# Optimizing the Honeycomb Spoke Structure of a Non-Pneumatic Wheel to Reduce Rolling Resistance
## 3 Rolling Resistance Analysis and Optimization
### Response Surface Model (RSM)  
The response surface model fits the design space using polynomial functions. The functional relationships are approximated relatively accurately on a local scale through fewer trials and are presented in simple algebraic expressions, which are computationally simple and bring great convenience to design optimization [26, 27]. With the experimental data obtained by DOE, a quadratic response surface model was created for subsequent parameter optimization of tyre rolling resistance. All possible combinations of terms were examined by exhaustive search to select the term that minimized the fitting error. Under the premise of ensuring the fitting accuracy, the expression was simplified to reduce the time of the subsequent optimization process. The simplified functional relationship between tyre rolling resistance and design variables is shown in Equation (3).  
\[F_{R}=112.43-28.69X4+1.72X3^{2}-0.086X1\cdot X3+0.070X1\cdot X4-0.14X2\cdot X3 +0.14X2\cdot X4+0.63X3\cdot X4 \tag{3}\]  
where \(F_{R}\) is the rolling resistance, \(X1\) represents angle \(\gamma\), \(X2\) represents angle \(\beta\), \(X3\) represents Tp spoke thickness, and \(X4\) represents Cp spoke thickness. In order to test the accuracy of the response surface model, an error analysis was carried out. The values predicted by the response surface model and the true values are shown in Figure 13a. The results of the error analysis are shown in Table 8. The mean value of the error between the predicted value and the true value was 0.10, the maximum value was 0.22, the root  
Figure 12: Interaction diagram.  
mean square value was 0.12, and the coefficient of determination was 0.86, and they were all within the acceptable range.  
The constraints should consider the radial stiffness constraints of the tyre in addition to the range of values of the four design variables. The radial stiffness of the tyre affects its load carrying capacity: too small will lead to large deformation of the tyre, and too large will affect the smoothness of the car, so the tyre stiffness needs to be constrained in the optimization procedure. A multivariate quadratic regression equation for radial stiffness was constructed by calculating the radial stiffness of the tyre from the 15 scenarios in the previous Latin hypercube sampling method, which was simplified by retaining the key terms. The expression for the radial stiffness of the tyre is as follows.  
\[TK=831.28-7.94X1+9.97X2-272.33X3-0.08X2^{2}+1.44X1\cdot X3+1.13X2\cdot X3+1.79X3 \cdot X4 \tag{4}\]  
where \(TK\) is the stiffness, \(X1\) represents angle \(\gamma\), \(X2\) represents angle \(\beta\), \(X3\) represents top spoke thickness, and \(X4\) represents middle spoke thickness. The values predicted by the regression equation and the true values are shown in Figure 13b. The results of the error analysis are shown in Table 8. The mean value of the error between the predicted value and the true value was 0.02, the maximum value was 0.06, the root mean square value was 0.03, and the coefficient of determination was 0.99, and they were all within the acceptable range. This proves that the regression equation has an excellent fit. The radial stiffness of the tyre was generally in the range of 150 to 400 kN/m.