Research Article An Ultra-Low Frequency Two DOFs Vibration ...

Research ArticleAn Ultra-Low Frequency Two DOFsrsquo Vibration IsolatorUsing Positive and Negative Stiffness in Parallel

Min Wang Xuedong Chen and Xiaoqing Li

The State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and TechnologyWuhan Hubei China

Correspondence should be addressed to Xiaoqing Li xqlihusteducn

Received 26 July 2016 Revised 27 October 2016 Accepted 3 November 2016

Academic Editor Roland Hildebrand

Copyright copy 2016 Min Wang 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

With the improvement of performance in the ultra-precision manufacturing engineering the requirements for vibration isolationhave become increasingly stringent In order to get wider effective bandwidth and higher performance of vibration isolation inmultiple DOFs system an ultra-low frequency twoDOFsrsquo vibration isolator with positive and negative stiffness in parallel (PNSP) isproposedThe two DOFsrsquo isolator which combines a positive stiffness (PS) air spring with a negative stiffness (NS) magnetic springin parallel and combines a PS flat spring with an NS inverted pendulum in parallel is designed to reduce the natural frequencyand broaden the effective bandwidth in horizontal and vertical direction Based on this structure stiffness models of differentcomponents in different directions are established Compared with a PS isolator it possesses the characteristic of high-static-low-dynamic stiffnessThe simulation curves also provide strong evidence Last a real-time active control system and a spectrum testingand analysis system are used for the contrast experiment between the mentioned PNSP structure and PS only The experimentalresults demonstrate that the isolator with PNSP can obviously reduce the natural frequency to 1Hz and simultaneously maintainthe stability of the system and consequently verify the validity and superiority of the mentioned structure

1 Introduction

With the improvement of performance in the ultra-precisionmanufacturing and measuring equipment the requirementsfor vibration isolation have become increasingly stringent [1ndash3] Due to these problems an effective control mechanism ora new structure is required to attenuate the vibration in orderto preserve structural integrity of such systems [4] An ultra-precision vibration isolation system with high performancehas to break the routine to meet the strict requirements Thenew system with low natural frequency and efficient activecontrol technology is put forward to realize multi-degreeof freedom (DOF) vibration isolation and wide effectivebandwidth

Absolute velocity feedback control can create a ldquosky-hookrdquo damping to reduce the amplitude of resonance peakeffectively without compromising high frequency perfor-mance [5] Feedforward control shows a significant effect bypredicting in advance against the vibration of the base plat-form and direct disturbance of the machine It can effectively

improve the performance of vibration isolation system [6 7]However the realization of ultra-low frequency still dependson the design of passive structure

Low stiffness can effectively reduce the natural frequencyof the vibration isolation system and broaden the effectivebandwidth of vibration isolation system However low stiff-ness is more likely to lead to instability of the vibrationisolation system Platus [8] from Minus K proposed anegative stiffness mechanism (NSM) based on the bucklingof prestressed bars and a vibration isolation system usingthe NSM was designed to obtain a lower natural frequencyFulcher et al [9] investigate the bistability and negativestiffness of a buckled beam mechanism in the vibration andshock isolation systems Mizuno [10 11] puts forward a novelisolator which combines a kind of zero power control NSmagnetic spring with a PS spring in series It shows a goodaspect to resist the effect of vibration isolation directly on thepayload direct disturbance but not quite fit the base vibration

Under the condition that the carrying capacity is con-stant the continuous reduction of the stiffness will lead

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3728397 15 pageshttpdxdoiorg10115520163728397

2 Mathematical Problems in Engineering

Payload Mp

Base mB

xo

xi

ckini

(a)

Payload Mp

Base mB

xo

c

xi

kini

kadd

(b)

Payload Mp

Base mB

xo

c

xi

xm

kini

kadd

(c)

Figure 1 Vibration isolation principle diagram (a) traditional (b) stiffness in parallel and (c) stiffness in series

to the formation of quasi-zero-stiffness (QZS) system [12ndash15] QZS mechanisms are generally achieved by combininga negative stiffness (NS) element with a positive stiffness(PS) element It has been found that to achieve a largeexcursion from the static equilibrium position such that thestiffness of the system does not exceed a prescribed valuethere is an optimum geometry and a corresponding optimumrelationship in the stiffness [14]

Based on QZS system Carrella et al [14 16] put for-ward the concept of the high-static-low-dynamic stiffness(HSLDS) mount which comprises two vertical mechanicalsprings and a permanent magnet A reduction in the naturalfrequency from 14Hz to 7Hz was achieved by using the mag-nets and the measured transmissibility from a displacementinput showed that the device behaved approximately linearlyRobertson et al [12] introduced theoretical analysis for afully HSLDS magnetic spring for vibration isolation wherethe payload was also supported by magnetism Shaw et al[17] presented dynamic analysis of HSLDS vibration isolationmounts to provide useful insight into the design of HSLDSmounts However there are some problems in the existingvibration isolators based on the efforts of QZS and HSLDSSpecific performances are as follows the natural frequencyof existed structure is still not competitive when facedwith a higher demand There are still some shortcomingsin the application of multi-DOFs vibration isolation systemTherefore one structure of urgent demands to solve theseproblems is becoming more and more necessary

In this paper to get wider effective bandwidth and higherperformance of vibration isolation in multiple DOFs systeman ultra-low frequency two DOFsrsquo vibration isolator withpositive and negative stiffness in parallel (PNSP) is proposedwhich can realize this requirement that ensures lower naturalfrequency and large capacity at the same time [12] and thefunction that isolation can be achieved simultaneously intwo DOFs Compared with a PS isolator it can also possessthe characteristic of HSLDS The ultra-low frequency twoDOFsrsquo isolator which combines a PS air spring with a NSmagnetic spring in parallel and combines a PS flat spring withan NS inverted pendulum in parallel is designed to reducethe natural frequency and broaden the effective bandwidthin horizontal and vertical direction The bearing capacity isdetermined by the PS structure which can effectively improve

the static stiffness and the stability of the vibration systemand maintain the natural frequency which is constant withthe changing of payload NS element is employed to reducethe dynamic stiffness and the natural frequency of vibrationsystem

To solve the problem of the existence of vibration isola-tion inmulti-DOFs at the same time three identical vibrationisolators proposed in this paper can be used together toestablish an ultra-precision vibration isolation system whichcan realize the ultra-low frequency vibration isolation in sixDOFs through the transform matrix of different coordinatesystems in this paper And it also can ensure the positioningaccuracy of the system simultaneously

2 Design of Two DOFsrsquo Ultra-Low FrequencyVibration Isolator

21 The Principle of PNSP The traditional passive vibrationisolation unit is composed of mass-spring-damping the sim-plified principle diagram is shown in Figure 1(a) Commonways of changing the system stiffness are the method of seriesconnection or parallel connection with additional stiffnessFigures 1(b) and 1(c) show the simplified principle diagram

Theway of series can only reduce the system stiffness butit cannot achieve the QZS systemThis can be improved uponby incorporating a negative stiffness element in parallel suchthat the dynamic stiffness ismuch less than the static stiffnessThe way to achieve this result of high-static stiffness andlow-dynamic stiffness is that the negative stiffness element isadded near the working position of the isolator to form thePNSP mechanism

According toNewton-Euler equation the dynamic math-ematical equations of the vibration isolation system withtraditional passive vibration isolation and with additionalstiffness in parallel are respectively written as

119872119901119900 + 119888 (119900 minus 119894) + 119896ini (119909119900 minus 119909119894) = 0 (1)

119872119901119900 + 119888 (119900 minus 119894) + 119896120578 (119909119900 minus 119909119894) = 0 (2)

where119872119901 is the mass of payload platform 119898119861 is the mass ofbase platform 119888 is the equivalent damping of the system 119909119900is the displacement deformation of the payload platform 119909119894

Mathematical Problems in Engineering 3

0 24 6

810

02000

40006000

800010000

0

20

40

60

80

100

times105

Nat

ural

freq

uenc

y120596n

(rad

s)

Payload mass Mp (kg) System stiffness k120578

(Nm)

(a)

minus1 minus05 0 05 10

5

10

15

20

25

30

times106

Nat

ural

freq

uenc

y120596n

(rad

s)

Additional stiffness kadd (Nm)

(b)

Figure 2 The simulation curves (a) the relationship between system stiffness (119896120578 isin (0 11198906)Nm) payload mass (119872119901 isin (0 11198904) kg) andnatural frequency 120596119899 (b) the relationship between additional stiffness (119896add isin (0 11198906)Nm) and natural frequency 120596119899 when initial stiffness(119896ini = 9281198905Nm) and payload mass (119872119901 = 3000 kg) are constantis the displacement deformation of the base platform 119896120578 isthe equivalent system stiffness after the change and 119896ini is theinitial equivalent stiffness

According to the principle of vibration system the naturalfrequency of vibration isolation system has the followingmathematical relations with the system stiffness and thesystem payload

120596119899 = radic 119896120578119872119901 = radic119896ini + 119896add119872119901119896add = 1198721199011205962119899 minus 119896ini

(3)

where120596119899 is the natural frequency of vibration system and 119896addis the additional stiffness

Equation (3) is simulated and analyzed and is shown inFigure 2 As can be seen from Figure 2(a) the natural fre-quency of the system is obviously changed with the payloadof the system (especially the initial payload mass segment)when the system stiffness is constant Reducing the naturalfrequency can be achieved bymeans of enlarging the payloadof the system and simultaneously maintaining low systemstiffness However the payload cannot always increase Andin some cases the payload is constant

Figure 2(b) shows that the relationship between the nat-ural frequency and the additional stiffness is nonlinear andsimilar to the inverse exponential curve when the payloadand initial stiffness are constant The natural frequency is thetrend of steep rise in the condition that the additional stiffnessis from negative to zero the natural frequency of the systemis into the trend of slow rise trend in the condition that theadditional stiffness is from zero to a greater positive So whenthe negative stiffness is adopted as the additional stiffness thelower the value is the more obvious the change is and theeasier it is to form a quasi-zero stiffness system

Base

PayloadPS for large

capacityNS for lower

natural frequencyNS

NS

Inve

rted

pen

dulu

m

Mag

netic

sprin

g

Airspring

Flat

sprin

g

Parallel

Horizontal stiffness in parallel

Vertical stiffness in parallel

Kpo Kne

Figure 3The structure sketch of passive isolator using PNSP

22The Structure of TwoDOFsrsquo Isolator The frequency rangeover which a linear passive vibration isolator is effective isoften limited by the mount stiffness required to support astatic load This can be improved upon by incorporating anegative stiffness element to form a high-static-low-dynamicstiffness mount which can be approximated to achieve thequasi-zero stiffness system and the structure sketch is shownin Figure 3

According to the structure sketch of Figure 3 theschematic diagram of stiffness in parallel is proposed sepa-rately from the horizontal and vertical direction The follow-ing can be seen from Figure 4

In vertical direction the structure which combines a PSair spring with a NSmagnetic spring in parallel is mentionedThe PS air spring can support the mass of payload platformand the NS magnetic spring is responsible for realizing ultra-low frequency vibration isolation The stiffness of the systemcan be changed by adjusting the position of the middlemagnet

In horizontal direction the structure which combines aPS flat spring with a NS inverted pendulum in parallel isused to attain ultra-low frequency vibration isolation TheNS


Mag

netic

sprin

g

Air spring


N

SN

S

Base

Payload

Kpo Kne

(a)

L

x fminusf

mgl

Invertedpendulum

Flatspring


Base

Payload

Flexiblejoint

KpoKne

(b)

Figure 4 The schematic diagram of stiffness in parallel (a) vertical direction (b) horizontal direction

inverted pendulum is realized through the reverse installingof the air spring When the payload is very large (about1000kg) the cross-sectional area of the air spring is relativelysmall which can be equivalent to simply supported beamstructure When one end of the simply supported beamis installed on the base platform and can be rotated theother end is connected with the payload platform and alsocan be rotated A pendulum structure that adopts invertedinstallation is formed

At the same time in order to eliminate the influence ofthe force from the other directions a flexible joint with lowerbending stiffness in theDOFs not required is adopted to servefor horizontal and vertical motion decoupling A two DOFsrsquoultra-low frequency vibration isolator is set up

After PNSP the stiffness of horizontal and vertical direc-tion can be expressed as

119896119881120578 = 119870119881 = 119870119881po + 119870119881ne = 119870as + 119870ms119896119867120578 = 119870119867 = 119870119867po + 119870119867ne = 119870fs + 119870ip (4)

where 119896119881120578 = 119870119881 is the system stiffness of vertical direction119870119881po is the positive stiffness of vertical direction 119870119881ne is thenegative stiffness of vertical direction 119870as is the stiffness ofair spring119870ms is the stiffness of magnetic spring 119896119867120578 = 119870119867 isthe system stiffness of horizontal direction119870119867po is the positivestiffness of horizontal direction 119870119867ne is the negative stiffnessof horizontal direction 119870fs is the stiffness of flat spring and119870ip is the stiffness of inverted pendulum

3 Stiffness Model

31 Vertical Stiffness Model In vertical direction the struc-ture combines a PS air spring with a NS magnetic spring inparallel The PS air spring can support the mass of payloadplatform and the NS magnetic spring is responsible forrealizing ultra-low frequency vibration isolation

311 PS Air Spring Asingle chamber air spring is designed tobe an inverted pendulum The stiffness model is established

Air chamber

Metalshell

Baseboard

Sealingmembrane

f

x

Figure 5 The structure diagram of air spring

to describe the dynamic characteristics of vertical passivevibration isolation element The structure diagram is shownin Figure 5

To establish stiffness model of the single chamber airspring shown in Figure 5 the dynamic process of metal cavityin the body is assumed as an ideal gas adiabatic process

119875119881120581 = 11987501198811205810 (5)

where 119875 is the pressure of the air chamber119881 is the volume ofthe air chamber 1198750 is the initial pressure of the air chamberunder static equilibrium state 1198810 is the initial volume of theair chamber under static equilibrium state 120581 is the adiabaticcoefficient and the value is about 1402

Under static equilibrium state of the air chamber theinitial volume and initial pressure satisfy the following math-ematical relations

1198750 = 119875atm + 119872119901119892119860 1198810 = 1198601199090 (6)

where 1199090 is the initial height of the air chamber 119860 is theeffective sectional area of the air chamber 119875atm is the pressure


The upper magnet

The middle magnet

The lower magnet

NS

N

S

h

h

Z

z

x y

J

J998400

Figure 6 The schematic of NS magnetic spring

of atmosphere and 119872119901 is the mass of payload under staticequilibrium state

When the amplitude of the air spring vibration is verysmall near the equilibrium position the volume of air cham-ber can be approximately considered as constant Equation(5) is differentiable at 119881 and is changed to

119889119875119889119881 = minus12058111987501198810 (1198810119881 )120581+1 asymp minus12058111987501198810 (7)

When there is an effect on payload from perturbation119891 the volume and pressure of the air chamber meet thefollowing relations

119860119889119875 = 119889119891119889119881 = 119860119889119909 (8)

According to the definition of stiffness and the abovemathematical formulas the stiffness of air spring can bederived as

119870as = minus119889119891119889119909 = 1205811198602 (119875atm +119872119901119892119860)1198810 (9)

The positive stiffness of vertical direction is expressed as

119870119881po = 119870as = 1205811198602 (119875atm + 119872119901119892119860)1198810 (10)

From (10) it can be seen that the stiffness of the singlechamber air spring is related to the mass 119872119901 of payload theeffective sectional area 119860 of the air chamber and the initialvolume 1198810 of the air chamber With the increase of mass thestiffness of air spring will also increase

312 NS Magnetic Spring The principle of NS magneticattraction spring is that different magnetic poles can attracteach other The schematic is depicted in Figure 6 There arethree rectangular magnets magnetized in the z-directionThemiddle magnet and the upperlower magnets are configuredin attractive interaction Due to the symmetry the middlemagnet maintains a balance in the expected initial position

But the balance is unstable Once a small perturbation acts onit the balance will be broken and cannot be restored withoutexternal force Thus the middle magnet acts as a NS spring[18]

It is assumed that themiddlemagnet canmove only in thez-directionThe other degrees of freedomwill be constrainedby applying extra mechanism As shown in Figure 6 theupper and the lower magnets are fixed with respect to eachother The distance of the outer magnet and the middlemagnet in the z-direction is ℎ The stiffness of the magneticspring attributes to magnetic force It can be adjusted bychanging the distance ℎ between the central magnet and theouter magnets

According to the research of rectangular permanent mag-net magnetic force analytical model from [18] the resultantforce of the middle magnet is from the effect of the upper andlower magnets in 119911 direction

119865119911 = 119865 (119911 minus ℎ) + 119865 (119911 + ℎ) (11)

For the magnets shown in Figure 6 we can obtain themathematical formula of attractive force between the twomagnets from [19]

119865 = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902Φ(119880119894119895 119881119896119897119882119901119902 119903) (12)

where 119869 and 1198691015840 are the magnetic polarization vectors and thevariables 119880 119881119882 and 119903 respectively are

Φ(119880119894119895 119881119896119897119882119901119902 119903) = minus119880119882 ln (119903 minus 119880)minus 119881119882 ln (119903 minus 119881)+ 119880119881 arctan (119880119881119903119882) minus 119903119882

119880119894119895 = 119910 + (minus1)119895 1198861015840 minus (minus1)119894 119886119881119896119897 = (minus1)119897 1198871015840 minus (minus1)119896 119887

119882119901119902 = 119911 + (minus1)119902 1198881015840 minus (minus1)119901 119888119903 = radic1198801198941198952 + 1198811198961198972 +1198821199011199022

(13)

where the geometric dimensions of the upperlower magnetsare 2119886 times 2119887 times 2119888 and the geometric dimensions of the middlemagnet are 21198861015840 times 21198871015840 times 21198881015840

According to the definition of stiffness the stiffness of NSmagnetic spring is expressed as

119870 = minus120597119865119911120597119911 = 119870 (119911 minus ℎ) + 119870 (119911 + ℎ) (14)


The mathematical expression of119870(119911) is119870ms (119911) = 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)

with 120593 (119880119894119895 119881119896119897119882119901119902 119903) = 2119903 + 119880 ln (119903 minus 119880)+ 119881 ln (119903 minus 119881) (16)

ditto the variables 119880 119881119882 and 119903The negative stiffness of vertical direction is expressed as

119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)

The stiffness of vertical direction can be derived as

119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)

32 Horizontal Stiffness Model In horizontal direction thestructure which combines a PS flat spring with a NS invertedpendulum in parallel is used to attain ultra-low frequencyvibration isolation The NS inverted pendulum is realizedthrough the reverse installing of the air spring

321 PS Flat Spring The horizontal PS structure is made upof two mutually perpendicular flat spring mechanisms Everyflat spring mechanism consists of two parallel flat springswhich is rectangular

Two mutually perpendicular flat spring mechanisms canbe equivalent to Figure 7 In the figure point119860 is the workingpoint of force When there is one unit force 119865 from arbitrarydirection (angle 120579 is arbitrary value) at point119860 the flat springwill produce the static deformation to point119861 in the directionof119865The static deformationwhich is generated by componentforce of 119865 in 1198961 and 1198962 direction is expressed as

Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)

Assume that the unit force 119865 is very small so the staticdeformation is very small Angle ang119862119861119863 is still approximatelyconsidered as the right angle the total deformation Δ119909 isindicated as

Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1

Figure 7 The schematic of two mutually perpendicular flat springmechanisms

The stiffness 1198961 and stiffness 1198962 are the same as thestiffness 119896 of flat spring the total stiffness of the mechanismin the direction of 120579 can be derived as

119896ℎ119901 = 119865Δ119909= 119865radic(119865 cos (45∘ + 120579) 1198961)2 + (119865 cos (45∘ minus 120579) 1198961)2= 119896 = 2119896119904

(21)

The stiffness of flat spring which is fixed at one end andfreed on the other end can be expressed as

119896119904 = 31198641198681198713 (22)

where 119864 is the elastic modulus of flat spring material 119868 isthe bending section moment of inertia and 119871 is the effectivebending length of flat spring

The positive stiffness of horizontal direction is expressedas

119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)

Given the arbitrariness of angle 120579 the stiffness of themechanism is all consistent in arbitrary direction

322 NS Inverted Pendulum The structure of invertedpendulum is not a real pendulum Instead the invertedpendulum is formed by the inverted installation of air springSealing membrane of air spring is fixed on the base platformWhen there is a small perturbation effect on the inverted airspring the special arrangement acts as a NS inverted pen-dulum The equivalent structure chart is shown in Figure 8A flexible joint is served for connective functions betweenthe payload and the air spring Owing to the flexible jointwith low stiffness the payload always keep in a horizontalstate Because of this reason the connection of the verticalmechanism can be considered as rigid connection


Under the undisturbed nature state inverted pendulum isin the unstable equilibrium state Once a small perturbationacts on it the state cannot be restored When there is adisturbance effect on payload a horizontal movement 119909deviated from the original unstable equilibrium state will beproduced We need to make a balance on the displacement 119909by applying a reverse force 119891 on this point The reverse force119891 can be expressed as

119891 = minus119898119892 119909radic1198972 minus 1199092 (24)

According to the definition of stiffness the mathematicalformula about the stiffness of NS inverted pendulum isexpressed as

119896 = 119889119891119889119909 = minus119898119892( 1radic1198972 minus 1199092 + 1199092radic(1198972 minus 1199092)3) (25)

Further when the horizontal amplitude 119909 is in a smallrange and it is far less than the effective length 119897 (25) can berewritten as

119896 = minus119898119892119897 (26)

The negative stiffness of horizontal direction is expressedas

119870119867ne = 119870ip = minus119898119892119897 (27)

From (27) it can be seen that the value of negativestiffness is related to the payload and the effective lengthof inverted pendulum and the inverted pendulum is char-acterized by negative stiffness However the characteristicdetermines that inverted pendulum must be used withpositive stiffness spring in parallel

The stiffness of horizontal direction can be derived as

119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)

4 Numerical Simulation

41 Numerical Simulation of StiffnessModel According to theabove stiffness model the simulation curves which give the

relationship between the stiffness of different component andassociated parameter values are shown in Figure 9

From Figure 9(a) it can be seen that the stiffness of airspring will decrease with the increase of volume This changeis more obvious when the initial volume of the air chamberis small As the value increases to a certain extent the effectbecomes less significant It also shows that the stiffness of airspring with different payload mass shows the same changetrend

Figure 9(b) shows that the amplitude of the stiffnessdecreases as the relative displacement 119910 increases within acertain vibration range and it reaches the maximum when119910 equals zero In particular the stiffness also depends on thedistance ℎ and it can be utilized to adjust the stiffness of theNS

According to (26) and Figure 9(c) the value of thestiffness is inversely related to the effective bending length119871 Therefore the value of the stiffness can be affected andadjusted by the effective bending length

The value of negative stiffness is related to the payloadand the effective length of inverted pendulum and theinverted pendulum is characterized by negative stiffnessin Figure 9(d) However the characteristic determines thatinverted pendulum must be used with positive stiffnessspring in parallel

42 Numerical Simulation of Vibration Isolation SystemAccording to Newton-Euler equation the dynamic math-ematical equations of the vibration isolation system withadditional stiffness in parallel separately from the horizontaland vertical direction are changed as

119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903)]]sdot (119909119900 minus 119909119894) = 0119872119901119900 + 119888119867 (119900 minus 119894) + [61198641198681198713 minus 119898119892119897 ] (119909119900 minus 119909119894) = 0

(29)

The natural frequency with PNSP is put into

120596119881119899 = radic 119896120578119872119901= radic 1205811198602 (119875atm +119872119901119892119860) 1198810 + (119869119869101584041205871205830)sum1119894=0sum1119895=0sum1119896=0sum1119897=0sum1119901=0sum1119902=0 (minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903)119872119901

120596119867119899 = radic 119896119867120578119872119901 = radic 61198641198681198713 minus 119898119892119897119872119901 (30)


Air chamber

Metalshell

BaseboardSealing membrane Natural state Vibration state

lll

m m m

xf

Figure 8 The equivalent structure chart of NS inverted pendulum

0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2

Initial volume V0 (m3)

times106

Payload 3000 kgPayload 2000 kgPayload 1000 kg

Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2

Effective bending length L (m)

times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0

Effective length of inverted pendulum l (m)

times105

Stiff

ness

Kip

(Nm

)

(d)Figure 9 The simulation curves (a) the relationship between stiffness 119870as of the PS air spring and the initial volume 1198810 of the air chamberunder different payload mass (119872119901 = 1000 2000 and 3000 kg) (b) the stiffness 119870ms of the NS magnetic spring with the following magnetparameters the upperlower 35mm times 14mm times 7mm the middle 30mm times 10mm times 10mm 119861119903 = 119869 = 134119879 a relative permeability 120583119900 equalto 1023 (c) the relationship between stiffness119870fs of the PS flat spring and the effective bending length 119871 of the flat spring with the bendingsection moment of inertia (119868 = 2193119890 minus 5 kgsdotm2) being constant and (d) the relationship between stiffness119870ip of the NS inverted pendulumand the effective length of inverted pendulum 119897 with the payload mass (119898 = 1000 kg) being constant Other unknown parameters are shownin Tables 1 and 2


=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q

120593(Uij VklWpq r)

+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)

Frequency (Hz)10minus1 100 101 102

PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)

Figure 10 Transmissibility curves of different stiffness model (a) vertical direction (b) horizontal direction

According to the simulation parameters of Tables 1 and 2the transmissibility curves of vibration isolation system arerespectively shown in Figure 10

From Figure 10(a) it can be seen that there are obviousdifferences under different methods in vertical directionWhere the green curve is the transmissibility curve ofadopting positive stiffness element alone the red line is thetransmissibility curve of using PNSP From the result it canbe seen clearly that the natural frequency with PNSP ismoved forward effective bandwidth of vibration isolation iswidened

According to the above theoretical analysis of horizon-tal direction Figure 10(b) shows the transmissibility curvesunder different methods Where the green curve is thetransmissibility curve of adopting positive stiffness elementalone the red line is the transmissibility curve of usingPNSP From the result we can clearly get that the naturalfrequency with PNSP is moved forward effective bandwidthof vibration isolation is widened

5 Experimental Results and Discussions

51 Principle Diagram and Layout of Experiment System Inorder to verify the effectiveness of the proposed structurethe principle diagram shown in Figure 11(a) is set up Thevibration isolation system consists of a payload platform abase platform three identical vibration isolators and a real-time position control system and a spectrum testing andanalysis system Two kinds of vibration isolator with differentstructures (PNSP or PS only) are adopted respectively forcomparing results A real-time position control system anda spectrum testing and analysis system (LMS SCADAS)are included in the experiment system Sensor signals frombase platform and payload platform are imported into LMS

Table 1 Simulation parameters of vertical direction

Parameters Name Values119872119901 Themass of payload 1000 kg

119860 The effective sectional area of the airchamber 00343m2

1198810 The initial volume of the airchamber 0002m3

120581 The adiabatic coefficient 1402119875atm The pressure of atmosphere 101325 kpa119886 The dimension of the upperlowermagnets (length) 0035m

119887 The dimension of the upperlowermagnets (width) 0014m

119888 The dimension of the upperlowermagnets (height) 0007m

1198861015840 The dimension of the middlemagnet (length) 0030m

1198871015840 The dimension of the middlemagnet (width) 0010m

1198881015840 The dimension of the middlemagnet (height) 0010m

ℎ The distance between the magnets 0012m119870119881po The positive stiffness of verticaldirection 309e5Nm

119870119881ne The negative stiffness of verticaldirection minus252e5Nm

119870119881 The system stiffness of verticaldirection 568e4Nm

SCADAS to compare the performance of the vibration isola-tor with PNSP or PS structure In the position control loop


LMSSCADAS

TestPC

1

2

3

Controller Valve Payloadplatform

Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)

Figure 11 The layout of the ultra-precision vibration isolation system

the six displacements fromhorizontal and vertical sensors areconverted to the logical axis signals by means of a transformmatrix firstlyThe error between the setting value of air springfloating and logical axis signal is input to the controller forposition control The output control signal is sent to the valveto control the vibration isolator with PNSP or PS structureand consequently realize the position control of the payloadplatform

The vibration isolators which are as isosceles triangle orequilateral triangle arrangement are installed on the baseplatform The payload platform and the vibration isolatorsare fixed and connected with each other through screwsThe layout of the ultra-precision vibration isolation system isshown in Figure 11(b)

52 Set-Up of Experiments According to the principle ofFigure 11(a) an experiment on the proposed isolator withPNSP structure as Figure 12 is built in super-clean lab Theexperiment system consists of three vibration isolators which

are as equilateral triangle arrangement installed on the baseplatform and a control system which can realize real-timeposition control and fast spectrum test and analysis

The real-time active control system proposed in paperconsists of a programmable NI controller (Model PXIE-8135) and a control cubicle It is used as a real-time signalacquisition and a real-time active control signal output Thespectrum testing and analysis system proposed in paperconsists of two servo velocity meters (Model VSE-15D6) anda vibration signal collecting and analysis instrument (ModelLMS SCADASIII SCM05) The vibration signal acquired bythe servo velocity meter is sent to the LMS SCADASIII fordata processing

53 Experiment Results The velocity admittance curves ofvibration isolation system in the logical axis coordinates aretested to verify the effect of the proposed PNSP structureThe experiment results are shown in Figure 13 and Table 3From the results we can see that the natural frequency of X


Servo velocity meters

Vibration isolator with PNSP

LMSSCADAS

Controlsystem

PNSP vibration isolation system

(a)

LMS SCADAS

PC controllerwindow

Controller

Control system

LMS data acquisition amp analysis window

NI controller

(b)

Figure 12 The experimental system (a) the principle diagram (b) the photo of the PNSP vibration isolation system

Table 2 Simulation parameters of horizontal direction

Parameters Name Values

119864 The elastic modulus of flat springmaterial 20611989011Pa

119868 The bending section moment ofinertia

2193119890 minus5 kglowastm2119871 The effective bending length of theflat spring 0074m

119898 The equivalent mass of invertedpendulum 1000 kg

119897 The effective length of invertedpendulum 0032m

119870119881po The positive stiffness of verticaldirection 1291198905Nm

119870119881ne The negative stiffness of verticaldirection minus8171198904Nm

119870119881 The system stiffness of verticaldirection 4771198904Nm

Table 3 The natural frequency comparison of vibration isolatorusing different structure

Naturalfrequency 119883 119884 119877119911 119885 119877119910 119877119909PS 286Hz 336Hz 437Hz 269Hz 241Hz 664HzPNSP 117Hz 119 Hz 194Hz 113 Hz 098Hz 195Hz

translation is reduced from 286Hz (PS) to 117Hz (PNSP)the natural frequency ofY translation is reduced from336 Hz(PS) to 119Hz (PNSP) the natural frequency of Z rotationis reduced from 437Hz (PS) to 194Hz (PNSP) the naturalfrequency of Z translation is reduced from 269Hz (PS)to 113Hz (PNSP) the natural frequency of Y rotation isreduced from 241Hz (PS) to 098Hz (PNSP) and the naturalfrequency of X rotation is reduced from 654Hz (PS) to195Hz (PNSP) It can be seen that the natural frequencyof each DOF has a substantial decline Besides above thereis a coupling peak in the direction of X translation due to

inadequate decoupling The second resonance peak maybeis the natural frequency of X rotation However comparedwith the PS the PNSP still can significantly reduce the naturalfrequency of the system

Figure 14 shows the analytical and experimental results ofacceleration vibration transmission of the vibration isolatordue to oscillation of the vibration isolation system Tables 4and 5 list the natural frequency and magnitude of vibrationtransmissibility curves in different directions separately Theanalytical results are calculated by (18) and (28) withoutregard to the damping of vibration isolation system PS (greensolid line) means that the vibration isolation system only usespositive stiffness mechanism PNSP (red solid line) meansthat the vibration isolation system adopts positive stiffnessmechanism and negative stiffness mechanism together

In Figure 14(a) and Tables 4 and 5 because of the low-dynamic stiffness of PNSP structure in the direction of X thefrequency of the resonance peak is moved from 279Hz (PSerror 036) to 117Hz (PNSP error 174) In the 279Hz theamplitude of the red solid line is minus1099 dB compared with1834 dB of the green solid line it drops by 2933dB In the10Hz the amplitude is also from minus1861 dB of the green solidline to minus3190dB of the red solid line it drops by 1329 dB

In Figure 14(b) and Tables 4 and 5 because of the low-dynamic stiffness of PNSP structure in the direction of Y thefrequency of the resonance peak is moved from 332Hz (PSerror 091) to 121Hz (PNSP error 083) In the 332Hz theamplitude of the red solid line is minus1145 dB compared with1836 dB of the green solid line it drops by 1948 dB In the10Hz the amplitude is also from minus1465 dB of the green solidline to minus3144 dB of the red solid line it drops by 1679 dB

In Figure 14(c) and Tables 4 and 5 because of the low-dynamic stiffness of PNSP structure in the direction of Z thefrequency of the resonance peak is moved from 275Hz (PSerror 179) to 114Hz (PNSP error 364) In the 275Hz theamplitude of the red solid line is minus1059 dB compared with1678 dB of the green solid line it drops by 2737 dB In the10Hz the amplitude is also from minus1824 dB of the green solidline into minus3478 dB of the red solid line it drops by 1654 dB

In order to further verify the high-static stiffness charac-teristics of PNSP structure and the stability of the payload


Table 4 The natural frequency of vibration transmissibility curves in different directions

Direction PS frequency(Calculation)

PS frequency(Experiment) Error () PNSP frequency

(Calculation)PNSP frequency(Experiment) Error ()

119883 280Hz 279Hz 036 115 Hz 117Hz 174119884 330Hz 332Hz 091 120Hz 121 Hz 083119885 280Hz 275Hz 179 110Hz 114Hz 364

Table 5 The magnitude of vibration transmissibility curves in different directions

Direction Frequency PS magnitude(Experiment)

PNSP magnitude(Experiment)

Decreasing amplitude PSresonance10Hz119883 PS resonance10Hz 1834 dBminus1861 dB minus1099 dBminus3190 dB 2933dB1329 dB119884 PS resonance10Hz 1836 dBminus1465 dB minus1145 dBminus3144 dB 1948dB1679 dB119885 PS resonance10Hz 1678 dBminus1824 dB minus1059 dBminus3478 dB 2737dB1654 dB

Table 6 The static stability of payload platform

Static stability H1 H2 H3 V1 V2 V3Positivedeviation (+)

25 120583m 10 120583m 20120583m 20 120583m 10 120583m 20120583mNegativedeviation (minus) 30 120583m 15120583m 20120583m 25120583m 10 120583m 15 120583m

platform six eddy current displacement sensor signals arecollected respectively The working position of the air springis set up to 1700120583m When the system is floating up to theworking position three displacement values in the horizontaland vertical physical axis are measured and recorded by thesensors The experiment results are given in Figure 15 andTable 6

Table 6 shows that the static stability of the payloadplatform canbe keptwithinplusmn30 120583mand is less than the activepositioning accuracy (8120583m 4120583m and 15120583m) with differentcontrol algorithms in [20] Despite decreasing of the dynamicstiffness of the vibration isolation system the static stiffnessis preserved and the system stability is still guaranteed Thehorizontal deviation and vertical deviation of 2 vibrationisolator are better than those of 1 and 3 in the static stabilityof the payload platform This result may be attributed to theproblems that the centroid of the isosceles structure is farfrom 2 isolator

54 Discussions On the condition of the open loop com-pared to the PS vibration isolator the PNSP vibration isolatorplays an important role in reducing the dynamic stiffnessand preserving the static stiffness of the system Fromthe experimental results of velocity admittance curves thenatural frequency of Z translation is reduced from 269Hzof PS to 113Hz of PNSP and the six directions of thesystem are significantly decreased The comparison results ofthe vibration transmissibility curves show that the vibrationtransmissibility of the new system is improved obviously withemploying the PNSP structure The results are basically the

same with the results of the simulation It also can be judgedfrom the stability experiment of the system simultaneouslyThe dynamic stiffness of the system is still preserved and thestability of the system is well guaranteed

The above-mentioned experimental results illustrate thatthe PNSP vibration isolator can effectively reduce the naturalfrequency of the system and further expand the ultra-lowfrequency of vibration isolation system At the same time italso provides the foundation for the further enhancement ofthe vibration attenuation ability in active control

6 Conclusion

A compact structure with PNSP is designed to reduce thenatural frequency and broaden the effective bandwidth ofvibration isolation system according to the principle of high-static-low-dynamic stiffnessThe proposed isolator combinesa PS spring with a NS spring in parallel in two DOFs ofhorizontal and vertical direction Different stiffness modelsof PS elements and NS elements for vertical and horizontaldirections are established And three identical vibration iso-lators can be used together to establish an ultra-low frequencyvibration isolation system which can realize the ultra-lowfrequency vibration isolation in six degrees of freedomA real-time active control system and a spectrum testingand analysis system are employed for contrast experimentbetween the proposed PNSP structure and PS structureThe velocity admittance results demonstrate that the naturalfrequency of Z translation is reduced from 269Hz of PS to113Hz of PNSP The transmissibility curves also prove thatthe frequency of the resonance peak ismoved forward 161Hz(from 275Hz to114Hz) In 275Hz the amplitude drops by2737 dB In the 10Hz the amplitude drops by 1654 dB Thestatic stability of the payload platform can be kept withinplusmn30120583m and is less than the active positioning accuracy(8120583m 4120583m and 15120583m) It means that PS with NS in parallelcan obviously reduce the natural frequency and broaden theeffective bandwidth of the system and simultaneously main-tain the stability of the system and further verify the validityand superiority of the mentioned structure Consequently


minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X

Frequency (Hz)100 101 102

(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)

Figure 13 The velocity admittance curves of logical axes (the red solid line for PNSP and the green solid line for PS) (a) X translation (b) Ytranslation (c) Z rotation (d) Z translation (e) Y rotation and (f) X rotation


minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)

Experiment-PNSP-XExperiment-PS-X

Calculation-PNSP-XCalculation-PS-X


(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)

Experiment-PNSP-YExperiment-PS-Y

Calculation-PNSP-YCalculation-PS-Y


(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)

Experiment-PNSP-ZExperiment-PS-Z

Calculation-PNSP-ZCalculation-PS-Z


(c)

Figure 14 The vibration transmissibility curves (the red solid line for PNSP the green solid line for PS) (a) X translation (b) Y translationand (c) Z translation

2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)

Figure 15 The static stability of payload platform (a) the horizontal displacement (b) the vertical displacement


this type of vibration isolator is suitable for applications onultra-precision manufacturing andmeasurement equipment

Competing Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFoundation of China (no 51435006 no 51421062)

References

[1] CGGordon ldquoGeneric vibration vriteria for vibration-sensitiveequipmentrdquo in Proceedings of the International Society forOptical Engineering (SPIE) Conference on Current Developmentsin Vibration Control for Optomechanical Systems E A DerbyEd International Society for Optics and Photonics pp 22ndash33Denver Colo USA July 1999

[2] E H Anderson and B Houghton ldquoELITE-3 active vibrationisolation workstationrdquo in Proceedings of the 8th Annual Inter-national Symposium on Smart Structures and Materials pp183ndash196 International Society for Optics and Photonics March2001

[3] Y-H Shin and K-J Kim ldquoPerformance enhancement of pneu-matic vibration isolation tables in low frequency range by timedelay controlrdquo Journal of Sound and Vibration vol 321 no 3ndash5pp 537ndash553 2009

[4] S Julai and M Tokhi ldquoActive vibration control of flexibleplate structures with distributed disturbancesrdquo Journal of LowFrequency Noise Vibration and Active Control vol 31 no 2 pp123ndash150 2012

[5] B Yan M J Brennan S J Elliott and N S Ferguson ldquoActivevibration isolation of a system with a distributed parameterisolator using absolute velocity feedback controlrdquo Journal ofSound and Vibration vol 329 no 10 pp 1601ndash1614 2010

[6] M Yasuda T Osaka and M Ikeda ldquoFeedforward control ofa vibration isolation system for disturbance suppressionrdquo inProceedings of the 35th IEEEConference onDecision andControlpp 1229ndash1233 Kobe Japan December 1996

[7] T H Yan H Y Pu X D Chen Q Li and C Xu ldquoIntegratedhybrid vibration isolator with feedforward compensation forfast high-precision positioning XY tablesrdquo Measurement Sci-ence and Technology vol 21 no 6 Article ID 065901 10 pages2010

[8] D L Platus ldquoNegative-stiffness-mechanismVibration IsolationSystemsrdquo in Proceedings of the SPIErsquos International Symposiumon Optical Science Engineering and Instrumentation Interna-tional Society for Optics and Photonics pp 98ndash105 1999

[9] B A Fulcher D W Shahan M R Haberman C C Seepersadand P S Wilson ldquoAnalytical and experimental investigationof buckled beams as negative stiffness elements for passivevibration and shock isolation systemsrdquo Journal of Vibration andAcoustics vol 136 no 3 Article ID 031009 12 pages 2014

[10] M E Hoque T Mizuno Y Ishino and M Takasaki ldquoA three-axis vibration isolation system using modified zero-powercontroller with parallel mechanism techniquerdquo Mechatronicsvol 21 no 6 pp 1055ndash1062 2011

[11] TMizunoMTakasakiDKishita andKHirakawa ldquoVibrationisolation system combining zero-power magnetic suspensionwith springsrdquoControl Engineering Practice vol 15 no 2 pp 187ndash196 2007

[12] W S Robertson M R F Kidner B S Cazzolato and AC Zander ldquoTheoretical design parameters for a quasi-zerostiffness magnetic spring for vibration isolationrdquo Journal ofSound and Vibration vol 326 no 1-2 pp 88ndash103 2009

[13] AValeevA Zotov and S Kharisov ldquoDesigning of compact lowfrequency vibration isolator with quasi-zero-stiffnessrdquo Journalof Low Frequency Noise Vibration and Active Control vol 34no 4 pp 459ndash474 2015

[14] A Carrella M J Brennan and T P Waters ldquoStatic analysis of apassive vibration isolator with quasi-zero-stiffness characteris-ticrdquo Journal of Sound and Vibration vol 301 no 3 pp 678ndash6892007

[15] P Alabuzhev A Gritchin L Kim G Migirenko V Chon andP Stepanov Vibration Protecting and Measuring Systems withQuasi-Zero Stiffness CRC Press Hemisphere New York NYUSA 1989

[16] A Carrella M J Brennan T P Waters and K Shin ldquoOnthe design of a high-static-low-dynamic stiffness isolator usinglinear mechanical springs and magnetsrdquo Journal of Sound andVibration vol 315 no 3 pp 712ndash720 2008

[17] A D Shaw S A Neild and D J Wagg ldquoDynamic analysis ofhigh static low dynamic stiffness vibration isolation mountsrdquoJournal of Sound and Vibration vol 332 no 6 pp 1437ndash14552013

[18] W Wu X Chen and Y Shan ldquoAnalysis and experiment of avibration isolator using a novel magnetic spring with negativestiffnessrdquo Journal of Sound and Vibration vol 333 no 13 pp2958ndash2970 2014

[19] G Akoun and J-P Yonnet ldquo3D analytical calculation of theforces exerted between two cuboidal magnetsrdquo IEEE Transac-tions on Magnetics vol 20 no 5 pp 1962ndash1964 1984

[20] T H Lee K K Tan S N Huang and H F Dou ldquoIntelligentcontrol of precision linear actuatorsrdquo Engineering Applicationsof Artificial Intelligence vol 13 no 6 pp 671ndash684 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Payload Mp

Base mB

xo

xi

ckini

(a)

Payload Mp

Base mB

xo

c

xi

kini

kadd

(b)

Payload Mp

Base mB

xo

c

xi

xm

kini

kadd

(c)

Figure 1 Vibration isolation principle diagram (a) traditional (b) stiffness in parallel and (c) stiffness in series

to the formation of quasi-zero-stiffness (QZS) system [12ndash15] QZS mechanisms are generally achieved by combininga negative stiffness (NS) element with a positive stiffness(PS) element It has been found that to achieve a largeexcursion from the static equilibrium position such that thestiffness of the system does not exceed a prescribed valuethere is an optimum geometry and a corresponding optimumrelationship in the stiffness [14]

Based on QZS system Carrella et al [14 16] put for-ward the concept of the high-static-low-dynamic stiffness(HSLDS) mount which comprises two vertical mechanicalsprings and a permanent magnet A reduction in the naturalfrequency from 14Hz to 7Hz was achieved by using the mag-nets and the measured transmissibility from a displacementinput showed that the device behaved approximately linearlyRobertson et al [12] introduced theoretical analysis for afully HSLDS magnetic spring for vibration isolation wherethe payload was also supported by magnetism Shaw et al[17] presented dynamic analysis of HSLDS vibration isolationmounts to provide useful insight into the design of HSLDSmounts However there are some problems in the existingvibration isolators based on the efforts of QZS and HSLDSSpecific performances are as follows the natural frequencyof existed structure is still not competitive when facedwith a higher demand There are still some shortcomingsin the application of multi-DOFs vibration isolation systemTherefore one structure of urgent demands to solve theseproblems is becoming more and more necessary

In this paper to get wider effective bandwidth and higherperformance of vibration isolation in multiple DOFs systeman ultra-low frequency two DOFsrsquo vibration isolator withpositive and negative stiffness in parallel (PNSP) is proposedwhich can realize this requirement that ensures lower naturalfrequency and large capacity at the same time [12] and thefunction that isolation can be achieved simultaneously intwo DOFs Compared with a PS isolator it can also possessthe characteristic of HSLDS The ultra-low frequency twoDOFsrsquo isolator which combines a PS air spring with a NSmagnetic spring in parallel and combines a PS flat spring withan NS inverted pendulum in parallel is designed to reducethe natural frequency and broaden the effective bandwidthin horizontal and vertical direction The bearing capacity isdetermined by the PS structure which can effectively improve

the static stiffness and the stability of the vibration systemand maintain the natural frequency which is constant withthe changing of payload NS element is employed to reducethe dynamic stiffness and the natural frequency of vibrationsystem

To solve the problem of the existence of vibration isola-tion inmulti-DOFs at the same time three identical vibrationisolators proposed in this paper can be used together toestablish an ultra-precision vibration isolation system whichcan realize the ultra-low frequency vibration isolation in sixDOFs through the transform matrix of different coordinatesystems in this paper And it also can ensure the positioningaccuracy of the system simultaneously

2 Design of Two DOFsrsquo Ultra-Low FrequencyVibration Isolator

21 The Principle of PNSP The traditional passive vibrationisolation unit is composed of mass-spring-damping the sim-plified principle diagram is shown in Figure 1(a) Commonways of changing the system stiffness are the method of seriesconnection or parallel connection with additional stiffnessFigures 1(b) and 1(c) show the simplified principle diagram

Theway of series can only reduce the system stiffness butit cannot achieve the QZS systemThis can be improved uponby incorporating a negative stiffness element in parallel suchthat the dynamic stiffness ismuch less than the static stiffnessThe way to achieve this result of high-static stiffness andlow-dynamic stiffness is that the negative stiffness element isadded near the working position of the isolator to form thePNSP mechanism

According toNewton-Euler equation the dynamic math-ematical equations of the vibration isolation system withtraditional passive vibration isolation and with additionalstiffness in parallel are respectively written as

119872119901119900 + 119888 (119900 minus 119894) + 119896ini (119909119900 minus 119909119894) = 0 (1)

119872119901119900 + 119888 (119900 minus 119894) + 119896120578 (119909119900 minus 119909119894) = 0 (2)

where119872119901 is the mass of payload platform 119898119861 is the mass ofbase platform 119888 is the equivalent damping of the system 119909119900is the displacement deformation of the payload platform 119909119894


0 24 6

810

02000

40006000

800010000

0

20

40

60

80

100

times105

Nat

ural

freq

uenc

y120596n

(rad

s)


(Nm)

(a)


5

10

15

20

25

30

times106

Nat

ural

freq

uenc

y120596n

(rad

s)


(b)




(3)




Base

PayloadPS for large


natural frequencyNS

NS

Inve

rted

pen

dulu

m

Mag

netic

sprin

g

Airspring

Flat

sprin

g

Parallel



Kpo Kne







Mag

netic

sprin

g

Air spring


N

SN

S

Base

Payload

Kpo Kne

(a)

L

x fminusf

mgl

Invertedpendulum

Flatspring


Base

Payload

Flexiblejoint

KpoKne

(b)







3 Stiffness Model



Air chamber

Metalshell

Baseboard

Sealingmembrane

f

x




119875119881120581 = 11987501198811205810 (5)



1198750 = 119875atm + 119872119901119892119860 1198810 = 1198601199090 (6)



The upper magnet

The middle magnet

The lower magnet

NS

N

S

h

h

Z

z

x y

J

J998400






119860119889119875 = 119889119891119889119881 = 119860119889119909 (8)


119870as = minus119889119891119889119909 = 1205811198602 (119875atm +119872119901119892119860)1198810 (9)


119870119881po = 119870as = 1205811198602 (119875atm + 119872119901119892119860)1198810 (10)






119865119911 = 119865 (119911 minus ℎ) + 119865 (119911 + ℎ) (11)


119865 = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902Φ(119880119894119895 119881119896119897119882119901119902 119903) (12)





(13)



119870 = minus120597119865119911120597119911 = 119870 (119911 minus ℎ) + 119870 (119911 + ℎ) (14)



sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)



119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)


119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)




Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)


Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1




(21)


119896119904 = 31198641198681198713 (22)



119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)









119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 24 6

810

02000

40006000

800010000

0

20

40

60

80

100

times105

Nat

ural

freq

uenc

y120596n

(rad

s)


(Nm)

(a)


5

10

15

20

25

30

times106

Nat

ural

freq

uenc

y120596n

(rad

s)


(b)




(3)




Base

PayloadPS for large


natural frequencyNS

NS

Inve

rted

pen

dulu

m

Mag

netic

sprin

g

Airspring

Flat

sprin

g

Parallel



Kpo Kne







Mag

netic

sprin

g

Air spring


N

SN

S

Base

Payload

Kpo Kne

(a)

L

x fminusf

mgl

Invertedpendulum

Flatspring


Base

Payload

Flexiblejoint

KpoKne

(b)







3 Stiffness Model



Air chamber

Metalshell

Baseboard

Sealingmembrane

f

x




119875119881120581 = 11987501198811205810 (5)



1198750 = 119875atm + 119872119901119892119860 1198810 = 1198601199090 (6)



The upper magnet

The middle magnet

The lower magnet

NS

N

S

h

h

Z

z

x y

J

J998400






119860119889119875 = 119889119891119889119881 = 119860119889119909 (8)


119870as = minus119889119891119889119909 = 1205811198602 (119875atm +119872119901119892119860)1198810 (9)


119870119881po = 119870as = 1205811198602 (119875atm + 119872119901119892119860)1198810 (10)






119865119911 = 119865 (119911 minus ℎ) + 119865 (119911 + ℎ) (11)


119865 = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902Φ(119880119894119895 119881119896119897119882119901119902 119903) (12)





(13)



119870 = minus120597119865119911120597119911 = 119870 (119911 minus ℎ) + 119870 (119911 + ℎ) (14)



sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)



119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)


119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)




Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)


Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1




(21)


119896119904 = 31198641198681198713 (22)



119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)









119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mag

netic

sprin

g

Air spring


N

SN

S

Base

Payload

Kpo Kne

(a)

L

x fminusf

mgl

Invertedpendulum

Flatspring


Base

Payload

Flexiblejoint

KpoKne

(b)







3 Stiffness Model



Air chamber

Metalshell

Baseboard

Sealingmembrane

f

x




119875119881120581 = 11987501198811205810 (5)



1198750 = 119875atm + 119872119901119892119860 1198810 = 1198601199090 (6)



The upper magnet

The middle magnet

The lower magnet

NS

N

S

h

h

Z

z

x y

J

J998400






119860119889119875 = 119889119891119889119881 = 119860119889119909 (8)


119870as = minus119889119891119889119909 = 1205811198602 (119875atm +119872119901119892119860)1198810 (9)


119870119881po = 119870as = 1205811198602 (119875atm + 119872119901119892119860)1198810 (10)






119865119911 = 119865 (119911 minus ℎ) + 119865 (119911 + ℎ) (11)


119865 = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902Φ(119880119894119895 119881119896119897119882119901119902 119903) (12)





(13)



119870 = minus120597119865119911120597119911 = 119870 (119911 minus ℎ) + 119870 (119911 + ℎ) (14)



sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)



119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)


119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)




Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)


Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1




(21)


119896119904 = 31198641198681198713 (22)



119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)









119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The upper magnet

The middle magnet

The lower magnet

NS

N

S

h

h

Z

z

x y

J

J998400






119860119889119875 = 119889119891119889119881 = 119860119889119909 (8)


119870as = minus119889119891119889119909 = 1205811198602 (119875atm +119872119901119892119860)1198810 (9)


119870119881po = 119870as = 1205811198602 (119875atm + 119872119901119892119860)1198810 (10)






119865119911 = 119865 (119911 minus ℎ) + 119865 (119911 + ℎ) (11)


119865 = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902Φ(119880119894119895 119881119896119897119882119901119902 119903) (12)





(13)



119870 = minus120597119865119911120597119911 = 119870 (119911 minus ℎ) + 119870 (119911 + ℎ) (14)



sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)



119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)


119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)




Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)


Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1




(21)


119896119904 = 31198641198681198713 (22)



119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)









119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (15)



119870119881ne = 119870ms = 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (17)


119870119881 = 119870119881po + 119870119881ne = 1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0

(minus1)119894+119895+119896+119897+119901+119902 120593 (119880119894119895 119881119896119897119882119901119902 119903) (18)




Δ1199091 = 119865 cos (45∘ + 120579)1198961 Δ1199092 = 119865 cos (45∘ minus 120579)1198962 (19)


Δ119909 = radicΔ11990921 + Δ11990922 (20)

a

C

a

D

FB

A Δx

120579

k2

k1




(21)


119896119904 = 31198641198681198713 (22)



119870119867po = 119870fs = 2119896119904 = 61198641198681198713 (23)









119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119896 = minus119898119892119897 (26)


119870119867ne = 119870ip = minus119898119892119897 (27)



119870119867 = 119870119867po + 119870119867ne = 61198641198681198713 minus 119898119892119897 (28)









119872119901119900 + 119888119881 (119900 minus 119894) + [[1205811198602 (119875atm +119872119901119892119860)1198810 + 119869119869101584041205871205830

sdot 1sum119894=0

1sum119895=0

1sum119896=0

1sum119897=0

1sum119901=0

1sum119902=0


(29)





Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Air chamber

Metalshell


lll

m m m

xf


0 0002 0004 0006 0008 0010

02

04

06

08

1

12

14

16

18

2


times106


Stiff

ness

Kas

(Nm

)

(a)

00120014

00160018

002

minus2minus1

01

2minus240000

minus200000

minus160000

minus120000

minus80000

minus40000

Displacement h (m)

Displacement y (m)

0 0180

01

Displ

Stiff

ness

(Nm

)

times10minus3

(b)

001 002 003 004 005 006 007 0080

05

1

15

2


times107

Stiff

ness

Kfs

(Nm

)

(c)

0 005 01 015 02 025 03 035minus10

minus8

minus6

minus4

minus2

0


times105

Stiff

ness

Kip

(Nm

)



=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=120581A2(Patm

V0

KV = KVpo + KV

ne

+JJ998400

41205871205830

1

sumi=0

1

sumj=0

1

sumk=0

1

suml=0

1

sump=0

1

sumq=0

(minus1)i+j+k+l+p+q


+ mgA)

KVpo =

120581A2(Patm

V0

+ mgA)


PS air springPNSP

Effective bandwidth

Natural frequency

Tran

smiss

ibili

ty (d

B)20

10

0

minus10

minus20

minus30

minus40

minus50

(a)

Effectivebandwidth

Naturalfrequency

Tran

smiss

ibili

ty (d

B)

20

10

0

minus10

minus20

minus30

minus40

minus50


PS flat springPNSP

KH = KHpo + KH

ne

KHpo =

6EI

L3

=6EI

L3minus

mg

L

(b)






















LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










LMSSCADAS

TestPC

1

2

3


Vibrationisolator

(PNSPPS)

Positionsensor

Transformmatrix

Logical axis

References Noise

Real sensor out

Real control out

x

y

(a)

1 vibrationisolator

2 vibrationisolator

3 vibrationisolator

Payloadplatform

Baseplatform

Feedforwardsensors

(b)











LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












LMSSCADAS

Controlsystem


(a)

LMS SCADAS

PC controllerwindow

Controller

Control system


NI controller

(b)







































6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























6 Conclusion



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

Am

plitu

de (d

B)

PNSP-XPS-X


(a)

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Am

plitu

de (d

B)

PNSP-YPS-Y


(b)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RzPS-Rz


(c)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-ZPS-Z


(d)

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

Am

plitu

de (d

B)

PNSP-RyPS-Ry


(e)

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Am

plitu

de (d

B)

PNSP-RxPS-Rx


(f)



minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus100

minus80

minus60

minus40

minus20

0

20

40A

mpl

itude

(dB)




(a)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(b)

minus100

minus80

minus60

minus40

minus20

0

20

40

Am

plitu

de (d

B)




(c)


2604

726

060

2608

0

2610

0

2612

0

2614

0

2616

0

2618

0

2620

026

210

1697000169750016980001698500169900016995001700000170050017010001701500170200017025001703000

Am

plitu

de

H1

H2

H3

(a)

V1

V2

V3

2627

7

2630

0

2632

0

2634

0

2636

0

2638

0

2640

0

2642

0

2644

416975001698000169850016990001699500170000017005001701000170150017020001702500

Am

plitu

de

(b)




Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Ultra-Low Frequency Two DOFs Vibration ...

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Transcript of Research Article An Ultra-Low Frequency Two DOFs Vibration ...