Bifurcation phenomena of the reversal behavior of an automobile wiper blade

Abstract

Reducing the noise generated by windshield wipers during reversal is desirable. As a fundamental first step in exploring the cause of this noise, the present study attempts to theoretically and experimentally clarify the dynamics of the behavior of the wiper blade near the reversal point. First, an experiment is conducted to observe the reversal behavior using the cross sectional model of the blade rubber. In order to theoretically explain the experimentally observed phenomena, an analytical link model of the wiper blade is introduced. The reversal behavior of a blade is theoretically investigated through the bifurcation analysis by considering Coulomb’s friction during reversal. Then we show the continuous variation of the angle of blade rubber, and predict a rapid variation of the normal force, which can cause reversal noise, acting to the blade rubber.

Appendices

Appendix A: Dimensional form of the equation of translation and the constraint equation

In Sect. 3.1.2, the dimensional form of the dimensionless equation of translation (7) is expressed as follows:

(30)

The dimensional form of the dimensionless constraint equation (9) is expressed as follows:

$$ x-l_s=l-l\cos{\theta}-d.$$

(31)

Appendix B: Geometric relationship between θ and ζ

The geometric relationship between θ and ζ, mentioned in Sect. 3.3, is determined when the blade reaches the endpoint of reciprocating motion in the positive y direction. Figure 12 shows the geometric relationship between θ and ζ at the y axis edge of reciprocating motion of wiper. In Fig. 12, the rigid bar represents the lip of the blade rubber. The tip of the rigid bar is rest during reversal as mentioned in Sect. 2.3. The purple arrow and light blue arrow in Fig. 12 correspond to the purple line and light blue line in Fig. 6, respectively. The purple dot represents the support side of the blade at the time when the blade starts to reverse. The orange dot represents the support side of the blade at the time when the blade is reversing. The difference between $y_{0}(\frac{\pi}{2\nu}+\frac{2n\pi}{\nu})$ and y ₀(t) is expressed using the angle of blade θ and θ ₁ as follows:

(32)

(33)

$\sin{(\frac{\pi}{2}+2n\pi)}$ in (33) is equal to 1, where n is integer, because the blade arrives at the endpoint of reciprocating motion in the positive y direction. Substituting $\sin{(\frac{\pi}{2}+2n\pi)}=1$ in (33), sinθ is expressed as follows:

$$ \everymath{\displaystyle}\begin{array}{l}\zeta(1-\sin{\nu t}) = \sin{(-\theta_1)}-\sin{(-\theta)} \\\sin{\theta} = \sin{\theta_1}+\zeta(1-\sin{\nu t}).\end{array}$$

(34)

Approximating the angle sinθ and sinθ ₁ by θ and θ ₁, respectively, (34) is expressed as follows:

$$ \theta=\theta_1+\zeta(1-\sin{\nu t}).$$

(35)

Fig. 12

(Color online) Geometric relationship between θ and ζ at the endpoint of reciprocating motion in the positive y direction. (The purple arrow and light blue arrow correspond to the purple line and light blue line in Fig. 6, respectively. The purple dot represents the support side of the blade at the time when the blade starts to reverse. The orange dot represents the support side of the blade during reversal)

Appendix C: Experimental method for finding the parameter used in the analytical model

In Fig. 8, the line density of the blade rubber m, the mass of the head of the blade rubber M, and the length of the lip of the blade rubber l are obtained from the actual measurement of the blade rubber. The spring constant of the vertical spring k ₁ is obtained from the actual measurement of the vertical spring. The spring constant of the rotational spring (it represents the stiffness of the blade) k ₂ and the coefficient of viscous damping c ₂ (it represents the damping of the blade rubber) are determined from the free vibration experiment of the blade rubber. The displacement of the swept surface d is obtained from the initial displacement of the vertical spring d ₀ mentioned in Sect. 2.2.

Appendix D: Reversal noise

We experimentally observe the relationship between the behavior of the blade and the noise as Fig. 13. It represents the behavior of the blade and the magnitude of the frequency component of the sound at each time, where the vertical and horizontal axes show the frequency of the sound, and the time, respectively. The magnitude of the frequency component is expressed by the colors. The high spectrum is colored red. Figure 13 confirms that the wiper generates the noise when the blade is erect. On the other hand, the main frequency component in the above experimental result corresponds to the natural frequency of the windshield and it is indicated in [1] that the reversal noise is caused from the impulsive force. Figure 11 derived by the link model in this study implies the generation of the impulsive force in the erection of the blade.

Fig. 13

(Color online) Experimentally observed relationship between the behavior of the blade and the noise. (The magnitude of the frequency component is expressed by the colors. The high spectrum is colored red)

Appendix E: Stability of the equilibrium angle of blade

Here, we analyze the stability around the equilibrium angle of blade. We estimate the order of each term in the left hand side of (24). Assuming that the angle of blade θ=Asinνt, where A is constant, the order of $\ddot{\theta}$ and $2\kappa\dot{\theta}$ are Aν ² and Aκν, respectively. In the present study, since ν is small enough, the order of $\ddot{\theta}$ and $2\kappa\dot{\theta}$ are smaller than that of (1−αδ)θ. Thus, $\ddot{\theta}$ and $\dot{\theta}$ are neglected. Also, assuming the frictionless condition in (24), $-\mu\alpha\delta \mbox{Sign}(\dot{y}_{1})$ is zero. In this condition, the angle determined by the balance between the restoring force of the blade rubber and the inertial force due to the reciprocating motion of the arm is defined as Θ. Θ is described as follows:

$$\everymath{\displaystyle}\begin{array}{l} (1-\alpha\delta){\varTheta}-\frac{3}{2}\zeta\nu^2\sin{\nu t}=0, \\{\varTheta}=\frac{3\zeta\nu^2}{2(1-\alpha\delta)}\sin{\nu t}.\end{array}$$

(36)

On the other hand, Δθ is defined as (37) from the difference between the angle of blade θ and the angle determined by the inertial force of the arm in the frictionless condition Θ.

$$ {\varDelta}\theta=\theta-{\varTheta}.$$

(37)

Δθ expresses the behavior due to the frictional force. We discuss the equilibrium states of Δθ and their stability. The equilibrium states of Δθ is expressed as follows:

$$ \everymath{\displaystyle}\begin{array}{l}(1-\alpha\delta){\varDelta}\theta=-\mu\alpha\delta \mbox{Sign}(\dot{y}_1), \\|{\varDelta}\theta|<\frac{-\mu\alpha\delta}{|(1-\alpha\delta)|}.\end{array}$$

(38)

The second term of (24), which represents the damping of blade, is neglected because of smallness of κ. Substituting (37) and (36) into (24), the linear differential equation about Δθ is derived as follows:

$$ {\varDelta}\ddot{\theta}+(1-\alpha\delta){\varDelta}\theta=-\mu\alpha\delta \mbox{Sign}(\dot{y}_1).$$

(39)

From (6), the linearized horizontal displacement of the tip of blade and the linearized velocity of the tip of blade are expressed as follows, respectively:

(40)

(41)

Substituting (37) and (41) into (39), and neglecting the velocity due to the reciprocating motion of the arm ζνcosνt, the following equation is obtained,

$$ {\varDelta}\ddot{\theta}+(1-\alpha\delta){\varDelta}\theta=-\mu\alpha\delta \mbox{Sign}({\varDelta}\dot{\theta}).$$

(42)

Using (28), the angle of the blade and the angular velocity of the blade during reversal are obtained as follows:

(43)

(44)

Based on (42), (43), and (44), the phase plane of (42) is given in Fig. 14. The red line indicates the region of the equilibrium solution of (42). The yellow-green line indicates the geometric relationship between Δθ and ${\varDelta}\dot{\theta}$ during reversal from (43) and (44). The light blue arrows show the vector field on the phase plane. The black lines show the integral curves of the solution of (42). The state variables in the hatched region are absorbed in the equilibrium region indicated by the red line. In this sense, we regard the equilibrium region as stable. The relationship between Δθ and ${\varDelta}\dot{\theta}$ is varied along the yellow-green line during reversal. The state variables on the yellow-green line are absorbed in the equilibrium region, and they do not grow with time.

Fig. 14

(Color online) Phase plane of (42). (The red line indicates the region of the equilibrium solution of (42). The yellow-green line indicates the geometric relationship between Δθ and ${\varDelta}\dot{\theta}$ during reversal. The light blue arrows show the vector field on the phase plane. The black lines show the integral curves of the solution of (42). The state variables in the hatched region are absorbed in the equilibrium region indicated by the red line)

Appendix F: Experimental method for identifying the frictional force acting to the blade

The experimental method is as follows: First, we measure the motion of the blade by the high-speed video camera and the video image analysis. Here, we filmed the reversal behavior, where the frame rate was 6,000 fps, and the resolution was 1,024 pixels × 512 pixels. We used the video image after the reversal. The velocity of $\frac{dy_{1}}{dt}$ is obtained through the central-difference scheme for y ₁(t) and y ₁(t+Δt), where Δt is 1/6000 second. The corresponding frictional force is calculated by (17), of which parameters, $\frac{d\theta}{dt}$, $\frac{d^{2}\theta}{dt^{2}}$, and $\frac{dy}{dt}$, are experimentally identified by the video image analysis through the central-difference scheme similar to the method for $\frac{dy_{1}}{dt}$. The values of m, l, k ₁, d are directly measured. The values of c ₂ and k ₂ are experimentally determined from the free oscillation of the blade rubber.