Analysis and enhancement of torsional vibration stopbands in a periodic shaft system

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 46 (2013) 145306 (7pp) doi:10.1088/0022-3727/46/14/145306

Analysis and enhancement of torsionalvibration stopbands in a periodic shaftsystemYubao Song, Jihong Wen, Dianlong Yu and Xisen Wen

Key Laboratory of Photonic and Phononic Crystals of Ministry of Education, and Laboratory of Scienceand Technology on Integrated Logistics Support, National University of Defence Technology, Changsha410073, People’s Republic of China

E-mail: wenjihong nudt1@vip.sina.com

Received 30 November 2012, in final form 2 February 2013Published 11 March 2013Online at stacks.iop.org/JPhysD/46/145306

AbstractThe band structure and the enhancement of stopbands in a periodic shaft are investigated. Theband structures created by a hard ring and a locally resonant (LR) ring are studied first. Auniform method for analysing the band structures of periodic shafts with various attachedstructures is proposed. Furthermore, the enhancement of the stopbands is discussed. Theproperties of a shaft with a hard ring and an LR ring in different positions are studied, and awide, combined gap is created at low frequencies. Additionally, continuous beams are alsointroduced to enhance the stopbands. Some helpful methods for analysis, prediction andincreasing the width of stopbands in a periodic shaft system are provided.

(Some figures may appear in colour only in the online journal)

1. Introduction

Over the last two decades, the propagation of elastic/acousticwaves in artificial periodic composites, known as phononiccrystals (PCs) or sonic crystals, has received considerableattention [1–19]. Among the most attractive characteristicsof PCs are the elastic wave band gaps, within which theelastic/acoustic waves cannot propagate freely.

Earlier studies have demonstrated that there are two typesof band gap mechanisms in PCs, known as Bragg scatteringand local resonance [6, 8, 9, 11, 13, 14]. Stopbands induced byBragg scattering are not practical for suppressing vibrations ornoise at low frequencies [2, 8, 11, 13, 16]. The locally resonant(LR) mechanism, which was first proposed by Liu et al [2],provides a new approach to this problem. Additionally, LRgaps can exist in a frequency range two orders of magnitudelower than the gaps resulting from Bragg scattering.

In recent years, the concepts of Bragg PCs and LR PCshave been introduced into the field of vibration engineeringand noise control. The vibrations in one-dimensional (1D) andtwo-dimensional (2D) phononic band gap structures (such asrods [16, 20], shafts [8], beams [18, 21] and plates [11, 14, 22])have been studied theoretically and experimentally. Torsional

vibration is one of the common phenomena in rotor shaftsand can affect the service life of the shaft and the reliableoperation of the machine. Some studies have also investigatedthe properties of phononic shafts. Yu et al [8] studied thetorsional vibration gaps in a shaft with LR structures. Anotherimportant study that is relevant to this work was performedby Li et al [23, 24]. A hybrid analysis method was proposed,and only the band gaps induced by soft, rubber rings and LRmultilayer rings attached to a shaft were investigated by them.However, compared with flexural and longitudinal vibration,very little work describing torsional band gaps exists in theopen literature.

In many cases, it will be very important to obtain awider stopband. Some studies have investigated increasingthe width of stopbands [15, 25–33]. For instance, Xiao et al[25] broadened LR gaps by introducing multiple resonators.In addition to widening the signal stopbands based on thecharacteristics of the two gaps, the combination of two gapsis an effective approach to solve this problem. Fuster-Garciaet al [29] obtained a broad, full stopband using mixed soniccrystal arrays that included resonators and rigid scatters inair. However, to the best of our knowledge, most of theearlier literature primarily analysed single gaps, and only a

0022-3727/13/145306+07$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

attached structures

uniformshaft

Kth Unit cell

qk,L(qk-1,R)

Fk,L(-Fk-1,R)

qk,R(qk+1,R)

Fk,R(-Fk+1,R)

qk,I

Fk,I

Figure 1. General model of a torsional shaft with periodicallyattached structures, which can be hard rings, LR rings or continuousbeams. The model of the unit cell is also illustrated.

few works studied the phenomena of two coexisting gaps inPC structures [12, 20, 21, 34]. In addition, widening stopbandsby combining different types of attached structures still needsto be studied in depth.

Additionally, in many studies, the introduction of localresonators is one of the main approaches for creating LR gaps.Traditional local resonators, which commonly consisted of asoft material and a heavy hard material and were simplifiedas a spring–mass system, usually induced only one LR gapin most earlier studies [8, 11, 16, 23, 24, 26, 34]. In addition,some studies indicated that the traditional resonators increasedthe weight cost in applications [24]. Recently, an LR phononicwaveguide, in which the added weight can be reduced becauseonly soft materials are attached, has aroused some interest[13, 14, 24, 35]. This waveguide provides a useful methodfor building LR structures and is a potential approach forenhancing stopbands.

In this paper, the properties and the enhancement ofstopbands created by various types of attached structures areinvestigated in a periodic shaft system. In section 2, a generalmodel of the periodic shaft is developed. In section 3, theband structures of a shaft with periodically attached hard ringsand LR rings are studied, and a uniform analysis method isdeveloped. In section 4, the stopbands of the periodic shaftare enhanced by adjusting the distance between the hard ringsand LR rings and by introducing continuous beams. Finally,section 5 summarizes the work.

2. General model and theory

In this paper, various attached structures are employed to createor enhance the stopband, including a hard ring, an LR ringand continuous beams. Thus, a general simplified model ofa 1D torsional shaft with periodically attached structures isdeveloped first, as shown in figure 1. The detailed structuresare illustrated in the relevant sections. As in [8, 24], the spacewhere the attached structures are mounted is assumed to bevery narrow, and the effects of the axial distribution of theattached structures are neglected. The finite element (FM)method [36], the transfer-matrix (TM) method [8] and Blochtheory are combined to calculate the band structures and theresponses. The main derivation process is given here.

The discrete dynamic equation of the kth unit cell obtainedfrom the FE model is

(KU + iωCU − ω2MU)qU = FU (1)

where KU, CU and MU are the stiffness, damping and massmatrices, respectively, FU = [Fk,L Fk,I Fk,R]T is the loadingvector, and qU = [qk,L qk,I qk,R]T is the vector of thedegrees of freedom. L, I and R represent the left, interiorand right boundaries of the shaft, respectively. Under theassumption that there are no external forces on the interiornodes (i.e. Fk,I = 0) and with the dynamic stiffness matrixDU = KU + iωCU − ω2MU, equation (1) gives the followingmatrix equation:

Dk,LL Dk,LI Dk,LR

Dk,IL Dk,II Dk,IR

Dk,RL Dk,RI Dk,RR

qk,L

qk,I

qk,R

=

Fk,L

0Fk,R

(2)

Furthermore, equation (2) can then be written as[Dk,LL Dk,LR

Dk,RL Dk,RR

] [qk,L

qk,R

]=

[Fk,L

Fk,R

](3)

where

Dk,LL = Dk,LL − Dk,LID−1k,IIDk,IL, Dk,LR

= Dk,LR − Dk,LID−1k,IIDk,IR,

Dk,RL = Dk,RL − Dk,RID−1k,IIDk,IL, Dk,RR

= Dk,RR − Dk,RID−1k,IIDk,IR.

With equation (3) and the displacement continuity andforce equilibrium constraints at the junction of adjacent unitcells given by qk,R = qk+1,L, and Fk,R = −Fk+1,L, the TMmodel of the unit cell is given as[qk+1,L

Fk+1,L

]=

[qk,R

−Fk,R

]

=[ −D−1

k,LRDk,LL D−1k,LR

−Dk,RL+Dk,RRD−1k,LRDk,LL −Dk,RRD−1

k,RL

] [qk,L

Fk,L

]

= TU

[qk,L

Fk,L

](4)

Based on Bloch theory, when a free wave propagates alongthe periodic structure, the displacements and forces at theconnections satisfy

[qk+1,L

Fk+1,L

]= λ

[qk,L

Fk,L

](5)

where λ = eµ is the propagation constant and µ = a + bi isa complex constant. The amplitude and phase of λ illustratethe amplitude attenuation and phase difference, respectively,of wave motion in a unit cell.

Free wave propagation can be described by theeigenproblem

TU

[qk,L

Fk,L

]= λ

[qk,L

Fk,L

](6)

where TU is the transfer matrix of the unit cell of the periodicshaft. λl = al + bl i (l = 1, 2, . . . 2n − 1, 2n), which come in

2

J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

Figure 2. Unit cells of the periodic shaft system, that is, the uniform shaft with attached (a) hard ring, which is connected to the shaft by ahard rib, (b) LR ring, composed of a soft ring enclosed with an outer hard ring, and (c) hard ring and LR ring in the same position.

reciprocal pairs, are the eigenvalues of the transfer matrix TU,and �l = [qk,L,l Fk,L,l]T is the corresponding eigenvector. If|λl| = 1 (al = 0), the wave propagates along the cell freely,which indicates a passband; and if |λl| �= 1 (al �= 0), thewave will be attenuated, which indicates a stopband. Thus,the band structure of the periodic structure can be describedby the amplitude of the propagation constant λ.

Additionally, the frequency response function (FRF),which indicates the ratio of the output response to the inputresponse, is also calculated for the finite periodic shaft to verifythe vibration gaps.

3. Uniform analysis of the torsional vibrationstopbands

Unit cells of the periodic shaft with a hard ring, an LR ringand a combination of hard and LR rings are shown in figure 2.The LR ring consists of a ring of soft material enclosed by anouter ring of hard material. The hard ring is modelled as a ringof hard material and is connected to the shaft or the LR ringby a hard rib. In the model, the length of the unit cell of theperiodic shaft is l. The radius, shear modulus and density ofthe uniform shaft are rs, Gs and ρs. In the structures shown infigures 2(a)–(c), the length of the rings are lr1, lr2 and lr3, andthe inner radius and outer radius of the hard ring and/or the LRring are rlri1, rlro1, rri2, rro2 and rri3, rro3, rlri3, rlro3, respectively.The length of the hard ribs are lri1, lri3. The shear modulus anddensity of the soft material are Gsr1 and ρsr1 in figures 2(a),and are Gsr3 and ρsr3 in figure 2(c). The shear modulus anddensity of the hard material are Ghr and ρhr.

The structure and the material parameters of the periodicshaft are l = 0.075 m, lr1 = lr2 = lr3 = 0.0012 m, lri1 = lri3 =0.0005 m, rs = 0.005 m, rlri1 = rri2 = rri3 = 0.008 m, rlro1 =rro2 = rro3 = 0.01 m, rlri3 = 0.012 m, rlro3 = 0.012 78 m,Gs = 1.59×108 Pa, ρs = 1180 kg m−3, Ghr = 1.49×1010 Pa,ρhr = 11 600 kg m−3, Gsr1 = 1.22 × 107 × (1 + 0.03i) Pa,ρsr1 = 1300 kg m−3, Gsr3 = 1.53 × 106 × (1 + 0.03i) Pa andρsr3 = 1300 kg m−3. The moment of inertia (MI) of thesoft ring and the rib is neglected. With these parameters, theadded mass in figures 2(a) and (b) is equal, and the resonantfrequencies of the LR rings in figures 2(b) and (c) are uniform.

(a)

(b)

0 2000 4000 6000 8000 100000

5

10

15

Frequency (Hz)

Am

plit

ude o

f λ

Hard ring

LR ring

Hard ring + LR ring

Negative MI

0 2000 4000 6000 8000 10000

-10

0

10

Frequency (Hz)

No

rma

lize

d M

I

Hard ring

LR ring

Hard ring + LR ring

Figure 3. (a) Band structure of the periodic shaft system withvarious attached structures, and (b) effective MI of a hard ring, anLR ring, and a hard ring and an LR ring at the same position; all ofwhich are normalized by the MI of the hard ring.

The equal added mass and uniform resonant frequencies areconvenient for the following comparison and analysis.

3.1. Band structures of the shaft with periodically attachedhard rings and LR rings

Figure 3(a) presents the band structures of the periodic shaftsystem for various cases. In this section, the observationswill be explained by the attached structure’s effective MI, bywhich the impedance mismatch is induced in the periodic shaftsystem. The effective MI is defined as the ratio of the torsionalmoment to the angular velocity; this effective MI is similarto the effective parameter of the medium used in studies onmetamaterials [2, 26, 33]. The effective MIs of a hard ring, an

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J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

LR ring and a hard ring added to an LR ring are illustrated infigure 3(b).

It can be seen that if only a hard ring is employed, thestopband is a Bragg gap and becomes wider and wider as thefrequency increases. If only an LR ring is employed, thennot only are LR gaps induced at low frequencies but Bragggaps are also generated. From the comparison of figures 3(a)and (b), LR gaps, within which the sharp attenuation is causedby the resonance effect, are found. At approximately theresonant frequency, the effective MI of the LR ring is huge,even approaching infinity, and can be negative. In otherregions in which no resonances are included, the effective MIis finite, and its changes are smooth. Thus, stopbands with thecharacteristics of Bragg gaps are generated.

We also find that, as the frequency increases, the Bragggaps induced by the LR ring become narrower and narrower;this narrowing corresponds to the effective MI of the LR ringgradually approaching zero. Thus, it is difficult to obtain widecollections of stopbands in a broad frequency range throughthe use of LR gaps alone. Another feature is that, in this model,the Bragg frequencies are the upper edges of the Bragg gapsinduced by the hard rings, but the lower edges of the Bragggaps are induced by the LR rings. This transition is causedby the difference in the effective MIs, whose sign is positivefor the hard ring and negative for the LR ring, as shown infigure 3(b).

It can also be observed from figure 3(a) that, if both thehard ring and the LR ring are employed, the band structure issimilar to the case with only an LR ring at low frequencies andis similar to the case with only a hard ring at medium to highfrequencies. That is, the stopbands caused by the hard ring andthe LR ring are combined, and the whole frequency range of thestopband increases. However, in the transition range betweenthe Bragg gaps and the LR gaps, only a narrow stopband isgenerated, and the attenuation amplitude is small; the narrowstopband and small attenuation amplitude are caused by thecounteracting effects of the LR ring and the hard ring. Asshown in figure 3(b), in this transition region, the effective MIis approximately zero.

Another feature shown in figure 3 is that the attenuationprofile in the LR gaps is nearly symmetric, and this profile isdifferent from that observed in most previous studies. Thisfeature was also found by Xiao et al [34]. In addition, thelower edge frequency of the LR gaps was usually believed tobe near the resonant frequency of the local resonator. Someprevious studies also indicated that the LR gaps correspondedto the frequency range of the negative mass density. However,figure 3 clearly shows that this correspondence may not bethe full story. Then, a new interesting problem is generated;that is, how do these gaps occur, or how do we explain thesephenomena. In section 3.2, a uniform analysis method for theband structures in the periodic shaft system is proposed, andthe question will be answered simply.

3.2. Uniform analysis of the stopband properties of theperiodic shaft with hard rings or/and LR rings

Based on figure 3, we find that the effects of differentattached structures can be represented by the effective MI,

Figure 4. Band structure of the periodic shaft for various attachedMIs. SB indicates the stopband regions.

which changes as a function of frequency and is negativein some frequency ranges. Thus, the properties of theband structures induced by various attached structures canbe analysed uniformly through the combination of the bandstructures created by the added MIs of various hard rings andthe effective MIs of these attached structures.

Figure 4 shows the band structure of the periodic shaftas a function of the attached MI. We find that, whether apositive or negative MI is induced, a stopband is generatedand a passband is always presented. The band structure createdby a purely negative MI is also shown in figure 3(a). Thus,there are no inevitable relations between the stopbands and thefrequency range for a negative MI or density. Additionally,in this model, in which the structures are mounted at a singleposition in each unit cell, the Bragg frequencies are the upperedges of the stopbands induced by positive MIs but are thelower edges of the stopbands induced by negative MIs. Thatis, the band structure is divided into uniform regions by theBragg frequencies. The other stopband edges are determinedby the coupling of the shaft and the attached MI. If we want toobtain a wider stopband in each region, a heavier attached MIis necessary. However, after the MI increases to some level,the increased width of the stopband is very small.

Thus, the advantage of introducing a negative MI is that avery-low-frequency stopband can be created only by a lessernegative MI. This effect occurs because the start frequency ofthe first stopband for a negative MI is the zeroth-order Braggfrequency (0 Hz). It should be noted that, if the negative MIis very small, the lowest stopband disappears. We also findthat the attenuation amplitude in the stopband is affected bytwo factors: the magnitude of the MI and the distance from theband edges.

To show the band edge frequency curves correspondingto figure 4 more clearly, they are illustrated in figure 5. Thefrequency is normalized by the first-order Bragg frequency(2447 Hz) of the periodic shaft. Some observations mentionedabove are validated. The effective MI of a hard ring added toan LR ring is also displayed. We can see that the effective MI isapproximately infinite, and shifts from positive to negative atthe resonant frequency. If the effective MI is large enough,there will be a stopband. In addition, there are no directrelations between the resonant frequency of the LR ring andthe start frequencies of the LR gaps.

Additionally, based on the uniform analysis, the design ofa stopband in a periodic shaft system can be divided into twosteps. One step is obtaining the analytical relation between

4

J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

Figure 5. Band edge frequency curves (bold solid, dashed anddotted–dashed lines) corresponding to figure 4, and the LR gappredicted by the uniform method. The thin line is the effective MI ofthe hard ring and the LR ring at the same position.

Figure 6. Unit cell of the periodic shaft system, that is, a uniformshaft with a hard ring and an LR ring attached in different positions.

the necessary effective MI and the desired stopbands of theperiodic shaft. The other is the realization of the necessaryMI, which can be achieved with continuous structures, LRrings, hard rings and so on. The method can also be used topredict the stopband ranges of the periodic shaft. That is, onlyif the effective MI of the attached structures is larger than thenecessary MI will the stopband be created.

4. Enhancement of the torsional vibration stopbands

4.1. Low-frequency stopband widening by the combination ofattached hard rings and LR rings

In addition to the simple combination of Bragg gaps and LRgaps introduced in section 3, we may be interested in how wecan increase the advantages of this combination. Therefore, thekey factors that influence widening of the combination gaps areinvestigated. When the geometrical parameters and the totalmass are retained, we find that the distance between the hardring and the LR ring significantly affects the band structures.Figure 6 shows the unit cell of the combined periodic shaft.The distance between the two attached structures is d, and theother parameters are the same as in section 3.

Figures 7(a) and (b) show the changes in the band structureof the system as a function of the distance between the hardring and the LR ring. It can be observed that the band structureschange significantly with the distance. The width of the

passband at low frequencies can be decreased by changing thedistance. In addition, a wide stopband that combines Bragggaps and LR gaps is induced, and only a very narrow passbandexists in the combined gap. If the distance is adjusted to somevalues, the LR gaps disappear. It can also be found that inmost cases, the Bragg frequencies are no longer one of the bandedges of the combined gaps or Bragg gaps. Both the upper andlower edge frequencies of the stopbands are determined by thecoupling of the system. Additionally, although the band edgefrequencies vary, the width of the passband is nearly constantat medium to high frequencies. Overall, the band structureis improved and the low-frequency stopband is effectivelywidened by adjusting the distance between the hard ring andthe LR ring.

To illustrate the band structure and FRF of the systemmore clearly, they are shown for d = l/3, d = l/2 andd = 0 in figures 7(c) and (d). We find that the wavemotion is significantly reduced in the combined stopband. Themagnitude of the attenuation is consistent with the propagationconstant λ. Compared with the case in which the hardring and the LR ring are in the same position, the stopbandis effectively widened at low frequencies when they are indifferent positions. Widening the combined gaps by adjustingthe distance is validated.

4.2. Enhancement of the stopbands of the periodic shaft withcontinuous beams

After understanding the band structures created by LRresonators in which only one LR gap is generated, we areinterested in how we can induce multiple LR gaps. This isimportant for reducing elastic waves in multiple frequencyregions and for widening the stopband. In this section,continuous beams are used to accomplish this purpose.

Figures 8(a) and (b) show sketches of the periodicshaft with periodically attached elastic beams. The structureparameters of the beams are lb = 0.0026 m, wb = 0.003 m andhb = 0.02 m in the axial, circumferential and radial directionsof the shaft, respectively. Two of the material parameters forthe four pairs of beams are defined. That is, Young’s modulusand the density are Eb0 = 1.5 × 109 × (1 + 0.01i) Pa andρb = 1300 kg m−3 for all the beams in one case, whereasYoung’s modulus is Eb = Eb0 × [0.5 1 2 4] for the four pairsof beams in the other case. The other parameters of the systemare the same as in section 3. The added mass of the beamsis the same as that of the LR ring. Figure 8(c) illustrates theinteraction effect between the beams and the shaft. The radialforces do not affect the torsional vibration of the shaft and arenot presented here. The effect of the beams is first representedin the analysis process by the effective MI attached to the shaft.

Figures 9(a) and (b) show the band structures and FRF ofthe periodic shaft with attached beams. The band structureand frequency response for the case with only LR ringsare also displayed for comparison. We find that, if LRrings are employed, large attenuation is generated mainly inthe first stopband in this model. If continuous beams areemployed, large attenuation is generated in multiple LR gapsand Bragg gaps, and the active regions are near the resonant

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J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

Figure 7. (a) 3D plot and (b) contours of the band structure as a function of the distance between the hard ring and the LR ring. (c) Bandstructure and (d) FRF of the shaft system for different distances: —�—, d = l/3; —◦—, d = l/2; –— · —–, d = 0 (rings at the sameposition).

Figure 8. (a) 3D plot of the periodic shaft system with four pairs of continuous beams periodically attached, (b) unit cell of the periodic shaftand (c) coupling of the continuous beams to the shaft. For different cases, the parameters for the four pairs of beams are uniform or different.

frequencies of the beams. Additionally, compared with thecase with uniform beams, the number of large attenuationregions has obviously increased because of the introductionof more resonant peaks from beams with differing parameters.Thus, the band structure can be enhanced by optimizing thedesign through combining the elastic beams. In the stopbanddesign, the use of continuous structures could also be a feasibleapproach to generating the desired MI.

Figure 9(c) illustrates the stopbands predicted by thecombination of the band edge frequencies and the effectiveMI of the uniform beams. The predicted stopbands agree verywell with the stopbands calculated by the TMM and Blochtheory and the FRF of the finite periodic shaft. Thus, using theuniform method proposed in section 3.1 to analyse and predictthe stopbands is valid.

5. Conclusions

In conclusion, the band structure and the enhancement ofstopbands are investigated in a periodic shaft system. Modelsof the shaft system with various attached structures are

developed. The band structures of a periodic shaft with hardrings and LR rings are studied first. The results show that,through the combination of hard rings and LR rings, boththe advantages of Bragg gaps and LR gaps are found in thecombined band structures. Next, the band structures createdby different structures attached to a periodic shaft are similarlyanalysed through the effective moment of inertia of theseattached structures. A uniform analysis and prediction methodis also developed.

Furthermore, the stopbands of the periodic shaft areenhanced by adjusting the distance between the hard ring andthe LR ring and by introducing continuous beams. For aperiodic shaft with a hard ring and an LR ring, if the distanceis adjusted to some values, a wide, low-frequency stopbandis generated, and only a very narrow passband exists insidethe combined gaps. In addition, when continuous beamsare used to induce the stopbands, better vibration attenuationcan be obtained. Continuous structures and combinations ofcontinuous structures can also be an effective approach forachieving the necessary moment of inertia and broadening thestopbands. Some of the methods and conclusions in this paperare also suitable for other periodic structures.

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J. Phys. D: Appl. Phys. 46 (2013) 145306 Y Song et al

Figure 9. (a) Band structure and (b) FRF for the periodic shaft withperiodically attached beams, (c) stopbands predicted by the uniformanalysis method (bold solid, dashed and dotted lines: band edgefrequencies, thin solid line: effective MI of the uniform beams). Theshaded areas show the stopband regions for the shaft with uniformbeams.

Acknowledgments

The authors are grateful for the support provided by theNational Natural Science Foundation of China (Grant Nos10902123, 51075392 and 51275519).

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