## 5 Structural Parameters Identification  
With the disassembly of the heavy load radial tire (Figure 11), some geometrical parameters can be measured directly; meanwhile, some material parameters of the proposed tire model cannot be measured directly and the intelligent algorithm [(24)] is utilized to identify the unknown material parameters. The parameters of the proposed tire model are listed in Table 1. Taking the error between the experimental modal resonance frequency (Figure 10a modal resonance frequency of heavy load radial tire with normal inflation pressure) and analytical modal resonance frequency (Equation (41)) as the object value, a genetic algorithm (GA) is utilized to identify the unknown structural parameters illustrated in Table 1.  
The optimization procedure is initialized with a population size of 20, 1000 generations, a generation gap of 0.9, a crossover rate of 0.7, and a mutation rate of 0.1. The major components of the GA are the encoding scheme, fitness evaluation, parent selection, crossover, and mutation operators, illustrated in Figure 12. \begin{table}
\begin{tabular}{c l c c c} \hline  & & **Parameters** & **Symbol** & **Unit** & **Value** \\ \hline \multirow{3}{*}{Geometrical parameters} & Belt width & \(b\) & m & 0.35 \\  & Inflation pressure & \(p\) & N/m\({}^{2}\) & \(8\times 10^{5}\) \\  & Tire radii & \(R\) & m & 0.65 \\ \hline \multirow{6}{*}{Structural parameters} & Density per rad of sidewall & \(m_{s}\) & kg/rad & 10 \\  & Density per line of belt & \(\rho A\) & kg/m & 19.64 \\  & Radial spring & \(k_{sr1}\) & N/m & unknown \\  & connecting the sidewall and belt & \(k_{sr2}\) & N/m & unknown \\  & Radial stiffness & \(k_{sr2}\) & N/m & unknown \\  & connecting the sidewall and wheel & \(k_{sr1}\) & N/m & unknown \\  & Tangential spring & \(k_{sr1}\) & N/m & unknown \\  & connecting the sidewall and belt & \(k_{sr2}\) & N/m & unknown \\  & Bending stiffness of flexible belt ring & \(EI\) & N/m & unknown \\  & Bending stiffness of continuous sidewall ring & \(E_{s}I_{s}\) & N/m & Unknown \\ \hline \end{tabular}
\end{table} 
Table 1: Geometrical and structural parameters of the GL073A tire. Figure 11: Disassembled tire test. Figure 12: Scheme diagram of the structural parameters identification utilizing GA. Figure 13 presents the structural parameters of the optimization procedure. The identified result utilizing the flexible ring on an elastic continuous foundation tire model is compared with that using the flexible beam on an elastic continuous sidewall beam tire model and the compared results are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Symbol**} & \multirow{2}{*}{**Unit**} & **Value Identified by Flexible** & **Value Identified by Flexible Beam** \\  & & **Ring on an Elastic Continuous** & **on an Elastic Continuous Sidewall** \\  & & **Sidewall Ring Tire Model** & **Beam Tire Model** \\ \hline \(k_{g\pi 1}\) & N/m & \(7.111\times 10^{6}\) & \(6.967\times 10^{6}\) \\ \(k_{g\pi 2}\) & N/m & \(3.957\times 10^{6}\) & \(3.325\times 10^{6}\) \\ \(k_{g\theta 1}\) & N/m & \(1.087\times 10^{6}\) & \(\times\) \\ \(k_{g\theta 2}\) & N/m & \(9.003\times 10^{6}\) & \(\times\) \\ \(EI\) & N/m & \(48.670\) & \(23.109\) \\ \(E_{s}I_{s}\) & N/m & \(38.479\) & \(48.753\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Identified parameters of the GL073A tire. Figure 13: Identification procedure of unknown structural parameters. (**a**) The compared object value between the flexible ring on an elastic continuous sidewall ring tire model and flexible beam on an elastic continuous sidewall beam tire model; (**b**) the bending stiffness of flexible belt ring and continuous sidewall ring; (**c**) the sectional radial stiffness of sidewall spring; and (**d**) the sectional tangential stiffness of sidewall spring. The identification results of Figure 13 and Table 2 imply that:  
1.