N UMERICAL SIMULATIONS OF FRICTION - INDUCED NOISE OF AUTOMOTIVE WIPER SYSTEMS
Océane Roure
Master of Science Thesis
Stockholm, Sweden, 2015
N UMERICAL SIMULATIONS OF FRICTION - INDUCED NOISE OF AUTOMOTIVE WIPER SYSTEMS
Océane Roure
Msc Thesis 2014 - 2015
Renault supervisor: Jean-Marc Duffal
KTH supervisors and examiners: Hans Bodén, Peter Göransson, Romain Rumpler
School of Engineering Sciences Royal Institute of Technology
100 44 Stockholm, Sweden
A CKNOWLEDGEMENT
Firstly, I would like to thank my Renault supervisor, Jean-Marc Duffal, for his help, his availability and his guidance all along the thesis. This master thesis allowed me to provide a solid knowledge of vibroacoustic numerical simulations and I am very grateful for that.
I would also like to thank Denis Ricot, as well as, Jean-Pascal Reille, for welcoming me in their respective team and department.
I would like to thank all the people in the “Fluid Mechanics, Vibration and Acoustic” team with whom I had the occasion to work, for their reception, the nice working environment and the good atmosphere they contributed to create in the office.
Besides, I would like to thank my KTH supervisors, Hans Bodén, Peter Göransson and Romain Rumpler, for reviewing my thesis and being my examiners.
A BSTRACT
Automotive parts may be the cause of very annoying friction-induced noise and the source of many customer complaints. Indeed, when a wiper operates on a windshield, vibratory phenomena may appear due to flutter instabilities and may generate squeal noise. As squeal noise generated by wiper system is a random and complex phenomenon, there are only few studies dealing with the wiper noise. The complexity of this phenomenon is due to the cinematic of the movement and to the various environmental parameters which have an influence on the appearance of the noise. This master thesis is a research and development project and presents a numerical simulation methodology used in the aim to reduce and eradicate squeal noise of wiper systems.
In the first part, the finite element model representing a wiper system and the numerical simulation methodology will be presented in detail.
In the second part, stability analysis will be carried out in nominal studies and in designs of experiments. Parametric studies will also be achieved to understand the behavior and the influence of each considered input parameters. Two wiper blades, with the same geometry but with different material, will be considered for the different studies. These two wiper blades will be examined to figure out when squeal noises appear.
Keywords:
Squeal noise, wiper system, numerical simulations, finite element model, mode-coupling, design of experiments, Farmer diagram
T ABLE OF CONTENTS
Acknowledgement ... 5
Abstract ... 6
Table of contents ... 7
Introduction ... 10
1 Overall presentation ... 11
1.1 Presentation of the company ... 11
1.1.1 Key figures ... 11
1.1.2 Master thesis location ... 12
1.2 Presentation of a wiper system ... 13
1.3 State of the art ... 14
1.3.1 Synthesis of previous studies ... 14
1.3.2 Origin of the squeal noise ... 15
2 Modeling ... 19
2.1 Preprocessing ... 19
2.1.1 Finite element model ... 19
2.1.2 Input parameters ... 20
2.1.3 Boundary conditions ... 23
2.2 Processing implementation ... 24
2.2.1 Methodology of calculations ... 24
2.2.2 Abaqus model ... 27
2.3 Verification of the model ... 28
2.3.1 Analytical calculation ... 28
2.3.2 Study of the different configurations ... 30
3 Stability studies ... 32
3.1 Nominal study of the two wipers ... 32
3.1.1 Stability analysis for the wiper A ... 32
3.1.2 Stability analysis for the wiper B ... 33
3.2 Parametric studies ... 35
3.2.1 Friction coefficient ... 35
3.2.2 Attack angle ... 36
3.2.3 Wiper arm load ... 38
3.3 Design of experiments ... 40
3.3.1 Presentation of the design of experiments ... 40
3.3.2 Comparison between the two wipers ... 41
Conclusion ... 46
References ... 47
Table of figures ... 48
List of tables ... 50
Appendix A: List of software ... 51
Appendix B: Customer complaints ... 52
Introduction
Automotive parts, like window seals, sunroofs, wipers, may be the cause of very annoying friction- induced noise like squeal, squeak and creak and the source of many customer complaints. Indeed, as automobiles become increasingly quieter, friction-induced noises become more noticeable by the customer and lead to huge warranty costs for car manufacturers and negative impacts on brand image. In this thesis, a specific automotive part is considered: the wiper system. Currently, experimental tests on noisy wipers are performed to reduce the squeal noise. However, these trial and error approaches are limited due to the little knowledge of the physical behavior of these noises and of the influential factors. Thus, the solutions to fix a noisy wiper are only curative: by changing the wiper geometry or material for example, but no predictive approaches are considered. It is therefore essential for car companies to use numerical simulations to eradicate these perceptible noises encountered in automotive systems by predicting the potential squeal behavior.
The objective of this thesis is to set up a numerical simulation methodology based on finite element model of realistic structures and showing an operation of wiping in the aim to understand and predict the squeal behavior of this automotive system. The relevance of this methodology will be then evaluated using two different wipers: one which experimentally makes a lot of squeal noise and one which experimentally makes less noise.
After presenting the origin of the squeal noise and the wipers used in this thesis, the finite element model and the methodology will be described in detail. A verification of the model will be carried out to check if the model is consistent with a real wiping operation. Then, stability studies will be performed to numerically compare the two wipers. To do that, a nominal study will be carried out with particular values of the considered input parameters. However, after realizing that it is hard to compare the two wipers in this nominal configuration, parametric studies will be achieved to understand the behavior and the influence of each input parameters on the characteristics of the system. Finally, the two wipers will be compared in a design of experiments with full factorial design using the variability of each input parameters.
1 Overall presentation
1.1 PRESENTATION OF THE COMPANY
Renault has been making cars since 1898. Today it is a multinational and multibrand automotive group that sold more than 2.7 million vehicles in 128 countries in 2014. It has 121,807 employees and 37 industrial sites, where it manufactures vehicles and powertrain parts. Directed by Carlos Ghosn, Renault Group is developing three complementary brands: Renault, Dacia and Renault Samsung Motors (RSM). It offers a range of vehicles that meet a variety of mobility requirements and are adapted to the specific demands of the different markets. Renault has diversified into selling both particular cars (87% of sales) and light commercial vehicles (13% of sales).
1.1.1 KEY FIGURES
The global revenues of the group are presented in the FIGURE 1.1. They are the amount generated by the sales of Renault Group automotive products (cars, parts, accessories...), services provided in conjunction with those sales (extended warranties, maintenance contracts, vehicle rentals…) as well as sales of automotive technologies and marketing rights.
FIGURE 1.1 – Renault group revenues (€ million)
The Renault group organizes its international presence into five large regions as shown in the following drawing. The bold figures represent the number of manufactured cars.
FIGURE 1.2 – Renault group global locations 38 971
42 628
41 270
40 932 41 055
2010 2011 2012 2013 2014
Renault Group has sold more than 2.7 million vehicles in 2014. Around 50% of its sales were made outside of Europe. The three main markets are France, Brazil and Russia.
FIGURE 1.3 – Renault group sales by region (€ thousands)
Founded in 1999, the Renault-Nissan Alliance is the world's fourth biggest automotive group behind brands like Toyota, General Motors and Volkswagen. The Alliance has helped Renault and Nissan outperform historic regional rivals, elevating both companies into an elite tier. Together, Renault and Nissan rank globally in the top three car groups.
1.1.2 MASTER THESIS LOCATION
The master thesis was done in the “Fluid Mechanics, Vibration and Acoustic” division which belongs to the “Methods and Models for Numerical Simulation” department. This takes place in the Renault Technocentre at Guyancourt near Paris, which is the biggest R&D center in France. This division is divided into several activities like aeroacoustics or vibroacoustic. It regroups eleven engineers and three PhD students.
FIGURE 1.4 – Hierarchy of the division 2 712
1 465
390 308 133 417
2 628
1 302
412 339 108 467
WORLD EUROPE EURASIA EUROMED
AFRICA
ASIA PACIFIC AMERICAS
Dec 2014 Dec 2013
Fluid Mechanics, Vibrations & Acoustic Methods & Models for Numerical Simulation
Digital mock-up, CAE & PLM CAE & Testing
Alliance Technology Development Competitiveness
RENAULT GROUP
1.2 PRESENTATION OF A WIPER SYSTEM
A wiper system is used to ensure a clear field of vision by removing the rain and debris from the windscreen. A conventional wiper generally consists of an arm, pivoting at one end and attached to the other end to a long rubber blade. The wiper is powered by an electric motor with gear reduction. A wiper blade has a lifetime of around 500,000 cycles, which corresponds to 800km of driving with the wipers.
FIGURE 1.5 – Wiper system
The wiper blades used in this thesis are the front wipers of the Laguna car and are not standard wiper blades but flat wiper blades. These types of blades feature a new style and technology and are becoming the new standard fit on new vehicles. On a conventional wiper blade, the force from the wiper arm is distributed with a series of 6 or 8 linkages, called whippletree, whereas in a flat blade, the force is distributed along the length of the blade with a constant pressure that gives a better clean of the windscreen.
FIGURE 1.6 – Pressure distribution on a) a standard and on b) a flat wiper blade
A flat wiper blade comprises a heel and a lip which are connected to one another by a portion of smaller cross section that forms a hinge. The heel contacts with the wiper arm while the lip surface is in contact with the windscreen. The design of the top of the heel allow for an aerodynamic shape which reduces noise contrary to the standard wiper blade. Two wiper blades, with the same geometry but with different material, will be considered. The first wiper (wiper A) is made of industrial rubber called EPDM, whereas the second wiper (wiper B) is bi-material: its heel and hinge are made of EPDM and its lip is made of natural rubber (NR).
a)
b)
Wiper arm
Wiper blade
FIGURE 1.7 – Design of a flat wiper blade
1.3 STATE OF THE ART
1.3.1 SYNTHESIS OF PREVIOUS STUDIES
There are only few articles dealing with the wiper noise and most of these articles treat of experimental sensitivity studies. Squeal noise generated by wiper system is a random phenomenon.
The system is subject to a lot of environmental parameters like the temperature, the quantity of water on the windscreen, the wear of the wiper blade for example. These parameters have an influence on the appearance of the noise and contribute to the complexity of the phenomenon.
EXPERIMENTAL APPROACH
Experimental studies have been carried out at Renault on wiper systems to identify the frequency ranges and the noise localization. They show that during the raising of the wiper, the frequencies are around 2000Hz whereas during the lowering of the wiper, the frequencies are around 250Hz. The FIGURE 1.8 shows the noise mappings during the two previous phases. It can be seen that during the raising, the noise is located at the top of the windshield; the noise appears consequently at the end of the raising. During the lowering, the noise is located at the bottom of the windshield; the noise appears consequently at the end of the lowering.
a) b)
FIGURE 1.8 – Noise mappings during the raising and the lowering Lowering Raising
Heel
Hinge Lip
Koenen and Sanon works [1] show that some conditions are very favorable to loud squeal noise. With specific values of environmental parameters, squeal noise occurs systematically. The noise can be either transitory or stay during the whole wiping operation. They also show that the quantity of water on the windscreen have an influence on the friction coefficient. The highest occurrence of noise appears during the half-dry phase where the friction coefficient is at its maximum value. The half-dry phase is the wetness condition between the wet and the dry phases. This observation is also perceived by Zhang [2] who has also revealed that the wiper was noisier during the raising phase.
Deleau explains, by a tribological study [3], the consequences of the friction and of the water retention on the vibroacoustic response of a contact between a wiper blade and a windscreen. He shows in his paper that the wiper blade length is not an influent parameter on the squeal noise. He demonstrates that the squeal noise obtained with a wiper blade of 1m is identical to the squeal noise obtained with a wiper blade of 3m. The problem can thus be considered in 2 dimensions. He also experimentally shows that the squeal noise appears for a frequency range between 300Hz and 3000Hz.
NUMERICAL APPROACH
They are few articles dealing with the wiper noise and using a numerical approach. In their respective works, Okura [4], Goto [5] and Grenouillat [6] represent the wiper by a mass-spring system in contact with a rigid plane to improve the comprehension of the phenomenon.
Other works have been carried out using finite element models representative of a realistic geometry of a wiper. For example, Chevennement-Roux et al. [7] calculate complex eigenvalues to study the stability of the system. They show that friction-induced noise is caused by mode-coupling instability at frequencies close to those experimentally observed.
Nowadays, there is no known numerical methodology to study the wiper system phenomenon. This is due to the complexity of the cinematic and the fact that it is a random phenomenon with a lot of environmental parameters. In this thesis, a methodology already employed at Renault will be applied.
This method has been already used to detect the squeal noise on the simple system of window seals.
1.3.2 ORIGIN OF THE SQUEAL NOISE
Friction-induced vibrations can be described by three mechanisms: stick-slip, sprag-slip and coupling modes [8]. In the stick-slip mechanism, the instability is engendered by the succession of adherence and sliding phases. In the sprag-slip mechanism, the instability is due to the variation of normal and tangential contact forces caused by the system geometry and deformation. The mode coupling instability may be defined as the observation of two different modes moving closer to each other until they become equal in the range of a controlled parameter. In the present case, the friction-induced vibrations and so the squeal noise generated by the wiper system is caused by mode-coupling instability, as shown by Chevennement-Roux [7].
To present the mode-coupling instability, the Hoffmann model [9] is considered in the FIGURE 1.9 below. In this self-excited mechanism, a conveyor belt with constant velocity 𝑉 is pushed with a constant normal force 𝐹𝑁 against a block modelled as a point mass 𝑚. The block is held in position by two linear springs 𝑘1 and 𝑘2. The linear spring 𝑘3 represents the normal contact stiffness between the block and the moving belt. A Coulomb-type friction force 𝐹𝐹 with a constant friction coefficient 𝜇, is assumed to take into account sliding friction.
FIGURE 1.9 – Mode coupling model (Hoffmann [9])
The equation of motion for this two degrees of freedom model is:
𝑚 0
0 𝑚 𝑥
𝑦 + 𝑘11 𝑘12 𝑘21 𝑘22 𝑥
𝑦 = 𝐹𝐹
𝐹𝑁 (1.1)
where the coefficients of the stiffness matrix can be obtained from elementary considerations as:
𝑘11 = 𝑘1cos2𝛼1+ 𝑘2cos2𝛼2
𝑘12 = 𝑘21 = 𝑘1cos 𝛼1sin 𝛼1+ 𝑘2cos 𝛼2sin 𝛼2 𝑘22 = 𝑘1sin2𝛼1+ 𝑘2sin2𝛼2+ 𝑘3
(1.2)
By writing the Coulomb relation and considering only small perturbations around the sliding steady- state, the resulting system of equation may be now written as a homogeneous system with non- symmetric stiffness matrix:
𝑚 0
0 𝑚 𝑥
𝑦 + 𝑘11 𝑘12− 𝜇𝑘3 𝑘21 𝑘22 𝑥
𝑦 = 0 (1.3)
This shows that an eigenvalue problem has resulted. In the previous equation, 𝑥 and 𝑦 denote the deviation from the equilibrium sliding steady-state.
By taking the same parameters than Hoffmann [9], which are 𝑚1= 1, 𝛼1= 150°, 𝛼2= 30°, 𝑘1=
2
3 2 − 3 , 𝑘2=2
3 2 + 3 , 𝑘3=4
3 and ∆= 𝜇𝑘3 , the analysis is restricted to the special case of:
1 0 0 1 𝑥
𝑦 + 2 1 − ∆ 1 2 𝑥
𝑦 = 0 (1.4)
To evaluate the stability of the system, an eigenvalue analysis is performed using the ansatz 𝑥 𝑦 𝑡 = 𝑒𝑠𝑡 where 𝑠 = 𝜆 + 𝑖𝜔 is the general complex eigenvalue such as:
𝑑𝑒𝑡 𝑠2 1 0
0 1 + 2 1 − ∆
1 2 = 0 (1.5)
The characteristic polynomial 𝑠4+ 4𝑠2+ 4 − 1 − ∆ = 0 has a discriminant 𝐷 equal to 4 (1 − ∆). The sign of this discriminant determines the system stability or instability. The FIGURE 1.10 shows the evolution of eigenvalues for a friction coefficient ranging from 0 to 1.
Three cases can be considered:
When 𝐷 > 0, i.e. ∆< 1, the eigenvalues are imaginary numbers equal to 𝑠1,2= 𝑖 2 ± 1 − ∆.
This corresponds to low values of the friction coefficient. The frequencies of the two associate modes are different. The real parts are null and the system is therefore stable.
When 𝐷 = 0, i.e. ∆= 1, the eigenvalues are coincident imaginary numbers equal to 𝑠1,2= 𝑖 2.
The frequencies of the two associate modes are equal. This is the frequency lock-in or Hopf bifurcation: when the friction coefficient increases, the frequencies move closer until they become equal for 𝜇 = 0.75 which is the critical value of the friction coefficient and the particular point of coalescence. The real parts are null and the system is consequently stable.
When 𝐷 < 0, i.e. ∆> 1, the eigenvalues have now real part and are equal to 𝑠1,2 = ± 3+∆
2 − 1 + 𝑖 3+∆
2 + 1. This corresponds to values of the friction coefficient superior to the critical friction coefficient, the frequencies of the associate modes are equal and the two real parts are opposite. The mode with the positive real part is unstable and the other with the negative real part is stable. When the friction coefficient increases, the frequencies stay the same and the absolute values of real parts increase.
FIGURE 1.10 – Eigenvalues evolution in function of friction coefficient
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.25 -0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20 0.25
1 , 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
1 , 2
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.9
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
1 , 2
1 , 2
a) b)
c)
In the FIGURE 1.10.a) and b), it can be seen that, for a low friction coefficient, the mode are stable.
Then there is a threshold at 𝜇 = 0.75 where the modes become unstable. There is a monotony of the phenomenon with respect to the friction coefficient. The FIGURE 1.10.c) shows the complex plan, the frequency, which corresponds to the imaginary part of the eigenvalue, is plotted in function of the real part. As say before, a mode is stable if the real part is negative or null and it is unstable if the real part is positive. For each unstable mode, there exists a stable one with the same frequency and with the opposite real part. This graph is therefore symmetric. This kind of graph will be used to perform the stability analysis in the different studies.
2 Modeling
2.1 PREPROCESSING
2.1.1 FINITE ELEMENT MODEL
The finite element model represents a portion of a wiper blade against a windshield during the operation of wiping. The wiper has a length of 650mm but for the simulation a length of 30mm is taken into account. It is considered that the pressure, the attack angle and the velocity are uniform along the portion. The system has been simplified and the dimensions have been reduced to save more computing time.
WIPERS
The finite element model of the wiper is obtained from its section 10X (ten times bigger than the real section) given by the wiping corporate technical leader.This section is digitized and extruded on Catia before being transferred to the CAE preprocessing tool Ansa. The volume is first removed to get a curve shape and after a surface. This 2D model is meshed and extruded of 30mm to get the final finite element model of the wiper. The wiper profile is extruded to take no notice of its curvature which is negligible considering the small size of the system. One material property (EPDM) has been set on the wiper A and two material properties (EPDM + NR) have been set on the wiper B. These successive steps are shown in the FIGURE 2.1.
SECTION 10X DIGITIZATION AND EXTRUSION
ON CATIA TRANSFER TO ANSA DESTRUCTION OF THE VOLUME
MESHING OF THE 2D MODEL EXTRUSION OF 30 MM
FIGURE 2.1 – Steps for the finite element model of the wipers
A B A B
Hexahedral elements have been chosen over tetrahedral elements as soon as it was possible because they allow more accurate results with a lower number of elements. The simulations are done with a fine mesh density; the element size in the section is varying from 0.10mm to 1mm and the element size in the extrusion direction is 3mm. The wipers contain 4520 elements.
WINDSHIELD
The windshield is modeled in the FIGURE 2.2 as a rectangular glass plate with the following dimensions 160x90x3mm and with only hexahedral elements. The element size in the section is 3mm and the element size in the extrusion direction is 1mm. The windshield contains 4770 elements.
FIGURE 2.2 – Finite element model of the windshield
MATERIAL PROPERTIES
Material properties of the two wipers and of the windshield are presented in the following table:
MATERIAL DENSITY 𝝆 YOUNG'S MODULUS 𝑬 POISSON RATIO 𝝂 WIPER A
EPDM 1070 kg/m3 7.2 MPa 0.45
WIPER B
NR 1090 kg/m3 5.1 MPa 0.45
WINDSHIELD GLASS 2530 kg/m3 69 GPa 0.2
TABLE 2.1 – Material properties
It can be seen that the material properties of the two wipers are close to each other. The main difference is between the Young's modulus where a difference of 30% is observed.
2.1.2 INPUT PARAMETERS
ATTACK ANGLE
The orientation of the wiper blade on the windshield surface is determined by an attack angle 𝐴. This angle is the most critical parameter for wipe quality and it is generally defined as the angle between the wiper blade symmetry plane and the vector normal to the outer glass surface. However, during this thesis, this parameter has been differently defined for more simple consideration: the angle 𝛼 between the wiper blade symmetry plane and the plane of the outer glass surface is now considered.
FIGURE 2.3 – Attack angle definition
According to the manufacturer datas, the attack angle A is varying from -7° to 7° equivalent to an angle 𝛼 varying from 83° to 97°. Those depend on the wiper localization on the windshield and it is considered that the attack angle 𝛼 is the same along the portion.
FIGURE 2.4–Variation of the attack angle
WIPER ARM LOAD
The load of the wiper arm is modeled by a nominal force 𝐹 equal to 10.4N according to the manufacturer data. This load is distributed with a constant pressure along the length of the blade equal to 650mm. The force per unit length is therefore 16N/m. During the simulation the length of the blade is reduced to 30mm which corresponds to a nominal force 𝐹 equal to 0.48N. However, it is necessary to take into account that the effect of the wiper arm on the blade is applying with a variation of 10%. Moreover, during the wiping operation, the pressure along the length of the blade is not really constant but a variation of 25% is observed. It is considered that the pressure is the same along the portion. These two previous variations have to be added to the nominal load to get the maximal and minimal loads apply to the blade.
MINIMAL LOAD NOMINAL LOAD MAXIMAL LOAD
0.48 ∗ 0.90 ∗ 0.75 = 0.324N 0.48N 0.48 ∗ 1.10 ∗ 1.25 = 0.660N
TABLE 2.2 – Values of the wiper arm load
In this thesis, the aerodynamic pressure of the wind applied to the wiper blade is not taken into account.
𝑨
𝜶
VELOCITY
The velocity 𝑉 of the wiper arm must also be taken into account as an input parameter to model the direction of the movement. During the different simulations, the velocity is applied on the windshield while the wiper remains motionless. The absolute value of the velocity has no effect on the eigenvalues of the system. Hence, to model the raising of the wiper, the velocity is equal to -1 (corresponding to the lowering of the windshield) and to model the lowering of the wiper, the velocity is equal to 1 (corresponding to the raising of the windshield). Zhang [2] has experimentally revealed that the wiper was noisier during the raising phase as shown in the FIGURE 2.6. The nominal value for the velocity will thus be taken as -1.
FIGURE 2.5 – Velocity definition
FRICTION COEFFICIENT
A friction coefficient 𝜇 between the outer glass surface and the wiper lip has to be considered as an input parameter. This coefficient takes into account the windshield moisture and cleanness as well as the characteristics of the rubber blade. It depends on the windshield wetness condition: wet, half-dry and dry. Wet condition is performed when the windscreen is wet during continuous rain. Half-dry condition, also called tacky condition, is defined as the conditions of wet windshield without flowing water. Dry condition is performed when the windshield is totally dry. Koenen and Sanon [1] and Zhang [2] have experimentally shown in their works that the wetness condition which causes the loudest squeal noise is the half-dry condition where the friction coefficient is at its maximum value. For the stability analysis, the values of the friction coefficient have to be decided. Taking into account experimental studies, the friction coefficient during the wet condition is taken varying from 0.2 to 0.5, it is taken varying to 0.6 to 1 during the dry condition and it is taken varying from 1.1 to 1.5 during the half-dry condition. The maximum value of the friction coefficient during the tacky condition corresponds to the nominal value of the friction coefficient.
FIGURE 2.6 – Experimental sound pressure level in the three wetness conditions [2]
𝑽 = +𝟏 𝑽 = −𝟏
Y Z
X
FIGURE 2.7 – Values of the friction coefficient during the three wetness conditions [1]
2.1.3 BOUNDARY CONDITIONS
The windshield is fixed on its lateral sides as shown in the FIGURE 2.8. The wiper is positioned in the middle and against the windshield surface with the attack angle 𝛼. The effect of the wiper arm is modeled by stiffness as shown in the FIGURE 2.9. The nominal value of the stiffness is 1000N/m. This value is randomly taken since the boundary conditions are hard to model. Gatti shows in his paper that this stiffness has almost no effect on the eigenvalues of the system [10].The stiffness is connected to the fictive node on one hand and to the wiper nodes in contact with the wiper arm on the other hand.
The load of the wiper arm is modeled by the force 𝐹 applied at the fictive node. As say before, the velocity is applied on the windshield nodes in contact with the wiper in the z-direction, these nodes are presented in the FIGURE 2.10. A surface interaction is created between the wiper lip and the windshield surface with the friction coefficient 𝜇. During the operations of wiper systems, other surfaces of the blade may contact. It is why, additional surface interactions, presented in the FIGURE
2.11, have to be created in addition to that defined between the blade and the windshield.
FIGURE 2.8 – Boundary conditions on the windshield
FIGURE 2.9 – Wiper nodes supporting the stiffness
X Z
Y
Y Z
X
𝑭
Fictive node
Figure 2.10 – Windshield nodes supporting the velocity
FIGURE 2.11 – a) Principal and b) additional surface interactions
2.2 PROCESSING IMPLEMENTATION
As say before, the methodology employed in this thesis is based on a methodology already used at Renault on the simple system of window seals.
2.2.1 METHODOLOGY OF CALCULATIONS
The wiper blade squealing is caused by mode-coupling instability. Consequently, to model this squealing, a modal approach studying the system stability around a particular point is used. This method consists in a nonlinear static calculation, where the sliding equilibrium position is investigated by preloading the system, following by a linear dynamic calculation, where the stability is studied by extracting the complex eigenvalues.
STATIC PRELOADING
The system has the particularity to be highly nonlinear because of the significant deformations of the wiper blades and because of the contact between the wiper and the windshield. To find the equilibrium position, the static calculation is performed with the Newton-Raphson method implemented in Abaqus
Y Z
X X
Z
Y
a) b)
and presented by Elmaian [8]. This method is used to solve the nonlinear equilibrium equations. Two preloading steps are executed, a normal preloading followed by a tangential preloading.
During the normal preloading, the wiper is piloted by applying the load at the fictive node in the y- direction while the windscreen stays motionless. This preload causes the wiper blade squashing against the windscreen in the normal direction, during this step the friction coefficient is equal to 0. The preload is applied iteratively to allow a better convergence for the calculations. The FIGURE 2.12 shows the wiper displacement after this step:
FIGURE 2.12 – Displacement after the normal preloading
A tangential preloading is then performed by applying the constant velocity in the z-direction at the windshield nodes in contact with the wiper blade. The friction coefficient 𝜇 is added at this interaction.
The velocity is defined using Lagrangian equations; the material of the windshield transported through the mesh is mobile while the nodes are motionless. The FIGURE 2.13 shows the wiper displacement after this step:
FIGURE 2.13 – Displacement after the tangential preloading
At the end of this static preloading, the wiper is at the sliding equilibrium position around which the stability of the system will be analyzed.
STABILITY ANALYSIS
The stability analysis is performed around the equilibrium position. This dynamic calculation consists in a linearization of the system and an extraction of the complex eigenvalues. The dynamic equation of the motion is according to [8]:
𝑀 𝑢 + 𝐶 𝑢 + 𝐾 𝑢 = 0 (2.1)
𝑭
where [𝑀] is the mass matrix, [𝐶] is the damping matrix, [𝐾] is the non-symmetric stiffness matrix and 𝑢 is the displacement. The non-diagonal elements of the matrix [𝐾] are due to the friction and cause the non-symmetry of [𝐾]. An eigenvalue problem has to be performed to determinate the complex eigenvalues. A method to solve this problem is to solve directly the EQUATION (2.1). However, this is a 𝑛𝑑𝑜𝑓-dimensional equation so this takes a long computational time. Another method to solve the eigenvalue problem is to project the equation on the real modal basis before extracting the complex eigenvalues.
In this case, the second method is used. Firstly, a real eigenvalues analysis is performed. Only the symmetric part [𝐾𝑠] of the stiffness matrix is considered by neglecting the friction (responsible of the non-diagonal elements of the matrix). The damping matrix [𝐶] is also neglecting and the EQUATION (2.1) becomes:
𝑀 𝑢 + 𝐾𝑠 𝑢 = 0 (2.2)
The real modes are calculated, the eigenvalues 𝑠 such as 𝑢 = 𝑒𝑠𝑡 are imaginary numbers and the eigenvectors are real. A modal basis [𝜙] composed with the eigenvectors is obtained and the displacements of the system in this basis are described by the vector 𝛾. To get the displacement 𝑢 of the system, a change of basis is executed:
𝑢 = 𝜙 𝛾 (2.3)
In the real modal basis, the new matrices are calculated:
𝐾′ = 𝜙 𝑇 𝐾 [𝜙]
𝐶′ = 𝜙 𝑇 𝐶 [𝜙]
𝑀′ = 𝜙 𝑇 𝑀 [𝜙]
(2.4)
Secondly, a complex eigenvalues analysis is performed by multiplying the EQUATION (2.1) by 𝜙 𝑇 and by using the EQUATION (2.4). The following equation is obtained:
𝑀′ 𝛾 + 𝐶′ 𝛾 + 𝐾′ 𝛾 = 0 (2.5)
The dynamic equation of the system is now a 𝑛𝑚𝑜𝑑𝑒𝑠-dimensional equation and is described in the real modal basis. The matrices dimension and the computational time are lower. The matrix [𝛤], containing the displacements 𝛾 of the system without friction in the real model basis, is determined. To obtain the displacements 𝑢 of the system with friction, the following equation is used:
[Ψ] = 𝜙 [Γ] (2.6)
where [Ψ] is the matrix containing the complex eigenvalues of the system with friction. With this method, the dynamic EQUATION (2.1) of the motion has been solved. The problem was solved using two steps, a big dimensional real eigenvalues analysis followed by a small dimensional complex eigenvalues analysis. Even if, this method uses two steps, it is more efficient than the method solving directly the dynamic equation since the computational time is lower.
VISUALIZATION OF THE RESULTS
Having the complex eigenvalues of the system, the stability can be studied plotting the complex plan presented in the FIGURE 2.14. For each mode, the frequency is plotted in function of its real part. The unstable modes, which have a positive real part, appear in red while the stable modes appear in black.
In the methodology of the window seals, it is said that the frequency range of interest to analyze the risk of noise is under 1500Hz. For each mode, the mode shape can also be observed as shown in the FIGURE 2.15.
FIGURE 2.14 – Example of a complex plan
FIGURE 2.15 – Example of a mode shape
2.2.2 ABAQUS MODEL
In Abaqus, the static and the dynamic steps are computed in succession as shown in FIGURE 2.16.
This solving methodology is presented by Gatti [10]. In some simulations, like in parametric studies or in factorial designs, several values of the input parameters are implemented. In that case, the normal preloading is applied only at the beginning of the calculations while the tangential preloading and the study analysis are repeated for each value of the parameter. Indeed, the possibility to realize multistep calculations in Abaqus is used. The calculation is subdivided into several steps during which the considered parameter is incremented. For each value of the parameter, the solver searches the equilibrium position and then a complex eigenvalues extraction is done. The FIGURE 2.16 is presenting by considering that the incremented parameter is the friction coefficient, which is the case in the design of experiments.
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500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Real Part
Frequency (Hz)
FIGURE 2.16 – Solving methodology according to [10]
2.3 VERIFICATION OF THE MODEL
2.3.1 ANALYTICAL CALCULATION
Firstly, an analytical calculation is computed to check if the model described above is reliable and consistent with a real wiping operation. The displacement ∆𝑥 of the wiper during the normal preloading is calculated using the Hooke’s law:
𝜎 = 𝐸 𝜖 (2.7)
where 𝜎 =𝐹
𝑆 is the stress, 𝐹 the wiper arm load, 𝐸 the Young’s modulus and 𝜖 =∆𝑥
𝐿 the deformation.
The displacement is thus equal to:
∆𝑥 =𝐹 𝐿 𝐸 𝑆= 𝐹
𝐾 (2.8)
The wiper is consequently considered as a spring of stiffness 𝐾. To calculate this stiffness, the wiper is approximate and divided into several parallelepipeds of length 𝐿𝑖, width 𝑙𝑖 and height 𝐻𝑖 as shown in the following figure:
Normal preloading with 𝜇 = 0
Definition of static boundary conditions
Tangential preloading with 𝜇 ≠ 0 Definition of static boundary conditions
Definition of dynamic boundary conditions
Complex eigenvalues extraction by projecting on the real basis Real eigenvalues extraction
𝜇 incrementing
FIGURE 2.17 – Approximation to calculate the stiffness
The length of the parallelepipeds is the same and it is equal to 𝐿𝑖 = 𝐿 = 30mm. The other dimensions of each parallelepiped are described in the TABLE 2.3.
Surface 1
Surface 2
Surface 3
Surface 4
Surface 5
Surface 6
Surface 7
Surface 8
Surface 9
Surface 10 Height 𝑯𝒊 (mm) 1,63 1,56 1,2 0,47 1 0,13 1 0,84 1,4 6,6
Width 𝒍𝒊 (mm) 0,77 1,4 3,57 0,5 7,74 3,28 7,74 1,72 7,74 6,5 TABLE 2.3 – Dimensions of the parallelepipeds
The parallelepipeds are considered like springs mounted in parallel and consequently the stiffness 𝐾 of the wiper is obtained from the following equation:
1
𝐾= 1
𝐾𝑖 (2.9)
where the stiffness of a parallelepiped 𝑖 is, according to the FIGURE 2.17, 𝐾𝑖 =𝐸 𝑆𝑖
𝐻𝑖 =𝐸 𝐿 𝑙𝑖
𝐻𝑖 . The total displacement ∆𝑥 of the wiper during the normal preloading is thus obtained from the EQUATION (2.8) where the wiper arm load is 𝐹 = 0.48 N. The analytical calculation of this displacement gives, for the wiper A, ∆𝑥 = 0.0144 mm. This value is compared to the numerical value of the displacement which is
∆𝑥 = 0.0142 mm. These two previous values are very close to each other, the model is consequently reliable. For this verification, the model has been simulated with an attack angle 𝛼 = 90° and a wiper arm load 𝐹 = 0.48N.
𝑙𝑖 𝐿𝑖
𝐻𝑖
F
1
2 3 4 6 8 9 10 5
7
FIGURE 2.18 – Numerical value of the displacement after the normal preloading
2.3.2 STUDY OF THE DIFFERENT CONFIGURATIONS
The behavior and the direction of the deformation of the wiper can also be checked in the different configurations to see if it is consistent with a real wiping operation. After the normal preloading, the deformation of the wiper can be observed in the FIGURE 2.19 for the two extreme attack angles 𝛼 = 83° and 𝛼 = 97°. It can be noticed that the lip angle is driven by the inclination of the plate. The deformation of the wiper can also be observed after the tangential preloading in the FIGURE 2.20 for an attack angle 𝛼 = 83° and in the FIGURE 2.21 for an attack angle 𝛼 = 97°. In each figure, the first configuration corresponds to a velocity equal to -1 and the second configuration corresponds to a velocity equal to +1. It can be noticed that the lip angle is, in that case, driven by the plate direction of displacement. It can also be observed that, for 𝛼 = 83°, there is a change in the lip inclination after the tangential preloading when the velocity is equal to +1. This observation can also be perceived for 𝛼 = 97°. Indeed, there is a change in the lip inclination after the tangential preloading for a velocity equal to -1. There is symmetry for the behavior between the two extreme attack angles. These different configurations are consistent with the reality. The model described above can consequently be used to perform the different stability studies.
a) b)
FIGURE 2.19 – Position of the wiper after the normal preloading for a) 𝜶 = 𝟖𝟑° and b) 𝜶 = 𝟗𝟕°
Maximum displacement
a) b)
FIGURE 2.20 – Position of the wiper after the tangential preloading for 𝜶 = 𝟖𝟑° and for a) 𝑽 = −𝟏 and b) 𝑽 = +𝟏
a) b)
FIGURE 2.21 – Position of the wiper after the tangential preloading for 𝜶 = 𝟗𝟕° and for a) 𝑽 = −𝟏 and b) 𝑽 = +𝟏
3 Stability studies
3.1 NOMINAL STUDY OF THE TWO WIPERS
In this chapter, the methodology of calculations seen before is applied in a nominal configuration with particular values of the input parameters. The attack angle 𝛼 shall be taken as 97° and the velocity 𝑉 as -1 to represent the wiper’s raising. The wiper arm load 𝐹 shall be taken as 0.48N and the friction coefficient 𝜇 as 1.5 which is its maximal value in the half-dry condition. For the stability analysis, the first 150 complex modes are extracted.
FIGURE 3.1 – Nominal configuration
3.1.1 STABILITY ANALYSIS FOR THE WIPER A
The final position of the wiper A, in the nominal configuration, is shown in the FIGURE 3.2. The maximal squashing of the wiper against the windshield at the end of the study is equal to 2.01 mm. The complex eigenvalues of the wiper A are illustrated in the FIGURE 3.3 which represents the frequencies in function of the real parts. The unstable modes, which have a positive real part, are shown in red while the stable modes appear in black. On the first 150 modes, there are 13 unstable ones. One unstable mode appears at a frequency equal to 1777Hz with a very low real part equal to 17.7. The other unstable modes are in a frequency range between 4000Hz and 5500Hz with real parts up to 3000. It can be noticed that there is no unstable modes under 1500Hz which is the frequency range of interest. The FIGURE 3.4.a) shows the mode shape associated to the first complex unstable mode.
Vibration nodes appear in blue whereas antinodes are represented in red. It can be observed that the first unstable mode at a frequency equal to 1777Hz is located at a part of the heel. It could be expected that this mode would be positioned at the lip as it is the part of the wiper blade which is in contact with the windshield. However, if this mode shape is observed with a different scale like in the FIGURE 3.4.b) it can be observed that the first unstable mode is also located in the wiper lip. Nodes and antinodes appear both in the profile and along the wiper. The main antinodes are situated at the end of the lip. During the visualization of the mode shape, it had been seen that it exists a mode of vibration at the hinge which moves to the end of the lip. With these results, it can be said that the stability analysis predicts a risk of noise for the wiper A when it is used in the nominal configuration.
F
𝜶 V
FIGURE 3.2 – Final position of the wiper A
FIGURE 3.3 – Complex plan for the wiper A
a) b)
FIGURE 3.4 – Mode shape of the first unstable mode of the wiper A
3.1.2 STABILITY ANALYSIS FOR THE WIPER B
The final position of the wiper B, in the nominal configuration, is shown in the FIGURE 3.5. For this wiper, the maximal squashing against the windshield after the normal and tangential preloading is equal to 2.13mm. In this case, the squashing is lower because the Young’s modulus of the lip of the wiper B is smaller than the Young’s modulus of the lip of the wiper A. The complex eigenvalues of the wiper B are illustrated in the FIGURE 3.6. As before, the unstable modes are shown in red while the stable ones appear in black. On the first 150 modes, there are 19 unstable ones which are in a frequency range between 3000Hz and 5500Hz with real parts up to 3300. It can be noticed, as for the wiper A, that there is no unstable modes under 1500Hz which is the frequency range of interest. The FIGURE 3.7 shows the mode shape associated to the first complex unstable mode. As before, vibration nodes appear in blue whereas antinodes are represented in red. It can be observed that the first unstable mode at a frequency equal to 3340Hz is located at the wiper lip and at parts of the heel close to the hinge. Nodes and antinodes appear both in the profile and along the wiper. The main antinodes
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1000 2000 3000 4000 5000
Real Part
Frequency (Hz)
are situated at the end of the lip. During the visualization of the mode shape, it had been seen, as for the wiper A, that it exists a mode of vibration at the hinge which moves to the end of the lip. With these results, it can be said that the stability analysis predicts a risk of noise for the wiper B when it is used in the nominal configuration.
FIGURE 3.5 – Final position of the wiper B
FIGURE 3.6 – Complex plan for the wiper B
FIGURE 3.7 – Mode shape of the first unstable mode of the wiper B
At the end of this study, it is difficult to compare the two wipers and say which one will be the loudest wiper. Indeed, considering only the frequency range from 3500Hz to 5500Hz, the wiper B has more unstable modes; nevertheless, the wiper A has one unstable mode in low frequency. It can be added that none of the wipers have unstable modes in the frequency range [0,1500Hz] which is the frequency range of interest. Since the wipers can not be compared in nominal configuration, designs of experiments with factorial design have to be implemented.
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500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Real Part
Frequency (Hz)
3.2 PARAMETRIC STUDIES
As say before, the squeal noise generated by the wiper is a random phenomenon. Studying this phenomenon at the nominal configuration is not sufficient, it is necessary to study the influence of the input parameters to see if trends emerge. Parametric studies are thus performed, the variability of each input parameter is taken into account and the influence on the main characteristics of the mode- coupling phenomenon is studied. In particular, the influence on the frequencies and on the real parts is considered. This analysis will be only done on the wiper A and when an input parameter is studied, the other parameters are at the nominal configuration. Only one calculation is realized in each analysis by incrementing the considered input parameter with the multistep option in Abaqus.
3.2.1 FRICTION COEFFICIENT
The influence of the friction coefficient on the stability of the modes of the system is first analyzed. The values of the friction coefficient are varying from 0 to 2 with an increment of 0.1. The other parameters are at the nominal configuration, i.e. the attack angle is taken as 97°, the velocity is taken as -1 and the wiper arm load is taken as 0.48N. The complex plan is first plotted in the FIGURE 3.8 in function of the friction coefficient. On this figure, it can be seen that there is an unstable mode at a frequency of 1107Hz with a very low real part. This unstable mode appears for a friction coefficient equal to 0.5 as it can be seen in the FIGURE 3.9, which represents the frequency of the first unstable modes in function of the friction coefficient. A second frequency range of interest is [1750,2400Hz] which corresponds to the frequency of the first unstable modes for the friction coefficients 𝜇 = 0.6, 𝜇 = 0.7 and 𝜇 = 1.2 to 𝜇 = 1.7. The unstable modes in these two frequency ranges are shown in detail in the FIGURE 3.8. The numbers in brackets on the FIGURE 3.8 and on the FIGURE 3.9 represent the frequency ranges of interest. The third frequency range of interest is [3500,5500Hz] where the majority of the unstable modes are grouped. The values of the real parts of these unstable modes can be high and increase in function of the friction coefficient until 𝜇 = 1.2 and then decrease, as it can be seen in the FIGURE 3.10.
The number of unstable modes also increases until 𝜇 = 1 and then decreases, as it is shown in the FIGURE 3.11. Looking at the FIGURE 3.9 it can be noticed that there is not monotony of the phenomenon. Taking into account a certain frequency range (for example [0,3000Hz]), when the friction coefficient increases, there is a first threshold at 𝜇 = 0.5 where the modes become unstable, then there is a second threshold at 𝜇 = 0.8 where the modes become stable and then a third threshold appears at 𝜇 = 1.2 where the modes become unstable. Thus, this sensitivity study show the appearance and the disappearance of unstable modes in the frequency range [1000,2500Hz].
FIGURE 3.8 – Complex plan in function of the friction coefficient
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2000 3000 4000 5000
Real Part
Frequency (Hz)
Friction
0 0.5 1 1.5 2
-200 -100 0 100 200
1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Real Part
Frequency (Hz)
Friction
0 0.5 1 1.5 2
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900 1000 1100 1200 1300
Real Part
Frequency (Hz)
Friction
0 0.5 1 1.5 2
[1]
[2]
[3]
FIGURE 3.9 – First unstable mode frequency in function of the friction coefficient
FIGURE 3.10 – Friction coefficient in function of the real parts
FIGURE 3.11 – Number of unstable modes in function of the friction coefficient
3.2.2 ATTACK ANGLE
The influence of the attack angle on the stability of the modes of the system is then analyzed. The values of the angle are varying from 80° to 100° with an increment of 2°. The other parameters are at
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 1000 2000 3000 4000 5000
Friction
First unstable mode frequency (Hz)
-3000 -2000 -1000 0 1000 2000 3000
0 0.5 1 1.5 2
Real Part
Friction
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 2 4 6 8 10 12 14 16 18
Friction
Number of unstable modes
[1] [2]
