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

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## Abstract

**:**

## 1. Introduction

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

#### 2.1. Analytical Two-Link Model

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

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

## 3. Numerical Calculation Method in Different States

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

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

## 4. Behavior of Wiper Blade

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

#### 4.2. Conditions of State Transitions

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

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

#### 4.2.3. Transition of the Rotational Stiffness

#### 4.3. Numerical Calculation Results

## 5. Experimental Results

#### 5.1. Experimental Apparatus and Procedure

#### 5.2. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## 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 |

${l}_{g1}$ | distance from the top to the center of gravity of the first link |

${l}_{g2}$ | 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 |

$\omega $ | frequency of the oscillator |

${h}_{d}$ | initial compression of the head spring |

$\theta $ | angle of the first link |

$\phi $ | angle of the second link |

${v}_{3}$ | displacement along the y-axis of the tip of the wiper blade |

${\theta}_{c}$ | angle of shoulder contact |

${\mu}_{d}$ | coefficient of dynamic friction |

${\mu}_{max}$ | maximum static friction |

$\alpha $ | 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

## Appendix B. Elements of Matrices B and Q

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

**Figure A2.**Two fixation methods of the experimental setup. (

**a**) The neck can be rotated. (

**b**) The neck is fixed.

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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 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. ${\theta}_{c}$ 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 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 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 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 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.

Parameter | Value | Units |
---|---|---|

${m}_{0}$ | $1.01\times {10}^{-2}$ | $\mathrm{kg}$ |

${m}_{1}$ | $6.4\times {10}^{-5}$ | $\mathrm{kg}$ |

${m}_{2}$ | $1.16\times {10}^{-5}$ | $\mathrm{kg}$ |

${I}_{1}$ | $6.8\times {10}^{-11}$ | ${\mathrm{kg}\mathrm{m}}^{2}$ |

${I}_{2}$ | $2.8\times {10}^{-12}$ | ${\mathrm{kg}\mathrm{m}}^{2}$ |

${l}_{0}$ | $2\times {10}^{-2}$ | $\mathrm{m}$ |

${l}_{1}$ | $3.15\times {10}^{-3}$ | $\mathrm{m}$ |

${l}_{2}$ | $1.53\times {10}^{-3}$ | $\mathrm{m}$ |

${l}_{{g}_{1}}$ | $1.37\times {10}^{-3}$ | $\mathrm{m}$ |

${l}_{{g}_{2}}$ | $7.64\times {10}^{-4}$ | $\mathrm{m}$ |

${k}_{0}$ | $3.92\times {10}^{1}$ | $\mathrm{kg}{/\mathrm{s}}^{2}$ |

${k}_{11}$ | $7.1\times {10}^{-4}$ | ${\mathrm{kg}\mathrm{m}}^{2}{/\mathrm{s}}^{2}$ |

${k}_{12}$ | $2.13\times {10}^{-2}$ | ${\mathrm{kg}\mathrm{m}}^{2}{/\mathrm{s}}^{2}$ |

${k}_{2}$ | $1.4\times {10}^{-3}$ | ${\mathrm{kg}\mathrm{m}}^{2}{/\mathrm{s}}^{2}$ |

${c}_{0}$ | $8.2\times {10}^{-2}$ | $\mathrm{kg}\mathrm{m}/\mathrm{s}$ |

${c}_{1}$ | $1.825\times {10}^{-7}$ | ${\mathrm{kg}\mathrm{m}}^{2}/\mathrm{s}$ |

${c}_{2}$ | $5.451\times {10}^{-8}$ | ${\mathrm{kg}\mathrm{m}}^{2}/\mathrm{s}$ |

a | $5\times {10}^{-2}$ | $\mathrm{m}$ |

$\omega $ | 2 × $\pi $$/2.5$ | $\mathrm{rad}/\mathrm{s}$ |

${h}_{d}$ | $5\times {10}^{-3}$ | $\mathrm{m}$ |

A | $0.4$ | |

B | 5 | $\mathrm{s}/\mathrm{m}$ |

E | $0.2$ | |

${\beta}_{1}$ | $0.01$ | |

${\beta}_{2}$ | $0.01$ |

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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