## 2 Analytical Two-Link Model and Equations of Motion  
### Analytical Two-Link Model  
As shown in Figure 1, a wiper blade consists of a head, neck, and lip. The head is clamped by the holder so that the wiper blade can be driven by the holder. In general, the thickness of the wiper blade below the head is not uniform. As in the figure, the thickness of the lip gradually decreases from top to bottom below the narrow neck. The shoulder is formed at the very top of the lip. The contact between the shoulder and the head leads to an increase in rotation stiffness, and this effect is taken into account when we model the wiper blade. The wiping surface corresponding to the windshield is angled with respect to the horizontal surface. The angle generated between the center line of the wiper blade and the normal vector of the wiping surface is defined as the attack angle. Figure 1: Cross-section of wiper system with attack angle. (**a**) Shoulder stays away from the head; (**b**) shoulder contacts the head. The two-link model of the wiper blade is shown in Figure 2. The first and second links (from top to bottom) correspond to the neck and lip, respectively. Their total length and the distance from the top to the center of gravity are indicated in the figure. The displacement of the head is limited in the horizontal direction by the holder, but it can move freely in the vertical direction. The mass, rotational inertia, stiffness, and damping of each component are also shown in the figure. The origin is set at the uppermost position of the entire model when both links are pointing vertically downward and the head spring \(k_{0}\) is at its original length. The \(x\)- and \(y\)-directions are perpendicular and parallel to the wiping surface, respectively; the \(x\)-axis is inclined from the vertical direction at \(\theta_{0}\). To apply the normal load to the wiper blade, the spring \(k_{0}\) can be compressed by a length of \(h_{d}\). The distance from the origin to the wiping surface is constant, so given the length \(h_{d}\), the position relationship of the entire model can be determined by the angles of the two links, \(\theta\) and \(\varphi\). Thus, this is a two-degree-of-freedom model in which the independent variables are the two link angles \(\theta\) and \(\varphi\). Our purpose is to clarify the characteristic dynamics under the transition between the dynamic and static friction of the wiper blade. We think that a simplified essential model with the smallest possible number of parameters is important to find the essential parameters governing the dynamics under the transition between dynamic and static frictions. The calculation results for the simple analytical model develop the physical understanding of the behavior of the wiper blade depending on the system parameters. As mentioned above, the rotation stiffness of the first link increases when the shoulder contacts the head. To take this effect into account, the restoring moment of the first link \(M_{k_{1}}\) is given as follows:  
\[M_{k_{1}}= \begin{cases}-k_{12}(\theta+\theta_{0})-(k_{12}-k_{11})\theta_{c},& \theta+\theta_{0}<-\theta_{c}\\ -k_{11}(\theta+\theta_{0}),&-\theta_{c}\leq\theta+\theta_{0}\leq\theta_{c}\\ -k_{12}(\theta+\theta_{0})+(k_{12}-k_{11})\theta_{c},&\theta+\theta_{0}>\theta _{c}\end{cases} . \tag{1}\]  
The parameter \(k_{11}\) represents the rotational stiffness of the first link without shoulder-head contact, and \(k_{12}\) represents the rotational stiffness of the first link with shoulder-head contact. The parameter \(\theta_{c}\) represents the angle of the first link when contact between the shoulder and the head occurs. Figure 2: Two-link model of wiper system. (**a**) Upright state; (**b**) buckled state. The relationship between \(\theta+\theta_{0}\) and \(M_{k_{1}}\) is illustrated in Figure 3. When the absolute value of the sum of the attack angle and the angle of the first link does not exceed \(\theta_{c}\), i.e., the shoulder does not contact the head, the rotational stiffness of the first link is \(k_{11}\).