Proceedings of Meetings on Acoustics

Volume 19, 2013 http://acousticalsociety.org/

ICA 2013 Montreal

Montreal, Canada

2 - 7 June 2013

Signal Processing in AcousticsSession 1pSPc: Miscellaneous Topics in Signal Processing in Acoustics (Poster Session)

1pSPc8. Comparative signal processing analyses of a speed-dependent problem asmotivated by brake judder problemOsman T. Sen*, Jason T. Dreyer and Rajendra Singh

*Corresponding author's address: Mechanical Engineering, Istanbul Technical University, Gumussuyu, 34437, Istanbul, Turkey,senos@itu.edu.tr The goal of this paper is to investigate a transient problem using several digital signal processing techniques. First, a simple linearmathematical model, where a point mass is connected to a roller through a contact interface, is developed and the dynamic interfacial force isanalytically calculated as a function of the speed. In this model, the contact interface is described with a linear spring and viscous damper, andthe system is excited with a base excitation, as defined by the undulations on the roller surface. Due to the time-varying speed characteristics ofthe roller, the resulting response is transient. Second, the dual-domain analyses of the calculated system response is carried out by using short-time Fourier and wavelet transforms, since single-domain representation leads to a loss of information due to signal's transient characteristics.Third, the Hilbert transform is applied and the envelope curves of the interfacial force response are successfully obtained. Finally, this problemis briefly linked to brake judder phenomenon and its source regimes are briefly explained.

Published by the Acoustical Society of America through the American Institute of Physics

Sen et al.

© 2013 Acoustical Society of America [DOI: 10.1121/1.4799005]Received 21 Jan 2013; published 2 Jun 2013Proceedings of Meetings on Acoustics, Vol. 19, 055030 (2013) Page 1

INTRODUCTION

Vehicle brake judder is a friction-induced resonant build-up problem as observed in automotive disc brake applications. Geometric distortions on the rotor surface lead to time-varying interfacial normal load at the pad-rotor interface. As a consequence, a multiple order friction torque is generated since rotor surface distortion profiles are complex waveforms [1-4]. Furthermore, the frequency of this friction torque excitation is proportional to the wheel speed, which decreases during the braking (deceleration) event. Dynamic (resonant) amplifications occur multiple times due to a multi-order excitation and such amplifications can be felt by the driver through the brake pedal, steering wheel or seat [1-5].

The chief goal of this article is to understand the source regimes of brake judder using alternate signal processing techniques. Hence, the main objectives are as follows: 1. Develop a simple yet representative linear mathematical model of the brake judder problem, and obtain an analytical solution for judder response; 2. Examine the non-stationary response signal in dual (time-frequency) domain by using short-time Fourier and wavelet transforms; 3) Estimate the envelope curve of the predicted judder response by using the Hilbert transform; and 4. Extend the results of a simpler example to the brake judder problem and explain its source regimes.

PROBLEM FORMULATION

A single degree of freedom linear system as depicted in Fig. 1 is developed to simulate the judder-like problem. In the proposed model, a mass (m) is attached to a roller with contact stiffness (k) and damping (c) elements. An undulated edge profile on the roller acts as a motion exciter, similar to the rotor surface distortions in the brake judder problem.

FIGURE 1. Single degree of freedom judder-like model with multi-order motion excitation from the roller

The edge profile of the roller y(θ) is defined with Fourier series expansion though truncated at the Nth order, and ξn and φn are assumed to be the magnitude and phase of the corresponding nth order displacement excitation, respectively. The governing equation of the system of Fig. 1 is:

mx + c x + kx t( ) = cy + ky t( )− Fp , (1)

where Fp is a constant preload applied on mass m to ensure a permanent contact between mass and roller. Define the following system parameters and nondimesinal variables:

ζ = c

2 km,

ω n =

km

, Δ =

Fp

k, τ =ω nt ,

X = x

Δ,

Y = y

Δ, (2a-f)

where ζ is the damping ratio, ωn is the natural frequency (rad/s), Δ is the initial deflection on the spring due to Fp, and τ, X and Y are the nondimensional time, nondimensional response and nondimensional excitation, respectively. Using these, Eq. (1) is then written in the nondimensional form as:

�(t)

m

k c

y(�)

x(t)

Fp

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Proceedings of Meetings on Acoustics, Vol. 19, 055030 (2013) Page 2

′′Z + 2ζ ′Z + Z τ( ) = 2ζ ′Y +Y τ( ) , (3)

where Z(τ) = X(τ) + 1, and ( )′ = d( ) dτ . Note that, Eq. (3) is the governing equation of the mass about its static equilibrium position.

ANALYTICAL SOLUTION OF INTERFACIAL FORCE RESPONSE

Assume that the roller angular motion θ(t) is defined with constant deceleration, i.e. θ(t) = −0.5Λt2+Ω0t+Ψ0, where Λ, Ω0 and Ψ0 are the deceleration rate, initial angular velocity and initial angular position of the roller, respectively. Applying Laplace transform with zero initial conditions (Z(0) = 0; ′Z (0) = 0 ) to Eq. (3):

Z s( ) = G1 s( )Y s( )Z1 s( )

−G2 s( )Y τ = 0( )

Z2 s( )

, (4)

where:

G1 s( ) = 2ζ s+1

s2 + 2ζ s+1,

G2 s( ) = 2ζ

s2 + 2ζ s+1,

Y τ( ) = Ξ n sin − nα

2τ 2 + nβ0τ +Φn

⎛⎝⎜

⎞⎠⎟n=1

N

∑ . (5a-c)

Note that, Y(τ) in Eq. (5c) is essentially the nondimensional form of roller edge profile which is defined with truncated Fourier series expansion, where Ξn = ξn/Δ, α = Λ /ω n

2 , β0 = Ω0/ωn, and Φn = nΨ0 +φn. Taking the inverse Laplace transform of Eq. (4) gives:

Z1 τ( ) = G1 τ − u( )Y u( )du

0

τ

∫ , (6)

Z2 τ( ) = −

2ζ exp −ζτ( )sin τ 1−ζ 2( )1−ζ 2

Ξ n sin Φn( )n=1

N

∑ . (7)

Taking the inverse Laplace transform of G1(s), after using Euler’s identity for Y(τ) and some exponential manipulations, the convolution integral given with Eq. (6) is written as:

Z1 τ( ) = Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

exp −ζτ + iτ 1−ζ 2 + iΦn( ) exp − inα2

u2 + inβ0 − i 1−ζ 2 +ζ( )u⎛⎝⎜

⎞⎠⎟

⎧⎨⎩⎪

⎫⎬⎭⎪

du0

τ

∫n=1

N

∑⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

exp −ζτ − iτ 1−ζ 2 + iΦn( ) exp − inα2

u2 + inβ0 + i 1−ζ 2 +ζ( )u⎛⎝⎜

⎞⎠⎟

⎧⎨⎩⎪

⎫⎬⎭⎪

du0

τ

∫n=1

N

∑⎛

⎝⎜⎜

⎞

⎠⎟⎟

. (8)

With the complex variable transformation v = inα / 2u , dv = inα / 2du , Eq. (8) becomes:

Z1 τ( ) = Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2inαn=1

N

∑exp iΦn( )

exp ζτ − iτ 1−ζ 2( ) exp −v2 +2 inβ0 − i 1−ζ 2 +ζ( )

inαv

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

dv0

ina2τ

∫⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2inαn=1

N

∑exp iΦn( )

exp ζτ + iτ 1−ζ 2( ) exp −v2 +2 inβ0 + i 1−ζ 2 +ζ( )

inαv

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

dv0

ina2τ

∫⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

. (9)

Define new parameters w1 = inβ0 +ζ − i 1−ζ 2( ) 2inα and

w2 = inβ0 +ζ + i 1−ζ 2( ) 2inα , and Eq. (9)

simplifies to:

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Z1 τ( ) = Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2inα

exp iΦn + w12( )

exp ζτ − iτ 1−ζ 2( ) exp − v − w1( )2( ){ }dv0

inα2τ

∫n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2inα

exp iΦn + w22( )

exp ζτ + iτ 1−ζ 2( ) exp − v − w2( )2( ){ }dv0

inα2τ

∫n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

. (10)

Finally, applying the coordinate transformations p1 = v − w1 and p2 = v − w2, the integrals in Eq. (10) simplify to the form of an error function definition [6]. Hence the solution of Z1(τ) is:

Z1 τ( ) = Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

π2inα

exp iΦn + w12( )

exp ζτ − iτ 1−ζ 2( )n=1

N

∑ erf τ inα2

− w1

⎛

⎝⎜

⎞

⎠⎟ − erf −w1( )

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

π2inα

exp iΦn + w22( )

exp ζτ + iτ 1−ζ 2( )n=1

N

∑ erf τ inα2

− w2

⎛

⎝⎜

⎞

⎠⎟ − erf −w2( )

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

. (11)

As previously given, Z(τ) = Z1(τ) + Z2(τ) and the dimensional dynamic interfacial force between mass and roller is calculated with

F(t) = c x − y( ) + k x(t)− y(t)( ) . In nondimensional form, this expression is written as:

F τ( ) = cω nΔ ′Z − ′Y −1( ) + kΔ Z τ( )−Y τ( )−1( ) . (12)

Note that ′Z1 and ′Z2 should be calculated to obtain F(τ). These are calculated with the direct derivatives of Eqs. (7) and (11) with respect to τ as follows:

Z1′ τ( ) = Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

π2inα

exp iΦn + w12( ) −ζ + i 1−ζ 2( )

exp ζτ − iτ 1−ζ 2( ) erf τ inα2

− w1

⎛

⎝⎜

⎞

⎠⎟ − erf −w1( )

⎛

⎝⎜

⎞

⎠⎟

n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ + 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

exp iΦn + w12( )

exp ζτ − iτ 1−ζ 2( )exp − τ inα2

− w1

⎛

⎝⎜

⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

π2inα

exp iΦn + w22( ) −ζ − i 1−ζ 2( )

exp ζτ + iτ 1−ζ 2( ) erf τ inα2

− w2

⎛

⎝⎜

⎞

⎠⎟ − erf −w2( )

⎛

⎝⎜

⎞

⎠⎟

n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

+ Im Ξ n ζ − 2ζ 2 −1

2i 1−ζ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

exp iΦn + w22( )

exp ζτ + iτ 1−ζ 2( )exp − τ inα2

− w2

⎛

⎝⎜

⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟n=1

N

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

. (13)

Z2′ τ( ) = −2ζ Ξ n sin Φn( )cos τ 1−ζ 2( )exp −ζτ( )

n=1

N

∑ + 2ζ 2

1−ζ 2Ξ n sin Φn( )sin τ 1−ζ 2( )exp −zτ( )

n=1

N

∑ . (14)

Incorporating Eqs. (7), (11), (13) and (14) to Eq. (12), dynamic interfacial force response F(τ) is analytically calculated, and normalized F is plotted in Fig. 2(a) as a function of normalized roller speed Ω(t) = θ(t) . Note that F and Ω are normalized with Fp and ωn, respectively, i.e.

F = F Fp and Ω =Ω ω n = −ατ + β0 . In order to verify the

analytical solution, Eq. (3) is numerically integrated, and predicted F is displayed and compared in Fig. 2(b). For the sake of completeness, numerical values of the key parameters used in the analytical and numerical solutions are given in Table 1.

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FIGURE 2. Dynamic interfacial force response between mass and roller. (a) Analytical solution; (b) Numerical integration. TABLE 1. Numerical values of parameters used in analytical and numerical solutions.

Parameter Numerical Value Ξ1 0.005 [-] Ξ3 0.005 [-] φ1 0 [rad] φ3 0 [rad] ζ 10-3 [-] ωn 100 [rad/s] α 10-4 [1/rad] β0 2 [-] Ψ0 0 [rad] Δ 0.2 [m]

First, observe an excellent match in Fig. 2 between analytical and numerical solutions. Second, the F < 0

condition is satisfied over the entire speed Ω interval; i.e. the system is always in contact, which validates the mathematical model since it does not include a loss of contact. Third, two distinct dynamic amplification regions are obtained, which are related to the two distinct orders (n = 1 and n = 3) of the roller edge profile y(θ). Since the excitations y(θ) and y(θ ) have instantaneous frequency characteristics (linearly decreasing frequency due to a constant deceleration), the resonant amplifications are at two speed values of Ω = 1/ n . For a detailed explanation of the analytical solution, the reader should go to reference [2].

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DUAL-DOMAIN INVESTIGATION OF DYNAMIC TRANSIENT RESPONSE

Since the dynamic interfacial force response F is a non-stationary signal with frequency decreasing with time, a time-frequency domain representation is required to obtain a better understanding of F. First, the short-time Fourier transform, as defined below, is applied to F(Ω) and the corresponding spectrogram is given in Fig. 3.

F τ ,ω( ) = F T( )w T −τ( )exp −iωT( )dT

−∞

∞

∫ , (15)

where w(T) is the sliding window function at the time instance of T. Even though an analytical solution of the integral of Eq. (15) is possible for certain window functions such as rectangular window, the short-time Fourier transform is implemented as a discrete-time event process in the current article with a Hamming window of 4096 data points and 97.6% overlap.

FIGURE 3. Short-time Fourier transform of signal F(τ).

As seen in Fig. 3, the natural frequency (ωn) of the system appears as a horizontal line at Ω = ωn. Two slanted lines represent the order lines of n =1 and n = 3, and as expected, these order lines also present linearly decreasing speed behavior. Note that, the slopes of these order lines are calculated with nα; hence they are 10-4 and 3×10-4 for n = 1 and n = 3, respectively. Time instances that the order lines cross the natural frequency line are calculated by solving the nondimensional roller angular speed equation nατ + nβ0 = 1 for n = 1 and n = 3, and they are predicted as τ = 10000 and τ = 16666 for n = 1 and n = 3, respectively. Obviously, these time instances correspond to the normalized speeds of 1/n, which are essentially the speed values corresponding to the dynamic amplifications in Fig. 2.

Since the length of w(T) in the short-time Fourier transform is constant, time and frequency resolutions of the resulting spectrogram are fixed though there is always a trade-off between them. In other words, there is a narrow frequency resolution but a poor time resolution. In order to overcome this problem, the wavelet transform is applied to F(τ), which is expressed as:

F s,λ( ) = 1

sF T( )ψ ∗ T − λ

s⎛⎝⎜

⎞⎠⎟

dT−∞

∞

∫ , (16)

where ψ(T) is the mother wavelet and * denotes the complex conjugate. Parameters s and λ are the scale and translation parameters, respectively. The parameter s is used to either compress (s < 1) or dilate (s > 1) the mother wavelet, and thus it can be thought of as a variable length window. At lower scale values (high frequencies) a good time (but a poor frequency) resolution is obtained, and at high scales (lower frequencies) wavelet transform provides a good frequency (but a poor time) resolution [7].

The wavelet transform of F(τ) is shown in Fig. 4. In the calculation, 250 scale values are used starting from 0.001 and with an increment of 0.05. In addition, the wavelet transform amplitudes in Fig. 4 are also plotted with respect to Ω and τ instead of s and λ in order to compare Figs. 3 and 4 easier. As seen in the figure, the wavelet and short-time Fourier transforms yield similar results though the order lines spread out in the high frequency section due to a

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Proceedings of Meetings on Acoustics, Vol. 19, 055030 (2013) Page 6

low resolution. Comparison of Figs. 3 and 4 suggests that the short-time Fourier transform provides better results than the wavelet transform. However, it should be noted that the fixed frequency resolution in Fig. 3 is 0.0163, and the frequency resolution in Fig. 4 varies from 3.1513e-4 to 1.7032. As seen, the order lines in the wavelet transform are still noticeable even with coarse frequency resolution. For the short-time Fourier transform, neither the order nor the horizontal system resonance lines would appear with a fixed frequency resolution of 1.0446 (window length of 64).

FIGURE 4. Wavelet transform of signal F(τ).

ENVELOPE CURVE ESTIMATION USING HILBERT TRANSFORM

Even though an analytical expression for the envelope curve of F(τ) is valid [2], the envelope curve is estimated next by using the Hilbert transform. Mathematically, this transform is given as follows:

F τ( ) = 1

πF T( )τ −T

dT−∞

∞

∫ . (17)

Equation 17 is essentially the convolution of F(τ) with 1/πτ; hence the spectral content of F(τ) is shifted by –π/2 while the amplitudes are kept intact. Here, it should be noted that Hilbert transform is defined as the Cauchy principal value of the integral of Eq. (17); hence the singularity at τ = T is eliminated.

The Hilbert transform of F(τ) is calculated and displayed in Fig. 5 along with the raw F(τ) signal. Observe that Hilbert transform successfully predicts the envelope curve. However, the very high frequency component of the raw signal couldn’t be filtered out. This is due to the algorithm used to calculate the Hilbert transform, which fails to perform at very low frequency (close to 0 Hz) and close to Nyquist frequency [8]. Nevertheless, the proposed method can be effectively used as a tool by practicing engineers, who usually collect raw data in the time domain, though they only need the local maximum amplitudes.

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Proceedings of Meetings on Acoustics, Vol. 19, 055030 (2013) Page 7

FIGURE 5. Predicted dynamic interfacial force response (F(τ)) and its envelope curve: F(τ); , envelope of F(τ).

CONCLUSION

The current article investigates the source regimes of brake judder by using a simplified yet representative mathematical model. First a single degree of freedom model with roller profile excitation is introduced, and an analytical solution of the interfacial normal force is obtained. Closed form solution compares well with the numerical estimation. Dynamic amplifications are successfully observed in the calculated time responses, and the proposed model does show symptoms of a brake judder-like problem. Second, the speed-dependent response analyzed in time-frequency domains with two mathematical transforms. From the dual domain representations of transformed signals, it is clearly observed that the response orders are the same as in the excitation, and a dynamic amplification occurs when the corresponding order frequency coincides with the system resonant frequency. Third, the Hilbert transform is applied using the interfacial force calculation, and its envelope curve is successfully predicted. Resonant amplifications at multiple time instances are successfully linked to the excitation due to rotor surface distortions.

REFERENCES

1. O. T. Sen, J. T. Dreyer, R. Singh, “Order domain analysis of speed-dependent friction-induced torque in a brake experiment”, Journal of Sound and Vibration 331, 5040-5053 (2012).

2. O. T. Sen, J. T. Dreyer, R. Singh, “Envelope and order domain analyses of a nonlinear torsional system decelerating under multiple order frictional torque”, Mechanical Systems and Signal Processing 35, 324-344 (2013).

3. H. Jacobsson, “Disc brake judder considering instantaneous disc thickness and spatial friction variation”, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 217, 325-431 (2003).

4. J. Kang, S. Choi, “Brake dynamometer model predicting brake torque variation due to disc thickness variation”, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 221, 49-55 (2007).

5. C. Duan, R. Singh, “Analysis of the vehicle brake judder problem by employing a simplified source-path-receiver model”, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 225, 141-149 (2010).

6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, 55, National Bureau of Standards, Washington DC, 1972.

7. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis”, IEEE Transactions on Information Theory 36, 961-1005 (1990)

8. M. Feldman, “Hilbert transform in vibration analysis”, Mechanical Systems and Signal Processing 25, 735-802 (2011).

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