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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 3578–3590

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Modeling and analysis of dual-chamber pneumatic spring withadjustable damping for precision vibration isolation

Huayan Pu, Xin Luo, Xuedong Chen n

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074,

Hubei Province, China

a r t i c l e i n f o

Article history:

Received 17 June 2010

Received in revised form

1 March 2011

Accepted 6 March 2011

Handling Editor: L.G. Thamthoroughly how the adjustable parameter of VOM affects the behavior of DCPS-AD,

Available online 2 April 2011

0X/$ - see front matter Crown Copyright & 2

016/j.jsv.2011.03.005

esponding author. Tel./fax: þ86 27 8755732

ail address: chenxd@mail.hust.edu.cn (X. Che

a b s t r a c t

Dual-chamber pneumatic spring with adjustable damping (DCPS-AD) employs a

variable orifice mechanism (VOM) to obtain the adjustable stiffness and damping

characteristics. These adjustable characteristics are aimed at improving the perfor-

mance of the pneumatic vibration isolation system (VIS). In order to understand

the model of DCPS-AD is derived analytically and validated experimentally. The

influence of VOM on the performance of DCPS-AD is analyzed quantitatively. All the

results demonstrate that VOM has the ability to vary complex stiffness distribution in

frequency domain. Based on these results, the approach optimizing the performance of

VIS is proposed, which is realized by adjusting VOM actively. Performance experimental

measurements of VIS in frequency domain are carried out under different payload

masses. The measurement results validate the proposed approach, which can optimize

the performance of VIS when some application conditions varying.

Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Ultra-precision vibration isolation systems (VIS) have been essential in the field of ultra-precision engineering, alongwith the progress of ultra-precision metrology equipment and ultra-precision manufacturing systems. The requirementsof lower natural frequency, higher attenuation ratio for ultra-precision vibration isolation systems have become more andmore stringent [1].

Of an ultra-precision VIS, pneumatic spring is one of the key elements. Compared to mechanical or electromechanicaldevices, a pneumatic spring has the ability to support large mass with small stiffness due to the volumetric compressibilityof air [2].

The demands for a pneumatic spring are softer stiffness and better damping characteristics. Optimal structure designand active control [3] are well-recognized as two effective ways to improve the vibration isolation performance of apneumatic spring. However, to achieve the best performance of both these ways, an accurate model of the pneumaticspring is essential.

Several researchers have investigated the performance of pneumatic spring involved in ultra-precision VIS theoreticallyand/or experimentally. The pneumatic spring using in precision vibration isolation has a small stroke and constanteffective piston area. Heertjes and van de Wouw [4] developed a nonlinear model of a single-chamber pneumatic spring.They presented that the single-chamber pneumatic spring shows obvious nonlinearity when the displacement of the

011 Published by Elsevier Ltd. All rights reserved.

5.

n).

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3579

payload is not small enough. Due to the lower stiffness and better damping characteristic, dual-chamber pneumatic springis used widely in the vibration isolation system for ultra-precision instruments.

DeBra [5] proposed a simple linear model of the pneumatic vibration isolator. He presented that air flow from theisolator to a tank can dissipate the oscillatory energy if it is passed through a restrictor. Erin and Wilson [6] employed thismodel to simulate the transmissibility of dual-chamber pneumatic isolator and measured the transmissibility experi-mentally. The existence of the discrepancy between the measured result and simulation was observed. A modified modelthat incorporates the effects of the diaphragm was proposed to improve the accuracy of analytical model. Lee and Kim [7]investigated the complex stiffness of the dual-chamber pneumatic spring experimentally. Except the frequencydependence, the experiment results exhibited significantly vibration amplitude dependent nonlinear behavior. Throughthe refined analysis, they pointed out that there were two phenomena attributed to this nonlinear behavior, one is theamplitude-dependent complex stiffness of the diaphragm, and another is the oscillating flow in capillary tube. Based onthe improved model, a transmissibility design method was proposed [8] to make the frequency of the maximum dampingcoincide with the resonance frequency of transmissibility. The flow restriction coefficient, chamber volume ratio anddiaphragm stiffness were optimized to achieve optimum transmissibility.

The performance of dual-chamber pneumatic spring (DCPS) is much better than the single chamber pneumatic spring.Nevertheless, it is still a passive vibration isolator with fixed characteristic parameters. While the application conditionvarying, such as the payload mass, frequency band of disturbance, the effectiveness of vibration isolation will be affected.For example, while payload mass of VIS varying, the natural frequency of the vibration isolation system changes. Due tovariation of natural frequency, the frequency of maximum damping of the optimal pneumatic spring does not coincidewith the resonance frequency of VIS anymore. The dual-chamber pneumatic springs in the previous studies have constantcharacteristic parameters, such as the restriction coefficient. This means that, once the pneumatic spring is engineered,these parameters cannot be adjusted any longer. Therefore, the optimal designed pneumatic spring obtains the optimumperformance only for the special application object under the special application condition. In order to obtain optimumperformance in a range of application, some characteristic parameters of pneumatic spring should be adjustable accordingto the characteristic of application condition.

The dual-chamber pneumatic spring with adjustable damping (DCPS-AD) can solve this puzzle, which utilizes a variableorifice mechanism (VOM) to vary the equivalent area of the orifice actively, which results in the variable stiffness anddamping adjustable ability. Hence, the behavior of DCPS can be varied actively according to the change of applicationcondition. Then, better effectiveness of vibration isolation can be obtained. To realize this, it is important to understandthoroughly how the adjustable parameter affects the behavior of DCPS-AD.

In this paper, the influence of VOM on the performance of DCPS-AD is analyzed quantitatively and the VIS performanceoptimal approach by adjusting VOM actively is proposed. The analytical model of DCPS-AD is investigated theoreticallyand experimentally. The effect of the variable orifice mechanism, along with other operating conditions (frequency ofvibration) on the characteristic of the pneumatic spring is studied. The paper comprises 6 sections. In Section 2, thestructure of the DCPS-AD is presented. The model of DCPS-AD is developed in Section 3. Computation simulation is carriedout to validate the analytical model in this section. The complex stiffness measurement is carried out to validate theanalytical model and simulation in Section 4. In Section 5, the relation between the variety of orifice area opening and thefrequency characteristic of complex stiffness is built based on the analytical model, and the performance of pneumaticvibration isolation system is analyzed and optimized. Conclusions are summarized in the end.

2. Structure of DCPS-AD

A DCPS-AD is presented as shown in Fig. 1. The DCPS-AD consists of a cylindrical metal body whose air volume isenclosed by a thin-walled, flexible and pressure resistant round membrane in the top end. A piston is seated on thismembrane which can provide high suspension force with low stiffness in the vertical direction. In order to obtain highdamping effect, the air space of the pneumatic spring is split into two chambers, named the Load Chamber (LC) and theDamping Chamber (DC), respectively. These two chambers are linked by a variable orifice mechanism (VOM), whichconsists of a pipe and an adjustable orifice valve. The stiffness of the dual-chamber pneumatic spring attributes to twophenomena: internal volume variation due to compression of air in the spring caused by an external force acting on it, andthe elastic stiffness of the membrane [9]. The damping attributes to air flow through the orifice, which controls the massflow rate between two chambers, and provides damping to the system. When the air fluid flows along a pipe, andencounters a constriction, for example an orifice, a pressure differentiation exists, which forces the fluid itself through theconstriction, causing a phenomenon that appears as damping [10]. The damping can be easily changed by the VOM.

3. Modeling and simulation of DCPS-AD

3.1. Some assumptions

The development of the model involves four parts: The first part deals with the motion of the piston in response to thedisplacement excitation; the second part establishes equations that govern thermodynamic behaviors of the LC and DC;

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903580

the third part represents the fluid dynamic behavior of air flow through the VOM; and the fourth part deal with the complexstiffness of the membrane. To analyze the characteristic of the pneumatic spring, the following assumptions are made:

Assumption 1. The air in both the static and dynamic equilibrium states can be considered to be ideal gas, since it is athigh temperature and low pressure with respect to its critical point values.

Assumption 2. The gas process that defines the relationship between pressure and volume in both LC and DC complieswith the adiabatic, isentropic relationship.

Assumption 3. The air flow through the VOM is frictionless, i.e., no energy loss due to friction, either in the fluid itself orbetween the fluid and the pipe walls.

Assumption 4. The amplitude dependent behavior of the dual-chamber pneumatic spring is neglected in the model ofDCPS-AD. The focus of this paper is to study the influence of the VOM on the characteristic of DCPS-AD, the vibrationamplitude assumes to be constant.

According to these assumptions, the pneumatic spring is simplified as two chambers connected by an orifice, whoseequivalent area is variable, as shown in Fig. 2.

3.2. Modeling of the DCPS-AD

3.2.1. Definition of the stiffness of the DCPS-AD

The stiffness of the dual-chamber pneumatic spring under preload K is given by Eq. (1), according to the definition of stiffness

K ¼�df

dx(1)

xf

Fig. 2. Simplified model of the DCPS-AD.

Damping chamber

Loadchamber

MembranePiston

Adjustableorifice

Metal body

pipe

Fig. 1. Schematic of the DCPS-AD.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3581

where df is the perturbation force acting on the piston, and dx is perturbation displacement of the piston. In static equilibriumstate, pressures in both chambers are constant and equal. The initial force acting on piston f is governed by

f ¼ Apðp0�patmÞ (2)

where p0 and patm are the absolute chamber pressure and the atmospheric pressure, respectively, and Ap is the piston area.When a small perturbation force df acts on the piton, the relation between the perturbation force and the instantaneous

variation of internal pressure of the LC dpL can be derived from the differentiation of Eq. (2) with respect to time t

df

dt¼ Ap

dpL

dt(3)

The volume of the LC VL is given by

VL ¼ Apx (4)

where x is the height of the LC.Differentiating Eq. (4) with respect to time t yields

dVL

dt¼ Ap

dx

dt(5)

Substituting Eqs. (3) and (5) into Eq. (1), the stiffness of the dual-chamber pneumatic spring is obtained as

K ¼�df

dx¼�A2

p

dpL

dVL(6)

Eq. (6) indicates that the stiffness of the DCPS-AD is related to the variations of both pressure (dpL) and volume (dVL)in LC. However, because of the existence of DC connected with LC by the VOM, the variations of both dpL and dVL in LC arestrongly coupled with those in DC. Therefore, the dynamic relations between LC and DC must be elaborately developed,in order to obtain stiffness characteristic of the pneumatic spring.

3.2.2. Modeling of the loading chamber

Perturbation force imposed on the piston causes variations of mass, pressure, volume and temperature changes in boththe LC and the DC. LC is analyzed as a control volume involving a moving boundary [11]. The mass flow rate of air into LC isas [6,7]

_mL ¼1

RT0

_pLVL0

k þpL0_V L

� �(7)

where pL, VL, mL, T are the pressure, volume, mass and temperature in LC, respectively, and k and R are the specific heatratio and the universal gas constant, respectively. The dot over a symbol is used to indicate time rate of change, and thesubscripts ‘0’ denotes static equilibrium.

3.2.3. Modeling of the Damping Chamber

The DC can be analyzed as the control volume with a fixed boundary, as there is no volume change in it. The mass flowrate of air into DC ð _mDÞ can be obtained by using the same method as LC

_mD ¼1

RT0

_pDVD0

k

� �(8)

where VD0 is the volume of DC, and _pD is the pressure change rate of DC.

3.2.4. Modeling of the variable orifice mechanism

The mass flow rates of air into LC and DC are connected by the mass flow rate of air through the VOM, which is given as

_mo ¼� _mL (9a)

_mo ¼ _mD (9b)

According to the empirical correlations [12], the mass flow rate of air through the orifice can be defined versus pressureat its ends

_mo ¼ Aopu

ffiffiffiffiffiffiffiffi2

RT0

sc (10a)

c¼

kk�1

pd

pu

� �2=k�

pd

pu

� �ðkþ1Þ=k� � 1=2

signðpL�pDÞ for pd

pu40:518

k2

2kþ1

� �ðkþ1Þ=ðk�1Þ� �1=2

signðpL�pDÞ for pd

pur0:518

8>>>><>>>>:

(10.b)

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903582

where Ao is the equivalent area of orifice. The pressures pu (p upstream) and pd (p downstream) are defined as

pu ¼maxðpL,pDÞ, pd ¼minðpL,pDÞ

Flow parameter c given in Eq. (10b) is related to the pressure ratio of the upstream to downstream. When the pressuredifferentiation between upstream and downstream is not too large, the pressure ratio pd/pu40.518, the flow is subsonic, c is afunction of the pressure ratio; otherwise, the flow is sonic, the flow parameter is constant. The pressure differentiation of DCPS-ADis induced by the vibration. Thereby, the pressure ratio in DCPS-AD is much bigger than 0.518 as the vibration amplitude is small.

3.2.5. Membrane model

The membrane is made of a rubber material. It works as a complex stiffness element parallel connecting with the DCPS-AD.For it is difficult to estimate the complex stiffness of the membrane analytically, the indirect method proposed by Lee and Kim[7] is used to measure the complex stiffness of membrane KM(o). The measurement result is shown in Appendix A.

3.2.6. Complex stiffness of the DCPS-AD in frequency domain

The relation between the variations of pressure and variation volume in LC can be obtained by combining Eqs. (7)–(10).Transforming these equations into frequency domain through the Fourier transformation, the frequency characteristics ofthe stiffness of the dual-chamber pneumatic spring can be obtained. The mass flow rates of air into LC and DC in frequencydomain are given by

~MLðoÞ ¼1

RT0p0joVLðoÞþ

VL0

k joPLðoÞ� �

(11)

~MDðoÞ ¼1

RT0

VD0

k joPDðoÞ� �

(12)

For the vibration amplitude is small compared to the height of the LC, the pressure differentiation betweentwo chamber is small. For the simplification of calculation, the nonlinear equations of mass flow rate through the VOM,i.e. Eq. (10), is linearized at the vicinity of the nominal operating point as

~MoðoÞ ¼ CoAo½PLðoÞ�PDðoÞ� (13)

where Co is defined as the flow coefficient of orifice, which can be obtained experimentally.Substituting Eqs. (9) and (11)–(13) into Eq. (6), the complex stiffness of DCPS-AD due to compression of air and air flow

between two chambers is obtained as

KcomplexðAo,oÞ ¼ kðVD0=RT0kÞo� �2

þð1þNÞðCoAoÞ2

ðVD0=RT0kÞo� �2

þð1þNÞ2ðCoAoÞ2þk

NðCoAoVD0=RT0kÞoðVD0=RT0kÞo� �2

þð1þNÞ2ðCoAoÞ2

j (14)

where N is the volume ratio of DC over LC (VD0/VL0), and k is the stiffness of LC as defined by Heertjes and van de Wouw [5], and

k¼kp0A2

p

VL0

The synthetical complex stiffness of the DCPS-AD can be represented by the sum of the complex stiffness due to compressionof air and air flow between two chambers and the complex stiffness due to the characteristic of membrane, it is given as

K�complexðAo,oÞ ¼ KcomplexðAo,oÞþKMðoÞ (15)

The real part (Kstorage) of the complex stiffness refers to the storage stiffness, which denotes the stiffness characteristicof the DCPS-AD; and, correspondingly, the imaginary part (Kloss) is the loss stiffness, which describes the dampingcharacteristic of the pneumatic spring.

3.3. Computational simulation

Computational simulation of the complex stiffness is carried out. Table 1 gives the values of all parameters used incomputational simulation.

The complex stiffness of DCPS-AD of Eq. (15) is computed numerically in the frequency domain. Fig. 3 shows curves ofcomputational simulations along frequency axis, each with a different orifice area, ranging from 9% opening to 100% opening.

From Fig. 3, it can be observed that, the storage stiffness has an ‘S’ shape along frequency axis as shown in Fig. 3(a),while the loss stiffness has a bell shape as shown in Fig. 3(b). If we separate frequency axis into five stages, says, still(less than 0.1 Hz), low-frequency stage (LF, up to about one Hertz), medium-frequency stage (MF, from several Hertz toten Hertz), and high-frequency stage (HF, ten Hertz to several ten Hertz, and more than 100 Hz), and higher-frequency(HRF, larger than 100 Hz), we can find that, no matter how much the orifice area opening is, the storage stiffness is almosta smaller constant at still and LF, and a bigger constant at HRF, while increases slightly along with frequency increasing inLF, and dramatically in MF. Meanwhile, the loss stiffness has a smaller value at still, increases slightly in LF, dramatically inMF till reaches the maximum (peak point). After the peak point, the value decreases dramatically in MF, and slightly in HF.

Table 1Parameters values used in simulation.

Symbol Value Unit Meaning

Patm 101.325 kPa The standard atmospheric pressure

Ap 3.0�10�3 m2 The piston area

k 1.4 – Specific heat ratio

R 286.9 J/kg K Universal gas constant

P0 4.6�102 kPa The initial pressure of Load and Damping chamber

x0 18.5�10�3 m The initial height of the Load Chamber

VL0 5.9�10�5 m3 The initial volume of Load Chamber

VD0 3.5�10�4 m3 The volume of Damping Chamber

N 6 – Volume ratio (VD0/VL0)

T0 295 K Temperature

Fig. 3. Computational simulation of the complex stiffness with different orifice area opening: (a) storage stiffness and (b) loss stiffness.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3583

It is also interesting to observe that, with increasing the opening area of the orifice, from 9% opening to 100% opening,the values at still and in HRF are nearly changed, but the interval in which the storage increases slightly is extended, aswell the interval in which the storage increases dramatically shrinks.

Although the peak value of the loss stiffness is nearly changed with the opening area of the orifice increase, the shapeis extended along the abscissa (frequency axis), and the peak point dramatically shrifts and appears at higherfrequency point. In our simulation investigation, by increasing the opening area of the orifice, the frequency point atwhich the maximum loss stiffness appears moves from 3 to 27 Hz.

4. Experiment results

4.1. Set-up of experiments

Experiment is undertaken to validate the effect of adjusting the opening area of the orifice on the dynamic behaviorof the DCPS-AD. The varying capability of equivalent area of orifice is generated by the adjustable orifice valve in VOM.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903584

There is a certain relationship between the spool position of orifice valve and orifice flow area. Through the spooldisplacement adjusting knob of the valve, the spool position can be easily adjusted to be the desired value and the desiredequivalent area of orifice is obtained. The sketch of experimental set-up is shown in Fig. 4.

The dual-chamber pneumatic spring is installed on a computer-controlled test facility, BOSE test instrument(ElectroForce 3500), which consists of the frame, linear motor, displacement sensor and load cell. The piston of thepneumatic spring is fixed to the mover of linear motor, which is used to exert the excitation. The bottom of the pneumaticspring is installed on the fixed part of the instrument. The input signal for stiffness test is a series of displacement sinewaves applied by the linear motor to the piston and measured by a linear variable differential transformer (LVDT). The testfrequency range is 0.2–20 Hz. The output is force wave which is measured by load cell. These signals are acquired by theWinTest PCI Control Electronics which is the BOSE Corporation’s digital control hardware, and the acquired data isanalyzed by the BOSE Corporation’s WinTest dynamic mechanical analysis of materials and devices (DMA) software forattaining the complex stiffness. The photo of the experimental set-up is shown in Fig. 5.

4.2. Results of experiments

The operational conditions in the experiment contain the frequency of the excitation sine signal and the orifice areaopening. The comparisons between analytical model and experiments are shown in Fig. 6 with four different equivalentorifice areas. For these experiments, the vibration amplitude is fixed at a constant value of 0.2 mm. It can be observed thatsimulation results closely resemble experimental results. It is noted that the frequency point of the maximum loss stiffness

Pneumatic spring

Load cell

Linear Motor

LVDT

ElectroForce 3500

Force Signal

DisplacementSignal

ControlSignal

WinTest PCI Control Electronics

DMA Softerware

Fig. 4. The sketch of experimental set-up.

Pneumaticspring

Load Cell

ElectroForce3500

VOM

Pipe

Adjustableorifice valve

spool displacement adjusting knob

Fig. 5. The experimental set-up for stiffness measurement of DCPS-AD.

100 101 1020

5

10

x 104

Frequency (Hz)

Sto

rage

Stif

fnes

s (N

/m)

Analytical ModelExperiment

100 101 1020

1

2

3

4

5x 104

Frequency (Hz)

100 101 102

Frequency (Hz)100 101 102

Frequency (Hz)

100 101 102

Frequency (Hz)100 101 102

Frequency (Hz)

100 101 102

Frequency (Hz)100 101 102

Frequency (Hz)

Loss

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

5

10

x 104

Sto

rage

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

1

2

3

4

5x 104

Loss

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

5

10

x 104

Sto

rage

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

1

2

3

4

5x 104

Loss

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

5

10

x 104

Sto

rage

Stif

fnes

s (N

/m)

Analytical ModelExperiment

0

1

2

3

4

5x 104

Loss

Stif

fnes

s (N

/m)

Analytical ModelExperiment

Fig. 6. Complex stiffness comparison between the analytical model and experiment; (a.1) storage stiffness under orifice area 9% opening; (a.2) loss

stiffness under orifice area 9% opening; (b.1) storage stiffness under orifice area 25% opening; (b.2) loss stiffness under orifice area 25% opening; (c.1)

storage stiffness under orifice area 64% opening; (c.2) loss stiffness under orifice area 64% opening; (d.1) storage stiffness under orifice area 100%

opening; and (d.2) storage stiffness under orifice area 100% opening.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3585

omax shifts according to different equivalent orifice areas. According to these results, it can be concluded that thedistribution of complex stiffness in the frequency domain can be actively adjusted by the VOM.

5. Analysis of complex stiffness and performance of VIS

5.1. Complex stiffness analysis of DCPS-AD

Through comparison between the experiment and computation, the fidelity of the analytical model is proved. Forfurther understanding the characteristic of the dual-chamber pneumatic spring in the frequency domain, the distributionof the complex stiffness and transmissibility of pneumatic vibration isolation system are analyzed.

The complex stiffness of membrane exhibits amplitude dependent and frequency dependent behavior. Nevertheless, itdoes not vary with the variant orifice area opening. Meanwhile, compared to complex stiffness due to compression of air

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903586

and air flow between two chambers, the complex stiffness due to dynamic of membrane is negligible. We prefer Eq. (14) toEq. (15) as the model describing the complex stiffness of DCPS-AD.

Eq. (14) shows that the complex stiffness of a dual-chamber pneumatic spring varies along with frequency o. Kstorage andKloss have some interesting properties at different frequencies. Three frequency regions are defined to simplify the analysis:

(1)

The low frequency region, in which the frequency obeys o5olow, where olow is defined as

olow ¼RTk

ffiffiffiffiffiffiffiffiffiffiffiffi1þNp

CoAoffiffiffi3p

VD0

In this frequency region, ðVD0=RT0kÞo� �2

5ð1þNÞðCoAoÞ2, the storage stiffness Kstorage is given by

Kstorageffikð1þNÞðCoAoÞðÞ

2

ð1þNÞ2ðCoAoÞ2¼

k

ð1þNÞ

The dual-chamber pneumatic spring is equivalent to a single chamber pneumatic spring, whose effective volumeequals to the sum of the Load and Damping Chamber.

(2)

The high frequency region, in which the frequency obeys ocohigh, where ohigh is defined as

ohigh ¼

ffiffiffi3p

RTkðNþ1ÞCoAo

VD0

In this frequency region, especially, o-1, ðððVD0Þ=ðRT0kÞÞoÞ244 ð1þNÞ2ðCoAoÞ2, the storage stiffness Kstorage is given by

KstorageffikððVD0=RT0kÞoÞ2

ððVD0=RT0kÞoÞ2¼ k

The dual-chamber pneumatic spring is equivalent to a single chamber pneumatic spring, whose effective volume equals tothe volume of LC.

(3)

The middle frequency region, in which the frequency o is in the range of [olow ohigh]

In this frequency region, storage stiffness increases dramatically as the frequency increasing. At the same time, the loss

stiffness is significant. Differentiation of Kloss with respect to frequency o yields the slope of loss stiffness dKloss/do.Making dKloss/do¼0, the frequency point of the maximum loss stiffness omax is obtained by

omax ¼RTkð1þNÞCoAo

VD0(16)

Fig. 7 shows the distribution of omax by increasing the orifice area from 9% to 100% opening. It is changed from 3 to27 Hz. It is noted that the distribution shows linear relationship between omax and the orifice area.

5.2. Performance analysis of VIS

The vertical pneumatic vibration isolation system consists of the dual-chamber pneumatic spring and the payloadmass, as shown in Fig. 8. The displacement transmissibility and force mobility are the two performance indices of theisolation system. The displacement transmissibility [4] between the base and payload mass can be expressed as

TðoÞ ¼ xðoÞxbðoÞ

¼KcomplexðM,Ao,oÞ

�Mo2þKcomplexðM,Ao,oÞ

0 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0

5

10

15

20

25

30

Orifice area opening

ωm

ax (H

z)

Fig. 7. Change in omax for varying orifice area.

Payload MassM

Base

xb

x

Dual-chamberPneumatic Spring

Kcomplex

Fig. 8. The sketch of the pneumatic vibration isolation system.

100 101−30

−25

−20

−15

−10

−5

0

5

10

15

Frequency (Hz)

Tran

sim

issi

bilit

y (d

B)

9% opening25% opening64% opening100% opening

Fig. 9. Transmissibility for varying orifice area.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3587

The force mobility of the system is defined by the payload acceleration response €xðoÞ and force acting on the payloadFðoÞ, which can be expressed as

YðoÞ ¼€xðoÞFðoÞ ¼

�o2

�Mo2þKcomplexðM,Ao,oÞ

The resonance frequency of the isolator system on is influenced not only by complex stiffness of the pneumatic spring,but also mass of the payload M. To optimize the transmissibility [8] and force mobility of the pneumatic vibration isolator,the frequency point of the maximum loss stiffness omax should be as close to the resonance frequency on as possible, inorder to reduce the peak value at resonance and obtain the large attenuation at high frequency (for the transmissibility).Both on and omax can be adjusted by changing the value of N, VD0 and Ao. Nevertheless, the value of the N and VD0 cannotbe changed after the pneumatic spring manufactured. Ao can be adjusted actively by the VOM. So the distribution ofcomplex stiffness in the frequency domain can be adjusted.

To validate the optimal approach by using VOM, the transmissibility and force mobility of the pneumatic VIS aremeasured. The instruments used in experimental measurement include the ultra-low frequency acceleration sensor(Model: PCB393B12, sensitivity: 9.8 V/g), modally tuned impulse hammer (model: PCB086d05, sensitivity: 0.25 mV/N).LMS Test.lab (LMS INTERNATIONAL Co.) is used to acquire and analyze signals to obtain the frequency response plots.

Figs. 9 and 10 show the transmissibility and force mobility of the pneumatic VIS for increasing the orifice area from 9% to100% opening. It is observed that the resonance frequency reduces continuously by increasing the orifice area from 9% to 100%opening. However, the peak value at resonance does not increase continuously. The minimum peak value at resonance existswhen the orifice area opening 25%. From the frequency distribution of omax and on shown in Figs. 7 and 9, it is noted that omax

is closed to on mostly when the orifice opening is 25%. However, omax is less than on when the orifice opening is 9%, and omax

is larger than on when the orifice opening 64% and 100%. To validate influence of VOM on the performance of VIS furthermore,

100 1010

0.5

1

1.5

2

2.5

3 x 10−3

Frequency (Hz)

Forc

e M

obili

ty (g

/N)

9% opening25% opening64% opening100% opening

Fig. 10. Force mobility for varying orifice area.

0 0.5 1 1.5 2−6

−4

−2

0

2

4

6

8 x 10−3

Time (s)

Impu

lse

resp

onse

(g)

9% opening25% opening64% opening100% opening

Fig. 11. Impulse response in time domain.

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903588

the acceleration response of payload under direct impulse disturbance is measured and shown in Fig. 11. It is obvious that thepayload response attenuated fastest when the orifice area opening is 25%.

While application condition of the pneumatic isolator changes, such as the change of the payload mass, on will change.The optimal orifice area opening of VOM should change along with the payload mass varying, which aims at keeping on

coincide with omax. To validate this, we decrease the payload mass and measure the force mobility of VIS with differentorifice area openings again. As shown in Fig. 12, the peak value at resonance with 25% orifice area opening is not minimalanymore. Along with the payload mass decreasing, the resonance frequency increases, which leads on not to coincide withomax for the 25% orifice area opening. Through increasing the orifice area opening, the new optimal performance of thesystem can be obtained when the orifice area opening is 30%.

From Figs. 9–12, it is noted that the resonance peak of the transmissibility and force mobility under the optimum orificeopening are not completely suppressed. The reason is that the optimal performance is limited by the fixed parameter ofDCPS, such as the volume ratio, the volume of the LC. The damping attributed to the air flow between two chambers hasthe maximum influence on the performance of DCPS-AD when omax coincides with on. However, changing the orifice areahas no effect on the degree of the maximum influence, which is reflected on the amplitude of maximum loss stiffness.Substituting the frequency of the maximum loss stiffness described by Eq. (16) into Eq. (14), the maximum amplitude ofthe loss stiffness Kloss_max in frequency domain is obtained as

Kloss_max ¼kRTKN

ð1þNÞððRTkÞ2þ1Þ

It is noted that the maximum amplitude of the loss stiffness does not change along with variety of orifice area.The maximum amplitude of the loss stiffness should be increased to suppress the resonance peak of transmissibility.

100 1010

1

2

3

4

5

6

7

8x 10−3

Frequency (Hz)

Forc

e M

obili

ty (g

/N)

9% opening25% opening30% opening64% opening100% opening

Fig. 12. Force mobility for varying orifice area (payload mass decreasing).

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–3590 3589

Redesign of the structure parameters of the DCPS [8] and pneumatic active control [13] are the two methods to suppressthe resonance peak completely.

6. Conclusions

This paper presents a dual-chamber pneumatic spring with a variable orifice mechanism, and focused on the analysis ofthe influence of VOM on the performance of the pneumatic spring. The complex stiffness model is developed to capture thedynamic characteristics of the pneumatic spring. Comprehensive computational simulation is carried out to describe thecomplex stiffness in the frequency domain with varying opening area of the orifice. Experiments were carried out tovalidate the analytical model. The results match the result of the computational simulation well. From these results, thefollowing conclusion can be drawn:

(1)

The VOM can adjust stiffness and damping characteristics of the dual-chamber pneumatic spring, bringing variety ofthe natural frequency and the peak value of the resonance frequency of the vibration isolation system.

(2)

The stiffness characteristic in the frequency domain changes with the variety of opening area of the orifice. Thefrequency point at which stiffness changes dramatically shifts to the high frequency region, when the opening area ofthe orifice is increased.

(3)

The damping characteristic in the frequency domain changes with the variety of opening area of the orifice. Thefrequency of the maximum damping increases when opening area of the orifice is increased.

(4)

The performance of pneumatic vibration isolator can be optimized by superposing the natural frequency and thefrequency point of maximum damping, through adjusting the VOM.

In the future work, we plan to install the VOM between two chambers to overcome the influence of the pipe.Furthermore, the orifice valve of VOM will be replaced by the solenoid valve, thereby; stiffness and dampingcharacteristics of the pneumatic spring can be adjusted in real time. The pneumatic VIS composed by dual-chamberpneumatic spring will become a semiactive system.

Acknowledgements

The work of this paper is partially supported by the Major Basic Research Program of China (973 Program)(No. 2009CB724205), 863 High-Tech R&D Program of China (No. 2009AA04Z148). We are grateful to Mr. Anatoly Burovfor providing kindly help in improvement of English writing.

Appendix A. Measurement of complex stiffness of membrane

A thin-walled, flexible and pressure resistant round membrane is involved to seal the Load Chamber from air leakage.For it is difficult to estimate the complex stiffness of the membrane analytically, the indirect method proposed by Lee andKim [7] is used to measure the complex stiffness of membrane KM(o). The measurement result is shown in Fig. A1. For thisexperiment, the vibration amplitude is 0.2 mm.

10−1 100 1010

0.5

1

1.5

2

2.5

3(a)

(b)

x 104

Frequency (Hz)

Sto

rage

Stif

fnes

s (N

/mm

)

10−1 100 1010

0.5

1

1.5

2x 104

Frequency (Hz)

Loss

Stif

fnes

s (N

/mm

)

Fig. A1. Complex stiffness of membrane: (a) storage stiffness and (b) loss stiffness .

H. Pu et al. / Journal of Sound and Vibration 330 (2011) 3578–35903590

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