Research Article Vibration Signal Analysis of Journal...

Research ArticleVibration Signal Analysis of Journal Bearing Supported RotorSystem by Cyclostationarity

Chang Peng and Lin Bo

State Key Laboratory of Mechanical Transmission Chongqing University 1510 Main Teaching Building Chongqing China

Correspondence should be addressed to Lin Bo bolin0001aliyuncom

Received 2 November 2013 Accepted 9 April 2014 Published 12 May 2014

Academic Editor Valder Steffen Jr

Copyright copy 2014 C Peng and L Bo This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Cyclostationarity has been widely used as a useful signal processing technique to extract the hidden periodicity of the energyflow of the mechanical vibration signature However the conventional cyclostationarity is restricted to analyzing the real-valuedsignal which is incapable of processing the constructed complex-valued signal obtained from the journal bearing supported rotorsystem operating with oil film instability In this work the directional cyclostationary parameters such as directional cyclic meandirectional cyclic autocorrelation and directional spectral correlation density are defined based on the principle of directionalWigner distribution Practical experiment has demonstrated the effectiveness and superiority of the proposed method in theinvestigation of the instantaneous planar motion of the journal bearing supported rotor system

1 Introduction

Journal bearings are one of the most widely used elementsin high speed rotary machines and their performance is ofgreat necessity to the normal operation of the rotor systemssupported by oil film bearings As such plenty of research onjournal bearing malfunction analysis has been conducted

Oil whirl and oil whip are common failure modes ofjournal bearings induced by oil film instability when therotating speed of the shaft exceeds the critical speed ofthe rotor system Based on the mathematical model of asymmetric rotor supported by one rigid journal bearing andone oil film journal bearingMuszynska [1 2] mathematicallyinterpreted the dynamic phenomena related to synchronousvibration and self-excited vibration Relying on a pair oforthogonally installed proximity sensors the orbit of theshaft centerline is usually monitored to help in analyzing thesystem operating condition In addition Moosavian et al [3]compared K-nearest neighbor (KNN) and artificial neuralnetwork (ANN) for the power spectral density techniquebased fault diagnosis of main journal bearings of internalcombustion (IC) engine under different conditions Yinget al [4] studied the system dynamics characteristics oftilting pad journal bearing rotor system operating aroundnatural frequency or on high-speed range by considering

the influence of the pad moment of inertia Dimond etal [5] reviewed strengths and weaknesses of identificationmethods for fluid film journal bearing static and dynamiccharacteristics particularly the bearing stiffness dampingand mass coefficients based on measured data of differentmeasurement systems The developments and trends inimproving bearing measurements were also documentedMoreover Zanarini and Cavallini [6] illustrated detailedexperimental results corresponding to oil whirl and oil whipmixed with misalignment unbalance and resonance

In statistics on random processes cyclostationary pro-cesses belonging to the classes of nonstationary processesare defined to represent the correlation features of peri-odic phenomenon As such it is necessary to analyze thecyclostationarity of the modulated vibration radiated bythe journal bearing supported system with its componentsoperating periodically Gardner firstly defined the conceptof cyclostationarity and foresaw the application of it ThenGardner et al [7] presented a concise review of the literatureon cyclostationarity based on the investigation of an extensivebibliography In addition Serpedin et al [8] attempted toprovide a detailed classification group which represented acomprehensive list of references on cyclostationarity and itsapplications Mccormick and Nandi [9] compared second

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 952958 16 pageshttpdxdoiorg1011552014952958

2 Shock and Vibration

order cyclostationarity with conventional spectral analysisand synchronous averaging in the condition monitoringof rolling element bearing They found that the formercan reveal significant fault features while the latter failedAntoni [10] presented a tutorial on cyclostationarity whichintroduced key concepts on actual mechanical signals andproved the superiority of cyclostationarity applied inmachinediagnostics identification ofmechanical systems and separa-tion of mechanical sources

Conditionmonitoring of journal bearing supported rotorsystemsusually relies on twoorthogonally installed proximitysensors The orbit signature of the center of the journal isthen qualitatively analyzed to investigate system operatingcondition Southwick [11 12] applied the full spectrumplot ascompared to the half-spectrum to diagnose diverse rotatingmachinery malfunctions They found that using the fullspectrum the orbit ellipticity and vibration precession canbe determined without being affected by probe orientationIn order to investigate the complex-valued signal of theinstantaneous lateral vibration of the mechanical structureLee et al [13] proposed the two-sided directional powerspectra complex-valued vibration signals for the diagnosis ofcylinder power faults of four-cylinder compression and sparkignition engines Furthermore Lee andHan [14 15] proposedthe directional Wigner distribution to effectively determinethe procession parameters of the planar motion both in time-frequency domain and in order-frequency domain

Based on the previous introduction in this work theconcept of directional cyclostationarity of complex-valuedsignal is defined and used to investigate the vibration of ajournal bearing supported rotor system which is influencedby oil film instability The structure of the paper is organizedas follows In Section 2 the theory responding to the firstand the second order cyclostationarity of real-valued signalis reviewed Furthermore the directionalWigner distributionis introduced Subsequently the proposed directional cyclo-stationarity is represented In Section 3 the experiments con-ducted on a test bench are described and the results obtainedfrom analysis compared and discussed using directionalcyclostationarity directional time frequency distributionand conventional cyclostationarity respectively In Section 4summary discussions and the conclusions are given

2 Directional Cyclostationarity

21 Cyclostationarity of Real-Valued Signals Since cyclosta-tionarity can indicate itself in statistics of any orders deter-mined by the degree of nonlinear operation there are variousstatistic parameters illustrated in the time the frequencyand the cyclic frequency domains [10] The first and secondorder of cyclostationarity are firstly reviewed for simplicitybut without losing accuracy

The mean value of a cyclostationary signal (cyclic meanCM) is defined as

CM (120572) = sum

120572isin119860

119864 [119909 (119905) sdot 119890minus2120587120572119905

] (1)

where 120572 is the cyclic frequency of the signal and the set 119860

contains all cyclic frequencies 119864[sdot] = lim119879rarrinfin

int119879

(sdot) 119889119905119879 isthe mean operator

Then the cyclic autocorrelation (CR) representing theenergy of the signal can be defined in as

CR (120572 120591) = sum

120572isin119860

119864 [119909(119905 minus120591

2) 119909 (119905 +

120591

2) sdot 119890minus2120587120572119905

] (2)

By taking the Fourier transform of CR with respect totime lag 120591 the spectral correlation density (SCD) is given by

SCD (120572 119891) = intinfin

minusinfin

CR (120572 120591) sdot 119890minus2120587119891120591

119889120591 (3)

Finally by transforming the cyclic frequency 120572 back tothe time domain the Wigner-Ville spectrum (WV) can beobtained in

WV (119905 119891) = sum

120572isin119860

SCD (120572 119891) sdot 1198901198952120587120572119905

(4)

22 Directional Wigner Distribution Lee and Han [14 15]developed a directional Wigner distribution (DWD) by sub-stituting complex-valued signals for the real-valued signalswhich could describe the instantaneous planar motion ofthe system In their work the signals 119909(119905) and 119910(119905) obtainedfrom the horizontal and the vertical proximity probes werecombined in

119901 (119905) = 119909 (119905) + 119895119910 (119905) = 119901119891

(119905) + 119901119887

(119905)

=1003816100381610038161003816100381611990311989110038161003816100381610038161003816119890119895(120596119905+120593

119891)

+1003816100381610038161003816100381611990311988710038161003816100381610038161003816119890119895(120596119905+120593

119887)

(5)

where 119901119891(119905) and 119901119887(119905) denote the forward and backwardrespectively procession of the rotor centerline

The 119901119891

(119905) and 119901119887

(119905) are given by

119901119891

(119905) =119901 (119905) + 119895119901 (119905)

2

119901119887

(119905) =119901 (119905) minus 119895119901 (119905)

2

(6)

where 119901(119905) is the Hilbert transform of 119901(119905)Then the auto-DWD is defined in

119882119889

119901(119905 120596)

= 119882119901119891 (119905 120596)

119882119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596120591119901119891lowast

(119905 minus120591

2)119901119891 (119905 +

120591

2) 119889120591 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596120591119901119887lowast

(119905 minus120591

2)119901119887 (119905 +

120591

2) 119889120591 for 120596 lt 0

(7)

where 119882119901119891(119905 120596) and 119882

119901119887(119905 120596) are respectively the forward

and the backward terms and 120596 is the circular rotatingfrequency

Shock and Vibration 3

The overall test apparatusJournal bearing pedestaland oil cup

Vibration sensor for obtainingthe acceleration signals

Shaft

Oil film journal bearingpedestal and oil cup

Figure 1 Test bench and its main components

0 10 20 30 40

10000

5000

0

rpm versus time during startup

Time

rpm

(a)

5

0

minus5

Rotor horizontal displacement versus time

ecx

0 10 20 30 40

Time

(b)

01

0

minus01

Oil film journal bearing horizontal acceleration

0 10 20 30 40

Time

ofjob

e x

(c)

5

0

minus50 10 20 30 40

Time

Rotor vertical displacement versus time

ecy

(d)

0

0 10 20 30 40

Time

002

minus002

Oil film journal bearing vertical acceleration

ofjob

e y

(e)

Figure 2 Vibration signal of journal bearing supported system during startup


Rotor horizontal displacement at 5500 rpm

0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecx

(a)

Power spectrum of x1 at 5500 rpmminus20

minus40

minus60

minus80

minus100

minus120

minus140

Frequency (times05Fs Hz)

ecx

0 01 02 03 04 05 06 07 08 09 1

(b)

Rotor vertical displacement at 5500 rpm

0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecy

(c)

0Power spectrum of y1 at 5500 rpm

minus20

minus40

minus60

minus80

minus100

minus120

minus140


ecy

0 01 02 03 04 05 06 07 08 09 1

(d)

Figure 3 The lateral vibration of the shaft and its power spectrum at 5500 rpm

3

2

1

0

minus1

minus2

minus3

times10minus3 Pedestal horizontal acceleration at 5500 rpm

0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)


minus80

minus100

minus120

minus140

minus160

Frequency (times05Fs Hz)0 01 02 03 04 05 06 07 08 09 1

dofjo

bex

(b)

8

6

4

2

0

minus2

minus4

minus6

times10minus4 Pedestal vertical acceleration at 5500 rpm

0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bey

(c)


Power spectrum of y2 at 5500 rpmminus60

minus80

minus100

minus120

minus140

minus1600 01 02 03 04 05 06 07 08 09 1

dofjo

bey

(d)

Figure 4 The lateral vibration of the pedestal and its power spectrum at 5500 rpm


0 05 1 15 2 25 3 35 4

Time (s)

ecx

1

05

0

minus05

minus1


(a)

minus20

minus40

minus60

minus80

minus100

minus120


ecx

0Power spectrum of x3 at 6200 rpm

0 01 02 03 04 05 06 07 08 09 1

(b)

2

15

1

05

0

minus05

minus1

minus15

minus2


0 05 1 15 2 25 3 35 4

Time (s)

ecy

(c)

0

minus20

minus40

minus60

minus80

minus100


ecy


0 01 02 03 04 05 06 07 08 09 1

(d)


003

002

001

0

minus001

minus002

minus003

Pedestal horizontal acceleration at 6200 rpm

0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)

minus40

minus60

minus80

minus100

minus120

minus140

Power spectrum of x4 at 6200 rpm

0 01 02 03 04 05 06 07 08 09 1


dofjo

bex

(b)

0

Pedestal vertical acceleration at 6200 rpm

0 05 1 15 2 25 3 35 4

Time (s)

times10minus2

1

05

minus05

minus1

dofjo

bey

(c)

minus60

minus80

minus100

minus120

minus140

minus160

Power spectrum of y4 at 6200 rpm

0 01 02 03 04 05 06 07 08 09 1


dofjo

bey

(d)



012

01

008

006

004

002

0

DCM of rotor vibration at 5500 rpm

dm1

minus600 minus400 minus200 0 200 400 600

Cyclic frequency (Hz)

1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0

times10minus4 DCM of pedestal vibration at 5500 rpm

dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+



2X2minus minus fowpminus

2X2+ minus fowp+fowpminusfowp+

(d)

Figure 7 Directional cyclic mean of the lateral vibration of the shaft and pedestal

DCR of rotor vibration at 5500 rpm


Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3

1X1minus 1X1+1X1minus + 0475X1minus

(a)

DCR of pedestal vibration at 5500 rpm12

10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus

2X1minus + 0475X1minus

2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus

DCR of rotor vibration at 6200 rpm01

008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6

4X2minus + fowpminus 4X2+ + fowp+

(d)

Figure 8 Directional cyclic autocorrelation of the lateral vibration of the shaft and pedestal


DSCD of rotor vibration at 5500 rpm

07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000

Frequency (Hz) Cyclic frequency (Hz)

800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000


DSCD of pedestal vibration at 5500 rpm

7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)

Figure 9 Directional spectral correlation density of the lateral vibration of the shaft and pedestal

The cross-DWD is given by

119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)

In addition the full spectrum (FS) another approach forshaft orbit signal analysis is developed by Southwick [11 12]defined in

FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)

23 Proposed Directional Cyclostationarity of Complex-ValuedSignals Originated from the cyclostationarity for real-valued

signal in this work the directional cyclostationarity forcomplex-valued vibration signal is developed according tothe principle of the previously introduced directionalWignerdistribution The proposed method can be applied not onlyto characterize the hidden periodicities of the vibrationradiated by the journal bearing supported system but also todetermine the procession directivity of the planar motion intime-frequency domain by analyzing the combined complex-valued vibration signal

The directional cyclic mean (DCM) can be defined in (10)according to (1) and (7)

DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)


DWD of rotor vibration at 5500 rpm0

05

1

15

2

25

3

35

15

10

5

1X1minus + 0475X1minus 1X1minus 1X1+

Circular frequency (Hz)

Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)

DWD of pedestal vibration at 5500 rpm

12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)

DWD of pedestal vibration at 6200 rpm12

10

8

6

4

2

times10minus3

2X2minus + fowp2X2+ + fowp

3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (

s)minus600 minus400 minus200 0 200 400 600


(d)

Figure 10 Directional Wigner distribution of the lateral vibration of the shaft and pedestal

10

8

6

4

2

0

times102 Full spectrum of rotor displacement at 5500 rpm

fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus

1X1minus minus 0475X1minus

(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0

Full spectrum of pedestal acceleration at 5500 rpm

fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)

Figure 11 Full spectrum of the lateral vibration of the shaft and pedestal


0 100 200 300 400 500 600

005

004

003

002

001

0

CM of rotor horizontal vibration at 5500 rpm


mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0

CM of rotor vertical vibration at 5500 rpm

my1

0 100 200 300 400 500 600


(b)

2

1

0

times10minus4 CM of pedestal horizontal vibration at 5500 rpm

mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0

times10minus5 CM of pedestal vertical vibration at 5500 rpm

my2

(d)

Figure 12 The cyclic mean of the lateral vibration of the shaft and pedestal at 5500 rpm

025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)



CR of rotor horizontal vibration at 5500 rpm10

8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)

CR of rotor vertical vibration at 5500 rpm001

0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)

CR of pedestal horizontal vibration at 5500 rpm2

15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


Interval = 1X1 minus 2 lowast 0475X1

(c)

CR of pedestal vertical vibration at 5500 rpm25

2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)

Figure 14 The cyclic autocorrelation of the lateral vibration of the shaft and pedestal at 5500 rpm


008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5

Interval = 1X2 minus 2fowp

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM

119887(120572 119905) are respectively the

forward and backward terms DCM is able to identify theperiodic components induced by the mean behavior of thesignal

The autodirectional cyclic autocorrelation (DCR) isdefined in (11) according to (2) and (7)

DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)

where DCR119901119891119901119891lowast(120572 120591) and DCR

119901119887119901119887lowast(120572 120591) are respectively

the forward and backward termsThe cross-directional cyclic autocorrelation is given in

(12) according to (2) and (8)

DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)

Both the auto- and the cross-directional cyclic autocorre-lation coefficients can indicate the energy of the signal

Subsequently taking the Fourier transform of the autodi-rectional cyclic autocorrelation with respect to the lag 120591

produces the auto directional spectral correlation densityfunction (DSCD) given by (13) according to (3) and (7)

DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)

In the particular case when the cyclic frequency 120572 = 0 (13)becomes the power spectrum of the signal 119901(119905)

The cross-directional spectral correlation density func-tion is given by (14) according to (3) and (8)

DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)

Both the auto- and the cross-directional spectral corre-lation density functions actually describe the density of thecorrelation of two spectral components spaced apart by 120572

around the central frequency 119891

Table 1 System operating states at different rotating speeds

Experiment Operating speed Observed state ofoperation

1 5500 rpm Oil whirl2 6200 rpm Oil whip

As the frequency 119891 in the conventional Wigner-Villespectrum does not represent the periodicity of the vibrationresponse as the cyclic frequency 120572 does the negative compo-nent of the frequency 119891 will be of no meaning Therefore bytransforming the cyclic frequency variable 120572 back to the timedomain no variable with physical meaning can be obtainedwhichmeans that the directionalWigner-Ville spectrumdoesnot exist

It is noted that all the significant symbols in the equationsare listed in the Nomenclature section

3 Experiment

31 Experimental Setup and Data Description In this workthe experiment is conducted to investigate the dynamics ofthe asymmetric journal bearing supported rotor system on atest rig illustrated in Figure 1 The experiment setup consistsof a rigid cylindrical shaft supported by two cylindricaljournal bearings with the supporting journal bearing on theright end near the driving motor and an oil film journalbearing at the left end for simulating oil film instability faultsTwo discs are mounted on the shaft with one at the midplanebetween the two bearings and the other near the left oil filmbearing Two pairs of accelerometers are used for measuringpedestal translational vibrations with one pair mounted onthe outboard of each of the two bearing pedestals In additiontwo proximity sensors aremounted just to the right-hand sideof the center disc to record the lateral vibrations of the rotorat that position

In order to simulate the oil whirl and oil whip phe-nomenon the rotor rotating in fluid lubricated cylindricaljournal bearings is lightly loaded with an unbalanced massWhen the shaft rotates with a slow rotating speed the stablesynchronous vibration with low amplitude is observed Withthe increasing rotation speed the system reaches the firstresonance with significant vibration amplitude at the firstnatural frequency Then the vibration returns to normalafter the resonance At higher speed the oil whirl appearsalong with higher vibration In addition the characteristicfrequency of oil whirl (around half of the rotating frequency)can be observed When the rotating speed approaches thedouble resonance speed the oil whirl pattern becomes theoil whip with the characteristic frequency remaining tothe first natural frequency of the system Furthermore theamplitude of the rotor under oil whip becomes higher thanthat of oil whirl The vibration signals of journal bearingsupported system during startup procedure are listed inFigure 2 Results obtained under different rotational speedswith various artificially simulated defects are listed in Table 1


SCD of rotor horizontal vibration at 5500 rpm

02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0

SCD of rotor vertical vibration at 5500 rpm

05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)

SCD of pedestal horizontal vibration at 5500 rpm

8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0

Frequency (Hz)Cyclic frequency (Hz)

0

00800600

600400400200 200

quency (Hz)

(c)

SCD of pedestal vertical vibration at 5500 rpm

4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)

Figure 16 The spectral correlation density of the lateral vibration of the shaft and pedestal at 5500 rpm

32 Vibration Signal Analysis The vibration response ofthe shaft and the pedestal when the system operated ata rotational speed of 5500 rpm and 6200 rpm was inves-tigated using directional cyclostationarity directional timefrequency distribution and conventional cyclostationarityrespectively In order to facilitate calculation the first 8192samples of the original signal and their power spectrumderived with sampling frequency 119891

119904= 2048Hz were selected

and shown in Figures 3 4 5 and 6

321 Directional Cyclostationarity of the Complex-ValuedSignal Firstly in this work the directional cyclostationaryparameters of the complex-valued vibration signal of therotor-bearing system were derived to help extract the hiddenperiodicity of nonstationary signature as well as investigatingthe instantaneous planar motion

In Figure 7 the positive (negative) horizontal axes rep-resent the first order cyclostationarity namely the cyclicmean of the forward (backward) vectors of the rotor lateralvibration In Figure 7(a) derived from the rotor vibration at5500 rpm the oil whirl frequency 0475119883

1and the rotating

frequency 11198831and its value modulated by the oil whirl

frequency 11198831minus0475119883

1can be clearly identified Figure 7(b)

characterizing the rotor vibration at 6200 rpm only displaysthe oil whip frequency 119891owp = 4934Hz but misses therotating frequency However Figures 7(c) and 7(d) derivedfrom pedestal vibration provide more abundant frequencyinformation The harmonics plusmn2119883

2of rotating frequency

dominates the horizontal axes in Figure 7(d) of pedestalvibration at 6200 rpm Furthermore the harmonics of rotat-ing frequency119883

2modulated by the oil whip frequency is also

clear In addition the developed directional cyclic mean canhelp determine the instantaneous procession orientation ofthe rotor centerline With 119877

11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+

the direction will be backward or opposite to the direction ofrotor rotation

In Figure 8 the positive (negative) horizontal axes givethe second order cyclostationarity namely the cyclic auto-correlation of the forward (backward) vectors of the shaftand pedestal lateral vibration It can be seen that the rotat-ing frequency modulated by oil whirl frequency (oil whipfrequency) appears periodically in the DCR plot derivedfrom rotor or pedestal vibration when system operates at5500 rpm (6200 rpm) while the harmonics of the singlerotating frequency keeps all the same In addition the



12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)


instantaneous procession orientation of the rotor centerlinecan be determined by comparing the color of the positive andnegative value which represents the radii of the orbit

As defined in [10] the SCD plots comprised of countablenonzero points show the first order cyclostationarity whilethe SCD plots comprised of parallel lines exhibit the secondorder cyclostationarity In Figure 9 the SCD plots of thepedestal vibration signal clearly represent that the density ofthe correlation continuously changes along several straightlines in the frequency cyclic frequency planeTherefore it canbe inferred that the case of vibrations induced by oil whirl oroil whip is more suitable to be interpreted as the second ordercyclostationarity In addition the DSCD plots of the pedestalvibration aremore suitable to interpret the case induced by oilwhirl or oil whip than that of the rotor vibration Besides theDSCD plots also can help determine the procession directionby comparing the amplitudes of the density of the correlationat positive and negative cyclic rotating frequency

322 Directional Wigner Distribution Subsequently direc-tional time-frequency distributions of the complex-valuedsignal derived from the lateral vibration of the journal bearingsupported rotor system are given in Figure 10 for comparison

As seen in Figure 10(a) derived from rotor displacementwhen system operates at 5500 rpm the orbit radius when 120596 =

11198831is much larger than that when 120596 = minus1119883

1 With 119877

11198831minusgt

11987711198831+

the instantaneous precession orientation of the rotorcenterline will be backward As the oil whip frequency com-ponent dominates the whole frequency band in Figure 10(b)of the rotor vibration the rotating frequency componentcannot be clearly seen However in Figures 10(c) and 10(d)derived from pedestal vibration the rotating frequency plusmn119883

2

the oil whip frequency plusmn119891owp and the modulating frequencyharmonics comprised of 119883

2and 119891owp can be clearly seen

In addition it also can be seen that the rotating frequencymodulated by oil whirl or oil whip appears periodically versustime while the harmonics of single rotating frequency existsall the time as the time changes

By further comparing the directionalWigner distributiondefined by (7) to (8) with the directional cyclic autocorrela-tion defined by (11) to (12) it is interesting to find that actuallythe directional Wigner distribution can be interpreted as thesecond order directional cyclostationarity (directional cyclicautocorrelation) characterizing the random behavior of thesystem Furthermore it is found that the full spectrum canbe interpreted as the first order directional cyclostationarity


WV of rotor horizontal vibration at 5500 rpmFr

eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)

WV of rotor vertical vibration at 5500 rpm

1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

WV of pedestal horizontal vibration at 5500 rpm

2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)

WV of pedestal vertical vibration at 5500 rpm

Dofj

obe y

1050minus05minus1minus15

times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)

Figure 18 TheWigner-Ville spectrum of the lateral vibration of the shaft and pedestal at 5500 rpm

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx

WV of rotor horizontal vibration at 6200 rpm

2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)



(directional cyclicmean) by comparing (9) and (10) as shownin Figures 11 and 7

323 Cyclostationarity of the Real-Valued Signal For com-pleteness the statistical parameters characterizing cyclosta-tionarity of real-valued signal of the journal bearing sup-ported system are also illustrated from Figures 12 13 1415 16 17 18 and 19 Similar conclusions to the directionalcyclostationarity plots can be found except for informationon the procession directivity of the planar motion

Compared with conventional Wigner-Ville distributionthe Wigner-Ville spectrum reduces the influence of theinterference terms on exhibiting how the signal energy flowsin the time and frequency domain [10] In Figure 18 onlythe WV Figures 18(a) and 18(b) derived from the rotorvibration are able to clearly describe the periodicity inducedby oil whirl fault In Figure 19 both the WV plots of therotor vibration and the pedestal vibration obviously show theperiodic phenomenon responding to oil whip fault

4 Discussions and Conclusions

In thework presented here the characteristics of the vibrationsignals obtained both from the rotor and the pedestal of arotor system supported by journal bearing were investigatedThe experiments were conducted on a test bench to simulatethe operation status of the rotor system suffering from oilwhirl and oil whip

The directional cyclostationary parameters such as direc-tional cyclic mean directional cyclic autocorrelation anddirectional spectral correlation density were defined andutilized to analyze the complex-valued lateral vibration signalof the rotor system The proposed method is able to identifythe oil whirl or oil whip fault In addition the informationon the procession directivity of the planar motion can alsobe obtained Furthermore it is found that actually the fullspectrum can be interpreted as the first order directionalcyclostationarity and the directional Wigner distributionas the second order directional cyclostationarity Thereforethe developed directional cyclostationarity integrates boththe advantages of conventional cyclostationarity and thedirectional Wigner distribution

Nomenclature

CM(120572) Cyclic mean at 120572CR(120572 120591) Cyclic autocorrelation at (120572 120591)SCD(120572 119891) The spectral correlation density at (120572 119891)

WV(119905 119891) The Wigner-Ville spectrum at (119905 119891)

120572 Cyclic frequency120591 Time lag119901(119905) Complex-valued vibration signal119909(119905) Real-valued vibration signal in 119909

direction119910(119905) Real-valued vibration signal in 119910

direction119901119891(119905) Forward complex-valued vibration

signal

119901119887(119905) Backward complex-valued vibration sig-

nal119882119889119901(119905 120596) Autodirectional Wigner distribution at

(119905 120596)

119882119889119901lowast119901(119905 120596) Cross-directionalWigner distribution at

(119905 120596)

FS(119905 120596) Full spectrum at (119905 120596)

DCM(120572) Directional cyclic mean at 120572DCR119901119901lowast(120572 120591) Autodirectional cyclic autocorrelation at

(120572 120591)

DCR119901119887lowast119901119891(120572 120591) Cross-directional cyclic autocorrelation

at (120572 120591)DSCD

119901119901lowast(120572 119891) Autodirectional spectral correlation

density function at (120572 119891)

DSCD119901119887lowast119901119891(120572 119891) Cross-directional spectral correlation


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no CDJZR12115501)

References

[1] A Muszynska ldquoWhirl and whip-Rotorbearing stability prob-lemsrdquo Journal of Sound and Vibration vol 110 no 3 pp 443ndash462 1986

[2] A Muszynska ldquoStability of whirl and whip in rotorbearingsystemsrdquo Journal of Sound and Vibration vol 127 no 1 pp 49ndash64 1988

[3] A Moosavian H Ahmadi A Tabatabaeefar and M Khaz-aee ldquoComparison of two classifiers K-nearest neighbor andartificial neural network for fault diagnosis on a main enginejournal-bearingrdquo Shock and Vibration vol 20 no 2 pp 263ndash272 2013

[4] J Ying Y Jiao and Z Chen ldquoNonlinear dynamics analysis oftilting pad journal bearing-rotor systemrdquo Shock and Vibrationvol 18 no 1-2 pp 45ndash52 2011

[5] T W Dimond P N Sheth P E Allaire and M He ldquoIdentifica-tion methods and test results for tilting pad and fixed geometryjournal bearing dynamic coefficientsmdasha reviewrdquo Shock andVibration vol 16 no 1 pp 13ndash43 2009

[6] A Zanarini and A Cavallini ldquoExperiencing rotor and fluid filmbearing dynamicsrdquo in Proceedings of the Quinta Giornata diStudio Ettore Funaioli pp 23ndash38 Bologna Italy 2011

[7] W A Gardner A Napolitano and L Paura ldquoCyclostationarityhalf a century of researchrdquo Signal Processing vol 86 no 4 pp639ndash697 2006

[8] E Serpedin F Panduru I Sari and G B Giannakis ldquoBibliog-raphy on cyclostationarityrdquo Signal Processing vol 85 no 12 pp2233ndash2303 2005

[9] ACMccormick andAKNandi ldquoCyclostationarity in rotatingmachine vibrationsrdquoMechanical Systems and Signal Processingvol 12 no 2 pp 225ndash242 1998


[10] J Antoni ldquoCyclostationarity by examplesrdquo Mechanical Systemsand Signal Processing vol 23 no 4 pp 987ndash1036 2009

[11] D Southwick ldquoUsing full spectrum plotsrdquoOrbit vol 14 pp 19ndash21 1993

[12] D Southwick ldquoUsing full spectrumplotsmdashpart 2rdquoOrbit vol 15pp 11ndash15 1994

[13] C-W Lee Y-S Han and J-P Park ldquoUse of directional spectrafor detection of engine cylinder power faultrdquo Shock and Vibra-tion vol 4 no 5-6 pp 391ndash401 1997

[14] C W Lee and Y-S Han ldquoThe directional Wigner distributionand its applicationsrdquo Journal of Sound and Vibration vol 216no 4 pp 585ndash600 1998

[15] Y-S Han and C-W Lee ldquoDirectional Wigner distribution fororder analysis in rotatingreciprocating machinesrdquo MechanicalSystems and Signal Processing vol 13 no 5 pp 723ndash738 1999

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order cyclostationarity with conventional spectral analysisand synchronous averaging in the condition monitoringof rolling element bearing They found that the formercan reveal significant fault features while the latter failedAntoni [10] presented a tutorial on cyclostationarity whichintroduced key concepts on actual mechanical signals andproved the superiority of cyclostationarity applied inmachinediagnostics identification ofmechanical systems and separa-tion of mechanical sources

Conditionmonitoring of journal bearing supported rotorsystemsusually relies on twoorthogonally installed proximitysensors The orbit signature of the center of the journal isthen qualitatively analyzed to investigate system operatingcondition Southwick [11 12] applied the full spectrumplot ascompared to the half-spectrum to diagnose diverse rotatingmachinery malfunctions They found that using the fullspectrum the orbit ellipticity and vibration precession canbe determined without being affected by probe orientationIn order to investigate the complex-valued signal of theinstantaneous lateral vibration of the mechanical structureLee et al [13] proposed the two-sided directional powerspectra complex-valued vibration signals for the diagnosis ofcylinder power faults of four-cylinder compression and sparkignition engines Furthermore Lee andHan [14 15] proposedthe directional Wigner distribution to effectively determinethe procession parameters of the planar motion both in time-frequency domain and in order-frequency domain

Based on the previous introduction in this work theconcept of directional cyclostationarity of complex-valuedsignal is defined and used to investigate the vibration of ajournal bearing supported rotor system which is influencedby oil film instability The structure of the paper is organizedas follows In Section 2 the theory responding to the firstand the second order cyclostationarity of real-valued signalis reviewed Furthermore the directionalWigner distributionis introduced Subsequently the proposed directional cyclo-stationarity is represented In Section 3 the experiments con-ducted on a test bench are described and the results obtainedfrom analysis compared and discussed using directionalcyclostationarity directional time frequency distributionand conventional cyclostationarity respectively In Section 4summary discussions and the conclusions are given

2 Directional Cyclostationarity

21 Cyclostationarity of Real-Valued Signals Since cyclosta-tionarity can indicate itself in statistics of any orders deter-mined by the degree of nonlinear operation there are variousstatistic parameters illustrated in the time the frequencyand the cyclic frequency domains [10] The first and secondorder of cyclostationarity are firstly reviewed for simplicitybut without losing accuracy

The mean value of a cyclostationary signal (cyclic meanCM) is defined as

CM (120572) = sum

120572isin119860

119864 [119909 (119905) sdot 119890minus2120587120572119905

] (1)

where 120572 is the cyclic frequency of the signal and the set 119860

contains all cyclic frequencies 119864[sdot] = lim119879rarrinfin

int119879

(sdot) 119889119905119879 isthe mean operator

Then the cyclic autocorrelation (CR) representing theenergy of the signal can be defined in as

CR (120572 120591) = sum

120572isin119860

119864 [119909(119905 minus120591

2) 119909 (119905 +

120591

2) sdot 119890minus2120587120572119905

] (2)

By taking the Fourier transform of CR with respect totime lag 120591 the spectral correlation density (SCD) is given by

SCD (120572 119891) = intinfin

minusinfin

CR (120572 120591) sdot 119890minus2120587119891120591

119889120591 (3)

Finally by transforming the cyclic frequency 120572 back tothe time domain the Wigner-Ville spectrum (WV) can beobtained in

WV (119905 119891) = sum

120572isin119860

SCD (120572 119891) sdot 1198901198952120587120572119905

(4)

22 Directional Wigner Distribution Lee and Han [14 15]developed a directional Wigner distribution (DWD) by sub-stituting complex-valued signals for the real-valued signalswhich could describe the instantaneous planar motion ofthe system In their work the signals 119909(119905) and 119910(119905) obtainedfrom the horizontal and the vertical proximity probes werecombined in

119901 (119905) = 119909 (119905) + 119895119910 (119905) = 119901119891

(119905) + 119901119887

(119905)

=1003816100381610038161003816100381611990311989110038161003816100381610038161003816119890119895(120596119905+120593

119891)

+1003816100381610038161003816100381611990311988710038161003816100381610038161003816119890119895(120596119905+120593

119887)

(5)

where 119901119891(119905) and 119901119887(119905) denote the forward and backwardrespectively procession of the rotor centerline

The 119901119891

(119905) and 119901119887

(119905) are given by

119901119891

(119905) =119901 (119905) + 119895119901 (119905)

2

119901119887

(119905) =119901 (119905) minus 119895119901 (119905)

2

(6)

where 119901(119905) is the Hilbert transform of 119901(119905)Then the auto-DWD is defined in

119882119889

119901(119905 120596)

= 119882119901119891 (119905 120596)

119882119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596120591119901119891lowast

(119905 minus120591

2)119901119891 (119905 +

120591

2) 119889120591 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596120591119901119887lowast

(119905 minus120591

2)119901119887 (119905 +

120591

2) 119889120591 for 120596 lt 0

(7)

where 119882119901119891(119905 120596) and 119882

119901119887(119905 120596) are respectively the forward

and the backward terms and 120596 is the circular rotatingfrequency




Shaft



0 10 20 30 40

10000

5000

0


Time

rpm

(a)

5

0

minus5


ecx

0 10 20 30 40

Time

(b)

01

0

minus01


0 10 20 30 40

Time

ofjob

e x

(c)

5

0

minus50 10 20 30 40

Time


ecy

(d)

0

0 10 20 30 40

Time

002

minus002


ofjob

e y

(e)




0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecx

(a)


minus40

minus60

minus80

minus100

minus120

minus140


ecx

0 01 02 03 04 05 06 07 08 09 1

(b)


0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecy

(c)


minus20

minus40

minus60

minus80

minus100

minus120

minus140


ecy

0 01 02 03 04 05 06 07 08 09 1

(d)


3

2

1

0

minus1

minus2

minus3


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)


minus80

minus100

minus120

minus140

minus160


dofjo

bex

(b)

8

6

4

2

0

minus2

minus4

minus6


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bey

(c)



minus80

minus100

minus120

minus140

minus1600 01 02 03 04 05 06 07 08 09 1

dofjo

bey

(d)



0 05 1 15 2 25 3 35 4

Time (s)

ecx

1

05

0

minus05

minus1


(a)

minus20

minus40

minus60

minus80

minus100

minus120


ecx


0 01 02 03 04 05 06 07 08 09 1

(b)

2

15

1

05

0

minus05

minus1

minus15

minus2


0 05 1 15 2 25 3 35 4

Time (s)

ecy

(c)

0

minus20

minus40

minus60

minus80

minus100


ecy


0 01 02 03 04 05 06 07 08 09 1

(d)


003

002

001

0

minus001

minus002

minus003


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)

minus40

minus60

minus80

minus100

minus120

minus140


0 01 02 03 04 05 06 07 08 09 1


dofjo

bex

(b)

0


0 05 1 15 2 25 3 35 4

Time (s)

times10minus2

1

05

minus05

minus1

dofjo

bey

(c)

minus60

minus80

minus100

minus120

minus140

minus160


0 01 02 03 04 05 06 07 08 09 1


dofjo

bey

(d)



012

01

008

006

004

002

0


dm1



1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0


dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+





(d)




Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3


(a)


10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus


2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus


008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6


(d)




07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Shaft



0 10 20 30 40

10000

5000

0


Time

rpm

(a)

5

0

minus5


ecx

0 10 20 30 40

Time

(b)

01

0

minus01


0 10 20 30 40

Time

ofjob

e x

(c)

5

0

minus50 10 20 30 40

Time


ecy

(d)

0

0 10 20 30 40

Time

002

minus002


ofjob

e y

(e)




0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecx

(a)


minus40

minus60

minus80

minus100

minus120

minus140


ecx

0 01 02 03 04 05 06 07 08 09 1

(b)


0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecy

(c)


minus20

minus40

minus60

minus80

minus100

minus120

minus140


ecy

0 01 02 03 04 05 06 07 08 09 1

(d)


3

2

1

0

minus1

minus2

minus3


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)


minus80

minus100

minus120

minus140

minus160


dofjo

bex

(b)

8

6

4

2

0

minus2

minus4

minus6


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bey

(c)



minus80

minus100

minus120

minus140

minus1600 01 02 03 04 05 06 07 08 09 1

dofjo

bey

(d)



0 05 1 15 2 25 3 35 4

Time (s)

ecx

1

05

0

minus05

minus1


(a)

minus20

minus40

minus60

minus80

minus100

minus120


ecx


0 01 02 03 04 05 06 07 08 09 1

(b)

2

15

1

05

0

minus05

minus1

minus15

minus2


0 05 1 15 2 25 3 35 4

Time (s)

ecy

(c)

0

minus20

minus40

minus60

minus80

minus100


ecy


0 01 02 03 04 05 06 07 08 09 1

(d)


003

002

001

0

minus001

minus002

minus003


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)

minus40

minus60

minus80

minus100

minus120

minus140


0 01 02 03 04 05 06 07 08 09 1


dofjo

bex

(b)

0


0 05 1 15 2 25 3 35 4

Time (s)

times10minus2

1

05

minus05

minus1

dofjo

bey

(c)

minus60

minus80

minus100

minus120

minus140

minus160


0 01 02 03 04 05 06 07 08 09 1


dofjo

bey

(d)



012

01

008

006

004

002

0


dm1



1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0


dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+





(d)




Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3


(a)


10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus


2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus


008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6


(d)




07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecx

(a)


minus40

minus60

minus80

minus100

minus120

minus140


ecx

0 01 02 03 04 05 06 07 08 09 1

(b)


0 05 1 15 2 25 3 35 4

03

02

01

0

minus01

minus02

minus03

minus04

Time (s)

ecy

(c)


minus20

minus40

minus60

minus80

minus100

minus120

minus140


ecy

0 01 02 03 04 05 06 07 08 09 1

(d)


3

2

1

0

minus1

minus2

minus3


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)


minus80

minus100

minus120

minus140

minus160


dofjo

bex

(b)

8

6

4

2

0

minus2

minus4

minus6


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bey

(c)



minus80

minus100

minus120

minus140

minus1600 01 02 03 04 05 06 07 08 09 1

dofjo

bey

(d)



0 05 1 15 2 25 3 35 4

Time (s)

ecx

1

05

0

minus05

minus1


(a)

minus20

minus40

minus60

minus80

minus100

minus120


ecx


0 01 02 03 04 05 06 07 08 09 1

(b)

2

15

1

05

0

minus05

minus1

minus15

minus2


0 05 1 15 2 25 3 35 4

Time (s)

ecy

(c)

0

minus20

minus40

minus60

minus80

minus100


ecy


0 01 02 03 04 05 06 07 08 09 1

(d)


003

002

001

0

minus001

minus002

minus003


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)

minus40

minus60

minus80

minus100

minus120

minus140


0 01 02 03 04 05 06 07 08 09 1


dofjo

bex

(b)

0


0 05 1 15 2 25 3 35 4

Time (s)

times10minus2

1

05

minus05

minus1

dofjo

bey

(c)

minus60

minus80

minus100

minus120

minus140

minus160


0 01 02 03 04 05 06 07 08 09 1


dofjo

bey

(d)



012

01

008

006

004

002

0


dm1



1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0


dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+





(d)




Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3


(a)


10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus


2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus


008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6


(d)




07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 3 35 4

Time (s)

ecx

1

05

0

minus05

minus1


(a)

minus20

minus40

minus60

minus80

minus100

minus120


ecx


0 01 02 03 04 05 06 07 08 09 1

(b)

2

15

1

05

0

minus05

minus1

minus15

minus2


0 05 1 15 2 25 3 35 4

Time (s)

ecy

(c)

0

minus20

minus40

minus60

minus80

minus100


ecy


0 01 02 03 04 05 06 07 08 09 1

(d)


003

002

001

0

minus001

minus002

minus003


0 05 1 15 2 25 3 35 4

Time (s)

dofjo

bex

(a)

minus40

minus60

minus80

minus100

minus120

minus140


0 01 02 03 04 05 06 07 08 09 1


dofjo

bex

(b)

0


0 05 1 15 2 25 3 35 4

Time (s)

times10minus2

1

05

minus05

minus1

dofjo

bey

(c)

minus60

minus80

minus100

minus120

minus140

minus160


0 01 02 03 04 05 06 07 08 09 1


dofjo

bey

(d)



012

01

008

006

004

002

0


dm1



1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0


dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+





(d)




Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3


(a)


10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus


2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus


008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6


(d)




07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










012

01

008

006

004

002

0


dm1



1X1minus 0475X1+

0475X1minus

1X1+ minus 0475X1+

(a)

07

06

05

04

03

02

01

0


dm3

fowpminus

fowp+



(b)

2

1

0


dm2

1X1minus1X1+

0475X1minus

0475X1+



(c)

3

25

2

15

1

05

0


dm4

2X2minus

1X2minus 1X2+

2X2+





(d)




Tim

e lag

s (s)


20

40

60

80

100

120

5

4

3

2

1

0

times10minus3


(a)


10

8

6

4

2

times10minus8


Tim

e lag

s (s)


20

40

60

80

100

120

2X1+2X1minus


2X1+ + 0475X1+

(b)


Tim

e lag

s (s)


20

40

60

80

100

120

1X2+1X2minus


008

006

004

002

0

(c)


Tim

e lag

s (s)


20

40

60

80

100

120


8

6

4

2

times10minus6


(d)




07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











07

06

05

04

03

02

01

01000

800600

400200

0 minus1000minus500

0500

1000


800600

400200 minus500

0500

1

Frequen (Hz)

(a)

01000

800600

400200

0 minus1000minus500

0500

1000



7

6

5

4

3

2

1

times10minus6

800600

400200 minus500

0500

1

Frequen (Hz)

(b)

01000

800600

400200

0 minus1000minus500

0500

1000



70605040302010

00600

400200 500

0500

1

re (Hz)

(c)

01000

800600

400200

0 minus1000minus500

0500

1000



6

5

4

3

2

1

times10minus4

800600

400200 minus500

0500

1

Freque (Hz)

(d)



119882119889

119901lowast119901

(119905 120596) = 119882119901119887lowast119901119891 (119905 120596)

= intinfin

minusinfin

119890minus119895120596120591

119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) 119889120591 forall120596

(8)


FS (119905 120596)

= FS119901119891 (119905 120596)

FS119901119887 (119905 120596)

=

intinfin

minusinfin

119890minus119895120596119905119901119891 (119905) 119889119905 for 120596 gt 0

intinfin

minusinfin

119890minus119895120596119905119901119887 (119905) 119889119905 for 120596 lt 0

(9)




DCM (120572)

=

DCM119891(120572)

DCM119887(120572)

=

sum

120572isin119860

119864 [119901119891 (119905) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905) sdot 119890minus2120587120572119905] for 120572 lt 0

(10)



05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











05

1

15

2

25

3

35

15

10

5



Tim

e lag

s (s)


(a)


1000

800

600

400

200

1X2minus 1X2+

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(b)


12

10

8

6

4

2

times10minus5

1X1minus + 0475X1minus1X1+ + 0475X1+

2X1+

2X1minus

0

05

1

15

2

25

3

35

Tim

e lag

s (s)



(c)


10

8

6

4

2

times10minus3


3X2+

3X2minus

0

05

1

15

2

25

3

35Ti

me l

ags (



(d)


10

8

6

4

2

0


fs1


Frequency (Hz)

0475X1+

1X1+ minus 0475X1+

1X1minus0475X1minus


(a)

10

8

6

4

2

0


fs3


Frequency (Hz)

fowpminus

fowp+

(b)

3

25

2

15

1

05

0


fs2


Frequency (Hz)

1X1minus

1X1+

0475X1minus

0475X1+

(c)

25

20

15

10

5

0


fs4


Frequency (Hz)

2X2minus

1X2minus1X2+

2X2+



(d)



0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 400 500 600

005

004

003

002

001

0



mx1

1X1

1X1 minus 0475X1

0475X1

(a)

007

006

005

004

003

002

001

0


my1

0 100 200 300 400 500 600


(b)

2

1

0


mx2

0 100 200 300 400 500 600


(c)

0 100 200 300 400 500 600


4

3

2

1

0


my2

(d)


025

02

015

01

005

0


mx3

0 100 200 300 400 500 600


fowp

1X2 minus fowp

1X2

(a)

04

03

02

01

0


my3

0 100 200 300 400 500 600


(b)

25

2

15

1

05

0


mx4

0 100 200 300 400 500 600


(c)

6

5

4

3

2

1

0


my4

0 100 200 300 400 500 600


(d)




8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











8

6

4

2

times10minus3

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


0008

0006

0004

0002

0

1X1

2X1 minus 0475X1

0475X1

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


15

1

05

times10minus7

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600



(c)


2

15

1

05

times10minus8

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



008

006

004

002

0

1X2

1X2 minus fowp

2fowp20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(a)


015

01

005

0

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(b)


3

2

1

times10minus5


20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(c)


3

25

2

15

1

05

times10minus6

20

40

60

80

100

120

Tim

e lag

s (s)

0 100 200 300 400 500 600


(d)



where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where DCM119891(120572 119905) and DCM




DCR119901119901lowast (120572 120591)

= DCR119901119891119901119891lowast (120572 120591)

DCR119901119887119901119887lowast (120572 120591)

=

sum

120572isin119860

119864 [119901119891 (119905 minus120591

2)119901119891lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 gt 0

sum

120572isin119860

119864 [119901119887 (119905 minus120591

2)119901119887lowast

(119905 +120591

2) sdot 119890minus2120587120572119905] for 120572 lt 0

(11)





DCR119901119887lowast119901119891 (120572 120591)

= sum

120572isin119860

119864 [119901119887

(119905 minus120591

2)119901119891

(119905 +120591

2) sdot 119890minus2120587120572119905

] forall120572(12)




DSCD119901119901lowast (120572 119891)

= DSCD

119901119891119901119891lowast (120572 119891)

DSCD119901119887119901119887lowast (120572 119891)

=

intinfin

minusinfin

DCR119901119891119901119891lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 gt 0

intinfin

minusinfin

DCR119901119887119901119887lowast (120572 120591) sdot 119890

minus2120587119891120591119889120591 for 120572 lt 0

(13)



DSCD119901119887lowast119901119891 (120572 119891) = int

infin

minusinfin

DCR119901119887lowast119901119891 (120572 120591) sdot 119890

minus2120587119891120591

119889120591 forall120572

(14)








3 Experiment





02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











02

015

01

005

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

Freque (Hz)

(a)

1000

1000800800600

600400400200 200

0 0


0


05

04

03

02

01

800800600

600400400200 200

Frequ )

(b)


8

6

4

2

times10minus6

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quency (Hz)

(c)


4

3

2

1

times10minus7

1000

1000800800600

600400400200 200

0 0


0

00800600

600400400200 200

quenc

(d)







1and the rotating









11198831minusgt 11987711198831+

or 11987711198832minus

gt 11987711198832+





12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











12

10

8

6

4

2

01000

1000800800600

600400400200 200

0 0


800800600

600400400200 200

eque )

(a)


40

30

20

10

01000

1000800800600

600400400200 2000 0


800800600

600400400200equ )

(b)


6

5

4

3

2

1

0

times10minus4

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

que )

(c)


2

15

1

05

0

times10minus5

10001000800

800600600400

400200 2000 0


800800600

600400400200 200

requ H )

(d)







1 With 119877

11198831minusgt

11987711198831+


2







eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











eque

ncy

(Hz)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

ecx

4

2

0

minus2

minus4

times10minus5

(a)


1

0

minus1

times10minus4

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)

(b)

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

050

minus05

Time (s)


2

1

0

minus1

minus2

times10minus9

times10minus3

Dofj

obe x

(c)


Dofj

obe y


times10minus10

10

minus1

Freq

uenc

y (H

z)0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

times10minus3

(d)


Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

10

minus1

Time (s)

ecx


2

1

0

minus1

minus2

times10minus3

(a)


20

minus2

5

0

minus5

times10minus3

ecy

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

(b)


0050

minus005

2

1

0

minus1

minus2

times10minus7

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)

Dofj

obe x

(c)

Dofj

obe y

Freq

uenc

y (H

z)

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0

200

400

600

800

1000

35 4

Time (s)


0010

minus001

2

1

0

minus1

minus2

times10minus8

(d)









Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Nomenclature






signal



(119905 120596)


(119905 120596)



(120572 120591)


at (120572 120591)DSCD







Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Vibration Signal Analysis of Journal...

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