research article parametric and internal resonances of an...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013 Article ID 919517 18 pageshttpdxdoiorg1011552013919517

Research ArticleParametric and Internal Resonances of an Axially MovingBeam with Time-Dependent Velocity

Bamadev Sahoo1 L N Panda2 and G Pohit3

1 Department of Mechanical Engineering International Institute of Information Technology Bhubaneswar 751003 India2Department of Mechanical Engineering College of Engineering and Technology Bhubaneswar 751003 India3 Department of Mechanical Engineering Jadavpur University Kolkata 700032 India

Correspondence should be addressed to G Pohit gpohitgmailcom

Received 10 May 2013 Accepted 27 August 2013

Academic Editor Abdelali El Aroudi

Copyright copy 2013 Bamadev Sahoo et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance isinvestigatedThe beam velocity is assumed to be comprised of a constantmean value alongwith a harmonically varying componentThe stretching of neutral axis introduces geometric cubic nonlinearity in the equation ofmotion of the beamThe natural frequencyof secondmode is approximately three times that of firstmode a three-to-one internal resonance is possibleThemethod ofmultiplescales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditionsThenonlinear steadystate response along with the stability and bifurcation of the beam is investigated The system exhibits pitchfork Hopf and saddlenode bifurcations under different control parameters The dynamic solutions in the periodic quasiperiodic and chaotic forms arecaptured with the help of time history phase portraits and Poincare maps showing the influence of internal resonance

1 Introduction

Band saws fibre textiles magnetic tapes paper sheets aerialtramways pipes transporting fluids thread lines and beltsare some technological examples classified as axially movingcontinua Analytical models for axially moving systems havebeen extensively used in the last few decades The vastliterature on axially moving continua vibration has beenreviewed by Wickert and Mote Jr [1] up to 1988 While alinear analysis provides natural frequencies mode shapesand critical speeds its validity regarding the response ofthe system diminishes as the vibration amplitude becomessufficiently large or as the critical speed is approached [2]In these cases one must resort to a nonlinear analysisWickert and Mote Jr [3 4] studied the transverse vibrationof axially moving strings and beams using an eigenfunctionmethodThey also studied the dynamic response of an axiallymoving string loaded suspendedmassWickert [5] presenteda detailed study of the nonlinear vibrations and bifurcationsof moving beams using the Krylov-Bogoliubov-Mitropolskyasymptotic method Chakraborty et al [6 7] investigated

both free and forced vibration of the nonlinear traveling beamusing complex normal modes

There are papers devoted to the analysis of the dynamicbehavior of traveling systems with time-dependent axialvelocity or with time-dependent axial tension force Oz andPakdemirli [8] investigated principal parametric resonancesand combination resonances of sum and difference types forany two modes for an axially accelerating beam using themethod of multiple scales They found that for combinationresonances instabilities occurred only for additive type butnot for difference type Oz et al [9] extended the work tononlinear transverse vibration and stability analysis Com-prehensive review of nonlinear modal interactions is therein [10ndash12] Using method of multiple scales Riedel and Tan[13] studied the coupled and forced behavior of an axiallymoving strip with internal resonance Ozkaya et al [14]investigated nonlinear transverse vibrations and 3 1 internalresonances of a beam with multiple supports and plottedfrequency response curves for different support numbersBagdatli et al [15] extended this work to find existence ofinternal resonance cases between different modes Chin and

2 Modelling and Simulation in Engineering

Nayfeh [16] investigated three-to-one internal resonances inparametrically excited hinged-clamped beams They studiedprincipal parametric resonance of the first or the secondmode and a combination parametric resonance of additivetype of these modes Panda and Kar [17 18] investigatedthe principal parametric resonance of first mode secondmode and combination parametric resonance in presenceof 3 1 internal resonance of a pipe conveying pulsatingfluid with pinned-pinned end conditions They studied theequilibria of these modulated equations and determinedtheir stability and then they did extensive dynamic studyat typical initial conditions Sze et al [19] used incremen-tal harmonic balance method for nonlinear vibration ofaxially moving beams They investigated the fundamentalsuperharmonic and subharmonic resonance in presenceof internal resonance Huang et al [20] used the samemethod to analyse the stability and bifurcation analysis of atraveling beam tuned to 3 1 internal resonancewith attentionto fundamental and subharmonic resonances Chen et al[21] investigated dynamic stability of an axially acceleratingviscoelastic beam undergoing parametric resonance Dinget al [22 23] used Galerkin methods for finding naturalfrequencies of high-speed axially moving beams with hybridboundary conditions Marynowski and Kapitaniak [24ndash26]introduced several internal dissipation mechanisms likeKelvin-voigt and Zener in the modeling traveling continuaPakdemirli and Oz [27] studied the transverse vibration ofsimply supported axially moving Euler-Bernoulli beam forinfinite mode analysis and truncation to resonant modesPonomareva and van Horssen [28] investigated transversalvibrations of axially travelling continua based on a stringmodel at the low frequencies and a tensioned beam modelat the higher frequencies Recently a systematic researchon travelling beam was pursued by Ghayesh et al [29ndash33]involving nonlinear dynamic phenomenon of a variety ofsystem models The forced dynamics of an axially movingviscoelastic beam was investigated in [29] The nonlinearvibrations and stability of an axially moving beam subjectedto a distributed harmonic excitation load were investigatedin [30] In addition the response of the system tunedto a three-to-one internal resonance was also examinedSubcritical dynamics of the system was examined [31] viathe pseudoarclength continuation technique while the globaldynamics was investigated using direct time integration

The present work aims to investigate the problem of asimply supported traveling beamunder parametric excitationdue to the harmonic pulsation in the travelling velocity Fora certain range of mean velocity the natural frequency ofthe second mode is approximately three times that of thefirst mode This relationship between natural frequencies offirst two modes of a system having cubic nonlinearity resultsin a condition of three-to-one autoparametric or internalresonance leading to energy exchange between the twoinvolved modes through nonlinear modal interaction In thepresent work the principal parametric resonance of the firstmode is considered The system behavior shows pitchforkHopf and saddle node bifurcations in steady state analysisDecreasing internal frequency detuning parameter affects theamplitude of directly excited first mode and the number of

(t)w

x

Figure 1 Schematic diagram of an axially traveling simply sup-ported beam with variable velocity

Hopf bifurcation points and shifts the occurrence of jumpphenomena Due to the influence of internal resonance thedynamic behavior of the system exhibits periodic quasiperi-odic and chaotic responses The results are presented in theform of time histories phase plane portraits fast Fouriertransforms (FFTs) and Poincare maps

2 Formulation of the Problem

For the present work a uniform horizontal beam simplysupported at both ends and travelling with a harmonicallyvariable velocity (Figure 1) is considered The assumptionstaken here are (1) the motion of the beam is planar (2) theuniform cross sections remain plane during the motion andthe beam behaves like an Euler-Bernoulli beam in transversevibration and (3) the type of nonlinearity is geometricdue to the midplane stretching effect of the beam Thenondimensional equation of transverse motion of the beamincluding the nonlinearity due to midplane stretching [5 9]and viscous damping [34] along with viscoelastic damping[35] is given by

+ 2V1015840 + V1199081015840 + (V2 minus 1)119908

10158401015840+ V2119891119908

1015840101584010158401015840

+ 2120576120572

1015840101584010158401015840+ 2120576120583 =

1

2

V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(1)

The nondimensional scheme used here is

119909 =

119909

lowast

119871

119905 = 119905

lowastradic

119875

120588119860119871

2 119908 =

119908

lowast

119871

V =

Vlowast

radic119875120588119860

2120576120572 =

119864

lowast

119871

2(

119868

119898119864

)

12

2120576120583 =

119862119871

2

radic119898119864119868

V119897=

radic

119864119860

119875

V119891

=

radic

119864119868

119875119871

2

(2)

where the variables with asterisk denote dimensional onesIn the EOM dot denotes derivatives with respect to time(119905) and the prime denotes derivatives with respect to spatialderivative (119909) 119898 is mass per unit length 120588 is density 119860

is cross-sectional area 119871 is length V119891is nondimensional

flexural stiffness and V119897is nondimensional longitudinal

stiffness of beam 119864lowast is the coefficient of internal dissipationof the beam material which is assumed to be Kelvin-Voigttype viscoelastic and 119862 is the external damping factor 120572 isnondimensional material damping and 120583 is nondimensionalviscous damping Reordering the transverse displacementwith the relation 119908 = radic120576119908

where 120576 lt 1 and putting itin the equation of motion (1) the system is converted into aweakly nonlinear one [9]

Modelling and Simulation in Engineering 3

minus20 minus15 minus10 minus5 0 5 10 15 200

05

1

15

2

25

3

35

4

45

5

(1 0)

(0 002)

(0 003)

(0 004)

1205902

1

(120583 = 0001 120572 = 0)

Figure 2 Trivial state boundary for different damping parametersValues of the nondimensional damping parameters (120583 120572) indicatedon the curves

For convenience the superscript ldquordquo is removed and theweakly nonlinear equation of motion becomes

+ 2V1015840 + V1199081015840 + (V2 minus 1)119908

10158401015840+ V2119891119908

1015840101584010158401015840

+ 2120576120572

1015840101584010158401015840+ 2120576120583 =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(3)

The variable velocity of the beam is

V = V0+ 120576V1sinΩ119905 (4)

where V0is mean velocity 120576V

1is the amplitude and Ω is

the frequency of the harmonically varying component Thisharmonic component of velocity which is the parametricexcitation term in the equation of motion gives variousparametric responses of the system Putting (4) in (3) we getthe equation of transverse motion

2120576120572

1015840101584010158401015840+ V2119891119908

1015840101584010158401015840+ [V20+ 2120576V0V1sinΩ119905 minus 1]119908

10158401015840

+ 120576V1Ω119908

1015840 cosΩ119905 + 2 (V0+ 120576V1sinΩ119905)

1015840

+ 2120576120583 + =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(5)

with boundary conditions

119908 (0 119905) = 119908 (1 119905) = 119908

10158401015840

(0 119905) = 119908

10158401015840

(1 119905) = 0 (6)

The linear natural frequencies for various modes vary withvariation in the mean velocity of beam For a consider-able range of mean velocity the first two natural frequen-cies become commensurable leading to modal interactionsthrough internal resonance The combination of internalresonancewith different kinds of parametric resonances givesthe system response in the form of directly excited modeand indirectly excited one because of the energy exchangebetween them [16ndash18]

3 Method of Analysis

An approximate solution to this weakly nonlinear distributedparameter system in the form of a first order uniformexpansion by using the direct perturbation technique ofmethod of multiple scales (MMS) [14ndash18] is aimed The timescale used here is 119879

119899= 120576

119899119905 119899 = 0 1 2 3 and the time

derivatives are

119889

119889119905

= 119863

0+ 120576119863

1+ sdot sdot sdot

119889

2

119889119905

2= 119863

2

0+ 2120576119863

0119863

1+ sdot sdot sdot

119863

119899=

120597

120597119879

119899

119899 = 0 1 2 3

(7)

Assuming an expansion of the form

119908 (119909 119905 120576) = 119908

0(119909 119879

0 119879

1) + 120576119908

1(119909 119879

0 119879

1) + sdot sdot sdot (8)

Substituting (7) and (8) into (5) and (6) and equatingcoefficients of like powers of 120576 on both sides we get

119874(120576

0) 119863

2

0119908

0+ 2V0119863

0119908

1015840

0+ (V20minus 1)119908

10158401015840

0+ V2119891119908

1015840101584010158401015840

0= 0

119908

0(0 119905) = 119908

0(1 119905) = 119908

10158401015840

0(0 119905) = 119908

10158401015840

0(1 119905) = 0

(9)

119874(120576

1) 119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ V2119891119908

1015840101584010158401015840

1+ (V20minus 1)119908

10158401015840

1

= minus2V0119863

1119908

1015840

0minus 2V1sinΩ119905119863

0119908

1015840

0minus 2119863

0119863

1119908

0

minus 2120572119863

0119908

1015840101584010158401015840

0minus 2120583119863

0119908

0minus 2V0V1sinΩ119905119908

10158401015840

0

minus V1Ω cosΩ119905119908

1015840

0+

1

2

V2119897119908

10158401015840

0int

1

0

119908

10158402

0119889119909 = 0

119908

1(0 119905) = 119908

1(1 119905) = 119908

10158401015840

1(0 119905) = 119908

10158401015840

1(1 119905) = 0

(10)

The solution of (9) may be written as

119908

0(119879

0 119879

1 119909) =

infin

sum

119899=1

120601

119899(119909) 119860

119899(119879

1) 119890

1198941205961198991198790

+ 119888119888 (11)

where 120601

119899is the mode shapes 120596

119899is the natural frequencies

and 119888119888 is complex conjugate The mode shapes are calculatedpreviously [8] as

120601

119899(119909) = 119862

1119899119890

1198941205731119899119909minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

119890

1198941205732119899119909

minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205733119899119909

+ [minus1 +

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

+

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205734119899119909]

(12)


minus50 0 50 100 150 2000

1

2

3

4

5

6

H Hopf bifurcationSN Saddle node bifurcation

StableSaddleUnstable

1205902

SN3

SN1

SN2

H1

H2

A1

(a)

minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2 H1 H2

A2

(b)

Figure 3 Frequency response curves as obtained by continuation algorithm for the first and second modes when the first mode isparametrically excited for the system parameters Chin and Nayfeh [16]

minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006

SN

H1

H2

H3

H4

H5H7H6



1205902

A1

(a)

minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006SN

H1

H2

H3

H4H5

H7H6



1205902

A2

(b)

Figure 4 Frequency response curves for the (a) first mode and (b) second mode when the first mode is parametrically excited for the systemparameters 120583 = 01 120572 = 0 V

1= 10 V

119897= 40 and 120590

1= 9239

where 120573in are eigenvalues which satisfy the dispersive relation(13) and support condition (14) [8]

V2119891120573

4

in minus (V20minus 1) 120573

2

in minus 2V0120596

119899120573in minus 120596

2

119899= 0

119894 = 1 2 3 4

(13)

(119890

119894(1205731119899+1205732119899)+ 119890

119894(1205733119899+1205734119899)) (120573

2

1119899minus 120573

2

2119899) (120573

2

3119899minus 120573

2

4119899)

+ (119890

119894(1205731119899+1205733119899)+ 119890

119894(1205732119899+1205734119899)) (120573

2

2119899minus 120573

2

4119899)

times (120573

2

3119899minus 120573

2

1119899) + (119890

119894(1205732119899+1205733119899)+ 119890

119894(1205731119899+1205734119899))

times (120573

2

1119899minus 120573

2

4119899) (120573

2

2119899minus 120573

2

3119899) = 0

(14)

For a fixed velocity the linear natural frequencies of thetravelling beam vary with flexural stiffness (V

119891) For specific

values of flexural stiffness the lower natural frequencies canbe commensurable for a range of mean travelling velocity ofthe beam These phenomena result in internal resonance in


minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H3

H4H5

H2

A1

(a)

0 100 200 300 400 5000

001

002

003

004

005

006

007

008

SN

1205902

H1

H3

H4H5

H2

A2

(b)


1= 15 V

119897= 40 and 120590

1= 9239

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4H5H6

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5H6

H2A2

(b)


1= 10 V

119897= 40 and120590

1= 9239

the system and nonlinear interaction between the involvedlower modes In the present investigation a three-to-oneinternal resonance (120596

2asymp 3120596

1) is considered for a range of

mean velocity of the beam Also it is assumed that there isno other commensurable frequency relationship with highermodes The case of principal parametric resonance of thefirst mode (Ω asymp 2120596

1) for subcritical flow velocities in

presence of 3 1 internal resonance is analyzed in the presentinvestigation These first two modes are not in internalresonancewith any highermodes so the highermodes exceptthe first two will decay with time due to the presence ofdamping and Coriolis terms present in the equation Hence

the first two modes will contribute to the long term systemresponse [10 11] Consequently we replace (11) with

119908

0(119879

0 119879

1 119909) = 119860

1(119879

1) 120601

1(119909) 119890

11989412059611198790

+ 119860

2(119879

1) 120601

2(119909) 119890

11989412059621198790

+ 119888119888

(15)

Now we write the frequency relations for the internal reso-nance and principal parametric resonance as

120596

2= 3120596

1+ 120576120590

1 Ω = 2120596

1+ 120576120590

2 (16)

where 120590

1and 120590

2are detuning parameters It is worthy to

note that Ω = 120596

2minus 120596

1+ 120576(120590

2minus 120590

1) a combination


SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)

Figure 7 Effect of internal detuning parameter on the frequency response of (a) first mode and (b) second mode for system parameters120583 = 01 120572 = 0 V

1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680

parametric resonance of the difference type is also activatedsimultaneously Substituting (15) and (16) into (10) we get

119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ

119899are defined in the Appendix section

NST stands for terms that do not produce secular or smalldivisor terms As the homogeneous part of (17) with itsassociated boundary conditions has a nontrivial solution thecorresponding nonhomogeneous problemhas a solution onlyif a solvability condition is satisfied [36 37] This requires theright-hand side of (17) to be orthogonal to every solutionof the adjoint homogeneous problem which leads to the

complex variable modulation equations for amplitude andphase

2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)

where the prime denotes the differentiation with respectto slow time 119879

1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890

119894are defined in

the Appendix section Overbar indicates complex conjugateThe terms in the previous equations involving the internalfrequency detuning parameter 120590

1are the contributions of the

internal resonance in the system


0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)

Figure 8 Effect of external damping parameter on the frequency response of first mode (a) and secondmode (b) for 120583 = 005 120572 = 0 V1= 10

V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)

minus2 minus1 0 1 2minus7

minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)

Figure 9 Phase portraits (a b) and time histories (c d) in the upper nontrivial stable branch of the frequency response curve of Figure 8 for120590

2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)

Figure 10 Phase portraits (a b) and time histories (c d) for 1205902= 752799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239

4 Stability and Bifurcations

The evolutions of the equilibrium solutions and their stabilityand bifurcation analysis for principal parametric resonanceof first mode are carried out from the modulation equation(18) The Cartesian transformation is used for the complexamplitude as

119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)

Putting this in (18) simplifying by trigonometric manipula-tions and separating the real and imaginary parts we getthe normalized reduced equations or the Cartesian form ofmodulation equations

119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)

minus0022 minus00218 minus00216 minus00214minus8

minus75

minus7

minus65times10minus3

q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)

Figure 11 Phase portrait (a) FFT power spectra (b) and Poincare maps (c d) for 1205902= 762799 120583 = 005 V

1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)

The previous equations are perturbed to evaluate the stabilityThe perturbed equation is

Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)

where 119879 denotes transpose and [119869

119888] is the Jacobian matrix

whose eigenvalues determine the stability and bifurcation ofthe systemThe stability boundary for trivial state is obtainedby setting 119901

1= 119902

1= 119901

2= 119902

2= 0 The nonlinear steady

state response behavior of the system is obtained from thenormalized reduced equation (20) by setting 119901

1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840

2= 0 and then solving the resulting set of nonlinear


6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)

minus15 minus10 minus5 0 5minus95

minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)

Figure 12 Phase portraits (a b) and Poincare maps (c d) for 1205902= 782799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239

algebraic equations The same set of equations is also usedfor the analysis of stability and bifurcation of trivial as wellas nontrivial solutions The analysis for dynamic solutionsis carried out by numerically integrating (20) with differentcombinations of system parameters

5 Results and Discussions

The natural frequencies of the beam are numerically evalu-ated at different mean velocities (V

0) with flexural stiffness

V119891

= 02 by simultaneous solution of dispersive relation(13) and support condition (14) The results are presented inTable 1 It is noticed that at nondimensional mean velocityV0= 0513 the natural frequency of second mode is approx-

imately equal to three times that of the first mode implyingthe existence of 3 1 internal resonance It is also noticed thatthere are no other commensurable frequency relationshipsinvolving higher modes Therefore nonlinear interactionamong higher modes is ruled outThe investigation is limited

to the case of principal parametric resonance of first modethat is Ω asymp 2120596

1 in presence of internal resonance in the

subcritical mean velocity regime of a travelling beamThe trivial state stability boundary shown in Figure 2 is

plotted in terms of principal parametric frequency detuning(1205902) and amplitude of fluctuating velocity component (V

1)

for system parameters V119891

= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596

2= 91403 and for different damping

values The book keeping parameter is taken as 120576 = 001 andthe corresponding internal frequency detuning parameter isassumed to be 120590

1= 9239 The region inside the boundary

denotes instability Higher values of damping have the effectof raising and narrowing the instability zones It is revealedthat the effect of material damping (120572) raises and narrows theinstability zone more compared to that of viscous damping(120583) on the trivial state stability boundary However thistrivial state stability plot may not reveal completely thereal system behavior because in the unstable zone of thetrivial stability plot the system may have stable nontrivial



0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239

equilibrium solution or stable dynamic solution like periodicor quasiperiodic solution In addition there is a possibility ofchaotic solutions or multiple stable solutions For this reasonit is required to carry out the dynamic analysis as well as thestability and bifurcation study of equilibrium solutions of thesystem

51 Stability and Bifurcations of Equilibrium Solutions Con-tinuation algorithm is used to determine the nonlinear steadystate response by solving the set of algebraic equationsgenerated after setting 119901

1015840

119894= 119902

1015840

119894= 0 in the normalized reduced

equation (20)The stability and bifurcation of the equilibriumsolutions are obtained from the eigenvalues of the Jacobeanmatrix at each point of the solution In order to validatethe results obtained by the present analysis the frequencyresponse and amplitude response curves of Chin et al [16]are generated once again using continuation algorithmTheyare shown in Figure 3 and the results are found to be ingood agreement Since the frequency and amplitude response

curves are symmetrical about 120590

2and V

1axes respectively

only positive sides of the response curves are shownFrequency response curves are obtained against variation

in frequency detuning parameter1205902for first and secondmode

for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239

and are shown in Figure 4 The normal continuous lines inthe figure represent stable equilibrium solutions the boldlines represent unstable foci and the dotted lines denotesaddles Different parameter values for characteristic pointson different branches of the plot are indicated in Table 2 Theresponse curves exhibit a hardening-spring type of nonlinear-ity With increase in 120590

2from a small value the trivial stable

solution loses stability at 1205902= minus22071 through supercritical

pitchfork bifurcation and results in a two-mode nontrivialstable equilibrium solution It is observed that the amplitudeof the first mode increases initially then decreases but theamplitude of second mode increases monotonically Whenthe value of 120590

2increases the equilibrium solution becomes

unstable at 1198671(120590

2= 206588) through Hopf bifurcation and


minus002 minus001 0 001minus003

minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)

Figure 14 Phase portraits (a b) time trace (c) and Poincare map (d) for 1205902= 1164201 120583 = 005 V

1= 10 and 120572 = 0 120590

1= 9239

out of two pairs of complex conjugate eigenvalues one paircrosses the imaginary axis from the left half of the complexplane to the right half With further increase of 120590

2 the same

state continues until a saddle node bifurcation occurs atSN (120590

2= 301873) where the system response jumps to

one of the two stable equilibrium branches one trivial andthe other nontrivial depending on the initial conditions asthe solution converges to the closer equilibrium state as perthe concept of region of attraction With further increase infrequency detuning parameter amplitude of the first modeincreasesmonotonically along the nontrivial branch whereasthe amplitude of second mode decreases continuously Thusthe amplitude of the indirectly excited secondmode is limitedto a fixed higher magnitude and for high values of 120590

2 it

becomes stagnant at fixed low amplitude while there is nosuch limitation for the directly excited first mode

When the detuning parameter decreases from a highvalue the system follows either trivial or nontrivial stableequilibrium path depending on the initial conditions If the

solution is nontrivial with decrease of 1205902value the nontrivial

stable branch loses stability via Hopf bifurcation at 1198672(120590

2=

190033) and regains stability via a reverse Hopf bifurcationat 119867

3(120590

2= 144502) When the detuning parameter is

further decreased again the system loses stability via Hopfbifurcation at 119867

4(120590

2= 12798) and regains stability via a

reverse Hopf bifurcation at 119867

5(120590

2= 80408) on the same

pathWith further decrease in frequency detuning parameterthe nontrivial stable equilibrium branch merges with stabletrivial equilibrium solution the system losing and regainingthe stability at 119867

6(120590

2= 49474) and 119867

7(120590

2= 44608)

respectively At 1205902= 14190 the trivial equilibrium solution

loses stability via subcriticalreverse pitchfork bifurcation andresults in a jump of the response to the stable nontrivialbranch of the solution Again with further decrease offrequency detuning the nontrivial stable solution branchloses stability through pitchfork bifurcation at 120590

2= minus22071

giving the way to trivial solution The directly excited firstmode dominates the indirectly excited second mode


0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)

minus003 minus002 minus001 0minus002

minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239

Figure 5 shows the frequency response curves for firstand second mode of the system for higher amplitude of thefluctuating velocity component (V

1) The system parameters

considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590

1= 9239 Even though the solution curves are similar

in shape to the curves obtained in case of lower amplitudeof excitation (V

1= 10) as shown in Figure 4 the jump

phenomena at the saddle node bifurcation (SN) occur athigher value of the detuning parameter (120590

2= 4577476)

compared to the previous case (1205902

= 3018730 (Figure 4))In addition unstable zone in trivial solution gets broadenedwhich is commensurate with the trivial state stability plot

Figure 6 shows typical frequency response curves for twomodes considering the effect of the internal damping forsystem parameters 120583 = 0 120572 = 0001 V

1= 10 V

119897=

40 and 120590

1= 9239 The strength of nonlinear interaction

due to internal resonance gets weakened due to internaldamping (Figure 6) compared to the case of external damping(Figure 4) The influence of internal detuning parameter(120590

1) on the frequency response is shown in Figure 7 It is

evident that the decrease in internal detuning parameter (1205901)

to 20320 (Figures 7(a) and 7(b)) and minus27680 (Figures 7(c)and 7(d)) from 92390 (Figure 4) weakens the strength ofnonlinear interaction due to three-to-one internal resonanceThe amplitude of the directly excited first mode decreasesmore pronouncedly than the indirectly excited secondmodeBeside this the number of Hopf bifurcation points on theupper nontrivial curve decreases from four for 120590

1= 9239

to two for 1205901= 20320 and totally vanishes for 120590

1= minus27680

and also there is decreasing trend in 120590

2value at which saddle

node bifurcation occurs Figure 8 shows the effect of decreaseof external damping (120583 = 005) on the frequency response ofthe systemThe nature of the shape of the curves is similar tothat of Figure 4 but the jump phenomena occur at a highervalue of parametric excitation frequency detuning parameter(120590

2= 3829953) and the amplitudes of both directly and

indirectly excited modes are amplified

52 Dynamic Solutions Frequency response and amplituderesponse plots reveal different stability and bifurcations of the


minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)

Figure 16 Phase portraits (a b) time trace (c) and FFT power spectra (d) in the lower nontrivial stable branch of the frequency responseplot of Figure 8 for 120590

2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239

equilibrium solutions with variation of control parametersDynamic analysis of the system which is dependent on initialconditions is studied in the form of periodic quasiperiodicand chaotic responses and some selected results are pre-sented

Figures 9(a)ndash9(d) show typical system response in termsof phase portraits (a b) and time traces (c d) at 120590

2= 682799

corresponding to the upper nontrivial stable branch of thefrequency response curve (Figure 8) for 120583 = 005 120572 = 0V1

= 10 and 120590

1= 9239 The response is periodic about

the nontrivial equilibrium solutionwhen the time integrationis started with the initial values 119901

1= 0002 119902

1= 00007

119901

2= 00067 and 119902

2= 00001 Further along the same branch

at 1205902= 752799 the response is quasiperiodic in both modes

being more prominent in the second mode as shown in thetwo-dimensional projections of the phase portraits onto the119901 minus 119902 planes in Figures 12(a) and 12(b) and the time traces inFigures 10(c) and 10(d) The response remains quasiperiodicin both modes for higher frequency detuning parameter

values typically at 1205902= 762799 as shown in the closed loop

Poincare maps and FFT power spectra in Figures 11(c) 11(d)and 11(b) respectively With further increase in the value ofdetuning parameter typically at 120590

2= 782799 we find the

closed loops of Poincare map get merged and give a way tochaotic response in second mode in Figure 12(d) Howeverin first mode the system response is still quasiperiodic asseen from phase portrait (Figure 12(a)) and Poincare map(Figure 12(c)) respectively

Figures 13(a)ndash13(d) show the typical system behavior at120590

2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239

in terms of phase portraits and time traces in Figures 13(a)13(b) 13(c) and 13(d) respectively The response is initiallychaotic and jumps to the nearby stable trivial attractor Withfurther increase in detuning parameter the system behaviorchanges drastically as shown in Figures 14(a)ndash14(d) at 120590

2=

1164201 illustrating quasiperiodic motion in the first modeand chaotic motion in second mode The dynamic responsein both modes becomes chaotic at detuning parameter


Table 1 Variation of natural frequencies for the first twomodeswithmean travelling velocity V

0showing 3 1 internal resonance for V

119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)

04500 33149 97032 99447 minus02415 minus7285304600 32969 96857 98907 minus02050 minus6218004700 32786 96677 98358 minus01681 minus5127204800 32599 96493 97797 minus01304 minus4000104900 32407 96305 97221 minus00916 minus2826505000 32211 96113 96633 minus00520 minus1614405100 32011 95917 96033 minus00116 minus0362405110 31991 95897 95973 minus00076 minus0237605120 31970 95878 95910 minus00032 minus0100105130 31950 95858 95850 00008 0025005140 31930 95838 95790 00048 0150305600 30948 94876 92844 02032 6565905800 30493 94431 91479 02952 9680906000 30020 93968 90060 03908 130180

120590

2= 1170201 as shown in phase portrait FFT power spectra

and Poincare maps in Figures 15(a) 15(b) 15(c) and 15(d)respectively The changes in system response from periodicin both modes to mixed mode that is quasiperiodic in firstand chaotic in second mode to chaotic in both modes asexplained above happen in the zone of frequency responseplot where all three kinds of curves stable saddle andunstable are in very close proximity and crossing each otherExistence of multiple branches is possible due to the presenceof internal resonance in the system The nonlinear modalinteraction influences simultaneously both the stable andunstable attractors which finally results in such varied systemresponses

Similar investigation is carried in the lower nontrivialstable branch of the frequency response plot of Figure 8 aswell For a point on the same branch at 120590

2= 2805201 the

system response exhibits one periodic and one quasiperiodicsystem behavior as shown in Figure 16 in terms of phaseportraits time trace and FFT power spectra The samebehavior is noticed for another point on the same branch at120590

2= 3105201 though the figures are not presented to avoid

repetition Thus due to the presence of internal resonancea wide range of dynamic behavior can be observed withvariation of control parameters

6 Conclusions

In the present investigation principal parametric resonanceof first mode in presence of 3 1 internal resonance of abeam moving with variable velocity is considered Stabilityboundaries of trivial state are obtained for different valuesof internal and external dissipations It has been observedthat higher values of damping have the effect of raising andnarrowing the instability zones Bifurcations of equilibriumsolutions are analyzed in the form response plots It has

been shown in frequency response plot that the nontrivialsteady state solutions bifurcate from trivial solutions throughsupercritical pitchfork bifurcations

Besides the pitchfork bifurcations the system also expe-riences Hopf bifurcation and saddle node bifurcation due tovariation of different system parameters Damping decreasesthe strength of nonlinear interaction due to internal res-onance Increasing amplitude of fluctuating velocity com-ponent broadens the range of trivial state instability andincreases the value of parametric frequency detuning atwhich jump phenomena occur Decreasing internal fre-quency detuning parameter affects the amplitude of directlyexcited first mode and number of Hopf bifurcation points Italso shifts the occurrence of jump phenomena

A detailed study is carried out to determine the influenceof different control parameters on dynamic behavior of thesystemThe dynamic solutions in the periodic quasiperiodicand chaotic forms are captured with the help of timehistory phase portraits and Poincare maps A wide arrayof dynamic behavior is noticed when nontrivial stable andsaddle branches are formed due to internal resonance andalso in the zone where the three branches are very close andcrossing each other

In case of conventional nontravelling beams with simplysupported boundary conditions occurrence of internal reso-nance is not possible due to vanishing of the nonlinear inter-action coefficients [10] However a varied system response ispossible in case of travelling beams due to nonlinear modalinteraction leading to simultaneous influence of both stableand unstable attractors

Appendix

We have the following

Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1


Table 2 Typical points on the different branches of the frequency response for 120583 = 01 120572 = 0 V1= 10 V

1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point

000500579 minus001569813 000004925 001964672 100minus002181265 minus000742674 001533298 000020989 1000010164208 000356793 001833426 000011845 100minus0004819409 001479703 000009386 001071881 100minus0007939 minus0003674 0042869 0006271 206588482 119867

1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)

times (minus119894120596

1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References

[1] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially moving materialsrdquo Shock andVibration Digest vol 20 pp 3ndash13 1988

[2] CDMote Jr ldquoOn the nonlinear oscillation of an axiallymovingstringrdquo Journal of AppliedMechanics vol 33 pp 463ndash464 1966

[3] J A Wickert and C D Mote Jr ldquoClassical vibration analysis ofaxially moving continuardquo Journal of Applied Mechanics vol 57no 3 pp 738ndash744 1990

[4] J A Wickert and C D Mote Jr ldquoTravelling load response of anaxially moving stringrdquo Journal of Sound and Vibration vol 149no 2 pp 267ndash284 1991

[5] J A Wickert ldquoNon-linear vibration of a traveling tensionedbeamrdquo International Journal of Non-Linear Mechanics vol 27no 3 pp 503ndash517 1992

[6] G Chakraborty A K Mallik and H Hatwal ldquoNon-linearvibration of a travelling beamrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 655ndash670 1999

[7] G Chakraborty and A K Mallik ldquoNon-linear vibration ofa travelling beam having an intermediate guiderdquo NonlinearDynamics vol 20 no 3 pp 247ndash265 1999

[8] H R Oz and M Pakdemirli ldquoVibrations of an axially movingbeam with time-dependent velocityrdquo Journal of Sound andVibration vol 227 no 2 pp 239ndash257 1999

[9] H R Oz M Pakdemirli and H Boyaci ldquoNon-linear vibrationsand stability of an axially moving beam with time-dependent

velocityrdquo International Journal of Non-LinearMechanics vol 36no 1 pp 107ndash115 2001

[10] A H Nayfeh and D T Mook Nonlinear Oscillations WileyNew York NY USA 1979

[11] A H Nayfeh and B Balachandran Applied NonlinearDynamics-Analytical Computational and ExperimentalMethods John Wiley amp Sons New York NY USA 1994

[12] A H Nayfeh and B Balachandran ldquoModal interactions indynamical and structural systemsrdquo Applied Mechanics Reviewsvol 42 pp 175ndash201 1989

[13] C H Riedel and C A Tan ldquoCoupled forced response ofan axially moving strip with internal resonancerdquo InternationalJournal of Non-LinearMechanics vol 37 no 1 pp 101ndash116 2002

[14] E Ozkaya S M Bagdatli and H R Oz ldquoNonlinear transversevibrations and 3 1 internal resonances of a beam with multiplesupportsrdquo Journal of Vibration and Acoustics vol 130 no 2Article ID 021013 11 pages 2008

[15] S M Bagdatli H R Oz and E Ozkaya ldquoNon-linear transversevibrations and 3 1 internal resonances of a tensioned beam onmultiple supportsrdquo Mathematical and Computational Applica-tions vol 16 no 1 pp 203ndash215 2011

[16] C Chin and A H Nayfeh ldquoThree-to-one internal resonancesin parametrically excited hinged-clamped beamsrdquo NonlinearDynamics vol 20 no 2 pp 131ndash158 1999

[17] L N Panda and R C Kar ldquoNonlinear dynamics of a pipe con-veying pulsating fluidwith parametric and internal resonancesrdquoNonlinear Dynamics vol 49 no 1-2 pp 9ndash30 2007

[18] L N Panda and R C Kar ldquoNonlinear dynamics of a pipe con-veying pulsating fluid with combination principal parametricand internal resonancesrdquo Journal of Sound and Vibration vol309 no 3ndash5 pp 375ndash406 2008

[19] K Y Sze S H Chen and J L Huang ldquoThe incrementalharmonic balance method for nonlinear vibration of axiallymoving beamsrdquo Journal of Sound and Vibration vol 281 no 3ndash5 pp 611ndash626 2005

[20] J L Huang R K L Su W H Li and S H Chen ldquoStabilityand bifurcation of an axially moving beam tuned to three-to-one internal resonancesrdquo Journal of Sound and Vibration vol330 no 3 pp 471ndash485 2011

[21] L Chen Y Tang andCW Lim ldquoDynamic stability in paramet-ric resonance of axially accelerating viscoelastic Timoshenkobeamsrdquo Journal of Sound and Vibration vol 329 no 5 pp 547ndash565 2010

[22] HDing andLChen ldquoGalerkinmethods for natural frequenciesof high-speed axially moving beamsrdquo Journal of Sound andVibration vol 329 no 17 pp 3484ndash3494 2010

[23] H Ding G C Zhang and L Q Chen ldquoSupercritical equilib-rium solutions of axially moving beams with hybrid boundaryconditionsrdquoMechanics Research Communications vol 38 no 1pp 52ndash56 2011

[24] KMarynowski and T Kapitaniak ldquoKelvin-Voigt versus Burgersinternal damping in modeling of axially moving viscoelasticwebrdquo International Journal of Non-Linear Mechanics vol 37 no7 pp 1147ndash1161 2002

[25] K Marynowski ldquoNon-linear vibrations of an axially movingviscoelastic web with time-dependent tensionrdquo Chaos Solitonsand Fractals vol 21 no 2 pp 481ndash490 2004

[26] K Marynowski and T Kapitaniak ldquoZener internal damp-ing in modelling of axially moving viscoelastic beam withtime-dependent tensionrdquo International Journal of Non-LinearMechanics vol 42 no 1 pp 118ndash131 2007


[27] M Pakdemirli and H R Oz ldquoInfinite mode analysis andtruncation to resonant modes of axially accelerated beamvibrationsrdquo Journal of Sound and Vibration vol 311 no 3ndash5 pp1052ndash1074 2008

[28] S V Ponomareva and W T van Horssen ldquoOn the transversalvibrations of an axially moving continuum with a time-varyingvelocity transient from string to beam behaviorrdquo Journal ofSound and Vibration vol 325 no 4-5 pp 959ndash973 2009

[29] M H Ghayesh ldquoNonlinear forced dynamics of an axially mov-ing viscoelastic beam with an internal resonancerdquo InternationalJournal ofMechanical Sciences vol 53 no 11 pp 1022ndash1037 2011

[30] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[31] M H Ghayesh ldquoCoupled longitudinal-transverse dynamics ofan axially accelerating beamrdquo Journal of Sound and Vibrationvol 331 pp 5107ndash5124 2012

[32] M H Ghayesh ldquoSubharmonic dynamics of an axially accelerat-ing beamrdquo Archive of Applied Mechanics vol 82 pp 1169ndash11812012

[33] M H Ghayesh and M Amabili ldquoSteady-state transverseresponse of an axially moving beam with time-dependent axialspeedrdquo International Journal of Non-Linear Mechanics vol 49pp 40ndash49 2013

[34] G Chakraborty and A K Mallik ldquoStability of an acceleratingbeamrdquo Journal of Sound and Vibration vol 27 no 2 pp 309ndash320 1999

[35] M P Paidoussis ldquoFlutter of conservative systems of pipe con-veying incompressible fluidrdquo Journal of Mechanical Engineeringand Science vol 17 no 1 pp 19ndash25 1975

[36] M Pakdemirli and H Boyaci ldquoComparision of direct pertur-bation methods with discretization-perturbation methods fornonlinear vibrationsrdquo Journal of Sound and Vibration vol 186pp 837ndash845 1985

[37] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



Nayfeh [16] investigated three-to-one internal resonances inparametrically excited hinged-clamped beams They studiedprincipal parametric resonance of the first or the secondmode and a combination parametric resonance of additivetype of these modes Panda and Kar [17 18] investigatedthe principal parametric resonance of first mode secondmode and combination parametric resonance in presenceof 3 1 internal resonance of a pipe conveying pulsatingfluid with pinned-pinned end conditions They studied theequilibria of these modulated equations and determinedtheir stability and then they did extensive dynamic studyat typical initial conditions Sze et al [19] used incremen-tal harmonic balance method for nonlinear vibration ofaxially moving beams They investigated the fundamentalsuperharmonic and subharmonic resonance in presenceof internal resonance Huang et al [20] used the samemethod to analyse the stability and bifurcation analysis of atraveling beam tuned to 3 1 internal resonancewith attentionto fundamental and subharmonic resonances Chen et al[21] investigated dynamic stability of an axially acceleratingviscoelastic beam undergoing parametric resonance Dinget al [22 23] used Galerkin methods for finding naturalfrequencies of high-speed axially moving beams with hybridboundary conditions Marynowski and Kapitaniak [24ndash26]introduced several internal dissipation mechanisms likeKelvin-voigt and Zener in the modeling traveling continuaPakdemirli and Oz [27] studied the transverse vibration ofsimply supported axially moving Euler-Bernoulli beam forinfinite mode analysis and truncation to resonant modesPonomareva and van Horssen [28] investigated transversalvibrations of axially travelling continua based on a stringmodel at the low frequencies and a tensioned beam modelat the higher frequencies Recently a systematic researchon travelling beam was pursued by Ghayesh et al [29ndash33]involving nonlinear dynamic phenomenon of a variety ofsystem models The forced dynamics of an axially movingviscoelastic beam was investigated in [29] The nonlinearvibrations and stability of an axially moving beam subjectedto a distributed harmonic excitation load were investigatedin [30] In addition the response of the system tunedto a three-to-one internal resonance was also examinedSubcritical dynamics of the system was examined [31] viathe pseudoarclength continuation technique while the globaldynamics was investigated using direct time integration

The present work aims to investigate the problem of asimply supported traveling beamunder parametric excitationdue to the harmonic pulsation in the travelling velocity Fora certain range of mean velocity the natural frequency ofthe second mode is approximately three times that of thefirst mode This relationship between natural frequencies offirst two modes of a system having cubic nonlinearity resultsin a condition of three-to-one autoparametric or internalresonance leading to energy exchange between the twoinvolved modes through nonlinear modal interaction In thepresent work the principal parametric resonance of the firstmode is considered The system behavior shows pitchforkHopf and saddle node bifurcations in steady state analysisDecreasing internal frequency detuning parameter affects theamplitude of directly excited first mode and the number of

(t)w

x

Figure 1 Schematic diagram of an axially traveling simply sup-ported beam with variable velocity

Hopf bifurcation points and shifts the occurrence of jumpphenomena Due to the influence of internal resonance thedynamic behavior of the system exhibits periodic quasiperi-odic and chaotic responses The results are presented in theform of time histories phase plane portraits fast Fouriertransforms (FFTs) and Poincare maps

2 Formulation of the Problem

For the present work a uniform horizontal beam simplysupported at both ends and travelling with a harmonicallyvariable velocity (Figure 1) is considered The assumptionstaken here are (1) the motion of the beam is planar (2) theuniform cross sections remain plane during the motion andthe beam behaves like an Euler-Bernoulli beam in transversevibration and (3) the type of nonlinearity is geometricdue to the midplane stretching effect of the beam Thenondimensional equation of transverse motion of the beamincluding the nonlinearity due to midplane stretching [5 9]and viscous damping [34] along with viscoelastic damping[35] is given by

+ 2V1015840 + V1199081015840 + (V2 minus 1)119908

10158401015840+ V2119891119908

1015840101584010158401015840

+ 2120576120572

1015840101584010158401015840+ 2120576120583 =

1

2

V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(1)

The nondimensional scheme used here is

119909 =

119909

lowast

119871

119905 = 119905

lowastradic

119875

120588119860119871

2 119908 =

119908

lowast

119871

V =

Vlowast

radic119875120588119860

2120576120572 =

119864

lowast

119871

2(

119868

119898119864

)

12

2120576120583 =

119862119871

2

radic119898119864119868

V119897=

radic

119864119860

119875

V119891

=

radic

119864119868

119875119871

2

(2)

where the variables with asterisk denote dimensional onesIn the EOM dot denotes derivatives with respect to time(119905) and the prime denotes derivatives with respect to spatialderivative (119909) 119898 is mass per unit length 120588 is density 119860

is cross-sectional area 119871 is length V119891is nondimensional

flexural stiffness and V119897is nondimensional longitudinal

stiffness of beam 119864lowast is the coefficient of internal dissipationof the beam material which is assumed to be Kelvin-Voigttype viscoelastic and 119862 is the external damping factor 120572 isnondimensional material damping and 120583 is nondimensionalviscous damping Reordering the transverse displacementwith the relation 119908 = radic120576119908

where 120576 lt 1 and putting itin the equation of motion (1) the system is converted into aweakly nonlinear one [9]



05

1

15

2

25

3

35

4

45

5

(1 0)

(0 002)

(0 003)

(0 004)

1205902

1

(120583 = 0001 120572 = 0)



+ 2V1015840 + V1199081015840 + (V2 minus 1)119908

10158401015840+ V2119891119908

1015840101584010158401015840

+ 2120576120572

1015840101584010158401015840+ 2120576120583 =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(3)


V = V0+ 120576V1sinΩ119905 (4)




2120576120572

1015840101584010158401015840+ V2119891119908

1015840101584010158401015840+ [V20+ 2120576V0V1sinΩ119905 minus 1]119908

10158401015840

+ 120576V1Ω119908

1015840 cosΩ119905 + 2 (V0+ 120576V1sinΩ119905)

1015840

+ 2120576120583 + =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(5)


119908 (0 119905) = 119908 (1 119905) = 119908

10158401015840

(0 119905) = 119908

10158401015840

(1 119905) = 0 (6)




119899= 120576

119899119905 119899 = 0 1 2 3 and the time

derivatives are

119889

119889119905

= 119863

0+ 120576119863

1+ sdot sdot sdot

119889

2

119889119905

2= 119863

2

0+ 2120576119863

0119863

1+ sdot sdot sdot

119863

119899=

120597

120597119879

119899

119899 = 0 1 2 3

(7)


119908 (119909 119905 120576) = 119908

0(119909 119879

0 119879

1) + 120576119908

1(119909 119879

0 119879



119874(120576

0) 119863

2

0119908

0+ 2V0119863

0119908

1015840

0+ (V20minus 1)119908

10158401015840

0+ V2119891119908

1015840101584010158401015840

0= 0

119908

0(0 119905) = 119908

0(1 119905) = 119908

10158401015840

0(0 119905) = 119908

10158401015840

0(1 119905) = 0

(9)

119874(120576

1) 119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ V2119891119908

1015840101584010158401015840

1+ (V20minus 1)119908

10158401015840

1

= minus2V0119863

1119908

1015840

0minus 2V1sinΩ119905119863

0119908

1015840

0minus 2119863

0119863

1119908

0

minus 2120572119863

0119908

1015840101584010158401015840

0minus 2120583119863

0119908

0minus 2V0V1sinΩ119905119908

10158401015840

0


1015840

0+

1

2

V2119897119908

10158401015840

0int

1

0

119908

10158402

0119889119909 = 0

119908

1(0 119905) = 119908

1(1 119905) = 119908

10158401015840

1(0 119905) = 119908

10158401015840

1(1 119905) = 0

(10)


119908

0(119879

0 119879

1 119909) =

infin

sum

119899=1

120601

119899(119909) 119860

119899(119879

1) 119890

1198941205961198991198790

+ 119888119888 (11)

where 120601




120601

119899(119909) = 119862

1119899119890

1198941205731119899119909minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

119890

1198941205732119899119909

minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205733119899119909

+ [minus1 +

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

+

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205734119899119909]

(12)


minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2

H1

H2

A1

(a)

minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2 H1 H2

A2

(b)


minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006

SN

H1

H2

H3

H4

H5H7H6



1205902

A1

(a)

minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006SN

H1

H2

H3

H4H5

H7H6



1205902

A2

(b)


1= 10 V

119897= 40 and 120590

1= 9239


V2119891120573

4


2

in minus 2V0120596

119899120573in minus 120596

2

119899= 0

119894 = 1 2 3 4

(13)

(119890

119894(1205731119899+1205732119899)+ 119890

119894(1205733119899+1205734119899)) (120573

2

1119899minus 120573

2

2119899) (120573

2

3119899minus 120573

2

4119899)

+ (119890

119894(1205731119899+1205733119899)+ 119890

119894(1205732119899+1205734119899)) (120573

2

2119899minus 120573

2

4119899)

times (120573

2

3119899minus 120573

2

1119899) + (119890

119894(1205732119899+1205733119899)+ 119890

119894(1205731119899+1205734119899))

times (120573

2

1119899minus 120573

2

4119899) (120573

2

2119899minus 120573

2

3119899) = 0

(14)





minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H3

H4H5

H2

A1

(a)

0 100 200 300 400 5000

001

002

003

004

005

006

007

008

SN

1205902

H1

H3

H4H5

H2

A2

(b)


1= 15 V

119897= 40 and 120590

1= 9239

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4H5H6

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5H6

H2A2

(b)


1= 10 V

119897= 40 and120590

1= 9239


2asymp 3120596






119908

0(119879

0 119879

1 119909) = 119860

1(119879

1) 120601

1(119909) 119890

11989412059611198790

+ 119860

2(119879

1) 120601

2(119909) 119890

11989412059621198790

+ 119888119888

(15)


120596

2= 3120596

1+ 120576120590

1 Ω = 2120596

1+ 120576120590

2 (16)

where 120590

1and 120590



2minus 120596

1+ 120576(120590

2minus 120590

1) a combination


SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)


1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680


119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ




2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)


1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890






0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

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

4

45

5

(1 0)

(0 002)

(0 003)

(0 004)

1205902

1

(120583 = 0001 120572 = 0)



+ 2V1015840 + V1199081015840 + (V2 minus 1)119908

10158401015840+ V2119891119908

1015840101584010158401015840

+ 2120576120572

1015840101584010158401015840+ 2120576120583 =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(3)


V = V0+ 120576V1sinΩ119905 (4)




2120576120572

1015840101584010158401015840+ V2119891119908

1015840101584010158401015840+ [V20+ 2120576V0V1sinΩ119905 minus 1]119908

10158401015840

+ 120576V1Ω119908

1015840 cosΩ119905 + 2 (V0+ 120576V1sinΩ119905)

1015840

+ 2120576120583 + =

1

2

120576V2119897119908

10158401015840int

1

0

119908

10158402119889119909

(5)


119908 (0 119905) = 119908 (1 119905) = 119908

10158401015840

(0 119905) = 119908

10158401015840

(1 119905) = 0 (6)




119899= 120576

119899119905 119899 = 0 1 2 3 and the time

derivatives are

119889

119889119905

= 119863

0+ 120576119863

1+ sdot sdot sdot

119889

2

119889119905

2= 119863

2

0+ 2120576119863

0119863

1+ sdot sdot sdot

119863

119899=

120597

120597119879

119899

119899 = 0 1 2 3

(7)


119908 (119909 119905 120576) = 119908

0(119909 119879

0 119879

1) + 120576119908

1(119909 119879

0 119879



119874(120576

0) 119863

2

0119908

0+ 2V0119863

0119908

1015840

0+ (V20minus 1)119908

10158401015840

0+ V2119891119908

1015840101584010158401015840

0= 0

119908

0(0 119905) = 119908

0(1 119905) = 119908

10158401015840

0(0 119905) = 119908

10158401015840

0(1 119905) = 0

(9)

119874(120576

1) 119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ V2119891119908

1015840101584010158401015840

1+ (V20minus 1)119908

10158401015840

1

= minus2V0119863

1119908

1015840

0minus 2V1sinΩ119905119863

0119908

1015840

0minus 2119863

0119863

1119908

0

minus 2120572119863

0119908

1015840101584010158401015840

0minus 2120583119863

0119908

0minus 2V0V1sinΩ119905119908

10158401015840

0


1015840

0+

1

2

V2119897119908

10158401015840

0int

1

0

119908

10158402

0119889119909 = 0

119908

1(0 119905) = 119908

1(1 119905) = 119908

10158401015840

1(0 119905) = 119908

10158401015840

1(1 119905) = 0

(10)


119908

0(119879

0 119879

1 119909) =

infin

sum

119899=1

120601

119899(119909) 119860

119899(119879

1) 119890

1198941205961198991198790

+ 119888119888 (11)

where 120601




120601

119899(119909) = 119862

1119899119890

1198941205731119899119909minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

119890

1198941205732119899119909

minus

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205733119899119909

+ [minus1 +

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205733119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

2119899) (119890

1198941205733119899

minus 119890

1198941205732119899)

+

(120573

2

4119899minus 120573

2

1119899) (119890

1198941205732119899

minus 119890

1198941205731119899)

(120573

2

4119899minus 120573

2

3119899) (119890

1198941205732119899

minus 119890

1198941205733119899)

119890

1198941205734119899119909]

(12)


minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2

H1

H2

A1

(a)

minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2 H1 H2

A2

(b)


minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006

SN

H1

H2

H3

H4

H5H7H6



1205902

A1

(a)

minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006SN

H1

H2

H3

H4H5

H7H6



1205902

A2

(b)


1= 10 V

119897= 40 and 120590

1= 9239


V2119891120573

4


2

in minus 2V0120596

119899120573in minus 120596

2

119899= 0

119894 = 1 2 3 4

(13)

(119890

119894(1205731119899+1205732119899)+ 119890

119894(1205733119899+1205734119899)) (120573

2

1119899minus 120573

2

2119899) (120573

2

3119899minus 120573

2

4119899)

+ (119890

119894(1205731119899+1205733119899)+ 119890

119894(1205732119899+1205734119899)) (120573

2

2119899minus 120573

2

4119899)

times (120573

2

3119899minus 120573

2

1119899) + (119890

119894(1205732119899+1205733119899)+ 119890

119894(1205731119899+1205734119899))

times (120573

2

1119899minus 120573

2

4119899) (120573

2

2119899minus 120573

2

3119899) = 0

(14)





minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H3

H4H5

H2

A1

(a)

0 100 200 300 400 5000

001

002

003

004

005

006

007

008

SN

1205902

H1

H3

H4H5

H2

A2

(b)


1= 15 V

119897= 40 and 120590

1= 9239

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4H5H6

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5H6

H2A2

(b)


1= 10 V

119897= 40 and120590

1= 9239


2asymp 3120596






119908

0(119879

0 119879

1 119909) = 119860

1(119879

1) 120601

1(119909) 119890

11989412059611198790

+ 119860

2(119879

1) 120601

2(119909) 119890

11989412059621198790

+ 119888119888

(15)


120596

2= 3120596

1+ 120576120590

1 Ω = 2120596

1+ 120576120590

2 (16)

where 120590

1and 120590



2minus 120596

1+ 120576(120590

2minus 120590

1) a combination


SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)


1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680


119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ




2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)


1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890






0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2

H1

H2

A1

(a)

minus50 0 50 100 150 2000

1

2

3

4

5

6



1205902

SN3

SN1

SN2 H1 H2

A2

(b)


minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006

SN

H1

H2

H3

H4

H5H7H6



1205902

A1

(a)

minus50 0 50 100 150 200 250 300 3500

001

002

003

004

005

006SN

H1

H2

H3

H4H5

H7H6



1205902

A2

(b)


1= 10 V

119897= 40 and 120590

1= 9239


V2119891120573

4


2

in minus 2V0120596

119899120573in minus 120596

2

119899= 0

119894 = 1 2 3 4

(13)

(119890

119894(1205731119899+1205732119899)+ 119890

119894(1205733119899+1205734119899)) (120573

2

1119899minus 120573

2

2119899) (120573

2

3119899minus 120573

2

4119899)

+ (119890

119894(1205731119899+1205733119899)+ 119890

119894(1205732119899+1205734119899)) (120573

2

2119899minus 120573

2

4119899)

times (120573

2

3119899minus 120573

2

1119899) + (119890

119894(1205732119899+1205733119899)+ 119890

119894(1205731119899+1205734119899))

times (120573

2

1119899minus 120573

2

4119899) (120573

2

2119899minus 120573

2

3119899) = 0

(14)





minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H3

H4H5

H2

A1

(a)

0 100 200 300 400 5000

001

002

003

004

005

006

007

008

SN

1205902

H1

H3

H4H5

H2

A2

(b)


1= 15 V

119897= 40 and 120590

1= 9239

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4H5H6

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5H6

H2A2

(b)


1= 10 V

119897= 40 and120590

1= 9239


2asymp 3120596






119908

0(119879

0 119879

1 119909) = 119860

1(119879

1) 120601

1(119909) 119890

11989412059611198790

+ 119860

2(119879

1) 120601

2(119909) 119890

11989412059621198790

+ 119888119888

(15)


120596

2= 3120596

1+ 120576120590

1 Ω = 2120596

1+ 120576120590

2 (16)

where 120590

1and 120590



2minus 120596

1+ 120576(120590

2minus 120590

1) a combination


SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)


1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680


119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ




2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)


1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890






0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H3

H4H5

H2

A1

(a)

0 100 200 300 400 5000

001

002

003

004

005

006

007

008

SN

1205902

H1

H3

H4H5

H2

A2

(b)


1= 15 V

119897= 40 and 120590

1= 9239

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4H5H6

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5H6

H2A2

(b)


1= 10 V

119897= 40 and120590

1= 9239


2asymp 3120596






119908

0(119879

0 119879

1 119909) = 119860

1(119879

1) 120601

1(119909) 119890

11989412059611198790

+ 119860

2(119879

1) 120601

2(119909) 119890

11989412059621198790

+ 119888119888

(15)


120596

2= 3120596

1+ 120576120590

1 Ω = 2120596

1+ 120576120590

2 (16)

where 120590

1and 120590



2minus 120596

1+ 120576(120590

2minus 120590

1) a combination


SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)


1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680


119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ




2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)


1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890






0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3 H4

H5

H2

A1

(a)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1

H3

H4

H5

H2

A2

(b)

SN

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006

H1H3

H2

A1

(c)

minus50 0 50 100 150 200 250 300 3501205902

0

001

002

003

004

005

006SN

H1

H3 H2

A2

(d)


1= 10 V

119897= 40 (a b) 120590

1= 20320 and (c d) 120590

1= minus27680


119863

2

0119908

1+ 2V0119863

0119908

1015840

1+ (V20minus 1)119908

10158401015840

1+ V2119891119908

1015840101584010158401015840

1

= Γ

1119890

11989412059611198790

+ Γ

2119890

119894(12059611198790+12059011198791)+ Γ

3119890

11989412059611198790

+ Γ

4119890

119894(12059611198790+12059011198791minus12059021198791)+ Γ

5119890

11989412059621198790

+ Γ

6119890

119894(12059621198790minus12059011198791)+ Γ

7119890

119894(12059621198790+12059021198791minus12059011198791)+ 119888119888 + NST

(17)

where the terms Γ




2119860

1015840

1+ 8119878

1119860

2

1119860

1+ 8119878

2119860

1119860

2119860

2+ 8119892

1119860

2

1119860

2119890

11989412059011198791

+ 2120583 119862

1119860

1+ 2120572119890

1119860

1+ 2119870

1119860

1119890

11989412059021198791

+ 2119870

2119860

2119890

119894(1205901minus1205902)1198791

= 0

2119860

1015840

2+ 8119878

4119860

2

2119860

2+ 8119878

3119860

1119860

2119860

1+ 8119892

2119860

3

1119890

minus11989412059011198791

+ 2120583119862

2119860

2+ 2120572119890

2119860

2+ 2119870

3119860

1119890

119894(1205902minus1205901)1198791

= 0

(18)


1and 119878

119894 119892

119894 119870

119894 119862

119894 and 119890






0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

001

002

003

004

005

006

0 100 200 300 400 5001205902

SNH1

H2

H3

H4

H5

H7

H6

A1

(a)

minus50 0 50 100 150 200 250 300 350 400 450 5000

001

002

003

004

005

006

1205902

SN

H1

H2

H3

H4H5

H7

H6

A2

(b)


V119897= 40 and 120590

1= 9239

69 7 71 720021

0021

00211

00211

00212

q1

p1

times10minus3

(a)


minus68

minus66

minus64

q2

p2

times10minus4

times10minus3

(b)

9985 999 9995 10000021

0021

00211

00211

00212

t

p1

(c)

9985 999 9995 1000t

minus7

minus68

minus66

minus64

p2

times10minus3

(d)


2= 682799 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239


00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00214

00216

00218

0022

65 7 75 8q1

p1

times10minus3

(a)


minus82

minus81

minus8

minus79

q2

p2

times10minus4

times10minus3

(b)

990 995 100000214

00216

00218

0022

t

p1

(c)

990 995 1000t

minus83

minus82

minus81

minus8

minus79

p2

times10minus3

(d)


1= 10 120572 = 0 and 120590

1= 9239



119860

119899=

1

2

[119901

119899(119879

1) minus 119894119902

119899(119879

1)] 119890

119894120582119899(1198791) 119899 = 1 2 (19)


119901

1015840

1= minus 120599

1119902

1minus 119878

1119877(119901

3

1+ 119901

1119902

2

1) minus 119878

1119868(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119901

1119901

2

2+ 119901

1119902

2

2) minus 119878

2119868(119902

1119901

2

2+ 119902

1119902

2

2)

minus 119892

1119877(119901

2

1119901

2minus 119901

2119902

2

1+ 2119901

1119902

1119902

2)

+ 119892

1119868(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2) minus 120583119862

1119877119901

1

minus 120583119862

1119868119902

1minus 120572119890

1119877119901

1minus 120572119890

1119868119902

1minus 119870

1119877119901

1

+ 119870

1119868119902

1minus 119870

2119877119901

2minus 119870

2119868119902

2

119902

1015840

1= 120599

1119901

1+ 119878

1119868(119901

3

1+ 119901

1119902

2

1) minus 119878

1119877(119901

2

1119902

1+ 119902

3

1)

minus 119878

2119877(119902

1119901

2

2+ 119902

1119902

2

2) + 119878

2119868(119901

1119901

2

2+ 119901

1119902

2

2)

+ 119892

1119877(2119901

1119902

1119901

2minus 119901

2

1119902

2+ 119902

2

1119902

2)

+ 119892

1119868(2119901

1119902

1119902

2+ 119901

2

1119901

2minus 119901

2119902

2

1) + 119870

1119877119902

1+ 119870

1119868119901

1

minus 119870

2119877119902

2+ 119870

2119868119901

2minus 120583119862

1119877119902

1+ 120583119862

1119868119901

1

minus 120572119890

1119877119902

1+ 120572119890

1119868119901

1

119901

1015840

2= minus 120599

2119902

2minus 119878

4119877(119901

3

2+ 119901

2119902

2

2) minus 119878

4119868(119902

3

2+ 119901

2

2119902

2)

minus 119878

3119877(119901

2

1119901

2+ 119901

2119902

2

1) minus 119878

3119868(119901

2

1119902

2+ 119902

2

1119902

2)


minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus1 0 1 2 38

82

84

86

times10minus4

times10minus3

q2

p2

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus75

minus7


q1

p1

(c)

828 83 832 834minus1

0

1

2

3q2

p2

times10minus4

times10minus3

(d)


1= 10 and 120572 = 0 120590

1= 9239

minus 119892

2119877(119901

3

1minus 3119901

1119902

2

1) + 119892

2119868(119902

3

1minus 3119901

2

1119902

1)

minus 119870

3119877119901

1minus 119870

3119868119902

1minus 120583119862

2119877119901

2minus 120583119862

2119868119902

2

minus 120572119890

2119877119901

2minus 120572119890

2119868119902

2

119902

1015840

2= minus 120599

2119901

2minus 119878

4119877(119902

3

2+ 119901

2

2119902

2) + 119878

4119868(119901

3

2+ 119901

2119902

2

2)

minus 119878

3119877(119901

2

1119902

2+ 119902

2

1119902

2) + 119878

3119868(119901

2

1119901

2+ 119901

2119902

2

1)

+ 119892

2119877(119902

3

1minus 3119901

2

1119902

1) + 119892

2119868(119901

3

1minus 3119901

1119902

2

1)

minus 119870

3119877119902

1+ 119870

3119868119901

1minus 120583119862

2119877119902

2+ 120583119862

2119868119901

2

minus 120572119890

2119877119902

2+ 120572119890

2119868119901

2

(20)

where

120599

1= 05120590

2 120599

2= 15120590

2minus 120590

1 (21)


Δ119901

1015840

1Δ119902

1015840

1Δ119901

1015840

2Δ119902

1015840

2

119879

= [119869

119888] Δ119901

1Δ119902

1Δ119901

2Δ119902

2

119879

(22)




1= 119902

1= 119901

2= 119902



1015840

1= 119902

1015840

1=

119901

1015840

2= 119902

1015840



6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6 7 8 9 100021

00215

0022

00225

0023

q1

p1

times10minus3

(a)


minus9

minus85

minus8

q2

p2

times10minus3

times10minus5

(b)

0021 00215 0022 00225 00236

7

8

9

10times10minus3

q1

p1

(c)


minus10

minus5

0

5

times10minus3

times10minus5

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239





V119891







1)


= 02 V119897

= 40 V0

= 07120596

1= 27388 and 120596







0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

5

q1

p1

times10minus5

times10minus4

(a)


minus1

0

1

2times10minus4

times10minus4q2

p2

(b)

160 170 180 190 200minus004

minus002

0

002

004

t

p1

(c)

160 180 200minus004

minus002

0

002

004

t

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



1015840

119894= 119902

1015840




2and V

1axes respectively



for 120583 = 01 120572 = 0 V1

= 10 V119897

= 40 and 120590

1= 9239






unstable at 1198671(120590




minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus002

minus001

0

q1

p1

(a)

minus001 0 0010015

002

0025

003

q2

p2

(b)

997 998 999 1000minus003

minus0025

minus002

minus0015

minus001

minus0005

t

p1

(c)

0018 002 0022 0024minus001

minus0005

0

0005

001

p2

q2

(d)


1= 10 and 120572 = 0 120590

1= 9239


2 the same




2 it





2=


3(120590



4(120590



5(120590



6(120590

2= 49474) and 119867

7(120590

2= 44608)



2= minus22071



0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0015

002

0025

003

minus002 minus001 0 001 002q1

p1

(a)

0 20 40 60 80minus100

0

100

200

f (rads)

psd

(dB)

(b)


minus001

0

001

q1

p1

(c)

0015 002 0025 003minus002

minus001

0

001

002

q2

p2

(d)


1= 10 120572 = 0 and 120590

1= 9239



considered are 120583 = 01 120572 = 0 V1

= 15 V119897

= 40 and120590





2= 4577476)




1= 10 V

119897=

40 and 120590






1= 9239


1= minus27680








minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus002

minus001

0

001

002

minus002 minus001 0 002001q1

p1

(a)


minus002

0

002

004

q2

p2

(b)

999 9995 1000minus002

minus001

0

001

002

t

p1

(c)

0 20 40 60 800

50

100

150

f (rads)

psd

(dB)

(d)


2= 2805201 120583 = 005 V

1= 10 120572 = 0 and 120590

1= 9239



2= 682799


= 10 and 120590



1= 0002 119902

1= 00007

119901

2= 00067 and 119902









2= 1157201 120583 = 005 120572 = 0 V

1= 10 and 120590

1= 9239


2=





119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119891=

02

V0

120596

1120596

23120596

1120576120590

1= 120596

2minus 3120596

1(120576120590

1120596

1)


120590




2= 2805201 the




6 Conclusions






Appendix


Γ

1= minus 2119894120596

1119860

1015840

1120601

1minus 2V0119860

1015840

1120601

1015840

1minus 2119894120583120596

1119860

1120601

1

minus 2119894120572120596

1119860

1120601

1015840101584010158401015840

1+

1

2

V2119897

times 2119860

2

1119860

1120601

10158401015840

1int

1

0

120601

1015840

1120601

1015840

1119889119909 + 119860

2

1119860

1120601

10158401015840

1

times int

1

0

120601

10158402

1119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

1

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

2119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

2119889119909

Γ

2=

1

2

V21198972119860

2

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

+119860

2

1119860

2120601

10158401015840

2int

1

0

120601

10158402

1119889119909

Γ

3= 119860

1V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894V0V1120601

10158401015840

1



1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1= 40 and 120590

1= 9239

119901

1119902

1119901

2119902

2120590

2Bifurcation point


1

minus0002453 minus0 004177 0 039646 0 039917 301873000 SN0032196 0010862 0005854 0000026 190033742 119867

2

0026628 0008991 0010665 0000036 144502843 119867

3

0022130 0007495 0015112 0000039 127980336 119867

4

0004908 0001822 0013291 0000245 80408029 119867

5

0000000 0000000 0000000 0000000 49474028 119867

6

0000000 0000000 0000000 0000000 44608834 119867

7

Γ

4= 119860

2V1120596

2120601

1015840

2minus

V1Ω

2

120601

1015840

2minus 119894 V0V1120601

10158401015840

2

Γ

5= minus 2 119894120596

2119860

1015840

2120601

2minus 2V0119860

1015840

2120601

1015840

2minus 2120583119894120596

2119860

2120601

2

minus 2120572119894120596

2119860

2120601

1015840101584010158401015840

2+

1

2

V2119897

times 119860

2

2119860

2120601

10158401015840

2int

1

0

120601

10158402

2119889119909

+ 2119860

1119860

1119860

2120601

10158401015840

2int

1

0

120601

1015840

1120601

1015840

1119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1

times int

1

0

120601

1015840

1120601

1015840

2119889119909 + 2119860

2

2119860

2120601

10158401015840

2

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + 2119860

1119860

1119860

2120601

10158401015840

1int

1

0

120601

1015840

2120601

1015840

1119889119909

Γ

6=

1

2

V2119897119860

3

1120601

10158401015840

1int

1

0

120601

10158402

1119889119909

Γ

7= 119860

1minus V1120596

1120601

1015840

1minus

V1Ω

2

120601

1015840

1+ 119894 V0V1120601

10158401015840

1

119878

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909

+int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

2= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

1119889119909

timesint

1

0

120601

1015840

2120601

1015840

2119889119909 + int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

1015840

1120601

1015840

2119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119878

3= (

1

8

V2119897int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

1120601

1015840

1119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909

timesint

1

0

120601

1015840

1120601

1015840

2119889119909 + int

1

0

120601

10158401015840

1120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119878

4= (

1

16

V21198972int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

1015840

2120601

1015840

2119889119909

+int

1

0

120601

10158401015840

2120601

2119889119909int

1

0

120601

10158402

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

119862

1=

minus119894120596

1int

1

0120601

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119862

2=

minus119894120596

2int

1

0120601

2120601

2119889119909

minus119894120596

2int

1

0120601

2120601

2119889119909 + V

0

1

int

0

120601

1015840

2120601

2119889119909

119890

1=

minus119894120596

1int

1

0120601

1015840101584010158401015840

1120601

1119889119909

minus 119894120596

1int

1

0120601

1120601

1119889119909 + V

0int

1

0120601

1015840

1120601

1119889119909

119890

2=

minus119894120596

2int

1

0120601

1015840101584010158401015840

2120601

2119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119892

1= (

1

16

V21198972int

1

0

120601

10158401015840

1120601

1119889119909int

1

0

120601

1015840

2120601

1015840

1119889119909

+int

1

0

120601

10158401015840

2120601

1119889119909int

1

0

120601

10158402

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1


119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119892

2=

(116) V119897

2int

1

0120601

10158401015840

1120601

2119889119909int

1

0120601

10158402

1119889119909

minus 119894120596

2int

1

0120601

2120601

2119889119909 + V

0int

1

0120601

1015840

2120601

2119889119909

119870

1= (

1

2

V1120596

1int

1

0

120601

1015840

1120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601

1119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

2= (

1

2

V1120596

2int

1

0

120601

1015840

2120601

1119889119909 minus

V1Ω

2

int

1

0

120601

1015840

2120601

1119889119909

minus119894V0V1int

1

0

120601

10158401015840

2120601

1119889119909)


1int

1

0

120601

1120601

1119889119909 + V

0int

1

0

120601

1015840

1120601

1119889119909)

minus1

119870

3= (

1

2

minusV1120596

2int

1

0

120601

1015840

1120601

2119889119909 minus

V1Ω

2

int

1

0

120601

1015840

1120601119889119909

+119894V0V1int

1

0

120601

10158401015840

1120601

2119889119909)


2int

1

0

120601

2120601

2119889119909 + V

0int

1

0

120601

1015840

2120601

2119889119909)

minus1

(A1)

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









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