# Evaluation of Wear of Disc Brake Friction Linings and the Variability of the Friction Coefficient on the Basis of Vibroacoustic Signals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{RMS}), the average value (A

_{AVERAGE}), the root value (A

_{SQUARE}), and the peak value of the vibration acceleration (A

_{PEAK}). From this analysis, two point measures were selected, i.e., A

_{RMS}and A

_{AVERAGE}, which, due to the smallest relative error in diagnosis, are best suited for the price and estimation of the wear value of rail brake linings. The least numerous group are those dealing with the use of vibrations of the braking system also in the assessment of the braking process [14,15]. The work [16] is the first attempt to present the dependence of the friction coefficient on vibrations of friction linings, which in the future may be used in the evaluation of the braking process. It was demonstrated only in the example of three frequency characteristics from braking from the speeds of 120, 160 and 200 km/h.

_{st}higher than the kinetic friction coefficient μ

_{k}, or the kinetic friction coefficient decreases with increasing velocity v

_{2}. Similar conclusions were contained in the work of Sinclair [6] from 1955 and Earles [18]. Other researchers, such as Mills, Bowden and Leben [19] conducted research on elastic friction systems, comparing them to stick-slip motion. The main conclusion from this research was that the vibroacoustic events have not been fully established. The most likely explanation of these phenomena is stick-slip motion occurring in the frictional coupling. The energy source of this motion is the change of the friction coefficient as a function of velocity [4]. However, the stick-slip model does not take into account the effect of system damping.

_{F}=

_{μk}N, the equilibrium of the system will be maintained on the basis of Equation (1) presented in [17]:

^{−1}(1/μ

_{k}), then the friction force T

_{F}will approach infinity. Spurr referred this particular case as spragging. This model was then refined by Jarvis and Earles, as shown in Figure 2b [7].

## 2. Materials and Methods

_{1}= 35 mm and two sets used up to a thickness of G

_{2}= 25 mm and G

_{3}= 15 mm. The friction lining weights were, respectively, m

_{G1}= 1.75 kg (new lining), m

_{G2}= 1.45 kg (lining worn to a thickness of 25 mm), and m

_{G3}= 1.02 kg (lining worn to a thickness of 15 mm). The vibroacoustic tests were carried out in parallel with the frictional (tribological) tests. One vibration transducer was attached to the brake system (right and left) as is shown in Figure 4a,b. The vibration transducers were bolted to the brake carrier plates to increase the vibration measurement range. The input quantities were the simulated braking initiation speed v

_{o}, the brake pad pressure on the brake disc N, the brake mass M and the friction lining thickness G. The output signals were the instantaneous tangential force F

_{t}related to the braking radius r

_{h}, the instantaneous pressure force on the brake disc F

_{b}, as well as the instantaneous acceleration value of vibrations a. Thanks to this, it was possible to observe the influence of changing the input parameters on the obtained output signal.

## 3. Results

_{1}—value of the measure determined for the G

_{1}cladding, A

_{2}—value of the measure determined for the G

_{3}or G

_{2}cladding.

_{t}and the pressure force on the brake disc F

_{b}were measured on the brake stand in order to determine the instantaneous coefficient of friction μ

_{a}. Then, the average coefficient of friction was calculated according to the relationship where s is the measured braking distance [26].

_{m}depending on the braking start speed, with the brake pad pressure to the disc equaling N = 25 kN and the brake mass being M = 5.7 t.

## 4. Regression Model of Friction Lining Wear and μ_{m} Coefficient Variability Based on the Analysis of Vibration Acceleration Signals in the Amplitude Domain

_{RMS}vibration accelerations described by the relationship (3), or the average value A

_{Average}(relationship (4)), are used. In the diagnostics of the braking system, the above-mentioned measures have the highest value of the dynamics of changes in the diagnostic parameter. However, in the scope of modeling the value of the average coefficient of friction of a disc brake, it should first be shown that there is a dependence of changes in the value of the vibration signal on the condition of the friction linings (their wear). Then, it would be possible to use these relationships when modeling the friction coefficient, which strongly depends on the wear of the friction material [41,42]. Figure 8 shows the dependence of the effective and mean value of vibration accelerations on braking at speeds of 50, 80, 120, 160 and 200 km/h. The results of the VA tests were adjusted in the form of the effective value and the mean value of vibration accelerations for the analyzed speeds using the least squares method. On this basis, regression models with a visible coefficient of determination R

^{2}were determined. Each point on the graph in Figure 8 is the average of eight measurements. The number of repetitions was based on previous tests carried out on a 40-braking test. These vibroacoustic tests were carried out on the vibrations recorded on the linings on the right and left side of the brake disc. During the preliminary tests, the mean value and variance (standard deviation) were calculated for each successive braking. Then, the coefficient of variation was calculated. It has been observed that the value of the coefficient of variation is below 10% with the braking. Then, the stationarity and ergodicity of the recorded vibration acceleration signals were checked. The study of the nature of the signals influenced the establishment of the later methodology of the main research.

^{2}. For the test points, the fit was checked with a linear function and various non-linear functions. The selection of the function depended on the highest value of the R

^{2}determination coefficient. Changes in the values of measures (A

_{RMS}and A

_{AVERAGE}) as a function of speed at the beginning of braking for different friction lining thicknesses were described by the following linear and non-linear functions:

_{AVERAGE}value. In the case of the RMS effective value, similar characteristics were obtained.

^{2}was approximated by the following linear functions for the beginning of braking at speeds from 50 to 120 km/h, and power functions for speeds of 160 and 200 km/h:

_{o}and G after verification of the parameters of the multiple regression model, is represented by the relationship:

^{2}for the μ

_{m}model before and after the verification of the model parameters.

_{AVERAGE}value of the acceleration of the linings vibration and the dependence of μ

_{m}on the thickness of the friction linings was used. The dependence of μ

_{m}on the thickness of the friction linings was presented in Figure 11.

_{RMS}example, Equation (25) is calculated based on two linear functions, and Equation (26) is the result of combining a non-linear function with a linear function.

- a
_{1}—multiplier of the linear model of the friction lining thickness as a function of A_{RMS}or A_{AVERAGE}for v_{0}= 50, 80 and 120 km/h; - b
_{1}—free term of the linear model of friction lining thickness in the A_{RMS}or A_{AVERAGE}function for v_{0}= 50, 80 and 120 km/h; - c
_{1}—directional coefficient of the non-linear (power) model of the friction lining thickness as a function of A_{RMS}or A_{AVERAGE}for v_{0}= 160 and 200 km/h; - d
_{1}—exponent of the non-linear model of friction lining thickness in the A_{RMS}or A_{AVERAGE}function for v_{0}= 160 and 200 km/h; - a
_{2}—multiplier of the linear model of the average friction coefficient as a function of the friction lining thickness for v_{0}= 50, 80 and 120 km/h; - b
_{2}—free term of the linear model of the average friction coefficient as a function of the friction lining thickness for v_{0}= 50, 80 and 120 km/h; - c
_{2}—multiplier of the linear model of the average friction coefficient as a function of the friction lining thickness for v_{0}= 160 and 200 km/h; - d
_{2}—free term of the linear model of the average friction coefficient as a function of the friction lining thickness for v_{0}= 160 and 200 km/h.

_{RMS}effective value and with the A

_{AVERAGE}value. During braking from v = 160 km/h, the error in estimating the average value of the coefficient of friction is about 4% for A

_{RMS}and 2.5% for A

_{AVERAGE}. Braking at v = 200 km/h causes an error in the estimation of 6% for both the average and the effective value.

## 5. Regression Model of Friction Lining Wear and μ_{m} Variability Based on the Analysis of Vibration Acceleration Signals in the Frequency Domain

_{m}value on the basis of the vibrations generated by the friction linings during braking was estimated. The main purpose of the spectral analysis of vibration signals was to determine common frequency bands related to the change in the lining thickness during the operation of the brake system in a wide braking range (speed from 50 to 200 km/h). Then, from a given frequency band, the effective value of vibration accelerations was determined in accordance with the relationship (4), and the parameter of dynamics of changes in dB was checked, in accordance with the relationship (5). Figure 14 shows examples of amplitude spectra of vibration acceleration signals generated during braking by the friction linings.

_{RMS}changes depend on the friction lining wear (Table 3).

_{RMS}exchange characteristics were determined (Figure 15).

_{RMS}) and μ

_{m}= f(G), in accordance with the general Equation (25), linear regression models of the average friction coefficient were determined based on the vibrations of the friction linings subjected to spectral analysis in the 1950–2000 Hz frequency band.

## 6. Conclusions

- Besides the assessment of the technical object condition, the changes in the average coefficient of friction as a function of the braking speed can be determined. This is due to the strong dependence of the diagnostic parameter on wear expressed by the dynamics of changes, exceeding 6 dB, as well as the dependence of the average coefficient of friction on the speed and friction linings wear. The combination of both functions enables the determination of linear (at low speeds) and non-linear (at higher braking speeds) regression models to estimate the value of the average coefficient of friction;
- The error in matching the regression model of the average coefficient of friction on the basis of the determined measures during braking only at some speeds of the beginning of braking reaches the value of 6%;
- It is not possible to find a common frequency band for a wide range of braking speeds (from 50 to 200 km/h). The common frequency band in the range of 1950–2000 Hz enables the determination of the A
_{RMS}dependence from the band for three cases of the friction material condition, and for medium and high speed when braking with the dynamics of changes exceeding 6 dB; - The error of adjusting the results of operational tests to the values determined from the regression linear models of the coefficient of friction, determined with the relative percentage error, does not exceed 1% for v = 120 km/h, 3% for v = 160 km/h, and 1.5% for v = 200 km/h;
- For amplitude analyses, the error of fitting the regression model to the test results exceeds 6% for v = 200 km/h, 2.4–3.7% for v = 160 km/h, depending on the measure used, and about 1–1.5% for the average (120 km/h) and low braking speeds (50–80 km/h).

_{m}exceeding the lower or upper tolerance, as a result of, i.e., damage or burning of the friction surface of the lining, a message appears on the control panel that the linings need to be replaced without visual inspection of the brake system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Brake system models: (

**a**) Model of an elastic friction system on the example of a conveyor belt (stick-slip phenomenon), k—spring stiffness, c—viscous damping coefficient, m—block mass, N—pressure force on the conveyor belt, T—friction, V—relative speed, v

_{1}—conveyor belt speed, v

_{2}—pad speed, µ

_{st}—static friction coefficient, µ

_{o}—kinetic friction coefficient, x—displacement; (

**b**) Model of the actual brake system 1Bg in the railway block brake.

**Figure 2.**Models of brake systems: (

**a**) Model of the sprag-slip phenomenon according to Spurr, 1—stiff rod rotating at point O, 2—stiff rotating disc; (

**b**) Model of the friction disc-slider system on the Jarvis and Earles angle cantilever, α—slope angle of cantilever in the horizontal plane, Z—bracket width in m, R—bracket rotation radius in m, Ω—plate rotation speed in rpm, θ—cantilever slope angle in the vertical plane, v

_{2}—slider linear speed in m/s, m

_{o}—weight of the cantilever in kg, m

_{d}—weight of the panel in kg, c

_{p}—cantilever damping, k

_{p}—cantilever stiffness, c

_{d}—disc damping, k

_{d}—disc stiffness.

**Figure 3.**Stand for testing railway disc brakes: (

**a**) driving part of the brake stand with rotating masses, (

**b**) brake disc type 610 × 110 mm mounted on the brake stand.

**Figure 4.**The tested brake disc mounted on the brake stand for testing railway disc brakes: (

**a**) view of the left friction pair—brake disc with lining and attached vibration transducer, (

**b**) view of the right brake holder and attached vibration transducer, (

**c**) view of the stand with vibration measurement equipment.

**Figure 6.**Signal of vibration accelerations registered on lining cladding for different thicknesses of linings during braking to stop (speed at beginning of braking v = 120 km/h).

**Figure 7.**Dependence of the average coefficient of friction μ

_{m}on the set braking start speed at N = 25 kN and M = 5.7 t at different friction lining thicknesses.

**Figure 8.**Dependence of the measure of vibration acceleration on the speed at the beginning of braking and the wear of the friction linings: (

**a**) A

_{RMS}effective value, (

**b**) average value A

_{Average}.

**Figure 9.**Dependence of the friction lining thickness on the effective value of vibration acceleration for the braking speed: (

**a**) v = 50, 80 and 120 km/h, (

**b**) v = 160 and 200 km/h.

**Figure 10.**Dependence of the average coefficient of friction μ

_{m}on the braking speed for N = 25 kN, M = 5.7 t and three friction lining thicknesses.

**Figure 11.**Dependence of the average coefficient of friction μ

_{m}on the thickness of the friction linings for the speed at the beginning of braking: (

**a**) v = 50 and 80 km/h, (

**b**) v = 120, 160 and 200 km/h.

**Figure 12.**Dependence of the average coefficient of friction μ

_{m}on the A

_{AVERAGE}function and velocity v

_{0}, obtained from the tests in relation to the regression model obtained from Equations (27)–(31) on the example of the spline function.

**Figure 13.**Algorithm of wear evaluation and estimation of the average coefficient of friction during braking at v = 120 km/h; MBS—perform service stopping braking, τ—increase of the braking time.

**Figure 14.**Dependence of the amplitude of vibration accelerations on the frequency for different friction lining thicknesses when braking from the speed: (

**a**) v = 120 km/h, (

**b**) v = 160 km/h, (

**c**) v = 200 km/h.

**Figure 15.**Characteristics of changes: (

**a**) A

_{RMS}= f (G), (

**b**) G = f (A

_{RMS}) for three braking speeds in the 1950–2000 Hz frequency band.

**Figure 16.**Algorithm of wear assessment and estimation of the average coefficient of friction during braking, for the speed v = 200 km/h using the spectral analysis; MBS—perform service stopping braking, τ—increase of the braking time.

**Table 1.**The values of the measures of the vibration acceleration of the friction linings depending on the speed of braking and the wear of the friction linings, along with the dynamics of changes.

Measure | Values of Measure m/s^{2} | Dynamics of Change dB | ||

Cladding Thickness G_{1} = 35 mm | Cladding Thickness G_{2} = 25 mm | Cladding Thickness G_{3} = 15 mm | ||

Braking start at
v_{0} = 50 km/h | ||||

A_{RMS} | 6.88 | 9.36 | 10.63 | 3.78 |

A_{AVERAGE} | 5.18 | 7.01 | 8.07 | 3.86 |

Braking start at
v_{0} = 80 km/h | ||||

A_{RMS} | 7.29 | 12.47 | 15.05 | 6.29 |

A_{AVERAGE} | 5.55 | 9.39 | 11.48 | 6.32 |

Braking start at
v_{0} = 120 km/h | ||||

A_{RMS} | 8.65 | 13.95 | 17.68 | 6.21 |

A_{AVERAGE} | 6.61 | 10.62 | 13.60 | 6.26 |

Braking start at
v_{0} = 160 km/h | ||||

A_{RMS} | 10.03 | 14.17 | 37.46 | 11.4 |

A_{AVERAGE} | 7.59 | 10.67 | 18.66 | 7.8 |

Braking start at
v_{0} = 200 km/h | ||||

A_{RMS} | 9.68 | 12.57 | 96.31 | 19.7 |

A_{AVERAGE} | 7.40 | 9.59 | 39.12 | 14.5 |

Before Verification | After Verification | |||

Coefficient | Coefficient Value | Value F * | Coefficient Value | Value F * |

λ_{1} | 0.00093858 | 0.01225 | 0.001393 | 0.001667 |

λ_{2} | 0.0043017 | 0.004541 | 0.005225 | 0.001726 |

λ_{3} | –1.035 × 10^{−5} | 0.00921 | –1.2173 × 10^{−5} | 0.005887 |

λ_{4} | –2.202 × 10^{−5} | 0.06318 | –4.0207 × 10^{−5} | 0.004797 |

λ_{5} | –0.0001027 | 0.005457 | –0.0001027 | 0.002679 |

λ_{6} | 2.778 × 10^{−8} | 0.009771 | 2.7788 × 10^{−8} | 0.01291 |

λ_{7} | –7.283 × 10^{−8} | 0.08624 | 1.0718 × 10^{−6} | 0.001060 |

λ_{8} | 1.072 × 10^{−6} | 0.001620 | ||

λ_{0} | 0.2885 | 4.9607 × 10^{−24} | 0.2654 | 3.3132 × 10^{−26} |

R^{2} | 0.996 | 1.04 × 10^{−7} * | 0.994 | 2.52 × 10^{−8} * |

**Table 3.**A

_{RMS}values from the frequency bands of the vibration acceleration of the friction linings depending on the speed of braking and the wear of the friction linings along with the dynamics of changes.

Frequency Hz | A_{RMS} Value m/s^{2} | Dynamics of Change dB | ||

Cladding Thickness G_{1} = 35 mm | Cladding Thickness G_{2} = 25 mm | Cladding Thickness G_{3} = 15 mm | ||

Braking start at v = 120 km/h | ||||

1950–2000 | 0.33 | 0.46 | 0.61 | 5.20 |

2450–2500 | 0.29 | 0.48 | 0.64 | 7.00 |

Braking start at v = 160 km/h | ||||

1950–2000 | 0.39 | 0.59 | 0.88 | 7.08 |

2050–2100 | 0.46 | 0.75 | 0.89 | 5.81 |

2450–2500 | 0.38 | 0.54 | 0.99 | 8.34 |

Braking start at v = 200 km/h | ||||

1950–2000 | 0.78 | 1.35 | 1.64 | 6.27 |

3400–3450 | 0.38 | 0.66 | 0.98 | 8.19 |

5050–5100 | 0.49 | 0.63 | 1.02 | 6.41 |

5300–5350 | 0.35 | 0.39 | 0.75 | 6.65 |

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Sawczuk, W.; Ulbrich, D.; Kowalczyk, J.; Merkisz-Guranowska, A.
Evaluation of Wear of Disc Brake Friction Linings and the Variability of the Friction Coefficient on the Basis of Vibroacoustic Signals. *Sensors* **2021**, *21*, 5927.
https://doi.org/10.3390/s21175927

**AMA Style**

Sawczuk W, Ulbrich D, Kowalczyk J, Merkisz-Guranowska A.
Evaluation of Wear of Disc Brake Friction Linings and the Variability of the Friction Coefficient on the Basis of Vibroacoustic Signals. *Sensors*. 2021; 21(17):5927.
https://doi.org/10.3390/s21175927

**Chicago/Turabian Style**

Sawczuk, Wojciech, Dariusz Ulbrich, Jakub Kowalczyk, and Agnieszka Merkisz-Guranowska.
2021. "Evaluation of Wear of Disc Brake Friction Linings and the Variability of the Friction Coefficient on the Basis of Vibroacoustic Signals" *Sensors* 21, no. 17: 5927.
https://doi.org/10.3390/s21175927