research article resonance analysis of high-frequency

Research ArticleResonance Analysis of High-Frequency Electrohydraulic ExciterControlled by 2D Valve

Guojun Pan1 Yan Ren2 and Jian Ruan3

1College of Information and Engineering Zhejiang Radio amp Television University Hangzhou 310030 China2College of Mechanical amp Electrical Engineering Wenzhou University Wenzhou 325035 China3Key Laboratory of Special Purpose Equipment and Advanced Machining Technology Ministry of Education amp Zhejiang ProvinceZhejiang University of Technology Hangzhou 310014 China

Correspondence should be addressed to Yan Ren rentingting211163com

Received 18 November 2014 Accepted 18 December 2014

Academic Editor Yanxue Wang

Copyright copy 2015 Guojun Pan 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

The resonant characteristic of hydraulic system has not been described yet because it is necessarily restricted by linear assumptionsin classical fluid theory A way of the resonance analysis is presented for an electrohydraulic exciter controlled by 2D valve Theblock diagram of this excitation system is established by extracting nonlinear parts from the traditional linearization analysis asa result the resonant frequency is obtained According to input energy from oil source which is equal to the reverse energy to oilsource load pressure and load flow are solved analytically as the working frequency reaches the natural frequency The analyticalexpression of resonant peak is also derived without damping Finally the experimental system is built to verify the theoreticalanalysis The initial research on resonant characteristic will lay theoretical foundation and make useful complement for resonancephenomena of classical fluid theory in hydraulic system

1 Introduction

Material fatigue or crack growth is a phenomenon commonlymet in mechanical engineering practice [1 2] Thus fatiguetesting becomes an important and wise means to objectivelydetermine the performance of a mechanical product workingunder vibrating condition [3] Recently high-cycle fatiguetesting is more and more of interest In Wright PattersonAir Force Base an electrodynamic shaker has been designedto study high-frequency characteristics of which workingfrequency is from 350Hz to 600Hz depending on thesystem itself And a magnetostrictive material as an actuatoris used in electromagnetic exciter in order that workingfrequency is up to about 2 kHz [4] In Southwest ResearchInstitute a piezoelectric driver is used in a new exciter forenhancing frequency bandwidth in a large scale Workingfrequency as a function of the specimen and grips canbe achieved near 2 kHz but it is limited to the resonancepoint [5] The vibrating enviroment is created artificially byan exciter which is driven by mechanical power electrical

power or hydraulic power [6] However in the case of heavypower an electrohydraulic exciter is commonly used Anelectrohydraulic exciter is designed by MTS specifically forhigh-cycle fatigue testing [7] It is controlled by a two-stageservo valve Though working frequency is improved greatlyflow capacity is limited because of the structure of nozzleflapper So a voice valve with higher flow rate and frequencyresponse is used in an electrohydraulic exciter designed byMTS Corporation This system is integrated for elastomericmaterial testing and crack growth testing in Michigan Tech-nological University This exciterrsquos operating frequency isincreased (1000Hz or above) but voice coil motor requiresadditional cooling device to reduce the temperature undera high current working environment A three-stage servovalve is produced by Rexroth [8] It has better dynamicresponse than nozzle-flapper structure but the bandwith ofbig flow valve is about 200Hz In addition to designing newelectromechanical transformer and two-stage or three-stagestructure of servo valve new control components have beentried to replace servo valve used in electrohydraulic exciter

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 590203 9 pageshttpdxdoiorg1011552015590203

2 Shock and Vibration

Hao and Zhang [9] used a rotary valve in exciter to improvevibration frequency By controlling this rotary valve the spoolwill make a reciprocating motion to switch pressurized flowrate to chambers of the cylinder or motor alternatively andthus a vibration is created to the piston of the cylinder andconnected load But vibration amplitude was dominated bythe oilrsquos pressure and thus had low accuracy and the offsetcontrol was difficult to achieve

The hydraulic exciter is much restrained by the dynamicperformance of the servo valve which is difficult to enhanceworking frequency to a very high level A novel scheme for anelectrohydraulic exciter controlled by a 2D valve is thereforeproposed to achieve higher frequency The frequency andthe amplitude of sinusoidal vibration wave are separatelycontrolled by circumferential rotation and axial sliding ofspool respectively [10] But the relationship of vibrationfrequency and the amplitudes is intercoupling real data isscarce Therfore a promising solution is increasing workingfrequency up to natural frequency of hydraulic systemWhen frequency of the excited vibration comes close tothe natural frequency resonance occurs and can improvevibration amplitude greatly This hydraulic resonance willbe researched and discussed through theoretical analysisand experimental system and finally some conclusions aregiven

2 Working Principle

2D valve as a necessary control component is used inhigh-frequency electrohydraulic exciter because of this valversquosspecial structure 2D valve has two degrees of freedom therotational and sliding motions of the spool Such structurenot only acts a two-stage valve with pilot stage and directposition feedback but also greatly improves the excitedfrequency by increasing spoolrsquos rotational speed The spool iscontinuously rotated by an electric motor and gear combina-tion and sliding by an eccentric mechanism driven by otherelectric motors as shown in Figure 1

The rotation of the 2D valve spool can produce thealternatively varied valve port area which is formed betweenthe grooves on a spool land and the rectangular windowson the sleeve There is a series of grooves symmetricallydistributed on the spools (central angle of every groove is120572) coordinated with the windows distributed uniformly onthe sleeve which create an ldquoalternately variedrdquo valve port areaas shown in Figure 2 Then geometrically the groove has 2120572angle difference with respect to one on any two adjacentshoulders Such arrangement will cause a waveform with180∘ phase angle between the meter-in valve port and themeter-out valve port The varied valve port area depends onrotary speed and axial sliding of the spool Consequentlythe electrohydraulic vibrator controlled by 2D valve couldrealize that the frequency and the amplitude of vibration aredominated respectively by the rotary speed and the slidingdisplacement of the spool of 2D valve

The combination of 2D valve and a piston is selected as ahydraulic power element to produce a reciprocating motionOne chamber of the biactuation cylinder is controlled by

1 2 34

5

6

7

Figure 1 2D valve with twomotors 1 valve body 2 sleeve 3 spool4 eccentric mechanism 5 stepping motor 6 electric motor and 7gearbox

3

1

2

120572

120596

Figure 2 Coupling pattern of 2D valve 1 groove on the spool 2window on the sleeve and 3 rectangular area

a hydraulic bridge formed by meter-in and meter-out alter-natively varied valve ports Another chamber is a hydraulicbridge with a 180∘ phase angle difference As the spool ofthe 2D valve makes a rotary motion the pressure inside twochambers of the cylinder will change alternatively to makethe piston output a reciprocating vibration by reference toFigures 3(a) and 3(b) Obviously it is easy to obtain high-frequency vibration by improving rotary speed of the spoolThe working frequency is also related to the number of thegrooves on a single spool land and the coupling patternbetween the grooves on the spool land and the windows onthe sleeve However this is achieved at the price of sacrificingthe magnitude of the 2D valve orifice area that decides theamplitude of vibration So in high-frequency section theexcited amplitude will be to a large extent limited

Shock and Vibration 3

TTBA P1

3

IIII

2 4 5

6

120596

p1p2

x

yp

(a)

PAB T

T1

6

IIIV

32 4 5

120596

p1p2

x

yp

(b)

Figure 3 Valve ports I and III open piston moves to the right (a) and valve ports II and IV open piston moves to the left (b) 1 cylinder 2sleeve 3 spool 4 window 5 groove and 6 excited object

3 Resonant Characteristics

The resonant principle of hydraulic fluid andmechanical sys-tem is thoroughly different because the elements of mechan-ical resonance including mass damping and spring aremutually independent and the relevance among them is veryweak However hydraulic resonance elements composed ofspring damping and flow force are almost made with flowpower transmission mediums except mass This hydraulicmedium that is working fluid not only acts as a force todrive cycle oscillation of the piston but also is a centeringspring of stiffness is a function of piston displacement andtakes into effect like a viscous damping because of a leakagepath acting to increase damping Therefore this hydraulicresonance will come out with peculiar resonant phenomenonand theoretical results

31 Resonance Process This exciter producing hydraulic res-onance is under a condition that working frequency shouldreach system natural frequency So acceleration of the pistonwill be greatly increased so that the piston still moves inthe left-hand direction even though passing by the centerposition of hydraulic cylinder when valve ports 1 and 3 areopen (the orifice areas are denoted by 119860V1 and 119860V3) whereoil is directed to the left cylinder chamber and then drainedfrom the right cylinder chamber to the tank (the flow ratesare resp119876V1 and119876V3) This leads to the fact that the volumeof the left chamber tapers off and then the pressure of theleft chamber 119901

1gradually increases As a result the oil in

this chamber is compressed until the pressure of the leftchamber119901

1is higher than the system pressure119901

119904 as shown in

Figure 4(a) When 1199011is high enough a phenomenon about

oil backward will appear because this pressure exceedingthe system pressure is so high that the oil retrogrades backinto the tank as shown in Figure 4(b) Then the compressedoil starts to expand gradually so that the inertia force isreduced to zero and then the piston is driven to make reverse

motion which could be obtained from arrowhead directionof displacement 119910

119901in Figure 4(c) This makes the volume

of left chamber be higher and the pressure of left chamberdecrease Until 119901

1becomes lower than 119901

119904 the oil stops

backing into the tank and repeats to enter the left cylinderchamber and exit from the right cylinder to drive the pistonmoving to the right as shown in Figure 4(d)When the pistonmoves to the center position of hydraulic cylinder valve ports1 and 3 are closed

As the spool of 2D valve spins the high-pressure oil flowsinto the right chamber of the cylinder through the valve port2 and out from the left chamber through the valve port 4(valve ports 2 and 4 are open the orifice areas are denotedby 119860V2 and 119860V4) the piston is driven to move rightwardunder the effect of inertia force The volume of the rightchamber decreases and then the pressure of the right chamber1199012gradually increases When this pressure 119901

2is higher than

the system pressure 119901119904 the phenomenon about oil backward

will appear again as shown in Figures 5(a) and 5(b) Thenthe compressed oil in the right chamber starts to expandgradually so that inertia force is reduced to zero and then thepiston is provided direction reversal This makes the volumeof right chamber become higher and the pressure of rightchamber decrease When 119901

2is lower than 119901


backing into the tank and flows into the right chamber andout of the left chamber (the flow rates are resp 119876V2 and119876V4) to drive the piston moving to the left It is illustratedby Figures 5(c) and 5(d) When the piston moves to thecenter position of hydraulic cylinder valve ports 2 and 4are closed and valve ports 1 and 3 are open simultaneouslyConsequently the piston of hydraulic cylinder is driven tomake reciprocating motion A vibration excited is achievedunder the condition of resonant frequency

32 Dynamic Analysis In making a dynamic analysis itis necessary that the nonlinear algebraic equations which


p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(a)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(b)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(c)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(d)

Figure 4 Resonance process of piston being in the left chamber (a) (b) (c) and (d)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(a)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(b)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(c)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(d)

Figure 5 Resonance process of piston being in the right chamber (a) (b) (c) and (d)

describe the pressure-flow curves be linearized A generalexpression for the load flow is

119876119871= 119870119902119883V minus 119870119888119875119871 (1)

where 119876119871is then flow through the load 119870

119902is the flow gain

119883V is the valve position 119870119888 is the flow-pressure coefficientand 119875

119871is the pressure drop across the load

In the combination of 2D valve and piston ideal geometryof 2D valve ports is completely analogous to matched andsymmetrical orifices of critical center servo valve So applyingthe continuity equation to each of the piston chambers yields

119876119871=

1198761+ 1198762

2

= 119896119894119888(1199011minus 1199012) +

1

2

119896119890119888(1199011minus 1199012) +

119860119901119889119910119901

119889119905

+

119881011198891199011

2119864ℎ119889119905

minus

119881021198891199012

2119864ℎ119889119905

+

119860119901119910119901

2

(

1198891199011

119889119905

+

1198891199012

119889119905

)

(2)

where11987611198762are forward and return flows119901

11199012are forward

and return pressures 119896119894119888is internal or cross-port leakage

coefficient of piston 119896119890119888

is external leakage coefficient ofpiston 119910

119901is displacement of piston 119860

119901is the area of piston

11988101is initial volume of forward chamber119881

02is initial volume

of return chamber and 119864ℎis the bulk modulus of oil

It is assumed that the piston is centered and the volumesof the piston chambers are equal that is 119881

01= 11988102= 1198811199052

This assumption is made that vibration amplitude is smallerin high working frequency so that 119860

119901sdot 119910119901is ignored here

Therefore

119876119871=

1198761+ 1198762

2

= 119896119905119888119901119871+

119860119901119889119910119901

119889119905

+

119881119905119889119901119871

4119864ℎ119889119905

(3)

where 119896119905119888is the total leakage coefficient of piston and119881

119905is the

total volume of fluid under compression in both chambersThe volume and continuity expressions can be Laplace-

transformed to yield

119876119871= 119860119901119904119884119901+ [119896119905119888+

119881119905

4119864ℎ

119904] 119901119871 (4)

The resulting force equation Laplace-transformed is

119901119871=

1

119860119901

[(1198981199042

+ 119861119901119904 + 119870119871) 119884119901+ 119865119871] (5)

where119898 is the totalmass of piston and load referred to piston119861119901is viscous damping coefficient of piston and load 119870

119871is

load spring gradient and 119865119871is arbitrary load force on piston

The three basic equations (1) (4) and (5) are representedin block diagram fashion in Figure 6


XKq

+ +

minus

minuspL

Ap

Yp

Aps

FL

1

ms2 + Bps + KL

1

ktc +Vt4Eh

s + Kc

Figure 6 Diagram based on obtaining piston position from flow

Nonlinear

++

minus

minus

Aps

Linear

A(120579)QL

f(pL)

pL 4Eh

Vts

4Eh

Vts

YpAps

1

ms2 + Bps + KL

Yp

Figure 7 Block diagram of electrohydraulic exciter controlled by 2D valve

The load flow as a function of valve position and loadpressure being linearized can give a complete description ofsteady-state valve performance However in electrohydraulicexcited system controlled by 2D the load pressure with high-frequency period variations has not been satisfied for linearconditions So the block diagram of this model is shown inFigure 7

The transfer function for this condition is

119884119901

119876119871

= (

4119864ℎ119860119901

(119881119905119870119871+ 4119864ℎ1198602

119901

)

)

times (119904(

1199042

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119898)

+

119904

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119861119901)

+ 1))

minus1

=

119870119881

119904 (1199042

1205962

0

+ (21205850119904) (120596

0) + 1)

(6)

Analysis of describing function equation (6) yields theresonant frequency

1205960=radic

119881119905119870119871+ 41198602

119901

119864ℎ

119881119905119898

(7)

Obviously the resonant frequency for electrohydraulicexciter controlled by 2D valve consists of hydraulic naturalpart and mechanical part

33 Resonant Peak There is resonance in this electrohy-draulic exciter system when working frequency is increasedto systemnatural frequencyThis phenomenon causes excitedoutput oscillating with a fixed amplitude and frequencyBecause excited waveform just contains a dominant fre-quency but no other harmonics it is assumed as follows

119910119901= minus119860

119901119891sin (119911120579) (8)

where 119860119901119891

is resonant peakAnd the pressure drop across the load under resonant

frequency is

119901119871= 119901119871119891sin (119911120579) (9)

where 119901119871119891

is amplitude of load pressureAs discussed in Section 31 flow equations are rewritten

as

119876119871=

sign (119901119904minus 119901119871) 1198620119860V1radic

1003816100381610038161003816119901119904minus 119901119871

1003816100381610038161003816

120579 isin [0 2120572]

minus sign (119901119904+ 119901119871) 1198620119860V2radic

1003816100381610038161003816119901119904+ 119901119871

1003816100381610038161003816120579 isin [2120572 4120572]

(10)


(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)

Figure 8 Photo of experimental system (a) and schematic diagram of experimental system of high-frequency electrohydraulic exciter (b)1 force cell 2 cylinder 3 displacement transducer 4 pressure sensor for the left chamber 5 pressure sensor for the right chamber 6 servovalve and 7 2D valve

where 1198620is a constant related to flow rate coefficient and oil

densityOn the basis of energy conservation input energy is equal

to output energy in a cycle that is

4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))

times 1198620119860V1radic

10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)

Equations (9) (10) and (11) can be solved simultaneouslyto obtain

119901119871119891= 119908119901119904 (13)

where 119908 is a pressure ratioAssuming mass force and spring force dominant force

equation on the piston is

119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)

Therefore these equations may then be combined to yieldthe resulting resonant peak

119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)

It is clear that resonant peak depends on electrohydraulicexciter system and is irrelevant to input parameters includingthe orifice areas or shapes of valve ports

4 Experiments and Results

The experimental system is illustrated by reference toFigure 8 which mainly consists of the combination of 2Dvalve and piston acquisition elements and a control partThis exciter is mostly applied to the high-cycle fatigue testingmachine so the load force can be considered as the frame inan axial direction of the cylinder A displacement transduceris chosen to be mounted inside the cylinder rod to measurethe displacement of the piston A force cell is placed betweenthe end of the cylinder rod and the rigid frame to measurethe output force of the cylinderrsquos piston And two pressuresensors respectively are used to acquire the pressures in theright and left chambers These signals are amplified and thenare sent to industrial computer to display save and furthertreatThe frequency and amplitude signals are sent separatelyto two motors to control the rotary and linear motion of 2Dvalversquos spoolThe bias signal is sent to a servo valve connectedin parallel with 2D valve to control the bias displacement ofoutput vibration

The partial experimental excited waveforms are shown inFigure 9 And the relationship between vibration amplitudeand working frequency is summarized and also presentedin this figure The vibration amplitude descends rapidly withthe working frequency increasing but tends to be flat as thisfrequency is much close to the natural frequency


0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)

Figure 9 Excited waveform 119891 = 40Hz (a) excited waveform 119891 = 300Hz (b) excited waveform 119891 = 800Hz (c) and amplitude-frequencycurve (d)

Primary resonant results are given in Figure 10 whichincludes the displacement waveform pressure waveform ineach chamber and frequency composition

By referring to Figure 10 it can be seen that vibrationamplitude is increased obviously at resonant frequency pointBut pressure in each chamber is not higher than systempressure which causes experimental resonant amplitudewhich is smaller than the theoretical result A major source isthat viscous force is completely ignored in resonant analysiseven though it is not dominant and is a soft quantity that isdifficult to be measured and computed

5 Conclusions

A 2D valve is adopted to control electrohydraulic exciterfor working frequency greatly increasing and even reachingthe resonant frequency Hydraulic resonance is a peculiarresonant phenomenon so it is necessary to analyse its charac-teristic especially within the context of energy conservationTheoretical resonant peak is a constant and its analyticalexpression is given which just depending on this excitersystem itself and not like mechanical resonant amplitudebecomes higher to destroy the whole system at resonant


0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)

Figure 10 Resonantwaveform119891 = 1800Hz (a) pressure of the left chamber (b) pressure of the right chamber (c) and frequency composition(d)

frequency The experimental system is built to verify thetheoretical analysis Though there is a coupling relationshipbetween vibration amplitude and working frequency thisamplitude is suddenly enlarged to resonant peak at resonantfrequency Consequently resonant energy can be used inhigh-cycle fatigue test to reduce the external input to this sys-tem and improve the vibration amplitude at high frequency

Conflict of Interests

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

Acknowledgments

The authors are grateful to the support from the Sci-ence Foundation of Department of Education of ZhejiangProvince (no Y201430397) and the Postdoctoral ScienceFoundation of China (no 2013M541798)

References

[1] W Schutz ldquoA history of fatiguerdquo Engineering Fracture Mechan-ics vol 54 no 2 pp 263ndash300 1996


[2] Y Huang S Liu and J Zhao ldquoOptimal design of two-dimen-sional band-gap materials for uni-directional wave propaga-tionrdquo Structural andMultidisciplinary Optimization vol 48 no3 pp 487ndash499 2013

[3] H Tian D Fielden M J Kirkham and P K Liaw ldquoControl ofnoise and specimen temperature during 1 kHz fatigue experi-mentsrdquo Journal of Testing and Evaluation vol 34 no 2 pp 92ndash97 2006

[4] D Lanning G K Haritos T Nicholas and D C MaxwellldquoLow-cycle fatiguehigh-cycle fatigue interactions in notchedti-6ai-4vrdquo Fatigue and Fracture of Engineering Materials andStructures vol 24 no 9 pp 565ndash577 2001

[5] D L Davidson J B Campbell and R A Page ldquoThe initiationand growth of fatigue cracks in a titanium aluminide alloyrdquoMetallurgical Transactions A vol 22 no 2 pp 377ndash391 1991

[6] X Du D Niu and W Liao ldquoDesign and experimental studieson the inverse control magneto-rheological damperrdquo Journal ofVibration and Shock vol 25 no 5 pp 49ndash53 2006

[7] J M Morgan and W W Milligan ldquoA 1 kHz servohydraulicfatigue testing systemrdquo in Proceeding of the Conference HighCycle Fatigue of Structural Materials pp 305ndash312 WarrendalePa USA 1997

[8] C-T Chen J-C Renn and Z-Y Yan ldquoExperimental identifi-cation of inertial and friction parameters for electro-hydraulicmotion simulatorsrdquoMechatronics vol 21 no 1 pp 1ndash10 2011

[9] J G Hao and Y C Zhang ldquoStudy on the properties of newelectro-hydraulic exciting systemrdquo Journal of TaiyuanUniversityof Technology vol 34 no 6 pp 706ndash709 2003

[10] Y Ren and J Ruan ldquoRegulating characteristics of an electro-hydraulic vibrator multiply controlled by the combination of atwo-dimensional valve and a standard servo valverdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 227 no 12 pp 2707ndash27232013

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Hao and Zhang [9] used a rotary valve in exciter to improvevibration frequency By controlling this rotary valve the spoolwill make a reciprocating motion to switch pressurized flowrate to chambers of the cylinder or motor alternatively andthus a vibration is created to the piston of the cylinder andconnected load But vibration amplitude was dominated bythe oilrsquos pressure and thus had low accuracy and the offsetcontrol was difficult to achieve

The hydraulic exciter is much restrained by the dynamicperformance of the servo valve which is difficult to enhanceworking frequency to a very high level A novel scheme for anelectrohydraulic exciter controlled by a 2D valve is thereforeproposed to achieve higher frequency The frequency andthe amplitude of sinusoidal vibration wave are separatelycontrolled by circumferential rotation and axial sliding ofspool respectively [10] But the relationship of vibrationfrequency and the amplitudes is intercoupling real data isscarce Therfore a promising solution is increasing workingfrequency up to natural frequency of hydraulic systemWhen frequency of the excited vibration comes close tothe natural frequency resonance occurs and can improvevibration amplitude greatly This hydraulic resonance willbe researched and discussed through theoretical analysisand experimental system and finally some conclusions aregiven

2 Working Principle

2D valve as a necessary control component is used inhigh-frequency electrohydraulic exciter because of this valversquosspecial structure 2D valve has two degrees of freedom therotational and sliding motions of the spool Such structurenot only acts a two-stage valve with pilot stage and directposition feedback but also greatly improves the excitedfrequency by increasing spoolrsquos rotational speed The spool iscontinuously rotated by an electric motor and gear combina-tion and sliding by an eccentric mechanism driven by otherelectric motors as shown in Figure 1

The rotation of the 2D valve spool can produce thealternatively varied valve port area which is formed betweenthe grooves on a spool land and the rectangular windowson the sleeve There is a series of grooves symmetricallydistributed on the spools (central angle of every groove is120572) coordinated with the windows distributed uniformly onthe sleeve which create an ldquoalternately variedrdquo valve port areaas shown in Figure 2 Then geometrically the groove has 2120572angle difference with respect to one on any two adjacentshoulders Such arrangement will cause a waveform with180∘ phase angle between the meter-in valve port and themeter-out valve port The varied valve port area depends onrotary speed and axial sliding of the spool Consequentlythe electrohydraulic vibrator controlled by 2D valve couldrealize that the frequency and the amplitude of vibration aredominated respectively by the rotary speed and the slidingdisplacement of the spool of 2D valve

The combination of 2D valve and a piston is selected as ahydraulic power element to produce a reciprocating motionOne chamber of the biactuation cylinder is controlled by

1 2 34

5

6

7

Figure 1 2D valve with twomotors 1 valve body 2 sleeve 3 spool4 eccentric mechanism 5 stepping motor 6 electric motor and 7gearbox

3

1

2

120572

120596

Figure 2 Coupling pattern of 2D valve 1 groove on the spool 2window on the sleeve and 3 rectangular area

a hydraulic bridge formed by meter-in and meter-out alter-natively varied valve ports Another chamber is a hydraulicbridge with a 180∘ phase angle difference As the spool ofthe 2D valve makes a rotary motion the pressure inside twochambers of the cylinder will change alternatively to makethe piston output a reciprocating vibration by reference toFigures 3(a) and 3(b) Obviously it is easy to obtain high-frequency vibration by improving rotary speed of the spoolThe working frequency is also related to the number of thegrooves on a single spool land and the coupling patternbetween the grooves on the spool land and the windows onthe sleeve However this is achieved at the price of sacrificingthe magnitude of the 2D valve orifice area that decides theamplitude of vibration So in high-frequency section theexcited amplitude will be to a large extent limited


TTBA P1

3

IIII

2 4 5

6

120596

p1p2

x

yp

(a)

PAB T

T1

6

IIIV

32 4 5

120596

p1p2

x

yp

(b)








119904 as shown in










2is higher than








p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(a)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(b)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(c)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(d)


p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(a)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(b)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(c)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(d)



119876119871= 119870119902119883V minus 119870119888119875119871 (1)






119876119871=

1198761+ 1198762

2

= 119896119894119888(1199011minus 1199012) +

1

2

119896119890119888(1199011minus 1199012) +

119860119901119889119910119901

119889119905

+

119881011198891199011

2119864ℎ119889119905

minus

119881021198891199012

2119864ℎ119889119905

+

119860119901119910119901

2

(

1198891199011

119889119905

+

1198891199012

119889119905

)

(2)


11199012are forward







02is initial volume



01= 11988102= 1198811199052



Therefore

119876119871=

1198761+ 1198762

2

= 119896119905119888119901119871+

119860119901119889119910119901

119889119905

+

119881119905119889119901119871

4119864ℎ119889119905

(3)


119905is the



119876119871= 119860119901119904119884119901+ [119896119905119888+

119881119905

4119864ℎ

119904] 119901119871 (4)


119901119871=

1

119860119901

[(1198981199042

+ 119861119901119904 + 119870119871) 119884119901+ 119865119871] (5)


119871is




XKq

+ +

minus

minuspL

Ap

Yp

Aps

FL

1

ms2 + Bps + KL

1

ktc +Vt4Eh

s + Kc


Nonlinear

++

minus

minus

Aps

Linear

A(120579)QL

f(pL)

pL 4Eh

Vts

4Eh

Vts

YpAps

1

ms2 + Bps + KL

Yp




119884119901

119876119871

= (

4119864ℎ119860119901

(119881119905119870119871+ 4119864ℎ1198602

119901

)

)

times (119904(

1199042

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119898)

+

119904

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119861119901)

+ 1))

minus1

=

119870119881

119904 (1199042

1205962

0

+ (21205850119904) (120596

0) + 1)

(6)


1205960=radic

119881119905119870119871+ 41198602

119901

119864ℎ

119881119905119898

(7)



119910119901= minus119860

119901119891sin (119911120579) (8)

where 119860119901119891


frequency is

119901119871= 119901119871119891sin (119911120579) (9)

where 119901119871119891


as

119876119871=


1003816100381610038161003816119901119904minus 119901119871

1003816100381610038161003816

120579 isin [0 2120572]


1003816100381610038161003816119901119904+ 119901119871

1003816100381610038161003816120579 isin [2120572 4120572]

(10)


(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)





4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)


119901119871119891= 119908119901119904 (13)



119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)


119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)






0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TTBA P1

3

IIII

2 4 5

6

120596

p1p2

x

yp

(a)

PAB T

T1

6

IIIV

32 4 5

120596

p1p2

x

yp

(b)








119904 as shown in










2is higher than








p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(a)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(b)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(c)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(d)


p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(a)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(b)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(c)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(d)



119876119871= 119870119902119883V minus 119870119888119875119871 (1)






119876119871=

1198761+ 1198762

2

= 119896119894119888(1199011minus 1199012) +

1

2

119896119890119888(1199011minus 1199012) +

119860119901119889119910119901

119889119905

+

119881011198891199011

2119864ℎ119889119905

minus

119881021198891199012

2119864ℎ119889119905

+

119860119901119910119901

2

(

1198891199011

119889119905

+

1198891199012

119889119905

)

(2)


11199012are forward







02is initial volume



01= 11988102= 1198811199052



Therefore

119876119871=

1198761+ 1198762

2

= 119896119905119888119901119871+

119860119901119889119910119901

119889119905

+

119881119905119889119901119871

4119864ℎ119889119905

(3)


119905is the



119876119871= 119860119901119904119884119901+ [119896119905119888+

119881119905

4119864ℎ

119904] 119901119871 (4)


119901119871=

1

119860119901

[(1198981199042

+ 119861119901119904 + 119870119871) 119884119901+ 119865119871] (5)


119871is




XKq

+ +

minus

minuspL

Ap

Yp

Aps

FL

1

ms2 + Bps + KL

1

ktc +Vt4Eh

s + Kc


Nonlinear

++

minus

minus

Aps

Linear

A(120579)QL

f(pL)

pL 4Eh

Vts

4Eh

Vts

YpAps

1

ms2 + Bps + KL

Yp




119884119901

119876119871

= (

4119864ℎ119860119901

(119881119905119870119871+ 4119864ℎ1198602

119901

)

)

times (119904(

1199042

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119898)

+

119904

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119861119901)

+ 1))

minus1

=

119870119881

119904 (1199042

1205962

0

+ (21205850119904) (120596

0) + 1)

(6)


1205960=radic

119881119905119870119871+ 41198602

119901

119864ℎ

119881119905119898

(7)



119910119901= minus119860

119901119891sin (119911120579) (8)

where 119860119901119891


frequency is

119901119871= 119901119871119891sin (119911120579) (9)

where 119901119871119891


as

119876119871=


1003816100381610038161003816119901119904minus 119901119871

1003816100381610038161003816

120579 isin [0 2120572]


1003816100381610038161003816119901119904+ 119901119871

1003816100381610038161003816120579 isin [2120572 4120572]

(10)


(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)





4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)


119901119871119891= 119908119901119904 (13)



119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)


119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)






0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(a)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(b)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(c)

p1 p2

A1A2

A3A4

yp

ps

Q1 Q3

(d)


p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(a)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(b)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(c)

p1 p2

A1 A2

A3A4

yp

ps

Q2Q4

(d)



119876119871= 119870119902119883V minus 119870119888119875119871 (1)






119876119871=

1198761+ 1198762

2

= 119896119894119888(1199011minus 1199012) +

1

2

119896119890119888(1199011minus 1199012) +

119860119901119889119910119901

119889119905

+

119881011198891199011

2119864ℎ119889119905

minus

119881021198891199012

2119864ℎ119889119905

+

119860119901119910119901

2

(

1198891199011

119889119905

+

1198891199012

119889119905

)

(2)


11199012are forward







02is initial volume



01= 11988102= 1198811199052



Therefore

119876119871=

1198761+ 1198762

2

= 119896119905119888119901119871+

119860119901119889119910119901

119889119905

+

119881119905119889119901119871

4119864ℎ119889119905

(3)


119905is the



119876119871= 119860119901119904119884119901+ [119896119905119888+

119881119905

4119864ℎ

119904] 119901119871 (4)


119901119871=

1

119860119901

[(1198981199042

+ 119861119901119904 + 119870119871) 119884119901+ 119865119871] (5)


119871is




XKq

+ +

minus

minuspL

Ap

Yp

Aps

FL

1

ms2 + Bps + KL

1

ktc +Vt4Eh

s + Kc


Nonlinear

++

minus

minus

Aps

Linear

A(120579)QL

f(pL)

pL 4Eh

Vts

4Eh

Vts

YpAps

1

ms2 + Bps + KL

Yp




119884119901

119876119871

= (

4119864ℎ119860119901

(119881119905119870119871+ 4119864ℎ1198602

119901

)

)

times (119904(

1199042

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119898)

+

119904

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119861119901)

+ 1))

minus1

=

119870119881

119904 (1199042

1205962

0

+ (21205850119904) (120596

0) + 1)

(6)


1205960=radic

119881119905119870119871+ 41198602

119901

119864ℎ

119881119905119898

(7)



119910119901= minus119860

119901119891sin (119911120579) (8)

where 119860119901119891


frequency is

119901119871= 119901119871119891sin (119911120579) (9)

where 119901119871119891


as

119876119871=


1003816100381610038161003816119901119904minus 119901119871

1003816100381610038161003816

120579 isin [0 2120572]


1003816100381610038161003816119901119904+ 119901119871

1003816100381610038161003816120579 isin [2120572 4120572]

(10)


(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)





4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)


119901119871119891= 119908119901119904 (13)



119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)


119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)






0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










XKq

+ +

minus

minuspL

Ap

Yp

Aps

FL

1

ms2 + Bps + KL

1

ktc +Vt4Eh

s + Kc


Nonlinear

++

minus

minus

Aps

Linear

A(120579)QL

f(pL)

pL 4Eh

Vts

4Eh

Vts

YpAps

1

ms2 + Bps + KL

Yp




119884119901

119876119871

= (

4119864ℎ119860119901

(119881119905119870119871+ 4119864ℎ1198602

119901

)

)

times (119904(

1199042

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119898)

+

119904

(119881119905119870119871+ 4119864ℎ1198602

119901

) (119881119905119861119901)

+ 1))

minus1

=

119870119881

119904 (1199042

1205962

0

+ (21205850119904) (120596

0) + 1)

(6)


1205960=radic

119881119905119870119871+ 41198602

119901

119864ℎ

119881119905119898

(7)



119910119901= minus119860

119901119891sin (119911120579) (8)

where 119860119901119891


frequency is

119901119871= 119901119871119891sin (119911120579) (9)

where 119901119871119891


as

119876119871=


1003816100381610038161003816119901119904minus 119901119871

1003816100381610038161003816

120579 isin [0 2120572]


1003816100381610038161003816119901119904+ 119901119871

1003816100381610038161003816120579 isin [2120572 4120572]

(10)


(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)





4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)


119901119871119891= 119908119901119904 (13)



119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)


119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)






0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

Amplifier

PCI 1714

BL2600

7

1 2 3

Frequencycontroller

6 T

T

5 4

Amplitudecontroller

Biascontroller

ps

ps

(b)





4

sum

1

119868119894= int

4120572

0

119876119871119901119871119889120579 = 0 (11)

where

1198681+ 1198682

= int

2120572

0

119901119871119891sin (119911120579) sign (119901

119904minus 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904minus 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

1198683+ 1198684

= int

4120572

2120572

minus119901119871119891sin (119911120579) sign (119901

119904+ 119901119871119891sin (119911120579))


10038161003816100381610038161003816119901119904+ 119901119871119891sin (119911120579)1003816100381610038161003816

1003816119889120579

(12)


119901119871119891= 119908119901119904 (13)



119860119901119901119871= 119898120596

2

0

1198892

119910119901

1198891205792

+ 119870119871119910119901

(14)


119860119901119891=

119881119905119901119871119891

4119860119901119864ℎ

(15)






0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 04 08 12 16 2

0

02

04

06

08

minus02

minus04

minus06

minus08

1205794120572

ypy

max

(a)

0 04 08 12 16 2

0

005

01

minus005

minus01

1205794120572

ypy

max

(b)

04 08 12 16 2

0

002

004

minus002

minus0040

1205794120572

ypy

max

(c)

0 200 400 600 800 10000

02

04

06

08

1

|yp|y

max

f (Hz)f (Hz)

(d)




5 Conclusions



0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 04 08 12 16 2

0

02

04

minus02

minus04

1205794120572

ypy

max

(a)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p1p

s(b)

0 04 08 12 16 20

01

02

03

04

05

1205794120572

p2p

s

(c)

0 2 4 6 8 100

02

04

06

08

1

|Ai|A

0

1205961205960

(d)





Acknowledgments


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article resonance analysis of high-frequency

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