Scuola di Dottorato “Leonardo da Vinci”
PhD Programme in
Mechanical Engineering
PhD Dissertation
Numerical and experimental investigation
on disc brake vibration phenomena
Romulo do Nascimento Rodrigues
Scuola di Dottorato “Leonardo da Vinci”
PhD Programme in
Mechanical Engineering
PhD Dissertation
Numerical and experimental investigation
on disc brake vibration phenomena
Author:
Romulo do Nascimento Rodrigues ...
Tutors:
Prof. Ing. Paola Forte ...
Prof. Ing. Francesco Frendo ...
SSD ING–IND/14 2017
Abstract
In this dissertation, the activities carried out during the PhD are comprehensively described. They covered, in parallel to a theoretical-numerical study on the main vibratory phenomena of disc brakes, the development of a simple experimental apparatus to investigate the "creep groan" generation mechanism. The experimental activity was conducted at the Tribology Laboratory of the Department of Civil and Industrial Engineering (DICI) of the University of Pisa with the support of Brembo SpA, which provided the materials (samples and discs) for the research.
The experimental characterization of the pad material was carried out on a pin on disc tribometer with the brake disc in place of the tribometer disc and the sample of pad fixed on a suitable interface in the place of the pin. The goal of this tribological characterization was to check the behavior of coefficient of friction between the sample and disc with varying of the operating (speed, load) and geometric (contact area) parameters.
After the tribological characterization, in order to determine the influence of the behavior of the coefficient of friction due to the variation of the operating and geometric parameters on the stability of brake system and in particular on its propensity to squeal, numerical simulations in ANSYS environment were carried out on the disc-pad system using the complex eigenvalue analysis.
Since it is generally accepted that brake creep groan is due to the stick–slip phenomenon an extended literature review was made on the relevant analytical models and simulations with some of the lumped parameter models were performed for a better understanding of the phenomenon.
Finally, experimental tests were carried out using a simple experimental apparatus set up to investigate the generation mechanism of the creep groan phenomenon where some operative parameters, such as angular velocity of the disc, load, as well as contact area, stiffness of material of the samples and system stiffness, were varied and their effects on the phenomenon observed.
Sommario
In questa tesi sono descritte le attività di ricerca condotte durante l’intero dottorato, che hanno riguardato, parallelamente ad uno studio teorico-numerico sui principali fenomeni vibratori di freni a disco, lo sviluppo di una semplice apparato sperimentale per indagare il meccanismo di generazione di "creep groan".L’attività sperimentale è stata svolta presso il laboratorio di tribologia del Dipartimento di Ingegneria Civile e Industriale (DICI) dell’Universitàdi Pisa grazie alla collaborazione di Brembo SpA che ha fornito i materiali (campioni e dischi) per la ricerca.
La caratterizzazione sperimentale del materiale della pasticca del freno è stata effettuata su un tribometro pin on disc con il disco del freno al posto del disco del tribometro e il campione di pasticca fissato su una apposita interfaccia al posto del pin.L'obiettivo di questa caratterizzazione tribologica è stato quello di verificare l’andamento del coefficiente di attrito tra il campione e il disco al variare dei parametri di funzionamento (velocità, carico) e geometrici (area di contatto).
Dopo la caratterizzazione tribologica, al fine di verificare l’influenza della variazione del coefficiente di attrito con i parametri di funzionamento e geometrici sulla stabilità del sistema frenante e sulla suscettibilità al fenomeno di squeal, sono state effettuate simulazioni numeriche in ambiente ANSYS del sistema disco-pasticca, utilizzando l'analisi agli autovalori complessi.
Dato che è generalmente accettato che il creep groan sia dovuto al fenomeno stick-slip è stato effettuata una revisione dei modelli di letteratura e sono state condotte simulazioni con alcuni dei modelli a parametri concentrati per una migliore comprensione del fenomeno.
Infine, sono state effettuate prove sperimentali utilizzando una semplice apparecchiatura sperimentale per studiare il meccanismo di generazione del fenomeno di crrep groan in cui alcuni parametri operativi, come la velocità angolare del disco, carico, così come la area di contatto, la rigidezza dei materiali dei campioni e del sistema, sono state variati e sono stato osservati i loro effetti sul fenomeno.
Contents
Introduction ... 10
Chapter 1 ... 12
Background and aims of the research ... 12
1.1. Automotive industry ... 12
1.2. Disc brake systems ... 14
1.3. Friction materials ... 16
1.3.1. Abrasives ... 17
1.3.2. Friction Producers / Modifiers ... 17
1.3.3. Fillers and Reinforcements ... 18
1.3.4. Binder (Matrix) Materials ... 20
1.3.5. Classification of the friction material ... 20
1.4. Automotive disc brake squeal ... 21
1.4.1. Low-frequency noises ... 22
1.4.2. Low frequency squeal ... 23
1.4.3. Squeal of high frequency ... 23
1.4.4. Other types of noise ... 24
1.5. Aims and objectives ... 24
1.6. Novel contributions ... 25
Chapter 2 ... 28
Tribological characterization of friction material for brake pads ... 28
2.1. Friction ... 28
2.2. Composite material friction ... 32
2.2.1. Adhesive friction ... 33
2.2.2. Deformative friction ... 34
2.3. Machines for tribological tests ... 35
2.4. Laboratory investigations ... 38
2.4.1. Materials and methods ... 38
2.4.2. Results and discussions ... 39
2.5. Summary ... 45
Chapter 3 ... 47
Numerical correlations between material and morphological parameters and disc brake squeal vibrations. ... 47
3.1. Finite element method (FEM) ... 47
3.2. The concept of stability of mechanical systems ... 48
3.3. Stability assessment using a finite element model ... 50
3.4. Analysis of squeal using a finite element model of the pad-disc set. ... 52
3.5. Summary ... 63
Chapter 4 ... 65
Theoretical models to explain the stick–slip behavior ... 65
4.1. The stick-slip phenomenon ... 65
4.2. Theoretical models ... 68
4.2.1. One degree of freedom model ... 68
4.2.2. The model by Shin et al. ... 70
4.2.3. The model by Hoffmann and Gaul ... 71
4.2.4. The model by Popp et al... 73
4.3. Simulation of theoretical models ... 74
4.3.1. Simulation of Ding's model ... 75
4.3.2. Simulation of Shing's model ... 79
4.4. Summary ... 84
Chapter 5 ... 86
Experimental investigation on the creep groan phenomenon using a pin-on-disc tribometer. ... 86
5.1. The creep groan phenomenon ... 86
5.2. Experimental details ... 88
5.3. Summary ... 99
Chapter 6 ... 102
Noise and vibration problems have become nowadays one of the main concerns connected with road vehicle brake systems. For example, low frequency brake noise has cost the industry US$100 million a year in warranty claims alone. Road vehicle vibration and brake noise can be classified according to its dominant frequency and its triggering conditions (such as decelerating or moving slowly while braking).Traditional categories are represented by judder, groan, moan and squeal, respectively corresponding to around 10 Hz, 100 Hz at very low vehicle speed (below 2 km/h), 100 Hz at higher speed (10-30 km/h) and more than 1 kHz vibration frequency phenomena. One way to understand, in order to eliminate the phenomenon, is finding more information about the interaction of various sensitive parameters and their respective contribution in an unstable vibration event.
In this dissertation, the activity is extensively described, starting from the environmental and key research motivations (Chapter 1). One of the most influential parameters on brake noise is indeed friction. Therefore a tribological characterization of the friction material employed in the tests (Chapter 2) was done in order to check the behavior of coefficient of friction between the sample and disc with varying operating (speed, load) and geometric (contact area) parameters. After the tribological characterization, numerical simulations in ANSYS environment were carried out in order to study the stability of the disc-pad system and its propensity to squeal, by using the complex eigenvalue analysis (Chapter 3).
As far as brake creep groan us concerned, which occurs commonly on vehicles with automatic transmission when moving off from a stopping position, it is generally accepted that it is due to a friction induced stick–slip phenomenon at the friction interface. Minimizing the creep groan phenomenon is of great importance, especially with an eye to markets where automatic transmissions dominate, such as the USA. Many researchers focused their efforts on developing theoretical (analytical or computational) models to explain the stick–slip behavior. A careful literature analysis was made on the models to study the stick–slip phenomenon and simulationswith some of the proposed models were performed for a better understanding of the phenomenon (Chapter 4).
Finally, in Chapter 5, experimental tests were carried out using a simple experimental apparatus set up to investigate the generation mechanism of the creep groan phenomenon where some operative parameters, such as angular velocity of the disc, load, as well as contact area, stiffness of material of the samples and system stiffness, were varied and their effects on the creep groan phenomenon observed.
Acknowledgements I would like to thank some people who supported me during the PhD period. First, I would like to thank my PhD tutors, prof. Paola Forte and prof. Francesco Frendo, for inspiring, encouraging and for having followed my activities daily with remarkable willingness, addressing my research. Many thanks to CAPES (Coordination for the Improvement of Higher Education Personnel of Brazil) for the financial support as regards the PhD scholarship. I acknowledge my sincere indebtedness and gratitude to my parents for their love, dream and sacrifice throughout my life.My wife has always believed in me and is an important part of my life. I would like to thank Ing. Andrea Cerutti and the Brembo Spa which provided the support material (samples and discs) important for the research. Finally, I would like to thank Mr. Dario Mondini from the University of Pisa, who contributed with technical support , as regards the experimental tests.
Chapter 1
Background and aims of the research
1.1. Automotive industry
The Ford Model T (colloquially known as the Tin Lizzie, T‑Model Ford, Model T, or T) is an automobile which was produced by Ford Motor Company from October 1, 1908, to May 26, 1927. It is generally regarded as the first affordable automobile, the car that opened travel to the common American middle-class; some of this was because of Ford's efficient fabrication, including assembly line production instead of individual hand crafting.The model T weighed 550 kg, had a 20-hp engine and a top speed of approximately 65 km/h (see Figure 1). It was equipped with a band brake system, a cotton textile band wound around a drum inside the planetary gearbox. The cotton band was lubricated with the oil from the gearbox and in order to avoid over-heating, the driver was instructed to apply the brake in short intervals only.
Figure 1 - Ford model-T, the first mass-produced car in history.
Eighty-three years later, Mercedes-Benz reintroduced the 600 S-class (the name was also used in the 60's). Designed to be the best and most comfortable car in the world, it weighed over 2 tons, with a 400-hp engine and an electronically limited top speed of 250
km/h (for safety reasons). The model–T’s single band brake was replaced by four disc brakes and no one even thought about giving the driver special instructions on how to take care of them. Nowadays, it is taken for granted that the brake systems should always work perfectly, despite careless users, extreme speeds and difficult environments.
The noise generated in the vehicle braking has troubled for long. By mid-1930, an American research found the brake noise among the ten most molestantes sounds of a city [2]. At the time, the result of this research showed the severity of the problem. The search for improvement and improvement of the systems that make up the vehicle, including braking systems, is something that keeps pace with the growth of the automobile market since its inception. By the late 60s, the passenger cars in the United States of America (USA) and light trucks used drum brakes on all four wheels and tarpaulins reinforced with asbestos fibers. The braking improvement requirements started in the late 60s with the braking performance tests promoted by Pure Oil Company in Daytona, USA, and resulted in the creation of the Federal Security Standard for Motor Vehicle No. 105 in the mid 1970, responsible for the transition to the front brake system disc and rear drum [1].
In recent years, brake manufacturers have especially focused on studies related to brake noise and vibration. It occurred mainly because it is associated as a symptom of failure in the brake system, which has led the owners of vehicles to present complaints against the automobile manufacturer [3,4]. This type of problem is generally considered a warranty expense by the automobile industry [5] and in many situations the brake material manufacturer ends up being responsible for the problem. So, it is relatively easy to understand why brake noise and vibration have become a major concern for brake manufacturers [6]. Moreover, the very quiet operation of the modern vehicles has significantly contributed to make brake noise and vibration an issue of great interest for industry [4,7]. The annual cost related to the noise and vibration issue in brakes systems [8] is estimated in more than U$100 million.
Due to the great complexity, elimination of noise and vibration in brake systems is a challenge that involved both the academia and the industry. While scientists seek to understand the phenomena that can cause the problem, the industry is responsible to find empirical solutions in their projects. A partnership of this knowledge has generated much more satisfactory results, in which the features found in the industry indicate the paths for the further development of studies and vice versa.
1.2. Disc brake systems
The disc brake has been developed since 1890 approximately, and the first patent came in 1902, recorded by British engineer Frederick William Lanchester (1868-1946).[9-12]In his 1902 patent [5,14], he described a disc brake consisting of a disc of sheet metal which is rigidly connected to one of the rear wheels of the vehicle. To slow the vehicle, the disc is pinched at its edge by a pair of jaws. The rotor (or disc) is rigidly mounted on the axle hub and therefore rotates with the automobile’s wheel. The pair of brake pad assemblies, which consist of friction material, backing plates and other components, are pressed against the disc in order to generate a frictional torque to slow the disc(and wheel’s) rotation (see Fig. 2). The caliper houses the hydraulic piston(s), which actuate the pad assemblies. It is attached by a caliper mounting bracket to the vehicle. The methods and points of attachment depend on the type of caliper.
Figure 2 - Components of an automotive brake. [9]
When a driver depresses the brake pedal, it effects an increase in hydraulic pressure in the pistons housed inside the caliper. The device which converts the brake pedal’s motion to hydraulic pressure is known as the master cylinder. It is connected by brake lines and hoses to the disc brake’s caliper. The master cylinder may be supplemented by another servo-device to boost the hydraulicpressure in the brake lines.
The brake discs may be divided into two categories: solid and with fins down as shown in Figure 3. In the majority of automotive disc brakes, the discs are made of grey cast
iron. This material is wear resistant and relatively inexpensive. According to Newcomb and Spurr [13], in order to protect the wheel bearings from the high temperatures induced in a braking action at the rotor-pad interface, the rotor is shaped like a top-hat. The hat section increases both the surface area (to improve cooling) and the length of the path that the heat must travel to affect the bearings. In vented discs, cooling is further enhanced by constructing the disc from two thinner discs connected by a series of thin fins (which are also known as ribs). A prime number (e.g., 31, 37 or 41) of fins is typically used in this case in order to inhibit symmetric modes of vibration in the disc. In high performance and customized disc brakes, the rotors are often slotted and/or drilled and are not necessarily composed of cast iron.[5]
Figure 3 - Brake discs solid (left) and disc with fins (right)
Braking converts most of the kinetic energy of a vehicle to thermal energy primarily within the pads and rotors (discs). Table 1 contains pertinent data for some representative production automobiles. As approximately 70–80% of a vehicle’s braking power is in the front brakes, it can easily be seen from data in Table 1 that, during a braking action, each front brake may be dissipating energy at a rate of over 50 kW at an interface of the size of a brake pad.Nearly all of this mechanical dissipation takes the form of heat generation at the interface.However, a small fraction of the energy finds its way to vibrational energy within the braking system which can even travel to the suspension of the vehicle. The vibratory energy follows a complex path, and the resulting radiation sound may involve any number of components of the brake system.
Table 1 - Dissipation rates in brakes for stops from 70 mph at impending brake lockup[5]
1.3. Friction materials
The friction materials are key elements to the brake system performance. Friction materials can be considered as composite materials, which Anderson [15] classifies, as organic, carbon-based, and metallic. Desplanques et al. [16], says that the friction materials must comply with technical standards to meet some basic needs, also cited by Eriksson [17] and Eriksson and Jacobson [18]:
high friction coefficient and mainly stable, disregarding environmental conditions such as temperature and humidity, time of use, degree of wear and corrosion, dirt and water spray from the road;
long useful life, presenting the least possible wear of the pad and disc;
high degree of comfort, limiting vibration and noise.
The brake pads usually consists of more than 20 different components. In the contact zone between pad and disc a highly complex tribological layer surface modulate the frictional behaviour of the system. The chemical composition of pads seems to be only a warehouse for the friction process to build up these structured layers in the contact zones. The connection between component mixture and friction layer and the connection between friction layer and friction behaviour of the system is not known yet. Therefore, every pad factory deals with their own formulas for pad production. Optimizations with respect to pad composition have to be done by trial and error [16,19,20]. One can group brake materials and additives based on their expected functions as follows:
Abrasives
Friction Modifiers
Binder MaterialsThere is a little ambiguity in this categorization [21]. Some of the additives can be placed into more than one category since they fulfill several functions. Consequently, there are some unavoidable overlaps in the tabular listings. In addition to the basic brake materials, some porosity (5-10% or more) is normally present. To analyze the role of additives in friction and wear control, it is insufficient to simply know their composition, since their form, distribution, and particle size can affect friction and wear behavior. For example, rounded beads of a hard, abrasive material can have a different effect from angular grits on the formation and stability of the friction-induced surface films that control stopping behavior.
1.3.1. Abrasives
Abrasives help maintain the cleanliness of mating surfaces and control the build-up of friction films (see Table 3). They also increase friction, particularly when initiating a stop (i.e., they increase “bite”)
Table 2 - Abrasive materials used by the automotive industry. [21]
1.3.2. Friction Producers / Modifiers
These materials lubricate, raise the friction, or react with oxygen to help control interfacial films (see Table 3).
Table 3 - Some friction modifiers materials used by the automotive industry. [21]
1.3.3. Fillers and Reinforcements
Fillers are used to maintain the overall composition of the friction material, and some have other functions as well. They can be metals, alloys, ceramics, or organic materials (see Table 4). Cashew-containing friction dust is said to have the ability to absorb the heat created by friction while retaining braking efficiency. It is a major export product of India and the Asian subcontinent. The supposed advantage of cashew resin, compared with plain phenolic resin, is that it produces a softer material, which is more efficient for wear when the brakes are relatively cold, as in temperatures generated by lower speed automobiles. Cashew friction dust is a granular, free flowing polymerised resin derived from Cashew Nut Shell Liquid (CNSL). The main component in processed cashew nutshell liquid (CNSL) is cardanol, a naturally occurring, 10 meta-substituted alkenyl phenol similar to nonylphenol. Cardanol is hydrophobic in nature and remains flexible and liquid at very low temperatures.[21]
1.3.4. Binder (Matrix) Materials
Typical binder materials are phenolic resins in the case of automotive and truck pads(see Table 5).
Table 5 - Binders materials used by the automotive industry.[21]
1.3.5. Classification of the friction material
After the banishment of friction material formulations with asbestos or asbestos chrysotile (Mg3Si2O3 (OH) 4) by major automakers between the 1980s and 1990s, two families of friction materials, Non-Asbestos Organic (NAO) and Semi - Metal (semi-TEM) have become predominant. They differ in their composition, as they seek to establish contacts of a different nature with the metal counterpart.
A semi-metallic brake pad consists of a lining that uses steel wool instead of non-asbestos organic (NAO) material as a reinforcing fiber. Most semi-metallic friction materials contain at least 60% steel by weight. The steel fibers are what act as the framework to lock the friction ingredients together. Semi-metallic pads provide better high temperature performance and wear characteristics than conventional non-asbestos linings. The lining gained in popularity with the introduction of front-wheel drive passenger cars whose typical braking characteristics call for approximately 80% of the braking to occur in front. This makes it critical to install a pad that is stable at higher operating temperatures and pressures, and non-asbestos organic and ceramic pads were not designed or substituted for vehicles originally equipped with semi-metallic pads.
Non-asbestos organic (NAO) brake pads consist of organic fibers that are used to reinforce the friction materials and provide strength to the brake pad. NAO friction material contains less than 30% steel by weight. NAO brake pads were designed to replace harmful
asbestos linings and were popular on pre-FWD vehicles. Typically, these were larger vehicles that shared a even more braking between the front and rear brakes.
1.4. Automotive disc brake squeal
Brake noise is classified according to its characteristic, frequency, and elements responsible for its radiation. During the last years various types of noise generated by the brake system were discovered and various terminologies were used. According to the dominant frequency of the noise, they can be classified into three categories: low noise, medium and high frequency. Akay mentions in his work that during his research found 25 or more designations to describe noise and vibration in automotive brake [2].
Some refer to their alleged mechanisms, and others describe the characteristics of the sounds.Figure 8 shows their approximate distribution in the frequency spectrum. Grunt, hum, groan, and moan have lower frequency content than the family of squeals. Not surprisingly, some of these sounds have similar generation mechanisms. For example, squeak describes a short-lived squeal, wire brush describes a randomly modulated squeal, and squelch describes an amplitude-modulated version of squeak noise.
Ouyang [22] classifies the noise according to the generation mechanism into three categories. Creep-squeal, which is caused by stick-slip movement between the friction material and the disc surface [23,24]. Hot Rumble Judder or that is caused by periodic imperfections in the disc surface that result in cyclic braking torques, [23,25,26] .The third type of noise is the squeal, noise with a dominant frequency above 1 kHz, or above the first vibrate mode off the disc plane [5.22]. Dessouki et al. [27] ranks squeal in three other categories: squeal induced by forceps, or insert induced by the disc. In this work the noise of automotive brakes are classified according to their frequency, and so will be described in the following sub-items.
Figura 4 -Different brake noises and their approximate spectral contents [2].
1.4.1. Low-frequency noises
Low-frequency noises occur mainly between 100 and 1000 Hz topics. Examples of this type of noise are: Moan, squeal and Judder. The generation mechanism for such noise is the heat of friction generated between the disc and the friction material, providing power to the system. This energy is then transmitted through a vibratory response brake assembly and coupled to the suspension and chassis components [28].
Brake moan noise is a friction induced noise occurring at very low vehicle speeds and brake pressures in the frequency range of 50 - 500 Hz. It is a geometric instability phenomenon caused by stick-slip excitation at the pads and rotor interfaces [24]. Unlike squeal, in which the resonance exists in one or more brake parts, brake moan only involves the resonant motions of chassis components while the brake moves as a rigid body [29]. It is reported that the magnitude of noises is affected by the resonance and the joint stiffness of various suspension components as well as the difference between static and dynamic friction.
The way the typical squeal vibration occurs at a range of deceleration between 0.15 to 0.62 g, the temperature range of 65-121 ± ° C, vehicle speed between 16-32 km / h and the noise is normally made throughout the arrest [29]. A typical groan has a spectrum between 10 to 30 Hz, with harmonics reaching 500 Hz. It occurs at low speeds and under
moderate braking conditions.A groan appears to result from a geometric instability of pads that gives rise to stick– slip, which, in turn, excites the low-frequency resonances in a brake system [2]. In particular, resonances of the rigid-body rotation mode of the caliper and the local suspension parts develop and radiate sound without the participation of the rotor. The position of the pads with respect to the rotor has a significant role: the higher the relative tilt between them, the higher the propensity for groan generation [30].
Judder is a low frequency vibration where its frequency is a multiple of the wheel
rotation speed [2]. This type of vibration is transmitted by the chassis and steering wheel, causing discomfort to the driver [31]. Also according to Akay, Judder is a consequence of non-uniformity of frictional force on the wafer-disc interface, and this non-uniformity may result from circumferential thickness variation, uneven coating the friction material, and variation in the finished surface.
1.4.2. Low frequency squeal
In the region of low frequency between 1 and 7 kHz, where the disc normally vibrates in axi-symmetric modes of 1 to 4 nodal diameters, low frequency squeal can be perceived.. The occurrence of this kind of noise may be associated with the phenomenon called modal locking, which is basically the coupling of two or more modes of the structures [29]. According to Kinkaid et al. [5] in the low frequency squeal region, the nodal spacing is greater than the length of the pad, thereby, treating the pad as a rigid member is acceptable.
1.4.3. Squeal of high frequency
A high-frequency squeal typically involves the higher order disc modes, with 5 to 10 nodal diameters. Nodal distance between these modes may be comparable or less than the length of the pad. Their frequencies range between 5–15 kHz.Usually this type of noise has a very characteristic frequency, which remains constant for a particular type of disc independent of other system components, so the disc is a decisive factor for this type of noise. According to Dunlap [29], the frequency of the noise, in many cases, coincides with the frequencies of the resonance of the brake disc.
Quaglia and Chen showed in their work that a drop occurs in the probability of squeal when uncoupled the in-plane modes (in the plane of the disc track) and out-of-plane
(the track bending modes). Kinkaid et al. [9] states in his work that Lang and Smales, for this type of squeal, showed that the nodal lines of drive modes are very close to each other and the bending of the wafer becomes important.
1.4.4. Other types of noise
The noises presented above are some of the noises found in practice, however, there are many other types of noise, or different designations may be found in literature. The noise called Screech or Wire Brush is a high frequency noise that normally occurs from 10 to 15 kHz. This noise is within the high frequency band squeal, however it has some features that differentiate it from squeal. The Wire Brush is a sound created by a quick unstable oscillation between different vibrational modes of high frequency [33].
Some terminology to differentiate certain types of noise are also found. For noise squeal and squeal there are peculiarities that make them be recognized by other names. In the case of squeal, two more names can be found: Chirp Chirp and Drag. The Chirp is nothing more than a squeal noise that occurs with interruptions. This type of squeal occurs briefly at each brake disc revolution and with applied pressure. The noise called Chirp Drag is a noise similar to Chirp, but this noise has no pressure applied to the brake system. In the case of the squeal noise, There are also some designations and variations of this type of noise, where can be mentioned dinamic squeal and the squeal squeal. Creep squeal occurs commonly on vehicles with automatic transmission when moving off from a stopping position. The brake creep groan may occur in vehicles when the driver slowly moves the car through some small distance and stops, such as in stop–go traffic, at traffic lights and in garage maneuvers. Thus the idle torque transmission can move the vehicle back. When this movement occurs, it is kept near brake pressure 3 bar and the vehicle being at a very low speed, the stick-slip phenomenon enters process. Consequently, this phenomenon that occurs between the pad and the disc may cause noise, which typically has a frequency less than 300 Hz between [3,34].
1.5. Aims and objectives
Creep groan is one of the most important for brake manufacturers [35] and remains an elusive problem [36] as it is still often found in applications.In order to investigate the effect
of various parameters of creep groan generation, a more sensitive experiment apparatus is necessary. [36] It was in this context that the PhD research project came into being. In particular it was aimed to find more information about the interaction of various sensitive parameters and their respective contribution in a creep groan event,through:
development of a simple support for tribometer that is sensitive to creep groan generation for creep groan analysis and to show its usefulness and potential for a fundamental analysis of creep groan generating mechanism;
to evaluate the influence of the tribological parameters in relation to the vibration generated during tests with the support to better understanding the phenomenon creep groan.
The project was developed in the Regione Toscana and involved the University of Pisa and Brembo Spa. The University of Pisa took part in the activity with their tribology laboratories from the Department of Civil and Industrial Engineering (Italian achronim DICI). The Brembo Spa took part in the activity with materials (samples and discs) for research.
1.6. Novel contributions
As we saw earlier, disc brake squeal is an irritating high-pitched sound which still remains a major prob- lem facing the automotive industry. The main concern over the squeal problem is that it can cause discomfort to passengers and pedestrians and, hence, reduce the overall acceptability of the vehicle. The overall aim of this project is to gain some new and better insight into the problem of disc brake squeal,especially on the phenomenon of stick slip and creep groan.The main conclusions drawn from the current research and the contributions to the subject can be listed..Simulations regarded both squeal and creep groan and showed that some parameters such as angular velocity of the disc, load, as well as contact area, system stiffness, stiffness of the friction material and difference between static and dynamic friction have influence on the brake system instability.
A finite element (FE) complex eigenvalue parametric analysis was performed on the brake assembly to evaluate the propensity to dynamic instability of brakes with multiple pads as a function of the number of pads and of geometrical and material parameters, obtaining useful indications. A simple experimental apparatus set up to investigate the generation mechanism of "creep groan" was presented. The experimental test rig proved to
be effective and versatile for a fundamental analysis of the creep groan generating mechanism in automotive brakes. Operating parameters, such as angular velocity of the disc, load, as well as contact area, stiffness of the friction material and system stiffness, in the tests were varied and their effects on the "stick-slip" phenomenon may be observed.
Chapter 2
Tribological characterization of friction
material for brake pads
Friction represents a universal attitude of matter, and has always raised challenges to humankind. When the word “friction” is mentioned, what first comes to mind is a picture of two surfaces rubbing against each other. This kind of friction is called contact or sliding friction, and is partly responsible for not letting everything drift apart, and for in general slowing things down. Another kind of friction is viscous friction. In general, irreversible frictional processes will accompany the macroscopic motion of bodies surrounded by an external medium. This results in converting the kinetic energy of the bodies into heat, and ultimately bringing the motion to a halt. We say that the energy is dissipated. Thus, the physics of friction is of fundamental interest both for engineering purposes and for a better understanding of the principles of physics. This thesis will focus on the first kind of friction, namely the friction between solid bodies in contact.
In this chapter we will present a short summary of the most important results that have led us to the present knowledge about friction and we will show the results for the tribological characterization of our study material.
2.1. Friction
It is known from experience that when two solid objects make contact they will resist to relative motion along the plane of the contact area. The first practical use of friction probably took place about 100 000 years ago, when frictional heating was used as a tool for making fire. A surviving portion of a potter’s wheel with a pivot hole smoothed with bitumen dated at 3500 B.C, shows that the “practical laws” of friction have been known for a long time.[37]
The motivation for exploring the different aspects of friction must have had its origin in solving practical problems. An illustration from 2400 B.C (figure 21) shows how
lubricants are used when a big statue is being pulled to reach its destination. The person who applies the lubricant (the tribologist) between the sledge and wooden planks is seen in front of the sledge. The lubricant consisted of water which has a great lubrication effect on wood.
Figure 5 - The picture shows the transporting of the statue of Ti (2400 B.C). [37]
Little academic interest in friction existed until the XV century. The classic friction theories have been developed starting from the XV to the XIX century, (figure 22). Leonardo da Vinci suggested that friction was proportional to the normal load, but independent of the apparent area of contact.In the 1699 Guillelme Amontons (1663-1705), carried out some experiences about contact of non-lubricated solids; the result can be resumed in the two Amontons laws:
the friction force is proportional to the normal load, with respect to the contact surface, acting to the bodies;
the proportionality constant does not depend both upon the load and upon the area of the contact surface.
Further these two laws can be resumed in a single equation:
where µ is the proportionality constant, and N is the normal load. Amontons explained the friction force as the interaction between the asperities of the surfaces of the bodies.
Figure 6 - Chronology of the main friction theories
The independence of the friction coefficient from the extent of the surface has been verified by Bowden and Tabor (1939). By measuring the electrical conductivity of the link between two bodies, Bowden and Tabor proved that the real contact surface is sensibly lower then the apparent contact surface, and its extent depends upon the contact pressure. Indeed when two surfaces are in contact, only the asperities are in real contact, and thus the real contact surface is the sum of the surfaces of the asperities, (figure 23). The number and the extent of the micro-surfaces of contact does not depend upon the extent of the apparent contact surface, but only upon the contact force.
In the XVIII century Coulomb observed that the friction force which is exchanged between two fixed surfaces is greater than that exchanged between two surfaces in relative motion. Thus, Coulomb distinguishes two different friction forces: the static friction force, and the kinetic friction force, which can be lower than the static one. This represents the well-known Coulomb's friction law. In terms of equations, the results obtained by Coulomb can be expressed as
(3.2) (3.3)
where two different proportionality constants appear, , the static friction coefficient, and , the kinetic friction coefficient. Coulomb's friction law affirms that when no relative motion appears between two surfaces, the friction force cannot exceed the value of the static friction force , while when relative motion arises, the value of the friction force is equal to the kinetic friction force . For what concerns the direction and the sign of the friction force, when relative movement persists, the direction is that of the relative velocity, while the sign is the opposite; on the other hand, when there is no relative motion, direction and sign of the friction force are such that the equilibrium is guaranteed, figure 24. Thus, we have:
(3.4)
Although it is widely used, the description of the friction force via the Coulomb's law results approximate, in particular when used for studying phenomena where the sliding speed is unsteady. More accurate research methods and experiments, carried out by Bowden and Leben (1939) and Sampson et al. (1943), showed that the dynamic friction coefficient cannot be considered independent of the relative velocity between the surfaces in contact, in particular when in the interface between the surfaces there is a lubricant.[38]
Figure 8 - The Coulomb's friction law.
2.2. Composite material friction
Friction between sliding surfaces results in complex molecular-mechanical interactions. These interactions are the deformation of the roughness, ploughing by debris and roughnesses of the harder counterpart and the adhesion between the contact surfaces. Since ploughing is a kind of deformation, the friction force can be divided into a deformative component and an adhesive component[39]:
(3.5)
where is the friction force, is the adhesive friction force and is the deformative
friction force.The friction coefficient can be expressed analogously by:
(3.6)
The value of each component depends on the conditions of the sliding surfaces, Tulipawhich can be influenced by the characteristics of the involved materials, the topography of the surfaces and the environmental conditions [40]. These can be summarised as three basic aspects, which influence the dry friction process (friction without liquid lubricants):
the real contact surfaces,
the shear strength of the contacts,
the way in which the material is sheared and fractured in and around the contact zone in the sliding process.
2.2.1. Adhesive friction
Examining the separating process of the contacts of the sliding partner, we get the following equation for the adhesive friction force [40]:
(3.7)
The shear strength of the contacts, , depends on the adhesive properties of the contacting surfaces. is the real contact surface. Analogously, this is the following
expression for the adhesive friction coefficient:
(3.8)
The determining factors for the adhesive friction coefficient are the adhesive interaction and the real contact surface. The acting radius of the different adhesive forces is small [41]. Therefore, the effective zone of adhesion includes only the real contact areas. The real contact area is the result of penetration of roughness under normal load. For an elastic contact the real contact area, , is approximately proportional to the normal force,
, and inversely proportional to Young's modulus E:
(3.9) where . The relation can be physically more precisely expressed by the following equation:
where is the relative contact area, which is defined as the quotient of the real contact area and the nominal geometric contact area, is the roughness coefficient and is the normal pressure. The equation above shows that the influence of Young's modulus and the normal pressure is very small for a large relative contact area.[41] The equation for the adhesive component of the friction coefficient can be then derived from equations (3.8) and (3.10):
(3.11)
The adhesive component of the friction coefficient can be reduced by increasing Young's modulus and by decreasing the shear strength of the contacts.
2.2.2. Deformative friction
The depth of penetration is an important influencing factor for both the adhesive and the deformative component of friction and for wear. For elastic deformation, there is the following relation between material property, normal load and depth of penetration [40]:
(3.12) where is the depth of penetration, is the maximal roughness, is the proportionality factor, is the normal pressure, is Young's modulus and , are exponential factors, 0
< , < 1. The equation for plastic deformation is [41]:
(3.13) where is the compression yield point.The materials properties which influence the depth
of penetration are Young's modulus, the compression yield point or the hardness. The maximal roughness of the harder counterpart has a stronger influence on the depth of penetration under a higher normal pressure than under a lower normal pressure.[40]
The discussion of depth leads to another important aspect of friction, namely the deformative component of friction, which results in the form-locking characteristic in the tangential direction.This resistance can be only overcomed by deformation or fracture of the material. Considering the fact that the deformation resistance is proportional to the depth of penetration, eq. (3.12), and that the friction represents a kind of dissipation of mechanical
energy, which is proportional to the mechanical loss factor, tan δ, the following relationship is suggested for the deformation component of friction [42]:
(3.14)
2.3. Machines for tribological tests
There are many types of machines in order to test the friction materials used in vehicle brakes. Each one of these machines aims to cater specifically one (or more) types of test, according to their characteristics or procedures. The main equipments used for tests involving friction materials for use in brakes are the FAST (Friction Assessment and Screening Test), the Chase, the inertial Dynamometer and Krauss. [21,35]
There is yet another machine named tribometer that has been used by many authors for basic research purposes with friction materials used in vehicle brakes. A tribometer is a machine to test friction and wear of materials widely used as a tool of research for the understanding of tribological phenomena, as well as involving friction materials for use in brakes, as shown by recent literature reviews [43,44]. Many tribometers are also used for testing with respect to sensitivity of the friction materials to the variation of certain parameters, as described in some studies [45,46].
Tribometers are able to simulate severe conditions of brakes, such as occur in train brakes; they were used in the work of Desplanques et al. [40, 48, 49], Cristol-Bulthé,et al. [47] and Siroux, et al. [50]. Desplanques et al. [48] mentions that the equipment is able to control precisely the rate of energy delivered to the test specimen through the deceleration control. Another feature of this machine is its ability to control the environmental conditions of temperature and humidity through a closed chamber, as highlighted in the work of Desplanques et al.[48] and Siroux et al.[50].
Some interesting results were produced from this tribometer, such as correlations between the flow body and the third dynamic real area of contact, the connection between the physical mechanisms of friction and thermal phenomena of the friction pair. Desplanques et al. [48] claims that the designed equipment is able to produce representative results of a real brake system.
Figure 9 - Schematic drawing of the braking tribometer. [48]
Another tribometer found in the literature was used in two important studies, the first with respect to the effect of a water film on the frictional behavior [51] and the second with respect to the effects of air humidity on the wear measurements of the material friction [52]. Figure 10 shows a schematic drawing of the tribometer highlighting details of instrumentation and systems installed.
According to Blau and McLaughlin [50], the tribometer has three-phase motor 10 HP, with control of rotation by optical sensor, and a system of water gun spray on the disc, composed of an injector nozzle and a metal wrapper that collects and drains the water resulting of the experiments. Test data are acquired using the LabView program and 5 parameters can be monitored: engine speed, pneumatic actuator pressure (normal force), frictional force, instantaneous coefficient of friction (calculated) and disc temperature (only under dry conditions due to limitations imposed by the pyrometer).
Figure 10 - Schematic diagram of the sub-scale brake material tester (SSBT). [51]
Another tribometer employed for brake material tests is that used in the two papers published by Bhabani and Bijwe [45,46]. In both studies, the authors vary the percentage and type of fiber, evaluating the wear of the material under different operating conditions. The equipment used is a horizontal tribometer, fitted with a 7.5 HP motor, which is capable of delivering a speed of 1400 rpm in the disc. The machine works with two samples of 25 mm x 25 mm placed diametrically opposed and also operates with inertia discs on the shaft. A hydraulic actuator produces the contact pressure between the sample and the disc reaches values from 0.1 to 6 MPa. It is possible to control various operating parameters, such as pressure, speed, duration of braking and the number of cycles through the specific program of the machine.The tibometer is shown schematically in Figure 11.
Figure 11 - Schematic of pad-on-disc type friction tester (tribo-test-rig): 1-5 HP AC Motor; 2-cross coupling; 3-flywheel; 4-haft housing; 5-spindle; 6-rotor disc; 7-pad specimens; 8-torque sensor; 9- linear guide rod; 10-hydraulic actuator; 11-pressure sensor; 12-hydraulic power pack; 13-controller with preset timer, cycle counter and indicators; 14-interchangeable flange; 15-base plate; and 16-anti-vibration mountings. [45]
relevant since focused mainly on the tests, offering little information about the characteristics of the equipment used.
2.4. Laboratory investigations
The coefficient of friction is one of the most important characteristics of the brake pads. In addition to the power dissipated in braking, it can affect vibration phenomena such as stick-slip and squeal that are evident with annoying noise in specific frequency ranges. The knowledge of the coefficient of friction of materials in contact, and also of its possible dependence on the morphology of the surfaces and the operating conditions is therefore essential to predict the behavior of the brake already in the design phase. Tests were aimed at measuring the friction coefficient of the pad samples and at investigating possible different responses to area (contact area), rotational speed and load variations using a tribometer.
2.4.1. Materials and methods
Experimental characterization of the pad material was carried out on a pin on disc tribometer with the brake disc in place of the tribometer disc and the sample of pad fixed on a suitable interface in place of the pin (Figure 28a). The tribometer was driven by a 2 kW electric motor with a maximum speed of 70 rpm of the disc. The tribometer is equipped with a DS Europe load cell, model 729-60 QA with a measuring range of 0-600 N. To acquire the signal of the load cell, a card of the data acquisition of National Instruments (NI-DAQ model MX8) was used, with Labview program for the acquisition and pre-processing. A commercial brake friction material, which was developed for the front disc brake of a model Brembo Spa., was used in this study. Cylindrical samples of the same material but with different diameters were used, to vary the nominal contact area (Figure 28b). To fix the samples on the tribometer two supports were made, one support for each sample size (Figure 28c).In this study two discs were used, one of steel and other of cast iron (model Brembo Spa.),as shown in figure 28d. The load was applied adding masses on top of the “pin” that can slide vertically with negligible friction in the bore of a support fixed on a load cell that was used to measure the tangential friction force.
The test apparatus did not allow for a continuous variation and acquisition of the parameters, therefore tests were carried out: (a) at 1.5 rads-1, 4.5 rads-1, 7 rads-1 rotational speed (disc average contact radius of 110 mm),(b) with 62.5 N, 102.5N, 156.5 N load (c) and with 176.5 mm2, 113 mm2, 78.5 mm2 and 50 mm2 sample area. Each test was performed three times to verify repeatability. The tests were carried out according to the relevant ASTM standard [53]. For each test data were collected for 500 s; at intervals of 100 s load was varied as follows: 0-100 s (63.5 N ), 100-200 s (102.5 N) 200-300 s (156.5 N), 300-400 s (102.5 N) and 400-500 s (63.5 N).
a) b)
c) d)
Figure 12 - a)the tribometer employed in friction tests; b) samples used in the tests; c)the two supports; d) steel and cast iron disc.
2.4.2. Results and discussions
As we said earlier, each test was performed three times to verify repeatability. The figure 13 show an example of the raw data obtained. Based on the measurements acquired in the three tests carried out on each 14 to 17 show the average friction coefficients obtained in a test for
wear of different composite materials. Results (Figures 29 to 32) show that friction coefficient has a tendency to increase almost linearly with sliding speed for tests with steel disc. With the increase in sliding speed, the frictional heat may decrease the strength of the materials and high temperature results in stronger or increased adhesion with pin [54,55]. The increase of friction coefficient with sliding speed can be explained by the more adhesion of counterface pin material on disc. These findings are in agreement with the findings of Mimaroglu et al. [54] and Unal et al. [56] for pure and glass fiber reinforced Poly-Ether-Imide on polymer [54] and some industrial polymers [56] both with steel counterface.
Figura 13 - Raw data obtained of friction coefficient for steel disc.
It is possible to observe in the results (Figures 14 to 17) that the friction coefficient has a tendency to decrease almost linearly with sliding speed for tests with iron cast disc.A similar behavior was observed by Chowdhury et al.[57] and Dezi [58] for aluminum and composite material, respectively. The decrease of friction coefficient with the increase of sliding speed may be due to the change in the shear rate which can influence in the way in which the material in and around the contacting regions is sheared and ruptured during sliding. The shear force, frictional heat and frictional thrust are increased with the increase in sliding speed and these increments accelerate the wear rate.[59] This decrease of friction coefficient with sliding speed can be explained by the higher wear rate of pin material on iron cast disc.The tests performed on iron cast disc, produced for all tests, a large amount of wear debris. Figure 18 shows the wear debris of samples on the iron cast disc. The wear debris forms a film between the sample and the disc that acts as a kind of lubricant. The
0.11 0.16 0.21 0.26 0.31 0.36 0 100 200 300 400 500 µ Time (s)
decrease of friction coefficient occurs due to formation of this film of debris between the disc and the sample.
Analyzing the results, we observe that the force is a parameter that had a positive influence with respect to the friction coefficient for all the sample and discs, i.e. the increase in force leads to an increase in friction.The increase of friction coefficient with the increase of normal load may be due to increase in the adhesion strength.In many composite materials pairs, the friction coefficient is low at low loads and a transition occurs to a higher value as the normal load is increased.At low loads, there is little or no true contact, hence the friction coefficient is low.At higher load conditions, there is a greater intimate contact, i.e. greater adherence, which is responsible for higher friction.
Figura 14 - Average friction coefficient of sample with area of 176.5 mm2. Left: steel disc, Right: cast iron disc.
Figura 15 - Average friction coefficient of sample with area of 113 mm2. Left: steel disc, Right: cast iron disc.
0.15 0.20 0.25 0.30 0.35 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.20 0.22 0.24 0.26 0.28 0.30 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.15 0.20 0.25 0.30 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.20 0.22 0.24 0.26 0.28 0.30 0.32 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
,
Figure 16 - Average friction coefficient of sample with area of 78.5 mm2. Left: steel disc, Right: cast iron disc.
Figure 17 - Average friction coefficient of sample with area of 50 mm2. Left: steel disc, Right: cast iron disc.
Figure 18 - The wear debris of samples on the iron cast disc.
0.15 0.20 0.25 0.30 0.35 63.5 N 102.5 N 15 6.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.15 0.20 0.25 0.30 0.35 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.15 0.20 0.25 0.30 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
1.5 rad/s 4.5 rad/s 7 rad/s
0.15 0.17 0.19 0.21 0.23 0.25 0.27 63.5 N 102.5 N 156.5 N 102.5 N 63.5 N µ Load (N)
The results show that the contact area did not have a significant influence on the friction coefficient.However, we know that this contact area is the apparent area and not the real area. According to Klaus [39, 41], the measurement or the accurate prediction of the real area of contact produced by the contact of two rough solids is problematical, if not impossible. Direct measurement is impracticable and predictions are uncertain. Looking at the results of all tests and discs, in summary, it can be seen that the samples maintain the same position relative to the average friction coefficient, with values between 0.20 - 0.32, 0.15 - 0.30,0.15 - 032 and 0.15 - 0.30 for the sample with 176.5 mm2, 113 mm2, 78.5 mm2 and 50 mm2,respectively. All the results obtained for different operating parameters were processed and the results are summarized in the 3D diagrams of figures 19 to 22 that show the variation of average friction coefficient of frequency as function of speed and load for each disc.The results of these diagrams confirm the results that we observed in the previous figures.
Figure 19 - Linear regression of friction to load x speed for samples with 153.5 mm2. Left: steel disc (R2=0.83), Right: cast iron disc (R2=0.80).
Figure 20 - Linear regression of friction to load x speed for samples with 113 mm2. Left: steel disc (R2=0.60), Right: cast iron disc (R2=0.81).
Figure 21 - Linear regression of friction to load x speed for samples with 78.5 mm2. Left: steel disc (R2=0.63), Right: cast iron disc (R2=0.81).
Figure 22 - Linear regression of friction to load x speed for samples with 50 mm2. Left: steel disc (R2=0.76), Right: cast iron disc (R2=0.83).
2.5. Summary
In this chapter we present a short summary of the most important results that have led us to the present knowledge about friction and we show the results for the tribological characterization of our study material. Experimental characterization of the pad material was carried out on a pin on disc tribometer with the brake disc in place of the tribometer disc and the sample of pad fixed on a suitable interface in place of the pin. In this study two discs were used, one of steel and other of cast iron (model Brembo Spa.). Then, the tests were performed varying some operative parameters such as rotational speed , load and sample area.Each test was performed three times to verify repeatability. The tests were carried out according to the relevant ASTM standard [53].
Results show that friction coefficient has a tendency to increase almost linearly with sliding speed for tests with steel disc and that the friction coefficient has a tendency to decrease almost linearly with sliding speed for tests with iron cast disc. Analyzing the results, we observe that the force is a parameter that had a positive influence with respect to the friction coefficient for all the sample and discs, i.e. the increase in force leads to an increase in friction.The results show that the contact area did not have a significant influence on the friction coefficient. All the results obtained for different operating parameters were processed and the results are summarized in the 3D diagrams that show the variation of average friction coefficient of frequency as function of speed and load for each disc and these diagrams confirm the results that we observed in the raw data obtained.
Chapter 3
Numerical correlations between material and
morphological parameters and disc brake
squeal vibrations.
In the literature, many models for squeal prediction can be found starting from analytical models with few degrees of freedom, linear and with concentrated masses, to more complex, non-linear and continuous models. Finite element models are also applied to the modeling of squeal phenomenon, embracing models with only a disc and a bar generating friction and models that include the suspension of the vehicle and its interactions with the brake system.
Analytical and numerical models are able to simulate different structures, compositions of materials and conditions of operation of the brake systems.The theoretical results can provide orientations for an experiment, helping in the interpretation of the experimental results findings, and can be used as a design tool.
3.1. Finite element method (FEM)
In the lasts years, due to the large computational advancement, many researchers have used the finite element method to deal with the noise phenomenon in automotive brakes. According to Ouyang et al. [22] the simulations and methods of squeal analysis can be divided basically into two broad categories: analysis of complex eigenvalues in the frequency domain and transient analysis in the time domain. In the analysis of complex eigenvalues some, or all of the complex eigenvalues can be found at once, while in a transient analysis the program should be run several times, until a limit cycle motion is found. Therefore, the analysis of complex eigenvalues usually induces lower computational costs. In a transient analysis, in theory, it is not necessary as many approximations such as: constant contact area between disc and pads, linear friction law and properties of materials independent of time. These and other advantages and disadvantages were remarked by
Liles [60] was one of the first to model the brake components through finite elements and validate these models experimentally, thereby generating greater interest of other researchers in this type of analysis. In this same model, the author considers friction by geometric coupling and the stiffness matrix in this coupling was built using the relative displacement between the contact surfaces. Nack [61] presents a detailed method for obtaining the stiffness matrix considering friction by geometric coupling. The author determined some necessary conditions so that the system becomes unstable and noted that the amplitude of a squeal event grows exponentially at the beginning and then stabilizes in some way. This can be explained considering a limit cycle for the phenomenon in a non-linear model.
It is important to note that in the models of Liles [60] and Nack [61] matching nodes in the contact surfaces are necessary. From these studies, other finite element models were created to analyze the instability of the system. Some authors also showed that when two modes are coupled under the influence of friction, the system becomes unstable. The formulation of a contact problem applied to a brake disc was presented by Yuan [62]. The formulation of Yuan [62] resulted in a lower stiffness matrix than that of Liles [60] for the same problem, converging to a better numerical conditioning. The results presented by Yuan [62] showed that with a stiffness of the contact spring of 109 N / m, the results of both authors, Yuan [62] and Liles [60] are very near.
Guan and Jiang [63] built a finite element model of each component of a disc brake system, building a system reduced multibody model. They calculated, from an analysis of the system eigenvectors, the factor of modal participation, in other words, the influence of the modes of components in certain squeal modes. This type of information can be useful in order to find ways to suppress certain undesirable modes, and thus reduce the propensity to squeal. The results showed that there are more than one dominant component in the occurrence of squeal, but the disc has great contributions in the phenomenon. The finite element method to study the noise phenomenon in automotive brake was established by Liles [60] and is widely used today, and we can mention other works by Trichês et al. [64], Liu et al. [65], Fritz et al. [66], Choi [67], Kang [68], Dai [69] and Nacivet [70] who use this method of study and assess the various parameters that influence the generation of squeal.
