Research Article Theoretical Vibration Analysis on...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013 Article ID 512674 11 pageshttpdxdoiorg1011552013512674

Research ArticleTheoretical Vibration Analysis on 600 Wh Energy StorageFlywheel RotormdashActive Magnetic Bearing System

Jing-na Liu1 Zheng-yi Ren12 Shan-wei Wu1 and Yin-long Tang1

1 College of Mechanical and Electrical Engineering Harbin Engineering University Harbin 150001 China2 Engineering Training Center Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Jing-na Liu liujingna2003163com

Received 24 January 2013 Revised 16 April 2013 Accepted 16 April 2013

Academic Editor Zuohua Huang

Copyright copy 2013 Jing-na Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper shows a theoretical vibration analysis regarding the controllerrsquos parameters and the gyroscopic effect based on asimplified rotordynamic model Combined with 600Wh energy storage flywheel rotor system mathematical model the Campbelldiagram of the rotor system was obtained by the calculation of the whirl frequency under different parameters of the controller inMATLAB to analyze the effect of the controller parameter on the whirl frequency and to limit the operating speed and accelerationor deceleration of the rotor The result of the analysis can be used to set the support position of the rotor system limit the ratioof transverse moment of inertia and the polar moment of inertia and direct the flywheel prototype future design The presentedsimplified rotordynamic model can also be applied to rotating machines

1 Introduction

Later in the 1970s flywheel energy storage was proposed as aprimary objective for electric vehicles and stationary powerbackup [1] With the improvements in materials magneticbearing technology and power electronics flywheel energystorage technology has large developments Compared withtraditional battery energy storage system flywheel energystorage system has many advantages such as higher energystorage density higher specific power and power densitylower risk of overcharge or overdischarge wide range of oper-ation temperature very long life cycle and environmentalfriendliness [2]

Many problems appear as the development of flywheelenergy storage and one of them is the bearing Besidesthe active magnetic bearing (AMB) implies that bearingforces are actively controlled by means of electromagnetsa well-designed closed control loop and other componentssuch as position sensors and power amplifiers [3] Thereforethe rotor of the AMB can be suspended to the predefinedpositions by the controlled electromagnetic forces withoutmechanical contact and friction between the magnetic bear-ing and the rotor [4] Based on the noncontact and friction-less characteristics the magnetic suspension of AMB offers

many practical and promising advantages over conventionalbearings such as longer life lower rotating frictional losseshigher rotational speed and elimination of the lubrication[5] Hence AMBs have been successfully and widely imple-mented in various high-performance applications includingthe rotating devices such as turbine engines [6] flywheelenergy and storage devices [7] bearingless motor [8] andvacuum pump [9] and the nonrotating devices such asmotion control stage [10] biomedical applications [11] andmanufacturing equipment [12] Furthermore the adjustablestiffness and damping characteristics make the AMB suitablefor elimination of vibration [13]

Many different kinds of excitations exist in rotor systemmdashfor example mechanical unbalance and misalignment of thecouplingmdashwhich may cause vibrations [14ndash18] Besides thesetypical excitations also specific excitations associated withthe type of the rotating machine occurThis paper focuses onvibrations of rotor system supported by active electromagnetbearing therefore also electromagnetic forces have to beconsidered which may cause vibrations by an eccentricityThe eccentricity can be divided into static eccentricity anddynamic eccentricity Static eccentricity is for examplecaused by production tolerances regarding concentricity and

2 International Journal of Rotating Machinery

Flywheel

RAMB1

Auxiliary bearing 1

TAMB

RAMB2

Motorgenerator

Auxiliary bearing 2

Shaft

119883

119884

119885

119874

Figure 1 Basic layout of a flywheel energy storage system

fitting tolerance between stator housing bearing housingand so on Dynamic eccentricity is caused if the rotor is bentor if the rotor core is eccentrically positioned on the rotorshaft and so on

The aim of this paper is to show a theoretical vibrationanalysis of the rotor system of 600Wh flywheel energystorage system based on a simplified rotordynamic modelThe result of the analysis can be used to set the supportposition of the rotor system limit the ratio of transversemoment of inertia and the polarmoment of inertia and directthe flywheel prototype future design

2 System Modeling

21 System Structure The basic layout of a flywheel energystorage system is depicted in Figure 1 that is the 600Whprototype system a rigid vertical rotor shaft with a rigidmotor and flywheel connected to shaft Flywheel energystorage system is a complex construction where energy isstoredmechanically and transferred to and from the flywheelby an integrated motorgenerator

Two radial active electromagnetic bearings (RAMBs)including upper and lower RAMBs (RAMB1 and RAMB2in Figure 1) and one thrust magnetic bearing (TAMB) arefixed on the platform to suspend and regulate the rotor inthe radial and axial DOF respectively The positions of therotor in five axes are defined as the displacements deviatedfrom the nominal air gapsTherefore two pairs of the perpen-dicular eddy-current position sensors are installed closely to

respective RAMBs to measure the respective radial positionsin 119909- and 119910-axes which are denoted as 119909

1and 119910

1of the upper

RAMB and 1199092and 119910

2of the lower RAMB Moreover one

eddy-current position sensor is installed closely to TAMB tomeasure the axial position in the 119911-axis which is denoted as119911 After the rotor positions in five axes are all measured andsent to the control core through the position signals line therequired electromagnet currents are generated from the drivesystem and they circulate the coils through the power lineTherefore the rotor can be regulated and stabilized in thecenters of the apertures of the two RAMBs and the thrust disccan be centered in the middle of the air gap of the TAMBrespectively

One 70 kg flywheel is mounted on the rotor whose posi-tion can be adjusted to modify the rotational characteristicsFurthermore one motorgenerator is equipped between thelower RAMB and flywheel and is responsible for the torquegeneration or electricity generation based on the rotationalspeed requirement However in the rotor system the rolling-element auxiliary bearings (auxiliary bearing 1 and auxiliarybearing 2 in Figure 1) are necessary to protect AMB statorsand stationary components along shaft in the even 119905 of AMBfailure or high transient loads Under normal operation ofAMBs the rotor maintains a positive clearance with auxiliarybearings which is less than a clearance with AMBs

22 AMB Modeling

221 Electromagnetic ForceModel Considering the diameterof the shaft and the loading capacity the RAMB with 8-pole legs was designed in the 600Wh prototype system Thestructural configuration of theRAMB is shown in Figure 2(a)including electromagnetic coils And the basic magneticbearing control loop is shown in Figure 2(b)

Taking the two pairs of poles in the 119910-axes of oneRAMB for example the electromagnetic force model will beestablished [19] As shown in Figure 2(b) the basic magneticbearing control loop includes differential driving positionsensor controller and power amplifier The included differ-ential driving is adopted in this study to obtain maximumrange of the force dynamic and good linearity of the controldynamic 120572 is the half of the angle between the two-pole legs1199100is the nominal air gaps of the RAMB in 119910-axes and 119910

1

is the deviation in 119910-axes So the upper available air gap iscalculated as (119910

0+ 1199101cos120572) and the lower one is calculated

as (1199100minus 1199101cos120572) 119894

0is the base current and the 119894

1199101is the

control current 1198940+1198941199101and 1198940minus1198941199101circulate the upper and the

lower coils in the 119910-axis respectively 1198911and 119891

2are the total

attractive electromagnetic forces in the 119910-axis which is givenas follows

1198911=

120583011986001198732

4

(1198940+ 1198941199101)

2

(1199100+ 1199101)2cos120572 (1)

1198912=

120583011986001198732

4

(1198940+ 1198941199101)

2

(1199100minus 1199101)2cos120572 (2)

International Journal of Rotating Machinery 3

Coils

Electromagnet

(a) Structural configuration of the RAMB with 8-polelegs

11991001199101

119884119900119900111989111198912

119883

120572

Poweramplifier

Position sensor

Controller

1198940 + 1198941199101

1198940 minus 1198941199101

+

+

+

1198941199101

1198940

1199100

1199101

minus

1199100 + 1199101 cos 120572

1199100 minus 1199101 cos 120572

(b) The basic magnetic bearing control loop and its elements

Figure 2 Structural configurations and the basic magnetic bearing control loop of the RAMB

where 1205830is the permeability 120583

0= 4120587 times 10

minus7Hm1198600is the

effective cross-sectional area of one electromagnet and 119873 isthe number of coils around the core

Additionally the total nonlinear attractive electromag-netic forces for the 119910-axis can be modeled as follows

119891119910= 1198911minus 1198912

=

119906011986001198732

4

[(

1198940+ 1198941199101

1199100+ 1199101

)

2

minus (

1198940minus 1198941199101

1199100minus 1199101

)

2

] cos120572(3)

Moreover by taking the Taylorrsquos expansions of (1) withrespect to its nominal operating position (119910

1= 0 119894

1199101=

0) the nonlinear electromagnetic forces can be representedby the following simplified linearized electromagnetic forcemodels

1198911199101cong 1198961198941198941+ 1198961199101199101 (4)

where 119896119910and 119896

119894are the displacement and current stiffness

parameters of the RAMB respectively and can be obtainedfrom the 119910-axis as follows

119896119910equiv

1205971198911199101(1199101 1198941199101)

1205971199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

= minus

4120583011986001198732cos21205721198942

0

1199103

0

119896119894equiv

1205971198911199101(1199101 1198941199101)

1205971198941199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

=

4120583011986001198732 cos120572119894

0

1199102

0

(5)

It is noted that since the coils in the 119909-axis and the 119910-axis arecirculated by the same bias currents 119894

0and the nominal air

gaps in the 119909-axis and the 119910-axis are also the same that is thedisplacement and current stiffness parameters 119896

1199091and 119896

1198941

obtained from the 119910-axis are the same as the ones obtainedfrom the 119909-axis (where the bearing has been assumed to beisotropic)

222 The Controller of the AMB The development of PIDcontrol has been around for 90 years and is still popu-lar for industries and academies nowadays [20] Due tothe complexity of the rotor vibration problem most engi-neers have turned to primitive control technologies such asthe proportional-integral-derivative (PID) controller whichpresently accounts for over 95 of all industrial controlapplications This is not because practicing engineers areunaware of recently developed control methods but becausethey find the advanced controllers difficult to tune and thatit requires years of training The overwhelming advantage inselecting PID over more advanced controllers is its ease ofuse with only three tuning parameters and applicability to avast range of plant models It is very important to reduce thecontroller complexity whenever possible and thus the overallcomplexity of the closed loop system

The block diagram of the total control system of the AMBis shown in Figure 3 which consists of electromagnet rotordisplacement sensor PID controller DA digital to analogconverter and AD analog to digital converter

The transfer function of the PID controller can be writtenas

119866 (119904) =

119880 (119904)

119864 (119904)

= 119870119901+

119870119894

119904

+

119870119889119904

1 + 119879119889119904

(6)

where 119870119901 119870119894 119870119889 and 119879

119889are the proportional parameter

the integral parameter the derivative parameter and the timeconstant of the PID controller respectively

Combined with the gains of the power amplifier andthe sensor the characteristic equation of the system can beobtained as follows

11988751199045+ 11988741199044+ 11988731199043+ 11988721199042+ 11988711199041= 0 (7)


PID controller Rotor

Position sensor

ElectromagnetPower

amplifier

AD

DA119878ref 119878119894 119891

Figure 3 Block diagram of the control system

where1198871= 119860119886119860119904119870119901119896119894minus 119896119909

1198872= 119896119894(119860119886119860119904119870119889+ 119860119886119860119904119870119901119879119889) minus 119896119909(119879119886+ 119879119889)

1198873= 119898 minus 119896

119909119879119889119879119886

1198874= 119898 (119879

119886+ 119879119889)

1198875= 119898119879119886119879119889

(8)

where 119896119909and 119896


parameters of the RAMB respectively 119860119886and 119879

119886are the

amplifiable parameter and the time constant of the poweramplifier respectively119860

119904and 119879

119904are the amplifiable parame-

ter and the time constant of the sensor respectivelyThe parameters 119870

119901 119870119889 and 119879

119889 of the controller can be

determined by the Routh stability criterion and the resultsare shown as follows (the detailed derivation is not listed inthis section)

0 lt 119879119886119879119889lt

119898

119896119909119909

(9)

(119870119889+ 119870119886119879119889) lt

119898 (119879119886+ 119879119889)

119860119886119860119904119879119886119879119889119896119894119909

(10)

119870119889gt (

1199042minus radic1199042

2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (11)

119870119889lt (

1199042+ radic1199042

2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (12)

119870119901ge (

119896119894119909

119860119886119860119904119896119909119909

+

(119898 minus 119896119909119909119879119886119879119889)

4119898119896119894119909119860119886119860119904119879119886119879119889

) (13)

119870119901gt

119896119909119909

119860119886119860119904119896119894119909

(14)

By the above determination the derivative time parameter119879119889can be determined firstly and then the appropriate value

can be selected similarly the ranges of the proportionalparameter 119870

119901and the differential parameter 119870

119889can also

be gained According to the effect of the actual control theintegral parameter119870

119894can be added to eliminate the lag error

The parameters of the bearing structure are set as 1205830=

4120587times10minus7 (Hm) and the initial bias current is selected as 119868

0=

015A Because of the air gap of the magnetic bearing 1198780=

3 times 10minus4m the scope of the actual changes of the rotor is in

the range of 0 to 06mm the change of the AD is in the rangeof 0 to 6000 unit and the equivalent gain119860

119904of the sensor can

be obtained as 1 times 107 unitmThe range of the differential time parameter 119870

119901deter-

mined by (9) is

0 lt 119879119889lt 08323 (15)

With 119879119889= 00001 the range of the proportional parameter

determined by (13) (14) and (9) can be carried out as

0154 lt 119870119901lt 4578 (16)

According to the optimal linear results the proportionalparameter119870

119901should be selected as119870

119901= 1 And the range of

the differential parameter119870119889determined by (11) and (12) can

be given as

5009 times 10minus5lt 119870119889lt 1921 times 10

minus1 (17)

Therefore combined with (9) the controllerrsquos parameters arefinally selected as follows

119870119901= 1 119870

119889= 2 times 10

minus3 119879

119889= 1 times 10

minus4 (18)

23 Rotor SystemModeling In this study the rotor is assumedto be a rigid and symmetric body It is assumed that allmagnets have identical structure For simplicity we neglectthe magnetic flux leakage the fringe magnetic flux the eddy-current loss the saturation and hysteresis of the magneticcore material and the coupling effects between the electro-magnets The relationship between the center of mass (119874) ofthe rotor and the five-DOF AMB is shown in Figure 4

1199001015840119909101584011991010158401199111015840 is the body fixed coordinate system with the 1199111015840-

axes overlapping the geometric center axis of the electronicbearing stator 119874119909119910119911 is space fixed coordinate system withthe 119911-axes overlapping the geometric center axis of the rotor119874 is the center of mass 119891

1199091and 119891

1199101are the upper RAMB

(RAMB1 in Figure 4) electromagnetic force of the rotorcorresponding to the 119909- and 119910-axes119891

1199092and1198911199102are the lower

RAMB (RAMB2 in Figure 4) electromagnetic force of therotor corresponding to the 119909- and 119910-axes 119891

119911is the TAMB

electromagnetic force of the rotor corresponding to the 119911-axes 119891

119887is the centrifugal force due to the static imbalance

1198971199111

and 1198971199112

represent the distances from the 119874 to the upperRAMB (RAMB1 in Figure 4) and lower RAMB (RAMB2 inthe Figure 4) respectively 119897

1198881and 1198971198882represent the distances


119891119911

1199101

1198911199101

1199102

1198911199102

1199091

1198911199091

1198971198881 1198971199111

1198971198882 1198971199112

1198911199092

1199092

120572 120573

1198973

120596119905 + 120579

120601

Position sensor 1RAMB1

RAMB2Position sensor 2

120596119905

119911(119911998400)

119910(119910998400)

119909(119909998400)

119874(119874998400)

Figure 4 Geometry relationships of rotor and AMB systems

from the 119874 to the upper position sensor and lower positionsensor respectively 119897

3represents the distances from the 119874 to

the eccentric mass point 1199091and 119910

1denote the displacement

output of the upper position sensor 1199092and 119910

2denote the

displacement output of the lower position sensor 120572 120573 and 120593denote the pitch yaw and spin angles displacements aroundthe 119909- 119910- and 119911-axes of the rotor 120579 is the initial phase of thecentrifugal force due to the static imbalance the rotationalspeed of the rotor can be denoted by 120596

119909 119910 119911 is the displacement of the rotor mass centerreferring to the absolute coordinate system The relationshipbetween the angular and translational motions of the masscenter and rotation stress and bearing is given as follows

1199091= 119909 + 119897

1198881120572

1199101= 119910 minus 119897

1198881120573

1199092= 119909 minus 119897

1198882120572

1199102= 119910 + 119897

1198882120573

119911 = 119911

120596119905 = 120593

(19)

119891119909= 1198911199091+ 1198911199092

119891119910= 1198911199101+ 1198911199102

119891119911= 119891119911

119879120572= 11989111990911198971199111minus 11989111990921198971199112

119879120573= 11989111991011198971199112minus 11989111991011198971199111

119879120593= 119879119889

(20)

Equations (19) are written in a matrix form for simpledescription as follows

1199091

1199101

1199092

1199102

119911

120596119905

=

[

[

[

[

[

[

[

[

1 1198971198881

0 0 0 0

0 0 1 minus1198971198881

0 0

1 minus1198971198882

0 0 0 0

0 0 1 1198971198882

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

119909

120572

119910

120573

119911

120593

(21)

Equation (21) is written as119884 = 119860119883119884 is the sensor outputsignal matrix and119883 is displacement of the rotor mass center

Equations (20) are also written in a matrix form for thesimple description as follows

119891119909

119879120572

119891119910

119879120573

119891119911

119879120593

=

[

[

[

[

[

[

[

[

1 0 1 0 0 0

1198971199111

0 minus1198971199112

0 0 0

0 1 0 1 0 0

0 minus1198971199111

0 1198971199112

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

1198911199091

1198911199101

1198911199092

1198911199102

119891119911

119879119889

(22)

Equation (22) is directly translated as 119866 = [119861]119865 where119866 denotes the forces vector of the mass center of the rotoras

119866 = [119891119909119879119886119891119910119879120573119891119911119879120593]

119879

(23)

119865 denotes the electromagnetic forces vector produced bythe AMBs as

119865 = [11989111990911198911199101

1198911199092

1198911199102

119891119911119879119889]

119879

(24)

It is noted that when the rotor is regulated perfectly(1199091= 1199101= 1199092= 1199102= 119911 = 0) the rotational speed

of the rotor can be denoted by 120596 As shown in Figure 3the control characteristics of the five-DOF AMB system arehighly nonlinear and time varying because of the systemparameter variations external disturbances and inherentnonlinearity such as the coupling effects among five axes andgyroscopic effects of rotation Therefore the dynamic statesthat is the rotor positions 119909

1 1199092 1199101 1199102 and 119911 of the five-

DOF AMB system are coupled and affected by more thanone force But observing from (22) the couple of the radialand axial motion of the system studied in this paper is sosmall that the dynamic model of the five-DOF AMB systemis decoupled as five independent subsystems including fourradial subsystems according to 119909

1 1199092 1199101 and 119910

2axes and

one axial subsystem according to the 119911-axis to achieve thedecentralized control So this study emphasizes the dynamicsbehavior analysis of the four radial subsystems

When the rotor is suspended steadily the relative coordi-nate system of the flywheel rotor coincides with the absolutecoordinate system of the AMBrsquos stator The potential energyof the stator in the origin119874 is zero119898 is the mass of the rotor119892 is the gravity constant 119911 are the coordinates of the 119874 Thepotential energy of the rotor is given as 119880 = 119898119892119911

The total kinetic energy of the system 119879 consists of thetranslational kinetic energy 119879

119901 rotational kinetic energy 119879

119911

and kinetic energy of eccentric mass 119879119887


119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910

Figure 5 Flywheel rotor with the eccentric mass

The expression for the translational kinetic energy 119879119901is

119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)

The expression for the rotational kinetic energy 119879119911is

119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)

where 119869119889is transverse mass moments of inertia of the rotor

and 119869119901is the polar moment of inertia of the rotor Due to the

actual machining error and the assembly error the rotor hasthe eccentric mass If the rotor has a residual static unbalancefollowing the ISO quality grade G63 it will produce thekinetic energy when the rotor is spinning As shown inFigure 5 119898

119887is the eccentric mass and 119903

119887is the eccentricity

so the expression for the kinetic energy of eccentric mass 119879119887

is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)

There is the displacement from the geometric center of therotor to the one of the stator 119887

119909 119887119910represent the displacement

corresponding to the 119909- and 119910-axes respectively as

119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)

Equation (28) is substituted to (27) as

119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)

So the total kinetic energy of the rotor is

119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)

According to the above analysis the axial motion and theradial one are approximately orthogonal and the influenceof the radial bearing on the rotor only is analyzed as follows

119902 = [119909 119886 119910 120573]

119879 denotes the displacement vector in themass center of the rotor corresponding to the 119909- and 119910-axes

119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]

119879 denotes the displacement vec-tor in the position of the RAMB corresponding to the 119909- and119910-axes

By means of Lagrange equations which state

119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)

where 119879 is the total kinetic energy 119880 is the total potentialenergy and 119876

119894is the generalized force for the coordinate 119902

119894

a system of four equations describing the dynamics of themodel is obtained which can be written as

(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101

are the upper RAMB electromagneticforce of the rotor corresponding to the 119909- and 119910-axes 119891

1199092

and 1198911199102

are the lower RAMB electromagnetic force of therotor corresponding to the 119909- and 119910-axesThe function of theelectromagnetic force is given as follows

1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)

where 1198961198941 1198961198942are the up and down current stiffness of the

active magnetic bearing respectively 1198961199091 1198961199092

are the up anddown displacement stiffness of the active magnetic bearing


respectively The equation is translated as matrix operationsfor observation as follows

[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)

where [119872] is the mass matrix Because of the small value theunbalance mass is neglected to make the simple calculation

119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)

where [119866] is the inertia matrix and 119861119911 is the length matrix

as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)

where 119865119911 is unbalanced force vector and 119865

119887 is the force

vector in bearing place as

119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)

The transfer function block diagram of the PID controller isshown in Figure 6

In Figure 6 the current output depends on the input ofthe referenced displacement which states

119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)

where 119894 = 119870119901119910119911+ 119870119901119879119889119910119911minus 119870119901119910 minus 119870

119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902

The differential equation of the system can be given as

119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D

Figure 6 Transfer function block diagram of the PID controller

where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911

When 119910119911= 0 that means that the rotor is running

steadily and the differential equation can be written as

119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)

where 119879119894is the differential coefficient of the PID controller

and 119887is the force as

119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)

Then in order to express this system in the state space form

= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868

minus119872minus1119885 minus119872

minus1119870 minus119872

minus1(120596119866 + 119863)

]

]

(43)

are definedFrom the above analysis119863119870 and119879 are directly involved

with the controllerrsquos parameters The model will be variantaccording to the variation of the controllerrsquos parameters Thedynamics characteristic of the system is given as follows

3 Simulation Result Analysis

31 Systems Parameters Suitable values for the parametersinvolved in themodel of (41) were determined Some of theseparameters will be varied with the goal of obtaining usefulinformation for the optimization of designs The parametersare shown in Table 1 However if not otherwise stated thefollowing values will be the ones used for analysis

32 Simulation Result Analysis A planer motion of a rotor iscalled awhirlingmotion or awhirl And a circularwhirl in thesame direction as the shaft rotation is called a forward whirland that in the opposite direction is termed a backward whirlPlot of these natural angular frequencies versus the rotationalspeed is called a natural angular frequency diagrams (orshortly natural frequency diagram) or a 119901 = Ω diagram [21]Positive and negative values of 119901 correspond to forward and


Table 1 Model parameters

Parameter Description Value Unit119898 Mass of rotor 70 kg119871 Length of rotor 0505 m1198711199111

Distance between 119874 and upper RAMB 03047 m1198711199112

Distance between 119874 and lower RAMB 02463 m1198711198881

Distance between 119874 and upper position sensor 03112 m1198711198882

Distance between 119874 and lower position sensor 02518 m119869119889

Transverse mass moments of inertia of rotor about the x- and y-axes 16426 Kg sdotm2

119869119901

Polar mass moment of inertia of rotor about the 119911-axis 19235 Kg sdotm2

119903119887

Distance between CG and external disturbances 0263 m1198780

Nominal air gaps in the x- and y-axes of RAMB 03 mm1205830

Electromagnet parameter 4120587 times 10minus7 Hm

1198940

Bias currents 015 A1199110

Nominal air gap in the 119911-axis of TAMB 03 mm120596max Design maximum speed 20000 rpm120596 Rated speed 5000sim16000 rpm

backwardwhirls respectivelyDiagramof natural frequenciesversus the rotational speed is also used Its natural frequencydiagram is illustrated in Figure 6 where natural angularfrequencies become horizontal straight lines in the diagramResonance mentioned later occurs at the angular frequencygiven by the cross-point of this straight line and the line 119901 =Ω A forced oscillation becomes large in the neighborhood ofthis resonance frequency Sometimes radii of these whirlingmotions are represented by relative sizes of circles in thediagram as shown in Figure 6 These diagrams are calledCampbell diagram which is used to analyze the transientdynamics characteristic of the rotor system

According to Table 1 the differential coefficient of theAMBrsquos controller is set as 0002 and the proportional coef-ficient is set as 1 Extracting the eigenvalues of 119860 for theparameters given above with the running speed 120596 varyingfrom 0 to 20000 RPM the Campbell diagram of the rotor isobtained

Examination of the results of the simulation using themodel of the rotor allows us to argument that rotationalmodes of vibration have natural frequencies that are consid-erably larger than those of translational modes The forwardwhirlingmodes that represent critical speeds that is the onesthat intersect with the line describing the frequencies equalto the running speeds in Figure 7 are all translational Therotational modes do not intersect with that line but withtheir backward directions (negative slopes) These naturalfrequencies increase rapidly with the running speed and donot represent critical speeds and thus stability problems arenot expected to occur in these modes

In Figure 7 we can know that the critical speed of therotor is about 2800 RPM the backward whirling frequency isabout 48Hz and the translational mode is kept as a constantvalue 42Hz So the stability problems are not expected tooccur in working running speeds It is needed to accelerate ordecelerate quickly over 2800 RPM so as to keep the system instability state

0 1 20

20

40

60

80

100

120

140

160

180

Forward whirling modes

Translational modes

Backward whirling modes

Critical speeds

05 15

Freq

uenc

y (H

z)

Running speed (rpm) times104

119875 = Ω

Figure 7 Campbell diagram of the system

In Figure 8 the integral parameter 119870119889is kept at 0002

and the proportional parameter 119870119901is set as 05 1 15 and 2

respectively With the increase of the proportional parameter119870119901 the forward whirling mode curve is away from the curve

119875 = Ω the cross-point between the backward whirling curveand the curve 119875 = Ω gradually rises and the critical speedincreases as well And the translational mode also increaseswhich distinctly showed that proportional parameter affectsthe stiffness of the rotor system that is the proportionalparameter plays a very important role in the stiffness of therotor system

In Figure 9 the proportional parameter 119870119901is kept at 1

and the differential parameter119870119889is set at 0001 0002 0003

and 0004 respectively With the increase of the differentialparameter119870

119889 the forward whirlingmode curve is away from

the curve 119875 = Ω the cross-point between the backward


119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


Figure 8 Campbell diagram of the system with the change of theproportional parameter

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004

Figure 9 Campbell diagram of the system with the change of thedifferential parameter

whirling curve and the curve 119875 = Ω gradually declined andthe critical speed decreases as well

The backward whirling modes change distinctly3200 RPM in the 119870

119889= 0001 and 2000RPM in the 119870

119889=

0004The effect of differential parameters on the whirlingmode

is distinct in running speed 0 sim 6000RPM but as the run-ning speed rises the effect of the differential parameter 119870

119889

becomesmore andmore smaller And the translational modealso increases which showed that the differential parameteralso affects the stiffness of the rotor system but the influenceis not very distinctly compared with the influence of theproportional parameter

The following analysis shows how the dynamics of thesystemwould change if the distance between the two RAMBschanged in the axial directionThe controller parameter119870

119901is

selected as 1 and 119870119889is set to 0002 the Campbell diagram of

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3

Figure 10 Campbell diagram of the system with the change of theRAMB positions

the systemwith the change of theRAMBpositions is shown inFigure 10 Curve 1 shows that the positions of RAMBs are setclose to the center 50mm respectively Curve 2 shows that thepositions of RAMBs are not be changed Curve 3 shows thatthe positions of RAMBs are set far away from the center ofthe system 50mm respectively Compared to the controllerrsquosparameters the change of the RAMB positions does notchange the systemrsquos translational mode but just changes thewhirlingmode of the systemwhich can verify that the changeof the RAMB positions does not change the overall stiffnessof the systemWith the positions of RAMBs far away from thecenter of the rotor the critical speed of the backward whirlingmode increases from 2400 rpm to 3800 rpmThe result of theanalysis which indicates that the shorter distance betweenthe locations of the twoRAMBs bearing can effectively reducethe whirling frequency of the rotor can be used to direct thedesign of the rotor system

The purpose of the flywheel storage machinery is to storeenergy asmuch as possibleThe polarmassmoment of inertiaof rotor determines the energy storage of the flywheel storagemachinery So the effect of the whirling modes of the systemis gained as follows The polar mass moment and transversemass moments of inertia of rotor depend on the geometryof the rotor Figure 10 is the Campbell diagram of the systemwith the inertia ratio of the polar moment and the transversemass moment

In Figure 11 the 119869119901119869119889of the curve 1 is 13 that of curve 2

is 12 that of curve 3 is 117 (the designed one of the rotorsystem) and that of curve 4 is 2 What we can see fromthis figure is that with the ratio of the moments of inertiaincreasing119869

119901119869119889from almost 13 to about 2 and the forward

rotationalmode has increased at the same time the backwardrotational mode has become smaller The main consequenceof this occurrence is that this mode now intersects (thecurve 1) with the line describing the frequencies equal to therunning speeds and thus becomes a critical speed with thepotential of becoming an unstablemode for a certain runningspeed Compared to the controllerrsquos parameters the change


0 1 205 15Running speed (rpm) times104

119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)

Figure 11 Campbell diagram of the system with the inertia ratio ofthe polar moment and the transverse mass moment

of the ratio of the moments of inertia also does not changethe systemrsquos translational mode but just changes the whirlingmode of the system which can verify that the change of theRAMB positions does not change the overall stiffness of thesystem

With a constantmagnetic force and a constant polar massmoment of inertia of rotor the transverse mass momentsof inertia of rotor can be changed just by the change of thedistance between the local center to the overall center of therotor to avoid the work speed in the strong whirling modewhich can greatly improve the stability of the system

4 Conclusions

Variation studies were conducted to assess the influence ofcontroller and the rotor geometry with running speed onrotor dynamic stability Each point on the lines (or surface)represents the threshold running speed above which therotor becomes unstable for a certain parameter configurationThis means that there is a distinctive operating speed belowwhich the system is always stable for a given parametricconfiguration

Considering the influence of controller the dynamicmodel of the rigidity flywheel rotor supporting by AMBs wasestablished to analyze the dynamics characteristic The influ-ence analysis of the controllerrsquos parameters to the dynamicscharacteristic was obtained

(1)The rotational mode and critical speed will reduce asthe proportional parameter increases And the translationalmode also increases which distinctly showed that propor-tional parameter affects the stiffness of the rotor system thatis the proportional parameter plays a very important role tothe stiffness of the rotor system

The rotational mode will increase as the differen-tial parameter increases And the translational mode alsoincreases which showed that the differential parameter alsoaffects the stiffness of the rotor system but the influence is

not very distinctly compared with the influence of the pro-portional parameter

(2) The change of the distance between the mass centerof the rotor to the position of the RAMB will change thecross-point of this straight line and the line 119875 = Ω but willnot change the trend of the total whirling frequency curveCompared to the controllerrsquos parameters the change of theRAMB positions does not change the systemrsquos translationalmode but just changes the whirling mode of the systemwhich can verify that the change of the RAMB positions doesnot change the overall stiffness of the system

(3) The ratio 119869119901119869119889is 117 and is close to critical value

1 There is not any cross-point of the curve of the forwardwhirling mode and the line 119875 = Ω These natural frequenciesincrease rapidly with the running speed and do not representcritical speeds and thus stability problems are not expectedto occur in these modes Compared to the controllerrsquosparameters the change of the ratio of the moments of inertiaalso does not change the systemrsquos translational mode but justchanges the whirling mode of the system which can verifythat the change of the RAMB positions does not change theoverall stiffness of the system

The result of the analysis can be used to set the supportposition of the rotor system limit the ratio of transversemoment of inertia and the polar moment of inertia anddirect the flywheel prototype future design The presentedsimplified rotordynamicmodel can also be applied to rotatingmachines

Acknowledgment

This work is supported by the National 863 Project of China(no 2013AA050802)

References

[1] J G Bitterly ldquoFlywheel technology past present and 21st cen-tury projectionsrdquo IEEE Aerospace and Electronic Systems Maga-zine vol 13 no 8 pp 13ndash16 1998

[2] R Hebner J Beno A Walls and M J Riezenman ldquoFlywheelbatteries come around againrdquo IEEE Spectrum vol 39 no 4 pp46ndash51 2002

[3] G Schweitzer and E H Maslen Magnetic BearingsmdashTheoryDesign and Application to Rotating Machinery Springer 2009

[4] A Chiba T Fukao O Ichikawa M Oshima M Takemotoand D G Dorrell Magnetic Bearings and Bearingless DrivesElsevier 2005

[5] A T A Peijnenburg J P M Vermeulen and J van Eijk ldquoMag-netic levitation systems compared to conventional bearingsystemsrdquoMicroelectronic Engineering vol 83 no 4ndash9 pp 1372ndash1375 2006

[6] E A Knoth and J P Barber ldquoMagnetic repulsion bearings forturbine enginesrdquo IEEE Transactions onMagnetics vol 24 no 6pp 3141ndash3143 1998

[7] C R Knospe ldquoActive magnetic bearings for machining appli-cationsrdquo Control Engineering Practice vol 15 pp 307ndash313 2007

[8] N Miyamoto T Enomoto M Amada et al ldquoSuspension char-acteristics measurement of a bearingless motorrdquo IEEE Transac-tions on Magnetics vol 45 no 6 pp 2795ndash2798 2009


[9] M D Noh S R Cho J H Kyung S K Ro and J K ParkldquoDesign and implementation of a fault-tolerant magnetic bear-ing system for turbo-molecular vacuum pumprdquo IEEEASMETransactions on Mechatronics vol 10 no 6 pp 626ndash631 2005

[10] O S Kim S H Lee and D C Han ldquoPositioning performanceand straightness error compensation of the magnetic levitationstage supported by the linear magnetic bearingrdquo IEEE Transac-tions on Industrial Electronics vol 50 no 2 pp 374ndash378 2003

[11] JH Lee P EAllaireG Tao J ADecker andX Zhang ldquoExper-imental study of sliding mode control for a benchmark mag-netic bearing system and artificial heart pump suspensionrdquoIEEE Transactions on Control Systems Technology vol 11 no 1pp 128ndash138 2003

[12] A A Hussien S Yamada M Iwahara T Okada and T OhjildquoApplication of the repulsive-type magnetic bearing for man-ufacturing micromass measurement balance equipmentrdquo IEEETransactions on Magnetics vol 41 no 10 pp 3802ndash3804 2005

[13] Y Sun Y S Ho and L Yu ldquoDynamic stiffnesses of activemagnetic thrust bearing including eddy-current effectsrdquo IEEETransactions on Magnetics vol 45 no 1 pp 139ndash149 2009

[14] LMaurice and J R AdamsRotatingMachinery Vibration CRCPress Taylor amp Francis Group Boca Raton Fla USA 2010

[15] M I Friswell J E T Penny S D Garvey and A W LeesDynamics of Rotating Machines Cambridge University PressCambridge UK 2010

[16] J M Vance F J Zeidan and B Murphy Machinery Vibrationand Rotordynamics John Wiley amp Sons Hoboken NJ USA2010

[17] U Werner ldquoTheoretical vibration analysis regarding excitationdue to elliptical shaft journals in sleeve bearings of electricalmotorsrdquo International Journal of Rotating Machinery vol 2012Article ID 860293 19 pages 2012

[18] T Dimond P Allaire S Mushi Z Lin Yoon and Y Y SeldquoModal tilttranslate control and stability of a rigid rotor withgyroscopics on active magnetic bearingsrdquo International Journalof Rotating Machinery vol 2012 Article ID 567670 10 pages2012

[19] J Crowe PID Control New Identification and Design MethodsSpringer London UK 2005

[20] F-J Lin S -Y Chen and M -S Huang ldquoIntelligent doubleintegral sliding-mode control for five-degree-of-freedom activemagnetic bearing systemrdquo IET Control Theory amp Applicationsvol 5 no 11 pp 1287ndash1303 2011

[21] Y Ishida and T Yamamoto Linear and Nonlinear Rotordynam-ics 2nd edition 2012

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Flywheel

RAMB1

Auxiliary bearing 1

TAMB

RAMB2

Motorgenerator

Auxiliary bearing 2

Shaft

119883

119884

119885

119874

Figure 1 Basic layout of a flywheel energy storage system

fitting tolerance between stator housing bearing housingand so on Dynamic eccentricity is caused if the rotor is bentor if the rotor core is eccentrically positioned on the rotorshaft and so on

The aim of this paper is to show a theoretical vibrationanalysis of the rotor system of 600Wh flywheel energystorage system based on a simplified rotordynamic modelThe result of the analysis can be used to set the supportposition of the rotor system limit the ratio of transversemoment of inertia and the polarmoment of inertia and directthe flywheel prototype future design

2 System Modeling

21 System Structure The basic layout of a flywheel energystorage system is depicted in Figure 1 that is the 600Whprototype system a rigid vertical rotor shaft with a rigidmotor and flywheel connected to shaft Flywheel energystorage system is a complex construction where energy isstoredmechanically and transferred to and from the flywheelby an integrated motorgenerator

Two radial active electromagnetic bearings (RAMBs)including upper and lower RAMBs (RAMB1 and RAMB2in Figure 1) and one thrust magnetic bearing (TAMB) arefixed on the platform to suspend and regulate the rotor inthe radial and axial DOF respectively The positions of therotor in five axes are defined as the displacements deviatedfrom the nominal air gapsTherefore two pairs of the perpen-dicular eddy-current position sensors are installed closely to

respective RAMBs to measure the respective radial positionsin 119909- and 119910-axes which are denoted as 119909

1and 119910

1of the upper

RAMB and 1199092and 119910

2of the lower RAMB Moreover one

eddy-current position sensor is installed closely to TAMB tomeasure the axial position in the 119911-axis which is denoted as119911 After the rotor positions in five axes are all measured andsent to the control core through the position signals line therequired electromagnet currents are generated from the drivesystem and they circulate the coils through the power lineTherefore the rotor can be regulated and stabilized in thecenters of the apertures of the two RAMBs and the thrust disccan be centered in the middle of the air gap of the TAMBrespectively

One 70 kg flywheel is mounted on the rotor whose posi-tion can be adjusted to modify the rotational characteristicsFurthermore one motorgenerator is equipped between thelower RAMB and flywheel and is responsible for the torquegeneration or electricity generation based on the rotationalspeed requirement However in the rotor system the rolling-element auxiliary bearings (auxiliary bearing 1 and auxiliarybearing 2 in Figure 1) are necessary to protect AMB statorsand stationary components along shaft in the even 119905 of AMBfailure or high transient loads Under normal operation ofAMBs the rotor maintains a positive clearance with auxiliarybearings which is less than a clearance with AMBs

22 AMB Modeling

221 Electromagnetic ForceModel Considering the diameterof the shaft and the loading capacity the RAMB with 8-pole legs was designed in the 600Wh prototype system Thestructural configuration of theRAMB is shown in Figure 2(a)including electromagnetic coils And the basic magneticbearing control loop is shown in Figure 2(b)

Taking the two pairs of poles in the 119910-axes of oneRAMB for example the electromagnetic force model will beestablished [19] As shown in Figure 2(b) the basic magneticbearing control loop includes differential driving positionsensor controller and power amplifier The included differ-ential driving is adopted in this study to obtain maximumrange of the force dynamic and good linearity of the controldynamic 120572 is the half of the angle between the two-pole legs1199100is the nominal air gaps of the RAMB in 119910-axes and 119910

1

is the deviation in 119910-axes So the upper available air gap iscalculated as (119910

0+ 1199101cos120572) and the lower one is calculated

as (1199100minus 1199101cos120572) 119894

0is the base current and the 119894

1199101is the

control current 1198940+1198941199101and 1198940minus1198941199101circulate the upper and the

lower coils in the 119910-axis respectively 1198911and 119891

2are the total

attractive electromagnetic forces in the 119910-axis which is givenas follows

1198911=

120583011986001198732

4

(1198940+ 1198941199101)

2

(1199100+ 1199101)2cos120572 (1)

1198912=

120583011986001198732

4

(1198940+ 1198941199101)

2

(1199100minus 1199101)2cos120572 (2)


Coils

Electromagnet


11991001199101

119884119900119900111989111198912

119883

120572

Poweramplifier

Position sensor

Controller

1198940 + 1198941199101

1198940 minus 1198941199101

+

+

+

1198941199101

1198940

1199100

1199101

minus

1199100 + 1199101 cos 120572

1199100 minus 1199101 cos 120572




0= 4120587 times 10




119891119910= 1198911minus 1198912

=

119906011986001198732

4

[(

1198940+ 1198941199101

1199100+ 1199101

)

2

minus (

1198940minus 1198941199101

1199100minus 1199101

)

2

] cos120572(3)


1= 0 119894

1199101=


1198911199101cong 1198961198941198941+ 1198961199101199101 (4)

where 119896119910and 119896



119896119910equiv

1205971198911199101(1199101 1198941199101)

1205971199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

= minus

4120583011986001198732cos21205721198942

0

1199103

0

119896119894equiv

1205971198911199101(1199101 1198941199101)

1205971198941199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

=

4120583011986001198732 cos120572119894

0

1199102

0

(5)




1199091and 119896

1198941





119866 (119904) =

119880 (119904)

119864 (119904)

= 119870119901+

119870119894

119904

+

119870119889119904

1 + 119879119889119904

(6)

where 119870119901 119870119894 119870119889 and 119879




11988751199045+ 11988741199044+ 11988731199043+ 11988721199042+ 11988711199041= 0 (7)



Position sensor

ElectromagnetPower

amplifier

AD

DA119878ref 119878119894 119891


where1198871= 119860119886119860119904119870119901119896119894minus 119896119909

1198872= 119896119894(119860119886119860119904119870119889+ 119860119886119860119904119870119901119879119889) minus 119896119909(119879119886+ 119879119889)

1198873= 119898 minus 119896

119909119879119889119879119886

1198874= 119898 (119879

119886+ 119879119889)

1198875= 119898119879119886119879119889

(8)

where 119896119909and 119896



119886are the


119904and 119879



119901 119870119889 and 119879



0 lt 119879119886119879119889lt

119898

119896119909119909

(9)

(119870119889+ 119870119886119879119889) lt

119898 (119879119886+ 119879119889)

119860119886119860119904119879119886119879119889119896119894119909

(10)

119870119889gt (


2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (11)

119870119889lt (

1199042+ radic1199042

2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (12)

119870119901ge (

119896119894119909

119860119886119860119904119896119909119909

+

(119898 minus 119896119909119909119879119886119879119889)

4119898119896119894119909119860119886119860119904119879119886119879119889

) (13)

119870119901gt

119896119909119909

119860119886119860119904119896119894119909

(14)




119889can also





0=






119901deter-

mined by (9) is

0 lt 119879119889lt 08323 (15)



0154 lt 119870119901lt 4578 (16)





be given as


minus1 (17)


119870119901= 1 119870

119889= 2 times 10

minus3 119879

119889= 1 times 10

minus4 (18)




1199091and 119891



1199092and1198911199102are the lower


119911is the TAMB



1198971199111

and 1198971199112




119891119911

1199101

1198911199101

1199102

1198911199102

1199091

1198911199091

1198971198881 1198971199111

1198971198882 1198971199112

1198911199092

1199092

120572 120573

1198973

120596119905 + 120579

120601



120596119905

119911(119911998400)

119910(119910998400)

119909(119909998400)

119874(119874998400)







2denote the



1199091= 119909 + 119897

1198881120572

1199101= 119910 minus 119897

1198881120573

1199092= 119909 minus 119897

1198882120572

1199102= 119910 + 119897

1198882120573

119911 = 119911

120596119905 = 120593

(19)

119891119909= 1198911199091+ 1198911199092

119891119910= 1198911199101+ 1198911199102

119891119911= 119891119911

119879120572= 11989111990911198971199111minus 11989111990921198971199112

119879120573= 11989111991011198971199112minus 11989111991011198971199111

119879120593= 119879119889

(20)


1199091

1199101

1199092

1199102

119911

120596119905

=

[

[

[

[

[

[

[

[

1 1198971198881

0 0 0 0

0 0 1 minus1198971198881

0 0

1 minus1198971198882

0 0 0 0

0 0 1 1198971198882

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

119909

120572

119910

120573

119911

120593

(21)



119891119909

119879120572

119891119910

119879120573

119891119911

119879120593

=

[

[

[

[

[

[

[

[

1 0 1 0 0 0

1198971199111

0 minus1198971199112

0 0 0

0 1 0 1 0 0

0 minus1198971199111

0 1198971199112

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

1198911199091

1198911199101

1198911199092

1198911199102

119891119911

119879119889

(22)


119866 = [119891119909119879119886119891119910119879120573119891119911119879120593]

119879

(23)


119865 = [11989111990911198911199101

1198911199092

1198911199102

119891119911119879119889]

119879

(24)



1 1199092 1199101 1199102 and 119911 of the five-


1 1199092 1199101 and 119910

2axes and





119911



119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910



119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)


119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)







is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)




119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)


119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)


119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)


119902 = [119909 119886 119910 120573]


119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]



119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)



119894


(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101


1199092

and 1198911199102


1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)






[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Coils

Electromagnet


11991001199101

119884119900119900111989111198912

119883

120572

Poweramplifier

Position sensor

Controller

1198940 + 1198941199101

1198940 minus 1198941199101

+

+

+

1198941199101

1198940

1199100

1199101

minus

1199100 + 1199101 cos 120572

1199100 minus 1199101 cos 120572




0= 4120587 times 10




119891119910= 1198911minus 1198912

=

119906011986001198732

4

[(

1198940+ 1198941199101

1199100+ 1199101

)

2

minus (

1198940minus 1198941199101

1199100minus 1199101

)

2

] cos120572(3)


1= 0 119894

1199101=


1198911199101cong 1198961198941198941+ 1198961199101199101 (4)

where 119896119910and 119896



119896119910equiv

1205971198911199101(1199101 1198941199101)

1205971199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

= minus

4120583011986001198732cos21205721198942

0

1199103

0

119896119894equiv

1205971198911199101(1199101 1198941199101)

1205971198941199101

100381610038161003816100381610038161003816100381610038161003816100381610038161199101=0 1198941199101=0

=

4120583011986001198732 cos120572119894

0

1199102

0

(5)




1199091and 119896

1198941





119866 (119904) =

119880 (119904)

119864 (119904)

= 119870119901+

119870119894

119904

+

119870119889119904

1 + 119879119889119904

(6)

where 119870119901 119870119894 119870119889 and 119879




11988751199045+ 11988741199044+ 11988731199043+ 11988721199042+ 11988711199041= 0 (7)



Position sensor

ElectromagnetPower

amplifier

AD

DA119878ref 119878119894 119891


where1198871= 119860119886119860119904119870119901119896119894minus 119896119909

1198872= 119896119894(119860119886119860119904119870119889+ 119860119886119860119904119870119901119879119889) minus 119896119909(119879119886+ 119879119889)

1198873= 119898 minus 119896

119909119879119889119879119886

1198874= 119898 (119879

119886+ 119879119889)

1198875= 119898119879119886119879119889

(8)

where 119896119909and 119896



119886are the


119904and 119879



119901 119870119889 and 119879



0 lt 119879119886119879119889lt

119898

119896119909119909

(9)

(119870119889+ 119870119886119879119889) lt

119898 (119879119886+ 119879119889)

119860119886119860119904119879119886119879119889119896119894119909

(10)

119870119889gt (


2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (11)

119870119889lt (

1199042+ radic1199042

2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (12)

119870119901ge (

119896119894119909

119860119886119860119904119896119909119909

+

(119898 minus 119896119909119909119879119886119879119889)

4119898119896119894119909119860119886119860119904119879119886119879119889

) (13)

119870119901gt

119896119909119909

119860119886119860119904119896119894119909

(14)




119889can also





0=






119901deter-

mined by (9) is

0 lt 119879119889lt 08323 (15)



0154 lt 119870119901lt 4578 (16)





be given as


minus1 (17)


119870119901= 1 119870

119889= 2 times 10

minus3 119879

119889= 1 times 10

minus4 (18)




1199091and 119891



1199092and1198911199102are the lower


119911is the TAMB



1198971199111

and 1198971199112




119891119911

1199101

1198911199101

1199102

1198911199102

1199091

1198911199091

1198971198881 1198971199111

1198971198882 1198971199112

1198911199092

1199092

120572 120573

1198973

120596119905 + 120579

120601



120596119905

119911(119911998400)

119910(119910998400)

119909(119909998400)

119874(119874998400)







2denote the



1199091= 119909 + 119897

1198881120572

1199101= 119910 minus 119897

1198881120573

1199092= 119909 minus 119897

1198882120572

1199102= 119910 + 119897

1198882120573

119911 = 119911

120596119905 = 120593

(19)

119891119909= 1198911199091+ 1198911199092

119891119910= 1198911199101+ 1198911199102

119891119911= 119891119911

119879120572= 11989111990911198971199111minus 11989111990921198971199112

119879120573= 11989111991011198971199112minus 11989111991011198971199111

119879120593= 119879119889

(20)


1199091

1199101

1199092

1199102

119911

120596119905

=

[

[

[

[

[

[

[

[

1 1198971198881

0 0 0 0

0 0 1 minus1198971198881

0 0

1 minus1198971198882

0 0 0 0

0 0 1 1198971198882

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

119909

120572

119910

120573

119911

120593

(21)



119891119909

119879120572

119891119910

119879120573

119891119911

119879120593

=

[

[

[

[

[

[

[

[

1 0 1 0 0 0

1198971199111

0 minus1198971199112

0 0 0

0 1 0 1 0 0

0 minus1198971199111

0 1198971199112

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

1198911199091

1198911199101

1198911199092

1198911199102

119891119911

119879119889

(22)


119866 = [119891119909119879119886119891119910119879120573119891119911119879120593]

119879

(23)


119865 = [11989111990911198911199101

1198911199092

1198911199102

119891119911119879119889]

119879

(24)



1 1199092 1199101 1199102 and 119911 of the five-


1 1199092 1199101 and 119910

2axes and





119911



119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910



119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)


119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)







is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)




119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)


119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)


119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)


119902 = [119909 119886 119910 120573]


119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]



119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)



119894


(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101


1199092

and 1198911199102


1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)






[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Position sensor

ElectromagnetPower

amplifier

AD

DA119878ref 119878119894 119891


where1198871= 119860119886119860119904119870119901119896119894minus 119896119909

1198872= 119896119894(119860119886119860119904119870119889+ 119860119886119860119904119870119901119879119889) minus 119896119909(119879119886+ 119879119889)

1198873= 119898 minus 119896

119909119879119889119879119886

1198874= 119898 (119879

119886+ 119879119889)

1198875= 119898119879119886119879119889

(8)

where 119896119909and 119896



119886are the


119904and 119879



119901 119870119889 and 119879



0 lt 119879119886119879119889lt

119898

119896119909119909

(9)

(119870119889+ 119870119886119879119889) lt

119898 (119879119886+ 119879119889)

119860119886119860119904119879119886119879119889119896119894119909

(10)

119870119889gt (


2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (11)

119870119889lt (

1199042+ radic1199042

2minus 411990411199043

21198601198861198601199041199041119896119894119909

+

119896119909119909(119879119886+ 119879119889)

119860119886119860119904119896119894119909

minus 119870119901119879119889) (12)

119870119901ge (

119896119894119909

119860119886119860119904119896119909119909

+

(119898 minus 119896119909119909119879119886119879119889)

4119898119896119894119909119860119886119860119904119879119886119879119889

) (13)

119870119901gt

119896119909119909

119860119886119860119904119896119894119909

(14)




119889can also





0=






119901deter-

mined by (9) is

0 lt 119879119889lt 08323 (15)



0154 lt 119870119901lt 4578 (16)





be given as


minus1 (17)


119870119901= 1 119870

119889= 2 times 10

minus3 119879

119889= 1 times 10

minus4 (18)




1199091and 119891



1199092and1198911199102are the lower


119911is the TAMB



1198971199111

and 1198971199112




119891119911

1199101

1198911199101

1199102

1198911199102

1199091

1198911199091

1198971198881 1198971199111

1198971198882 1198971199112

1198911199092

1199092

120572 120573

1198973

120596119905 + 120579

120601



120596119905

119911(119911998400)

119910(119910998400)

119909(119909998400)

119874(119874998400)







2denote the



1199091= 119909 + 119897

1198881120572

1199101= 119910 minus 119897

1198881120573

1199092= 119909 minus 119897

1198882120572

1199102= 119910 + 119897

1198882120573

119911 = 119911

120596119905 = 120593

(19)

119891119909= 1198911199091+ 1198911199092

119891119910= 1198911199101+ 1198911199102

119891119911= 119891119911

119879120572= 11989111990911198971199111minus 11989111990921198971199112

119879120573= 11989111991011198971199112minus 11989111991011198971199111

119879120593= 119879119889

(20)


1199091

1199101

1199092

1199102

119911

120596119905

=

[

[

[

[

[

[

[

[

1 1198971198881

0 0 0 0

0 0 1 minus1198971198881

0 0

1 minus1198971198882

0 0 0 0

0 0 1 1198971198882

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

119909

120572

119910

120573

119911

120593

(21)



119891119909

119879120572

119891119910

119879120573

119891119911

119879120593

=

[

[

[

[

[

[

[

[

1 0 1 0 0 0

1198971199111

0 minus1198971199112

0 0 0

0 1 0 1 0 0

0 minus1198971199111

0 1198971199112

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

1198911199091

1198911199101

1198911199092

1198911199102

119891119911

119879119889

(22)


119866 = [119891119909119879119886119891119910119879120573119891119911119879120593]

119879

(23)


119865 = [11989111990911198911199101

1198911199092

1198911199102

119891119911119879119889]

119879

(24)



1 1199092 1199101 1199102 and 119911 of the five-


1 1199092 1199101 and 119910

2axes and





119911



119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910



119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)


119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)







is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)




119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)


119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)


119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)


119902 = [119909 119886 119910 120573]


119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]



119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)



119894


(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101


1199092

and 1198911199102


1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)






[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119891119911

1199101

1198911199101

1199102

1198911199102

1199091

1198911199091

1198971198881 1198971199111

1198971198882 1198971199112

1198911199092

1199092

120572 120573

1198973

120596119905 + 120579

120601



120596119905

119911(119911998400)

119910(119910998400)

119909(119909998400)

119874(119874998400)







2denote the



1199091= 119909 + 119897

1198881120572

1199101= 119910 minus 119897

1198881120573

1199092= 119909 minus 119897

1198882120572

1199102= 119910 + 119897

1198882120573

119911 = 119911

120596119905 = 120593

(19)

119891119909= 1198911199091+ 1198911199092

119891119910= 1198911199101+ 1198911199102

119891119911= 119891119911

119879120572= 11989111990911198971199111minus 11989111990921198971199112

119879120573= 11989111991011198971199112minus 11989111991011198971199111

119879120593= 119879119889

(20)


1199091

1199101

1199092

1199102

119911

120596119905

=

[

[

[

[

[

[

[

[

1 1198971198881

0 0 0 0

0 0 1 minus1198971198881

0 0

1 minus1198971198882

0 0 0 0

0 0 1 1198971198882

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

119909

120572

119910

120573

119911

120593

(21)



119891119909

119879120572

119891119910

119879120573

119891119911

119879120593

=

[

[

[

[

[

[

[

[

1 0 1 0 0 0

1198971199111

0 minus1198971199112

0 0 0

0 1 0 1 0 0

0 minus1198971199111

0 1198971199112

0 0

0 0 0 0 1 0

0 0 0 0 0 1

]

]

]

]

]

]

]

]

1198911199091

1198911199101

1198911199092

1198911199102

119891119911

119879119889

(22)


119866 = [119891119909119879119886119891119910119879120573119891119911119879120593]

119879

(23)


119865 = [11989111990911198911199101

1198911199092

1198911199102

119891119911119879119889]

119879

(24)



1 1199092 1199101 1199102 and 119911 of the five-


1 1199092 1199101 and 119910

2axes and





119911



119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910



119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)


119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)







is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)




119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)


119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)


119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)


119902 = [119909 119886 119910 120573]


119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]



119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)



119894


(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101


1199092

and 1198911199102


1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)






[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119910998400119910

119903119887

119898119887

120596119905

119900 119887119909

119909998400

119909119900998400

119887119910



119879119901=

119898

2

[(

119889119909

119889119905

)

2

+ (

119889119910

119889119905

)

2

] =

119898

2

(2+ 1199102) (25)


119879119911=

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573) (26)







is

119879119887=

1

2

119898119887(1198872

119909+1198872

119910+ 12059621199032

119887)

+ 119898119887(119887119909120596119903119887cos120596119905 minus

119887119910120596119903119887sin120596119905)

(27)




119887119909= 119909 + 119897

3120572

119887119910= 119910 minus 119897

3120573

(28)


119879119887=

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(29)


119879 = 119879119901+ 119879119911+ 119879119887

=

1

2

119898 (2+ 1199102) +

1

2

119869119889(2+

1205732) +

1

2

119869119901120596 (120572

120573 minus 120573)

+

1

2

119898119887[( + 119897

3)2

+ ( 119910 minus 1198973120573)

2

+ 12059621199032

119887]

+ 119898119887( 119910 minus 119897

3120573) 120596119903119887cos120596119905 minus 119898

119887( + 1198973) 120596119903119887sin120596119905

(30)


119902 = [119909 119886 119910 120573]


119902119911= [1199091199111

1199101199111

1199091199112

1199101199112]



119889

119889119905

(

120597119879

120597 119902119894

) minus

120597119879

120597119902119894

+

120597119880

120597119902119894

= 119876119894 (31)



119894


(119898 + 119898119887) + 119898

1198871198973 = 1198911199091+ 1198911199092+ 1205962119903119887sin120596119905

1198981198871198973 + (119869

119889+ 1198981198871198972

3) minus 119869

119901120573120596

= 11989111990911198971199111minus 11989111990921198971199112+ 12059621199031198871198973sin120596119905

(119898 + 119898119887) 119910 minus 119898

1198871198973120573 = 119891

1199101+ 1198911199102+ 1205962119903119887cos120596119905

minus 1198981198871198973119910 + (119869119889+ 1198981198871198972

3)120573 + 119869119901120596

= 11989111991021198971199112minus 11989111991011198971199111+ 1205962119903119887119897119887sin120596119905

(32)

where 1198911199091

and 1198911199101


1199092

and 1198911199102


1198911199091= 11989611989411198941+ 11989611990911199091

1198911199101= 11989611989411198941+ 11989611990911199101

1198911199092= 11989611989421198942+ 11989611990921199092

1198911199102= 11989611989421198942+ 11989611990921199102

(33)






[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[119872] 119902 + 120596 [119866] 119902 = [119861119911] 119865119911 + 119865

119887 (34)


119872 =

[

[

[

[

119898 + 119898119887

119898119887119897119887

0 0

119898119887119897119887

119869119889+ 1198981198871198972

1198870 0

0 0 119898 + 119898119887

minus119898119887119897119887

0 0 minus119898119887119897119887

119869119889+ 1198981198871198972

119887

]

]

]

]

asymp

[

[

[

[

119898 0 0 0

0 1198691198890 0

0 0 119898 0

0 0 0 119869119889

]

]

]

]

(35)


as

119866 =

[

[

[

[

0 0 0 0

0 0 0 minus119869119901

0 0 0 0

0 1198691199010 0

]

]

]

]

119861119911=

[

[

[

[

1 0 1 0

1198971199111

0 minus1198971199112

0

0 1 0 1

0 minus1198971199111

0 1198971199112

]

]

]

]

(36)


119887 is the force


119865119885= (

1198911199091

1198911199101

1198911199092

1198911199102

) =(

11989611989411198941+ 11989611990911199091

11989611989411198941+ 11989611990911199101

11989611989421198942+ 11989611990921199092

11989611989421198942+ 11989611990921199102

)

119865119887= (

1205962119903119887sin120596119905

12059621199031198871198973sin120596119905

1205962119903119887cos120596119905

minus12059621199031198871198973cos120596119905

)

(37)



119868 (119904) = 119870119901(1 + 119879

119889119904) [minus119884 (119904) + 119884

119887(119904)] (38)


119901119879119889119910 119910 =

[1199091119910111990921199102]

119879

119861119911119865119911= 119861119911119896119894119870119901[(minus119862119911119902 + 119910119911) + 119879119889(minus119862119911119902 + 119910119911)] + 119861

119911119896120575119902119887

(39)

where 119862119911= (

1 11989711988810 0

0 0 1 minus1198971198881

1 minus11989711988810 0

0 0 1 1198971198881

) 119902119887= 119861119879

119911119902


119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887+ 1198651 (40)

Rotor 119868(119904) 119884(119904)

+

+

++

minus

119884119903(119904)

P

I

D


where 119863 = 119896119894119870119901119879119889119861119911119862119911 119870 = 119896

119894119870119901119861119911119862119911minus 119896120575119861119911119861119879

119911 119879 =

119870119901119896119894119861119911119862119911119879119894 and 119865

1= 119896119894119870119901119861119911119910119911+ 119896119894119870119901119879119889119861119911119910119911



119872 119902 + (120596119866 + 119863) 119902 + 119870 119902 + 119879119902 = 119887 (41)



119887= (

1199031198871205963 cos120596119905

1199031198871198971198871205963 cos120596119905

minus1199031198871205963 sin120596119905

1199031198871198971198871205963 sin120596119905

) (42)


= 119860119909

119909 = [119902 119902 119902]

119879

119860 =[

[

0 119868 0

0 0 119868



minus1(120596119866 + 119863)

]

]

(43)














119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119869119901


119903119887




1198940







0 1 20

20

40

60

80

100

120

140

160

180


Translational modes


Critical speeds

05 15

Freq

uenc

y (H

z)


119875 = Ω












119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119870119875 = 2

119875 = Ω

119870119875 = 15

119870119875 = 1

119870119875 = 05

0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)



0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

119870119889 = 0001

119870119889 = 0002

119870119889 = 0003

119870119889 = 0004




119889= 0001 and 2000RPM in the 119870

119889=



119889



119901is


0 1 20

20

40

60

80

100

120

140

160

180

05 15

Freq

uenc

y (H

z)


119875 = Ω

12

3










119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875 = Ω

0

20

40

60

80

100

120

140

160

180

1234

Freq

uenc

y (H

z)




4 Conclusions










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





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Research Article Theoretical Vibration Analysis on...

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Transcript of Research Article Theoretical Vibration Analysis on...