# Optimization on Linkage System for Vehicle Wipers by the Method of Differential Evolution
## 4 Discussion  
In this article, we have explained how to improve the _steadiness_ of a linkage system for vehicle wipers without weakening its normal function. This improvement is based on an optimization process on the maximal absolute values of _angular acceleration_ of the output-links of a linkage system. We have demonstrated how to optimize the maximal magnitude of _angular acceleration_ of the output-links of a commercially available center driven linkage system (CDLS) for vehicle wipers using a Differential Evolution type search method. This improvement on the _steadiness_ of a linkage system for vehicle wipers could reduce the possible material fatigue of a linkage system. It also improves the torque-efficiency of the input-link. This optimization problem is related to another problem which we now discuss. Reducing the unpleasant rubber wiper noise on vehicle windshield has been an interesting research topic over the past few decades [28; 29; 40; 41; 42]. Some simple spring-mass models for the vibration of rubber wiper were presented for simulations before [39; 41; 32]. Improvement on the electrical motor of a linkage system for rubber wipers was discussed in [42]. Since rubber wipers are made of the almost _hyper-elastic_ material \"rubber\" [43; 44; 45], it is natural to expect that the mathematical theories of _Elasticity_ Mechanics [46; 47; 48], and \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\underset{0\leq\theta_{1}\leq 2\pi}{\max}\boldsymbol{\alpha_{4}}\) & \(\underset{0\leq\theta_{2}\leq 2\pi}{\min}\boldsymbol{\alpha_{4}}\) & \(\underset{0\leq\theta_{2}\leq 2\pi}{\max}\boldsymbol{\omega_{4}}\) & \(\underset{0\leq\theta_{2}\leq 2\pi}{\min}\boldsymbol{\omega_{4}}\) \\ \hline Before DE & \(1.254\) (rad/s\({}^{2}\)) & \(-0.690\) (rad/s\({}^{2}\)) & \(0.674\) (rad/s\({}^{2}\)) & \(-0.769\) (rad/s\({}^{2}\)) \\ After DE & \(1.120\) (rad/s\({}^{2}\)) & \(-0.819\) (rad/s\({}^{2}\)) & \(0.683\) (rad/s\({}^{2}\)) & \(-0.730\) (rad/s\({}^{2}\)) \\ Change & \(-10.69\)\% & \(18.70\)\% & \(1.34\)\% & \(-5.07\)\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(\alpha_{4}\) and \(\omega_{4}\) of the driver side output-link of CDLS before/after DE optimization. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\underset{0\leq\theta_{1}\leq 2\pi}{\max}\boldsymbol{\mu_{4}}-\frac{\pi}{2}\) & \(\underset{0\leq\theta_{1}\leq 2\pi}{\min}\boldsymbol{\mu_{6}}-\frac{\pi}{2}\) \\ \hline Before DE & \(0.8379\)(rad) & \(0.8232\)(rad) \\ After DE & \(0.8377\)(rad) & \(0.8205\)(rad) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The deviation of \(\mu_{4}\) and \(\mu_{6}\) respectively from \(\pi/2\) before/after DE optimization. the mathematical theories of Partial Differential Equations [49; 50; 51; 52], should be helpful for us to understand the complicated vibration of rubber wiper. In our earlier research [21], we seriously analyze the _Elasticity_ Mechanics and Physics of rubber wiper on _convex_ windshield through mathematical theories of Partial Differential Equations and through physical aspects of vibration [46; 47; 48; 49; 50; 51; 52].