Mathematical Formulae for the Vibration Frequencies of Rubber Wiper on Windshield
- 1 -
Mathematical Formulae for the Vibration Frequencies of Rubber Wiper on Windshield
Ying-Ji Hong and Tsai-Jung Chen Department of Mathematics, National Cheng-Kung University, Taiwan Department of Vehicle Engineering, National Pingtung University of Science and Technology, Taiwan Corresponding author: Ying-Ji Hong, Department of Mathematics, National Cheng-Kung University, No.1, University Road, Tainan City 70101, Taiwan Email: [email protected] Abstract
Automotive engineers want to reduce the noise generated by the vibrations of rubber wiper on the windshield of an automobile. To understand the vibrations of wiper noise, certain spring-mass models were presented by some specialists, over the past few years, to simulate the vibrations of rubber wiper on windshield. In this article, we will give precise mathematical formulae for the vibration frequencies of rubber wiper on windshield. Comparison of our model predictions with experimental data confirms the accuracy of our mathematical formulae for the vibration frequencies of wiper on windshield. In fact, our model predictions are in almost perfect agreement with experimental data. These mathematical formulae for the vibration frequencies of rubber wiper on windshield are derived from our analysis of a 3-dimensional elastic model with specific boundary conditions. These specific boundary conditions are set up due to mechanical and mathematical consideration. Our mathematical formulae can be used to test the quality of wiper design.
Keywords
Wiper, friction, sliding, vibration, frequency, rubber, elasticity, mechanics
1. Introduction
Automotive engineers want to reduce the noise generated by the vibrations of rubber wiper on the windshield of an automobile. To understand the vibrations of wiper noise, certain spring-mass models were presented by some specialists, over the past few years, to simulate the vibrations of rubber wiper on windshield (Stein et al., - 2 - denote the density of the rubber wiper. Let l denote the length of the rubber wiper. Our analysis shows that there exist two classes of vibration frequencies of rubber wiper. For the vibration frequencies of Class I, we predict that the vibration frequencies of rubber wiper will locate around Hz22 ln (Class I) (1) where n is a nonnegative integer. For the vibration frequencies of Class II, we predict that the vibration frequencies of rubber wiper will locate around Hz2 ln (Class II). (2) Here and are Lame coefficients. Lame coefficients are material constants of the rubber wiper. These material constants are related to the Young's modulus E and the Poisson's ratio as follows (Chaichian et al., 2012): )21()1( E and )1(2 E . (3) Experimental data highly supports the effectiveness of our model. In fact, our model predictions are in almost perfect agreement with experimental data. - 3 - Figure 1. V ibration frequencies of rubber wiper before reversal. For the material parameters of our rubber wiper, our calculation shows that the vibration frequencies of Class I will locate around 240Hz, 480Hz, 720Hz, 960Hz, 1200Hz, 1440Hz, 1680Hz, 1920Hz, 2160Hz, 2400Hz, etc. Our calculation shows that the vibration frequencies of Class II will locate around n . (33.6Hz). The predicted peaks of frequencies are indicated on the FFT (Fast Fourier Transform) diagram Figure 1 of our experimental data. In this FFT diagram, magnitude of FFT for frequencies below 25,600Hz can be trusted. Details of our calculation will be explained in Section 3.
2. Physical model for the vibrations of rubber wiper on windshield
In this section, we discuss our 3-dimensional physical model for the vibrations of rubber wiper on windshield. We will introduce the Lame system of partial differential equations with specific boundary conditions on a 3-dimensional rectangular solid plate P in Subsection 2.1. There are two wave equations with different wave speeds associated with the Lame system. Assume that the smooth solutions of the wave equation with higher wave speed are collected in Class I. Assume that the smooth solutions of the wave equation with lower wave speed are collected in Class II. It is known that the solutions of Class I and the solutions of Class II can be added to generate all smooth vector-valued solutions of the Lame system (Hetnarski et al., 2004; Hong et al., 2020). On the other hand, it is known that all the smooth solutions of a one-dimensional wave equation can be expressed as the infinite sums of special solutions for this wave - 4 - equation found through the Fourier method (Strauss, 2008; Serov, 2017). Thus, in Subsection 2.2 and Subsection 2.3, we will respectively construct special solutions of Class I and Class II for the Lame system with specific boundary conditions on the rectangular solid plate P . The vibration frequencies of these special solutions of Class I and Class II will be shown explicitly. In Subsection 2.4, we will discuss the physical significance of some important parameters appearing in our construction of the special solutions of Class I and Class II. We will then explain how to arrive at the mathematical formulae (1) and (2) through physical consideration. When a rubber wiper blade moves on the windshield of an automobile, there are 4 different forces acting on it: pressure on the wiper blade, support force from the windshield, drag force, and the (kinetic) friction force. Figure 2 is a 2-dimensional force diagram for the rubber wiper blade.
Figure 2.
Two-dimensional force diagram for the rubber wiper blade on windshield. To understand the vibrations of rubber wiper blade, it is natural to consider the vibrations of wiper lip. Thus we consider, for simplicity, a 3-dimensional rectangular solid plate P with length m l , width m w , and thickness m h . See Figure 3. - 5 - Figure 3.
Rectangular plate with length m l , width m w , and thickness m h . On this hyper-elastic plate, all the forces acting on wiper lip will be considered as distribution of traction/stress force. In fact, it is known that the "friction force" should be considered, from a physical point of view, as "distribution of intermolecular forces" (Yang et al., 2008). Thus the traction/stress distribution on the boundary of our 3-dimensional elastic model could be discontinuous. Let bottom Z , ll Z , and rl Z denote respectively the bottom, left lateral, and right lateral rectangles of the boundary surface of P defined as follows. wxlxhxx bottom Z . (4) xhlxxx ll Z . (5) xhlxxwx rl Z . (6) As shown in Figure 2 and Figure 3, the traction force acts on the rectangles bottom Z , ll Z , and rl Z of the boundary surface of P . Let top Z , anterior Z , and posterior Z denote respectively the top, anterior, and posterior rectangles of the boundary surface of P defined as follows. wxlxxx top Z . (7) xhwxxxl anterior Z . (8) xhwxxx posterior Z . (9) As shown in Figure 3, the traction force does not act on the rectangles top Z , anterior Z , and posterior Z of the boundary surface of P . To determine the vibrations of rubber wiper blade, we consider the vector-valued displacement function - 6 - ),(),,(),,(),( xxxxu tututut with Px ),,( xxx . (10) The dynamics of rubber wiper blade is completely determined by the vector-valued displacement function ),( xu t . This displacement function ),( xu t must satisfy the following vector-valued Lame system (of 3 interdependent partial differential equations) )( xuxuxut u xxx uuu u div)( u (11) (Frankel, 2011; Chaichian et al., 2012). Since the distribution of shear stress on the boundary surface of the solid plate P is not continuous , we will only require, due to mathematical consideration, the following boundary conditions (11), (12), and (13). xuxuxuxu when 0 x . (12) xuxuxuxu when lx . (13) xuxuxuxu when x . (14) Assume that ),(),,(),,(),( xxxxu tututut is the displacement function. We define the component functions of ),( xu t as follows. ),( x tu )sin()sin(cos)sin( xxal xntkln . (15) ),( x tu )sin()cos(sin)sin( xxal xntk . (16) ),( x tu )cos()sin(sin)sin( xxal xntk . (17) Here the set ),,,( kn of parameters must satisfy
222 222 lnk (18) in which n is an integer . It can be checked readily that the vector-valued function ),( xu t defined by (15), (16), and (17) is a solution for (11), (12), (13), and (14). The - 7 - frequency of this solution ),( xu t is lnk . (19) When and are relatively small, we have ln . (20) In general, we may consider ),( xu t ),(),,(),,( xxx tututu defined as follows. ),( x tu )()(cos)sin( xgxafl xntkln . (21) ),( x tu )()('sin)sin( xgxafl xntk . (22) ),( x tu )(')(sin)sin( xgxafl xntk . (23) Here n must be an integer . We discuss the choice of f and g in what follows. ● Case I : sin f and sin g . In this case we arrive at the displacement function ),( xu t defined by (15), (16), and (17), with the set ),,,( kn of parameters satisfying (18). ● Case II : sin f and sinh g . In this case the set ),,,( kn of parameters must satisfy
222 222 lnk . (24) With (24) being satisfied, it can be checked readily that the vector-valued function ),( xu t defined by (21), (22), and (23) is a solution for (11), (12), (13), and (14). The frequency of this solution is lnk . (25) ● Case III : sinh f or cosh f with sin g .In this case the set ),,,( kn of parameters must satisfy
222 222 lnk . (26) With (26) being satisfied, it can be checked readily that the vector-valued function - 8 - ),( xu t defined by (21), (22), and (23) is a solution for (11), (12), (13), and (14). The frequency of this solution is lnk . (27) ● Case IV : sinh f or cosh f with sinh g . In this case the set ),,,( kn of parameters must satisfy
222 222 lnk . (28) With (28) being satisfied, it can be checked readily that the vector-valued function ),( xu t defined by (21), (22), and (23) is a solution for (11), (12), (13), and (14). The frequency of this solution is l nk . (29) When and are relatively small, we have the following estimate for the vibration frequency of ),( xu t : ln (30) for all the cases discussed above. Remark 1.
When the requirement k is not satisfied in any of the last three cases,
222 222 lnk , (31) so that k is purely imaginary , the displacement function ),( xu t defined by (21), (22), and (23) is still a solution for (11), (12), (13), and (14). However, the corresponding displacement function ),( xu t would grow exponentially and could lead to failure of rubber wiper. Assume that ),(),,(),,(),( xxxxu tututut is the displacement function. We define the component functions of ),( xu t as follows. ),( x tu )sin()sin(cos)sin( xxal xntk . (32) - 9 - ),( x tu )sin()cos(sin)sin( xxal xntk . (33) ),( x tu )cos()sin(sin)sin( xxal xntk . (34) Here the sets ),,( and ),,,( kn of parameters must satisfy respectively ln (35) and
222 222 lnk (36) in which n is an integer . It can be checked readily that the vector-valued function ),( xu t defined by (32), (33), and (34) is a solution for (11), (12), (13), and (14). The frequency of this solution is lnk . (37) In general, we may consider ),( xu t ),(),,(),,( xxx tututu defined as follows. ),( x tu )()(cos)sin( xgxafl xntk . (38) ),( x tu )()('sin)sin( xgxafl xntk . (39) ),( x tu )(')(sin)sin( xgxafl xntk . (40) Here n must be an integer . We discuss the choice of f and g in what follows. ● Case I : sin f and sin g . In this case we arrive at the displacement function ),( xu t defined by (32), (33), and (34), with the sets ),,( and ),,,( kn of parameters respectively satisfying (35) and (36). ● Case II : sin f and sinh g . In this case the sets ),,( and ),,,( kn of parameters must respectively satisfy ln and
222 222 lnk . (41) With (41) being satisfied, it can be checked readily that the vector-valued function ),( xu t defined by (38), (39), and (40) is a solution for (11), (12), (13), and (14). The frequency of this solution is - 10 - lnk . (42) ● Case III : sinh f or cosh f with sin g . In this case the sets ),,( and ),,,( kn of parameters must respectively satisfy 0 ln and
222 222 lnk . (43) With (43) being satisfied, it can be checked readily that the vector-valued function ),( xu t defined by (38), (39), and (40) is a solution for (11), (12), (13), and (14). The frequency of this solution is lnk . (44) ● Case IV : sinh f or cosh f with sinh g . In this case the sets ),,( and ),,,( kn of parameters must respectively satisfy 0 ln and
222 222 lnk . (45) With (45) being satisfied, it can be checked readily that the vector-valued function ),( xu t defined by (38), (39), and (40) is a solution for (11), (12), (13), and (14). The frequency of this solution is lnk . (46) When and are relatively small, we have the following estimate for the vibration frequency of ),( xu t : ln (47) for all the cases discussed above. Remark 2.
When the requirement k is not satisfied in any of the last three cases,
222 222 lnk (48) - 11 - so that k is purely imaginary , the displacement function ),( xu t defined by (38), (39), and (40) is still a solution for (11), (12), (13), and (14). However the corresponding displacement function ),( xu t would grow exponentially and could lead to failure of rubber wiper. It has been explained in Remarks 1 and 2 that unfavorable solutions of Class I or Class II might lead to malfunctioning of rubber wiper on windshield. Now we discuss the physical significance of the parameters and appearing in the construction of soultions of Class I defined by (15), (16), (17), and (18). Assume that ),(),,(),,(),( xxxxu tututut is a displacement function of Class I defined by (15), (16), (17), and (18). It can be shown, through calculation using the Elasticity Mechanics (Frankel, 2011; Chaichian et al., 2012), that the ),( xx shear stress of this solution is )cos()cos(sin)sin( xxal xntk . (49) It can be observed readily that, when and are large, the variation of the distribution of ),( xx shear stress on the wiper lip becomes large. This means that the friction force on the wiper lip becomes large, if the pressure on wiper, the drag force, and the support force from the windshield remain unchanged. Similarly, it can be shown, through calculation using the Elasticity Mechanics, that the ),( xx shear stress of this solution is )cos()sin(cos)sin( xxal xntkln . (50) Besides, the ),( xx shear stress of this solution is )sin()cos(cos)sin( xxal xntkln . (51) It can be observed readily that, when and are large, the friction force on the wiper lip becomes more uneven , if the pressure on wiper, the drag force, and the support force from the windshield remain unchanged. Thus we conclude that, when the friction force acting on the wiper lip is not highly irregular, the vibration frequencies of solutions of Class I, discussed in Subsection 2.2, would locate around ln . (52) - 12 - Similarly we expect that the vibration frequencies of solutions of Class II, discussed in Subsection 2.3, would locate around ln . (53)
3. Comparison of model predictions with experimental data
The material constants of our rubber wiper are listed as follows: /10 mKg , Pa105 E , and . (54) The length of our rubber wiper is 24" ≒ Hz240Hz221 l and Hz6.33Hz21 l . (55) Thus the vibration frequencies of Class I should locate around Hz240 n = n . Our experiments are made for rubber wiper on slightly wet windshield. Since the dynamics of rubber wiper around reversal, where the wiper reverses its moving direction, is much complicated, we only compare our model predictions with the experimental data of the vibration frequencies of rubber wiper away from reversal. Assume that the wiper starts a "side-to-side cycle" at time T . Assume that the wiper reverses its moving direction at time T . Assume that the wiper comes back to its starting position at time T , to start another cycle. See Figure 4. - 13 - Figure 4.
Frequencies recorded depending on the time variable. Let TTA . The FFT (Fast Fourier Transform) diagram of the fequencies recorded between AT )25.0( and AT )75.0( is shown in Figure 5. In this FFT diagram Figure 5, magnitude of FFT for frequencies below 25,600Hz can be trusted. It can be observed in Figure 5 that the predicted peaks of frequencies, 240Hz, 480Hz, 720Hz, 960Hz, 1200Hz, 1440Hz, 1680Hz, 1920Hz, 2160Hz, 2400Hz of Class I, appear manifestly. - 14 - Figure 5.
Vibration frequencies of wiper before reversal. Figure 6 shows the FFT diagram of background /machine/white noise recorded between AT )25.0( and AT )75.0( . In Figure 6, magnitude of FFT for frequencies below 25,600Hz can be trusted. Figure 6.
Frequencies of "background/machine/white noise" before reversal. Let TTB . The FFT diagram of the frequencies recorded between - 15 - BT )25.0( and BT )75.0( is shown in Figure 7. In this FFT diagram Figure 7, magnitude of FFT for frequencies below 25,600Hz can be trusted. In Figure 7, the predicted peaks of frequencies, 33.6Hz, 240Hz, 720Hz, 1440Hz, 1680Hz, appear manifestly. Here the highest peak appears at the frequency 33.6Hz of Class II. Figure 7.
Vibration frequencies of wiper after reversal. Figure 8 is the FFT diagram of "background/machine/white noise" recorded between BT )25.0( and BT )75.0( . In Figure 8, magnitude of FFT for frequencies below 25,600Hz can be trusted. - 16 - Figure 8.
Frequencies of "background/machine/white noise" after reversal.
4. Conclusion and discussions
Comparison of our model predictions with experimental data shows the accuracy of our mathematical formulae for the vibration frequencies of rubber wiper on windshield. As is explained in Subsection 2.4, our model predictions will be in excellent agreement with experimental data, when the friction force acting on the wiper lip is not highly irregular. It turns out that our mathematical formulae can be used to test the quality of wiper design on the evenness of friction force. Further implications of our analysis will appear elsewhere.
Acknowledgements
We are very grateful to Professors Chien-Hsiung Tsai, Hsi-Wei Shih, Min-Hung Chen, Yu-Chen Shu, and Chyuan-Yow Tseng for many stimulating discussions.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or - 17 - publication of this article.
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