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June, 2021 Geometric Analysis of the Vibration of Rubber Wiper Blade
Tsai-Jung Chen, Ying-Ji Hong
Taiwanese J. Math. 25(3): 491-516 (June, 2021). DOI: 10.11650/tjm/201206

## Abstract

The purpose of this paper is to work out the theoretical aspects of the vibration problem of rubber wiper blade on convex windshield. Over the past 20 years, some $2$-dimensional spring-mass models were presented in engineering science to simulate the vibration of rubber wiper blade on windshield. In this paper, we will consider the elasticity perspective on this $3$-dimensional vibration problem. Our theoretical analysis suggests that there should exist two classes of vibration frequencies corresponding to “$\ast$-exact deformations (Class I)” and “$\ast$-closed deformations (Class II)”. We prove mathematical theorems on the characterization of deformations of Class I. We also explain how elementary deformations of Class II can be constructed. We then deduce two mathematical formulas, for the vibration problem of rubber wiper blade on convex windshield, from our theoretical analysis. Our theoretical predictions are in almost perfect agreement with experimental data. One of the crucial steps of our analysis is a decomposition theorem motivated by the de Rham Cohomology and the Hodge Theory.

## Acknowledgments

The authors are indebted to the reviewer of this paper for providing insightful comments and important suggestions on an earlier version of this paper.

## Citation

Tsai-Jung Chen. Ying-Ji Hong. "Geometric Analysis of the Vibration of Rubber Wiper Blade." Taiwanese J. Math. 25 (3) 491 - 516, June, 2021. https://doi.org/10.11650/tjm/201206

## Information

Received: 15 November 2020; Revised: 2 December 2020; Accepted: 3 December 2020; Published: June, 2021
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.11650/tjm/201206

Subjects:
Primary: 35J05, 53C07, 58A14, 70S05

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 3 • June, 2021