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1 Copyright © 2012 by ASME

Proceedings of the ASME 2012 11th Biennial Conference On Engineering Systems Design And Analysis

ESDA2012

July 2-4, 2012, Nantes, France

ESDA2012-82142

GEAR RATTLE ANALISYS BASED ON WAVELET SIGNAL DECOMPOSITION

R. Brancati1, E. Rocca, S. Savino and F. Farroni

Università di Napoli “Federico II”

Dipartimento di Meccanica ed Energetica

Napoli, Italy

1Contact Author

ABSTRACT

The “gear rattle” phenomenon is a research topic of great

interest for the NVH (Noise, Vibration and Harshness)

automotive sector, concerning driveline noise and vibrations

coming from the manual gear boxes. It is due to the internal

combustion engine variable torque producing impacts and

rebounds, and consequently noise, between the teeth of the

unloaded gear pairs of the gear box because of the

unavoidable presence of backlashes.

Discrete Wavelet Transform (DWT) is used to decompose the

angular relative motion signal and the wavelet decomposition

details are adopted to analyze the dynamic behaviour of gears

under rattle conditions. The DWT has been chosen because it

is a particularly suitable instrument to recognize

discontinuities, such as jumps or instantaneous changes in the

slope of the signals, due to their localization in the time

domain. Wavelet analysis allows to characterize an event by

the localization, the kind, and the amplitude of the event.

This technique enables, moreover, to define new indices for

metrics of gear rattle especially useful in order to conduct

comparisons among different operative conditions. Some

examples of application of the proposed technique are

reported in the paper. The theoretical investigations regard

comparative analysis with respect to the amplitude and to the

frequency of speed fluctuations and to the gears lubrication

regime.

INTRODUCTION

Gear rattle is a vibro-acoustic phenomenon, originated in the

automotive gear box, due to repeated impacts between the

teeth of unloaded gears. It’s caused by the angular speed

fluctuations typical of the internal combustion engines

because of periodic torque oscillations. Impacts occur on

both the surfaces of the tooth and may be of different kind

and severity in dependence of various factors, such as the

moment of inertia of the gears, the backlashes, the applied

driving torque, and so on. The rattle level depends on the

quality of these impacts, and so little variations in the gear

system parameters could influence the dynamic behavior of

the gear [1,2].

For this reason it would be useful to dispose of an instrument

able to reveal this changing, and also to allow a comparative

analysis based on the severity of the impacts.

In unloaded gears the operative conditions may configure a

‘one-sided’ rattle, in which impacts and rebounds between

the gears teeth regard mainly one of the two teeth flanks. In

other cases it may arise a more severe ‘double sided’ rattle

when the speed fluctuations amplitudes are more significant.

In this last case the gap between the teeth is entirely and

alternately crossed during the gears relative motion. Impacts

and tooth contact phases are alternate on both the tooth

surfaces, and consequently they produce a noise qualitatively

different from that produced in the case of one-sided rattle.

Many efforts have been made in the last years in order to

evaluate the quality of rattle to define methods and

procedures for both the theoretical and experimental

analyses.

The theoretical models generally consist in an interaction

between the torsional behaviour of the driveline (constituted

by flywheel, clutch, synchronizers, input and output shafts)

and the coupled vibrations of the unloaded gear pairs [2].

With regards to the vibrations of a single gear pair, a one

degree of freedom model can be adopted, that considers a

non linear stiffness between the meshing teeth with the

presence of gear “dead spaces”. The presence of oil, inserted

between the gear teeth, produces a damping effect that is

appreciable especially during the impacts. The oil squeeze

depends on many tribological characteristics but, particularly,

on the oil quantity and on the lubricant temperature and

viscosity [3].

As regard to the experimental analysis, generally, methods

are based on the relief of the noise by acoustic microphones

or on the relief of vibrations by accelerometers. Another

technique, recently applied, is based on the use of optical

encoders mounted on each gear [4-6].

The present paper aims to propose a theoretical method,

based on the wavelet decomposition, that could be able to

2 Copyright © 2012 by ASME

detect the gear rattle produced by the teeth impacts. The

method can be of help in discriminating different rattle levels

in terms of quality.

Wavelet analysis, commonly used in the industrial faults

diagnosis, is a flexible and powerful analytical tool for

processing of non-stationary signals, widely adopted for the

gears faults monitoring [7-9].

Wavelet decomposition analysis can be considered an

alternative technique, respect the FFT, to analyze vibration

and it presents better performance than the Fourier analysis

because enables a time localization for the errors [10-15].

Moreover the technique is particularly suitable to detect

discontinuities, such as jumps or impacts, and can help in the

definition of indices and metrics suitable for rattle

quantization.

GEAR RATTLE MODEL

As regard to the theoretical model it is possible to obtain the

relative gear motion described by a single degree-of-freedom

lumped parameter model, considering the driven gear forced

to vibrate by a motion imposed on the driving gear [3].

Motion is considered along the line of contacts. Model keeps

in count the squeeze actions due to the oil interposed between

the gears teeth. Figure 1 shows the physical model of the gear

pair where the subscript 1 indicates the driving gear.

m

r

2

k

r

1

a)

X

Oxm

Ox

k

Omx

θ1

θ

2

b) c)

FIGURE 1: THE PHYSICAL MODEL OF THE GEAR PAIR

The mathematical model is then:

⎪

⎩

⎪

⎨

⎧

<<−

−≤

≥

=

−=++

bxhbxS

hbxxxS

bxxXK

xxF

XmfxxFxm r

)( if

)( if ,)(

if ,)(

),(

,),(

minmax

min

with

&

&

&

&&

&&&

(1)

where m is the mass of the driven gear, x = r2

θ

2 - r1

θ

1 is the

relative displacement of the gears evaluated along the line of

contacts, X = r1

θ

1 is the absolute motion of the driving gear, b

is the backlash, and hmin represents the film thickness of the

oil layer that remains adsorbed to the tooth surface during the

contact phase. Moreover to keep in count the resistant actions

arisen in the bearings supporting each unloaded gear idle on

its shaft, a constant drag torque applied to the driven gear has

been here considered, corresponding in eq. (1) to a constant

force fr at the pitch point, calculated for stationary operating

conditions in terms of loads, speed and oil parameters [3,4]

During the contact phases, a non-linear elastic force acts on

the driven gear (Fig. 1: case a, c). Such elastic force is given

by the sum of the contributes due to the n teeth pairs that are

in contact at the same time. Each teeth pair contributes with a

stiffness term [16]:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛−

=

3

zα

z

api Xε1.125

2/XεX

Cexpk(X)K (2)

where ε and εα are the total contact ratio and the transverse

contact ratio respectively and Xz is the transverse base pitch.

For a complete meshing cycle, X starts from 0 and ends at

ε

Xz. Finally, kp indicates the stiffness at the pitch point that

depends on the tooth parameters along with the Ca

coefficient. The elastic force is consequently the sum of n

periodic functions shifted each other of a transverse base

pitch Xz. The contacts can occur either on the driving or on

the driven side of the tooth.

During the approach phase when the teeth are separated

through an oil layer (Fig. 1: case b), the oil squeeze effect

gives a non-linear damping force

()

xxS &, with:

()

()()

()

2

22/3

2

22

23

arctan

3

xRax

xR

a

xRaxRxRaa

ZRxS +

⎥

⎦

⎤

⎢

⎣

⎡⎟

⎠

⎞

⎜

⎝

⎛

++−

−=

μ

(3)

in which

μ

is the absolute viscosity, Z indicates the axial

width of the gear pair, R is the relative curvature radius of the

teeth and a denotes the semi-length of the oil film along the

tooth profile related to the lubrication conditions [3].

In the (1) Smax = S(hmin) represents the saturation value of the

squeeze damping coefficient.

The time history of the relative angular motion, defined in

literature as "Transmission Error", TE =

Δθ

(t) = xr2 = θ2 -

θ1r1/r2, obtained from the (1) has a characteristic pattern of a

square wave, how reported in figure 2.

00.2 0.4 0.6 0.8 1

-3

-2

-1

0

1

2

3x 10

-4

Time [s]

[rad]

FIGURE 2: THE TIME HISTORY OF THE GEAR RELATIVE

MOTION UNDER RATTLE CONDITIONS

In this figure a “bilateral” (or double-sided) rattle case is

reported, with alternate contacts along the two sides of the

teeth. By means of the wavelet technique it is possible to

3 Copyright © 2012 by ASME

discriminate the different kind of impacts that happen in the

two sides, giving so useful information about the quality of

the rattle phenomenon. In this figure it is possible to observe

two different contact phases; the lower side represents the

“correct” side of contact, where the driving gear pushes the

driven one. Vice-versa, in the upper side there is the

“incorrect” contact, where the driven gear pushes the driving

one. The teeth impacts on these two sides of contacts are

qualitatively different because they are due to different

causes. The correct impacts are produced by the drive torque

applied at the driving gear, while the incorrect impacts are

mainly due to the moment of the inertia forces of the driven

gear. These circumstances should be highlighted in a

qualitative analysis of the gear rattle, and for this reason the

wavelet decomposition analysis is proposed.

WAVELET ANALYSIS

The wavelet is a waveform of limited duration that has an

average value of zero. It differs from the sine wave, at the

basis of Fourier analysis, which is not time limited

oscillating, between a minimum and maximum, ad infinitum.

A wavelet transform is the representation of a function by

wavelets. The wavelets are scaled and translated copies

(daughter wavelets) of a finite-length or fast-decaying

oscillating waveform, known as the "mother wavelet" [17-

18]. The wavelet transforms, compared to the traditional

Fourier transform, has a better ability to analyze the functions

that are characterized by discontinuities and peaks, as well as

non-periodic and /or non-stationary signals [12].

A wavelet can be used also to divide a given function or

continuous-time signal into different scale components,

assigning a frequency range to each scale component. In

numerical analysis and functional analysis, a discrete wavelet

transform (DWT) is any wavelet transform for which the

wavelets are discretely sampled. As with other wavelet

transforms, a key advantage it has over Fourier transforms is

temporal resolution: it captures both frequency and location

information (location in time). By means of the discrete

wavelet transform, it is possible to perform a decomposition

of a given signal s with n samples. This multiresolution

decomposition consists of log2(n) stages at most. Starting

from s, the first stage produces two sets of coefficients:

approximation coefficients CA1, and detail coefficients CD1.

These vectors are obtained by convolving s respectively with

a low-pass filter Lo_D for approximation, and with a high-

pass filter Hi_D for detail, followed by a dyadic decimation

(downsampling) [18].

It should be noted that, being fs the sampling frequency, the

CAi coefficients approximate a lowpass filter of the signal

CAi-1 in the frequency interval [0, fs/2i+1], while the detail

coefficients CDi approximate a bandpass filter of the signal

CAi-1 in the frequency interval [fs/2i+1, fs/2i], being CA0 = s.

By means of coefficients CAi and CDi, it is possible to

reconstruct the single level i of wavelet decomposition

obtaining two vectors Ai and Di, that are the components of

vector Ai-1. In figure 3 is shown an example of wavelet

decomposition of a TE signal at level 5.

FIGURE 3: WAVELET DECOMPOSITION OF A TE SIGNAL AT LEVEL 5.

4 Copyright © 2012 by ASME

The reconstruction is obtained by inserting zeros in the

coefficients vectors CAi and CDi (upsampling), and by

convolving the results with a low-pass filter Lo_R for

approximation Ai, and with a high-pass filter Hi_D for detail

Di. The low-pass and high-pass filters depend on the adopted

wavelet family.

Hence, the signal s can be reconstructed as a sum of (n+1)

components, at most:

121 .... DDDDAs nnnn

+

++++= −− (4)

ANALISYS OF RATTLE BY THE WAVELET

DECOMPOSITION

For the rattle detection, the Haar wavelet transform was

adopted. This kind of wavelet is conceptually simple and fast,

and is particularly suitable for the analysis of signals with

sudden transitions, such as the TE time history for a

“bilateral” (or double-sided) rattle case.

The Haar wavelet's mother function ψ(t) can be described as

[18]:

()

⎪

⎩

⎪

⎨

⎧

≤<→−

≤≤→

=ψ

otherwise 0

1t211

21t01

t (5)

The proposed technique consists of the analysis of the sum of

detail vectors Di of the wavelet decomposition conducted on

the TE signal, neglecting the approximation vector. This

permits to highlight the impacts with respect to entire signal.

By the DWT the s signal can be decomposed at level n,

writing so:

n

nnnnn DADDDDAs 1

121 .... +=+++++= −− (6)

where:

∑

=

=n

i

i

nDD

1

1 (7)

In figure 4, with reference to a double sided rattle test, an

example of a details vectors sum 5

1

D of a wavelet

decomposition is shown.

In this figure it is possible to observe that in the time history,

during the “jump” phases, in which there isn’t contact

between the gear teeth, there is an oscillation with a time

duration equal to the crossing period of the gap between the

teeth.

The analysis of details vectors sum shows that there is a

relation between the oscillations and the gap crossing relative

velocity. In fact, the most intense and sudden impacts in TE

time history are preceded by faster oscillations with greater

amplitude.

In order to analyze the oscillations of 5

1

D, two parameters

can be adopted:

- ΔA: the maximum value of the oscillation amplitude,

obtained in correspondence of the teeth impact;

- ΔT: the time duration of the oscillation that coincides with

the entire duration of the “jump” phase in the gears relative

motion.

As shown in figure 4, it is possible to observe two

qualitatively different kinds of impact with different energy.

When the driven gear teeth impinge on the driving gear ones,

the impact can be considered as “incorrect”, being caused by

5.55 5.6 5.65 5.7 5.75 5.8

-0.01

-0.005

0

0.005

0.01

time (s)

[rad]

5.55 5.6 5.65 5.7 5.75 5.8

-0.001

-0.005

0

0.005

0.001

time (s)

[rad]

incorrect

impact

flight

phase

flight

phase

correct

impact

ΔΑ ΔΑ

ΔΤ

ΔΤ

Sum of detail vectors

TE

FIGURE 4: EXAMPLE OF VECTOR SUM OF WAVELET DECOMPOSITION DETAILS

5 Copyright © 2012 by ASME

the moment of inertia forces of the driven gear; vice-versa

when the driving gear teeth impinge on the driven gear ones,

the “correct” impact is caused by the engine torque applied at

the driving shaft.

The vector 5

1

D shows a difference in the trend in the two

kinds of oscillations, both in terms of period (ΔT) and in

terms of amplitude (ΔA).

These two parameters have been used to analyze some

different operating condition of rattle phenomenon.

RESULTS

A series of simulations have been conducted for a gear pair,

coming from an actual gear box, with a transmission ratio ε =

z2/z1=33/37=0.89, under two different lubrication conditions:

an absence of lubrication (No-oil), and a “boundary

lubrication” regime with an oil absolute viscosity value

μ

=0.1 (Pa s).

The parameter a in the eq. 3 indicates the oil film extension

along the tooth profile, and so it is connected with the

lubrication regime. In order to correlate its value among

experiments and theory, many comparisons have been

conducted. For “boundary lubrication” regimes the a value

has to be less than 10% of the tooth height. Values greater

than 10% indicate, in practice, a “full film” lubrication

regime. The value adopted in the present analysis is: a=5% of

tooth height.

The gear is mounted with a mean angular backlash value of

0.013 rad. The driving gear is forced to rotate by the

following law of speed:

)2sin()( ftt m

π

ω

ΔΩ+Ω= (8)

where

Ω

m is the speed mean value,

ΔΩ

is the speed

fluctuation amplitude and f represents the rattle frequency.

The comparisons have been conducted referring to a

variation in the speed fluctuation amplitude, in the frequency

of the rattle cycle, and in the lubrication regime.

In the figure 5 and 6 the lubrication conditions are compared

for “correct” and “incorrect” impacts. The ΔA values are

reported by varying the speed fluctuation amplitude

respectively for the No-oil and for the boundary lubrication

regime. The speed mean value is 500 rpm and the rattle

frequency f is equal to 4 Hz. The speed fluctuation

amplitudes range in the interval 50 ÷ 100 rpm.

How it can be seen in the diagrams the oil lubricated series

are characterized by ΔA values smaller than the No-oil series.

The positive contribute of the lubricant in reducing the rattle

problems is due to a “damping” action exerted, during the

phase that precedes the impact, by the oil interposed between

teeth.

Another characteristic can be noted in the diagrams: the ΔA

values for the “correct” impacts of fig. 5 are greater than the

“incorrect” impacts ones in figure 6. This fact is confirmed

also by an experimental analyses as regard to the rattle

quality in term of impacts on the two different sides of

contacts [1]. The different behaviour mainly depends on the

comparison between the moment of inertia of the driven gear

and the driving gear acceleration.

50 60 70 80 90 100

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2x 10

-3

ΔΩ

ΔΑ

no oil

boundary lubrication

[rpm]

FIGURE 5: ΔΑ MEAN VALUES FOR THE “CORRECT”

IMPACTS FOR TWO LUBRICATION CONDITIONS

50 60 70 80 90 100

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2x 10-3

ΔΩ

ΔΑ

no oil

boundary lubrication

[rpm]

FIGURE 6: ΔΑ MEAN VALUES FOR THE “INCORRECT”

IMPACTS FOR TWO LUBRICATION CONDITIONS

By varying the rattle frequency it can be noted that ΔΑ values

increase as frequency increases, independently from the type

of contacts, either correct and incorrect.

Figure 7 reports the comparisons for two values of the rattle

frequency: 4 and 5 Hz. The figure refers to correct impacts in

the case of boundary lubrication regime.

6 Copyright © 2012 by ASME

50 60 70 80 90 100

1.4

1.5

1.6

1.7

1.8

1.9

2x 10

-3

ΔΩ

ΔΑ

4 Hz

5 Hz

[rpm]

FIGURE 7: ΔΑ MEAN VALUES FOR THE “CORRECT”

IMPACTS FOR TWO RATTLE FREQUENCY VALUES

The max amplitude of the wavelet details ΔA seems to give

useful information regards the rattle quality due to impacts.

In the figure 8 a diagram of the ΔT parameter for the rattle

frequency values of 4 and 5 Hz is reported. This parameter

shows a trend that goes in the opposite direction respect the

ΔA parameter. In fact it decreases as the ΔΩ values increase.

50 60 70 80 90 100

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

ΔΩ

ΔΤ

5 Hz

4 Hz

FIGURE 8: ΔΤ MEAN VALUES FOR THE “CORRECT”

IMPACTS FOR TWO RATTLE FREQUENCY VALUES

Additionally it could be helpful to adopt, as index of rattle

quality, the ratio I between the max amplitude ΔA value

respect the ΔΤ time duration, for a single impact.

This index could be significant for the analysis of the details

5

1

D, because it can synthesize the information correlated

with ΔA and ΔT. As an example of application, in the figure

9, a diagram of I is reported for a boundary lubrication

regime. The trend of the index is comparable with that seen,

in figure 7, for the ΔΑ parameter.

50 60 70 80 90 100

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

ΔΩ

I

5 Hz

4 Hz

FIGURE 9: INDEX I VERSUS ΔΩ FOR 4 HZ AND 5 HZ

RATTLE FREQUENCY

In fact it increases as ΔΩ increases, independently from the

rattle frequency value.

CONCLUSION

In the paper a theoretical procedure, based on the Wavelet

decomposition, in order to study the gear rattle generated by

the teeth impacts for lightly loaded gears has been proposed.

The study has been conducted with a theoretical model that

keeps in count the effect of the oil lubricant squeezed

between the impacting teeth. The parameters considered for

the evaluation of the quality of rattle derive from the details

of the wavelet decomposition, obtained adopting as mother

wavelet an Haar function. The quality of the teeth impacts

has been evaluated by the wavelet details in the “jump” phase

that precedes the impact. In particular, the maximum

oscillation amplitudes of the wavelet details ΔA and the time

duration ΔT of the jumps have been considered.

By the proposed theoretical methodology the different kind

of impacts happening on the two sides of teeth (correct and

incorrect) can be well discriminated.

By the present technique the different dynamic behaviour of

the gears has been analyzed, with regard to variations in

terms of speed fluctuation amplitude, frequency, and gears

lubrication conditions.

The conclusions of this satisfactory study are that “gear

rattle” can be opportunely investigated by adopting this

methodology.

To validate the goodness of the theoretical results an

experimental analysis is in course at the Dime laboratory of

the University of Naples.

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7 Copyright © 2012 by ASME

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