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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 α∈Rand norms p∈R. 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) = (iω)nXeiωt .(2)

In references (1, 2, 3, 4) Lahdelma et al. have deﬁned real order derivatives and integrals of

function (1) by replacing n∈Nwith a real number α∈R

x(α)(t) = (iω)αXeiωt ,(3)

where (iω)α=|ω|αi·sign(ω)α=|ω|αeiα π

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+iβ where α, β ∈R(5,6)

x(z)(t) = (iω)zXeiωt ,(4)

and only the principal value of the function (iω)zis considered, which gives us (iω)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)e−i2πνt dt. (5)

Its inverse transform is deﬁned as

F−1{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 −pν) 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 −pν +απ

2dpdν. (8)

We deﬁne diﬀerintegrals x(z)for such functions xwhose Fourier transforms Xand inverse

transforms of (i2πν)zXexist. Let z∈Cand Xis the Fourier transform of x. The

diﬀerintegral of order zis

x(z)(t) = F−1(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 (iω)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, . . . , xN−1and we can approximate the

signal’s Fourier transform with the Discrete Fourier transform (DFT)

F{xn}=Xk=1

N

N−1

X

n=0

xne−i2πkn/N .(10)

Its inverse transform (IDFT) is

F−1{Xk}=xn=

N−1

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 t≤0

0.51−cos πt

T / ,if 0< t < T/

1,if T/ ≤t≤T /2

w1(T−t),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 t≤0

RT/

0eτ(τ−T/)−1

dτ−1

Rt

0eτ(τ−T/)−1

dτ, if 0< t < T /

1,if T/ ≤t≤T /2

w2(T−t),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 mk∈Z. 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/2≤k≤N/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 z∈Cfor 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) =

N−1

X

k=0

Xkei2π(k+mkN)t/T ,

4

which at the sample points tnreceives the values xnfor all mk∈Z, 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 0≤k < N/2, equation (14) implies mk= 0 and when 0≤k < 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

2−N)t/T i

=XN/2ueiπN t/T + (1 −u)e−iπN 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=XN−kand

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 +e−iπN 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 +XN−ke−i2πkt/T +XN/2cos πNt

T

=X

0≤k<N/2

Xkei2πkt/T +X

−N/2<k<0

XN+kei2πkt/T +XN/2cos πNt

T.

(15)

Let us diﬀerintegrate interpolation (15) z∈Ctimes with the deﬁnition (9)

x(z)(t) = X

0≤k<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

0≤k<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+e−iπn−iz 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= 2n−1, 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,0≤k≤N-1of the sequence xn,0≤n≤N-1with the

FFT algorithm

Xk=F{xn}.

2. Calculate a new sequence Gk,0≤k≤N-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=F−1{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 α∈R, 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)

N∈RN. The lpnorm of this sequence is

x(α)

p= N

X

n=1 x(α)

n

p!1/p

,(17)

where 1≤p < ∞. As pgrows, the norms lpbecome smaller (16)

x(α)

p≥

x(α)

q,if p < q.

Norm l∞is 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 p≥1. 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 p→0and 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 0≤p < 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 p∈R\{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 n∈N. 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

α1,α2,...,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 2∗νmesh 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 0≤p≤10

and −2≤α≤6using 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 0≤p≤10 and

−2≤α≤6using 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 0≤p≤10 and −2≤α≤12 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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