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# Condition monitoring of the front axle of a load haul dumper with real order derivatives and generalised norms

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## Abstract and Figures

Fractional calculus and generalised norms provide a powerful toolkit for analysing vibration from rotating machines. They have been used effectively in the condition monitoring of immobile machines. A harsh environment and varying operating conditions complicate the reliable condition monitoring of mobile machines in underground mining industry. In this paper, we focus on the condition monitoring of the front axle of a load haul dumper and the numerical calculation of complex order derivatives and integrals. Measurements are performed with accelerometers, which measure horizontal and vertical vibrations near the planetary gearboxes. A tachometer records the rotational speed of the drive shaft, which is essential for recognising different operation stages of the machine. We describe how the mentioned difficulties can be overcome and what real order derivatives and generalised norms can reveal about the condition of the axle. An improved algorithm for the numerical calculation of complex order derivatives and integrals is given.
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Condition monitoring of the front axle of a load haul dumper
with real order derivatives and generalised norms
1Juhani Nissilä, 2Sulo Lahdelma and 1Jouni Laurila
1Mechatronics and Machine Diagnostics Research Group, Faculty of Technology
P.O. Box 4200, FI-90014 University of Oulu, Finland
E-mail: juhani.nissila@oulu.ﬁ, jouni.laurila@oulu.ﬁ
2Engineering Oﬃce Mitsol Oy
Tirriäisentie 11, FI-90540 Oulu, Finland
E-mail: sulo.lahdelma@mitsol.inet.ﬁ
Abstract
Fractional calculus and generalised norms provide a powerful toolkit for analysing vibra-
tion from rotating machines. They have been used eﬀectively in the condition monitoring
of immobile machines. A harsh environment and varying operating conditions complicate
the reliable condition monitoring of mobile machines in underground mining industry. In
this paper, we focus on the condition monitoring of the front axle of a load haul dumper
and the numerical calculation of complex order derivatives and integrals. Measurements
are performed with accelerometers, which measure horizontal and vertical vibrations near
the planetary gearboxes. A tachometer records the rotational speed of the drive shaft,
which is essential for recognising diﬀerent operation stages of the machine. We describe
how the mentioned diﬃculties can be overcome and what real order derivatives and gen-
eralised norms can reveal about the condition of the axle. An improved algorithm for the
numerical calculation of complex order derivatives and integrals is given.
Keywords: load haul dumper, LHD, axle, planetary gearbox, condition monitoring, di-
agnostics, real and fractional order derivatives, lpnorms, MIT indices
1. Introduction
This paper provides a detailed discussion of the numerical calculation of real and com-
plex order derivatives and integrals using the discrete Fourier transform. The improved
algorithm is utilised to analyse vibration measurements from the front axle of a load
haul dumper (LHD). These machines operate in harsh conditions where failures may be
diﬃcult to repair. The machine under study is a Sandvik LH 261 working underground in
the Pyhäsalmi mine. Four accelerometers were mounted on its front axle housing. They
measure horizontal and vertical vibrations near the planetary gearboxes on either side. A
tachometer records the rotational speed of the drive shaft. The measurements included
in this study cover a period of 271 days and more data are still being collected. Signals
that were recorded when the LHD was moving at constant speed and without a load
proved to be useful for monitoring condition. The generalised norms are calculated using
diﬀerent orders of derivatives αRand norms pR. These are divided by reference
values when the machine is in good condition in order to form a three dimensional surface
to reveal which of the norms have changed the most. Trend analysis of diﬀerent norms is
performed for the whole measurement period.
2. Real and complex order derivatives and integrals
2.1 Deﬁnitions
Functions of the form
x(t) = Xeiωt,(1)
where Xand ωare constants and i=1can be diﬀerentiated ntimes and we obtain
the expression
x(n)(t) = ()nXeiωt .(2)
In references (1, 2, 3, 4) Lahdelma et al. have deﬁned real order derivatives and integrals of
function (1) by replacing nNwith a real number αR
x(α)(t) = ()αXeiωt ,(3)
where ()α=|ω|αi·sign(ω)α=|ω|αeπ
2sign(ω). The function sign is the signum function
that extracts the sign of a real number. Equation (3) is further generalised to derivatives
and integrals of complex order z=α+where α, β R(5,6)
x(z)(t) = ()zXeiωt ,(4)
and only the principal value of the function ()zis considered, which gives us ()z=
ezln(|ω|)+iz π
2sign(ω). In the case of vibration signals the constant Xrepresents the ampli-
tude at the angular frequency ω= 2πν in the signal’s spectrum. This can be stated
mathematically with the Fourier transform X(ν)of the signal x(t)
F{x(t)}=X(ν) = Z
−∞
x(t)ei2πνt dt. (5)
Its inverse transform is deﬁned as
F1{X(ν)}=Z
−∞
X(ν)ei2πνt dν. (6)
In suitable function spaces the inverse transform returns the original signal. If this is the
case, then the original function can be written as a double integral, which is called the
Fourier integral theorem.
Fourier derived in 1822 in his book Théorie analytique de la chaleur (7) a form of
integral theorem (here shown with modern notation)
x(t) = 1
2πZ
−∞
x(ν)Z
−∞
cos(pt ) dpdν, (7)
2
and used it to present the real order derivatives and integrals of function xin the form
dα
dtαx(t) = 1
2πZ
−∞
x(ν)Z
−∞
pαcos pt +απ
2dpdν. (8)
We deﬁne diﬀerintegrals x(z)for such functions xwhose Fourier transforms Xand inverse
transforms of (i2πν)zXexist. Let zCand Xis the Fourier transform of x. The
diﬀerintegral of order zis
x(z)(t) = F1(i2πν)zX(ν)
=Z
−∞
(i2πν)zX(ν)ei2πν t dν, (9)
where (i2πν)z=ezln(2π|ν|)+iz π
2sign(ν).
In many textbooks on fractional calculus this is not presented as a deﬁnition of a
diﬀerintegral, but rather as a way of representing (and calculating) diﬀerintegrals based
on other deﬁnitions. Similar deﬁnitions are also used in (8,9)
, for example. For Re(z)<0it
is convenient to assume that the signal has zero mean value, i.e. R
−∞ x(t) dt=X(0) = 0,
or otherwise the zero frequency component would be ampliﬁed into a Dirac delta function.
A closely related deﬁnition was considered by Hermann Weyl in 1917 (10) using Fourier
series.
The usefulness of deﬁnition (9) in vibration mechanics and other physical applications
is that the object may vibrate with any angular frequency ω= 2πν Rand the eﬀect
of these frequencies on fractional derivatives and integrals is plainly seen. Even more
importantly it does not include a built-in ansatz, as most of the deﬁnitions of fractional
calculus in the time domain do (11)
. For vibration signals this means that with the Fourier
deﬁnition one takes into account the whole past history of the vibration and does not
assume that it was zero at some exact time in the past. In the article (1) Lahdelma has
studied the transition of the real order derivatives and integrals ()αXeiωt in the complex
plane as the order of diﬀerentiation and integration changes.
In practice, signals are sampled with sample time tand the length of the signal
is ﬁnite. Then we have a sampled sequence x0, . . . , xN1and we can approximate the
signal’s Fourier transform with the Discrete Fourier transform (DFT)
F{xn}=Xk=1
N
N1
X
n=0
xnei2πkn/N .(10)
Its inverse transform (IDFT) is
F1{Xk}=xn=
N1
X
k=0
Xkei2πkn/N .(11)
Discrete Fourier transform and its inverse are complex-valued sequences of equal
length. They are also N-periodic and the inverse returns the original sequence. DFT
is therefore a discrete approximation of the signals spectrum but also contains all the
information needed to return the original sequence.
3
The problem with the DFT and IDFT is that they assume the signal to be periodic.
This can be improved by multiplying the signal with a suitable window function. When
calculating real order derivatives and integrals it is desirable that the signal is distorted
in the time domain as little as possible. Therefore, the window function should have quite
rapid ascent and descent.
A simple window function is constructed from the Hann window by using it in the
ascent and descent parts of the window
w1(t) =
0,if t0
0.51cos πt
T / ,if 0< t < T/
1,if T/ tT /2
w1(Tt),if t > T/2,
(12)
where is the portion of Tfor ascent and descent. In the books (12, 13) examples are given
of = 10.
Lahdelma and Kotila introduced in the article (6) a window function
w2(t) =
0,if t0
RT/
0eτ(τT/)1
dτ1
Rt
0eτ(τT/)1
dτ, if 0< t < T /
1,if T/ tT /2
w2(Tt),if t > T/2,
(13)
and used the value = 8 in their calculations. It can easily be seen that all the derivatives
of w2are continuous and, therefore, it preserves the continuity properties of the original
signal.
2.2 Numerical algorithm
For discrete samples one cannot straightforwardly calculate diﬀerintegrals according to
deﬁnition (9). However, one can form the discrete components Xkof DFT. Then the IDFT
formula (11) deﬁnes a trigonometric interpolation of the function that can be diﬀerinte-
grated in the frequency domain by multiplying the DFT components Xkwith (i2πk/T )z
and taking IDFT of this new sequence. The numbers kare problematic, because due to
the N-periodicity of IDFT the interpolant gets the same values at discrete points even if
kwas replaced with k+mkN, where mkZ. Bigger k(higher frequency) creates more
oscillation between the sample points, which has a major impact on the diﬀerintegrals. If
we want the interpolant to be as smooth as possible, the best choices should be from the
interval N/2kN/2. In the reference (14) this diﬀerentiation of the triconometric
interpolant has been studied with respect to the 1st and 2nd derivatives of the function.
In this article the order of diﬀerintegration is allowed to be zCfor the sake of com-
pleteness, although only real order diﬀerintegrals are utilised in the experimental part.
Trigonometric interpolation is a function of the continuous time variable t
x(t) =
N1
X
k=0
Xkei2π(k+mkN)t/T ,
4
which at the sample points tnreceives the values xnfor all mkZ, where tn= ∆t·n,
n= 0,1, . . . , N-1and T= ∆t·N. We want the interpolation to oscillate as little as
possible between the sample points. In the reference (14) 1
TRT
0|x0(t)|2dtwas minimised,
but it was also mentioned, that the same result can be obtained by limiting the frequencies
|k+mkN| ≤ N
2.(14)
When 0k < N/2, equation (14) implies mk= 0 and when 0k < N/2, we get
mk=1. Therefore we get the values N/2< k < N/2. The problem is solved if Nis
odd. However if Nis even, the term N/2is problematic. Frequency limitation (14) gives
mk= 0 or mk=1. Let’s divide it between both of these frequencies
XN/2huei2πN
2t/T + (1 u)ei2π(N
2N)t/T i
=XN/2ueiπN t/T + (1 u)eiπN t/T ,
where ucould be any complex number, since at the sample points tn= ∆t·n=T
Nnwe
have e±iπn = (1)nand the right IDFT component is regained
XN/2(1)n[u+ (1 u)] = XN/2(1)n.
In the reference (14) the number uwas decided by minimising 1
TRT
0|x0(t)|2dtbut a sim-
pler way is to demand that an interpolant of a real-valued signal must be real-valued
everywhere. For real-valued signals the DFT values have the symmetry Xk=XNkand
therefore X0and XN/2are real-valued. These symmetric terms form in pairs cosine and
sine terms. Therefore, the multiplier of the lone XN/2must be real, which is true if u= 1/2
XN/2
1
2eiπN t/T +eiπN t/T =XN/2cos πNt
T.
We ﬁnally have the trigonometric interpolation of minimal oscillation (14)
x(t) = X0+X
0<k<N/2Xkei2πkt/T +XNkei2πkt/T +XN/2cos πNt
T
=X
0k<N/2
Xkei2πkt/T +X
N/2<k<0
XN+kei2πkt/T +XN/2cos πNt
T.
(15)
Let us diﬀerintegrate interpolation (15) zCtimes with the deﬁnition (9)
x(z)(t) = X
0k<N/22πki
Tz
Xkei2πkt/T +X
N/2<k<02πki
Tz
XN+kei2πkt/T
+πN
Tz
XN/2cos πN t
T+zπ
2,
and compute it at the sample points tn=T
Nn
x(z)
n=X
0k<N/22πki
Tz
Xkei2πkn/N +X
N/2<k<02πki
Tz
XN+kei2πkn/N
+πN
Tz
XN/2cos πn +zπ
2.
5
The last term has to be written in the form
πN
Tz
XN/2
1
2eiπn+iz π
2+eiπniz π
2
=πN
Tz1
2iz+ (i)zXN/2(1)n
=πN
Tz
cos zπ
2XN/2(1)n
to get the expression in the form of IDFT. Apart from this last term all the values Xk
get multipliers (2πki/T )z,N/2< k < N/2and for XN/2we get the multiplier
πN
Tz
cos zπ
2.(16)
This multiplier is zero only at the zeroes of the cosine function at z= 2n1, n N.
Previous calculations were performed because of a single problematic term in the
DFT. If Nis odd, none of those calculations matter. However, to utilise the speed of the
FFT algorithm we need even N(15)
.
Algorithm 1. Diﬀerintegration in frequency space
1. Calculate the DFT Xk,0kN-1of the sequence xn,0nN-1with the
FFT algorithm
Xk=F{xn}.
2. Calculate a new sequence Gk,0kN-1and G0= 0
Gk=2πki
Tz
Xk,0< k < N/2
GN+k=2πki
Tz
XN+k,N/2< k < 0
GN/2=πN
Tz
cos zπ
2XN/2(if Nis even).
3. Calculate the diﬀerintegrated sequence x(z)
nwith the IFFT
x(z)
n=F1{Gk}.
To provide a simple example of the usefulness of the window functions and the eﬀec-
tiveness of the Algorithm 1, we diﬀerentiate the function cos(πt)at the interval [0,7]. The
periodic continuation of this function is not continuous, since cos(0) = 1 6=1 = cos(7π).
Figure 1 shows the convergence of the absolute error of the ﬁrst derivative at the point
6
t= 3.5as the number of sample points increase. Values of the window function w2(13)
have been integrated numerically with the trapezoidal rule and the Simpson’s rule, but
these two perform almost identically. Their convergence seems to be exponential (i.e.
faster than linear on the log-log scale) whereas with w1(12) and without window func-
tions the convergence is only polynomial (i.e. linear on the log-log scale).
The complexity of phases 1 and 3 in the Algorithm 1 is ONlog2(N)and for phase 2
O(N). Therefore, the complexity of the whole algorithm is determined by the complexity
of the FFT algorithm ONlog2(N).
Real order diﬀerintegrals of real-valued functions should in general be real-valued as
well but numerical calculations may cause small imaginary parts for these solutions. They
may be ignored or their amplitudes added to the corresponding real parts. Complex order
diﬀerintegrals of real-valued functions are in general complex-valued (5, 6)
.
It is good to realise that Matlab functions ﬀt and iﬀt, for example, have the division
by Nthe other way around, but this has no impact on the algorithm.
Figure 1. Convergence of the absolute error of the ﬁrst derivative as the
number of sample points increase
3. Generalised norms and MIT indices
3.1 lpnorms and generalised norms
We will focus on diﬀerintegrals of order αR, although the following norms could also
be calculated for x(z)CN. The diﬀerintegrated sequence is represented as a vector
7
x(α)=x(α)
1, x(α)
2, . . . , x(α)
NRN. The lpnorm of this sequence is
x(α)
p= N
X
n=1 x(α)
n
p!1/p
,(17)
where 1p < . As pgrows, the norms lpbecome smaller (16)
x(α)
p
x(α)
q,if p < q.
Norm lis the limit as p→ ∞ and turns out to be
x(α)
= max
n=1,...,N x(α)
n.
Norm (17) can be generalised with weight factors, which we choose to be 1/N (3,4)
.Gen-
eralised norm lpis 1
Nweighted lpnorm
x(α)
p, 1
N
= N
X
n=1
1
Nx(α)
n
p!1/p
=1
N1/p
x(α)
p(18)
Norm lpis also called the power mean or Hölder mean after Otto Hölder and it has the
opposite order of growth than lpnorms (17)
x(α)
p, 1
N
x(α)
q, 1
N
,if p < q,
but the limit p→ ∞ is the same
x(α)
,1
N
= max
n=1,...,N x(α)
n,
which is the peak value. Norm p= 2 is the root mean square (rms)
x(α)
2,1
N
= 1
N
N
X
n=1 x(α)
n
2!1/2
.
Peaks and rms values are examples of traditional features calculated from vibration sig-
nals.
Clearly there is no reason to limit our attention to so few features, when lpnorms
can be calculated for any p1. If we do not mind violating the axioms of norms, we
can extend these calculations even further. If p < 1, the triangle inequality is no longer
satisﬁed and the following features are called quasinorms.
Quasinorm l0is meaningful as a limit p0and turns out to be the geometric mean
x(α)
0,1
N
= N
Y
n=1 x(α)
n!1/N
.
Lahdelma et al. (18) have utilised the cases 0p < 1with good results. With p < 0
none of the vector values can be 0. Let us collect these generalisations under one deﬁnition,
8
which is the Hölder mean or generalised norm lp. The word ’generalised’ then refers not
only to weights 1/N but also to the inclusion of quasinorms
x(α)
p, 1
N
=
PN
n=1
1
Nx(α)
n
p1/p if pR\{0}
QN
n=1 x(α)
n1/N if p= 0
maxn=1,...,N x(α)
nif p=
minn=1,...,N x(α)
nif p=−∞.
(19)
From the results above we can see that lpnorms form a continuous measurement
scale for the magnitude of vector signals. Large pvalues amplify the biggest elements of
vectors and small pvalues the smallest ones. Compare this with diﬀerintegrals, where
diﬀerentiation ampliﬁes high frequencies and integration lower frequencies.
3.2 MIT index
Lahdelma presented in 1992 (19) the measurement index, or MIT index, utilising rms
values of displacement and its derivatives and integrals of order nN. Later it has been
generalised to lpnorms and real order derivatives and integrals (3, 4)
. Thus it is formulated
as
τM IT p1,p2,...,pn
α12,...,αn=1
n
n
X
k=1
bαk
x(αk)
pk
kx(αk)kpk0
,(20)
where Pn
k=1 bαk= 1,τis signal length and the signals with lower index 0 are reference
values of the machine in good condition. M I T index can also be combined with other
quantities that are related to its condition, such as temperature, pressure or some statis-
tical features of diﬀerent signals. The weight factors bαkare used to take into account the
severity of diﬀerent faults occurring at diﬀerent αkand pk. The inverse of MIT index is
deﬁned as condition index SOL
SOL =1
MIT .(21)
When the machine is in good condition, the two indices are equal, i.e. MIT =SOL = 1.
As its condition becomes weaker, growth of the MI T index and decrease of the SOL
index usually occur. The word ’usually’ is necessary, since, for example, minor wear on
new parts may cause smoother operation at the beginning of their usage and hence
temporarily reduce M I T .
4. Data acquisition
Four SKF CMPT 2310 accelerometers were mounted externally onto the LHD’s front
axle housing to measure horizontal and vertical vibrations near the planetary gearboxes
on either side. These four vibration measurements together with a tachometer pulse from
the drive shaft are recorded with a National Instruments CompactRIO 9024 data logger
into a solid-state drive (SSD) as ﬁles of one minute length. Signals are recorded with
sampling frequency 12800 Hz, and a built-in antialising ﬁlter guarantees that there are
9
no aliases at frequencies that are less than 0.45 ·12800 Hz = 5760 Hz. More information
on the measurements can be found from an article by Laukka et al. (20)
.
The measurement points are right vertical (RV), left vertical (LV), right horizontal
(RH) and left horizontal (LH). During the ﬁrst month of the measurements, the ac-
celerometer cables of LV and LH were broken. The accelerometers were replaced and the
cables have remained intact since. Two SSDs also broke down after six months of service,
which stopped the whole measurement for a month and a half. A third accelerometer at
RH broke down during this stoppage and was replaced. At the beginning, measurements
were always recorded when the LHD was operating. After the stoppage the program was
modiﬁed to record only two hours of data after the starting of the LHD.
5. Results
For calculations 106 signals were selected from the beginning of most measurement days,
when the rotational frequency of the drive shaft was approximately 13.5 Hz. From each
signal a 4 second sample was multiplied with the window function (13) using = 10
and this new signal was diﬀerintegrated with the Algorithm 1. High-pass ﬁltering was
performed with an ideal ﬁlter at 3 Hz to remove unreliable low-frequency components,
and diﬀerent low-pass ﬁlterings at cut-oﬀ frequencies 2000Hz, 3000 Hz and 5000 Hz were
also performed with an ideal ﬁlter. From each end of the signal 20% was rejected and the
remaining 2.4 second signal was used in the calculation of lpnorms. All the calculations
were performed with Matlab.
Figure 2 shows the 4 s samples from the ﬁrst and last days of the measurements
and their amplitude spectra from the point RV. The vibration level has risen from ap-
proximately 12 m/s2to 16 m/s2and on the ﬁrst day the two biggest components in the
spectrum were approximately at frequencies 1610 Hz and 1880 Hz and on the last day at
1990 Hz and 2540 Hz. Figure 3 shows the corresponding signals from the point LV. Here
the vibration level has risen from approximately 12m/s2to 23 m/s2and on the ﬁrst day
the two biggest components in the spectrum are approximately at frequencies 1070Hz
and 1500 Hz and on the last day at 1440 Hz and 1670 Hz.
The diﬀerential of the front axle has a driving pinion with 9 teeth and a crown wheel
with 46 teeth. The planetary gearboxes consist of a stationary ring gear (104 teeth), three
planetary gears (39 teeth) and a sun gear (19 teeth). Drive is provided via the sun gear
and the planet carrier provides the output to the front wheel. Assuming then that the
rotational frequency of the drive shaft is νdrive = 13.5Hz and that the LHD is not turning,
we get the rotational frequency of the sun gear νsun =9
46 νdrive and the planetary gear
mesh frequency (21,22)
νmesh =19 ·104
19 + 104νsun 42.4Hz.
This and its second harmonic 2νmesh can be found from the spectra, but they are
insigniﬁcant when compared to higher frequency components that dominate the spectra
of Figures 2 and 3.
10
Figure 2. Signals and their amplitude spectra from the point RV at the
beginning and end of the measurement period
Figure 3. Signals and their amplitude spectra from the point LV at the
beginning and end of the measurement period
11
Figure 4. 2.4M I T p
αsurface from the point RV in the frequency range
3 - 5000 Hz
Figure 4 shows a surface that has 2.4M IT p
αvalues from the point RV with 0p10
and 2α6using a step of 0.1. The values for 2.4MI T p
αare obtained using the last
and ﬁrst day measurements. The measurements from the ﬁrst day are used as reference
measurements in all the calculations in this paper. Signiﬁcant changes have taken place.
Although these surfaces vary slightly from day to day, the ridge at approximately α= 4.4
has constantly been getting bigger with time. The two peaks that occur at the high values
of pand αare not the best choices for trend analysis, because the norm values vary too
much between days for those values. One also notices that at α= 2, which are values
of the original acceleration signal, sensitivity is fairly low. This is also seen in the trend
analysis of the whole measurement period. Figure 5 shows the trend of right vertical
measurements using the values 2.4M I T 2
2. The variance in the values is most probably due
to the changing terrain and minor ﬂuctuations in the speed of the LHD. The values fall
at the beginning but then they bounce around 1 for a long time. Only some time after
200 days the 2.4M IT 2
2values stay above 1.2. The highest value is almost 1.8.
Figure 6 shows another trend of the same signals but this time using the derivative
x(4.4) and p= 10. Now the values start to rise very quickly and after 75 days they are
always bigger than 1.2. At the end of the measurement period the values almost reach
2.6. This is probably a sign of wear on the gearbox components. The 2.4M IT 10
4.4values
increase almost linearly with time.
12
Figure 5. Trend of 2.4M I T 2
2from the point RV in the frequency range
3 - 5000 Hz
Figure 6. Trend of 2.4M I T 10
4.4from the point RV in the frequency range
3 - 5000 Hz
13
Figure 7. 2.4M I T p
αsurface from the point LV in the frequency range
3 - 5000 Hz
Figure 7 shows a surface of 2.4M IT p
αvalues from the point LV with 0p10 and
2α6using a step of 0.1. Its shape diﬀers quite a lot from that of Figure 4, but
a similar prominent ridge is present, this time around α= 3.6. Integrated values have
also risen but they are not suitable for trend analysis because there is too much variation
between the measurement days. Figures 8 and 9 show trends from the point LV with
rms of the original acceleration signal and x(3.6) respectively. Using α= 3.6in the trend
analysis improves sensitivity in the end, but does not quicken the observation of wear. It
looks like the MIT indices rise exponentially with time.
Since the biggest frequency components from the point RV were below 3000 Hz and
from the point LV below 2000 Hz, the calculations were repeated in the frequency ranges
3 - 3000 Hz and 3 - 2000 Hz. Figure 10 shows a surface of values 2.4M I T p
αfrom the point
RV with 0p10 and 2α12 using a step of 0.2in the frequency range 3 -
3000 Hz and Figure 11 shows the corresponding values calculated from the point LV in
the frequency range 3 - 2000 Hz. In both the cases there is a ridge centered at α= 10.
Since the order of norm seems to have little eﬀect, the trend analysis is performed with
easy to calculate p= 1 values. The trend of 2.4MIT 1
10 values for these measurement points
are plotted in Figures 12 and 13. The 2.4M IT 1
10 values from the point RV are doubled
after only 50 days and the highest value reached is over 6. For the point LV the highest
value reached is over 4, but again the wear is clearly revealed only after 200 days.
14
Figure 8. Trend of 2.4M I T 2
2from the point LV in the frequency range
3 - 5000 Hz
Figure 9. Trend of 2.4M I T 2
3.6from the point LV in the frequency range
3 - 5000 Hz
15
Figure 10. 2.4M I T p
αsurface from the point RV in the frequency range
3 - 3000 Hz
Figure 11. 2.4M I T p
αsurface from the point LV in the frequency range
3 - 2000 Hz
16
Figure 12. Trend of 2.4M I T 1
10 from the point RV in the frequency range
3 - 3000 Hz
Figure 13. Trend of 2.4M I T 1
10 from the point LV in the frequency range
3 - 2000 Hz
17
Horizontal measurements have not changed as much. This can be seen from the trends
of the rms values of horizontal acceleration signals in Figures 14 and 15. Values 2.4M IT 2
2
from the point RH seem to have risen at the beginning, but they return back to around 1
at the end of the measurement period. No clear trend is imminent and 2.4M IT p
αsurfaces
also vary considerably from day to day.
Figure 14. Trend of 2.4M I T 2
2from the point RH in the frequency range
3 - 5000 Hz
Figure 15. Trend of 2.4M I T 2
2from the point LH in the frequency range
3 - 5000 Hz
18
6. Conclusions
Real order derivatives and integrals are simple and fast to calculate in the frequency
domain. Together with Algorithm 1 and window function (13), the results are also very
reliable, although the signiﬁcance of this window function and the multiplier for the
N/2term in the Algorithm 1 becomes less important when Nis big. Combined with
generalised norms, one can plot a surface using the values of the MIT index. This surface
is suitable for searching the optimal values for αand pin fault detection.
It is diﬃcult to monitor the condition of a mobile vehicle. In this case the LHD
is fortunately operated without load at almost constant speed nearly every morning.
Vertical vibrations measured during constant speed and in the case of no load increased
during the 271 days. The diﬀerentiation of the signals improves sensitivity. The order
of norm only seems to have little eﬀect in this study. In the frequency range 3 - 3000 Hz
the 2.4M IT 1
10 index from the point RV has doubled only after 50 days and the highest
value reached is over 6. In the frequency range 3 - 2000 Hz, the 2.4M IT 1
10 index from the
point LV has exceeded 4, but has changed signiﬁcantly only after 200 days. This probably
means that the gearbox components on right side have constantly suﬀered wear during
the whole measurement period but on the left side they have suﬀered considerable wear
only recently.
Measurements are still running and the damage in the axle has not yet been inspected.
To utilise signals from the working stages of the LHD, measurements should incorporate
more data from the LHD, especially its load. Otherwise the signals are hardly comparable.
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20
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Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory. © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
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The book presents a concise introduction to the basic methods and strategies in fractional calculus and enables the reader to catch up with the state of the art in this field as well as to participate and contribute in the development of this exciting research area. The contents are devoted to the application of fractional calculus to physical problems. The fractional concept is applied to subjects in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy and quantum field theory and it will surprise the reader with new intriguing insights. This new, extended edition now also covers additional chapters about image processing, folded potentials in cluster physics, infrared spectroscopy and local aspects of fractional calculus. A new feature is exercises with elaborated solutions, which significantly supports a deeper understanding of general aspects of the theory. As a result, this book should also be useful as a supporting medium for teachers and courses devoted to this subject. © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.