**3.3 Rolling resistance analysis**

Rolling resistance force in tyre is mainly generated by friction force, drag force, and hysteresis loss. This study will only discuss rolling resistance force generated by hysteresis loss inside the rubber and cord. The analysis was performed in two main steps, those are static tyre simulation (footprint and radial stiffness) and calculation of strain energy loss to find rolling resistance force.

With review of a number of tyre rolling resistance simulations, it is found that rolling resistance calculation is based on the strain energy loss during a traveled distance. Aldhufairi et al. [9] used a script of Abaqus to extract the 3D tyre model data as input and an analytical rigid road drum with a straight and smooth surface was added to the model, equivalent to that used in the experiment, due to the limitation of the testing machine the travel speed was limited to 30 km/h. Ghosh et al. [10] suggested a method that implements a steady state rolling simulation using Abaqus software to obtained the strain energy and principal strains, together with the loss factors (Tan d) of the material obtained separately in the laboratory, are used to estimate the energy dissipation of a rolling tyre through post processing. The internal code was developed to perform such a task.

Lind [11] suggested three sequential steps for solving the rolling resistance model; inflation, footprint and rolling. The last rolling step was performed using a dynamic solver setting where the center node was moved in the x-direction with a prescribed acceleration up to a target speed. The rolling resistance was from the FE-simulation result computed in two different ways. The first method uses the contact forces from each node multiplied with its distance from the wheel centre; the second method uses the reaction forces from the constrained middle node and computes the rolling resistance. The result presented for the material model and for the rolling resistance does not aim toward representing any specific tyre rubber compound or tyre.

While the others used FE tyre model without tread, Cho et al. [12] Included the tread in FE tyre model. The hysteretic loss during one revolution was computed with the maximum principal value of the half-amplitudes of six strain components, and the temperature distribution of tyre was obtained by the steady-state heat transfer analysis. The static tyre deformation analysis is performed by ABAQUS/ Standard in the deformation module and the strain and stress results are input into the in-house dissipation module where the hysteretic loss, rolling resistance and heat generation rate are computed.

In this study, the footprint analysis was carried out with patterned tyre model and for rolling resistance simulation used tyre model without pattern for the sake of computing time. However, the accuracy is a little sacrificed but still acceptable i.e. 6.2% error as describe in the Section 3.4.

The rolling resistance analysis was based on hysteresis of rubber and cord where the phase of stress lags behind the strain as it is shown in **Figure 8**. The hysteretic loss ΔW per unit volume during a period Tc = 2*π=*ω is:

$$
\Delta \mathbf{W} = \int\_0^{T\_\varepsilon} \sigma(\tau) \frac{d\varepsilon(\tau)}{d(\tau)} d\tau = \int\_0^{T\_\varepsilon} \sigma\_0 \varepsilon\_0 \sin \left( \alpha \tau + \delta \right) \cos \left( \alpha \tau \right) d\tau. = \pi \sigma\_0 \varepsilon\_0 \sin \delta \tag{3}
$$

where *σ*<sup>0</sup> and *ε*<sup>0</sup> being the stress and strain amplitudes and *ω* being the excitation frequency.

In engineering application, as suggested by Cho et.al [9] 3D viscoelastic bodies are subjected to more complicated multi-axial cyclic excitations, so the time histories of strains and stresses are neither one-dimensional nor sinusoidal.

**Figure 8.** *Stress - strain phase.*

Therefore, the hysteretic loss is expressed in a generalized form:

$$
\Delta \mathbf{W} = \int\_0^{T\_\varepsilon} \sigma\_{\vec{\eta}}(\mathbf{r}) \frac{d e\_{\vec{\eta}}(\mathbf{r})}{d(\mathbf{r})} d\mathbf{r} \tag{4}
$$

The hysteretic loss can be converted to the heat generation, and the heat generation rate Q per unit volume during a cycle is:

$$\mathbf{Q} = \frac{\Delta W}{T\_c} = \frac{1}{T\_c} \int\_0^{T\_c} \sigma\_{\vec{\eta}}(\mathbf{r}) \frac{d\varepsilon\_{\vec{\eta}}(\mathbf{r})}{d(\mathbf{r})} d\mathbf{r} \tag{5}$$

In order to calculate the energy loss during deformation, curve interpolation and FFT function were developed. Abaqus python contains NumPy which can do FFT. A python scripting is used to read signal curve, perform the FFT and create a new curve, i.e. amplitude vs. frequency, for plotting in Abaqus.

```
def interpolation(curve):
   myCurve = []
   i=0
   n = len(curce)
   myCurve.append(0.0,curve[0])
   while i < n:
     myCurve.append(myAngle[i], curve[i])
     i = (i + 1)
   myCurve.append(360.0,curve[-1])
   i=0
   n = len(myCurve)
   NewCurve = []
   while i < (n - 1):
     angle_A = (myAngle[i + 1][1] - myAngle[i][1]) /
                (myAngle[i + 1][0] - myAngle[i][0])
     yo = myAngle[i][1]
     xo = myAngle[i][0]
     j = myAngle[i][0]
     while j < myAngle[(i + 1)][0]:
   newAngle.append(yo + (angle_A * (j - xo)))
   j = (j + delta)
   i = (i + 1)
   if i == (n - 1) and newAngle.append(myAngle[i][1]):
           pass
```

```
return newAngle
def fourier(sigma, epsilon):
   FFT1 = 2 * abs(fft.fft(sigma)) / len(sigma)
   FFT2 = 2 * abs(fft.fft(epsilon)) / len(epsilon)
   k=0
   total = 0
   while k < (len(FFT1) / 2):
     total = total + (FFT1[k] * FFT2[k]) * k
     k=k+1
   return total
```
The input for sigma and epsilon are the interpolated stress and the interpolated strain respectively.

In a rolling tyre, the rubber compounds exhibit the complicated 3D dynamic viscoelastic deformation. The strains and stresses are constituted in terms of the complex modulus G\* = G' + iG". In this case, G' is called the storage modulus and G" is the loss modulus. The complex modulus is a function of the strain amplitude *εo*, frequency f, and temperature T. The correlation between storage and loss modulus in terms of the phase difference *δ* as follows:

$$\text{Tan } \delta = \frac{G''}{G'} . \text{and } \mathbf{G''} = \mathbf{G} \* \sin \delta \tag{6}$$

In Abaqus simulation the complex modulus G\* can be obtained by extracting axisymmetric element data and therefore the heat dissipation from energy loss G" can be calculated by multiplying G and sin *δ*, and in terms of Python coding is written as follows:

$$\text{heat\\_dissipation} = \text{energy} \ast \sin\left(\text{tand}\right) \tag{7}$$

Where energy is extracted from previous axisymmetric simulation element data and tand is from input data.

The rolling resistance force generated by the hysteretic loss is computed as the total hysteretic loss of the rolling tyre during one revolution divided by the traveling distance of tyre during the same period of time, hence:

$$\mathbf{F}\_{\rm RR} = \frac{\mathbf{W}}{2\pi r} \tag{8}$$

where *<sup>W</sup>* <sup>¼</sup> <sup>Ð</sup> *<sup>Ω</sup>ΔW dV*

r = effective radius of tyre

*Ω* = material volume of tyre

Rolling resistance coefficient Cr is the indication of how large the rolling resistance is for a given load upon which it is rolling and is calculated by:

$$\mathbf{C\_r} = \frac{\text{Total force } (N) \ge 1000}{Load \, (N)} \text{ N/kN} \tag{9}$$

Total force is meant the sum of force caused by hysteresis loss in each tyre component material.

To analyze the force produced by tyre component materials, a Python code was developed as plugin in Abaqus software. The process of analyzing the rolling resistance is describe in the following steps and is shown in **Figure 9**.

**Figure 9.** *Rolling resistance analysis process using Abaqus plugin.*

1.Prepare input files, i.e.:


Abaqus job = full oldjob = axi cpus = 4

*Rolling Resistance Estimation for PCR Tyre Design Using the Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.94144*

	- Copy axi, inp and rename it to axi\_heat.inp
	- Change the tyre element type from cgax into dcax
	- Delete input of tyre\_coord and rim
	- Delete all properties in each material and replace with: \*conductivity: 0.2
	- Delete all existing steps and boundary conditions and replace with steps and boundary conditions necessary for rolling resistance simulation
	- a. Input files:
		- axi.inp
		- axi-heat.inp
		- sequence.inp

b. Odb files:


Below is tan-δ example of tread compound.

	- Select how energy is interpolated from coordinate element to bulk elements
	- Define speed of tyre [km/h]
	- Define error limit for heat transfer [%]
	- Define interpolation parameter [deg]
	- Define parameter for tyre radius calculation.

$$\mathbf{C\_{r}} = \frac{Total\ force \ (N) \ \mathbf{x} \ \mathbf{1000}}{Load\ (N)} \ \mathbf{N/kN}$$

The example of the simulation result is shown below: Results:

Force produced by material I40 is 2.08360116975 N Force produced by material A02 is 1.29144915874 N Force produced by material T61 is 18.8669933222 N Force produced by material BW08 is 0.0 N Force produced by material T61 is 1.67943517041 N Force produced by material Z80 is 1.13411378677 N Force produced by material S70 is 0.41902012456 N Force produced by material S70 is 3.82462665603 N Force produced by material R50 is 1.72655457713 N Force produced by material N20 is 0.883588825178 N Force produced by material C32 is 1.62535580812 N Total force is 33.5347385989 N

Since load index of the tyre is 82, the maximum tyre load is equal to 475 kg. According to ETRTO standard, the tyre load for rolling resistance calculation is 80%

**Figure 10.** *Temperature distribution of 2 groove tyre.*

*Rolling Resistance Estimation for PCR Tyre Design Using the Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.94144*

of maximum load which is 380 kg or 3728 N, and then the rolling resistance coefficient is equal to:

$$\mathbf{C\_{r}} = \frac{\mathbf{33.5347 \times 1000}}{\mathbf{3728}} = \mathbf{9 N/kN}$$

Using the same calculation for tyre B and C, we obtain the following result: tyre A produces Cr = 9 N/kN tyre B produces Cr = 8.77 N/kN tyre C produces Cr = 8.4 N/kN

#### **3.4 Validation of rolling resistance simulation result**

The rolling resistance simulation result obtained from an Abaqus plugin code need to be validated by comparing the result with the actual test result. The actual test has been carried out using 14 different tyres that has been tested on RR machine conducted by certified bodies, such as TUV, and the results are compared with the RR result from simulation, as shown in **Figure 11**.

In average, the simulation result is higher than the actual testing result by 0.46 or 6.2%.

#### **3.5 Rolling resistance on different radial stiffness and crown radius**

The radial stiffness of tyre significantly affects the rolling resistance. **Table 4** shows that smaller stiffness of sidewall (indicated by higher R deflection) resulted in higher rolling resistance. This phenomenon explain that to deform a higher stiffness material needs more energy, meaning that the energy loss is higher and eventually the rolling resistance is also higher.

Tyre tread contour has a great influence on rolling resistance. To study this, the simulation was performed on three tyres with different crown radiuses, i.e.:

Tyre A: R1 = 250 mm and R2 = 150 mm,

Tyre B: R1 = 550 mm and R2 = 300 mm, and

Tyre C: R1 = 900 mm and R2 = 300 mm, as it is shown in **Figure 12**.

**Figure 11.** *Rolling resistance coefficient test result.*


#### **Table 4.**

*Correlation between radial stiffness and rolling resistance.*

**Figure 12.** *Crown radius relation with footprint and rolling resistance [7].*
