Finite Element Analysis on Squeal-Noise in Railway Applications

In this paper, we present the results of an experimental investigation on squeal noise emitted by a disc brake for railway applications. The measurement, that show in an evident manner the onset of instability forms during the braking phase were made on a test-rig in real scale on which we measured by means of a sound intensity probe the noise emitted by the disc brake. By using a Fast Fourier Transform (FFT) we identified the strongest detected frequencies. The aim of this paper is to analyse such phenomena by conducting a Finite Element Analysis (FEA) of the 3D CAD model of the disk modelled in Ansys Workbench FEM software, in order to, better understand the path that leads a stable system to unstable behaviour. The system analysed is composed of a steel disk and four pads made of an array of cylinders made of frictional material. Such pad system has been designed in order to simplify the simulation, assuming that it acts on the disk surface just in some points. The numerical results agree with the experimental ones.


INTRODUCTION
Brakes are one of the most important components in terms of safety and performance in vehicles, and due to increasing requirements related to stability and comfortability, have become more complex [1][2][3].Noise and vibrations are more and more important for the automotive industry, especially for vehicle manufacturers and component suppliers, and is generally regulated by noise pollution legislation, while the interior noise and sense of vibrations are much strictly evaluated by customers themselves [4,5].The increasing problem of noise generation during braking assumes an economic and technical importance in the transport industry [6,7].The generation of noise and its supp-ression have become important considerations to be assessed during the design phase and execution of the brake systems.In fact, as noted by Abendroth and Wernitz [11,12], many pads manufacturers spend up to 50% of their budget in engineering activities on noise and vibration problems [13][14][15][16].Much progress has been made and many solutions have been suggested, for example by reducing the causes, adding constrained layer (shims) or by moving damping modal coupling elements that constitute the brake assembly [17][18][19][20][21].Not all of these solutions contribute to the improvement of the vehicle braking qualities [22][23][24].Due to these undesirable consequences, friction materials used in automotive braking systems must have wear resistance and high mechanical strength, low thermal conductivity, lubrication components with a role of increasing seizure resistance [25][26][27].An interaction between a brake disc and friction material of automotive brake is characterised by a number of dry contact phenomena.These phenomena are influenced by brake operation conditions (applied pressure, speed, and brake interface temperature) and material characteristics of a friction couple.The coefficient of friction should be relatively high and keep a stable level irrespective of temperature change, humidity, age, degree of wear and corrosion, presence of dirt and water spraying from the road [28,29].The complex dynamic behaviour of such mechanical systems can be studied within the multibody framework [30][31][32][33][34][35].Multibody systems are mechanical systems composed of continuum bodies and kinematic joints which can be rigid and/or deformable.Several examples of mechanical systems that can be modelled using the multibody approach can be found in different engineering applications [36][37][38][39][40][41][42][43][44][45][46].Examples of the use of multibody modelling can be the study of robotic systems that are based on systems present in nature [47], and studies on the measurements and mitigation of vibrations in structures for comfort improving [48,49].Therefore, advanced analysis approaches and computational procedures are necessary for performing reliable dynamic simulations [50][51][52][53][54][55][56][57][58].Control strategies for rigid-flexible multibody systems also are particularly challenging and require computational approaches capable of handling nonlinear behaviour of a general multibody systems [59][60][61][62][63][64][65][66].In the last decades, a considerable amount of scientific literature has been produced on the possibility to eliminate noise [67].Brake noise can be categorised according to the frequencies range in which the phenomenon occurs.We talk about low-frequency noise for frequencies between 100 and 300 Hz, while for frequencies above 2000 Hz but below the first in plane mode of the disc we talk about low-frequency squeal while high-frequency squeal is the noise produced by self-induced vibration [68][69][70].There are two theories that try to describe squeal phenomenon.One theory ascribes the increase of vibration amplitudes to the stick-slip mechanism.The other assigns it to geometrical instability of the brake assembly.In spite, both approaches agree that squeal phenomenon depends above all on a variation of the friction force.In this paper, we chose to keep constant the friction coefficient µ, since it is possible to obtain variables friction forces due to the presence of variables closing forces.This is the assumption underlying the studies carried out by Spurr [71], Earles and Soar [72,73].In this paper, we show that if two modes are geometrically coupled, for increasing values of friction coefficient, frequencies tend to coincide [74][75].Once the two modes couple, they become unstable.The paper is organised as follows.In section 2 we describe the mathematical model used for understanding the pathway that leads to unstable behaviour.In section 3 we focus the attention on the finite element analysis conducted on the meshed 3D geometry of the disk-pads created with SolidWorks 3D software and simulated in Ansys Workbench FEM Software, while section 4, we report an experimental activity conducted on disk brake for recording sound pressure levels during braking.The final section presents the conclusions.

MATHEMATICAL MODEL
The system analysed in this paper consists of a SEE (mod.10-9007302) steel disk mechanically coupled to a system of four pads used in railway applications.Each pad has six pairs of cylinders, made of friction material, connected by a cantilever.The mechanical behaviour of such cantilevers is shown in Fig. 1.

Figura 1. Mechanical scheme for cantilevers
In this way, the disk is always in contact with the 48 cylinders.In Fig. 2  Given the geometry and the materials used for the pad, for the numerical analysis, we considered the four sub-systems rigid except cantilevers and disk modelled as flexible bodies.In Fig. 4, we have reported the meshed assembly realised for the subsequent FEM analysis carried out.In such figure, it is possible to distinguish the parts considered flexible ( meshed parts ) from the ones considered rigid ( no-meshed parts ).From the use of Lagrange equation reported: we developed the differential equations of motion of the system as follows: with M and K mass and stiffness matrix of the disk-pad system, respectively; Q pad is the force field between disk and pads and Q v is the quadratic velocity vector resulting from the differentiation of the kinetic energy with respect to time and with respect to the body coordinates.This quadratic velocity vector contains the gyroscopic and Coriolis force components.
( ) ( ) Q pad can be seen as a force field proportional to the friction coefficient present between disc and pad.

( ) ( )
; Thus, (2) can be rewritten as: The eigenvalue problem of (5) allows us to predict instability forms of the system.The modal parameters of the disk-pad system have been calculated by using ANSYS FEM software.

NUMERICAL SIMULATION
Several simulations were conducted under ANSYS Workbench, for fixed values of closing force and angular velocity of the disk.The friction coefficient µ is considered constant for each simulation and caused to vary from the value 0 to 1 in order to study the influence of dry friction on the occurrence of forms of instability.For simplicity, a Coulomb law characteristic is applied.The upper limit of the frequency range to be studied is fixed at 8000 Hz.To define the contact behaviour between the cylinders and disc, frictional contact has been selected leaving liberty to separate in their normal direction while in the tangential direction, sliding occurs only when the shear stress exceeds the value multiplying the normal stress by the constant friction coefficient µ.For ensuring contact compatibility at the interface preventing penetration of the contacting point, Augmented Lagrange formulation has been used providing the normal force F n = k n x n + λ with λ contact pressure, whenever a contacting point penetrates normally target face.In Fig. 5 are indicated the real and imaginary part of the eigenvalues which are relative to unstable modes.Besides the first form of instability that is recorded for values greater than zero of the coefficient of friction, the first unstable mode is recorded around 7500 Hz for values of µ > 0,18 as shown in the top left diagram of Fig. 5. Five modes tend to become unstable.In Fig. 6 are indicated the natural frequencies versus nodal diameters for the disk-brake analysed.By the figure it can be observed that the pad force field generates modes which become unstable to increase of µ only for ND = 0.

EXPERIMENTAL INVESTIGATION
To verify the results obtained from the finite element analysis, we confronted the numerical results with the sound pressure levels recorded during an experimental investigation conducted on a test-bench during braking, from 70 km=h to zero.The experimental apparatus is normally used for braking tests for railway systems.In our case, the benchmark consists of a 640 mm diameter and 45 mm in thickness disk and four sub-systems pads.Pressurized air pushes a piston connected through a mechanical linkage to the brake shoes in order to rub on the disk, using the resulting friction force to slow the wheel.
In 4 seconds the calliper closing force reaches its maximum value (1000 daN) and then keeps constant.Two microphones were placed one meter close to the disk: one in front of the pads and the other on the opposite side along the disk diameter, as illustrated in Fig. 7.In Fig. 8, the sound pressure frequency spectrum recorded during braking is reported.Some peak frequencies increase as caliper closing force grows up, as it can be seen.This can be due to stress-stiffening phenomenon.Furthermore, it can be noted that: 1) less than 4000 Hz frequency peaks appear as soon as braking starts; 2) 6000 Hz frequency peak damps in 6 seconds; 3) 7500 and 9300 Hz frequency peaks appear after first 6 seconds.The results of the numerical simulations fairly match experimental data.In particular, the adopted model can predict geometric instability incoming at frequencies close to 1300, 2100, 3100, 5500 and 6800 Hz, especially in first braking moments.Contrariwise, the model does not predict one important peak close to 9300 Hz.Considering that the first transversal pad-subsystem beam resonance frequency is close to that value, one can deduce that such value is due to a form of dynamic instability such as stick-slip nor to geometric instability.For this reason, the adopted model is not able to predict it.This work confirmed squeal to be a very complicated phenomenon in which dynamic and geometric instability take place simultaneously.

CONCLUSIONS
A theoretical study followed by an experimental activity is presented in this paper.Furthermore, we presented a mathematical model to describe the dynamic behaviour of a disk brake system for railways applications.In this system, the pad is composed of arrays of pad sub-system elements composed of cylinders mounted at the end of elastic beams.The pad model proposed was realised in order to facilitate variable force fields between disc and pad systems in order to analyse the influence of a variable friction force on the dynamic behaviour of the whole system.The numerical analysis and the subsequent experimental investigation allows to make the following consideration: 1) five modes became instable to increase of the friction coefficient µ; 2) the instable modes are (0,3), (0,4), (0,5), (0,7) and (0,8); 3) the instability relative to the (0,5) mode occurs for value of the parameter µ > 0,5 and hence it isn't significative for this problem; 4) The experimental data are in a good agreement with the theoretical results; 5) Every instability form predicted by the model occurs in the experimental analysis; 6) Nevertheless in the experimental results appears a frequency component of about 9300 Hz which is not predicted by the theoretical model.This component could be generated by stick-slip phenomenon.In fact, the pad sub-system beam has a transversal natural frequency close to the foregoing frequency.This work is the first step of a wide research project on the selfexcited vibrations induced by dry friction.

NOMENCLATURE q
Lagrangian coordinates vector (a) and 2(b) are indicated respectively the disk and one of the four sub-systems composing the pads.The disk has a diameter of 640 mm, a thickness of 45 mm and geometrical characteristic indicated in the same figure.The twelve cylinders, which form the pad, have a diameter of 35 mm and a height of 20 mm.In Fig. 3 is reported the assembly model developed in Dassault Systèmes SolidWorks 3D software.
Figure 2. Physical system

Figure 4 .
Figure 4. Mesh of brake assembly