## Appendix A Process of Deriving the Equations of Motion  
The model of the wiper blade can be divided into a free body diagram, as shown in Figure 11.  
The equations of motion for each part are as follows:  
\[m_{0}\ddot{u} =F_{u_{1}}-(F_{s}+F_{d})\cos\theta_{0}+F_{0}\sin\theta_{0}, \tag{11}\] \[m_{0}\vartheta =F_{v_{1}}+(F_{s}+F_{d})\sin\theta_{0}+F_{0}\cos\theta_{0},\] (12) \[m_{1}\ddot{u}_{1} =-F_{u_{1}}+F_{u_{2}},\] (13) \[m_{1}\ddot{v}_{1} =-F_{v_{1}}+F_{v_{2}},\] (14) \[m_{2}\ddot{u}_{2} =-F_{u_{2}}-N,\] (15) \[m_{2}\ddot{v}_{2} =-F_{v_{2}}+f. \tag{16}\]  
Combining Equations (11) and (12) and eliminating the internal forces through Equations (13)-(16), we obtain  
\[(m_{0}\dot{u}+m_{1}\ddot{u_{1}}+m_{2}\ddot{u_{2}})\cos\theta_{0}+(m_{0}\dot{v} +m_{1}\ddot{v_{1}}+m_{2}\ddot{v_{2}})\sin\theta_{0}+N\cos\theta_{0}+f\sin \theta_{0}=F_{s}+F_{d}. \tag{17}\]The equations of rotation of the two links are as follows:  
\[\begin{split} I_{1}\ddot{\theta}=&-F_{u_{1}}l_{S_{1}} \sin\theta+F_{v_{1}}l_{S_{1}}\cos\theta-F_{u_{2}}\big{(}l_{1}-l_{S_{1}}\big{)} \sin\theta+F_{v_{2}}\big{(}l_{1}-l_{S_{1}}\big{)}\cos\theta\\ &+M_{k_{1}}-c_{1}\dot{\theta}+k_{2}(\varphi-\theta+\theta_{0})+c_ {2}\big{(}\dot{\varphi}-\dot{\theta}\big{)},\end{split} \tag{10}\]  
\[\begin{split} I_{2}\dot{\varphi}=&-F_{u_{2}}l_{S_{2}} \sin\varphi+F_{v_{2}}l_{S_{2}}\cos\varphi+N\big{(}l_{2}-l_{S_{2}}\big{)}\sin \varphi+f\big{(}l_{2}-l_{S_{2}}\big{)}\cos\varphi\\ &-k_{2}(\varphi-\theta+\theta_{0})-c_{2}\big{(}\dot{\varphi}- \dot{\theta}\big{)}.\end{split} \tag{11}\]  
The distance from the origin to the wiping surface is fixed at a certain constant value. This distance, named \(C\), is defined as  
\[C=(l_{0}+l_{1}+l_{2}-h_{d})\cos\theta_{0}. \tag{12}\]  
The variation of the length of the spring \(k_{0}\) can also be expressed as  
\[\Delta l=l_{1}\bigg{(}1-\frac{\cos\theta}{\cos\theta_{0}}\bigg{)}+l_{2}\bigg{(} 1-\frac{\cos\varphi}{\cos\theta_{0}}\bigg{)}-h_{d}. \tag{13}\]  
Thus, the restoring force \(F_{s}\) and damping force \(F_{d}\) can be expressed as  
\[F_{s}=k_{0}\Delta l=k_{0}\bigg{[}l_{1}\bigg{(}1-\frac{\cos\theta}{\cos\theta_{ 0}}\bigg{)}+l_{2}\bigg{(}1-\frac{\cos\varphi}{\cos\theta_{0}}\bigg{)}-h_{d} \bigg{]}, \tag{14}\]  
\[F_{d}=c_{0}\dot{\Delta l}=c_{0}\bigg{(}\dot{\theta}\frac{l_{1}\sin\theta}{\cos \theta_{0}}+\dot{\varphi}\frac{l_{2}\sin\varphi}{\cos\theta_{0}}\bigg{)}. \tag{15}\]  
The wiper is driven by the reciprocating motion of the head. The reciprocating motion is given as a simple harmonic oscillation of frequency \(\omega\) and amplitude \(a\), i.e.,  
\[y=a(1-\cos\omega t). \tag{16}\]  
The geometric relationships of each coordinate are as follows:  
\[u=C-l_{1}\cos\theta-l_{2}\cos\varphi, \tag{17}\]  
\[v=y-(C-l_{1}\cos\theta-l_{2}\cos\varphi)\tan\theta_{0}, \tag{18}\]  
\[u_{1}=C-l_{1}\cos\theta-l_{2}\cos\varphi+l_{S_{1}}\cos\theta, \tag{19}\]  
\[v_{1}=y-(C-l_{1}\cos\theta-l_{2}\cos\varphi)\tan\theta_{0}+l_{S_{1}}\sin\theta, \tag{20}\]  
\[u_{2}=C-l_{2}\cos\varphi+l_{S_{2}}\cos\varphi, \tag{21}\]  
\[v_{2}=y-(C-l_{1}\cos\theta-l_{2}\cos\varphi)\tan\theta_{0}+l_{1}\sin\theta+l_{ S_{2}}\sin\varphi. \tag{22}\]  
Substituting Equations (14), (15), and (17)-(22) into Equations (10)-(11), we obtain Equations (2)-(4).