Modeling and Sensitivity Analysis

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Modeling and sensitivity analysis of a pneumatic vibration isolation systemwith two air chambers

Jun-Hee Moon a , , Bong-Gu Lee ba Department of Mechatronics, Daelim University College, 526-7, Bisan-dong, Dongan-gu, Anyang-si, Gyeonggi-do, 431-715, South Koreab Department of Mechanical Engineering, Daelim University College, 526-7, Bisan-dong, Dongan-gu, Anyang-si, Gyeonggi-do, 431-715, South Korea

a r t i c l e i n f o a b s t r a c t

Article histo ry:Received 17 August 2009Received in revised form 12 August 2010Accepted 14 August 2010Available online 17 September 2010

This paper aims at accurate modeling and sensitivity analysis for a pneumatic vibration isolationsystem (PVIS) as a foundation for practical design. Even though the PVIS is widely used for itseffective performance in vibration isolation, its design has depended largely on trial-and-errormethods. In previous studies, nonlinear characteristics of the diaphragm and the air owrestrictor, which signi cantly affect the performance of a PVIS, have been investigated. However,several hurdles,such as the absence of a mathematical model for the diaphragm, still remain withregard to the model-based predictionof performance. Therefore, a fractional derivative model forthe diaphragm and a quadratic damping model for the air ow restrictor are newly developedbased on the careful examination of previous studies. Then, sensitivities of vibration isolationperformance indices with regard to major design variables are analyzed and new approximationformulas are created based on the dynamic characteristics of the PVIS. Our models with atransmissibility-computing algorithm are veri ed by comparison with experimental data. The

sensitivityanalyses andapproximationformulasare expectedto be usefulfor practical PVISdesignowing to their simplicity and accuracy. 2010 Elsevier Ltd. All rights reserved.

Keywords:Pneumatic vibration isolationDiaphragmAir ow restrictorFractional derivativeEquivalent mechanical systemSensitivity analysis

1. Introduction

As high precision industries suchas semiconductorproduction, precisionmetrology, optics, and microbiology continue to grow,higher performance vibration isolation systems are needed to meet the corresponding vibration tolerance requirements [1 3]. Toachieve vibration isolation for local precision equipment, a pneumatic vibration isolation system (PVIS) is widely used because itneeds no energysupplyandno control unit, andperformsstableandeffective vibrationattenuation acrossa wide frequency range.Even though a PVIS is very useful, its design for better vibration isolation has depended largely on trial and error methods.

Vibration isolation performance enhancement of the PVIS has been attempted by a variety of ways such as reshaping of elastomeric diaphragm [4], usageof anair ow restrictor of porous media [5], energy dissipation by a gimbal piston in oil chamber[6], parallelization with a negative-stiffness device [7] and adoption of active control schemes [8 11] . In all those attempts, thebasic work is the modeling of components of the PVIS since effects of design variables on vibration isolation performance can bepredicted only with accurate mathematical models and corresponding computational techniques.

In this paper, our goal is accurate modeling and vibration isolation performance evaluation of a PVIS such that our results maybe used to predict the performance for practical PVIS design. We examined previous studies and determined that three additionalefforts are required.

The rst is to make a mathematical model of the diaphragm. Some studies pointed out that the diaphragm has an importantrole in the elastic and damping characteristic of a PVIS [4,12,13] . However, its nonlinear properties have been thus far neglected in

Mechanism and Machine Theory 45 (2010) 1828 1850

Corresponding author. Tel.: +82 31 467 4687; fax: +82 31 467 4869.E-mail address: [email protected] (J.-H. Moon).

0094-114X/$

see front matter 2010 Elsevier Ltd. All rights reserved.doi: 10.1016/j.mechmachtheory.2010.08.006

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j o u r n a l h o me p age : www. e l s ev i e r. co m/ l o c a t e / mech mt
http://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006http://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006http://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006mailto:[email protected]://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006http://www.sciencedirect.com/science/journal/0094114Xhttp://www.sciencedirect.com/science/journal/0094114Xhttp://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006mailto:[email protected]://dx.doi.org/10.1016/j.mechmachtheory.2010.08.006


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Nomenclature

A areab1 , b2 , b3 tuned coef cient for approximation formulasc E 0 , c E 1 , c E 3 material constants of the diaphragmC damping coef cientd diameter

E elastic modulus f force variationF Fourier transform of f fr friction coef cient g acceleration of gravityG complex nonlinear mappingh heightH transmissibilityi imaginary unitI mechanical impedanceK stiffnessL loss coef cientm air mass in the chamber

M tabletop massN volume ratio of damping chamber to spring chamberP pressureRe Reynolds numbert timeu uid velocityV volume x displacement X Fourier transform of xy ordered pair of complex variables acceleration parameter E shift factor exponent of fractional derivative

the speci c heat ratio (=1.4) variation strain loss factor viscosity density stress time constant angular velocity gradient operator

Superscripts

maximum minimum time derivative averaged

Subscripts0 static or averagea airat atmosphereb basec capillary tube or air ow restrictord diaphragm

1829 J.-H. Moon, B.-G. Lee / Mechanism and Machine Theory 45 (2010) 1828 1850


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many cases, since the diaphragm have been treated as a linear element in most analyses [14 17] or the properties of thediaphragm have taken the form of lookup table even in some studies that regarded it as a nonlinear element [12,13] .The mathematical modeling of the diaphragm is essential since it makes transfer functions to be analytic so that thetransfer functions might be used not only for transmissibility calculation but also for sensitivity analysis and performanceoptimization.

The second is to consider the air ow restrictor as a nonlinear damper, and the air volume through the restrictor as asimultaneous independent variable. Some previous studies regarded the ow restrictor as a linear damper, in which case thetransfer function of the PVIS is easily formulated [12,14,18] . However, the nonlinearity of the damping characteristics of the owrestrictor was demonstrated in many studies [11,19 21] . Leeand Kim [13] advanced the analysis of a PVIS by considering the owrestrictor as a nonlinear damper.

They made the approximation that the air volume passing through the restrictor is proportional to tabletop displacement.However, since tabletop displacement is not proportional to base displacement of vibration, it is easily inferred that the air volumecoupled with both of them is not proportional to either one of them. The last is to analyze sensitivities of vibration isolationperformance indices to design variables based on nonlinear dynamic characteristics of the PVIS. Many design strategies weredeveloped using the assumption that nonlinear models can be approximated as linear models [5,6,14,18,22] . However, the designstrategies could not cope with the nonlinear characteristics of a PVIS.

Therefore, the sensitivity analyses using nonlinear models are required for accurate performance prediction and for ef cientdesign of a PVIS. Consideration of the rst two issues, which is presented in Section 2 , results in the transfer functions of a PVISconsisting of two simultaneous nonlinear complex equations. To derive and calculate the transfer functions, the equal energydissipation method is adopted, and an equivalent mechanical system and a recursive numerical method are devised in Section 3 .The models and the PVIS transfer functions are veri ed by experimental data in Section 4 , in which the discrepancies between theprevious air ow restrictor models and experimental data lead to an adjustment of the model. Based on the dynamiccharacteristics of the PVIS, sensitivities of PVIS performance indices to design variables are analyzed and approximation formulasare created in Section 5 . Our concluding remarks are given in Section 6 .

In exploring the modeling of the PVIS, this research will be limited to considering only vertical rigid-body mode vibration of thetabletop because horizontal vibration and tabletop exural modes can be controlled by very different mechanisms, which shouldbe subjects of distinct research [23] .

2. Modeling of pneumatic vibration isolation system with two air chambers

A typical pneumatic vibration isolation system (PVIS), whose schematic is shown in Fig. 1, includes a piston, diaphragm, airchambers, air ow restrictor and wire pendulum. The piston is attached to, and moves along with, the tabletop mass. Thediaphragmseals air in the air chambers andallowssmoothmovementof the piston. Theair chambers arenamed according to theirfunctions in the PVIS: the spring chamber acts as a mechanical spring, and the damping chamber acts both as a spring and as adamper together with the air ow restrictor. The air ow restrictor hampers air ow across the air chambers. The wire pendulumisolates horizontal vibration and will not be mentioned further because horizontal vibration isolation is beyond the scope of thisstudy.

2.1. Air chambers

To simplify the analysis, thermodynamic condition of the air chambers is assumed as follows [4,12 14] : pressure andtemperature in an air chamber is uniform, and the thermal process in an air chamber is adiabatic.

The pressure variation induced by the volume variation in the damping chamber is derived from the gas law for adiabaticprocesses as follows:

P D = P 01 + V D

V D

1 1

e experimentdp experimental data for the diaphragmD damping chamberi inertialmp minimum resonant peakn iteration number p piston

S spring chambert transmittingT experimental data for the total PVIS



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where P D and V D are the pressure and volume of the damping chamber, respectively, and P 0 is the average air pressure. The airvolumethat has left the damping chamber, V D, is equal to the air volume passed through the ow restrictor, and is positive whenair leaves the damping chamber as seen in Fig. 1.

The pressure variation of the spring chamber is derived from the gas law for adiabatic processes involving volume variations of the spring and damping chambers as follows:

P S = P 0 1 + V S V D

V S

1

2

where P S and V S are the pressure and volume of the spring chamber, respectively, and V S is the volume variation of the springchamber, which results in lifting up the piston, as seen in Fig. 2.

2.2. Diaphragm

The diaphragm in the PVIS is a thin ber-reinforced elastomeric membrane that prevents air leakage in the air chambers andallows smooth movement of the piston as detailed in Fig. 2. The elasticity and damping properties of the diaphragm cannot bemeasured directly because the shape of the diaphragm in the de ated condition is quite different from that in the in atedcondition, and the de ated diaphragm is very dif cult to handle due to its exibility. Therefore, the stiffness of the diaphragm isobtained from experimental data, by subtracting the stiffness of an air chamber from the total stiffness of a PVIS having a single airchamber [13] .

Fig. 2. Schematic of diaphragm: ; static: , moved.

S

D

D

p

S

b

Fig. 1. Schematic of pneumatic vibration isolation system.



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We developed a novel fractional derivative model for the diaphragm, which has not been reported in the literature as far aswe know. Following Eq. (3) is a typical form of the four-parameter fractional derivative Zener model for viscoelastic materials[24 26] :

t + d

dt t = t

E d

dt t 3

where ( t ) is the stress, ( t ) is the strain,

E and are the maximum and minimum of elastic modulus, respectively, is the timeconstant, and is theexponent of the fractional derivative. Based on the fact that force is proportional to stress anddisplacement isproportional to strain, Eq. (3) can be rewritten as

f d + 1 d

d

dt f d =

K d x

K d1

d d

dt x 4

where f d is the force acting on the diaphragm, x is the displacement resulting from the rolling motion of the diaphragm,

K d and

K dare the maximum and minimum of the stiffness of the diaphragm, respectively, and d is the characteristic angular velocity of thediaphragm. Applying a Fourier transform to Eq. (4) yields

F d X =

K d +

K d i = d

1 + i = d 5

The effect of amplitude can be brought into Eq. (5) using a shift factor E as follows [27]:

F d = K d ; j X j X 6with

K d ; j X j =

K d +

K d i = d E j X j

1 + i = d E j X j 7

log 10 E x = c E 1 xc E 0 c E 2 + xc E 0 8

where c E 0 , c E 1 , and c E 3 are material constants which can be determined experimentally from measured data. The stiffness of thediaphragm is the real part of K d, and its loss factor is the ratio ofthe imaginary part to the real part of K d, which will be calculated forveri cation of the diaphragm model in Section 4.1 .

2.3. Air ow restrictor

The air ow restrictor of PVIS in this research is a capillary tube placed between the spring chamber and the damping chamberas shown in Fig. 3.

Fig. 3. Schematic of air ow restrictor.

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We propose a simple model for the air ow restrictor described in Eq. (9) , which improves upon the previous models as will beshown by comparing with experimental data (the proof will be provided in Section 4.2 since it uses a new analysis technique to beexplained in Section 3 ).

P c = L a2 jua jua 9

where L is the loss coef cient, a is the air density and ua is the air velocity.For comparison, a linear model and an nonlinear model, which were developed in previous research that adopted a capillarytube as an air ow restrictor [12,13] , are summarized in Appendix A .

2.4. Constitutive equations describing interactions between components

To describe the complete PVIS system, the constitutive equations expressing the relationships between the models arerequired.

The average pressure P 0 can be obtained from the static equilibrium condition for the tabletop mass (i.e., the gross mass on thePVIS including the piston and all mass attached to it such as the platform and payload):

P 0 = P at + Mg A p

10

where P at is the ambient atmospheric pressure, M is the tabletop mass, g is the acceleration of gravity, and A p is the effectivebottom area of the piston. By using the geometric compatibility between the piston movement and the spring chamber volume asshown in Fig. 1, the volume variation of the spring chamber V S is described as

V S = A p x p xb 11 where x p and xb are the displacements of the piston and the base, respectively. The volumetric rate of air ow from the dampingchamber is

V D = Ac ua 12

where Ac is the cross-sectional area of capillary tube or dc 2 /4. Variation of the supporting force to the piston f S is caused bypressure variation in the spring chamber:

f S = A pP S 13

External force variations on the tabletop mass include force variation caused by the variation of air pressure in the springchamber and force variation caused by diaphragm deformation:

Mx:: p = f d + f S 14

where the dead weight effect is removed naturally since only variations from equilibrium are considered. Pressure drop due to theow restrictor is equal to the pressure difference between the two air chambers:

P c = P S P D 15

3. Transfer functions and equivalent mechanical system

In this research, transmissibility from base vibration to tabletop motion is used to evaluate the vibration isolation performanceof the PVIS. To obtain transmissibility analytically, transfer function should be developed in advance.

3.1. Variable conversion and linearization that keeps original properties

For consistent physically-meaningful notation, variables and coef cients in the models, which have been derived from variousmechanics such as dynamics, thermodynamics, uid mechanics, and viscoelasticity, are converted into equivalent forms in

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dynamics. The conversion facilitates construction of the equivalent mechanical system that will be introduced in Section 3.3 .Equivalent displacement of volume variation in the damping chamber xD is de ned as

xD = V D = A p 16

As for the air ow restrictor, the equivalent damping forces caused by the capillary tube is de ned as

f c = A pP c 17By substituting Eqs. (12) and (16) into Eq. (9) and inserting the result into Eq. (17) , we obtain

f c = a LA

3 p

2 A2c

xDj j

xD 18

Quadratic damping is shown in Eq. (18) . In order to insert the model into transfer function, we should apply the Fouriertransform, which processes only a linear function with a single term.

Thus linearization that keeps original nonlinear property is required. Therefore, theequal energydissipation methodis adopted[29,30] . The basic concept of the method is as follows: the energy dissipated by quadratic damping during a cyclic motion is equalto that dissipated by the equivalent linear damping coef cient C c as described in Eq. (19) :

C c

xDdxD = a LA

3 p

2 A2c j

xD j

xDdxD 19

By letting xD =

xD sin t , we obtain

f c = C c ;

xD

xD 20

with

C c ;

xD = 4a LA

3 p

3 A2c

xD 21

Even though Eqs. (20) and (21) constitute a linear form appropriate to be Fourier-transformed, they virtually conceivequadratic damping since the equivalent damping coef cient has the product of frequency and amplitude.

As for the air chambers, the models described by Eqs. (1) and (2) are linearized by Taylor series expansion only up to the

rstorder term as follows:

P D = P 0 V DV D 22

P S = P 0 V S V DV S 23

Since the ratio of volume variation to original volume or V =V is less than 0.01 for a typical PVIS, the difference betweennonlinear Eq. (1) and corresponding linearized Eq. (22) is less than 1.22%. Thus, the linearized Eqs. (22) and (23) can besubstituted for Eqs. (1) and (2) with negligible error.

By multiplying both sides of Eqs. (22) and (23) by A p and substituting Eqs. (11) and (16) into the result, the equivalent elasticforces caused by the damping and spring chambers can be expressed as

f D = K D xD 24

f S = K S x p xb xD 25where the equivalent stiffnesses are given by

K D = P 0 A

2 p

V D26

K S = P 0 A

2 p

V S 27

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3.2. Transfer functions and computation

The following are the transfer functions derived in Appendix B :

X p X b

= 1

1 + 2 M K d ; X b X p = X b1j j +

11K S

+ 1K D + i C c ; X Dj j

28

X D =K S X b X p = X b1 K S + K D + i C c ; X Dj j 29

We see that Eqs. (28) and (29) with the coef cient equations of Eqs. (7), (8), (21), (26) and (27) constitute the PVIS transferfunctions (Eqs. (28) and (29) are the same as Eqs. (B.13) and (B.14) , respectively. Eqs. (B.10), (B.3) and (B.13) are the exact formsof Eqs. (7), (8) and (21) for calculation, respectively). The term the PVIS transfer functions will be used to reference theseequations in subsequent analyses.

In previous studies, the computations of transfer functions of the PVIS needed no iteration because the transfer functions werelinear [12 14] . Since our transfer functions are a complex nonlinear system that cannot be solved with one-step calculation, acomputational algorithm, which is composed of the xed-point iteration and under-relaxation, is newly devised and applied. Thedetails are described in Appendix C .

3.3. Equivalent mechanical system

Among the PVIS transfer functions, Eq. (28) is the major equation that renders transmissibility. Terms in Eq. (28) can beclassi ed by their physical meaning as follows:

H = 11 + I iI t

= I t I i + I t

30

where

H = X p = X b 31

I i = 2 M 32

33

Here, H , I i and I t denote the transmissibility, inertial mechanical impedance and transmitting mechanical impedance, respectively(mechanical impedance, also referred to as dynamic modulus in Ref. [28], represents force per unit displacement [29]).

The corresponding equivalent mechanical system is depicted in Fig. 4, where mechanical impedances are connected in serial orin parallel according to positions of corresponding terms in Eqs. (30) (33) . This equivalent mechanical system clari es the

physical meanings of components and their effects on the vibration isolation performance of the PVIS, and will be used to verifythe model of the air ow restrictor in Section 4.2 .

4. Experimental veri cation of the models and PVIS transfer functions

4.1. Experimental veri cation of the diaphragm model by stiffness comparison

The curve evaluated using the diaphragm model of Eqs. (7) and (8) is tted for the experimental data of Ref. [13] by tuning thecoef cients in the model. The experimental data and tted curves for displacements of 0.05 mm, 0.15 mm and 0.5 mm are shownin Fig. 5. The tuned coef cients are as follows:

K d = 1 :0 103 N m 1 ,

K d = 2 :5 104 N m 1 , d =6.310

4 rad s 1 , =0.15,c E 0 =5.010

3 m, c E 1 =4.0, and c E 0 =1.010 3 m.

The closeness of the curve and the experimental data indicates the validity of the model. Since the curve tting is performedmainly with the real part and complementarily with the loss factor, the loss factor differs from the experimental data. Because the

difference between the loss factor from the experimental data and that of the model is about 50% as seen in Fig. 5, and the ratio of the damping force from the diaphragm to that of the entire PVIS in experimental data, i.e. ( K dp dp ) / ( K T T ) is about 10% near the

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natural frequency, the overall damping force of the PVIS differs from the experimental results by about 5% (=50 % 10 % ).Accordingly, the in uence of this difference on the overall damping performance of the PVIS is so small.

4.2. Experimental veri cation of the air ow restrictor model by mechanical impedance comparison

The schematic of the experimental setup of Ref. [13] is the same as Fig. 4 except that their system contains no payload, and thedisplacement of the base is xed or X b =0. Thus, the total stiffness of the dual-chamber pneumatic spring in Ref. [13] is the sameas

10 1 10 0 10 10

2000

4000

6000

8000

10000

12000

14000

16000

Frequency [Hz]

10 1 10 0 10 1

Frequency [Hz]

R e a l p a r t o

f K

d [ N / m ]

0.05 mm (experiment)0.15 mm (experiment)0.5 mm (experiment)0.05 mm (model)0.15 mm (model)0.5 mm (model)

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

L o s s

F a c t o r o

f K

d

0.05 mm (experiment)0.15 mm (experiment)0.5 mm (experiment)0.05 mm (model)0.15 mm (model)0.5 mm (model)

(b)

Fig. 5. Experimental data and tted curve for mechanical impedance of diaphragm K d: (a) real part of K d, (b) loss factor of K d.

Fig. 4. Equivalent mechanical system of PVIS with two air chambers.



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the transmitting mechanical impedance of our PVIS model described in Eq. (33) in terms of physical meaning, because both implythe ratio of supporting force variation to base displacement variation.

By subtracting K d from both sides of Eq. (33) and replacing the transmitting mechanical impedance I t and the complexdiaphragm stiffness K d with K T (1+i T ) and K dp (1+i dp ), respectively, we obtain

K T 1 + i T K dp 1 + i dp = 1

1K S

+ 1K D + i C c ; X Dj j 34

where K T and K dp are real parts of mechanical impedances of the total PVIS and of the diaphragm stiffness, respectively, and T and dp are loss factors of the total PVIS and of the diaphragm, respectively.

For comparison between experiment and model, we divided both sides of Eq. (34) into Eqs. (35) and (36) :

I a = 1

1K S

+ 1K D + i C c ; X Dj j 35

I a;e = K T 1 + i T K dp 1 + i dp 36where I a denotes mechanical impedance caused by two air chambers and an air ow restrictor, and I a , e denotes the correspondingexperimental mechanical impedance. By setting X b =0 in Eq. (29) for the calculation of C c ,

X D = K S X p

K S + K D + i C c ; X Dj j 37

Since Eq. (37) is a recursive nonlinear equation, X D is evaluated by the xed-point iteration method with under-relaxation,mentioned in Section 3 .

Among the coef cients of Eq. (35) , K S and K D, which are determined by Eqs. (26) and (27) , respectively, are well-de ned andhave been accepted in numerous prior studies of air chambers [5,6,12,13,18] . Thus, Eq. (35) together with Eq. (37) can be used forthe veri cation of only one coef cient, C c . Therefore, the models are accessed by graphical comparison between Eqs. (35) and (37)(determined by the air ow restrictor model) and Eq. (36) (determined by experimental data).

For comparison between models, two air ow restrictor models developed in previous studies is applied to the equivalentlinear damping coef cient of air ow restrictor C c to be inserted into the PVIS transfer functions. By the equal energy dissipationmethod and variable conversion explained in Section 3 , Eqs. (A.1) and (A.2) become corresponding equivalent linear damping

coef cients, respectively, as follows:

C c ;1 = 128 a hc A

2 p

d4c 38

C c ;2 = hc

dc fr + L 4a A

3 p

3 A2c X Dj j 39

where the friction coef cient fr is calculated using Eqs. (A.3) (A.5) with the constitutive Eq. (12) and the dimension conversionformula of Eq. (16) .

In Figs. 6 8, the shape of the mechanical impedance of Erin and Wilson's model of Eq. (38) is very different from experimental

results and is independent of the base vibration amplitude because the model is linear.In Fig. 6, Lee and Kim's model of Eq. (39) with the loss coef cient L=0.7, and our model of Eq. (B.3) , which is the exact form of Eq. (21) for calculation, with L=1.2, coincide very well with experimental results. As the base vibration amplitude decreases, theplot based on Lee and Kim's model shows discrepancy with experimental results as seen in Fig. 8, which is the case when air owrate through the capillary tube is slow ( Reb 2300). The discrepancy is thought to originate from following two de ciencies.

The rst de ciency is that they assumed that the amplitude of air ow passing through the capillary tube

ua is proportional tothe amplitude of tabletop displacement X p. Although this assumption made the transfer function of the PVIS simple enough toneed only a single step calculation,

ua is found not to be proportional to X p either in magnitude and phase when

ua is taken as avariable independent of X p, as in our analysis.

The second de ciency is that they applied uid mechanics for internal ow in a fully developed region to a short capillary tube.Entrance length, where the fully developed region starts, ranges from 18 dc (for turbulent ow) to138 dc (for laminar ow) [31,32] .However, the length of capillary tube hc is about 13.3 dc in Lee and Kim's speci cation, so fully developed ow does not occur foralmost all ow rates. Therefore, we improved the capillary tube model by removing the pressure drop by internal ow term from

Lee and Kim's model of Eq. (A.2) . The criterion between laminar ow and turbulent owof Eq. (A.3) is not necessary for our modeldue to the removal of the term.

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The fact that our model shows good agreement with experimental results validates our inferences. The loss coef cient for theair ow restrictor Lis tuned to1.2 for the best t. The value is quitereasonable because the value is in the interval ofthe sum oflosscoef cients of entrance and exit: the capillary tube has both an entrance and exit, and the loss coef cient ranges from 0 to 1 forvarious shapes of entrance, and is about 1 for exit regardless of shape [31] .

Dynamic characteristics of the mechanical impedance of the air chambers and the capillary tube in Figs. 6 8 can be explainedusing Eq. (35) . The mechanical impedance of air chambers and capillary tube I a changes from 1/(1/ K S +1/ K D) to K S as C c

increases, and C c increases quadratically as frequency or amplitude of vibration increase according to Eq. (21) . This corresponds tothe fact that the mechanical impedance changes approximately from 1 = 1 = K S + 1 = K D = 8 :39 10 3 N m 1 to

K S =2.39104 N m 1 as frequency or amplitude of the base vibration increases, both in the analytic results and in the

experimental results. This implies that the effect of the spring chamber is dominant due to high damping by the capillary tube athigh frequency or large amplitude, and that the two air chambers act as one due to low damping at low frequency or smallamplitude.

4.3. Experimental veri cation of the PVIS transfer functions by transmissibility comparison

Experimental transmissibility data that were obtained from the research of Ref. [12] , are used to verify the PVIS transferfunctions by transmissibility comparison. The following are the speci cations and conditions of Ref. [12]: V S =7.3210

5 m 3 ,V D =4.1810

4 m 3 , A p =1.8510 3 m 2 , M =110 kg, a =1.82410

5 Pa s, hc =7.2710 3 m, and dc =6.1010

4 m 2 .There are seven coef cients to be determined in the diaphragm model of Eqs. (7) and (8) . Since d , c E 0 , c E 1 , c E 2 , and are

material constants, and is the angular velocity of movement or ffiffiffiffiffiffiffiffiffiffiffiK S + K d = M p , only

K and

K d are tuned to

t the model usingthe experimental data. Thus, we obtain

K = 4 :4 10 3 N m 1 and

K d = 12 :5 103 N m 1 .

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

x 10 4

R e a

l P a r t o f

I a [ N / m ]

(a)

0

0.1

0.2

0.3

0.4

0.5

L o s s

F a c t o r o

f I a

(b)

10 1 10 0 10 1

Frequency [Hz]

10 1 10 0 10 1

Frequency [Hz]

Fig. 6. Comparison of mechanical impedances caused by air chambers and capillary tube I a at X b =0.5 mm: (a) real part of I a , (b) loss factor of I a : +, experiment; , Erin and Wilson; , Lee and Kim ( L=0.7); , Moon and Lee ( L=1.2).

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With all the model coef cients identi ed, transmissibility is calculated by the PVIS transfer functions at X b =210 6 m and is

compared graphically in Fig. 9 with the experimental results. The good agreement validates the PVIS transfer functions andcalculation algorithms. The second resonance in Fig. 9 can be ignored in our analysis since it is from a rocking mode caused bymultiple isolator legs [12] .

5. Sensitivity analysis of vibration isolation performance

Our aim in using sensitivity analysis is to know how the major design variables affect the major performance indices of thePVIS, and to extract simple approximation formulas for design. The fundamental tasks required in a sensitivity analysis are toidentify a model of the object, and to screen out major inputs (or design variables) and outputs (or performance indices) of themodel [33,34] .

The selected design variables are the spring chamber volume V S , tabletop mass M , effective piston area A p, volume ratio of airchambers N , and capillary tube cross-sectional area Ac because these values have a strong effect on the vibration isolationperformance of a PVIS as will be shown in this section. The selected performance indices are the frequency and magnitude of theresonance peak since a PVIS uses the attenuation of displacement which occurs at a frequency range higher than the resonantfrequency, and resonant peaks should be suppressed if the frequency of base vibration is close to resonant frequency.

The speci cations of PVIS for the transmissibility calculation are the same as in the experiment of Ref. [13] as follows:

V S =8.110 4

m3

, V D =1.510 3

m3

, A p =5.310 3

m2

, M =212 kg (calculated by Eq. (10) with P 0 =4.93105

Pa),P 0 =4.9310 5 Pa, a =5.97 kg m 3 , a =1.7910

5 Pa s, hc =1.210 2 m 2 and dc =9.010

4 m. The tuned coef cients

0

0.1

0.2

0.3

0.4

0.5

L o s s

F a c t o r o

f I a

(b)

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

x 10 4

R e a

l P a r t o f

I a [ N / m ]

(a)

10 1 10 0 10 1

Frequency [Hz]

10 1 10 0 10 1

Frequency [Hz]


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0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

x 10 4

R e a

l P a r t o f

I a [ N / m ]

(a)

0

0.1

0.2

0.3

0.4

0.5

L o s s

F a c t o r o

f I a

(b)

10 1 10 0 10 1

Frequency [Hz]

10 1 10 0 10 1Frequency [Hz]


Frequency [Hz]

T r a n s m

i s s i

b i l i t y

10 1

10 0

10 0

10 1

10 1

Fig. 9. Comparison between transmissibility of experimental data and transmissibility evaluated by the PVIS transfer functions: +, experiment; , model.



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were determined in Section 4 as follows: L=1.2,

K d = 1 :0 103 N m 1 ,

K d = 2 :5 104 N m 1 , d =6.310 4 rad s

1 , =0.15, c E 0 =5.010

3 m, c E 1 =4.0, and c E 0 =1.010 3 m. If speci cations and tuned coef cients are not mentioned in the

remainder of this paper except for a certain value of design variable, the remaining design variable values are the same as those just cited. The frequency range is selected as 0.1 Hz to 100 Hz, and the amplitude range is selected from 0.1 m to 1 mm, becausethese values represent the general condition of base vibration [4,12] .

Xb=1 mm

Xb=0.1 mm

Xb=0.01 mm

(a)

Amplitude [m]Frequency [Hz]

T r a n s m

i s s i

b i l i t y

(b)

T r a n s m

i s s i

b i l i t y

10 1

10 1

10 0

10 0 10 1 10 2 10-10

10 -510 0

10 1

10 2

10 0

10-2

10 -4

Frequency [Hz]10 0 10 1

10 -710 -1 10 0 10 1 10 2

10 -6

10 -5

10 -4

10 -3

Frequency [Hz]

A m p

l i t u

d e

[ m ]

(c)

A

B

C

Fig. 10. Transmissibility of the PVIS: (a) transmissibility curve, (b) transmissibility surface, (c) contour view: A, one-chamber region; B, two-chamber region;C, minimum resonant peak.



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5.1. Dynamic characteristics of pneumatic vibration isolation system

Transmissibility of the PVIS is shown in Fig. 10(b) in the form of a surface in 3-dimensional space. Fig. 10(c) is a contour view,which is viewed from the transmissibility axis. As shown in Fig. 10, the transmissibility depends on amplitude of base vibration aswell as frequency, whereas the transmissibility of a linear second-order system depends only on frequency, and not on amplitude.The nonlinear dynamic characteristic originates mainly from the damping force or restriction force by the air ow restrictor asveri ed in Section 4.2 .

Consequently, at high frequencies or large amplitudes of vibration in the region labeled A in Fig. 10(c), the spring chamberworks alone as a mechanical spring, and the damping chamber nearly does not work, because the ow restriction force is strongenough to signi cantly block air ow between air chambers. Thus, for the remainder of this paper, we call this the one-chamberregion. The corresponding equivalent mechanical system is illustrated in Fig. 11.

At low frequencies or small amplitudes of vibration in the region labeled B in Fig. 10(c), the two air chambers act as a singlechamber, since the ow restriction force is weak enough to freely allow the air ow across air chambers. Thus, for the remainder of this paper, we call this the two-chamber region. The corresponding system is illustrated in Fig. 12.

5.2. Sensitivity of resonant frequency to air chamber volume, tabletop mass, and piston area

Eqs. (26) and (27) can be consolidated to the effective stiffness of air chambers K a as follows:

K a = P 0 A

2 p

V a40

where the effective volume of air chambers V a is assigned to V S for the one-chamber region and to V S + V D for the two-chamberregion. By inspection of the equivalent mechanical systems shown in Figs. 11 and 12 , the natural frequency of the PVIS with theeffective air volume becomes

f n = 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK d + K aM r 41

MPayload

Base Excitation

Tabletop Motion

Spring Chamber

Damping Chamber

Diaphragm Kd

KS

KD

Xp

Xb

Fig. 12. Equivalent mechanical system for the two-chamber region.

Fig. 11. Equivalent mechanical system for the one-chamber region.

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Thus the resonant frequency of the PVIS can be derived by substituting Eqs. (10) and (40) into Eq. (41) as follows:

f n = 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK dM + A2 pP at MV a + A p g V as 42

The approximation formula of Eq. (42) cannot be used for the transition region between the one-chamber region and the two-chamber region because the effective air chamber volume V a cannot be de ned in the transition region. As seen in thisapproximation formula, it was found that the PVIS resonant frequency depends on effective air chamber volume V a , tabletop massM , and piston area A p. Among these, M and A p are dependent on each other as described by Eq. (10) . Considering the maximumgauge pressure is about 5.5 bar (80 psi), which is general practice in industry, we con ne the ranges of V a , M and A p in thissubsection so that P 0 in Eq. (10) may be 2 through 6.5 bar in absolute pressure.

The solid line in Fig. 13 represents results calculated using the PVIS transfer functions by following procedure: 1) search aresonant peak of PVIS transmissibility curve for a given amplitude of base vibration by the golden section search method [35]; 2)store the frequency at the found peak; 3) repeat the procedure by varying chamber volume; and 4) draw the graph of resonantfrequency with respect to chamber volume. The solid lines in Figs. 14 and 15 can be obtained by applying the same procedure tothe tabletop mass and piston area, respectively.

The dashed lines in Figs. 13 15 are evaluated by the approximate formula of Eq. (42) . Since the resonant frequency for a basevibration amplitude of X b =1 m is in the two-chamber region, as seen in Fig. 10, the approximate resonant frequency should becalculatedusing Eq. (42) with V

a= V

S + V

Dand K

d =

K d +

K d =

2. Similarly, since the resonant frequency at X b

=1 mm isin the

1 2 3 4 5 6 7 8 9

x 10 3

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

Air chamber volume, V S+VD [m3]

R e s o n a n

t f r e q u e n c y ,

f n [ H z ]

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10 3

1

1.5

2

2.5

3

3.5

4

Air chamber volume, V S [m3]

R e s o n a n


f n [ H z ]

(b)

Fig. 13. Resonant frequency with respect to air chamber volume: (a) X b =1 m, (b) X b =1 mm:

, calculated by the PVIS transfer functions;

, calculated byapproximation formula of Eq. (42) .

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one-chamber region, the approximate resonant frequency should be calculated with V a = V S and K d =

K . The volume ratio waskept constant both in analytic and approximate solutions since the ratio between air chambers affects the shape of thetransmissibility surface.

By inspection of Figs. 13 and 14 , the resonant frequency decreases and approaches a horizontal asymptote as air chambervolumeor payload mass increases. Such phenomenaare also found in commercial air springs [36] . From Fig. 15, it is found that theresonant frequency increases as the piston area increases. The approximation Eq. (42) explains the reasons as follows: resonantfrequency f n approaches

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK d =M

p =2 as the air chamber volume V a goes to in nity; resonant frequency f n approaches

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A p g = V ap =2 as the tabletop mass M goes to in nity; resonant frequency f n increases as the effective piston area A p increases.Since the supplied pneumatic pressure generally has an upper bound in factories, there is a limitation on the enhancement of

PVIS attenuation performance resulting from the lowering of resonant frequency by increasing tabletop mass or reducing pistonarea, as described by Eq. (10) . Moreover, the diaphragm stiffness determines the lower bound of the PVIS resonant frequency

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK d =M p =2 that can be decreased by no more than the increase in air chamber volume. Therefore, making the diaphragm asexible as possible is an important factor with which to lower PVIS resonant frequency.5.3. Sensitivity of resonant peak magnitude to volume ratio of air chambers

In Fig. 16, the magnitude of the resonant peak with respect to the volume ratio N is depicted at X b =1 m, which causes thetransmissibility curve to lie in the two-chamber region. Since the resonant peak in the one-chamber region is maintained, only themagnitude in the two-chamber region is shown.

As shown in Fig. 16 by the solid line calculated using the PVIS transfer functions, the resonant peak magnitude decreased as thevolume ratio increased. However, when N is large, the curve levels off.

50 100 150 200 250 3001

1.5

2

2.5

3

3.5

Tabletop mass, M [kg]

50 100 150 200 250 300

Tabletop mass, M [kg]

R e s o n a n


f n [ H z ]

(a)

1.6

1.8

2

2.2

2.4

2.6

R e s o n a n


f n [ H z ]

(b)

Fig. 14. Resonant frequency with respect to tabletop mass: (a) X b =1 m, (b) X b =1 mm: , calculated by the PVIS transfer functions; , calculated byapproximation formula of Eq. (42) .

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1 2 3 4 5 6 7 8 9 104

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Volume ratio, V D /VS

R e s o n a n

t p e a

k m a g n

i t u d e ,

| X p

/ X b

|

Fig. 16. Resonant peak magnitude with respect to volume ratio of air chambers at X b =1 m:

, calculated by the PVIS transfer functions;

, calculated byapproximation formulas of Eqs. (43) and (48) .

0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.021.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Piston area, A p [m2]

0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02Piston area, A p [m2]

R e s o n a n

t f r e q

u e n c y ,

f n [ H z ]

(a)

1.5

2

2.5

3

3.5

4

4.5

R e s o n a n


f n [ H z ]

(b)

Fig. 15. Resonant frequency with respect to piston area: (a) X b =1 m, (b) X b =1 mm: , calculated by the PVIS transfer functions; , calculated byapproximation formula of Eq. (42) .

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To develop an approximation formula, we exploit damping characteristic of the secondorder mechanical system. The resonantpeak magnitude can be expressed with the loss factor as follows [37] :

H peak = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + 2q 43Meanwhile, the magnitude of damping force from the air ow restrictor is proportional to the square of the equivalent

displacement of the volume of air oscillating through air ow restrictor as described in Eqs. (B.2) and B.3 ). Thus,

F Dj j X Dj j2 44

Because, at resonant frequency, air ows through the ow restrictor at a high rate, it is assumed that the air in the springchamber and the air in the damping chamber is compressed by the same ratio at resonant peak. Then, the volume variations inboth air chambers can be expressed as

1 + N : N V S : V D 45

By substituting Eqs. (11) and (16) into Eq. (45) , we obtain

X D N N + 1 X p X b

1 X b 46By using Eq. (46) and the fact that transmissibility X p / X b is proportional to X b at resonant peak and that X b is an input variable,

Eq. (44) becomes

F Dj j N N + 1

247

Since the loss factor is related to system damping by the air ow restrictor and the diaphragm, the following can be extractedfrom Eq. (47) and from the fact that the damping force of the diaphragm F d is independent of hN.

= b1 N N + 1 2 + b2 48where the rst term is related to the damping force caused by the air ow restrictor, and the second term is related to the dampingforce caused by the diaphragm. By curve- tting Eqs. (43) and (48) to the resonant magnitude calculated by the PVIS transferfunctions, b1 =0.115 and b2 =0.03 are obtained, and the tted curve is displayed in Fig. 16 by a dashed line. Since coef cients b1and b2 are also dependent on the other design variables such as air chamber volume, piston area, tabletop mass, and capillary tubearea, theusage of approximation formulas of Eqs. (43) and(48) is limited only to predict the tendencyof peak magnitude variationwith respect to volume ratio variation.

5.4. Sensitivity of vibration amplitude at minimum resonant peak to air ow restrictor area

As seen in the transmissibility surface of the PVIS, it is found that the resonant peak reached its minimum, whose position is

labeled C in Fig. 10(c), at a certain value of the base vibration amplitude. In Fig. 17, the solidline in the graph represents the resultscalculated by the PVIS transfer functions using the following procedure: 1) search for the minimum point among resonant peaks of the transmissibility surface of the PVIS for a given capillary tube area by the golden section search method; 2) store the basevibration amplitude at the found minimum peak; 3) repeat the procedure by varying the area; and 4) draw the graph of basevibration amplitude of minimum resonant peak with respect to the area.

The plot of the base vibration amplitude minimizing the resonant peak magnitude X b , mp is approximately linear with respect tothe capillary tube area Ac in the log log graph shown in Fig. 17, so it can be expressed by

X b;mp = Ac

b3 2:3

49

where coef cient b3 remains constant as Ac changes while theother designspeci cations areheld constant. Thecurve evaluatedby

Eq. (49) , which is

tted to the exact value evaluated by the PVIS transfer functions, is shown by the dashed line in Fig. 17, and thetuned coef cient value is b3 =2.610 5 m 2 .

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A useful approximation formula can be derived from Eq. (49) as follows:

Ac ;2 = Ac ;1 X b;mp;2 X b;mp;1 !

0:435

50

This approximation formula is supposed to be very helpful in suppressing theampli cation of vibration in following case: thebasevibration amplitude X b, mp , 1 and the capillary tube area Ac , 1 are known only for a small range of the frequency and amplitude of basevibration because of the limitation of actuation and/or measurement instruments. The vibration amplitude of the base X b , mp , 2 wherethe object of vibration isolation stands is known. In addition, the frequency of the base vibration is close to the resonant frequency.With this known information, the capillary tube area Ac , 2 can be determined in order to minimize resonant peak magnitude by usingEq. (50) . This is equivalent to placing the resonant peak on point C in Fig. 10(c).

6. Conclusions

The nonlinear properties of the diaphragm of a PVIS have been described using look-up tables in previous studies since it isvery dif cult to model the stiffness and damping, which depend on the amplitude and frequency of deformation. In this paper,a fractional derivative model for the diaphragm was newly developed and was shown to agree very well with experimentaldata.

The capillary tube in the PVIS, referred to as an air ow restrictor, has previously been regarded as a linear damper or as anonlinear damper with its air ow volume being proportional to the tabletop displacement. We have described the limitations of these assumptions of air ow restrictors and improved the model for the air ow restrictor. In our model, the damping force isproportional to the square of the ow rate and the air volume through the capillary tube is de ned as an independent variable.

Based on comparisons with experimental data, the new air

ow restrictor model has proved to be more accurate than previousmodels.The PVIS transfer functions were derived from our new models by the equal energy dissipation method and the conversion to

equivalent dynamic variables. To calculate the obtained transfer functions that cannot be calculated in one step due to thenonlinearity, xed-point iteration with under-relaxation was applied. The computational algorithm was proven to be valid bycomparison of its results with experimental ndings. The algorithm enabled sensitivity analysis as well as transmissibilitycalculations.

The equivalent mechanical system, which was newly proposed in the formulation of the transfer functions, clari ed thephysical meanings of components of the PVIS and was useful in adjusting the air ow restrictor model. The dynamic characteristicof the PVIS, namely, that the transmissibility surface should be divided into two characteristic regions according to the ranges of the frequency and amplitude, could be explained using the equivalent mechanical system.

For the twocharacteristic regions, the sensitivities of vibration isolation performance indices to design variables were analyzedusing the PVIS transfer functions and the golden-section search method. In order to facilitate ef cient design, approximation

formulas were created based on the dynamic characteristics of the PVIS. The results agreedvery well with the results of sensitivityanalyses through the tuned coef cients.

10 -610 -7 10 -6

10 -5

10 -4

10 -3

Capillary tube area, A c [m2]

B a s e a m p

l i t u

d e a

t m

i n i m u m

p e a

k , X

b , m

p [ m ]

Fig. 17. Base vibration amplitude at minimum peak magnitude with respect to capillary tube area: , calculated by the PVIS transfer functions; , calculatedby approximation formula of Eq. (49) .

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Acknowledgement

We wish to express our gratitude for the support of the Ministry of Education, Science and Technology (BK21) and the KoreaScience and Engineering Foundation (ERC: Micro Thermal System Research Center). We are also grateful to Prof. Heui Jae Pahk, Dr.Byung-Hoon Lee and Dr. Jeung-Hoon Lee for their helpful comments.

Appendix A. Previous models of air ow restrictors

The Erin and Wilson's model [12] of the air ow restrictor is

P c = 32 a hc d2c ua A:1

where P c is the pressure drop across the capillary tube, a is the viscosity of the air, hc is the height of the capillary tube, dc is thediameter of the capillary tube, and ua is the velocity of air.Subsequently, Lee and Kim [13,22] adopted a nonlinear model for thecapillary tube as follows:

P c = hc dc fr + L a2 uaj jua A:2where L is the loss coef cient and a is the air density. The friction coef cient fr is a function of Reynolds number Re for laminarow and turbulent ow as follows:

fr =

64Re

for Re 2300 laminar flow 0:3164Re1 =4

for 4000 b Re105 turbulent flow 8>>>:

A:3

with

Re = aua dc a

A:4

where the cycle-averaged air velocity ua is given by

ua = 2

2 = 0 uaj j t =

2

ua A:5

with ua =

ua sin t .

Appendix B. Derivation of PVIS transfer functions

The pressure relation in Eq. (15) can be converted into the force relation in Eq. (B.1) by multiplying both sides by A p:

f c = f S f D B:1

To obtain the transfer function of the PVIS, Fourier transforms are performed on the models and constitutive equations. Fouriertransforms of variables and functions are signi ed by capital letters as follows: x X ,

x j X j ,

x i X , x::

2 X , and f F .Accordingly, Eqs. (20), (21), (24), (25), (14), and (B.1) are Fourier-transformed into Eqs. (B.6) through (B.7) , respectively.

F c = i C c ; X Dj j X D B:2

C c ; X Dj j = 4a LA

3 p

3 A2c X Dj j B:3

F D = K D X D B:4

F S = K S X p X b X D B:5 2 MX p = F d + F S B:6

F S = F D + F c B:7

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Thediaphragmmodeldescribedin Eqs. (6), (7) and (8) shouldbe adjusted into Eqs. (B.8), (B.9) and(B.10) , because the relativedisplacement of the piston to the base X p X b is the effective rolling displacement, as shown in Fig. 1:

F d = K d ; X p X b X p X b B:8K d ; X p X b

=

K d +

K d i = d E X p X b

1 + i = d E X p X b B:9

log 10 E X p X b =c E 1 X p X b c E 0 c E 2 + X p X b c E 0

B:10

By substituting Eqs. (B.2) and (B.4) into Eq. (B.7) and rearranging for X D, we obtain

X D = F S K D + i C c ; X Dj j B:11

By substituting Eqs. (B.11) into Eq. (B.5) and rearranging for F S ,

F S = 1

1K S

+ 1K D + i C c ; X Dj j X p X b B:12

Finally, by inserting Eqs. (B.8) and (B.12) into Eq. (B.6) and rearranging, we obtain the transmissibility from base vibration totabletop motion or X p / X b as follows:

X p X b

= 1

1 + 2 M K d ; X b X p = X b1j j +

11K S

+ 1K D + i C c ; X Dj j

B:13

The equivalent displacement of the air volume passing through the air ow restrictor X D is obtained by substituting Eqs. (B.2),(B.4) and (B.5) into Eq. (B.7) as follows:

X D =K S X b X p = X b1 K S + K D + i C c ; X Dj j B:14

Appendix C. Computational algorithm for the PVIS transfer functions

The PVIS transfer functions can be abstracted as follows.

y = Gy ; y = X p = X b ; X D C:1where G is the complex nonlinear mapping, and the ordered pair of y are complex numbers. We applied the xed-point iterationfor calculation of the PVIS transfer functions. When the xed-point iteration method was directly applied to the PVIS transferfunctions, the speed of contraction was very slow. Thus, the under-relaxation method is applied to hasten convergence bydamping out oscillations of y as follows [38]:

y n + 1 = 1 y n + y n + 1 ; y = X p = X b; X D C:2where is the acceleration parameter and subscript n is the number of iterations. Conditions for numerical calculation of transmissibility of the PVIS are as follows: the initial values are X p =0and X D =0; input values are the base vibration amplitude X band the base vibration frequency ; and iteration stops when the product of relative errors of variables between two adjacentiteration steps is less than relative error criterion of 10 6 . The acceleration parameter is set to 0.75, since iteration convergence isthe fastest at this value.

References

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27.[2] C.G. Gordon, Generic criteria for vibration-sensitive equipment, Proceedings of SPIE, vol.1619, Nov 1991, pp. 71 85, San Hose, CA.

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of Sound and Vibration 286 (2005) 615 636.[29] F.S. Tse, I.E. Morse, R.T. Hinkle, Mechanical Vibrations: Theory and Applications, second ed.Allyn and Bacon, 1978.[30] J.H. Moon, Analysis and optimal design of a pneumatic vibration isolation system, PhD Thesis, Seoul National University, 2002.[31] F. White, Fluid Mechanics, third ed.McGraw-Hill, 1994.

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Jun-Hee Moon received the B.S., M.S. and Ph.D. degrees in the Department of Mechanical Design and Production Engineering fromSeoul National University, South Korea, in 1992, 1994, and 2002, respectively. From 2000 to 2004 he was a Research Engineerat SNUPrecision Co., Ltd. From 2005 to 2008 he was a Senior Researcher at the Micro Thermal System Research Center. Since 2009 he hastaught in the Department of Mechatronics at Daelim University College, South Korea. His research interests include the analysis andcontrol of precision systems using air pressure, piezoelectric materials, and mechanical linkages.

Bong-Gu Lee received the B.S., M.S. and Ph.D degrees in the Department of Mechanical Engineering from Yonsei University, SouthKorea, in 2000, 2003, and 2009, respectively. He has been working as a Senior Researcher at the Korea Institute of IndustrialTechnology (KITECH) from 2000 to 2002. He is currently a Professor in the Department of Mechanical Engineering at DealimUniversity College. His main research interests are micro machining and material processing using lasers and ultrasonic vibration.

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Modeling and Sensitivity Analysis

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