# Optimizing the Honeycomb Spoke Structure of a Non-Pneumatic Wheel to Reduce Rolling Resistance
## 3 Rolling Resistance Analysis and Optimization
### Design Variables and Design of Experiment (DOE)  
When there are many influencing factors, some factors that do not have a significant effect on the response increase the amount of computation and waste computational resources. Therefore, in this work, orthogonal experiments were used to pick out the significant influencing factors as design variables to continue with the optimization design. The orthogonal experiment scheme and simulation results are shown in Table 4. FR represents the rolling resistance, and t1, t2, t3, and t4 represent the thicknesses of the Tp, Cp, Lp, and Bp, respectively, which are shown in Figure 2.  
The polar analysis of orthogonal experimental results is shown in Table 5. K represents the sum of the test results for each level of each factor. K-avg represents the average of the test results for each level of each factor, which is equal to K divided by the number of levels of the corresponding factor. R represents the polar value of factors. The magnitude of its value reflects the degree of influence of the factors on the experimental results. The larger the R of the factors, the deeper the influence on the experimental results is approximately. The effects of the factors on the results of the experiment were, in descending order, t1, t2, t3, and t4. The relatively better levels corresponding to each factor were 7, 7, 7, and 5 by using the variance analysis. Despite its low workload and intuitive convenience, the extreme variance analysis method cannot distinguish between fluctuations in test data caused by changes in test conditions during the test and fluctuations in data caused by test errors. Therefore, the experimental data were analyzed by ANOVA, and the results are shown in Table 6. The SS stands for the sum of squared deviations, df stands for degrees of freedom; MS stands for mean square, which is equal to SS divided by df; and F is the statistic of the test, which is equal to the mean square of each item divided by MSD. The symbol \"*\" means that the factor has a significant effect on the results of the test with a 95 per cent confidence level. The aggregate represents the total sum of the squared deviations. Since the sum of the squares of intergroup deviations at the t4 factor was small, it was considered the sum of the squares of errors to test the significance of the t1, t2, and t3 factors. The degree of influence of the factors on the results of the experiment was consistent with the analysis of extreme variance method. Factors t1 and t2 had a significant effect on the test results, while factors t3 and t4 had no significant effect on the test results. Combining the analysis of polarity and analysis of variance of the test results, the top and middle spoke thicknesses were selected as the optimal design variables in terms of spoke thickness. Considering that the change in the spoke angle affects the length of the spoke, which will have an effect on the rolling resistance, the angles \(\beta\) and \(\gamma\) were comprehensively selected as the optimal design variables in terms of the spoke angle. On the basis of the above four optimal design variables, 15 sets of experimental scenarios were designed using the Latin hypercube sampling method, and the experimental scenarios and simulation results are shown in Table 7, where FR represents rolling resistance, TK represents vertical stiffness, X1 represents angle \(\gamma\), X2 represents angle \(\beta\), X3 represents the thickness of Tp, and X4 represents the thickness of Cp. The Latin hypercube sampling results were analyzed using the SIMULIA light software 2020. The correlations of the four design variables were first analyzed, and the results are shown in Figure 9. High correlations between multiple independent variables may lead to multicollinearity problems. Severe multicollinearity problems may lead to a loss of significance of variables, failure of the predictive function of the model, unreasonable parameter estimation, and so on.