pa] (c) 10−2 10−1 100 101 102 103 104 105 106 0 1 2 3 4 5 6 7 8 9x 10 5 Frequency [Hz] Im(E)[Pa] (d) 10−2 10−1 100 101 102 103 104 105 106 1 1.5 2 2.5 3 3.5x 10 6 Frequency [Hz] Re(E)[Pa] (e) 10−2 10−1 100 101 102 103 104 105 106 0 1 2 3 4 5 6 7 8 9x 10 5 Frequency [Hz] Im(E)[Pa] (f)
Figure 5: Scaled storage (left) and loss (right) modulus from [7] (thick line) and Prony series fit (thin line) for (a)-(b) original fit [9], (c)-(d)τ = 0.75τoriginal, (e)-(f)τ = 0.5τoriginal.
Table 2: Parameters of the second order Prony series.
E0 p1 p2 τ1 τ2
3.3e6 [Pa] 0.487 [-] 0.137 [-] 9.96e-5 [s] 1.20e-3 [s]
It is clear from the figures that the second order Prony series with τ = 0.75τoriginalis also a reasonably good
fit of the experimental data. Furthermore, although the elastic modulus obtained with τ = 0.5τoriginalmight
seem too high, a temperature increase of 2 or 3 degrees would already lead to such a shift in frequency.
3.2 Rolling resistance versus traveling velocity
A steady state rolling analysis is performed with the FE model shown in figure 2 at vehicle speeds, v, ranging from 20 to 140 km/h and the three different parameter sets for the Prony series discussed in the previous section. The rolling resistance coefficient versus travelling velocity is shown in figure 6, where the solid line corresponds to the original Prony series parameter set from [9], the dash-dotted line to the parameter set with τ = 0.75τoriginaland the dashed line to the parameters set with τ = 0.5τoriginal.
20 40 60 80 100 120 140 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Vehicle velocity [km/h]
Rolling resistance coefficient [−]
τfit
0.75τfit
0.5τ
fit
Figure 6: Rolling resistance coefficient as a function of vehicle velocity for three different Prony parameter sets: τoriginalsolid, 0.75τoriginaldash-dotted, 0.5τoriginaldashed.
Obviously these results do not agree at all with the expectation of a rolling resistance coefficient which is independent of the vehicle speed, as is found experimentally [6]. It can also be seen that the calculated rolling resistance coefficient is very sensitive to the parameters of the Prony series and that the rolling resistance coefficient decreases as the curve elastic modulus versus frequency shifts to higher frequencies. This gives an indication that more realistic results could be achieved if the change in loss and storage modulus due to the temperature increase as the vehicle speed increases could be included in the FE tyre model.
4
Influence of damping on the road texture-induced rolling
resis-tance
4.1 Influence of belt damping
The tyre/road interaction model described in section 2.4 and schematically drawn in figure 3 is used to determine the road texture-induced rolling resistance. In this model the belt response provides a boundary condition for the tread layer. This boundary condition is expressed as Green’s functions which are derived following the M-ALE approach. The matrix Dmod from (3) can in general be a non-diagonal matrix, but
here Rayleigh damping is considered. In general Rayleigh damping is defined as
D = αM + βK (9)
where M and K are the mass and stiffness matrices of the system. Rayleigh damping is clearly a form of proportional damping which is also frequency dependent, since the term proportional to the mass matrix de-creases with increasing frequency and the term proportional to the stiffness matrix inde-creases with increasing frequency. Therefore the parameters α and β provide some flexibility to shape the frequency dependency of the damping matrix. Although Rayleigh damping is a straight-forward way to introduce frequency depen-dent damping, it has been shown that it is not possible to describe the damping of a tyre with a single set of parameters [17]. Nevertheless Rayleigh damping is applied here as a pragmatic first approximation to the belt damping.
The stiffness parameter is set to zero, β = 0, and three different values of the mass parameter α = 30, 100, 300 are considered. In figure 7 the calculated rolling resistance coefficient as a function of RMS texture depth for the 30 different test tracks described in section 2.5 is shown for α = 30 (figure 7(a)) and α = 300 (figure 7(b)). 0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC 0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC (a) (b)
Figure 7: Calculated rolling resistance versus RMS texture depth on 30 different test tracks at 80 km/h and d = 5 [Ns/m] for, (a)α = 30, (b)α = 300 from [9]
It can be seen that the parameter α in the Rayleigh damping model has a negligible influence on the pre-dicted influence of RMS texture depth on the rolling resistance coefficient. To clarify this point the slope of the regression line and the correlation coefficients for all three α values and for the experimental data are summarized in table 3.
The high values of the correlation coefficient found for all three predicted rolling resistance coefficient data sets and the experimentally determined rolling resistance coefficient indicate a significant correlation of the rolling resistance coefficient with the RMS texture depth. The present Rayleigh damping model with β = 0
Table 3: Slope of the regression line and correlation coefficient for several values of α and d = 5 [Ns/m]. α Slope of fitted line Correlation coefficient
30 1.22 0.79
100 1.21 0.80
300 1.18 0.85
Experiments 0.81 0.90
shows a very limited influence of the parameter α on the predicted increase of rolling resistance coefficient with RMS texture depth. This leads to the conclusion that the amount belt damping included in the M-ALE model is probably not the reason for the discrepancy between experiments and simulations found in section 2.5. However more simulations with other combinations of β and α values are needed in order to definitely establish this conclusion.
4.2 Influence of tread damping
In the implementation described in [9], the tread layer is modeled with a line of tread blocks which are in turn combinations of spring-damper elements in parallel. Each tread block is modeled as seven spring-damper systems with spring k = 0.15e5 [N/m] and damping d = 5 [Ns/m]. These parameters are chosen to let the overal tread stiffness and damping match the data provided in [7].
In order to assess the influence of tread damping on the predicted relationship between rolling resistance coefficient and RMS texture depth the following values of the damping coefficient are considered: d = 0.5, 1, 5, 10, 25, 50 [Ns/m]. The slope of the regression line and the correlation coefficients found for all damping values considered are summarized in table 4 together with the results corresponding to the exper-imental data. Additionally, in figure 8 the calculated rolling resistance coefficient as a function of RMS texture depth for the 30 different test tracks described in section 2.5 is shown for d = 0.5 [Ns/m] (figure 8(a)), d = 5 [Ns/m (figure 8(b)), d = 25 [Ns/m (figure 8(c)) and d = 50 [Ns/m (figure 8(d)).
Table 4: Slope of the regression line and correlation coefficient for several values of the tread damping coefficient and α = 300.
Damping coefficient [Ns/m] Slope of fitted line Correlation coefficient
0.5 1.42 0.83 1 1.39 0.83 5 1.18 0.85 10 1.19 0.80 25 1.42 0.65 50 – – Experiments 0.81 0.90
The high values of the correlation coefficient found for the predicted rolling resistance coefficient data sets corresponding to the four lower damping values indicate a significant correlation of the rolling resistance coefficient with the RMS texture depth. However the correlation coefficient decreases for d = 25 [Ns/m] and the linear relationship between rolling resistance coefficient and RMS texture depth is completely lost for d = 50 [Ns/m].
0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC (a) 0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC (b) 0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC (c) 0 0.5 1 1.5 2 2.5 x 10−3 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 RMStex [m]
Rolling Resistance Coefficient [−]
ISO SMA DAC TSF PAC DPAC (d)
Figure 8: Calculated rolling resistance versus RMS texture depth on 30 different test tracks at 80 km/h and α = 300 for, (a)d = 0.5 [Ns/m], (b)d = 5 [Ns/m] from [9], (c)d = 25 [Ns/m], (d)d = 50 [Ns/m]
If only the lower four damping values are considered, the slope of the regression line increases as damping decreases. In other words the increase in rolling resistance coefficient with increasing RMS texture depth is larger as the damping decreases. This seems odd at first, since one would expect the rolling resistance coefficient to increase with increasing damping. A detailed analysis of these results is needed in order to explain this apparent contradiction.
It can be concluded that the tread damping has a significant influence on the predicted relationship between rolling resistance coefficient and RMS texture depth. However the discrepancy between simulation results and experimental data found in section 2.5 cannot be explained as being due to the tread damping, since none of the damping values considered lead to a decrease in slope of the regression line.
At this stage of the presented research the most likely explanation for the overestimation of the slope of the regression line found in the model is the fact that the interaction model is a 2D model, where the tread blocks are in fact extruded across the width of the contact patch. This leads to a larger contact area compared to a fully 3D model, where the gaps between the tread blocks across the contact patch are also included. Converting the tyre/road interaction model into a full 3D model is the subject of current ongoing research.
5
Conclusion
In this paper a comprehensive analysis of the influence of material damping properties on the prediction of the road texture and tread pattern contribution to rolling resistance has been presented. The total rolling resistance is approximated as the sum of two components: smooth road rolling resistance and road texture induced rolling resistance. The influence of tyre damping on each of this components has been studied. The predicted smooth road rolling resistance is very sensitive to the viscoelastic material parameters consid-ered and a more accurate FE description of the viscoelastic properties, including temperature dependence, is needed in order to produce realistic values of the rolling resistance coefficient.
For the road texture-induced rolling resistance it is found that the belt damping has a negligible influence on the relationship between rolling resistance coefficient and RMS texture depth. While, on the other hand, tread damping has a significant influence on this relationship. However the discrepancy found between simulations and experiments cannot be explained by differences in damping properties. At this stage of the research the most likely explanation for this discrepancy is the use of a 2D tyre/road interaction model. Extended the interaction model to 3D is the subject of current research.
Acknowledgements
The author wishes to thank Stijn Boere and Ard Kuijpers for their contribution to this work.
References
[1] European Environmental Agency, Greenhouse gas emission trends and projections in Europe, (2008). [2] Forum of European National Highway Research Laboratories (FEHRL), FEHRL study SI2.408210
Tyre/Road Noise, (2006).
[3] Soci´et´e de Technologie Michelin, The Tyre - Rolling resistance and Fuel Savings, (2003).
[4] H. Bendtsen SILVIA PROJECT REPORT, Rolling Resistance, Fuel Consumption and Emissions: A Lit-erature Review”, Danish Road Institute, Technical Note,(2004).
[5] U.S.I. Sandberg Road macro- and megatexture influence on fuel consumption, ASTM Special Technical Publication, No. 1031, (1990), pp. 460-479.
[6] G. Descornet Road-surface influence on tire rolling resistance, ASTM Special Technical Publication, No. 1031, (1990), pp. 401-415.
[7] M. Fraggstedt, Vibrations, damping and power dissipation in car tyres, PhD Thesis, MWL, KTH, Stock-holm (2008).
[8] SIMULIA, Steady-state transport analysis, Section 6.4.1 of the Abaqus Analysis User’s Manual, (2008). [9] S.W. Boere, I. Lopez, A.H.W.M. Kuijpers, H. Nijmeijer, Prediction of road texture influence on rolling resistance and tyre/road noise, in Proceedings of Euronoise 2009, Edinburgh, United Kingdom, 2009 October 26-28, Edinburgh (2009).
[10] S.W. Boere, I. Lopez, A.H.W.M. Kuijpers, H. Nijmeijer, Tyre/road interaction model for the prediction of rolling resistance due to texture induced tyre vibrations, submitted (2010).
[11] I. Lopez, R. Blom, N. Roozen, H.Nijmeijer, Modelling vibrations on deformed rolling tyres - a modal approach, Journal of Sound and Vibration, Vol. 307, No. 3-5, Academic Press (2007), pp. 481-494.
[12] S.W. Boere, G.J. v. Blokland Influence of road surface properties on rolling resistance of car tyres, Internal report, M+P.DVS.08.17.1, (2008).
[13] I. Lopez, R.R.J.J. van Doorn, N. Roozen, H.Nijmeijer, Frequency loci veering due to deformation in rotating tyres, Journal of Sound and Vibration, Vol. 324, No. 3-5, Academic Press (2009), pp. 622-639. [14] I. Lopez, S.H.M. Kersjes, N.B. Roozen, A.J.C. Schmeitz, H. Nijmeijer, Green’s Functions for a
Rotat-ing Tyre: A Semi-Analytical Approach, in ProceedRotat-ings of Euronoise 2006, Tampere, Finland, 2006 May 30 - June 1, Edinburgh (2009).
[15] P.B.U. Andersson, W. Kropp, Time domain contact model for tyre/road interaction including nonlin-ear contact stiffness due to small-scale roughness, Journal of Sound and Vibration, Vol. 318, No. 1-2, Academic Press (2008), pp. 296-312.
[16] F. Yang, Indentation of an incompressible elastic film, Mechanics of Materials, Vol. 30, No. 4, (1998), pp. 275-286.
[17] P. Kindt, Structure-borne tyre/road noise due to road surface discontinuities, PhD Thesis, Katholieke Universiteit Leuven, Leuven (2009).
