Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
Time Domain Analysis and Spec al Me hods o
De e mining Ro a ional Speed o Ro a y Machines
Gábo Gá donyi
Depa men o Mecha onics Op ics and Enginee ing
In o ma ics,
Budapes Uni e si y o Technology and Economics
Budapes , Hunga y
[email p o ec ed]me.hu
K isz ian SAMU PhD
Depa men o Mecha onics Op ics and Enginee ing
In o ma ics,
Budapes Uni e si y o Technology and Economics
Budapes , Hunga y
[email p o ec ed]me.hu
Abs ac — Accu a e es ima ion o o a ional speed o o a y
machines has usually high p io i y in echnical applica ions. This
in o ma ion should be calcula ed o many diagnos ic algo i hms,
con ol o egula ion p ocesses. Inco ec ly es ima ed alues
could occu se ious dis u bances in he ope a ion o machines.
Addi ional ins umen a ion o en may be obs uc ed due o lack
o space, bu he cons uc o he machine may also a ec he
accu acy o measu emen . In such cases, ib a ion diagnos ic
ools can be he disposal o di icul y. Moun ing an accele a ion
senso on o he ou e su ace o he measu ed de ice is no a
majo challenge. In mos cases using ime, equency o
que ency domain analysis, i is possible o es ima e he o a ional
speed o he analysed o a y machine. The calcula ed spec a and
ceps a can help o de e mine he o a ional speed mo e easily
and mo e accu a e han he ime domain me hods. This pape
p esen s he compa ison o hese me hods in e ms o hei
usabili y and o a ional speed es ima ion accu acy. A possible
e o o adi ional op ical measu emen due o misalignmen
and bene i s o he o he me hods a e illus a ed in his a icle ia
measu ed da a se ies o a B ushless DC (BLDC) mo o d i en
sys em.
Keywo ds — ceps um, diagnos ic, o a y machine, o a ional
speed, spec um
I. INTRODUCTION
This pape is conce ned wi h he de elopmen o me hods
o calcula e o a ional speed o o a y machines using encode
o ib a ion signals. Ro a y machines a e an in eg al pa o
ou e e yday li e, e en i we do no pe cei e i di ec ly in all
cases. Bu i we hink abou i : o a ing machines a e in ou
ehicles, in many household appliances, child en’s oys,
mode n obo s, p oduc ion equipmen , and e en mobile
phones con ain o a ing pa s. Ro a ional speed should ideally
be measu ed o a high numbe o diagnos ic algo i hms,
con ol o egula ion p ocesses. Inco ec alues could cause
se ious dis u bances in he ope a ion o machines.
The e a e plen y o common o a ional speed measu ing
solu ions o sol e his p oblem. One can easily use elec ical,
op ical p inciple based echniques, obus magne ic [1] o
combined [2] echniques as well. O he speci ic solu ions
which ha e been published o da e include image p ocessing
echniques [3] and elec os a ic senso s [4]. The ques ion may
a ise how one could measu e he key pa ame e s o an
ope a ing machine, pa icula ly he o a ing speed, i he
enginee s ha e no pe o med hese measu emen s in he ea ly
s ages o design. Moun ing senso s in o a de ice a a la e
s age is usually no an easy ask, especially i i equi es he
modi ica ion o a o a ing componen – gea ansmission
sys em o o o – which canno be obse ed di ec ly by he
use . The di icul y is mainly caused by he compac ness o
mode n de ices. These pa s a e usually no di ec ly accessible
o use s and he e ends o be insu icien ee space nea he
d i e. In hese ins ances, ib a ion diagnos ics ha uses
addi ional accele a ion senso can be he bes al e na i e
solu ion. Vib a ion signal analysis is nowadays he mos
commonly used me hod o he oubleshoo ing and he
condi ion moni o ing o o a y machines. In ecen yea s,
esea ch on ib a ion based o a ional speed de e mina ion
me hods has become e y popula . Many applica ions, u he
esea ches and al e na i e calcula ion me hods can be ound in
he ollowing e e ences [5]-[10].
II. VIBRATION MEASUREMENT OF ROTARY MACHINES
Moun ing accele ome e s on o he de ice su ace akes no
a majo challenge. Fu he mo e he e a e also a ailable
non-con ac ib a ion elocime e s, which ha e much less
ins alla ion equi emen s. Nowadays in he case o mos
ib a ion measu emen solu ions piezoelec ic accele ome e s
a e used because o hei small physical size, he good
sensi i i y and hei especially wide dynamic ange. P obably
he main eason o his choice could be ha accele ome e s do
no equi e an independen e e ence poin . Ins ead o ixing i
on o he su ace, hese senso s can also be used ia special
p obe- ip. I mo e accu a e measu emen is equi ed, magne ic
as ening o sc ewing is ecommended. I he su ace is
unsui able o hese ype o ixings, moun ing by special
beeswax can be also a easible solu ion o he p oblem. Fu he
in o ma ion abou accele ome e s and hei moun ing can be
ound in [11].
The o a ional speed can be easily ob ained om ib a ion
signals ha has been sampled in ime. In addi ion i is also
possible o ge de ailed in o ma ion abou condi ion o he
measu ed sys em. Using ad anced diagnos ic he da e o
o a y machine b eakdown can be well p edic ed as well. We
can so diagnos ic algo i hms in o se e al ca ego ies: ime
domain analysis, equency o que ency domain analysis,
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
highe o de spec al analysis, o de analysis, wa ele analysis
and many kind o join domain analysis me hods as ime-
spec a o ime-ceps a. In his a icle he accu acy o
es ima ed o a ional speed calcula ed ia me hods in ime and
equency domain and he ceps a a e analysed.
A. The analyzed sys em con igu a ion
The es sys em consis ed o a BLDC se omo o
(Faulhabe 3557-K024-CS) connec ed o a hys e esis b aking
mo o (HB-20M-2). A wheel wi h adial o ien ed black and
whi e s ipes was ixed o he end o he assembly’s d i en
sha . Thanks o his s iped wheel he angula mo emen
could be de ec ed wi h a single e lec i e op ical senso . The
piezoelec ic accele ome e (PCB 356A33) was moun ed on o
he housing su ace o he BLDC se omo o wi h special
beeswax.
III. POSSIBLE ERROR OF OPTICAL ROTATIONAL SPEED
MEASUREMENT
Conside ing he layou (Fig. 1), i becomes clea ha
moun ing o an accele ome e akes much less ime and e o
as o he al e na i es. An ad an age o he axle moun ed
s iped wheel is ha he o a ional speed can be ob ained wi h
simplis ic signal p ocessing me hods wi h good accu acy. Due
o he mechanical s uc u e – i he bea ing is no sui able he
sys em may lea e he p oposed ope a ional ange – in ensi e
ib a ion o he analysed sys em could be obse ed. Because
o his phenomenon, a a e o g oss e o o measu emen has
occu ed. I he s iped wheel ha e le he ope a ional ange
o he used op ical senso , he e would ha e been unde ec able
s ipes. Howe e , he e a e se e al co ec ion me hods which
could be used o epai he calcula ed esul . The esul a e
he co ec ion canno be so accu a e, as i would be i a be e
measu emen assembly we e used.
Fig. 2 shows he ickleness o op ical me hod abo e
2000 pm ( e olu ions pe minu e) o a ional speed due
o he non-de ec ed s ipes o he wheel. In such
ins ances he aul has caused a le el o up o 60%
ela i e e o in he esul .
A sec ion o he sou ce signal om a s eady s a e
measu emen in his high-speed ange is in e p e ed in
Fig. 3. This igu e shows he measu ed accele a ion
and he ol age signal o he op ical encode in ime.
The Figu e clea ly shows ha he e we e cyclically
missing signal changes a a pa icula angula ange. In
hese cases we can o ce co ec ion echniques, bu
speed es ima ion based on ib a ion is also possible.
Hiba! A hi a kozási o ás nem alálha ó.
Fig. 1. Op ical o a ional speed measu emen (le ) and ib a ion accele a ion measu emen
( igh ) o he analysed sys em con igu a ion
Fig. 2. Resul s o op ical measu emen o a ying o a ional speed
Fig. 3. Measu ed signals o op ical senso (yellow) and he 3D accele ome e (blue, ed, g een) a s eady s a e o he sys em
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
IV. ROTATIONAL SPEED ESTIMATION BY TIME SIGNAL
PROCESSING
A. Compa ison me hod o il e ed ime signal
A special digi al cha can be ob ained om he sampled
ime signal. The sou ce o cyclical changing ib a ion and he
o ien a ion o accele ome e mus be aken in o accoun a his
calcula ion me hod. The calcula ed esul desc ibes he
o a ional speed.
Fo de ec ion o e en s in ime signal, signal p ocessing
was pe o med in wo main s eps. Fi s ly smoo hing was
applied wi h a simple unweigh ed MA (Mo ing A e age) low
pass il e (Eq. 1). Eq. 2 ep esen s he ecu si e o mula o
he unweigh ed MA calcula ion. Secondly we compa ed he
calcula ed da a se ies o a speci ied h eshold le el (Eq. 3). In
he ollowing equa ions xi is he i h sample o he inpu da a
se ies, yi is he i h sample o he ou pu da a and M ma ks he
applied ke nel size. TH ep esen s he h eshold le el used o
compa ison. I is impo an o no e ha in he case o his
expe imen MA il e ing has been enough o each he
expec ed esul s. In many cases unning RMS (Roo Mean
Squa e) le el calcula ion o o he s a is ical alues as
skewness, ku osis, e c. could be mo e e ec i e.
   



 1
0
1M
j
jix
M
iy
(1)
     



 1
0
1
1M
j
jix
M
iyiy
(2)
   
 





THixi
THixi
iy 0
1
(3)
The calcula ed esul s o he second measu emen channel
o he analysed assembly ( e ical accele a ion) is shown in
Fig. 4. In his way we ge a esul ha is simila o he op ical
encode ou pu . The disad an age o his me hod is i s low
esolu ion. Usually he e is only one o wo de ec able
h eshold c ossing pe e olu ion, so he speed can be
de e mined jus once o a ew imes pe e olu ion. In
addi ion, his me hod does no p oduce any in o ma ion abou
he possible luc ua ion wi hin one e olu ion.
In Fig. 4 e e y second ising edge, which occu s h eshold
c ossing indica es a ull pe iod o ope a ion. One ull
e olu ion akes app oxima ely 13 o 14 ms ime ha equals o
abou 4500 pm.
B. Using Au oco ela ion unc ion o de e mining o a ional
speed
The ACF (Au o Co ela ion Func ion) shows he a e aged
epe i ion ime o he analysed sec ion o he da a se ies. The
sou ce o he epea ed phenomenon can be an unbalanced pa
o he assembly, cyclically inc eased dissipa i e e ec – such
as bea ing ic ion – o he une en o que o he d i ing mo o .
All o hese easons p oduce pe iodic de ia ion in he
measu able ib a ion accele a ion.
The co ela ion is a ma hema ical ool o inding epea ing
pa e ns, such as p esence o pe iodic signal componen s
obscu ed by andom noise. ACF is he co ela ion o a signal
wi h i sel a di e en poin s in ime. I shows how simila is
he analysed da a se ies o i sel and gi es in o ma ion abou
pe iod ime o simila i y. P ominen ea u e o ACF is ha he
calcula ion o pe iodic signal esul s pe iodic ou pu . By
con as he unco ela ed componen s such noises ha e been
elimina ed by he ans o ma ion.
The mos commonly used es ima e o he heo e ical ACF
o a WSS (wide-sense s a iona y) andom p ocess is he
so-called biased es ima e. I can be compu ed by he ollowing
equa ion:
     


 1
0
1hN
k
XX hnxnx
N
hR
(4)
whe e
 
hRXX
is he h h sample o he au oco ela ion
unc ion, h is he lag and * deno es he complex conjuga e.
The inpu da a is a leng h N ealiza ion o he andom p ocess.
The main p ope ies o he esul o Eq. 4 a e he ollowings:
 The au oco ela ion unc ion is an e en unc ion.
 The calcula ed alue o
 
0
XX
R
is p opo ional o
he signal ene gy.
 Global maximum o he unc ion is always in 0.
Fig. 4. Smoo hed accele a ion signal (abo e) and he compa ed bea diag am (below)
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
Fig. 5 shows he ACF ha was calcula ed o he e ical
ib a ion accele a ion o he analysed sys em. The second
p ominen peak o he unc ion a e he ze o loca ion indica es
he sea ched ime pe iod. An impo an p ope y o his
me hod is ha i p oduces he a e aged o a ional speed o he
ela ed ime sec ion. This p ope y causes ha he me hod is
e ec i e jus in case o he s eady s a es o he examined
sys em. A p ominen ad an age agains he p e iously
p esen ed ime based me hod is ha he esul is much less
in luenced by ansien de ia ions and andom noises.
V. ROTATIONAL SPEED ESTIMATION IN FREQUENCY
DOMAIN
The adi ional spec al analysis is conce ned wi h he s udy
o how he powe o a signal is dis ibu ed in he equency
domain. A e decomposing he o iginal da a se ies in o
sinusoidal componen s, i is ela i ely easy o de ec a powe
con en ha co esponds o he o a ional equency [10]. The
alue o his peak on he X-axis ep esen s he main
ope a ional equency. In o de o app ecia e all o he
de ec ed equency peaks one has o gain a ho ough
unde s anding how he analysed assembly wo ks. In he case
o a simple o a y machine he o a ing speed can be
de e mined di ec ly om spec al esul s. Be o e any kind o
equency analysis is unde aken i is necessa y o cla i y
which pa o he assembly is he dominan sou ce o
ib a ion.
The equa ion ha desc ibes o a ing speed in pm
(Re olu ions pe Minu e) is as ollows:
1
60 n 
(5)
whe e n is he o a ing speed and 1 shows he main equency
in Hz.
Fo example o in e nal combus ion engines he
c anksha speed can be wo ked ou acco ding o he i s main
ha monic o de when he numbe o cylinde s is known.
Assuming i/2 is he equency o he i s main ha monic o de
k=i/2, 1 is he o a ing equency o he engine, he engine
speed n will be [8]:
i k n ii /120/60 2/2/ 
(6)
A. Spec um analysis
Maybe he mos commonly used me hod in his ca ego y is
he so-called Au o Powe Spec um (APS) calcula ion. To
compu e he APS, he DFT (Disc e e Fou ie T ans o ma ion)
o he signal is compu ed, and hen mul iplied by i s complex
conjuga e [15]. Hence he magni ude o an APS is equal o he
squa e magni ude o a DFT. This is one me hodology, bu
APS can be calcula ed a se e al ways. The APSD (Au o
Powe Spec al Densi y) is he APS no malized o a 1 Hz
bandwid h. Tha means APSD is he APS di ided by he
in e al be ween equency da a poin s. The APSD can be also
p oduced as he DFT o he ACF.
The equa ions o DFT and IDFT (In e se Disc e e Fou ie
T ans o ma ion) a e he ollowings:
 
 
   




 1
0
2
1N
n
kn
N
i
enx
N
kXnxDFT

(7)
 
 
   



 1
0
2
N
k
kn
N
i
ekXnxkXIDFT

(8)
     
 
















1
0
1
0
2
sin
2
cos N
n
N
n
kn
N
nxikn
N
nxkX

(9)
The o mula o powe spec um calcula ion is as ollows:
   
 
 
 
22 ImRe kXkXkP 
(10)
whe e
 
kP
is he APS,
 
kX
is he Fou ie ans o med
 
nx
signal and
ep esen s he equency in Hz.
Du ing spec al analysis, i s ly signals a e usually cu in o
smalle pieces, whe e o e lapping and windowing can be
applied. A e he spec al calcula ion has been comple ed o
e e y sepa a e da a se ies, i becomes possible o d aw mo e
eliable conclusions based on he a e aged esul .
Al e na i ely, spec al analysis can also be calcula ed wi hou
a e aging. Howe e , a e aging is a e y e ec i e me hod o
Fig. 5. ACF o he measu ed ib a ion accele a ion
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
elimina ing andom noises in case o s ochas ic p ocesses and
ge ing a smoo he ou pu . In he ollowing s udy 8192 wide
Hanning window has been used o compu ing he a e aged
spec a. Fo he esul ing plo , see Fig. 6.
As can be seen in Fig. 6, he me hod gi es us a
ep esen a i e esul . A e he DC componen (0 Hz) he i s
p ominen spec um peak belongs o he base ha monics o
ib a ion. The loca ion o his peak on he equency axis is
he o a ional equency. The spec al analysis gi es a esul
wi h linea equency esolu ion. In his case he measu emen
da a se ies has go a sampling a e o 51.2 kHz, he spec um
has been calcula ed wi h 8192 samples long window size.
These de ails esul a equency esolu ion o Δ =6.25 Hz. The
loca ion o he peak is a 75 Hz ha means a o a ional speed
o 4500 pm. Howe e he esolu ion causes 8.3% ela i e
unce ain y. The eliabili y becomes be e by highe speed.
Spec um based es ima ion is possible based on uppe
ha monics as well. Acco dingly he loca ion o a speci ic
ha monic componen can be de ined wi h be e ela i e
accu acy. F om his equency alue he na u al equency can
be calcula ed by a simple di ision. The ad an age o he uppe
ha monic based me hod has been he much lowe eading
e o , howe e his me hod has go i s disad an ages as well.
As highe he uppe ha monic’s equency as lowe he SNR
(Signal o Noise Ra io). I has esul ed p oblema ic ha monic
peak de ec ion. The measu ed noise ype can conside as whi e
noise. I means ha e e y equency componen is equally
con ained ega ding he ene gy o noise. In spi e o his he
ene gy le el o ib a ion componen s om mechanical
ope a ion is as lowe as highe equency is examined. The
easonable consequence o he p e iously desc ibed ac s is
he inc easingly low SNR.
B. Analysis by Sho -Time Fu ie T ans o ma ion
As men ioned abo e, he ou pu o spec um and ceps um
can be smoo hed using a e age calcula ion me hods. Whils
linea weigh ing usually p oduces a be e esul du ing a
s eady s a e analysis, loga i hmic weigh ing has been much
p e e ed o uns eady s a es o o eal- ime applica ions.
Using ano he me hod, he spec a o ceps a is calcula ed
o e e y windowed signal sec ion as opposed o a e aging.
This pa ial esul can be used also o analysing he
coe icien s’ changes o e ime. This conside a ion has
b ough o he Sho -Time Fou ie T ans o ma ion (STFT).
Eq. 11 demons a es he calcula ing me hod o STFT o
disc e e ime signals.
 
   
   





m
km
N
j
emnwmxknXnxSTFT

2
,
(11)
whe e
 
nx
is he sou ce signal,
 
mnw 
ep esen s a eal,
e en window and
 
knX ,
is conside ed o be he
Sho -Time Fou ie T ans o m.
The disad an age o he abo e me hod is ha , lacking
a e aging, he esul is noisie han he adi ional spec al
calcula ion’s ou pu . Ne e heless, he g ea ad an age o
STFT is ha i gi es e aluable esul s e en in he case o
a iable o a ional speed. In such ins ances he a e aged
spec um p oduces smea ed peaks. These can be mo e
dis o ed because o he mo ing uppe ha monics and o he
noises. The in e p e abili y o he esul s ob ained h ough
STFT is always dependen on a comp omise. Wi h a la ge
Fig. 6. APS o he measu ed ib a ion accele a ion signal
Fig. 7. Resul o Sho -Time Fou ie T ans o ma ion

Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
window size we ge a high equency esolu ion. The la ge
he equency esolu ion used, he mo e accu a e he o a ional
speed which can be de e mined using he ime- equency
ep esen a ion. On he o he hand, when using a small window
size he esolu ion along he equency axis becomes lowe .
Howe e , he localizabili y o e ime becomes be e . I
becomes possible o obse e o analyse ansien s a e
p ocesses due o he be e esolu ion o e ime. Mo e de ailed
in o ma ion abou STFT and he esul s o u he
imp o emen esea ches ha e been published unde he
ollowing e e ences: [12], [16]-[18]
The ou pu o STFT is a wo- a iable dis ibu ion in he
ime- equency domain. We can display his wo- a iable da a
se ies as an image and also pos -p ocess his esul wi h
common image p ocessing me hods. Thus we can de ec
idges, edges o apply digi al con olu ion il e s. The ou pu o
idge de ec ion is shown in Fig. 8. The i s idge is a 75 Hz,
which equals o 4500 pm o a ional speed. The applied
window leng h and he ou pu s esolu ion is he same as in he
case o he u he p esen ed a e aged PSD me hod.
C. Ro a ional speed es ima ion ia Ceps um analysis
Ceps um was o iginally de ined as he powe spec um o
he powe spec um’s loga i hm. La e , a newe de ini ion was
coined; ceps um being he in e se ans o m o he powe
spec um’s loga i hm [13]-Hiba! A hi a kozási o ás nem
alálha ó., as exp essed ma hema ically as:
   
 
kPIDFTnc log
(12)
In he li e a u e ou basic kinds o ceps al ep esen a ions
can be ound. These a e he eal-, complex-, powe - and phase
ceps ums.
 Real ceps um:
 
][log][ kXIDFTnc 
(13)
 Complex ceps um:
 
][log][
ˆkXIDFTnc 
(14)
 Powe ceps um:
 
2
2][log][ kXIDFTncp
(15)
 Phase ceps um:
 
 













][Re
][Im
a c an][ kX
kX
IDFTncF
(16)
A de ailed s udy o he calcula ion me hodology used o
disc e e da a se ies and he pa icula s o he ceps al esul s
can be ound in [13]. Fo a numbe o u he s udies on he
his o y and he applica ion o ceps um analysis could be
ound in Hiba! A hi a kozási o ás nem alálha ó.-
[15][18]-[21].
Fig. 9 ep esen s he eal ceps um o he analysed da a
se ies. The illus a ion shows he ou pu s calcula ed o all 3
channels o he 3D accele ome e .
This me hod con e s he signal in o so-called que ency
componen s. One o he ea lies applica ions o he ceps um
heo y can be ound in he s udy o signals con aining echoes
Fig. 8. STFT ep esen a ion a e image p ocessing
Fig. 9. Resul o Ceps um analysis
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
and in he s udy o speech analysis. The applica ion discussed
in hese s udies was aimed a de ec ing he ha monic s uc u e
o measu ed sound. Such ha monic s uc u es so-called
ha monic amilies can be de ec ed by gea box analysis o
al e na i ely by any kind o analysis o o a y machines. The
ceps um calcula ion o a signal only esul s an e aluable
ou pu i i s spec um con ains ha monic amilies, as i
sea ches o pe iodic changes in he spec um. Fo example, a
clea one-componen sinusoidal signal can be examined well
by powe spec um; howe e , i s ceps um p oduces much less
use ul in o ma ion.
As i has been shown p e iously by spec al analysis,
windowing and a e aging me hods should be used also by
ceps um calcula ion. The p esen ed esul (Fig. 8) was made
wi h he same basic pa ame e s as by he powe spec um
calcula ion. I means 8196 sample was he wid h o he used
Hanning window. In his case he i s dominan peak a e he
noisy sec ion close o 0 que ency belongs o he o a ional
speed o he so-called main ha monic componen . The
que ency loca ion o his peak shows he pe iod leng h o one
ull e olu ion in seconds. The peak can be ound a 0.0133s
along he que ency axis and he da a se ies has he same
esolu ion in que ency as he sou ce signal had in ime. I
means a que ency s ep size o 0.0195 ms because o he
applied 51.2 kHz sampling a e. These p ope ies esul a
possible equency es ima ion e o aised om he esolu ion
o he disc e e se ies. The s ep size be ween wo da a poin s
can be exp essed ela i e o he ac ual que ency alue along
he x axis. In he case o he ep esen ed esul s he me hod
gi es he expec ed 4500 pm o a ional speed. A his poin he
ela i e s ep size o he se ies was 0.146%.
By lowe que encies he esul is usually much noisie .
Tha is why his o a ional speed es ima ing me hod has
become un eliable a e y high speeds.
D. Ro a ional speed es ima ion by Time-Ceps um Analysis
The ela ionship be ween APS and STFT has been
discussed in he p e ious chap e abo e. By analogy, we can
de ine an algo i hm o examine he ceps um changes o e
ime. Du ing his me hod, sec ions o he ime signal a e
windowed and ceps um is hen compu ed o hese pieces. I
we use hese pa ial esul s wi hou a e aging hem, i is
possible o ep esen he ceps um changing o e ime in h ee
dimensional g aphs. These g aphs a e usually called
ceps og ams.
A g ea ad an age o ceps um o ceps og am
ep esen a ion is ha he esolu ion along he que ency axis is
no in luenced by he window size. I solely depends on he
sampling a e o he sou ce signal. Howe e , speci ying he
sui able block size is no a clea ly de inable ask. I we choose
an o e ly sho sec ion he pe iod ime alls ou side o he
esul ’s que ency ange. In his case e alua ing he TCA
(Time-Ceps um Analysis) p oduced ou pu is p oblema ic.
Inc easing he applied window size has a nega i e e ec on
he localizabili y o e ime. The same mechanisms o ac ion
has been discussed o STFT abo e.
The TCA me hod gi es he same esul by o a ing speed
es ima ion as he a e aged ceps um ou pu . I means we ge
he expec ed 4500 pm o a ing speed in he examined s eady
s a e sec ion wi h p ominen ela i e accu acy.
VI. CONCLUSION
The p esen ed me hods o o a ional speed es ima ion
pose a g ea con enience i he examined o a y machine
hasn’ go any buil -in senso o o a ing speed measu ing
pu pose. I is possible o comple e he assembly wi h o he
senso s, bu i causes o en di icul y because o he
complexi y o de ices. Fu he mo e he g oss e o o a widely
used solu ion wi h op ical senso was es ed as well. Fo such
a p oblem many o ib a ion diagnos ic me hods can supply
solu ion. The easies way is o de e mine he pe iodici y o
ib a ion signal di ec ly in he ime domain. The usabili y o
signal compa ison is be e a e signal condi ioning. We can
also de e mine pe iodici y om he calcula ed au oco ela ion
unc ion. The e a e se e al me hods o ans e signal in o
o he domains using spec a o ceps a calcula ion. The
ela i e s ep size o he disc e e se ies by o a ional speed
es ima ion depends on he me hod, he applied block size, he
sampling a e o he sou ce signal and he ac ual o a ional
speed. All o he p esen ed me hods a e able o show us he
main p ope ies o he ope a ion. The spec al analysis has he
be e ela i e esolu ion as highe he main equency is.
Howe e he ceps um has an in e se beha iou . In case o
bo h me hods he SNR became wo se a high speeds. The e
Fig. 10. Resul o Time-Ceps um Analysis
Recen Inno a ions in Mecha onics (RIiM) Vol. 3. (2016). No. 1-2.
DOI: 10.17667/ iim.2016.1-2/8.
a e also a ailable me hods o speed es ima ion by a ying
o a ional speed, hese a e he discussed STFT and he TCA.
These solu ions ha e almos he same p ope ies abou ela i e
accu acy as he spec um o he ceps um calcula ions.
I is possible o de elop such hyb id me hods in digi al
signal p ocessing, which can combine he abo e discussed
ways o calcula ion. Such a hyb id calcula ion me hod should
be based on he obse a ion o signal componen s, SNR le el,
es ima ed main equency e c. The ela i e accu acy o se e al
me hods can be es ima ed om he esul o p e ious i e a ions
and he hyb id logic should make he decision, which heo y
could gi e he bes esul .
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