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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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