**Gearbox
Diagnostics Fault Detection**

By
Walter Bartelmus

Abstract

The paper deals
with the method of gearbox diagnostics fault detection, and
shows that using: design, production technology, operational,
change of condition (DPTOCC) factors analysis leads
to inference diagnostic information.

In the paper is a review of the current
possibilities for using mathematical modeling and
computer simulation for investigating the dynamic properties
of gearbox systems. Computer simulation of dynamic behavior
of gearboxes is a powerful tool for supporting
diagnostic inference.

The paper shows that for gearbox fault detection,
many different ways of signal analysis should be done.
In the paper it is suggested that for fault detection:
time trace, spectrum, cepstrum and time-frequency spectrogram
examination has to be used. It has been shown that
using these ways of vibration signal analysis there
are possibilities to detect signal faults and distributed
faults in gearboxes. A signal fault is caused by a tooth
crack/fracture and breakage, a spall in a gearing or
in an inner or outer race of a bearing, a spall on
a rolling element of a bearing; distributed faults
are caused by uneven wear (pitting, scuffing, abrasion,
erosion).

Computer simulation enables one to infer
that the cepstrum not only detects single gearing faults,
but also distributed faults. It has been pointed that
for explicit detection of a tooth fracture and breakage
there is a need to use a cepstrum and a time-frequency
spectrogram.

The given consideration is also based on
industrial experience of the author of using condition
monitoring for double stage gearboxes. The gearboxes
of power 630 and 1000 kW are used for driving belt
conveyors.

**1. ** **Introduction **

Diagnostics is understood as identification
of a machine's condition/faults on the basis of symptoms.
Diagnosis requires a skill in identifying machine's
condition from symptoms. The term diagnosis is understood
here similarly as in medicine.

It is generally thought that vibration is
a symptom of a gearbox condition. Vibration generated
by gearboxes is complicated in its structure but gives
a lot of information. We may say that vibration is
a signal of a gearbox condition. To understand information
carried by vibration one have to be conscious/ aware
of a relation between factors having influence to vibration
and a vibration signal.

Let me refer to some publications giving
more details on the discussed subject. In [1] the factors
having influence to a vibration signal are divided
into four groups: design, production technology, operational,
change of condition (DPTOCC). As it is suggested
in [1]:

**Design
factors** include specified stiffness of the gear components,
especially flexibility of gearing, and specified
machining tolerance (errors) of components. Design
factors are given in element/part drawings.

**Production
technology** **factors **include deviations from specified design
factors acquired during machining and assembly of
the gearboxes.

**Operational
factors** include peripheral speed (pitch line velocity) and
its change, and outer load and its change

**Condition
change factors** include condition of bearings and gearbox gearing
and shafts. For gearing we have signal faults (cracking/fracture,
breakage) and distributed faults, pitting, scuffing,
erosion. For bearings we have signal faults (spalling,
pitting, erosion) occurring on bearing's races and
rolling elements. The bearing elements undergo also
abrasive wear causing bearing elements dimensions
change.

**2. ** **Influence of DPTOCC factors to vibration diagnostic signals **

All design factors are specified in design
drawings. The design factors connected with gear dimensions
are transformed into dynamic model parameters like
it is given for example in Fig.1 for a two-stage gearing
system with electric motor and driven machine. The
system consists of: rotor inertia *I*_{s},
gear inertia of first stage *I*_{1p}, I_{2p},
gear inertia of second stage *I*_{3p}, I_{4p},
driven machine inertia *I*_{m}, gearing
stiffness *k*_{z1} k_{z2}_{ }and
shaft stiffness *k*_{1}; k_{2}; k_{3},
damping coefficient of a flexible couple *C*_{1}.

The manifestation of gear dimensions (model
parameters) in diagnostic signals are vibration components
coming from natural vibration of a gearbox system.
Frequencies of these components are independent of
a system rotation. When the frequency of natural vibration
equals to the frequency of excitation caused for example
by machining/ gearing a system runs at resonance. Some
natural components in a vibration time trace signal
are manifested during a starting of gearbox systems.

As it is given in Fig.2 in the period (1)
in which a gear rotation increases from 0 to 980rpm. Fig.2
shows in the period (time within 0 -1.7s) local maxims
of a vibration signal. The plot given in Fig.2 can
be divided into four periods: first period, a starting
of a system and connected with it change of system
rotation from 0 to 980rpm, within time 0 - 1.7s; second
period within time 1.7 - 2.1s, free system rotation;
third period within time 2.1 - 2.6s, linear increase
of outer gearbox load; fourth period within time 2.6
- 5s, run of the system under steady outer load.

Fig.1 System with two stage gearbox

Fig.2 gives relative reference acceleration signal
for gear errors fulfilled design specification.

Fig.2 Relative reference acceleration [m/s^{2}] is a function of time
[s] at gearing condition given by error function *E(0.5,10,0)* and_{ } =0.02; 1 - period of
system rotation increase; 0 - 1.7s; 2 - free rotation
of system, 1.7 - 2.1s; 3 - period of linear outer load
increase 2.1 - 2.6s; 4 - rotation under steady load
2.6 - 5s

In the gearing drawings limits for machining
errors of gearing are specified, for condition simulations
error function/mode is defined as three parameter function *E(a,
e, r)*. The error trace for two co-operating gears
- gearing is given in Fig3. In Fig.3 we see influence
of tooth to tooth error and influence of wheel eccentricities.
Fig.4 gives the time trace of tooth to tooth errors
used in mathematical modeling and computer simulation.
In Fig.4 meaning of *a, e, r* parameters are clarified.
Parameters *a* and *r* can
be chosen between 0 and 1. As it is seen in Fig.3 that
the gearing error function is a sum of error function *E(**a,
e, r)* plus wheel's eccentricities and it may be
expressed by an error function

*E*_{b} = E(a_{ux},a,e,r)
+ b_{1}sin(_{ } _{2}) + b_{2}sin(_{ } _{3}) (1)

where:

*b*_{1},
b_{2} wheel eccentricities in [m]

_{ } _{2}, _{ } _{3} rotation angels
in [rad].

An example of influence of design and condition change
factors described by parameters of the error function *a,
e* is given in Fig.5. Influence of error function
parameters *a, e, r* and a damping coefficient *C*_{1} is
given in [3].

Investigations on influence of operational
factors on vibration are given in Fig.6 and 7. An operational
factor may be rotation speed in rpm of gear wheels.
So the system may run under resonance, at resonance,
an over resonance as it is given in Fig.6. In Fig.8
is given influence of gearing errors (production technology
factor) to inter-tooth force which have direct influence
to vibration generated by a gearing. The influence
is caused by change of max error value *e* from 10 to 15_{ } m. Results given in
figures 7 to 8 were obtained for a system with one
stage gearbox.

Further results of computer simulations
for investigation on influence condition change factors
on the signal generated by gearing and the results
of signal analysis taken by measurements from real
gearboxes are given in publications [2] to [5]. The
condition of a gearing is characterized by signal faults
given by a tooth fracture or breakage or distributed
faults caused by gearing and bearing wear: pitting,
scuffing, erosion. Here will be given some analysis
of condition change of a system with double stage gearboxes
using computer simulation and for vibration measurements
more details are given [5].

Fig.3 Gearing co-operation
errors for new gearing a), and for failed gearing by
pitting b).

Fig.4 a) Error function for error mode E(0.5,
10, 0), a=0.5, e_{1}=10mm., r=0; b) Plot of error function for E(0.5,
10, 0.3), a=0.5, e_{1}=10mm, r=0.3, [3]

a) b)

Fig.5
Coefficient K_{d} as function of inter-tooth
error and error mode: a ? E(0.1,
e, 0), b ? E(0.5, e, 0), c ? E(0.5, - e, 0), according
to [3].

a) b)

Fig.6a) Inter-tooth force measurements at over resonance,
under resonance and resonance, F(t) ? current inter-tooth
force value, F ? constant rated inter-tooth force value;
measurements according to R.Rettig.

b) Function of K_{d} for under-resonance operation
of gearing, obtained by computer simulation for error
mode E(0.5, 10, 0), according
to [2].

** **

Fig.7 Function of K_{d} for resonance
operation of gearing, obtained by computer simulation
for error mode E(0.5, 10, 0), according to [2].

Fig.8 Function of K_{d} for unstable operation
of gearing, obtained by computer simulation for error
mode E(0.5, 15, 0), according
to [2].

The best way of mathematical presentation
of influence of design, production technology and change
of condition factors is a statement describing an inter-tooth
force *F=k*_{z}(a_{ux},g)(max(r_{1}_{ } _{2}-r_{2}_{ } _{3}-l_{u}+E_{b} ,min(r_{1}_{ } _{2} - r_{2}_{ } _{3} +l_{u}+E_{b},0))) (2)

where *E*_{b} is
given by (1)

l_{u} inter
tooth backlash

functions *max* and *min* are defined as follow

*min(**a,b) for which if a<b then min=a else min=b *

*max(**a,b) for which if a>b then max=a else min=b*

*a*_{ux} -
auxiliary value gives position of a wheel.

Substituting E_{b} into equation for F we obtain
the description of influence mentioned factors on the
inter-tooth force generating vibration which is a signal
of a gearing condition.

For modeling of a local fault the time
trace of the stiffness change for the case of local
fault caused by the crack is given in Fig.9 together
with its spectrum and
detail of the spectrum Fig.10.

a) b)

Fig.9 Time stiffness trace [s] with local fault (tooth breakage) with stiffness
drop to 0.5k_{z} and its spectrum in
[Hz]

The detail from the spectrum given in Fig.9 is presented
in Fig.10.

Fig.10 Detail of spectrum given in Fig.9

In Fig.9 and 10 are seen frequency components connected
with a shaft rotation *f*_{o1} its harmonics
and

*z*_{1} f_{o1} mashing frequency its components, where
z_{1} - pinion teeth number*.* The normal
maximum value of stiffness is *k*_{z} and
normal fluctuation of stiffness is *0.06k*_{z} the
drop of stiffness caused by a crack is *0.5k*_{z}.
The parameter of stiffness function change *g=0.06* is
given in (2) by *k*_{z}(*a*_{ux},g). Parameter *g *is a measure of stiffness drop caused by two teeth and
one tooth gearing co-operation. A time trace vibration
signal received from the first stage with the local
fault *D** k*_{z1} =0,5k_{z1}_{ }and
reference errors *E(**a*_{1}=0.5;
e_{1}=20; r_{1}=0) is given in
Fig.11. The time trace acceleration [m/s^{2}]
signal is a function of time [s].

Fig.11
Time [s] acceleration [m/s^{2}] trace of signal
for reference gearing error *E(**a*_{1}=0.5;
e_{1}=20; r_{1}=0) and local fault *D** k*_{z1} =0,5k_{z1}_{ }

a) b)

Fig.12a)
Linear spectrum [Hz] from signal caused by reference
gearing error *E(a*_{1}=0.5; e_{1}=20;
r_{1}=0) and local fault *D** k*_{z1} =0,5k_{z1}_{ }b)
Decibel spectrum from signal caused by reference gearing error *E(a*_{1}=0.5;
e_{1}=20; r_{1}=0) and local fault *D** k*_{z1} =0,5k_{z1}

A linear spectrum from signal caused by a reference
gearing error *E(**a*_{1}=0.5; e_{1}=20; r_{1}=0) and
local fault *D** k*_{z1} =0,5k_{z1} together with its decibel spectrum is given
in Fig.12.a) and b). For local fault detection is recommended
cepstrum analysis given in Fig.13. The cepstrum is
a function of time [s] is given in Fig.13 for three
different gearing conditions defined by a local fault
and the reference error function. In Fig.13 it is seen
that with increase of a fault/crack capstrum components
are increasing. In a gearing may also occur distributed
gearing faults caused by gearing wear, pitting, scuffing, erosion.
The case of distributed faults is considered in [5].
Computer simulation investigations on influence of
distributed faults to cepstrum show that cepstrum is
also a measure of distributed faults.

a) b)

c)

Fig.13
a) Signal's cepstrum for reference gearing error *E(**a*_{1}=0.5;
e_{1}=20; r_{1}=0) and local *fault **D** k*_{z1} =0,5k_{z1}, b) Signal's cepstrum for reference gearing
error *E(a*_{1}=0.5; e_{1}=20; r_{1}=0) and
local fault *D** k*_{z1} =0,1k_{z1}, c)
Signal's cepstrum for reference gearing error *E(a*_{1}=0.5;
e_{1}=20; r_{1}=0) and local fault *D** k*_{z1} =0,9k_{z1}

a) b)

Fig.14
Time frequency spectrograms for a) for signal without signal
fault and b) with signal fault.

For explicit identification of signal gearing
fault there is a need of additional signal analysis
as is given in Fig.14. Fig.14 a) shows a time frequency
spectrogram for signal without a signal fault. Fig
14 b) shows a spectrogram with a signal fault. In Fig.14
a) there are only seen horizontal lines showing meshing
frequencies and its components. Fig.14 b) shows additionally
vertical lines separated by a time period equivalent
a rotation time of a gear wheel with a signal fault.

Taking into consideration given example
and results given in [5] the influence of condition change factors of gearing
for double stage gearboxes is given for local faults,
for wheel eccentricities and distributed faults.

In the spectrum given in Fig.12 we see spectrum
components connected with gearing frequency nz/60 and
its harmonics where *n* [rpm]; *z *- number
of teeth. As considered these harmonics are the measure
of gearing stiffness change and the measure of tooth
errors and with their increase we can see change in
harmonic intensity. In the spectrum of a gearing fault
function Fig.9b we see that first harmonic has the
biggest intensity. In vibration spectrum Fig.12a we
see transformation of components and the biggest intensity
has third spectrum component. This transformation is
caused by influence of design factors, in a system
with a gearbox by parameters (I_{s}, I_{1p},
I_{2p} and so on look Fig.1.

Almost all considered factors are modeled by modification of inter-tooth force given by (2).
The modeling of bearing conditions, described in [5],
is obtained by application of additional forces caused
by bearing faults.

The chapter shows us a relation between
DPTOCC factors and a vibration signal generated by
a gearbox system. This relation leads to inferring
diagnostic information.

The inferring diagnostic information is used for gearbox
condition assessment.

**3
Measured signal analysis final consideration**

In the case of a tooth crack a series of peaks is visible in the signal
trace as in Fig.11 or 15. The peaks are separated by
equal time period T. Peaks indicate a single gear fault
as it is described in chapter 2.

Fig.15 Acceleration
[m/s^{2}] signal time [s] trace with series
of peaks (peaks marked with arrows) [5]

A spectrum for a gearing with a local fault shows meshing frequency
f_{z} and its multiples and noise-like components
due to a local fault Fig.12. Using a cepstrum analysis
we obtained results as given in Fig.16. In Fig.16 there
are given three results obtained from three signals
received from three different points on gearbox housing.
The results are similar because a cepstrum analysis
eliminates influence of a signal path to the results.

More detailed examination of noise like components are made as suggested
in [5] by a time frequency spectrogram and is given
in Fig.14. Horizontal lines representing meshing frequency
components can be seen in a spectrogram for a gearing
without a local fault, Fig.18a. In
the spectrogram for the signal with a local fault also
vertical lines with a time period equivalent to the
period of gear fault revolution/repetition are visible,
Fig.17b.

This chapter shows that results obtained by computer simulation helped
in interpretation of results of real signals.

a) b)

c)

Fig.16 Cepstrums for signal received from three different points
for gearbox mark 5

a) b)

Fig.17 Time-frequency
[s]-[Hz] spectograms a) spectogram of signal without
regular peaks b) spectogram of signal with regular
peaks.

**4. ** **Gearbox diagnostic method and final consideration**

Following description given in [1] first
an overview of a diagnostic method used for double-stage gearboxes is described here next the advanced diagnostic method
will be developed. The method giving in [1] is based
on wide band signal analysis. This method was used
when only some rough description of relation between
the DPTOCC factors and vibration signal was available
and some simple instruments for vibration measurements
were used.

The
signal of vibration is divided into three spectrum
bands. According to chapter [1] the following attributes
have been taken for gear diagnostics:

Operating conditions of the high rotation
shaft of the gear should be reflected by a wall vibration
averaged velocity attribute v mm/s in frequency range
(10-100Hz).

Operating conditions of gearing should be
determined by averaged values of velocity v [mm/s]
and acceleration a m/s^{2} within
the frequency range (3.5-10kHz)

Operating conditions of rolling bearings
should be reflected by acceleration attribute a [m/s^{2}]
within the frequency range (3.5-10kHz).

The
given above method can give only the rough evaluation
of gearing condition. There is no possibility to evaluate
the early change of gearing condition caused by the
early stage of wear caused by pitting or scuffing and
is no possibility to detect a local fault caused by
a tooth cracking/fracture. By using the mentioned method
we can not separate gearing faults coming from two
deferent gearbox stages. We can't separate local faults
and distributed faults. So a new more advanced method
based on knowledge gained from computer simulations
and real measured signal analysis is needed.

The
new method takes into account considerations given
in chapter 2 and in publications [2] to [4] and monograph
[5]. The wide band method is easy to use for on-line
condition monitoring. The detailed condition assessment
of a gearbox may be done using off-line additional
signal analysis.

The
method has to take into consideration the scenario
of gearbox condition change here is taken typical scenario
occurring in surface mines. The scenario of the gearboxes
deterioration is as follows: Gearboxes work in heavy
environment conditions and very fine particles of sand
get inside housings of the gearboxes and cause abrasive
wear of gearbox bearings.

Fig.18 Interection of gearbox elements condition and influence of environment

The bearings in that condition have influence to the
gearing co-operation and cause increase of inter-teeth
vibration (acceleration). For the additional evidence
of condition change caused the bearings abrasive wear y coefficient given by (3) should be calculated.
The check consists in determining the slope of a straight
line (look Fig.5) in the linear relation by calculating
the coefficient

_{ } = (_{ } - _{ } )/(*I*_{1} -
I_{2}) m/(s^{2}A)* *(3)

where _{ } , _{ } ? mean values
of accelerations for signal within a band of 100¸3500 Hz for respectively: a load of about
0.85*I*_{n} and 0.5*I*_{n}; *I*_{n} -
rated electric current value [A]; *I*_{1}, *I*_{2} ? values
corresponding to the smaller and greater load, respectively.

The interaction of gearbox element condition
is according to Fig.18. The sequence of condition change
is as follows: influence of mine environment, abrasive
wear of bearings, change of
co-operation in gearing, change of gearing condition,
pitting or scuffing of teeth. The abrasive wear of
bearings has also influence to a high rotation/speed
shaft (shaft driven by an electric motor) working condition.
In the former method gearing condition is described
by the gearing co-operation classification, in succession
B, C look [1]. Further dramatic change of the gearbox
condition is ?knocking out' rolling bearing seatings,
unstable run of gearing. Function of K_{d} given in Fig.8 is equivalent to gearing condition
co-operation for class D. In [1] is recommended that
replacement of a gearbox ought to be done when gearing
condition is in class B (averaged vibration within
30 and 45 m/s^{2}). Replacement of a gearbox
when vibration is within class B is recommend as ?economic
replacement'. As it was mentioned the method described
in [1] can not detect the early change of gearing condition.

The early stages of gearing condition change
can be detected by new ways of signal analysis obtained
by cepstrum. Full description of use of cepstrum analysis
is given in chapter 2 and in [5]. The gearing condition
change is caused by pitting or scuffing, erosion. The
pitting scuffing and erosion cause distributed faults.
The separated rahmonics in cepstrum can be distinguished
for gearbox stages. Besides of distributed faults there
are signal faults caused by a tooth crack/fracture
or breakage of a tooth Fig.13. For explicit detection
of a tooth crack/fracture or breakage cepstrum and
time-frequency spectrogram signal analysis in an advanced
method is used and given in Fig.14 and 17.

In the former method [1] there is not possibility
to detect such change of a gearing condition. There
was not clear interpretation/relation between gearing
condition and vibration signal. There was a tendency
of the diagnostic system user to make a replacement
decision when a vibration level is for gearing condition
in class C (averaged vibration within 45 and 70 m/s^{2}), it
is termed as necessary replacement of a gearbox. This
tendency causes increase of funds for gearbox repairs.

Finally we may come to conclusion that joined
inference which coming from consideration given in
the method description [1] and new findings presented
here and more in the monograph [5] should be taken
during a gearbox system diagnostic assessment. The
condition of gearboxes may be interpreted as it was
given in the wide band vibration method and using new
ways of signal interpretation more evidence for gearing
condition is obtained.

It is recommended to use the wide
band method for on-line condition monitoring and off-line
condition monitoring may be used when signal comes
to the level of class B for detailed evaluation of
a gearbox condition change.

One should mention that using recommendations
given in ISO 3945, measurement of velocity within 10
-1000Hz and Canadian standard CDA/MS/NYSH 107 measurement
of acceleration within 10 -10000Hz we may have some
troubles with gearbox condition evaluation. It is recommend using
signal filtration in mentioned above bands.

Literature

[1]
Bartelmus W.: Vibration
condition monitoring of gearboxes. **Machine
Vibration** 1992 nr1 s.178-189.

[2] Bartelmus W.: Transformation of gear inter teeth
forces into acceleration and velocity. Conference
Proceedings of The 7th International Symposium on Transport
Phenomena and Dynamics of Rotating Machinery Hawaii USA 1998. i w **International
Journal of Rotating Machinery** 1999 Vol.5. No.3
s.203-218

[3] Bartelmus W.: Mathematical Modelling of Gearbox Vibration
for Fault Diagnosis, **International Journal of COMADEM,** Vol.3,
no.4, 2000

[4] Bartelmus W. Mathematical Modelling
and Computer Simulations as an Aid to Gearbox Diagnostics. **Mechanical Systems and Signal Processing** 2001 Vol.15, nr5, s.
855-871

[5] Bartelmus W.: Computer-aided multistage gearbox diagnostic
inference by computer simulation. **Scientific
Papers of the ****Institute**** of ****Mining**** of ****Wroclaw**** ****University**** of Technology**. No.100, 2002