Brake noise

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Brake noise

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An investigation of multi-tonal brake noise _{| V. Wiese}

Abstract

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Acknowledgements

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1 Introduction

1.1 Background

For a brake to be operational it needs two parts, a rotor and a stator, when the stator applies a force on the rotor, the rotor slows down, thus reducing the speed of the vehicle equipped with the system. This is a simple system, but never the less a system where non-trivial problems may arise. From an acoustical perspective it is obvious that the problematics regarding noise are of the greatest interest. Brake noise can be classified in several ways, but two general classifications exist, the low frequency, rigid body motion known as ’groan’ or ’mooing’, or the medium to high frequency sound known as squeal, squeal usually occur at a single, mid to high, frequency at moderate to low speeds. Groan can often be felt by the passengers of a vehicle as vibrations in the vehicles operating controls, whereas squeal is perceived as sound. Since it can not be felt, the squeal does not a↵ect the perceived braking performance of a vehicle, but the high pitch sound is known to create discomfort, noise pollution and reduce the overall impression of the vehicle. On top of this there are many other sounds that can arise for example: ’clunking’ is a low pitch noise that can occur when the brakes are initially applied, and ’wire-brush’ is a stochastic sound in the mid frequency region. Since there exist a vast unstandardized terminology it is of the utmost importance to explicitly define the brake noise in terms of frequency occurrence and character. For the vehicle industry, brake noise is a major problem since it causes warranty problems as well as reducing the customers perceived quality of the brand.

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1.2 Aim

This report will investigate the multi-tonal behavior of a motorcycle, single disc, brake system equipped with anti-lock technology. Since alternations to the brake disc and tonal wheel alone has been proven to alter the squeal, and even eliminate it, this report will revolve mainly around the understanding of these subsystems. Therefore the energy source will be simplified, as proposed by S.K. RHEE, [2], the problematics regarding the friction induced vibrations can be disregarded by view-ing the excitations as hammer excitations, e.a. the source excites all frequencies at once.

Previous remedies to eliminate the noise have been tried out and reported, the tested alterations include a brake disc with an altered geometry, shims placed between the tone wheel and brake disc, and backing plates between the brake pads and brake caliper piston. The brake disc alteration consisted of milling out material of the friction ring, e.a. the outer rim of a brake disc that come in contact with the friction material of the brake pads. The resulting disc has a optimized shape, where only the material that come in contact with the brake pads remain, this alteration has proven to minimize the radiated sound and eliminate the multi-tonal behavior. Another successful remedy has been installing shims in between the tone wheel and brake disc, in tests shims of a thickness of just 0,1mm has been proven to eradicate the squeal. Among the things that have previously been tried out, with little to no e↵ect, is to incorporate a backing plate between the brake pad and caliper. The idea being that the plate dampens the deflection-shapes of the brake pad and thus counteract any eventual modal-modal couplings between the pad and brake disc.

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1.3 Measurement object

The project was carried out on KTM’s Racing Competition 390, (from here on-wards RC390) as portrayed in Fig.1. The RC390 is, as the name would suggest, a single cylinder performance bike, aimed mainly at the first time motorcycle buyer and those with a restricted A2 drivers license, it is also the ground for a bespoke competition series, the RC390 cup. It features a front brake system consisting of a single, non-floating, solid 320 mm brake disk, an external tone wheel for speed pickup, which is required for the anti brake lock system, a radially mounted four piston brake caliper coupled to a master cylinder, both items of the brand Bybre. In order to cope with the thermal expansion without any risk of warping, solid brake discs feature a spoke design that allows the friction ring to expand and con-tract radially with minimal axial deformation. The tone wheel is mounted to the wheel in order for a hall e↵ect sensor, mounted on the fork, to record the wheel speed for the ABS. A schematic of the ABS is presented in Fig. 2.

In order to achieve a distinct feeling with ample initial bite in the brake and distinguish the RC 390 from the smaller bikes in the RC-range, the caliper is fitted with sintered brake pads from BREMBO. This type brake pads generally o↵ers more feel and bite than their organic counterpart, as well as higher wear resistance. Sintered pads are composed of a steel backing plate with the brake material consisting of powdered metals, mainly copper, that are pressed together at high pressure and temperature. The exact composure of copper and other additives are di↵erent for di↵erent manufacturers and are a closely guarded secret. Organic brake pads uses brake material that are made from fibrous materials and abrasives molded together using a resin, are generally not as resistive to wear and lacks the distinct feel that sintered pads are known for, but have the benefit of a higher material damping making the system less prone to squeal. The tone wheel, the brake discs and brake pads can be viewed in Fig.3.

2 Experimental techniques

2.1 Measurement systems

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(a) Original brake disc (b) Tone wheel (c) Modified brake disc

(d) Sintered brake pads

Figure 3: Brake system parts

overview of the bikes measurement systems is presented in table.1.

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Table 1: Test vehicle measurement system Bike #1 Bike #2 Brake temp

Tellert compact line type: TH2M

Thermocouple

Tellert compact line type: TH2M

Thermocouple Brake pressure sensor Natec type: P444 Natec type: P444 Wheel speed sensor Bosch DF11 hall sensor Bosch DF11 hall sensor Microphone G.R.A.S. type 46AE N/A

Accelerometer Endevco Model 65 -10 N/A Data Logger Tellert compact line

DL16CAN

Tellert compact line DL16CAN

Recording system Head acoustics

SQuadriga II N/A

(a) Front wheel setup (b) Remote control

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2.2 Field measurements

Good measurement data is essential when analyzing brake squeal, since neither a numerical simulation, nor a experimental setup is able to replicate brake conditions in a reliable way, or determine the seriousness of the real world problem. In order to reproduce the multi-tonal squeal, and thus investigate if the recorded sound was an isolated incident, three, unused, identical brake discs of the type BR1 was sourced as well as a new corresponding tone wheel. To ensure safe, reproducible measurement conditions and to minimize the background sound, the high risk field measurements (measurements at high speeds as well as overheated system) where carried out at Applus Idiada, a proving ground for the automotive industry. The low risk measurements (low speeds, temperature within specified working window) where performed outside KTM’s on site workshop. Two bikes where used, equipped with identical brake calipers and master cylinders with the only di↵erence between the two being the bottom of the forks. The plots presented of these measurements are not taken on the same day, nor in the same conditions, as such, caution needs to be taken when comparing the overall sound pressure levels from one measurement to another as the results are not directly comparable. Measurement conditions and test layout are presented in table 3

The following setups was evaluated and measured: • Original brake system.

• KTM proposal brake disc.

• Original brake system with spacers installed between the tone wheel and brake disc.

• Added mass on spokes, both setups.

• Adhesive mounting of the tone wheel, original brake system. • Abrasive measure on tone wheel, original system.

Table 2: Brake setups used in the field measurements Setup Setup Tone wheel

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An investigation of multi-tonal brake noise _{| V. Wiese} Table 3: Test layout

Measurement Setup Modification Vehicle Temp Humidity

1 1 N/A 1 18o_C _48,3% 2 2 N/A 2 18o_C _48,3% 3 3 N/A 1 7,4o_C _51,5% 4 4 N/A 2 11,1o_{C 44,8%} 5 3 0,1mm spacer 1 15,4o_{C 21,3%} 6 3 Adhesive mounting 1 15,4oC 21,3% 7 4 Added mass 2 5,4o_C _44,6% 8 1 Heat up test 1 8,3o_C _37,1% 9 1 200o_{C disc} ₁ _8,3o_C _37,1% 10 1 200o_{C+ disc} ₁ _8,3o_C _37,1% 11 1 Pressure test 1 8,3o_C _37,1%

12 1 Added mass 1 N/A N/A

13 1 Abrasive measure 1 13,2o_{C 29%}

2.3 Modal analysis

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Figure 5: Measurement setup for tangential excitation. 2.3.1 Laser vibrometer

The LDV is an instrument that measures the velocity of an object using the Doppler shift in the reflected light, more precisely, the interference between the emitted light and the reflected light, [11]. A schematic of the basic principle for the laser head in presented in Fig.6.

The total intensity of the laser beam at its detector, I, is given by the equation:

I( ) = Imax

2 (1 + cos( )) (1)

Where is the phase angle between the contributions at the detector and is given by:

= 2⇡(r1 r2) (2)

In which is the wavelength of the emitted light, r1 and r2 are the di↵erent path

lengths for the laser beam, r1 is represented by the dotted line coming out of

the laser in Fig.6 and is constant unless there is a thermal expansion in the laser head. The solid line represents r2 and this is the beam that is reflected o↵ the

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Figure 6: Basic principle for the Polytec laser vibrometer. The laser paths are represented as follows; solid line - Path to measurement object, fine dotted line - beam reflected from object, coarse dotted line - reference beam for phase-shift determination

a measurement r2is changing in time. This change is used to calculate the velocity,

v(t) of the point where the laser beam is directed as: dL(t)

dt = 2v(t) (3)

Where L(t) = r1 r2(t), the factor 2 stems from the fact that both the incident

and reflected wave is a↵ected by a Doppler shift, the magnitude of the velocity is thus given by:

dL(t)

dt = 2⇡ d

dt = 2|v(t)| (4) The signal generated at the detector in the laser head due to interference between the reference beam and the measurement beam is equal to the frequency shift of the beam reflected back from the measurement surface, the frequency shift is equal to the Doppler shift and is given by the equation:

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The superpositioning of the reflected beam and the reference beam creates an optical interference at the detector, this interference signal is the same whether the test specimen is moving towards, or away from the laser head, meaning that there is no way of determine which direction the measurement point is moving. Since the LDV is used to extract the test specimens deflection shapes, and therefore needs information on the relative movement between measurement points a small frequency shift is introduced in the reference signal using a Bragg cell. A Bragg cell uses an acoustic field to di↵ract the light and introduce a shift in frequency. Since the frequency shift of the Bragg cell is known and constant, the direction of the measurement object can be determined. The accuracy of the PSV is investigated by comparing its results to a analytical solution, this report is presented in appendix B.

2.3.2 Modal loss factor

The severity of found modes can be classified by calculating the modal loss factor, ⌘, witch is a ratio on how much energy is dissipated per harmonic cycle, [15]. For a steady state oscillation this can be defined as:

⌘ = D

2⇡W (6)

Where D is the amount of energy dissipated per cycle and W is the total amount of energy stored in the vibrational movement. An ⌘ of one would imply that all vibration energy is dissipated over just one single cycle, whereas a value of zero would imply that the vibration could carry on unchanged in eternity. The measurement of the modal loss factor can be done in several di↵erent ways, with the half-power bandwidth method being one of the most commonly used due to its simplicity. The half power bandwidth is defined as:

f = f3dB+ f3dB (7)

Where f3dB is the frequency value on the mobility slope 3dB before the peak and

f3dB+ is the frequency value 3dB after. From this the modal loss factor is given

as:

⌘ = f

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2.4 Working window

Brake squeal, in most recorded cases, is a phenomenon that occurs at relatively low vehicle speeds. RHEE argues, [2], in his hammering excitation theory, means that the brake-pads skips on top of the amplitude peaks of the brake discs surface at high speeds and therefore the rocking motion that excites the vibrations is prohibited. In his dissertation, Philippe Du↵our, [4], presents an extensive literature study, compiling previously discovered data and theories regarding critical speed of a rotating system. Critical speed is the maximum speed where squeal can occur, e.a. above this speed no brake squeal can be recorded.

Whatever the physical reasoning behind the rotational velocity dependence may be, it is important to classify a parametric window in which the problems occur in order to determine its severity. When it comes to brake squeal, three parameters can be singled out as the most important; brake temperature, brake line pressure and vehicle speed. Surface roughness has also proven to be a major factor in the loudness of the radiated sound, [6]. In an experimental pin on disc setup, di↵erent surface roughnesses, H1 & H2, of the pin have showed a sound pressure

level di↵erence of:

Lp = 20 log ✓ H1 H2 ◆% (9) When using the same pressure on the pin and rotational speed of the disc. In equation 9, % is an experimentally validated factor of 0.8 for the overall sound pressure level, and 1.2 for the peak sound pressure level. As neither the material of the brake pads nor the brake disc are changed the surface roughness becomes an irrelevant parameter to test. Never the less, this shows the importance of a proper run in period as every wheel-change upsets the surface of the brake pads and causes irregularities on the friction surface.

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An investigation of multi-tonal brake noise _{| V. Wiese} 2.4.1 Temperature dependence

Heat is a crucial factor in brake systems since the friction coefficient changes with temperature. Brake manufacturers develop di↵erent compounds for di↵erent cir-cumstances, a racing pad will have a higher specified optimal temperature than a brake pad designed for street use. During the measurement, the bike was taken up to a control speed of 35 kph between brake events, and pressure was regulated as accurate as possible by the rider. Since the test rider is the one controlling the brake pressure, a element of human error is introduced. From the test data it was decided that a brake pressure deviation of 2,5 bar between the measurement was within tolerance. Five brake events was deemed to be within tolerance, all taken in the same conditions on the same day, from each measured event, two seconds was cut out and used to calculate an averaged sound pressure. The total sound pressure level at di↵erent brake pad temperatures is presented in Fig. 7. It is im-portant to note that the temperatures specified are the temperatures recorded at the beginning of each brake event, and that this temperature is rapidly increased during braking.

2.4.2 Brake line pressure influence

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2.4.3 Vehicle velocity dependence

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(a) Brake event from 56 km/h

(b) Brake event from 75 km/h

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3 Numerical simulation

3.1 Simulation-model

A simulation-model of the disc and tone wheel was built in COMSOL by importing the geometry of the tone wheel and the brake discs from the RC390 parts library. It is important to note that COMSOL lacks the ability to model contact between parts in a frequency domain, thus any eventual contact between the brake disc and the tone wheel in reality is neglected in the simulated model. This is especially important for the lower modes since the displacement of objects generally is greater at lower frequencies and therefore any contact restricting movement will have a large influence of the behavior of the system. Due to the inability to model contact problems, the brake discs and tone wheel has been evaluated separately.

Comsol Multiphysics is a software that uses finite element method, FEM for short, to approximate real physical problems that might be hard to solve analytically. Real problems can often be described as a set of partial di↵erential equations, PDE’s, FEM is used to formulate a set of linear equations using test functions, Ui,

that approximates the real solution over a discretized domain. The discretization in controlled by a procedure called meshing, which divides an object in to a set of finite elements, each element is connected to one another with n number of nodes. A node can be viewed as an infinitesimal connecting point. Each element is solved for using the solution for the node next to it as boundary conditions, [12], or mathematically, coefficients, a, as:

Ureal⇡Uapprox = n

X

i=1

aiUi (10)

Where Ureal is the real solution, and Uapprox is the approximated solution of the

real problem.

There are several di↵erent finite element methods, [13], for simplicity we go through an example using the direct approach to show the base principal behind finite element modeling. To illustrate the direct approach we consider a beam, rigidly mounded on both sides, as displayed in Fig.10. The beam has length L, cross sectional area A, and a Young’s modulus of E. An external force F is applied 1/3L from its right end.

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Figure 10: Top: Visualization of the clamped-clamped beam. Bottom:1-D FEM discretization.

each, numbered E1 E3. The discretization does not need to be of uniform size, but in this example the elements are homogeneous for simplicity. The elements are connected with four nodes, numbered N 1 N 4, In a 1-D case, the number of nodes are the same as the number of global degrees of freedom as each node only can move in one direction and lacks rotational freedom. Next, we formulate the elemental sti↵ness in matrix form, since the beam is axisymmetrical the expression is the same for all three elements:

k1,2,3 = 3EA L  1 1 1 1 (11)

The local sti↵ness matrices k1,2,3 are then put together in a global matrix, K, as:

K = 3EA L 2 6 6 4 1 1 0 0 1 2 1 0 0 1 2 1 0 0 1 1 3 7 7 5 (12)

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u = 2 6 6 4 u1 u2 u3 u4 3 7 7 5 (13)

The force is also written in vector form as:

F = 2 6 6 4 F1 F2 F3 F4 3 7 7 5 (14)

The boundary conditions needs to be implemented, node 1 & 4 are rigidly attached, thus no displacement is possible then: u1 = u4 = 0. The known external force

is applied at node 3, node 1 & 2 are subject to reaction forces at the mounting, this means that F2 = 0, F3 = Fext. Now the equation system can be set up as:

KU = F . Using the boundary conditions the equation system simplifies to:

KU = 3EA L 2 6 6 4 1 1 0 0 1 2 1 0 0 1 2 1 0 0 1 1 3 7 7 5 2 6 6 4 0 u2 u3 0 3 7 7 5 = F = 2 6 6 4 F1 0 Fext F4 3 7 7 5 (15)

Which is a system of four equations and four unknowns, thus the displacement for each node is given as:

2 6 6 4 u1 u2 u3 u4 3 7 7 5 = 2 6 6 4 0 FextL/6EA FextL/3EA 0 3 7 7 5 (16)

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From this simple example of a 1-D problem it is easy to see that the equation sys-tems have a tendency to fast become too large to solve by hand. For a 3-D problem each node has six degrees of freedom, three translational and three rotational e.a. six equations needs to be solved for for each node. A FEM-simulation of a brake disc needs to solve for a number of degrees of freedom in the order of magnitude of 106_{, for this reason a adequate simulation software is of the utmost importance}

in order to build a model that can accurately represent real problems.

3.2 Simulation - Boundary conditions

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4 Results and discussion

4.1 Original brake system

The original brake system consists of the 320mm Bajaj brake disc and a 2mm thick tone wheel. Three identical setups where tested, two on bike number 1 and one on bike number 2. The new brakes where coupled to unused brake pads and broken in during 2h where the brake temperature was kept under 200 degrees in order not to overheat the disc. After the run-in period, the brakes where allowed to cool down to ambient temperature before the measurements was started. In Fig.11 -13 the sound pressure level, as well as the x-axis acceleration of the brake caliper over time and frequency is displayed for all three discs. From the displayed results it is evident that the multi-tonal brake squeal characteristics is present in all three setups, and thus, a highly reproducible phenomenon.

Figure 11: Measurement with setup 1. Left plot; Sound pressure level. Right plot; Acceleration.

4.2 KTM proposal brake disc

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Figure 12: Measurement with setup 2, sound pressure level.

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pads was mounted on bike number two, together with the corresponding tone wheel, and broken in in the same manner as the original brake setup. Due to the reduced friction ring, the brakes had a tendency to heat up rapidly, leading to the 200 degree temperature barrier being broken on a handful occasions. A visual inspection of the brake disc and pads showed no signs of critical overheating (no color changes of the surface of disc or pads). The sound pressure level was then measured via a pass-by measurement, see Fig.14. The measurement confirmed the previous observations as the disc showed no signs of a multi-tonal behavior.

Figure 14: Measurement with setup 4, sound pressure level.

4.3 Decoupling of brake disc and tone wheel

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Figure 15: Measurement with setup 3 with installed 0,1mm shims. Left plot; Sound pressure level. Right plot; Acceleration.

4.4 Added mass

By adding mass to the spokes of the brake disc a discontinuity is introduced. In general, a discontinuity causes a part of the incident wave to be reflected back thus isolating the hub of the wheel, leading to a reduction of vibration in the tone wheel. The added mass will also alter the modal behavior of the brake disc as well as reducing the vibration amplitude. A total of 300g of led weights where evenly spaced along the brake discs, 25g per spoke, and fastened using a rigid glue specially designed for acoustic purposes. The setup is displayed in Fig.16. For this measurement brake both setup 1 (original disc) and 4 (KTM proposal design) where used, no extra break-in was deemed required.

Setup 4 was mounted on test vehicle 2, a slight high-pitch squeal could be observed objectively. This squeal was not reproducible, due to this inconsistency no squeal could be recorded in the pass-by measurement.

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Figure 16: Mounting of external weights on brake disc

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4.5 Abrasive measure

In order to separate the outer edges of the tone wheel from the brake disc and achieve a smaller gap than the separation created using shims, the outer edges of the tone wheel where cut o↵ using a fine sandpaper. The separation was checked during the material removal using a feeler gauge until the gap measured a uniform 0,05mm. The tone wheel from brake setup number 1 was modified and mounted to and tested on vehicle number 1.

Empirically, the sound is vastly reduced, but regains its distinctive character, this notion is confirmed by studying the test results, Fig.18.

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4.6 Adhesive mounting

To prevent any eventual relative motion between the tone wheel and brake disc the tone wheel was coated with a thin layer of X-60 glue before being mounted. The X-60 is a two component glue especially made for mounting of accelerometers, and thus very sti↵ and brittle. During the break-in period no squeal could be perceived. A number of heavy braking events before the measurement caused the glue to shatter, thus resulting in that no conclusive data could be recorded. A visual inspection showed that the remaining glue caused irregularities in the contact surface between disc and tone wheel, Fig. 19 shows the corresponding measurement results. The introduced irregularities causes the sound to take on a more stochastic nature.

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4.7 The tone wheel

The tone wheel is, as previously discussed, a toothed steel ring stamped out of sheet metal. Since a spacer between the brake disc and tone wheel has proven to completely eradicate the multi tonal behavior, and the abrasive measure has indi-cated that even a small gap on the tone wheels outer rim vastly reduced radiated sound as well as out of plane vibrations, the characteristics of the tone wheel is of great interest.

In this chapter the modal behavior of the original tone wheel, as well as modified versions, is investigated numerically and experimentally. All modifications to the tone wheel are tested in the field and the relevant results are presented.

4.7.1 Simulation

Comsol multiphysics was used to find the mode-shapes of the tone wheel. The model was set up according to section 3.2 and the material parameters was chosen from Comsol’s built in material library to steel (AISI 4340). The mesh was set up as a fine tetrahedral with a altered minimum element size in order to mesh the narrow regions of the geometry. The bolts where kept in the FEM-model as they can be regarded as sti↵ in comparison to the tone wheel and the bolt heads provide a good surface to impose a rigid domain boundary condition.

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(a) 1/2 wavelength between bolts, 1606Hz

(b) 1/2 wavelength between bolts, 1650Hz

(c) 1 wavelength between bolts, dis-torted, 3370Hz

(d) 1 wavelength between bolts, 3840Hz

(e) 3/2 wavelengths between bolts, 5300Hz

(f) 3/2 wavelengths between bolts, 7576Hz

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An investigation of multi-tonal brake noise _{| V. Wiese} 4.7.2 Experimental modal analysis

For the tone wheel it is of the utmost importance to have a closer look at the modal behavior of the mounted system as the simulation is unable to model contact problems in the frequency domain. In total three studies was made, one with the tone wheel mounted according to original specifications, one where the tone wheel’s inner surface was disconnected to the brake disc using the same spacers as discussed in section 4.3, and one using the KTM proposal brake disc. The setup used is the one previously presented in section 2.3, a total of 96 points where measured along the outer and inner rim of the tone wheel.

By taking the results from section 4.1 in to Matlab the frequencies where the loudest sound is radiated can be identified, from the problem frequencies the closest eigenmodes can be identified, both are presented in table 4.

Table 4: Critical frequencies

Lp max region Frequency Closest identified mode

1 1825Hz 1883Hz

2 3753Hz 3766Hz

3 5572Hz 5569Hz

4 7526Hz 7623Hz

5 9314Hz 9308Hz

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An investigation of multi-tonal brake noise _{| V. Wiese} 4.7.3 Altered thickness

Both the experimental modal analysis and the field testing give indications that the tonal behavior of the radiated noise stems from the tone wheel. By reducing its thickness the sti↵ness is reduced, this will have a impact on its modal behavior by reducing the frequency of the eigenmodes. In order to confirm this, an eigenfre-quency study was performed in COMSOL on two altered geometries, one with a thickness of 1,75mm and a second study of a 1,5mm wheel. Out of plane bending modes where identified from the original system by studying the results from the previous simulation and experimental analysis. By doing so, a linear frequency ratio between the simulated eigenmodes of the di↵erent thicknesses was identified. Table 5: Frequency shift for the identified simulated out of plane modes of the 1,5mm tone wheel. Modal frequencies Mode 2mm 1,5mm Ratio 1 1613 Hz 1246,5 Hz 1,294 2 3859,2 Hz 2995,5 Hz 1,288 3 5572 Hz 4128,3 Hz 1,350 4 7616,6 Hz 5963,5 Hz 1,277 5 9675 Hz 7457,8 Hz 1,297 Avg Ratio 1,301

Table 6: Frequency shift for the identified simulated out of plane modes of the 1,75mm tone wheel. Modal frequencies Mode 2mm 1,75mm Ratio 1 1613 Hz 1432,8 Hz 1,126 2 3859,2 Hz 3434,5 Hz 1,128 3 5572 Hz 4738 Hz 1,176 4 7616,6 Hz 6809,9 Hz 1,119 5 9675 Hz 8580,8 Hz 1,128 Avg Ratio 1,134

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order to test if the tonal shift exist in practice and investigate if a linear ratio could be identified in reality, one tone wheel was machined down to 1,75mm from its original thickness of 2mm and mounted on bike number 1. The tonal behavior in the sound signal was weakened, but still evident in the acceleration signal, this is displayed in Fig.22.

Figure 22: Measurements with 1,75mm tone wheel. Top left; Recorded sound. Bottom left; recorded acceleration. Top right; recorded wheel speed. Bottom Right; recorded brake pressure and temperature.

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Table 7: Measured frequency shift and local peaks in acceleration and sound pres-sure level for 2mm and 1,75mm tone wheel

Lp 2mm Acc 2mm Lp 1,75mm Acc 1,75mm Ratio

1825Hz - 1444Hz - 1,26 3753Hz - 2941Hz - 1,28 - 5317Hz - 4315Hz 1,23 - 7608Hz - 5690Hz 1,34 - 8892Hz - 7151Hz 1,24 Avg Ratio 1,27

4.8 Brake disc

Besides separating the tone wheel from the brake disc by using spacers, a previous remedy which has proven e↵ective to combat the noise is to alter the geometry of the brake disc itself. This solution was discussed and tested in section 3.2, but the physical reasons behind why it works is unknown. This chapter investigates the change of behavior in the disc when its geometry is altered, as well as the modal behavior and sound propagation of the original disc and KTM proposal.

4.8.1 Sound propagation

For a vibration to radiate as sound there must be a fluid-acoustic boundary, the larger the boundary, the more efficient the sound radiation. In most brake squeal cases, this boundary is made up out of the brake rotors area. In order to calculate how much the area reduction in the modified disc would impact the sound prop-agation, a simplified FEM-model was set up in COMSOL. Simplified, the brake discs friction ring can be viewed as a beam bent to a circular shape, using this assumption, a FEM model can be made in a 2-D environment.

By setting the depth of the 2-D model the friction rings circumference can be modeled, for a 320mm disc, the circumference corresponds to roughly 1m. The beam is modeled as a cross section and is given the depth 1m and the width 4,5mm, the height is adjusted through a parametric sweep to vary the size of the friction ring between 0,027m to 0,037m in the same study and in that way get a good approximation of the propagation of both discs. The model geometry is presented in Fig.23.

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Figure 23: 2-D geometry for sound propagation simulation.

eled as a circular fluid domain with a diameter of 0,7m surrounded by a perfectly matched layer (PML) which is used to prevent reflections and simulate free field conditions. The mesh for the boundary layer as well as the attenuation of sound pressure is presented in Fig.24. The beam is modeled as a linear elastic material in COMSOL’s solid mechanics interface whereas the fluid domain and the PML is modeled using the pressure acoustics module, the two are linked together via a acoustic-structure boundary defined at the outer edges of the beam. The material selection for the solid domain was set to steel and for the fluid domains, air. The model’s ambient temperature and pressure are set to 293 K and 1 atm. The speed of sound in the acoustics module is set at c = 343 m s 1 _{and is considered as valid}

for most real-world scenarios. The standard reference power of Wref = 10 12W is

used for sound power level calculations (LW = 10log10(W/Wref)).

The frequency range of interest is given by the measured range of the multi-tonal sound, e.a. 1, 5 15 kHz, giving the shortest wavelength of interest at min =

343 m s 1_{/15000 Hz = 0.0229 m. The same rules regarding sampling that applies}

for Fourier transformation of a frequency spectrum also applies when modeling a wave patterns, that is the Nyquist-Shannon theorem. Thus, in order to achieve an accurate resolution of the wavefronts the number of elements per wavelength needs to be higher than 2,56, but even more elements is recommended. As a rule of thumb it is considered good practice to define a maximum element size as 1/6 of the minimum wavelength of interest. Fig.25 shows the pressure distribution in the surrounding fluid for COMSOL’s extra fine mesh preset compared to the pressure distribution generated using a custom mesh.

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(a) Mesh transfer region for PML (b) PML attenuation

Figure 24: (a) shows the mesh for the boundary region of the PML (blue), quadratic mesh is used to minimize reflections. (b) shows the SPL attenuation across the PML domain.

(a) Extra fine preset (b) Custom mesh

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the averaged sound pressure, p, along its boundary for every calculated frequency, this data was then transfered to Matlab and the sound power level calculated as:

LW = 10log10

p2_S

⇢cWref

(18) Where S is the area of the fluid domain boundary, ⇢ is the density of air which is set to 1, 2kg/m3_{, and c is the speed of sound (343m/s for the models ambient}

conditions). The simulation results are shown in Fig. 26.

Figure 26: Sound power level for the two di↵erent heights of the friction ring. In Fig. 26 it is clear that the size of the friction ring has an impact on the sound generated as the smaller friction ring exhibits a lower sound power level throughout the spectrum. This di↵erence, although significant in the mid frequency region around 2-5000 Hz, can not alone account for the elimination of the radiated multi-tonal noise.

4.8.2 New modified disc

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design was developed. This new proposal reduces the thickness of the spokes rather than making them longer, thus achieving less sti↵ness while retaining most of the mass, moment of inertia and acoustic-structure boundary area. The sti↵ness and deformation was investigated using Comsol in order to create a disc that deforms similar to the KTM-proposal disc while still being strong enough to withstand the forces applied during heavy breaking so that field testing can be performed in a safe manner.

The RC390 weighs in at 165,3 kg ready to ride with measurement equipment installed, the weight distribution without rider is 52/48 distributed over front/rear. The test rider (the writer), have a mass of 90 kg with rider gear. When maximum brake force is applied the back wheel lifts o↵ the ground, a simplification that gives ample safety margin would be to say that the brake should be able to hold the bike level with just the front wheel touching the ground, using this simplification the maximum load is given by:

Fedge =

xflbMtotg

rbrake

(19) Where xf is the weight ratio, lb is the wheelbase, Mtot is the total mass, g is the

gravitational constant and rbrake the brake disc radius. This equates to a edge load

of 10kN, which is applied to the disc in Comsol as a boundary load in cylindrical coordinates. The resulting deformation is investigated by defining a point probe on the outer rim of the disc, the thickness of the spokes where then changed in order to get the deformation as close to the KTM proposal as possible while still retaining enough material to make the disc safe to use, and keep its weight as close to the original as possible, resulting in a design with 2mm thinner spokes.

The simulation shows a maximum value of the first principal stress of 192 MPa, e.a. the stress in the plane normal to the plane where the shear stress in zero, which is lower than the materials yield stress of 206 MPa. The torsional displacement for the new geometry under load is 5_·10 4_{rad which compares well to the KTM designs}

displacement of 5, 28· 10 4_{rad, for reference, the original disc has a displacement}

under load of 4, 12· 10 4_{rad. Stress distribution in the disc is shown in Fig. 27.}

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Figure 27: Stress distribution in new geometry during maximum load.

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An investigation of multi-tonal brake noise _{| V. Wiese} 4.8.3 Insertion loss

In a previous section, the influence of the greater surface area of the original brake disc compared to the suggested modified disc with a smaller friction ring was discussed and calculated. Even though the simulations suggest that there is a sound propagation attenuation coupled to the size of the structure-acoustic boundary, this attenuation is not great enough to alone account for the elimination of multi-tonal brake squeal. To completely explain the losses in sound generation some other physical phenomenon need to be at work. One possible explanation could be that the longer spokes of the modified variant causes higher losses in the brake disc, transferring less energy to the tone wheel, thus reducing its vibration amplitude to the point where it is not able to excite the out of plane vibrations necessary for the sound to occur. In order to test this, the new design brake disc was used, this variant has thinner spokes but retains the over-sized friction ring, thus, the moment of inertia, as well as the acoustic structure boundary remains close to the original brake disc.

In all structures there are losses, without it, resonances would never die out. One of the most popular models is the dissipation function presented by Lord Rayleigh, (Rayleigh damping), this function assumes a viscous damping in solids that is proportional to the linear combination of the mass and sti↵ness of a solid body. Viscous damping is directly proportional to the velocity of the system, [3], for harmonic oscillations higher velocity in a certain frequency directly implies a higher displacement in the same frequency. By removing material on the friction ring, the spokes of the disc are elongated leading to a reduced torsional sti↵ness thus damping the vibrations allowed to propagate to the center of the disc. Reduced torsional sti↵ness implies a higher initial displacement in the tangential direction and could therefore lead to higher losses in the system. In order to investigate if this is the case, the mobility of the tone wheel was measured using the PSV. To replicate the forces introduced by the friction of the brake pads the shaker was mounted tangential to the brake disc, and connected via a steel connecting rod fastened to the edge of the disc using a purpose maid two component glue, as seen in Fig.5. In total, 96 measurement points where defined on the tone wheel, 48 placed on the wheels outer rim, and the remaining 48 along its inner rim.

To see if the measurement is trustworthy, the coherence has to be controlled. Coherence is a measure of the relation between input and output signal, and is defined as, [8]:

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Where GAA and GBB, is the averaged autospectrum of the input (A) and output

(B) signal respectively which is defined as:

GAA = lim N!1 N P i=1 GAiG⇤Ai N (21)

Where GAi is the FFT of time record i at A, ⇤ denotes complex conjugate and N

the number of averages. For all practical purposes, N is approximate by a finite number of averages, (time records). The averaging is done to reduce the influence of noise thats always present in any kind of real world measurement. GAB denotes

the average cross spectrum between input and output signals, this is defined as:

GAB = lim N!1 N P i=1 GAiG⇤Bi N (22)

GAB eliminates all contributions in the output that is not correlated to the input,

so that if the output does not correlate to the input, GAB is zero.

All three discs was measured using the same settings, the averaged coherence for all measurement points between the force transducer and PSV is shown in Fig.29, the mobility is shown in Fig.30. From Fig.29 it is clear that the input and output correlate well up to 2 kHz, after this coherence is reduced. The drop in coherence is clearly evident on the measurement points on the tone wheels outer rim, this is believed to be due to the non-linearity caused by the interaction between the tone wheel and brake disc, [8], previously discussed in 3.2.

The general formula for insertion loss reads:

DIL = 20 log

YM + YI + YF

YM + YF

(23) In this case the added isolator mobility, YI, is added by the material from the

original brake disc, yielding an expression for the measured insertion loss as:

DIL = 20 log

Yoriginal

Ymodif ied

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Figure 29: Average coherence during mobility test for all three designs.

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Figure 31: Insertion loss for vibration isolation of the tone wheel.

By comparing the out of plane modes for the tone wheel and the measured sound spectra with the original disc to the insertion loss spectrum it is clear that the altered geometries reduce the amplitude of the tone wheel at these frequencies. The total insertion loss for the KTM proposed geometry compared to the original disc is 3,9 dB, and for the new design 8,2 dB.

4.8.4 Mode shapes

The mode shapes of the brake disc where investigated numerically according to section 3.1 after an experimental validation of the FEM-model by comparing the real mode shapes of the original brake disc mounted on a wheel to the results generated by the FEM-model, three of these mode shapes are presented in Fig.32. The validation of the FEM-model shows a maximum deviation of frequency of 2,77% within the region of interest, the deviation is presented in table 8.

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Table 8: Occurrence of bending modes in the frequency range of interest, experi-mentally gathered values

Bending mode PSV Comsol Deviation 5 1395Hz 1369Hz -1,86% 6 1966Hz 1928Hz -1,93% 7 3574Hz 3507Hz -1,87% 8 7516Hz 7308Hz -2,77%

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No critical out-of-plane bending modes for the brake discs could be identified in the higher end of the frequency spectra in the simulations, nor the experimental modal analysis. This indicates that the multi-tonal sound can not be a product of the friction rings eigenfrequencies.

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An investigation of multi-tonal brake noise _{| V. Wiese} 4.8.5 Pre-stressed analysis

When a object is under load, its modal characteristics change. Think of a guitar, where it is possible to tune the instrument to di↵erent scales without changing the strings. A guitar string can be modeled in 1D via the the simple transverse wave equation: d2_w dx2 m0 T d2_w dt2 = 0 (25)

Where m0 _{is the mass per unit length, T is the tension of the string, w is the}

deflection perpendicular from its original state, x denoted coordinate and t is time. The first resonant frequency is then defined via the length of the string, L, as:

f = c

2L (26)

Where c =pT /m0 _{is the group velocity of the bending wave. T varies with}

changing load, [3].

Equation 25 is by no means sufficient when considering a solid, finite plate, with bending sti↵ness. Assuming that flexural waves are the dominating contributor to the radiated noise, a fourth order partial di↵erential equation is needed to make a analytical estimation, [5]. The complex geometry of the brake disc means that severe simplifications has to be made. Furthermore, this analytical solution would not account for the damping introduced by the pressure of the pads, resulting in a solution with no practical use. Instead, a numerical simulation is performed in Comsol by performing a two stage study, first a stationary pre-stressed study, where force is added at the brake disc/pad interface in a tangential direction, as well as the pressure in axial direction. The pressure acting on each side of the disc is calculated from the brake line pressure as:

ppad= pbl

2⇡r2 p

Apad

(27) Where ppad is the pressure of the pad acting on the brake disc, pbl = 30Bar is the

pressure measured in the brake line during a hard brake event, Apad = 0, 0017m2

is the contact area between the brake disc and brake pad, and rp = 0, 0125m is

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plausible for a sintered brake pad. The result is a pressure acting on either side of the brake disc of ppad = 17, 58Bar and a tangential force on either side of 1620N .

This study was then followed up by a eigenfrequency study, the results of which are presented in table 9.

Table 9: Eigenmode occurrence under load compared to unloaded case Bending mode Unloaded disc Loaded disc

1 214,78Hz 214,11Hz 2 216,5Hz 219,99Hz 3 285,15Hz 283,81Hz 4 507,5Hz 504,03Hz 5 883,34Hz 879,12Hz 6 1369,1Hz 1364,6Hz

From table 9 it is evident that only minor changes in the frequency occurrence of eigenmodes are to be expected under load. This matches the field measurement results discussed in section 2.4, as the frequency content of the noise does not change with varying brake pressure over the critical limit.

4.8.6 Field test of modified disc

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4.9 Relative motion

To investigate if there could be actual movement between the brake disc and the tone wheel, the relative motion between the two was analyzed using the PSV. Points along the edge of the tone wheel where defined together with nearby point directly on the brake disc. The system was then exited via a periodic chirp as previously discussed in section 2.3, and measured with the laser vibrometer. Since the PSV registers the phase and directional movement, the relative motion could be extracted by subtracting the movement of the measurement point on the brake disc from the point on the tone wheel. If the subtraction yields a value of zero, the disc and tone wheel move in unison and no hitting can occur. The phase is extracted by the Polytec software by registering the time di↵erence between input and output. For example, if the first point has an out of plane mobility function at frequency f1 of magnitude x1 with phase ei⇡/y, where y is an arbitrary real

number, and the second point at the same frequency has a mobility function with magnitude x2 with phase ei2⇡/y, then the total relative velocity between the two

at f1 is given by:

ˆ

v = real(x1ei⇡/y x2ei2⇡/y) (28)

If the distance between the points, L, is small, and fulfills L << /2 where is the wavelength of the deflection shape, then ˆv gives the total relative velocity between the edge of the tone wheel and the brake disc.

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Figure 36: Relative velocity and phase between a point on the edge of the tone wheel and a nearby measurement point on the brake disc

5 Modal coupling

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6 Conclusions

Brake noise remains a complex subject, especially since the fundamental friction source is poorly understood. This report aimed to investigate a special brake noise phenomena and is not to be viewed as a guide on how to come to grips with more general brake squealing.

Several methods of investigating the characteristics of the brake system has been used and all evidence points towards the interaction between brake disc and tone wheel being at the core of the problem, as disconnecting, or even altering the tone wheel eliminates or drastically changes the characteristics of the radiated sound. The tested remedies have been investigated to understand the physics on behind why they work. Some, as the decoupling of the tone wheel are easy to understand, whereas others, like the altered geometry of the brake disc itself have proven to be more of a challenge. By designing a new brake disc with retained size of the friction ring, as well as performing a simulation of the impact of the friction rings surface area on the radiated sound, it can be concluded that the surface area is not the main reason behind the sound elimination. This is not to say that the surface area is without some kind of impact, the simulation shows a reduced sound level throughout the frequency range, but it is not found to be the primary cause of sound elimination.

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of the bike as the number of parts would stay the same. Unfortunately, this change alone is not enough to completely eradicate the sound but it is well worth to keep in mind for future projects. Knowing that the behavior changes with the thickness of the tone-wheel presents the question what would happen if a thicker tone-wheel where to be used, test shows that indications from simulation has so far proven to be trustworthy. If this remains the case with a thicker tone-wheel, the tones would be spaced further apart in frequency. There is also a possibility that the displacement of the modes would be reduced in amplitude, leading to a reduction in radiated sound.

However, the easiest, and most e↵ective change, would be to decouple the surfaces by pointing the tone wheel in the opposite direction, towards the center of the wheel. This change would require a redesign of the tone wheel and a repositioning of the hall sensor, but would eliminate the risk of the problem reoccurring in any future production.

7 Suggested future studies

In this report the sound has been assumed to radiate from the friction ring of the brake disc, this assumption is backed up by previous research on brake noise but not tested. To ensure that this is the case, a test using a acoustic camera would be the natural next step, as this would highlight which part are responsible for the sound radiation. As well as controlling the radiation, a modal analysis using a rolling test bench to see the deflection shapes of the brake under real load, this would provide a better understanding of the boundary conditions and thus help develop a more accurate numerical model to predict all sorts of squealing noise in future projects.

Since a characteristic dimension could be found as the distance between the tone wheels mounting points, yet another modification could be tested on the current tone wheel. This modification would be to remove half of the fastening points, thus making the characteristic dimension longer. The longer bar could make the system weaker, and in this way counteract the interaction between the brake disc and tone wheel.

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References

[1] Nagi Elabbasi: Natural frequencies of immersed beams,

https://www.comsol.com/blogs/natural-frequencies-immersed-beams [2] S.K.Rhee, P.H.S. Tsang, Y.S. Wang Friction-induced noise and vibrations of

disc brakes Wear, 133:39–45, 1989.

[3] A.C. Nilsson Vibroacoustics part 1-2 KTH Vehicle engineering, MWL, 2000 [4] Philippe Du↵our Noise generation in vehicle brakes PhD dissertation,

Cam-bridge University, 2002.

[5] Daniel G Gorman, Chee K Lee, Ian A Craighead, Jarom´ır Hor´aˇcek Transverse vibration analysis of a prestressed thin circular plate in contact with an acoustic cavity Engineering Mechanics, 12:417–427, 2005.

[6] M. Yokoi, M. Nakai A Fundamental study on friction noise Bulletin of the JSME, Vol. 25, No. 203, 1982.

[7] Eskil Lindberg A Vibro-Acoustic Study of Vehicle Suspension Systems: Experi-mental and Mathematical Component Approaches Doctoral Thesis in Technical Acoustics, KTH Vehicle engineering, MWL, 2013.

[8] Ulf Carlsson Experimental Structure Dynamics Lecture Notes, KTH Vehicle engineering, MWL, 2009.

[9] H.P. Wallin, U. Carlsson, M. ˚Abom, H. Bod´en, R. Glav Sound and Vibration KTH Vehicle engineering, MWL, 2014

[10] Leping Feng Acoustical Measurements Lecture Notes, KTH Vehicle engineer-ing, MWL, 2014.

[11] Polytec Theory Manual; Polytec Scanning Vibrometer PSV Theory, software 9.1.

[12] Comsol Comsol Multiphysics user’s guide

[13] Mats G. Larson, Fredrik Bengtzon The Finite Element Method: Theory, Im-plementation, and Practice Springer, 2010.

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[15] Monica Carfagni, Eoardi Lenzi, Marco Pierini The loss factor as a measure of mechanical damping Proc. SPIE Vol. 3243, Proceedings of the 16th Interna-tional Modal Analysis Conference., p.580, 1998.

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A

MATLAB Code

1 %% a c c e l e r a t i o n maximas i n o r d e r t o e x t r a c t r a t i o f o r x a c c e l e r a t i o n ( out o f p l a n e ) 2 3 l o a d(’ 20170302 A Bremsen Mattighofen MT mr02 #1 D#2 B a j a j 1 7 5 m m r e p 3 1 . mat ’) ; 4 l o a d(’ f r e q . mat ’) ;

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An investigation of multi-tonal brake noise _{| V. Wiese} 33 f o r n=1:5 34 fm ( n ) =(x ( n )+f t ( n ) )⇤ f r e q d i s c ; 35 end 36 37 %% r a t i o , u t s t r a l a t l j u d 38 39 l o a d(’ 20170302 A Bremsen Mattighofen MT mr02 #1 D#2 B a j a j 1 7 5 m m r e p 1 1 . mat ’) ; 40 l o a d(’ f r e q . mat ’) ;

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70 fmp ( n ) =(x ( n )+f t ( n ) )⇤ f r e q d i s c ; 71 end

72

73 %% 2mm d i s c 74

75 l o a d(’ 20170206 A Bremsen Idiada MT mr04 #1 BajajD# I I I B I 3 VW 1 1 . mat ’) ;

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An investigation of multi-tonal brake noise _{| V. Wiese} 107 %% c l e a r t o n e s : fmp2 ( 1 : 2 ) . / fmp ( 1 : 2 ) o n l y two t o n e s a r e r e a l l y c l e a r i n t h e 1 , 7 5mm r e c o r d i n g 108 109 Accrat =[fm2 ( 1 ) fm2 ( 3 ) fm2 ( 5 ) ] . / [ fm ( 1 ) fm ( 3 ) fm ( 5 ) ] 110 111 p r e s s r a t=fmp2 ( 1 : 2 ) . / fmp ( 1 : 2 ) 112 113 %r i g h t column : f r e q , l e f t column : (m/ s ) /N , f o r c o h e r e n c e , t h i s i s i n t h e 114 %r i g h t column i n s p e c i f i e d t x t f i l e s 115 116 l o a d(’ B a j a j t r a n s v e r s e m s N . t x t ’) ; 117 B a j a j t r a n s v e r s e m s N=B a j a j t r a n s v e r s e m s N ’ ; 118 119 f r e q=B a j a j t r a n s v e r s e m s N ( 1 , 1 :end) ; 120 BajajMag=B a j a j t r a n s v e r s e m s N ( 2 , 1 :end) ; 121 v r e f =10ˆ 9; %m/ s 122 L v B a j a j =10⇤l o g 1 0( ( BajajMag . ˆ 2 ) / ( v r e f ˆ 2 ) ) ; 123 L v B a j a j t o t =10⇤l o g 1 0(sum( ( BajajMag . ˆ 2 ) / ( v r e f ˆ 2 ) ) ) ; 124 125 l o a d(’ KTM transverse msN . t x t ’) ; 126 127 KTM transverse msN=KTM transverse msN ’ ; 128

129 KTMMag=KTM transverse msN ( 2 , 1 :end) ; 130 Lv KTM=10⇤l o g 1 0( (KTMMag. ˆ 2 ) / ( v r e f ˆ 2 ) ) ; 131 DIL KTM2=20⇤l o g 1 0(abs( BajajMag . /KTMMag) ) ;

132 Lv KTM tot=10⇤l o g 1 0(sum( (KTMMag. ˆ 2 ) / ( v r e f ˆ 2 ) ) ) ; 133

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145

146 DIL ND=Lv Bajaj Lv ND ; 147 DIL KTM=Lv Bajaj Lv KTM ; 148

149 DIL ND tot=L v B a j a j t o t Lv ND tot 150 DIL KTM tot=L v B a j a j t o t Lv KTM tot 151 152 l o a d(’ B a j a j t r a n s v e r s e C o h e r e n c e . t x t ’) ; 153 l o a d(’ KTM transverse Coherence . t x t ’) ; 154 l o a d(’ n e w d e s i g n t r a n s v e r s e c o h e r e n c e 2 . t x t ’) ; 155 BC=B a j a j t r a n s v e r s e C o h e r e n c e ’ ; 156 KTMC=KTM transverse Coherence ’ ; 157 NDC=n e w d e s i g n t r a n s v e r s e c o h e r e n c e 2 ’ ; 158 BC=BC( 2 , 1 :end) ; 159 KTMC=KTMC( 2 , 1 :end) ; 160 NDC=NDC( 2 , 1 :end) ; 161 162 f i g u r e( 1 ) 163 t i t l e( ’ M o b i l i t y ’)

164 p l o t( f r e q ,KTMMag, f r e q , BajajMag , f r e q , NewDesignMag , ’ k ’) 165 l e g e n d(’ ktm ’ , ’ b a j a j ’, ’New Design ’) 166 g r i d on 167 y l a b e l(’ (m/ s ) /N ’) 168 x l a b e l(’ f r e q u e n c y ( Hz ) ’) 169 170 f i g u r e( 2 ) 171 t i t l e( ’ Coherence ’)

172 p l o t( f r e q , smooth (KTMC) , f r e q , smooth (BC) , f r e q , smooth (NDC) , ’ k ’

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184 t i t l e( ’ I n s e r t i o n l o s s ’)

185 p l o t( f r e q , smooth (DIL KTM) , f r e q , smooth ( DIL ND ) ) 186 l e g e n d(’ ktm ’ , ’New Design ’) 187 g r i d on 188 y l a b e l(’dB ’) 189 x l a b e l(’ f r e q u e n c y ( Hz ) ’) 190 191 f i g u r e( 5 ) 192 t i t l e( ’ I n s e r t i o n l o s s ’)

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223 l o a d(’ mr02 . mat ’) 224 225 Ptot=A Brakesqueal mr02 RC390 WW ; 226 227 l o a d(’ f r e q r a n g e ’) 228 Ptot=r e a l( Ptot ) ;

229 %Lptot =20⇤ l o g 1 0 ( Ptot ./(2⇤10ˆ 5) ) ; %conver t to Lp 230 Ptot1=Ptot ( 1 :end, 7 3 : 9 3 ) ; 231 Ptot2=Ptot ( 1 :end, 1 8 3 : 2 0 3 ) ; 232 Ptot3=Ptot ( 1 :end, 2 9 3 : 3 1 3 ) ; 233 Ptot4=Ptot ( 1 :end, 4 0 3 : 4 2 3 ) ; 234 Ptot5=Ptot ( 1 :end, 5 9 3 : 6 1 3 ) ; 235 %s u r f ( Lptot ) 236

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An investigation of multi-tonal brake noise _{| V. Wiese} 263 p l o t( temp , Lp ) 264 x l a b e l(’ temperature , c e l c i u s ’) 265 y l a b e l(’ Lp [ dB ] ’) 266 t i t l e( ’ Temperature dependence ’) 267 g r i d on 268 s u b p l o t( 2 , 1 , 2 )

269 p l o t( temp , p r e s s , temp , sp eed ) 270 x l a b e l(’ temperature , c e l c i u s ’) 271 l e g e n d(’ Brake p e s s u r e [ bar ] ’ ,’ V e l o c i t y [ km/h ] ’) 272 %y l a b e l ( ’ B r a k e l i n e p r e s s u r e [ Bar ] ’ ) 273 g r i d on 274 275 l o a d(’ S R m o b i l i t y m a g b a j a j . t x t ’) ;

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bandwidth f u n c t i o n 302 bw1=powerbw (SRmag ( 6 4 : 9 6 ) )⇤ f r e q d i s c 2 ; 303 bw2=powerbw (SRmag ( 1 1 5 2 : 1 1 8 4 ) )⇤ f r e q d i s c 2 ; 304 bw3=powerbw (SRmag ( 1 2 8 0 : 1 3 1 2 ) )⇤ f r e q d i s c 2 ; 305 bw4=powerbw (SRmag ( 1 3 7 6 : 1 4 0 8 ) )⇤ f r e q d i s c 2 ; 306 bw5=powerbw (SRmag ( 1 6 0 0 : 1 6 3 2 ) )⇤ f r e q d i s c 2 ; 307 bw6=powerbw (SRmag ( 1 8 2 4 : 1 8 5 6 ) )⇤ f r e q d i s c 2 ; 308 bw7=powerbw (SRmag ( 2 2 4 0 : 2 2 7 2 ) )⇤ f r e q d i s c 2 ; 309 bw8=powerbw (SRmag ( 3 6 8 0 : 3 7 1 2 ) )⇤ f r e q d i s c 2 ; 310 311 BW=[bw1 bw2 bw3 bw4 bw5 bw6 bw7 bw8 ] ; 312 313 mode=[1622 3320 3523 3655 4030 4383 5009 7 2 7 0 ] ; 314

315 e t a=BW. / mode %modal damping 316

317 %% 3 dB bandwidth , c a l c u l a t e d with b u i l t i n h a l f power

bandwidth f u n c t i o n 318 bw1=powerbw (SRmag ( 2 0 0 : 2 3 5 ) )⇤ f r e q d i s c ; 319 bw2=powerbw (SRmag ( 1 4 3 5 : 1 4 7 0 ) )⇤ f r e q d i s c ; 320 bw3=powerbw (SRmag ( 2 6 0 0 : 2 6 2 5 ) )⇤ f r e q d i s c ; 321 bw4=powerbw (SRmag ( 3 9 0 0 : 3 9 5 0 ) )⇤ f r e q d i s c ; 322 bw5=powerbw (SRmag ( 4 9 8 0 : 5 0 1 0 ) )⇤ f r e q d i s c ; 323 324 BW=[bw1 bw2 bw3 bw4 bw5 ] ; 325 326 mode=[1883 3766 5569 7623 9 3 0 8 ] ; 327

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When performing a modal analysis, there are three di↵erent approaches, an-alytic, numerical and experimental. When building a numerical model, it is often a good idea to validate the model experimentally or analytically before any conclusions are drawn. In this appendix, a simple example consisting of the first three out of plane modes of a cantilever aluminum beam is investi-gated using all three approaches, this is done to further increase the authors understanding of the measurement tools available, and validate the accuracy the built-in studies in Comsol.

2 Measurement object

The measurement object consisted of a flat aluminum beam rigidly clamped at one end. The data for the beam is presented in table 1.

Beam parameters

Material Aluminium Length 0.78 m Width 0.003 m Height 0.032 m Young’s modulus 69 GPa Density 2700 kg/m3

Table 1: Parameters of the measurement object.

3 Analytic model

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I = hb

3

12 (1)

Figure 1: Schematic of a cantilever beam. The boundary conditions for such a beam are;

z(x = 0) = 0 (2) dz(x = 0) dx = 0 (3) d2_{z(x = L)} dx2 = 0 (4) d3_{z(x = L)} dx3 = 0 (5)

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The eigenfunction for the out of plane flexural vibrations, n can be derived

using the boundary conditions, which yields:

n = cosh(nx) cos(nx)

(cos(nL) + cosh(nL))(sinh(nx) sin(nx))

sin(nL) + sinh(nL)

(7) Where n = 1, 2, 3... Using the boundary conditions the eigenvalues, n, can

now be obtained from:

cosh(nL)cos(nL) = 1 (8)

The natural frequencies corresponding to eigenvalues n are given by:

!n = (nL)2 s EI ⇢AL4 (9) fn = !n 2⇡ (10)

4 Numerical model

A numerical model was created in an 3-D environment using the in-built functions in Comsol Multiphysics. All calculations where made in the solid mechanics physics module. The geometry was set up as a block, using the same dimensions as the real beam, the resulting domain was defined as a linear elastic material and given the properties of aluminum, using Comsols material library. To create a cantilever beam, one end of the beam received a fixed constraint boundary condition whereas all other surfaces was defined as free. Gravitational e↵ects where assumed to be of no importance and thus neglected.

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this setting and the extra fine setting. An eigenfrequency study was selected and the number of desired modes was set to 20 in order to cover all the desired out-of-plane modal shapes. The shapes of the first three bending modes from the numerical simulation is presented in Fig.2.

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An experimental modal analysis was performed at KTM entwicklungszen-trum, located in Mattighofen, Austria using a Polytec PSV-400 scanning vibrometer. The measurement object was set up on a rigid, steel table and connected to a LDS-shaker via a sti↵ connecting rod glued to its surface. In order to monitor the outgoing signal, a force transducer was mounted be-tween the connecting rod and the shaker. The shaker-connection was placed as close to the rigid mount as possible in order to minimize its structural influence on the system. Amplitude was adjusted using a external amplifier. The measurement arrangement is shown in Fig.3.

Figure 3: Measurement arrangement.

A Polytec doppler laser head was set up 1.5m from the objects surface, the laser head can be seen in Fig.4. The laser works by sweeping the object with a laser and registering di↵erences in the wavelength reflected light, to get an optimal signal it is therefore of great importance to prepare the surface. This was done by rubbing the beams surface with a fine sandpaper to get rid of the beams initial mirror-finish.

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Figure 4: PSV-400 laser vibrometer head.

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6 Conclusions

By viewing the resulting frequencies for each mode in table 2 it is clear that both an numerical or analytic approach can be used in order to get a somewhat accurate representation of reality. The lowest flexural mode at 4 Hz are missing from the experimental analysis due to overestimation of the frequency range. However, for more intricate problems, the analytic solution becomes to complex for any real world purposes.

Modal frequencies

Mode 1 2 3

Analytic 4 Hz 25.2 Hz 70.7 Hz Numeric 4 Hz 25.3 Hz 70.9 Hz Experimental NA 26.25 Hz 70 Hz

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A

Matlab Code

1 c l c 2 c l e a r a l l 3 4 %% Parameters 5 L= 0 . 7 8 ; % [m] l e n g h t o f beam 6 h = 0 . 0 0 3 ; % [m] depht o f beam 7 b = 0 . 0 3 2 ; % [m] h e i g h t o f beam 8

9 E=69e9 ; % [N/mˆ 2 ] Youngs modulus f o r aluminium 10 rho =2700; % [ kg /mˆ 3 ] d e n s i t y 11 12 %% A n a l y t i c a l s o l u t i o n f o r a c a n t i l e v e r beam 13 % C a l c u l a t i o n 14 I =(b⇤h ˆ3) / 1 2 ; % Second moment o f c r o s s s e c t i o n a l a r e a 15 A=b⇤h ; % [mˆ 2 ] c r o s s s e c t i o n a l a r e a o f t h e beam 16 17 knL = [ 1 . 8 7 5 1 0 4 1 4 . 6 9 4 0 9 1 1 7 . 8 5 4 7 5 7 4 1 0 . 9 9 5 5 4 0 7 1 4 . 1 3 7 1 6 8 4 ] ; %Numerical s o l u t i o n s t o t h e t r a n s c e n d e n t a l eq ’ s o f t h e f i r s t bending modes 18

19 omega=(knL . ˆ 2 ) .⇤s q r t( (E⇤ I ) /( rho ⇤A⇤Lˆ4) ) ; 20

21 f r e q=omega /(2⇤p i) 22

Brake noise

Brake noise

Acknowledgements

Contents

1

Introduction

1.1

Background

1.2

Aim

1.3

Measurement object

2

Experimental techniques

2.1

Measurement systems

2.2

Field measurements

2.3

Modal analysis

2.4

Working window

3

Numerical simulation

3.1

Simulation-model

3.2

Simulation - Boundary conditions

4

Results and discussion

4.1

Original brake system

4.2

KTM proposal brake disc

4.3

Decoupling of brake disc and tone wheel

4.4

Added mass

4.5

Abrasive measure

4.6

Adhesive mounting

4.7

The tone wheel

4.8

Brake disc

4.9

Relative motion

5

Modal coupling

6

Conclusions

7

Suggested future studies

References

A

MATLAB Code

2

Measurement object

3

Analytic model

4

Numerical model

6

Conclusions

A

Matlab Code