## 4 Behavior of Wiper Blade
### Numerical Calculation Results  
Following the method described above, a numerical calculation program was developed in Matlab(r), and the differential equation solver ODE45 was used to obtain solutions. The parameter values in the program were obtained from the actual wiper blade used in subsequent experiments. The maximum speed of the head, which is determined by \(a\) and \(\omega\), was found to be about 125 mm/s. The dynamic friction in the velocity region from 0-125 mm/s exhibits strongly nonlinear characteristics. The initial condition for the numerical calculation was the stick state. The calculations terminated at the end of one round trip of the wiper blade. The parameters used in the program are listed in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline \(m_{0}\) & \(1.01\times 10^{-2}\) & kg \\ \(m_{1}\) & \(6.4\times 10^{-5}\) & kg \\ \(m_{2}\) & \(1.16\times 10^{-5}\) & kg \\ \(I_{1}\) & \(6.8\times 10^{-11}\) & kg m\({}^{2}\) \\ \(l_{2}\) & \(2.8\times 10^{-12}\) & kg m\({}^{2}\) \\ \(l_{0}\) & \(2\times 10^{-2}\) & m \\ \(l_{1}\) & \(3.15\times 10^{-3}\) & m \\ \(l_{2}\) & \(1.53\times 10^{-3}\) & m \\ \(l_{g_{1}}\) & \(1.37\times 10^{-3}\) & m \\ \(l_{g_{2}}\) & \(7.64\times 10^{-4}\) & m \\ \(k_{0}\) & \(3.92\times 10^{1}\) & kg/s\({}^{2}\) \\ \(k_{11}\) & \(7.1\times 10^{-4}\) & kg m\({}^{2}\)/s\({}^{2}\) \\ \(k_{12}\) & \(2.13\times 10^{-2}\) & kg m\({}^{2}\)/s\({}^{2}\) \\ \(k_{2}\) & \(1.4\times 10^{-3}\) & kg m\({}^{2}\)/s\({}^{2}\) \\ \(c_{0}\) & \(8.2\times 10^{-2}\) & kg m/s \\ \(c_{1}\) & \(1.825\times 10^{-7}\) & kg m\({}^{2}\)/s \\ \(c_{2}\) & \(5.451\times 10^{-8}\) & kg m\({}^{2}\)/s \\ \(a\) & \(5\times 10^{-2}\) & m \\ \(\omega\) & \(2\times\pi/2.5\) & rad/s \\ \(h_{d}\) & \(5\times 10^{-3}\) & m \\ \(A\) & \(0.4\) & \\ \(B\) & \(5\) & s/m \\ \(E\) & \(0.2\) & \\ \(\beta_{1}\) & \(0.01\) & \\ \(\beta_{2}\) & \(0.01\) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and values used in the program.  
The numerical results in Figure 7 show the relationship between the attack angle and the jump phenomenon. Figure 8 shows enlarged views of Figure 7c,f around the jump phenomenon. Figure 7a,b show the changes in the angles \(\theta\) and \(\varphi\), while Figure 7d,e show the change in the normal force \(N\) of the wiper blade over one reciprocal motion with attack angles of 0 and 5 degrees, respectively. From the figures, it can be seen that the wiper blade does not jump under these conditions. Figure 7c shows the changes in \(\theta\) and \(\varphi\) with an attack angle of 10 degrees. In this condition, the normal force \(N\) changes direction, i.e., the jump phenomenon occurs, as shown in Figures 7f and 8b. Figures 7c and 8a confirm that the two angles become 0 at the same time when the jump phenomenon occurs, which does not happen in other conditions. Around the reversal point, both angles become 0 at the same time, as shown in Figure 7d-f. However, the normal force \(N\) does not become 0, and so, the jump phenomenon does not occur. Therefore, the occurrence of the stick-slip motion is a necessary condition for producing the jump phenomenon.  
Figure 8: Enlarged view of Figure 7c,f from 1.64–1.72 s. Panels **(a)** and **(b)** correspond to Figure 7c,f, respectively.  
Figure 7: Behavior of wiper blade zero and nonzero attack angle. **(a)** Angular displacements of two links with zero attack angle. **(b)** Angular displacements of two links with 5-degree attack angle. **(c)** Angular displacements of two links with 10-degree attack angle. **(d)** Normal force of wiper blade with zero attack angle. **(e)** Normal force of wiper blade with 5-degree attack angle. **(f)** Normal force of wiper blade with 10-degree attack angle.