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Article

# Dynamic Analysis of a Wiper Blade in Consideration of Attack Angle and Clarification of the Jumping Phenomenon

by
Zihan Zhao
1,
Hiroshi Yabuno
1,* and
Katsuya Kamiyama
2
1
Mechanical Systems Laboratory, Degree Programs in Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8573, Japan
2
MITSUBA Co., Ltd., Kiryu 376-0013, Japan
*
Author to whom correspondence should be addressed.
Appl. Sci. 2022, 12(9), 4112; https://doi.org/10.3390/app12094112
Submission received: 14 March 2022 / Revised: 13 April 2022 / Accepted: 15 April 2022 / Published: 19 April 2022

## Abstract

:
Automobile windshields are typically curved, creating an oblique angle of attack between the wiper blade and the windshield. This attack angle means that the wiper may jump off the windshield while wiping, causing a chattering noise and preventing the rainwater from being fully wiped off the windshield. Thus, it is important to examine the dynamics of the wiper blade under friction. In this study, the relationship between the attack angle and the jumping phenomenon is clarified through dynamic analysis. We introduce an analytical two-link model corresponding to an actual wiper blade that considers the exchange of dynamic and static friction between the windshield and the blade. The dynamic friction is assumed to be negatively correlated with the relative velocity, and the static friction is described by a set-valued function. As the motion transitions from the stick state to the slip state, the equation to be solved changes. Hence, the initial condition after a transition must agree with the final condition before the transition. Because the governing equations are nonlinear and the solution is highly dependent on the initial condition, the transition time and corresponding state variables are vital. The slack variable method is used to obtain the exact transition time and initial conditions. The sign of the normal force acting on the blade from the windshield determines the occurrence of the jump phenomenon. A larger attack angle makes the jump phenomenon more likely. However, the jump phenomenon does not occur when the motion of the blade reverses. Experimental observations support the theoretical description of the wiper blade.

## 1. Introduction

To ensure excellent aerodynamic performance and aesthetics, automobile windshields are generally constructed to have some integral curvature. Thus, an angle is generated between the symmetry plane of the wiper blade and the normal vector of the windshield surface. This is defined as the attack angle. The presence of the attack angle means that the wiper blade may jump away from the windshield during the wiping process, which causes a chattering noise and incomplete wiping. These issues negatively affect the driving experience and pose a safety hazard. In developing the optimal wiper blade design that does not suffer from the jumping phenomenon, it is essential to understand the dynamics of the wiper blade in consideration of the attack angle.
In the study of the tribology between rubber and glass, the change in the coefficient of dynamic friction under wet conditions is particularly critical. Deleau et al. [13] investigated the tribological behavior of the contact between the rubber blade and the windshield. On a wet windshield, the rubber blade has the same actual contact area as on a dry windshield under extremely slow velocity. However, as the sliding velocity increases, the dry area gradually decreases, and a thin film of water can be formed. The lubrication effect of the thin film of water in between the rubber blade and windshield leads to a reduction in the coefficient of dynamic friction. Bódai et al. [14] clarified that the film arises because the pressure of the fluid gradually increases as the sliding velocity increases. Due to the presence of the film, the friction is mainly controlled by the fluid film friction. Thus, the friction coefficient is significantly reduced. However, due to the low viscosity of water, the reduction of the friction coefficient at low sliding velocity cannot be explained by the hydrodynamic effect.
Several studies have taken the attack angle into account. Chevennement-Roux et al. [15] developed a finite element model considering the attack angle to analyze the dynamics of the wiper system. The stability of the system was analyzed for different values of the normal pressure and attack angle. The validity of this model was then verified through a series of experiments. Min et al. [16] built an experimental setup to simulate the waves on the windshield. The waves have the effect of changing the contact angle of the wiper blade, which corresponds to the so-called attack angle effect. As the liquid on the windshield is removed by the wipers, the coefficient of dynamic friction becomes greater than 1. The waviness changes the wiping velocity of the wiper blade and the coefficient of dynamic friction, leading to noise.
In the present study, all of the above-mentioned factors are considered in analyzing the dynamics of the wiper blade. These factors include the attack angle, independent dynamic and static friction models, the lubrication effect of the thin film of water, and the change in rotation stiffness brought about by the shoulder–head contact. We first introduce a two-degree-of-freedom analytical model for the wiper blade. Dynamic friction is considered to be negatively related to the relative velocity of the motion, and static friction is considered to be a set-valued function that keeps the system in equilibrium. Unlike the case where the attack angle is not considered, the normal force cannot be obtained independently, because the friction force affects the normal force. A simulation algorithm was established based on the relevant equations. In the numerical simulations, we used different equations of motion to solve the corresponding motion states. As the governing equations in each state are nonlinear, the initial condition has a significant effect on the simulation result. To ensure the accuracy of the result, we used the slack variable method to determine the exact time at which the transition occurs between the different states. The purpose of this study is to analyze the dynamics of the wiper blade in the range where the normal force is positive, i.e., until the jump phenomenon occurs, and to detect the state of the wiper blade at this point. Through numerical simulations with different values of the attack angle, it is shown that a large attack angle makes the jumping phenomenon more likely to occur. The configuration of the blades required to produce the jump is theoretically determined. We also conduct experiments using a real wiper blade. The experimental results are in qualitatively good agreement with those from our theoretical system.

## 2. Analytical Two-Link Model and Equations of Motion

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 $θ 0$. 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, $θ$ and $φ$. Thus, this is a two-degree-of-freedom model in which the independent variables are the two link angles $θ$ and $φ$. 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:
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 $θ c$ represents the angle of the first link when contact between the shoulder and the head occurs.
The relationship between $θ + θ 0$ 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 $θ c$, i.e., the shoulder does not contact the head, the rotational stiffness of the first link is $k 11$. The restoring moment varies within a narrow range. However, when the absolute value of the sum of the attack angle and the angle of the first link is greater than $θ c$, i.e., the shoulder contacts the head, the rotational stiffness of the first link becomes $k 12$, which is much greater than $k 11$. In this condition, the restoring moment changes from the maximum value in the non-contact case, and the rate of change with respect to the angle will increase.

#### 2.2. Equations of Motion of the Model

From the model, we can obtain the following governing equations of motion (a detailed derivation is given in Appendix A).
There are four unknown quantities in these three equations, namely the angles of the two links $θ$ and $φ$, the normal force N, and the friction force f. One additional equation is required to obtain the solution. This is obtained next by considering the different states of motion.
Unlike the case where the attack angle is not considered , it is impossible to obtain an equation for the normal force independently. If the attack angle is zero, Equation (2) can be deformed as follows:
Equation (5) is an independent equation that determines the normal force N. The friction force f can also be derived from the relationship with N. We can then obtain the solution by substituting N and f into Equations (3) and (4). The normal force N has no effect on the horizontal restraining force on the head, because its direction is always perpendicular to this force. However, in this study , the restraining force in the horizontal direction of the head must be balanced in the x- and y-directions. This force is simultaneously influenced by the normal force N and the friction force f in the x- and y-directions. As a result, we obtain an equation that appears to show that the normal force N is determined by the friction force f. We numerically calculate these equations in a unique way.

#### 2.3. Friction Models in the Slip and Stick States

There are two states in which the motion of the wiper blade is governed by different kinds of friction. The slip state corresponds to the case where the relative velocity between the tip of the wiper blade and the windshield is nonzero. The friction acting on the tip of the wiper blade is dynamic in this case. The wiper blade is in the stick state when the relative velocity is continuously zero, in which case static friction acts on the blade.
The dynamic friction is uniquely determined by the relative velocity. As long as the stick state is maintained, the absolute value of the static friction can be taken arbitrarily in the range from zero to the maximum static friction. Hence, different models are required for the slip and stick states.
The dynamic friction in the slip state is modeled as a function of the relative velocity. The displacement along the y-axis of the tip of the wiper blade, which corresponds to $v 3$ in Figure 2, can be expressed as follows:
where C is a constant representing the distance from the origin to the wiping surface. The relative velocity can be expressed as follows:
In the motion of the rubber in contact with the glass, the change in the coefficient of dynamic friction under wet conditions due to the thin film of water in between the rubber blade and glass is particularly critical. A thin film of water can be formed between the wiper blade and glass because the pressure of the fluid gradually increases as the sliding velocity increases [14]. The lubrication effect of the thin film of water leads to a reduction in the coefficient of dynamic friction [13]. Thus, the relationship between the relative velocity of the tip of the second link $v ˙ 3$ and the coefficient of dynamic friction $μ d$ is assumed to be as follows in the case $v ˙ 3 ≠ 0$:
where the coefficient of dynamic friction is assumed to be equal to the coefficient of the maximum static friction when $v 3 ˙$ is zero and has a negative correlation with $v 3 ˙$. The parameters A and E define the coefficient of the maximum static friction $μ m a x$ as follows:
$μ m a x = A + E .$
The parameter B reflects the magnitude of the negative correlation. A, B, and E are all positive constants, and sign is the signum function, expressed as
The dynamic friction $f d$ is then expressed as
$f d = − μ d N .$
The normal force N is always positive, except when the jump phenomenon occurs. Therefore, the dynamic friction $f d$ always acts in the direction opposite the relative velocity $v 3 ˙$.
In the stick state, the relative velocity remains at zero continuously and the equilibrium should be maintained. The absolute value of the friction force should be less than the maximum static friction and balance the tangential force of the wiper blade. Considering the above characteristics, the static friction is defined as follows:
where is a set-valued function expressed as
By combining the two models for dynamic and static friction, the relationship between the coefficient of friction and the relative velocity is as shown in Figure 4. Given the relationship between the normal force N and the friction force f for the different states of motion, the behavior of the two links can be analyzed. For the four unknown quantities, in addition to Equations (2)–(4), the other equations in the slip and stick states are Equations (11) and (12), respectively.
By introducing the representative time $T = I 2 + m 2 l g 2 2 k 2$ and the representative length $L = a$, the dimensionless forms of Equations (2)–(4) are expressed as follows:
where the dots denote derivatives with respect to the dimensionless time $t *$. The dimensionless parameters are as follows:

## 3. Numerical Calculation Method in Different States

#### 3.1. Numerical Calculation Method in the Slip State

From hereon, the superscript ∗ representing dimensionless quantities is omitted for simplicity. The unknown friction force f can be eliminated by substituting Equation (11) into Equations (14)–(16). The equations for the unknown variables $θ$, $φ$, and N become
$A 1 θ ¨ + B 1 φ ¨ = C 1 + N H ,$
$A 2 θ ¨ + B 2 φ ¨ = C 2 + N l 1 G 1 ,$
$A 3 θ ¨ + B 3 φ ¨ = C 3 + N l 2 G 2 ,$
where $A 1$, $A 2$, $A 3$, $B 1$, $B 2$, $B 3$, $C 1$, $C 2$, $C 3$, H, $G 1$, and $G 2$ are
$A 1 = − m d l 1 sin θ cos θ 0 + m a l g 1 sin ( θ + θ 0 ) + m b l 1 tan θ 0 cos ( θ + θ 0 ) ,$
$B 1 = − l 2 sin φ cos θ 0 + m b l g 2 sin ( φ + θ 0 ) ,$
$H = cos θ 0 − μ d sin θ 0 ,$
$G 1 = sin θ − μ d cos θ ,$
$G 2 = sin φ − μ d cos φ .$
Using Equation (20), the normal force N is expressed in terms of $θ$ and $φ$ as
$N = A 3 θ ¨ + B 3 φ ¨ − C 3 l 2 G 2 .$
Substituting Equation (33) into Equations (18) and (19) yields
Equations (34) and (35) can also be written in matrix form as follows:
$a 11 a 12 a 21 a 22 θ ¨ φ ¨ = P 1 P 2 ,$
where $a 11$, $a 12$, $a 21$, $a 2$, $P 1$, and $P 2$ are
$a 11 = A 1 − A 3 H l 2 G 2 ,$
$a 12 = B 1 − B 3 H l 2 G 2 ,$
$a 21 = A 2 − l 1 A 3 G 1 l 2 G 2 ,$
$a 22 = B 2 − l 1 B 3 G 1 l 2 G 2 ,$
$P 1 = C 1 − C 3 H l 2 G 2 ,$
$P 2 = C 2 − l 1 C 3 G 1 l 2 G 2 .$
The differential equations for $θ$ and $φ$ can be rewritten as follows using Equation (36):
$θ ¨ = a 22 P 1 − a 12 P 2 a 11 a 22 − a 12 a 21 ,$
$φ ¨ = − a 21 P 1 + a 11 P 2 a 11 a 22 − a 12 a 21 .$
The state equation can then be written as Equation (45), and the Runge-Kutta method can be used for the numerical calculation of $θ$ and $φ$.
$d d t θ θ ˙ φ φ ˙ = θ ˙ θ ¨ φ ˙ φ ¨ .$

#### 3.2. Numerical Calculation Method in the Stick State

Unlike the slip state, a definite relationship between the friction force f and the normal force N cannot be established in the stick state. Static friction acts at the tip of the blade as a constraining force to balance the external forces acting on the wiper blade and keep the tip stationary. The holonomic constraint that the displacement of the tip cannot vary must be satisfied. A constant $α$ is introduced to indicate the stationary position of the tip. In the stick state, the distance $Φ$ between the tip and $α$ should always be 0. This condition can be expressed as follows:
$Φ = v 3 − α = 0 ,$
where the y-coordinate position of the tip $v 3$ is given by Equation (6). In the stick state, $v 3$ remains constant at $α$. Equation (46) is an independent equation describing the relationship between $θ$ and $φ$. The dynamic behavior of the wiper blade in the stick state becomes resolvable by associating Equation (46) with Equations (14)–(16). However, because Equation (46) is not a differential equation, a set of algebraic and differential equations (ADEs) consisting of Equations (14)–(16) and (46) must be solved.
To solve this set of ADEs, Baumgarte’s stabilization method [20] is employed to modify the ADEs into a set of differential equations. The second-order differential equation:
$Φ ¨ + β 1 Φ ˙ + β 2 Φ = 0$
is introduced instead of Equation (46). The parameters $β 1$ and $β 2$ are both constants. The solution $Φ$ of Equation (47) can be maintained close to 0 if $β 1$ and $β 2$ in Equation (47) have suitable values. Equation (47) can be written in dimensionless form as follows:
Combining the differential equations and rewriting them in matrix form yields
$B θ ¨ φ ¨ N f = Q ,$
where the matrices $B$ and $Q$ are
$B = b 11 b 12 b 13 b 14 b 21 b 22 b 23 b 24 b 31 b 32 b 33 b 34 b 41 b 42 0 0 ,$
$Q = Q 1 Q 2 Q 3 Q 4 .$
The elements of $B$ and $Q$ are given in Appendix B.
The differential equations for $θ$ and $φ$ are then obtained from Equation (49) by multiplying the inverse of matrix $B$ from the left side.

## 4. Behavior of Wiper Blade

#### 4.1. Employment of the Slack Variable Method to Obtain the Transition Time and State Variables

There are two state transitions—the transition of the friction state from slip to stick or from stick to slip and the transition of the stiffness of the first link.
Because of the nonlinearity of the governing equations, it is important to find the exact transition time and the state variables of the wiper blade at this point. In nonlinear systems, even small errors in the initial conditions may cause significant errors in the calculation results. However, as shown in Figure 5, the transient time and conditions of the wiper blade at the exact transition time $t e$ cannot be obtained from calculations using a discrete time step size $Δ t$.
We introduce the slack variable method [21] to detect the transition time and calculate the conditions at that point accurately. First, the slack variable s is introduced. This variable satisfies Equation (52).
Differentiating Equation (52) with respect to s gives
$d s d s = 1 = d R d s = d R d t d t d s .$
Deforming Equation (53) yields
Regarding s as an independent variable and substituting Equation (54) into the state Equation (45) yield
In the calculations of using the discrete time, we consider the case in which the state transition occurs in the interval $[ t n , t n + 1 ]$, as shown in Figure 6. The calculation then starts again from $t n$ using s as the independent variable and t as the dependent variable. The exact transition time $t e$ and the conditions of the wiper blade at this time can be obtained by integrating Equation (55) from $s 0$ to $s e$, where $s e$ is defined later.

#### 4.2.1. Transition from Slip to Stick State

When applying the slack variable method, it is essential to apply the appropriate R function for different state transitions.
In the slip state, the relative velocity $v ˙ 3$ is assumed to be nonzero. The state transition from slip to stick occurs when $v ˙ 3$ becomes zero. Hence, the R function for the state transition from slip to stick is as follows:
The derivative of t with respect to s is
The exact transition time is obtained by substituting Equation (57) into Equation (55) and integrating the result with respect to s from $s 0$ to $s e$.

#### 4.2.2. Transition from Stick to Slip State

In the calculations for the stick state, if the absolute value of the static friction force f is greater than the maximum friction force $μ m a x N$, the force equilibrium of the wiper blade will be destroyed. Hence, when the static friction force f reaches the maximum friction force $μ m a x N$, the state transitions from stick to slip. The R function for the state transition from stick to slip is as follows:
$R = | f | − μ m a x N .$
The derivative of t with respect to s is
which can also be obtained by differentiating Equation (49). The exact transition time is obtained by substituting Equation (59) into Equation (55) and integrating from $s 0$ to $s e$, as described above. Here, $s 0$ is the value of $| f | − μ m a x N$ at $t n$ and $s e$ is the value of $| f | − μ m a x N$ at $t e$, which is zero.

#### 4.2.3. Transition of the Rotational Stiffness

In addition to the above two transitions, there is also a state transition in the rotational stiffness, because the first link is piecewise linearly related to the shoulder contact, as shown in Figure 3. The state transition occurs when the angle of the first link $θ$ passes through $θ c$ while in motion. The R function for this transition can be set as follows:
$R = θ .$
The derivative of t with respect to s can be easily computed as
The exact transition time is determined through the same process as for the stick–slip transitions, except that the start and end points of the integration are different. The calculation starts at $s 0$, which is the angle $θ$ at $t n$, and ends at $s e$, which is the angle at which shoulder contact occurs, $θ c$.

#### 4.3. Numerical Calculation Results

Following the method described above, a numerical calculation program was developed in Matlab®, 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 $ω$, 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.
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 $θ$ and $φ$, 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 $θ$ and $φ$ 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 Figure 7f and Figure 8b. Figure 7c and Figure 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.

## 5. Experimental Results

#### 5.1. Experimental Apparatus and Procedure

To verify the accuracy of the theoretical analysis, the experimental setup shown in Figure 9 was created. There is a spring with a spring constant of $1.43$ N/mm and an original length of $0.2$ m above the holder. A pressure force is supplied to the holder by rotating the screw to change the length of the spring. This is equivalent to determining the value of $h d$. The entire system can move liberally within the stroke of the linear bearing. The wiper blade is clamped by the holder and driven by the actuator to perform a reciprocating motion on the wiping surface. Before each experiment, water was sprayed on the wiping surface.
An enlarged view of the wiper blade is shown in Figure 10. There are three white dots marked on the neck, the center of rotation of the second link, and the tip of the wiper blade.
The motion of the wiper blade was filmed by a high-speed video camera (FASTCAM-APX RS 250 K, Photron Inc., San Diego, CA, USA) with a frame rate of 3000 fps at a 1024 × 512 pixel resolution on a screen measuring $3.165$ × $10 − 2$ m × $1.583$ × $10 − 2$ m. Thus, each pixel had dimensions of $3.091$ × $10 − 5$ m. We conducted two experiments for attack angles of 0, 10, and 18 degrees. In the first experiment, the camera recorded the location at which the jump phenomenon occurs during the wiping process. In the second experiment, the filming position was located around the reversal point of the wiper blade.

#### 5.2. Experimental Results

After filming, we used image analysis software (Dipp-Motion V, DITECT Corp., Toronto, ON, Canada) to track the markers on the wiper blade in advance. From the tracking results, we obtained the angular changes of the two links and the change in displacement of the tip of the wiper blade in the direction perpendicular to the wiping surface.
Figure 11 shows the experimental results obtained during the wiping process; Figure 12 shows an enlarged view of Figure 11c,f near the occurrence of the jump phenomenon. Figure 11d,e confirm that attack angles of 0 and 10 degrees do not produce the jump phenomenon. When the attack angle is 18 degrees, however, the jump phenomenon occurs, as shown in Figure 11f and Figure 12b. We can also confirm from Figure 11c and Figure 12a that the angles $θ$ and $φ$ converge to 0 at the same time when the jump phenomenon occurs. In turn, these two angles do not converge to 0 at the same time when no jump phenomenon occurs, as shown in Figure 11a,b.
The experiments also suggested that the stick–slip motion does not occur around the reversal point. Thus, even if the angles become 0 at the same time, there is no jump phenomenon. Figure 13 shows the experimental results obtained around the reversal point with different attack angles. Figure 13d–f confirm that the wiper blade does not jump away from the wiping surface around the reversal point at the different attack angles. This is despite the two angles becoming 0 at the same time in this process, as shown in Figure 13a–c.
There are quantitative discrepancies between the theoretical and experimental results. For example, in the theoretical results, the angles of the two links vary periodically also in the state where the jump phenomenon does not occur, as in Figure 7a,b. However, the angles of the two links do not change significantly in the experiments in such a state, as in Figure 11a,b. Besides, although it was theoretically and experimentally shown that the jump phenomenon becomes more likely as the attack angle increases, the theoretical and experimental values of the attack angle where the jump phenomenon starts to occur are different quantitatively. In the theoretical result, a 10-degree attack angle produces the jump-up phenomenon, as shown in Figure 7c,f. However, there is no jump phenomenon with a 10-degree attack angle in the experimental result, as shown in Figure 11b,e. In addition, in the theoretical results where the jump phenomenon occurs, the angles of the two links are very close to 0 at about 1.66 s and 1.68 s before they become 0 at the same time, as in Figure 8a. However, the angles of the two links are not so close to 0 at about 0.05 s, 0.06 s and 0.07 s before they become 0 at the same time in the experimental results shown in Figure 11c. The actual wiper blade is an infinite-degree-of-freedom elastomer rather than the low-degree-of-freedom two-link model used in the theoretical analysis. This is the reason for the quantitative discrepancies between the theoretical and experimental results.

## 6. Conclusions

In this study, we analyzed the occurrence of the jump phenomenon of a wiper blade depending on the attack angle. We introduced a two-degree-of-freedom model of a wiper blade and considered effects such as dynamic and static friction, nonlinear rotation stiffness, and attack angle on the dynamic behavior of the wiper blade. In the slip state, we considered the negative slope between the dynamic friction and the relative velocity. In the stick state, we used Baumgarte’s stabilization method for the numerical calculations. As the numerical simulation results of nonlinear systems are highly dependent on the initial conditions, we also used the slack variable method to determine the exact time of each state transition and the state of motion at that point. When considering the attack angle, the normal force N is no longer independent, but is influenced by the friction force f. We established a simulation program based on these considerations. The simulation results indicated that a larger attack angle made the jump phenomenon more likely. The reason for the jump phenomenon can be attributed to the angle of the two links becoming 0 at the same time during the stick–slip motion of the wiper blade. In contrast, if the wiper blade is not in stick–slip motion, such as during the process of reversal, the jumping phenomenon does not occur even if the two angles become 0 at the same time. The results of our theoretical analysis were confirmed by experiments conducted using an actual wiper blade. The theoretical analysis for the model of wiper blade considering a constant attack angle can be proven to be effective. The attack angle varies continuously on the practical windshield of the car. The present theoretical approach can be easily applicable to such a practical situation.

## Author Contributions

Conceptualization, methodology, project administration, supervision, and writing—review and editing, H.Y.; conceptualization, investigation, software, visualization, and writing—original draft, Z.Z.; Funding acquisition, resources, K.K. All authors have read and agreed to the published version of the manuscript.

## Funding

This work was supported by MITSUBA Co., Ltd.

Not applicable.

Not applicable.

## Data Availability Statement

Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

## Acknowledgments

We thank Stuart Jenkinson for editing a draft of this manuscript.

## Conflicts of Interest

The authors declare no potential conflict of interest with respect to the research, authorship, and/or publication of this article.

## Nomenclature

 $m 0$ weight of the head $m 1$ weight of the first link $m 2$ weight of the second link $I 1$ moment of inertia of the first link about the center of gravity $I 1$ moment of inertia of the second link about the center of gravity $l 0$ original length of the head spring $l 1$ length of the first link $l 2$ length of the second link $l g 1$ distance from the top to the center of gravity of the first link $l g 2$ distance from the top to the center of gravity of the second link $k 0$ spring constant of the head $k 11$ rotation stiffness of the first link without shoulder contact $k 12$ rotation stiffness of the first link with shoulder contact $k 2$ rotation stiffness of the second link $c 0$ damping of the head $c 1$ damping of the first link $c 2$ damping of the second link a amplitude of the oscillator $ω$ frequency of the oscillator $h d$ initial compression of the head spring $θ$ angle of the first link $φ$ angle of the second link $v 3$ displacement along the y-axis of the tip of the wiper blade $θ c$ angle of shoulder contact $μ d$ coefficient of dynamic friction $μ m a x$ maximum static friction $α$ y-direction coordinate of the tip after transition from the slip state to the stick state N normal force acting on the tip f friction force acting on the tip

## 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 A1.
Figure A1. Free body diagram of the model.
Figure A1. Free body diagram of the model.
The equations of motion for each part are as follows:
$m 1 u ¨ 1 = − F u 1 + F u 2 ,$
$m 1 v ¨ 1 = − F v 1 + F v 2 ,$
$m 2 u ¨ 2 = − F u 2 − N ,$
$m 2 v ¨ 2 = − F v 2 + f .$
Combining Equations (A1) and (A2) and eliminating the internal forces through Equations (A3)–(A6), we obtain
The equations of rotation of the two links are as follows:
The distance from the origin to the wiping surface is fixed at a certain constant value. This distance, named C, is defined as
The variation of the length of the spring $k 0$ can also be expressed as
Thus, the restoring force $F s$ and damping force $F d$ can be expressed as
The wiper is driven by the reciprocating motion of the head. The reciprocating motion is given as a simple harmonic oscillation of frequency $ω$ and amplitude a, i.e.,
The geometric relationships of each coordinate are as follows:
$u = C − l 1 cos θ − l 2 cos φ ,$
$u 1 = C − l 1 cos θ − l 2 cos φ + l g 1 cos θ ,$
$u 2 = C − l 2 cos φ + l g 2 cos φ ,$
Substituting Equations (A12), (A13), and (A15)–(A20) into Equations (A7)–(A9), we obtain Equations (2)–(4).

## Appendix B. Elements of Matrices B and Q

$b 11 = − m d l 1 sin θ cos θ 0 + m a l g 1 sin ( θ + θ 0 ) + m b l 1 tan θ 0 cos ( θ + θ 0 ) ,$
$b 12 = − l 2 sin φ cos θ 0 + m b l g 2 sin ( φ + θ 0 ) ,$
$b 13 = − cos θ 0 ,$
$b 14 = − sin θ 0 ,$
$b 23 = − l 1 sin θ ,$
$b 24 = − l 1 cos θ ,$
$b 33 = − l 2 sin φ ,$
$b 34 = − l 2 cos φ ,$

## Appendix C. Experimental Identification of the Parameters Expressing the Stiffness and Damping in the Analytical Model

The parameters related to the rotational stiffness and damping of the two links at the two joints in the analytical model were experimentally identified by using the free oscillations of the actual wiper blade.
First, we set the wiper blade upside down, as shown in Figure A2. We experimentally obtained the free oscillations in the cases when the neck part is free and fixed. These two cases correspond to the states where only the first link and only the second one can be rotated, respectively.
Figure A2. Two fixation methods of the experimental setup. (a) The neck can be rotated. (b) The neck is fixed.
Figure A2. Two fixation methods of the experimental setup. (a) The neck can be rotated. (b) The neck is fixed.
The free vibrations were recorded using a high-speed video camera. Then, the markers made on the wiper blade in advance were traced. From the tracking data, the angle variation of the free vibrations in both cases can be obtained. In turn, the natural frequency and damping ratio of the free vibrations can be derived. The inverted wiper blade can be regarded as the link vibrating about a point as shown in Figure A3.
Figure A3. Model of inverted wiper blade. (a) Static inverted state. (b) Rotating state.
Figure A3. Model of inverted wiper blade. (a) Static inverted state. (b) Rotating state.
The mass of the link is m. The moment of inertia of this rigid body around the origin is I. The distance from the center of gravity of this rigid body to the origin is $l g$. The rotation stiffness and damping of this link is k and c, respectively. The equation of motion for this system is
$I θ ¨ + c θ ˙ + k θ − m g l g sin θ = 0 .$
This equation can be rewritten as
$θ ¨ + 2 ζ ω n θ ˙ + ω n 2 θ = 0 ,$
where $ζ$ and $ω n$ are, respectively,
$ω n = k − m g l g I .$
When $ζ < 1$, the solution of Equation (A40) is
$θ = A e − ζ ω n t cos ( ω d t + φ ) ,$
where $ω d = ω n 1 − ζ 2$. A and $φ$ in Equation (A43) are determined by the initial condition. The solution corresponds to the time history of the experimentally obtained free oscillation. Since the experimentally obtained damping ratio is about 0.1, the frequency of damped free vibration $ω d$ is almost the same as the natural frequency $ω n$. Regarding the experimentally obtained frequency as $f n$, the following equation can be obtained:
$ω n = 2 π f n = k − m g l g I .$
The rotation stiffness k can be derived from Equation (A44).
$k = ( 2 π f n ) 2 I + m g l g .$
Substituting Equation (A44) into Equation (A41) yields
In turn, the damping c can be derived as
$c = 4 π f n I ζ .$

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Figure 1. Cross-section of wiper system with attack angle. (a) Shoulder stays away from the head; (b) shoulder contacts the head.
Figure 1. Cross-section of wiper system with attack angle. (a) Shoulder stays away from the head; (b) shoulder contacts the head.
Figure 2. Two-link model of wiper system. (a) Upright state; (b) buckled state.
Figure 2. Two-link model of wiper system. (a) Upright state; (b) buckled state.
Figure 3. Restoring moment of the first link. $k 11$ and $k 12$ are the rotation stiffness of the shoulder with or without head contact, respectively. $θ c$ is the angle at which the shoulder begins to contact the head.
Figure 3. Restoring moment of the first link. $k 11$ and $k 12$ are the rotation stiffness of the shoulder with or without head contact, respectively. $θ c$ is the angle at which the shoulder begins to contact the head.
Figure 4. Relationship between coefficient of friction and relative velocity of wiper blade and wiping surface.
Figure 4. Relationship between coefficient of friction and relative velocity of wiper blade and wiping surface.
Figure 5. Exact state transition time and discrete calculation time. $t e$: exact transition time; $t n − 1$, $t n$, $t n + 1$: discrete calculation times.
Figure 5. Exact state transition time and discrete calculation time. $t e$: exact transition time; $t n − 1$, $t n$, $t n + 1$: discrete calculation times.
Figure 6. Process of slack variable method.
Figure 6. Process of slack variable method.
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.
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.
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 8. Enlarged view of Figure 7c,f from 1.64–1.72 s. Panels (a) and (b) correspond to Figure 7c,f, respectively.
Figure 9. Experimental apparatus (a) without attack angle and (b) with an 18-degree attack angle.
Figure 9. Experimental apparatus (a) without attack angle and (b) with an 18-degree attack angle.
Figure 10. Enlarged view of the wiper blade.
Figure 10. Enlarged view of the wiper blade.
Figure 11. Experimental results during wiping process with zero and nonzero attack angles. (a) Angular displacements of two links with zero attack angle. (b) Angular displacements of two links with 10-degree attack angle. (c) Angular displacements of two links with 18-degree attack angle. (d) Variation of y-coordinate of the tip with zero attack angle. (e) Variation of y-coordinate of the tip with 10-degree attack angle. (f) Variation of y-coordinate of the tip with 18-degree attack angle.
Figure 11. Experimental results during wiping process with zero and nonzero attack angles. (a) Angular displacements of two links with zero attack angle. (b) Angular displacements of two links with 10-degree attack angle. (c) Angular displacements of two links with 18-degree attack angle. (d) Variation of y-coordinate of the tip with zero attack angle. (e) Variation of y-coordinate of the tip with 10-degree attack angle. (f) Variation of y-coordinate of the tip with 18-degree attack angle.
Figure 12. Enlarged view of Figure 11c,f from 0.09–0.1 s. Panels (a) and (b) correspond to Figure 11c,f, respectively.
Figure 12. Enlarged view of Figure 11c,f from 0.09–0.1 s. Panels (a) and (b) correspond to Figure 11c,f, respectively.
Figure 13. Experimental results around the reversal point with zero and nonzero attack angles. (a) Angular displacements of two links with zero attack angle. (b) Angular displacements of two links with 10-degree attack angle. (c) Angular displacements of two links with 18-degree attack angle. (d) Variation of y-coordinate of the tip with zero attack angle. (e) Variation of y-coordinate of the tip with 10-degree attack angle. (f) Variation of y-coordinate of the tip with 18-degree attack angle.
Figure 13. Experimental results around the reversal point with zero and nonzero attack angles. (a) Angular displacements of two links with zero attack angle. (b) Angular displacements of two links with 10-degree attack angle. (c) Angular displacements of two links with 18-degree attack angle. (d) Variation of y-coordinate of the tip with zero attack angle. (e) Variation of y-coordinate of the tip with 10-degree attack angle. (f) Variation of y-coordinate of the tip with 18-degree attack angle.
Table 1. Parameters and values used in the program.
Table 1. Parameters and values used in the program.
ParameterValueUnits
$m 0$$1.01 × 10 − 2$$kg$
$m 1$$6.4 × 10 − 5$$kg$
$m 2$$1.16 × 10 − 5$$kg$
$I 1$$6.8 × 10 − 11$$kg m 2$
$I 2$$2.8 × 10 − 12$$kg m 2$
$l 0$$2 × 10 − 2$$m$
$l 1$$3.15 × 10 − 3$$m$
$l 2$$1.53 × 10 − 3$$m$
$l g 1$$1.37 × 10 − 3$$m$
$l g 2$$7.64 × 10 − 4$$m$
$k 0$$3.92 × 10 1$$kg / s 2$
$k 11$$7.1 × 10 − 4$$kg m 2 / s 2$
$k 12$$2.13 × 10 − 2$$kg m 2 / s 2$
$k 2$$1.4 × 10 − 3$$kg m 2 / s 2$
$c 0$$8.2 × 10 − 2$$kg m / s$
$c 1$$1.825 × 10 − 7$$kg m 2 / s$
$c 2$$5.451 × 10 − 8$$kg m 2 / s$
a$5 × 10 − 2$$m$
$ω$2 × $π$$/ 2.5$$rad / s$
$h d$$5 × 10 − 3$$m$
A$0.4$
B5$s / m$
E$0.2$
$β 1$$0.01$
$β 2$$0.01$
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MDPI and ACS Style

Zhao, Z.; Yabuno, H.; Kamiyama, K. Dynamic Analysis of a Wiper Blade in Consideration of Attack Angle and Clarification of the Jumping Phenomenon. Appl. Sci. 2022, 12, 4112. https://doi.org/10.3390/app12094112

AMA Style

Zhao Z, Yabuno H, Kamiyama K. Dynamic Analysis of a Wiper Blade in Consideration of Attack Angle and Clarification of the Jumping Phenomenon. Applied Sciences. 2022; 12(9):4112. https://doi.org/10.3390/app12094112

Chicago/Turabian Style

Zhao, Zihan, Hiroshi Yabuno, and Katsuya Kamiyama. 2022. "Dynamic Analysis of a Wiper Blade in Consideration of Attack Angle and Clarification of the Jumping Phenomenon" Applied Sciences 12, no. 9: 4112. https://doi.org/10.3390/app12094112

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