A complex eigenvalue approach to brake squeal performance: Comparing commercial and Coconut shell-reinforced friction materials in vented railway brake discs

Abstract

Natural fibers, like coconut shells, offer an eco-friendly and renewable alternative for brake pads, providing promising performance in terms of friction and durability. Research has shown that these reinforced materials can perform comparably to commercial options, particularly in areas such as wear resistance and mechanical stability. While many studies have focused on chemical and mechanical properties, there is still limited understanding of their impact on noise phenomena in braking systems. Further exploration is needed to evaluate their potential in reducing noise. Train noise pollution is a significant issue, largely due to vibrations and high-pitched sounds caused by friction in braking systems. This noise impacts passengers and arises from complex friction interactions where excess energy is converted into vibrations and sound. Advances in computing have enabled the application of the Finite Element Method to analyze these vibrations and their connection to noise phenomena in braking systems. This study proposes an iterative method to correlate brake component properties with instability parameters through finite element and complex eigenvalue analyses. The objective is to compare the performance of commercial and coconut shell-reinforced friction materials in mitigating brake squeal, with the aim of reducing vibrational instability and improving braking efficiency in train systems. The results demonstrate that the coconut shell-reinforced material outperforms the commercial material in reducing instability and noise across various parameters. The commercial material exhibits more unstable points and higher noise, while the reinforced material maintains better stability, lower noise, and more consistent performance in terms of friction, stiffness, and thickness variations. Overall, the reinforced material shows superior noise reduction and stability compared to the commercial option under the conditions studied.

Keywords

Railway brake disc acoustic emission brake squeal eco-friendly composites stability analysis

Introduction

A noise pollution associated with trains is a significant issue, largely due to the vibrations and high-pitched sounds produced by friction in the railway braking system. The noise produced by railway disc brakes is a common source of discomfort for passengers, affecting both those aboard the trains and those waiting at stations.^1,2 Interfacial friction events are known to be quite complex, and brake squealing can be triggered when a rotating brake disc rubs against the brake pad under certain operating conditions. Due to the friction generated between the pads and the disc, the kinetic energy of the train is primarily dissipated as heat, causing the train to lose speed.³ The difference between static and dynamic friction coefficients is well established in scientific knowledge, with the first two laws proposed by Amonton in 1699.⁴ According to Akay,³ when the friction condition provides more energy than the system is able to dissipate in the form of thermal energy, part of this energy is converted into dynamic instability (vibrations) which in turn generate sound radiation. The discontinuity in the friction regime is closely related to the occurrence of vibrations in brake systems, as it causes energy dissipation peaks. However, as living standards improve and environmentally friendly regulations are established, consumers are increasingly prioritizing noise pollution reduction and comfort levels in trains.²

Engineers and researchers have classified brake noise into several categories, with squeal being the most significant due to its friction-induced nature.^3,5–8 Squeal is an auto-excited noise that creates self-sustained vibrations for a short period.^9–11 Two definitions of squeal exist: Low-frequency squeal, with spectrum bands varying from 1 to 5 kHz, and high-frequency squeal, occurring between 6 and 20 kHz.^3,8,12,13 Currently, there are three explanations for the occurrence of brake squeal: Stick-slip, sprag-slip, and mode coupling.^3,8,13 The stick-slip explanation is primarily based on the assumption that the coefficient of friction between the brake pad and the brake disc decreases with increasing sliding speed. In systems where the coefficient of friction declines with increased speed, negative damping factors can occur, leading to unstable oscillation modes and resulting in brake squeal.^14–16 The sprag-slip explanation arises from the geometrical deformation of the brake components, primarily the friction material. This deformation leads to variations in normal and tangential forces, making the system prone to brake squeal.¹⁷ The third explanation, mode coupling, occurs when the vibration modes of the brake components coincide. It has been shown that this can happen even if the friction coefficient remains constant.¹⁸ It has also been demonstrated that a brake squeal occurs only when at least two of these events occur simultaneously.^3,14

Experimental studies aimed at understanding the dynamic instability of brake systems due to various parameters often face significant challenges, primarily related to budget constraints. In recent years, due to great advances in computing, researchers have used the Finite Element Method to deal with the phenomenon of vibration generation.^19–24 Liles²⁵ was one of the first to use the finite element method to try to understand the phenomenon of vibrational instabilities in brake systems. According to Liles,²⁵ knowing the unstable modes of the brake system facilitates several courses of action during design; the modal frequencies could be moved by changing the components or damping could be added so that the unstable mode in question becomes stable.

According to Ouyang,¹⁴ the simulations and analysis methods for the Squeal of the brakes can be divided into two broad categories: Analysis of complex eigenvalues in the frequency domain and transient analysis in the time domain. In the complex eigenvalue analysis, it is possible to find some or all of the eigenvalues at once, while the transient analysis program must be run several times, until a cycle limit movement is found. Therefore, the complex eigenvalue method yields lower computational costs.¹⁴ Over the past thirty years, numerous numerical studies have been conducted across a wide range of brake systems, aiming to better understand the underlying phenomena and the factors that influence them.^{6,7,16,19–24,26–29} Although research on the prediction and suppression of squeal noise has steadily increased, most of the focus has been on the automotive industry, with brake noise in the railway sector receiving far less attention in the existing literature.^2,30,31

Given the critical role brake pads play in both noise generation and overall system performance, understanding their material composition and behavior becomes essential, especially in addressing brake squeal across different sectors. Brake pads are fundamental components in any braking system, playing a crucial role in vehicle safety and, consequently, ensuring a safe driving experience. They are responsible for generating the friction needed to slow down or completely stop the vehicle when the brake is applied, converting kinetic energy into heat. The quality, durability, and performance of brake pads directly influence the efficiency of the braking system, as well as the comfort of the driver and passengers, since they are also related to the reduction of unwanted noise and vibrations during the braking process. Brake pads are made from complex composites that incorporate various materials critical for enhancing wear resistance and friction performance. Using natural fibers as reinforcement in these composites presents a promising, environmentally friendly alternative due to their biodegradability and renewable properties. These sustainable materials have broad potential for application across various fields, thanks to their environmentally friendly properties.³²

Abutu et al., in their study, produced friction material for brakes using reinforcement fibers. In this study, non-hazardous local materials, such as coconut shell, were used to manufacture brake pads, optimized through grey relational analysis (GRA). The results showed that the coconut shell-reinforced pads perform comparably to commercial ones.³³ In another study, Abutu et al.³⁴ used seashells as a non-hazardous reinforcement material to produce brake pad composites. The performance of the new brake pads compared favorably with commercially available ones. It was found that changes in process parameters (molding pressure, molding temperature, curing time, and heat treatment time) affect the material properties, as all the brake pads developed with varying process parameters exhibited different performance values. The literature presents various studies in which different materials are used as reinforcement for brake pads, such as banana peels, palm kernel fibers, wood powder, cow bone, sisal fiber, sugarcane fiber (bagasse), among others.^35–38 While these studies offer extensive chemical and mechanical analyses of the new reinforced materials compared to commercial ones, their performance in relation to noise phenomena in braking systems remains underexplored.

Building on the existing approaches to vibrational instability, this work introduces an iterative method to correlate brake component material properties—such as Young’s modulus and coefficient of friction—and geometric features with instability parameters like frequency, instability modes, and their intensity. By utilizing the finite element method and complex eigenvalue analysis, this method identifies the key factors influencing brake squeal. The objective is to compare the performance of commercial and coconut shell-reinforced friction materials, aiming to reduce vibrational instability and improve braking performance in train systems.

Nomenclature

The main notations adopted in this document are reported in in Table 1.

Table 1.

Parameter nomenclature.

Symbol	Name
Pressure	P
Contact pressure	p
Static pressure at interface	p₀
Maximum pressure	p_max
Normal contact force	F_n
Frictional contact force	F_f
Normal stress	σ _x
Shear stress	τ _xy
Elongation or compression offset of the component	dL
Length of the component	L
Friction coefficient	μ
Disc speed	ω
Force	F
Disc Young’s modulus	E_Disc
Back plate Young’s modulus	E_{Back Plate}
Pad material Young’s modulus	E_Pad
Disc density	γ _Disc
Back plate density	γ _{Back Plate}
Pad material density	γ _Pad
Disc Poisson’s ratio	ν _Disc
Back plate Poisson’s ratio	ν _{Back Plate}
Pad material Poisson’s ratio	ν _Pad
Disc thickness	h_Disc
Back plate thickness	h_{Back Plate}
Pad material thickness	h_Pad
Disc diameter	D_Disc
Pad material contact area	A_pad
Complex eigenvalues	λ _i
Real part eigenvalue	σ _i
Imaginary part eigenvalue	ω _i
Nodal normal force row matrix	{f_n}
Nodal friction force row matrix	{f_f}
System mass square matrix	[M]
System rigidity square matrix	[K]
System damping square matrix	[C]
System friction matrix	[K_f]
System contact stiffness matrix	[K_c]
System contact-friction matrix	[K_fc]
Eigenvector rectangular matrix	[V]^T
System nodal acceleration row matrix	{ $\ddot{U}$ }
System nodal velocity row matrix	{ $\dot{U}$ }
System nodal displacement row matrix	{U}
System applied resulting force row matrix	{F}
Nodal braking force	{F_brake}
Displacement boundary conditions	B.C

Methodology

The process driven by this analysis (Figure 1) begins with the development of railway disc brakes finite element model. Five parts are considered, the disc, two back plates and two brake pads. For the analysis, a Vented Railway Disc Brakes was considered (see Figure 2). This model include, as input parameters, the material modulus of elasticity (E) and friction coefficient (μ) owing to contact between the brake pad and the disc during its rotation. The brake disc is fixed to the hub that is modeled with a fixed support (Sticker A on Figure 2). The brake pad (Friction Material and Back Plate) are modeled with the remote displacement (Sticker B on Figure 2), though with activation force on the opposite extremity (Sticker A on Figure 2). The support fixes the body for translational and rotational movements while providing freedom for displacement normal to the disc. The brake disc mesh comprises 5439 elements and 13,310 nodes for the ventilated disc. The elements of contact between the friction material and disc are CONTA174 and TARGE170, respectively. According to Ansys APDL Theory Reference CONTA174 is a 3-D and 8-node element located on the surfaces of solids. It adopted quadrilateral elements on friction material surfaces through the command KEYOPT (4). For the disc internal surface, TARGE170 permits the rotational displacement input (ω) on the elements using CMROTATE command. Furthermore, it is performed by the following processes, (1) a static analysis of the brake disc operation, without disc rotation, to achieve the pre-stress conditions, and (2) a frequency calculation of the complex eigenvalues during disc rotation while the brake pad are coupled on it. Moreover, with all calculations finished, the output parameters are extracted. These are consisted by unstable vibration modes, that is, the complex eigenvalues with a positive real part and its frequencies. Hence, these data are used for the generation of the bar chart, relating input and output parameters, dot plot chart, and relating the imaginary part and real part of the eigenvalue and the response surface graphs that are statistical representations of the output parameter behaviors in relation to input parameters.

The brake disc design procedure is applied on models illustrated by Figure 2. The small circle illustrates the contact between the pad material and the disc being F_n the normal force, F_f is the friction force and K_c the contact stiffness. Hence, considering that disc is rotating with constant velocity, unidirectional sliding, and permanent contact, then according to Coulomb’s law of friction:

f_{t} = - μ f_{n}

(1)

There are two stresses generated in section of contact, therefore, Coulomb’s law can also be written in terms of shear stress τ and normal stress σ as follows:

τ = - μ p

(2)

σ = p

(3)

The equivalent stress is evaluated by ANSYS using von Mises theory:

σ^{'} = \sqrt{σ_{x}^{2} + 3 {τ^{2}}_{x y}}

(4)

Finally, it is known that Young’s modulus E is a relation between stress (σ) and strain (ϵ); thus, pad material Young’s modulus can be expressed by:

E_{p a d} = \frac{σ^{'}}{ϵ} = \frac{\sqrt{σ_{x}^{2} + 3 {τ^{2}}_{x y}}}{d L / L_{0}}

(5)

To model the contact between the pad material and disc, an interface of finite elements of mass, mechanical springs, and dash-pots is used. The rigidity of the contact k_c is estimated by:

k_{c} = \frac{E_{p a d} A_{p a d}}{h_{p a d}}

(6)

Hence, substituting the E_pad (equation (5)) in equation (6), it can be observed that p impacts on contact rigidity:

k_{c} = \frac{p L_{0} A_{p a d} \sqrt{1 + 3 μ^{2}}}{d L h_{p a d}}

(7)

The pad material is discretized into a set of spring-modified elements of rigidity k_c and friction coefficient μ.⁵ The equation below

f_{n} = k c u n \to f_{t} = - μ f_{n} = - μ k c u n

(8)

results in elementary matrix equations (9) and (10). Equation (9) is given as

[K_{c}] [\begin{matrix} U_{2 n} \\ U_{1 n} \end{matrix}] = [\begin{matrix} F_{2 n} \\ F_{1 n} \end{matrix}]

(9)

Expanding to the node canvas and adding [K_c] and [K_f] to form the contact-friction element [K_cf] results in

[K_{f c} = K_{c} + K_{f}] [\begin{matrix} U_{2 n} \\ U_{2 t} \\ U_{1 n} \\ U_{1 t} \end{matrix}] = [\begin{matrix} F_{2 n} \\ F_{2 t} \\ F_{1 n} \\ F_{1 t} \end{matrix}]

(10)

After K_fc definition, the approach to model the system without approximation on the pad and to create an FEM model of the pad and join it to the FEM model of the back plate is the Lagrange’s multipliers. Therefore, the coupling model between the brake pad and the disc and the contact-friction model are acquired by the full system matrix equation

[M] {\ddot{U}} + [C] {\dot{U}} + ([K] + [K_{f c}]) {U} = {F b r a k e} + B . C .

(11)

Equation (11) can be solved including F_brake = F(t), and this type of analysis is reserved for full transient studies. The pseudo-rotating hypothesis that will be adopted consists in considering a fixed mesh for the disc. It is important to mention that the pad material and back plate mesh is also fixed. The rotating part has to be axisymmetric, and the contact nodes stay opposite. In addition, the disc rotation is considered for displacement, velocity, and acceleration of it. This affects the manner that centrifugal forces and gyroscopic damping matrix could be added to the equation (11)^6,19,20,21. From this, a homogeneous equation is obtained removing F_brake from the second member of it. However, F_brake is internally involved at the contact. Hence, before removing it, a static computation is carried out which gives p₀, the static pressure at the interface, for a given value of the force, F_brake = F₀. The contact and friction stresses now are reformulated:

p n = p 0 + k c u n, τ = - μ p = - μ k c u n

(12)

This results in [K_fc] that is modified to [ K_fc]+[ K_p0], where [K_p0] is the pre-stressed contact-friction stiffness matrix for a given load F₀. Therefore,the homogeneous equation to be studied is written as follows:

[M] {\ddot{U}} + [C] {\dot{U}} + ([K] + [K f c] + [K f c (F 0)]) {U} = 0

(13)

Static structural analysis

The static structural analysis is a simulation of these brake systems in which the pad material, fixed on the back plate, initiate a frictional contact with the disc surface without rotation (Figure 2). The inertial and rigidity influences of components are considered for the eigenvalue and eigenvector calculations.²²

([K] + [K f c]) {U} = {F b r a k e} + B . C .

(14)

Simulation process starts with a static structural analysis, in which the disc is considered immobile and the brake pressure is applied, imposing the contact between the pads and the disc. This situation is represented by the following boundary conditions: The internal face of the fixing hole in the disc were considered as fixed supports (Figure 2), the outer faces of the back plates were constrained in all degrees of freedom, except for translation in the direction normal to the contact surface (Figure 2), and a distributed force is also applied to the surface of the outer faces of the back plates (Figure 2). The brake disc operation (Figure 2) begins with pressure of 0.5 MPa³⁹ applied on back plate (F sticker); The Ansys contact element terminology is CONTA 174 - TARGET 170, a sophisticated contact based on Lagrange’s multipliers.

Complex modal analysis

This analysis uses the static structural stress field and results. The disc rotates at constant 5 rad/s speed,³⁹ while the 0.5 MPa pressure³⁹ pushes the brake pad against the disc. In this analysis, the movement equation internal dissipation (damping) is neglected without drastic penalties to the results.²³ Hence, the dynamic equation of equilibrium is given by

[M] {\ddot{U}} + [K n s] {U} = {F}

(15)

The contact-friction elements are used to integrate the incompatibility of the meshes and also to take into account the pre-loads and friction that lead to an unsymmetrical matrix. Therefore, the term [K_cf] is renamed to [K_ns] owing to non-symmetry of the contact-stiffness matrix. The asymmetry arises from the pad material and disc contact, with relevant influence of their material characteristics. The eigenvalue problem to be solved is written as follows:

[M] {\ddot{U}} + [K n s] {U} = 0

(16)

As [K_ns] is non symmetric contact stiffness, the system will exhibit complex eigenvalues,λ_i, written as

λ i = σ i \pm j ω i

(17)

The extraction of the complex eigenvalues and eigenvectors is fulfilled through the Ansys QR damped method that results in a reduced system of equations:

[k n s] = {[V]}^{T} [K n s] [V]

(18)

Hence, the complex eigenvectors and eigenvalues of the reduced system are extracted allowing the study of the stability looking for the sign of the real part of the eigenvalues (Figure 3 ). The brake instability occurs when eigenvalues with positive real parts are generated.²⁵ Generally, the real part of the complex eigenvalues is plotted against the associated natural frequencies, in order to visualize the general occurrence of instability. A preliminary implementation of the model, with the reference values from Table 2, was carried out to evaluate the consistency of the model. The material properties were taken from the works of Maciel et al.,⁸ Ahn et al.^39,40 and Abutu et al.³³ The operational parameters (Coefficient of friction, Pressure and speed of rotation of the disc) were taken from Cascetta et al. and Sha et al.^41,42 The geometric parameter data, encompassing measurements such as thickness, diameter, and additional factors, were extracted from Cascetta et al. and Sha et al.^41,42

Table 2.

Nominal vented railway brake disc parameters.

Parameter name	Notation	Value
Friction coefficient	μ	0.30
Disc speed	ω	0.5 rad/s
Pressure	P	0.5 MPa
Disc Young’s modulus	E_Disc	1.20 E + 11 Pa
Back plate Young’s modulus	E_{Back Plate}	2 E + 11 Pa
Pad Young’s modulus 1^a	E_Pad	5.2 E + 9 Pa
Pad Young’s modulus 2^b	E_Pad	7.57 E + 9 Pa
Disc density	γ _Disc	7550 kg/m³
Back plate density	γ _{Back Plate}	7850 kg/m³
Pad material density 1^a	γ _Pad	2516 kg/m³
Pad material density 2^b	γ _Pad	1073 kg/m³
Disc Poisson’s ratio	ν _Disc	0.3
Back plate Poisson’s ratio	ν _{Back Plate}	0.3
Pad material Poisson’s ratio 1^a	ν _Pad	0.25
Pad material Poisson’s ratio 2^b	ν _Pad	0.25
Disc thickness	h_Disc	108 mm
Back plate thickness	h_{Back Plate}	6 mm
Pad material thickness	h_Pad	30 mm

^aCommercial material.

^bCoconut shell-reinforced material.

As can be seen from Figure 3, both pad materials generated several modes of instability. The results indicate that the commercial material exhibits 6 unstable modes in the low squeal zone and 114 unstable modes in the high squeal zone, with a total number of unstable frequency (TUF) of 120 unstable points. (Table 3). For the Coconut Sheel-reinforced material, the results show 5 unstable modes in the low squeal zone and 60 unstable modes in the high squeal zone, with a total number of unstable frequency (TUF) of 120 unstable points. (Table 3) These initial simulations demonstrate that, in the reduction of unstable modes, the coconut shell-reinforced material appears to be more effective. It exhibits fewer unstable modes in both zones compared to the commercial material, especially in the high squeal zone. This suggests that the coconut shell-reinforced material may be more efficient in reducing squeal, particularly at higher frequencies. However, not all of these modes result in squeal. Vibrational modes that are slightly unstable in theory may never become unstable in reality due to dissipative damping in an actual brake system.⁴³ These unstable modes, while not generating noise, can cause other issues in the structure such as gaps, structural damage, and increased wear on components, ultimately reducing their useful life.

Table 3.

Instability occurrence on vented disc for commercial and coconut sheel-reinforced material.

	Coconut sheel-reinforced	Commercial
Low squeal	5	8
High squeal	37	79
TUF	42	87

This article presents the design of a rotor brake that applies braking force to railway brake discs with commercial and coconut shell-reinforced friction materials. The modal analysis was performed to extract the complex eigenvalues for the vibrating modes of the disc brake system up to 25 kHz. When there are modes coupled at the same frequency, one of them becomes unstable. The unstable modes can be identified during complex eigenvalue analysis, because the real parts of the complex eigenvalues are positive (3). Those modes are prone to generate instabilities.²⁴

Results and discussion

The disc brake system instability prediction model addressed in this paper is partially disturbed modal analysis. Figure 1 is a flowchart of the process adopted in order to investigate the influence of the input parameters on instability generation (sensitivity analysis). A variation of the input parameters there should be some effects on the output parameters. This analysis allows the creation of a parameter sensitivity. Table 4 summarizes all input values for this analysis, divided in two main categories: geometric parameters (disc, pad and back plate thickness), and materials parameters (Disc and Back Plate Young’s modulus). Since the friction coefficient between the pads and the disc generates energy that, in excess, can be converted into dynamic instability (vibrations), its value was varied to examine its influence on the phenomenon. This also tested the efficiency of the numerical model in representing the phenomenon. Understanding the relationship between geometric and material parameters, along with the coefficient of friction in the brake squeal phenomenon, is fundamental when seeking to improve brake system design and reduce the effects of squeal.^44–46

Table 4.

Range values of input parameters for parametrization.

Parameter name	Value range
Friction coefficient	[0.15, 0.3 and 0.45]
Disc Young’s modulus	[60, 120 and 240] GPa
Back plate Young’s modulus	[100, 200 and 400] GPa
Disc thickness	[88,108 and 128] mm
Back plate thickness	[5,6 and 7] mm
Pad material thickness^a	[20, 30 and 40] mm

^aThis thickness was used for both the commercial and the coconut shell-reinforced materials.

To evaluate the performance of vented railway brake discs, three objectives were established.

1. Maximize first unstable frequency FUF, min ω i for σi positive;

2. Minimize TUF, the total number of unstable frequency, ∑_i for σ_i positive (in the given frequency range of analysis);

3. Minimize NI, noise index. In terms of the noise index (NI), an brake system may exhibit many unstable vibration modes within the audible frequency range. However, not all of these modes result in squeal. Vibrational modes that are slightly unstable in theory may never become unstable in practice due to dissipative damping in a real brake system. To compare the squeal propensity among unstable vibrational modes, the magnitude of the instability has traditionally been used as a noise index. In this study, the noise index was defined as Yuan^43,47 for each vibration mode;

N I = \frac{ω i}{\sqrt{{ω i}^{2} + {σ i}^{2}}} \times 100 % (i = 1,2 \dots)

(19)

where, σ_i is the instability of the system (real part) and ω _i is the natural frequency of the system (imaginary part). The greater the noise index the more likely the corresponding mode was considered to cause audible squeal noise.

Freio a disco de moto

Disc and friction material form the frictional pair of the disc brake system. Selection of materials for the disc is of vast importance. There is a set of limitations to development of special materials for disc brake applications, cost being the primary. The disc, besides braking, is responsible for heat dissipation in the braking system.⁷ Friction material purpose is to control the movement by deceleration of the vehicle, through transformation of kinetic energy in heat, via friction, and dissipate the heat to the medium. The disc brake pads consist of two parts, friction plates which are made of organic composite material and back plates, made of steel. The main function of the back plate is to absorb the vibration and consequently to reduce the instability.^7,44 Hence, it becomes imperative to assess the impact that the material’s properties of this components exert on the instability of the brake system.

Brake squeal is generally defned as a friction-induced instability phenomenon. Since friction is the main cause of instability, it must always be considered for brake noise analysis.^3,7,14 Table 5 and Figure 4 present a comparison of braking performance between a commercial material and a coconut shell-reinforced material under different coefficients of friction. In the low squeal zone for the commercial material, the results demonstrate an increase in the number of unstable points as the coefficient of friction increases, suggesting that, with more friction, the material tends to generate more noise and unstable points in this zone. For the coconut shell-reinforced material, in the low squeal zone, unstable points are non-existent at the lowest coefficient of friction (0.15), but slightly increase at higher coefficients, indicating that this material has better resistance to generating low squeal noise (Table 5).

Table 5.

Analysis of the variation in the coefficient of friction in vented disc for commercial and coconut sheel-reinforced material.

	0.15	0.30	0.45
Commercial material
Low squeal	5	8	11
High squeal	39	79	132
TUF	44	87	143
FUF	3458.2 Hz	3362.9 Hz	2416.4 Hz
NI	0.6	2.2	6.1
Coconut shell-reinforced material
Low squeal	0	5	7
High squeal	13	37	62
TUF	13	42	69
FUF	8485.5 Hz	2417.1 Hz	2417.1 Hz
NI	0.02	2.2	3.8

In the high squeal zone for the commercial material, the unstable points increase significantly with the rise in friction, from 39 to 132. This suggests that the commercial material is prone to generating high squeal noise at higher friction coefficients. In the same zone, the reinforced material also shows an increase with the coefficient of friction, but the rise is less pronounced compared to the commercial material. This suggests that the reinforced material is acoustically more stable in relation to high squeal noise. The TUF (Total Under Force) for both materials increases with friction; however, for the reinforced material, the increase is lower than that of the commercial material. For all three parameters (low squeal, high squeal, and TUF), the coconut shell-reinforced material exhibited significantly fewer unstable points than the commercial material (Table 5).

The FUF for the commercial material decreases from 3458.2 Hz to 2416.4 Hz, showing that the material vibrates at lower frequencies as friction increases. For the reinforced material, the FUF is initially high (8485.5 Hz) at 0.15, but reduces to 2417.1 Hz for the other coefficients. In almost all cases, the commercial material achieved lower FUF values compared to the reinforced material (Table 5). The NI for the reinforced material is extremely low at a friction coefficient of 0.15 but grows more moderately compared to the commercial material, indicating that the reinforced material generates less overall noise, especially at lower friction levels (Table 5). The commercial material also shows a significant increase in noise with rising friction, reflecting an overall increase in noise levels. For all friction values analyzed, the reinforced material had lower or equal NI values compared to the commercial material.

The Table 6 and Figure 5 compares the braking performance of a commercial material and a coconut shell-reinforced material under varying values of the disc’s Young’s modulus. In the commercial material, the Low squeal remains stable around 8 at lower Young’s modulus values and slightly decreases to 7 at 240 GPa, suggesting little variation in the generation of low noise as the disc stiffness increases. The reinforced material shows a different behavior, decreasing from 9 to 4 with increasing stiffness, indicating that the reinforced material performs better in reducing low noise with stiffer discs.

Table 6.

Analysis of the variation in the Disc Young’s modulus in vented disc for commercial and coconut sheel-reinforced material.

	60 GPa	120 GPa	240 GPa
Commercial material
Low squeal	8	8	7
High squeal	99	79	80
TUF	107	87	87
FUF	2800.9 Hz	3362.9 Hz	3404.1 Hz
NI	1.9	2.2	2.3
Coconut shell-reinforced material
Low squeal	9	5	4
High squeal	37	37	85
TUF	46	42	89
FUF	3983.4 Hz	2417.1 Hz	4829.8 Hz
NI	2.1	2.2	2.0

In the High squeal zone, the commercial material shows an initial decrease from 99 to 79 at 120 GPa, but slightly rises to 80 at 240 GPa, indicating that increasing stiffness helps to reduce high squeal noise at intermediate stiffness values, but the effect stabilizes at higher levels. In the reinforced material, the High squeal remains stable at 37 for the first two modulus values but rises sharply to 85 at 240 GPa, indicating a higher generation of high squeal noise under extreme stiffness conditions (Table 6). The TUF decreases from 107 to 87 between 60 and 120 GPa for the commercial material, stabilizing at 240 GPa, suggesting that increasing stiffness reduces the total number of unstable points but reaches an equilibrium. For the reinforced material, the TUF increases from 46 to 89, following the increase in stiffness, indicating that the reinforced material experiences greater instability in stiffer discs. In all three parameters, the reinforced material exhibited lower instability values compared to the commercial material (Table 6).

Both materials show an increase in FUF with stiffness, although the reinforced material shows more significant variation, suggesting a more sensitive response to stiffness changes. The FUF increases from 2800.9 Hz to 3404.1 Hz with increasing stiffness for the commercial material, indicating that it vibrates at higher frequencies in stiffer discs. The reinforced material exhibits variable behavior, with a higher frequency at 60 GPa (3983.4 Hz), a drop at 120 GPa (2417.1 Hz), and a significant increase at 240 GPa (4829.8 Hz), indicating more unpredictable acoustic responses compared to the commercial material (Table 6).

The NI slightly increases from 1.9 to 2.3 for the commercial material (Table 6), indicating a rise in overall noise levels with increased stiffness. For the reinforced material, it remains stable, ranging from 2.1 to 2.0, suggesting that stiffness has little impact on the overall noise levels for the reinforced material. The NI values for the reinforced material were higher than those of the commercial material at 60 GPa (2.1 for the reinforced vs 1.9 for the commercial). At the other two Young’s modulus values (120 GPa and 240 GPa), the NI values are similar or lower for the reinforced material.

Table 7 and Figure 6 present the analysis of the variation in the Young’s Modulus of the back plate in vented discs for commercial material and coconut shell-reinforced material. The results reveal significant differences in acoustic behavior and stability between the two materials with increasing backing plate stiffness. In the commercial material, Low squeal remains stable around 7-8, indicating little variation in low noise generation with increasing stiffness from 100 GPa to 400 GPa. For the reinforced material, Low squeal also remains stable at 5, suggesting that this material is more effective in preventing low noise regardless of stiffness (Table 7).

Table 7.

Analysis of the variation in the back plate Young’s Modulus in vented disc for commercial and coconut sheel-reinforced material.

	100 GPa	200 GPa	400 GPa
Commercial material
Low squeal	7	8	8
High squeal	94	79	92
TUF	101	87	100
FUF	3344.5 Hz	3362.9 Hz	3372 Hz
NI	3.2	2.2	1.9
Coconut shell-reinforced material
Low squeal	5	5	5
High squeal	37	37	50
TUF	42	42	55
FUF	2417.1 Hz	2417.1 Hz	2417.1 Hz
NI	2.2	2.2	2.0

Regarding High squeal, the commercial material shows an initial decrease from 94 to 79 at 200 GPa but increases slightly to 92 at 400 GPa. This suggests that while intermediate stiffness helps reduce high noise, this effect is lost at higher stiffness values. In contrast, the reinforced material remains stable at 37 up to 200 GPa, increasing slightly to 50 at 400 GPa, indicating that the reinforced material generates less high noise but its stability decreases at extreme stiffness (Table 7). The TUF for the commercial material follows a similar pattern, decreasing from 101 to 87 at 200 GPa but increasing to 100 at 400 GPa, suggesting that increasing stiffness improves stability to a point. The reinforced material shows more stable behavior, with TUF increasing slightly from 42 to 55, indicating that this material remains more stable in stiffer discs. The reinforced material consistently had lower values of Low squeal, High squeal, and TUF compared to the commercial material when varying the Young’s modulus of the backing plate (Table 7).

The FUF for the commercial material varies little, remaining around 3344.5 Hz to 3372 Hz (Table 7), indicating that the stiffness of the plate has little effect on the vibration frequency of the material. In the reinforced material, FUF remains constant at 2417.1 Hz for all stiffness levels, suggesting that this material is less sensitive to stiffness variation. The commercial material exhibited higher FUF values than the reinforced material in all cases. The NI in the commercial material decreases from 3.2 to 1.9 as stiffness increases, suggesting that stiffer discs tend to generate less overall noise. The reinforced material remains stable around 2.2 to 2.0, with smaller variations, indicating that increasing stiffness has less impact on overall noise levels in this material. The reinforced material did not consistently have lower NI values; it was lower only at 100 GPa (Table 7).

Geometric parameters

Much like the significance of the material properties associated with the disc, back plate, and pad, the geometric attributes hold equal importance.^45,46 These characteristics wield direct influence not only over the overall mass of the brake system but also over its inherent rigidity. The thickness of the disc brake is reduced when it comes into contact with the brake pad due the effect of wear and tear and this is an important parameter for safety.⁴⁵ The first result that will be analyzed in this topic is disc thickness (Figure 7 and Table 8). Low squeal decreases from 9 to 7 as the thickness increases from 88 mm to 128 mm for the commercial material, and decreases from 7 to 5 with increasing thickness for the reinforced material, indicating a reduction in low noise with thicker discs for both materials.

Table 8.

Analysis of the variation in the disc thickness in vented disc for commercial and coconut sheel-reinforced material.

	88 mm	108 mm	128 mm
Commercial material
Low squeal	9	8	7
High squeal	106	79	81
TUF	115	87	88
FUF	3062.1 Hz	3362.9 Hz	3548 Hz
NI	2.2	2.2	2.0
Coconut shell-reinforced material
Low squeal	7	5	5
High squeal	51	37	29
TUF	58	42	34
FUF	4336.5 Hz	2417.1 Hz	4457.9 Hz
NI	1.7	2.2	2.2

For the commercial material, High squeal shows an initial decrease from 106 to 79 with increasing thickness, but then slightly increases to 81, suggesting that thicker discs initially reduce high noise but the effect is less effective at greater thicknesses. For the reinforced material, High squeal decreases from 51 to 29 with increasing thickness, suggesting that thicker discs are more effective at minimizing high noise (Table 8). TUF decreases from 115 to 88 for the commercial material, indicating improved stability with increasing thickness, and shows a significant decrease from 58 to 34 for the reinforced material, indicating that the reinforced material remains more stable with increasing thickness compared to the commercial material. Although the commercial material often shows higher values than the reinforced material for Low squeal, High squeal, and TUF, this is not always the case, especially as the disc thickness increases (Table 8).

FUF increases from 3062.1 Hz to 3548 Hz, suggesting that thicker discs vibrate at higher frequencies for the commercial material. FUF varies more for the reinforced material, increasing from 4336.5 Hz to 4457.9 Hz, with a drop to 2417.1 Hz at 108 mm, reflecting variability in vibration frequencies. The commercial material had lower FUF values compared to the reinforced material for 88 mm and 128 mm thicknesses, and a higher FUF value for the 108 mm thickness (Table 8).

NI is slightly lower for the coconut shell-reinforced material, with values of 1.7 for the smaller thickness (88 mm), reflecting better overall noise performance for thinner discs. For greater thicknesses (108 mm and 128 mm), NI remains stable at 2.2, indicating that despite the increase in thickness, noise performance does not improve significantly. In comparison, the commercial material shows a relatively stable NI around 2.2, decreasing slightly to 2.0 with increasing thickness, reflecting a slight improvement in overall noise levels (Table 8). The reinforced material had a lower NI only for the 88 mm thickness. For 108 mm and 128 mm thicknesses, the NI of the reinforced material was the same as that of the commercial material.

Table 9 and Figure 8 present the analysis of the variation in back plate thickness. For the commercial material, Low squeal decreases from 9 to 6 as the thickness increases from 5 mm to 7 mm, indicating a reduction in low noise with increased thickness. For the reinforced material, Low squeal increases from 4 to 6 with increasing thickness, suggesting that thicker discs generate more low noise. The commercial material shows variability in High squeal, decreasing from 84 to 79 and then increasing to 89, indicating that thicker discs initially reduce high noise, but the effect is not consistent. For the reinforced material, High squeal increases from 35 to 42 with thickness, indicating that thicker discs generate more high noise. TUF for the commercial material decreases from 93 to 87 and then increases to 95, suggesting an improvement in stability with intermediate thicknesses but with a peak at the maximum thickness (Table 9). The reinforced material shows a constant increase from 39 to 48, indicating worsening stability with increased thickness. The coconut shell-reinforced material consistently showed lower values for Low squeal, High squeal, and TUF compared to the commercial material.

Table 9.

Analysis of the variation in the back plate thickness in vented disc for commercial and coconut sheel-reinforced material.

	5 mm	6 mm	7 mm
Commercial material
Low squeal	9	8	6
High squeal	84	79	89
TUF	93	87	95
FUF	3413.5 Hz	3362.9 Hz	3326.9 Hz
NI	3.6	2.2	1.7
Coconut shell-reinforced material
Low squeal	4	5	6
High squeal	35	37	42
TUF	39	42	48
FUF	4799.8 Hz	2417.1 Hz	2417 Hz
NI	2.2	2.2	1.6

For the commercial material, FUF varies slightly from 3413.5 Hz to 3326.9 Hz, reflecting a slight decrease in vibration frequency with increased thickness. In the reinforced material, FUF varies significantly, with high values ranging from 4799.8 Hz to 2417 Hz, indicating that the reinforced material has a more variable acoustic response with plate thickness. The FUF for the reinforced material was not always lower; it had higher values for the 5 mm thickness and lower values for the 6 mm and 7 mm thicknesses (Table 9).

The NI for the commercial material decreases from 3.6 to 1.7 with increasing thickness, reflecting a significant improvement in noise performance with greater thicknesses. The reinforced material maintains NI relatively stable at 2.2 and 1.6. The NI for the reinforced material is consistently lower compared to the commercial material, especially for the 5 mm thickness, where NI is 2.2 for the reinforced material and 3.6 for the commercial material. The reinforced material shows a slight advantage in noise performance across all thicknesses (Table 9).

Analyzing the Variation in Pad Thickness (Table 10 and Figure 9), for the commercial material, the Low Squeal remains stable at 9 for thicknesses of 20 mm and 40 mm, with a slight decrease to 8 at 30 mm. In contrast, the reinforced material shows an increase from 2 at 20 mm to 4 at 40 mm, suggesting that greater thicknesses lead to more Low Squeal in the reinforced material. The commercial material exhibits a significant variation, with High Squeal increasing from 119 at 20 mm to 196 at 40 mm. The reinforced material also shows increasing values, starting at 63 for 20 mm and rising to 125 at 40 mm, indicating that greater thicknesses increase High Squeal in both materials. For the commercial material, the TUF increases from 128 at 20 mm to 205 at 40 mm. The reinforced material starts with lower values (65 at 20 mm) and increases to 129 at 40 mm, indicating that the commercial material experiences more instability with increased thickness. The coconut shell-reinforced material consistently showed lower values for Low Squeal, High Squeal, and TUF compared to the commercial material (Table 10).

Table 10.

Analysis of the variation in the pad thickness in vented disc for commercial and coconut sheel-reinforced material.

	20 mm	30 mm	40 mm
Commercial material
Low squeal	9	8	9
High squeal	119	79	196
TUF	128	87	205
FUF	2417 Hz	3362.9 Hz	3410.5 Hz
NI	1.8	2.2	3.5
Coconut shell-reinforced material
Low squeal	2	5	4
High squeal	63	37	125
TUF	65	42	129
FUF	4850.5 Hz	2417.1 Hz	4818.3 Hz
NI	1.2	2.2	3.4

The FUF for the commercial material shows a slight increase from 2417 Hz at 20 mm to 3410.5 Hz at 40 mm, suggesting an increase in vibration frequency with thickness. The reinforced material, on the other hand, presents higher values (4850.5 Hz at 20 mm) and varies to 2417.1 Hz at 30 mm before rising again to 4818.3 Hz at 40 mm, reflecting a more variable acoustic response. The FUF for the reinforced material varied, with some values higher and others lower compared to the commercial material (Table 10).

For the commercial material, the NI increases from 1.8 at 20 mm to 3.5 at 40 mm, showing an overall increase in noise with greater thicknesses. The reinforced material maintains lower NI values, starting at 1.2 for 20 mm and rising to 3.4 at 40 mm. This indicates that the reinforced material generally has a lower noise index compared to the commercial material, despite a slight rise with increased thickness (Table 10). The NI was consistently lower for the reinforced material compared to the commercial, reflecting better acoustic performance.

Overall, the coconut shell-reinforced material consistently performs better than the commercial option in reducing instability and noise under various conditions. It demonstrates enhanced stability and generates less noise. This finding aligns with the experimental results of Abutu et al., which indicated that the optimized coconut shell-reinforced brake pad performed comparably to commercial samples and effectively reduced brake noise and vibration during use.³³

Conclusion

This work proposes an iterative method using the analysis of complex eigenvalues, which correlates brake component material characteristics, coefficient of friction, and geometric parameters to instability parameters such as frequency, number of instability modes, and their intensity. To evaluate commercial and coconut Shell-reinforced Material performance, three objectives are established: Maximize first unstable frequency FUF, minimize TUF, the total number of unstable frequency and Minimize NI, noise index. The results obtained in the simulations demonstrate that:

• Coefficient of friction - In the low squeal zone, the commercial material shows more unstable points as friction increases, while the coconut shell-reinforced material is more stable and generates fewer points. For high squeal, both materials increase, but the reinforced material is more stable. TUF increases for both, but less for the reinforced material. FUF varies more in the reinforced material, and NI is generally lower or equal in the reinforced material, indicating less overall noise compared to the commercial material.

• Disc Young’s modulus - For the commercial material, Low squeal remains stable or decreases slightly with increased stiffness, while the reinforced material significantly reduces Low squeal. High squeal initially decreases for the commercial material but rises for the reinforced material under high stiffness. TUF decreases and stabilizes for the commercial material, whereas it increases for the reinforced material, indicating greater instability. FUF rises with stiffness for both materials, but with more variation for the reinforced material. NI slightly increases for the commercial material but remains stable for the reinforced material, which generally shows better noise performance.

• Back plate Young’s modulus - With increasing backing plate stiffness, the commercial material shows stable Low squeal but increased High squeal and variable TUF. The reinforced material maintains lower Low squeal and stable TUF, with less variation in FUF and a more stable NI. The reinforced material generally outperforms the commercial one in noise reduction and stability.

• Disc Thickness - With increasing disc thickness, Low squeal decreases for both materials. High squeal initially decreases and then slightly increases for the commercial material, while the reinforced material shows a more consistent reduction. TUF improves with thickness, with the reinforced material maintaining greater stability. FUF increases with thickness, showing more variation for the reinforced material. NI is slightly lower for the reinforced material at smaller thicknesses but stabilizes at similar levels to the commercial material at larger thicknesses.

• Back plate Thickness - For the commercial material, Low squeal decreases with thickness, while the reinforced material shows an increase. High squeal varies for the commercial material but increases for the reinforced material. TUF improves initially for the commercial material but worsens with thickness for the reinforced material. FUF varies slightly for the commercial material and more significantly for the reinforced material. NI improves with thickness for the commercial material and remains stable for the reinforced material, which generally shows better noise performance.

• Pad material Thickness - For the commercial material, Low Squeal is stable with slight decreases as thickness increases. The reinforced material shows higher Low Squeal with increased thickness. Both materials exhibit rising High Squeal with thickness. TUF increases significantly for the commercial material and more moderately for the reinforced material. FUF increases slightly for the commercial material and varies more for the reinforced material. NI is lower for the reinforced material, indicating better noise performance overall.

In general, for the material values used in the study, the coconut shell-reinforced material generally outperforms the commercial one in reducing instability and noise under various conditions. It maintains better stability and lower noise.

The findings of this study highlight the potential of coconut shell-reinforced materials in improving brake stability and reducing noise. Future research should focus on experimental validation of the numerical results, considering real operating conditions such as temperature and humidity variations. Additionally, extending the proposed method to different braking systems, including railway and heavy-duty applications, could provide further insights into the material’s performance across various scenarios.

Footnotes

Acknowledgments

The authors would like to acknowledge the support given by Centre of Vehicles for Sustainable Mobility of Faculty of Mechanical, Czech Technical University, Prague, Czech Republic.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Rômulo do Nascimento Rodrigues

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