# Empirical Models for the Viscoelastic Complex Modulus with an Application to Rubber Friction
## 3 Proposed Empirical Models
### Fitting Accuracy  
Several published moduli from the literature were used to assess the quality of fitting of the polynomial method. The storage and loss moduli, and the loss tangent, were fitted separately, and the relevant coefficients are provided in Table 1. For all three materials, three linear parts were combined with two fourth-order polynomials, to obtain the overall curves. In Table 1, any subscript indicates the role of the parameter in Equation (9), and a superscript indicates the sequential number of the respective linear or fourth-order polynomial. Clearly, any two of the three complex modulus curves are sufficient for fully describing any tested material. The correlations between measured and fitted curves for three different rubbers using the polynomial method are shown in Figures 5 and 6. Figure 5 shows that the piecewise polynomial method is capable of following the curvature of the curves faithfully. A quantitative assessment of the quality of fit is provided via the relative percentage error in Figure 6, where it is shown that the maximum relative error for the storage modulus is 0.7%, and for the loss modulus the maximum relative error is 1.49%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Coefficient** & \multicolumn{3}{c}{**Loss Modulus**} & \multicolumn{3}{c}{**Storage Modulus**} \\ \hline  & F14 [22] & L13 [26] & P01 [11] & F14 [22] & L13 [26] & P01 [11] \\ \hline \(a^{1}\) & 0.202 & 0.089 & \(-\)0.167 & 0.003 & 0.058 & 0.072 \\ \(b^{1}\) & 5.059 &  
The same dataset was used for assessing the quality of the fit using the LBEM approach. The MATLAB function \"lsqonlin\" with the default trust-region-reflective fitting algorithm [28] was used to identify the optimal parameter values of the proposed formulation by minimising the non-linear cost function that effectively consists of the sum of the squared differences between measured and estimated loss tangent and storage modulus. The resulting parameter values for three example datasets have been tabulated in Table 2.  
Figure 7 shows a comparison between the measurements and the fitted LBEM models. The results are very satisfactory, as the formula is able to describe all the measured characteristics and is continuous at all times. From Figure 8 it can be appreciated that the relative error for the LBEM is higher than that of the polynomial formulation, as expected. This is because the LBEM approach uses a generic formulation with fewer parameters that can, however, be easily linked to the typifying quantities of the data. For the storage modulus \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Acronym** & **r\(\omega\)**0** & **r\(\gamma\)**0** & **L** & **k** & **t\(\omega\)**0** & **\(\delta\)\(\gamma\)\(\gamma\)\(\gamma\)**1** & **\(\delta\)\(\gamma\)\(\gamma\)2** & **\(r_{1}\)** & **\(r_{2}\)** & **p** \\ \hline F14 [22] & 6.613 & 6.341 & 2.394 & 1.084 & 6.719 & 0.020 & 0.000 & 1.258 & 1.129 & 2.181 \\ L13 [26] & 5.490 & 7.039 & 2.338 & 0.524 & 5.216 & 0.105 & 0.051 & 1.834 & 1.838 & 0.686 \\ P01 [11] & \(-\)1.785 & 6.204 & 3.209 & 0.555 & \(-\)2.780 & 0.014 & 0.000 & 1.347 & 2.612 & 0.825 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values for the LBEM curves shown in Figure 7.  
Figure 5: Storage (**left chart**) and loss (**right chart**) moduli for three rubber compounds. Experimental data (markers) vs. polynomial model (lines). Figure 6: Storage (**left chart**) and loss (**right chart**) moduli relative error for three rubber compounds. the maximum relative error increased to 1.4%. The loss tangent shows a higher error of up to 27%, but such discrepancies happen at a limited number of discrete points. The fitting quality of the polynomial and LBEM models was also evaluated by means of the normalised root mean squared error (NRMSE) and the coefficient of determination, denoted \(R^{2}\) for both the storage modulus and the loss tangent. The results are summarised in Tables 3 and 4.