http://jvc.sagepub.com/ Journal of Vibration and Control http://jvc.sagepub.com/content/early/2014/01/31/1077546313520026 The online version of this article can be found at: DOI: 10.1177/1077546313520026 published online 31 January 2014 Journal of Vibration and Control Huijie Shen, Jihong Wen, Dianlong Yu and Xisen Wen Stability of clamped-clamped periodic functionally graded material shells conveying fluid Published by: http://www.sagepublications.com can be found at: Journal of Vibration and Control Additional services and information for http://jvc.sagepub.com/cgi/alerts Email Alerts: http://jvc.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: What is This? - Jan 31, 2014 OnlineFirst Version of Record >> at National University of Defense Technology on February 26, 2014 jvc.sagepub.com Downloaded from at National University of Defense Technology on February 26, 2014 jvc.sagepub.com Downloaded from
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http://jvc.sagepub.com/Journal of Vibration and Control
http://jvc.sagepub.com/content/early/2014/01/31/1077546313520026The online version of this article can be found at:
DOI: 10.1177/1077546313520026
published online 31 January 2014Journal of Vibration and ControlHuijie Shen, Jihong Wen, Dianlong Yu and Xisen Wen
Stability of clamped-clamped periodic functionally graded material shells conveying fluid
Published by:
http://www.sagepublications.com
can be found at:Journal of Vibration and ControlAdditional services and information for
at National University of Defense Technology on February 26, 2014jvc.sagepub.comDownloaded from at National University of Defense Technology on February 26, 2014jvc.sagepub.comDownloaded from
The dynamic behavior of cylindrical shells conveyingfluid is of substantial practical applications, forinstance, in heat exchangers, hydraulic systems, powerplants and nuclear reactor systems, etc. It is not surpris-ing, therefore, that the dynamic problems of fluid-conveying shells have been extensively investigatedboth theoretically (Tang and Paıdoussis, 2009;Ahmed, 2010, 2012; Sofiyev and Kuruoglu, 2013) andexperimentally (Zhang et al., 2000; Liu et al., 2009;Chebair et al., 1989; Karagiozis et al., 2005, 2008).
In general, research on the dynamics of fluid convey-ing shells may be categorized into two aspects: (i)dynamic responses, including both the transient (Kimand Lee, 2002; Han et al., 2002; Jafari et al., 2005) andthe frequency responses (Ming et al., 2001; Sorokin andErshova, 2006; Sorokin et al., 2008), and (ii) dynamicstability (Chen, 1987; Paıdoussis, 2004). The presentwork focuses on the second aspect, as the dynamic sta-bility of the system is quite fascinating and engineeringimportant. Paıdoussis (2004) and Chen (1987) pre-sented an excellent review related to the dynamic
stability of shells and pipes interacted with the flowingfluid. Amabili et al. (2002, 2006, 2009) performed asystematically study on the nonlinear dynamics and sta-bility of simply supported, circular cylindrical shells.Zhou (2012) analyzed the vibration and stability ofring-stiffened shells conveying fluid and examined theeffects of fluid velocity, Young modulus, number of thering stiffeners on the natural frequency and the instabil-ity characteristics of the shell. Recent studies have beenon the effects of periodic structures on the stability offluid-conveying cylindrical shells (Ruzzene, 2004;Aldraihem, 2007). Ruzzene (2004) placed circumferen-tial stiffeners periodically along the axial direction asmeans to enhance the stability of the considered class
Laboratory of Science and Technology on Integrated Logistics Support,
National University of Defense Technology, Hunan, China
Corresponding author:
Jihong Wen, Laboratory of Science and Technology on Integrated
Logistics, Support, National University of Defense Technology, Changsha
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of shells. Aldraihem (2007) presented a periodic pipethat consisted of identical substructures or cells; resultsdemonstrated that the collar-stiffened periodic pipeexhibits unique stability characteristics when comparedto a uniform pipe. The attraction of using periodicstructures may be due to its unusual dynamical charac-teristics, e.g., the existence of wave bands within whichthe propagation of elastic waves is forbidden over someselected frequency ranges (Wen, 2006; Yu et al., 2008;Wen et al., 2009; Shen et al., 2009, 2012; Song et al.,2013), and the superior characteristics of dynamicalstability. Nevertheless, in such periodic structures, aserious shortcoming is found, that is, the stress concen-tration produced by the material/geometric discontinu-ities, which can lead to damage in the form of matrixcracking, adhesive bond separation, and so on.
Functionally graded materials (FGMs) are a class ofcomposites that have a smooth and continuous vari-ation of material properties from one surface toanother and thus can alleviate the aforementionedstress concentration problem. Thus, with the applica-tion of FGM in shells, the stress concentration problemthat existed in the laminated composites can be alle-viated, yet retaining the superior stability characteris-tics of periodic shells. Extensive research has beencarried out on this new class of composites and itsapplications (Abrate, 2006; Yang and Shen, 2007;Tornabene, 2009). However, to the best of our know-ledge, not much research has been done on the appli-cation of FGM, such that material properties varycontinuously in the axial direction, into the area of sta-bility enhancement for fluid-conveying shell systems.The present work is motivated by the aforementioneduseful properties of FGM and periodic structures.
The paper is organized into four sections and oneappendix. In section 1, a brief introduction is given.Section 2 presents a periodic FGM (PFGM) shell toenhance the stability for the system; also a finite elementmodel (FEM) algorithm is developed here. In section 3, thecharacteristics of stability for the clamped-clampedPFGMshell are investigated, as well as an analysis of some keyparameters and the impact of FGMon the stability for thesystem. Finally, a brief summary of the conclusions andsuggestions for future work are outlined in section 4.
2. Governing equations andFEM method
The investigated fluid-conveying shell system issketched in Figure 1, and the flow velocity is U. Thelongitudinal, circumferential and radial displacements,i.e., u, v and w, are assumed to be constant across thethickness according to the Flugge shell theory (Wanget al., 2005). The basic equations of this shell theory aregiven in Appendix A. When the number of circumfer-ential wave n is equal to 1, the equations are reduced tothe case of beam-mode motion for the shell. The beam-mode dynamics of fluid-conveying shells has receivedconsiderable attention and lots of references have beencarried out on the dynamic analysis of this mode (Lakisand Sinno, 1992; Wang et al., 2005). The beam-modestability is what the current paper will focus on, becausethe beam-mode dynamics is frequently observed in lotsof applications (e.g. piping systems and thick shellsystems such as shell systems as nuclear reactors, air-craft jet engines, heat exchangers and jet pumps ofunderwater vehicles). Particularly, because considerablesimplification is thereby introduced in the
(b)
(a)
Figure 1. Sketch of the PFGM shell structure: (a) the basic uniform FGM shell cell and (b) the shell structure with the basic
geometric parameters and the deflection variables.
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computations needed to obtain the characteristics ofwave propagation for this mode. Also, the basicdesign methodology and analysis method for thismode may be applicable to other modes or even otherstructures such as beams and plates. Nevertheless, itshould immediately point out that the shell-modeinstability does occur in this PFGM shell system ifthe flowing fluid is sufficiently high, although we didnot present any corresponding results for the shellmode in the current work.
For this mode, it is reasonable to assume, that nodistortion of the cross-sectional shape occurs in thiscase and displacements are linked with each other asv ¼ �w and u ¼ �@w=@� (Sorokin et al., 2008). Thenthe governing equation for the beam-type motion ofthe fluid-conveying shell can be further simplified to asingle differential equation (Sorokin et al., 2008; Usukiand Yogo, 2009), as follows
B@4w
@x4þ �f�R
2U2 @2w
@x2þ 2�R2�fU
@2w
@x@tþm
@2w
@t2¼ 0 ð1Þ
in which B¼�EhR(12R2þ h2)/(12(1��2)) and m¼
�(2�sRhþ �fR2); �f is the density of internal fluid. It
is obvious that the second term is associated with cen-trifugal forces; the third term is associated with Corioliseffects; and the last term represents the inertial force ofthe fluid-filled shell. To apply the FEM technology tothe investigation of the beam-mode stability of the peri-odic shell, the weak form is used (Paıdoussis, 2004;Aldraihem, 2007; Sadeghi and Karimi-Dona, 2011),for a finite shell with length L; its equation is
Z L
0
B@2w
@x2@2ð�wÞ
@x2� �f�R
2U2 @w
@x
@ ð�wÞ
@x
�
þ�R2�fU@w
@t
@ ð�wÞ
@x�@2w
@x@t�w
� �þm
@2w
@t2�w
�dx
þmfU@wL
@tþUw0L
� ��wL ¼ 0 ð2Þ
wherein wL¼w(L) and mf¼��fR2. Utilizing the one-
dimensional (1D) element to discretize the shell system,approximating the displacements of the shell elementsvia Galerkin method and Hermite cubic interpolationfunctions, the discrete differential equation for thebeam mode of the cylindrical shell are obtained, asfollows
XNe¼1
Me €ve þ ðCfe þ CendÞ_ve þ ðKe þ Kfe þ KendÞve� �
¼XNe¼1
Fe ð3Þ
in which
Me ¼
Z Le
0
mNTNd�, Ke ¼
Z Le
0
BN00TN00d� and
Kfe ¼
Z Le
0
�mfU2N0
TN0d�; Kend ¼ 0,
except for Kendð2n� 1, 2nÞ ¼ mfU2; Cfe ¼R Le
0 mfU NTN0 �N0TN
� d� and Cend ¼ 0, except for
Cendð2n� 1, 2n� 1Þ ¼ mfU; N is the shape function(Aldraihem, 2007); Fe is the external loading force,and ve is the vector of radial deflection w and slope w0
of a discrete shell element; the primes denote spatialderivatives with respect to x. Furthermore, equation(3) can be rewritten into a more simple form, as follows
M€vþ C_vþ Kv ¼ F ð4Þ
in which M, C and K are the global mass, damping andstiffness matrices, respectively; F and v are the globalexternal force and deformation vector, respectively.The total degree of freedoms (DOFs) of the shellsystem 2N is reduced to 2N�Nb, if the boundary con-ditions are imposed, wherein Nb is the DOFs of theconstraints; for the clamp-clamped shell, the con-straints at the two ends are w1¼w01¼ 0 and w2N¼
w02N¼ 0, thus Nb¼ 4; accordingly, appropriate rowsand columns of the stiffness, mass and damping matri-ces are deleted to satisfy these constraints.
3. Results and discussion
As sketched in Figure 1, the PFGM shell is consisted ofa series of basic FGM shell cells which are connectedend-to-end and repeated periodically along the axialdirection of the cylindrical shell. Parameter distributionfor a basic FGM shell cell is
�FG¼
�min, 05x5 la,
�minKFGðx� laÞþ�min, la5x5 laþ lf,
�max, laþ lf5x5 laþ lfþ lb,
�minKFGðlaþ lfþ lb�xÞ
þ�max, laþ lfþ lb5x5 lc� lf,
�min, lc� lf5x5 lc
8>>>>>>>><>>>>>>>>:
ð5Þ
wherein KFG¼ (�max��min)/(lf�min), lc¼ lbþ 2lfþ 2la,and �FG represents the Young’s modulus E or thedensity �s. When the length of the FGM shell sectionlf equals to zero, the PFGM shell turns into the binarymaterial/geometric periodic shell (i.e., the PS shell)(Ruzzene, 2004; Sorokin and Ershova, 2006). In thisanalysis, epoxy is chosen as the reference material for
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the FGM shell, i.e., shell sections A, whose parametervalues (i.e., Young’s modulus Emin and density �smin),are the minimum in the shell system; and shell sectionsB takes on the maximum. The average radius R and thethickness h are 0.125m and 0.01m, respectively; thelength of the basic FGM shell cell is lc¼ 1m, thisvalue is kept unchanged during the investigation. Thefluid velocities are normalized as u¼UL(mf/B)
1/2,wherein L¼ 5m is the length of the shell; andthe dimensionless fluid density is defined as �¼mf/(mfþ 2��sminRh). The shells are exposed to flowingfluids traveling at a constant velocity u from the fixedend (x¼ 0) toward the free end (x¼L).
As a conservative system, the clamped-clamped shellmay lose stability by divergence, as illustrated inFigure 2, for a uniform shell with �¼ 0.1. A perfectagreement of this plot with that of Paıdoussis (2004)is seen, hence the correctness of the present FEMcode is validated. For 0< u< 2�, the eigenfrequencies! are purely real and they are diminished with increas-ing u; at ucr1 ¼ 2�, the eigenfrequency of the first modebecome zero, and thereafter are purely imaginarybefore the eigenfrequency of the second-mode decreasesto zero, thereby the system is unstable in the form offirst-mode divergence. In the PFGM shell system, thiscritical velocity ucr1 is raised, for instance, in Figure 3,ucr1 is raised from 2� to 7.75 by the PFGM shell. If the
flow velocity u keeps rising, then both of these twoshells (i.e., the uniform and the PFGM shells) will besubjected to the so-called Paıdoussis-type (coupled-mode) flutter (Paıdoussis, 2004). The characters ofthis case is that the bifurcation originates directlyfrom a divergent state (as shown in Figure 3(c));hence, at the onset of flutter (ucr3 ¼ 11.28 for thePFGM shell), the frequency of oscillation is zero, thatis Re(!)¼ 0, and then for u> ucr3 , Re(!) becomes non-zero. Apparently, in the PFGM shell system, the criticalvelocities for either the flutter- or the divergence- typeinstability are all enhanced, to various extents, as com-paring with those for the uniform shell system. In whatfollows, this conclusion will be further validated forhigher �.
For higher �, the clamped-clamped shells also losestability by divergence firstly; however, since suchclamped-clamped systems are gyroscopic that theymay then regain stability, but be further subjected toinstabilities as the flow velocity is increased. Figures 4and 5 present the eigenfrequencies !, as functions of thedimensionless flow velocity u, for the uniform and thePFGM shells, respectively; and �¼ 0.604. Similar withthat of �¼ 0.1, in the uniform shell, again loses stabilityin the form of the first-mode divergence at ucr1 ¼ 2�, for�¼ 0.604; whereas in the PFGM shell, this critical valueis approximately 7.75. When the flow velocity reaches
0
20
40
60
80
100
120
Re(
ω)
0 2 4 6 8 10 12–40
–20
0
20
Flow velocity u
Im(ω
)
1st mode
3rd mode
1st–mode divergence
1st and 2nd coupled–mode flutter
2nd–mode divergence
u1cr=6.28
2nd and 3rdmodes combined
u3cr=9.38
u2cr=9
2nd mode
Figure 2. The real and imaginary components of the eigenfrequencies, !, as functions of the dimensionless flow velocity, u, for the
lowest three modes of a clamped-clamped uniform shell; �¼ 0.1.
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ucr2 (ucr2 & 9), instead of inducing a second-mode diver-gence, the first-mode divergence of the uniform shellsystem is re-stabilized; and it lasts to u& 9.4, thereaftercoupled-mode flutter occurs via a HamiltonianHopf bifurcation. The distinguish features of thePaıdoussis-type and Hamiltonian Hopf bifurcation flut-ter are that (i) for u< ucr, there is no damping in thesystem and (ii) for u> ucr the coalescence of the twomodes has resulted in two eigenfrequencies, respectivelypositively and negatively damped, whilst, Re(!) is non-zero in this case, as illustrated in Figure 5(c), similarlyto the ordinary Hopf bifurcation, except here morethan one mode is involved. For the PFGM shellsystem, the Hamiltonian coupled-mode (1st and 2ndcoupled-mode) flutter occurs approximately at 11.3,
and the third mode develops divergence at u& 15.04.Obviously, these values of the flutter- or the divergence-instability for the PFGM shell are all higher than thoseof the uniform shell.
On examination of Figures 2–5, we can see that thedivergence-critical velocities for the same shell systembut with different �, are identical. In fact, the criticalvelocities ucr for the divergent form of instability isindependent of �. This is so because � is always asso-ciated with velocity-dependent terms in the equation ofmotion, while, divergence represents a static loss of sta-bility. That is to say, divergence is a static rather thandynamic form of instability, hence, the dimensionlesscritical flow velocity for divergence can be obtainedby estimating the value of jKj: if jKj admits the zeroor negative solution at a certain flow velocity ucr, thatis, now at this flow velocity, the overall stiffness of thesystem vanishes, then the system is statically unstable.Thus, to enhance the stability of divergence, an effectiveway is to increase the overall stiffness of the shellsystem. Specifically, for the uniform shell, the particu-larly simple results for the dimensionless critical flowvelocity of divergence is
2ð1� cos uÞ � u sin u ¼ 0 ð9Þ
0
20
40
60
80R
e(ω
)
0 5 10 15 20
–50
0
50
Flow velocity u
Im(ω
)
u1cr=7.75
1st mode
2nd mode
3rd mode
u2cr=11.04
u3cr=11.28
u4cr=15.04
2nd and 3rdmodes combined
1st–mode divergence
2nd–mode divergence
3rd–mode divergence
1st and 2nd modescombined flutter
(a)
(b)
12.5
12.5
15
15
17.5
17.5
0 10 20 30 40 50 60 70
–100
–50
0
50
100
2.5 2.510
10
16
22.5
22.5
25
25
25
25
16
20
20
1214
6
Im(ω
)
Re(ω)
3rd–mode2nd–mode1st–mode
1st and 2nd modescombined flutter
7.75
11.04
15.04
11.28
(c)
Figure 3. The real and imaginary components of the eigenfre-
quencies, !, as functions of the dimensionless flow velocity, u, for
the lowest three modes of a clamped-clamped PFGM shell;
parameters for this PFGM shell are: �max¼ 8�min, la¼ 0.275lc,
lb¼ 0.15lc, lf¼ lb; �¼ 0.1. The inset is the amplified view for the
dashed-square frame.
0
20
40
60
80
100
120
Rea
lfre
quen
cyH
z
1st mode
2nd mode
3rd mode
1st and 2ndmodes combined
2nd and 3rdmodes combined
0 5 10 15 20 25–100
–50
0
50
Fluid velocity uIm
agfr
eque
ncy
Hz
1st–modedivergence
1st and 2nd coupled–modeflutter
2nd and 3rd coupled–modeflutter
6.28 9.0
9.4 12.97
14.75
(b)
(a)
13.04
Figure 4. The eigenfrequencies of a clamped-clamped uniform
shell, !, as functions of the dimensionless flow velocity, u, for the
lowest three modes; �¼ 0.604. (a) and (b) are the real and
imaginary components of the eigenfrequencies, respectively.
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It is obtained by considering only the time-independent terms in the governing equation and theclamped-clamped boundary condition. However, forthe PFGM shell, the derivation of such a simple formfor determining the critical velocity seems impossible.
Figure 6 addresses another diagram of the eigenfre-quencies !, for another PFGM shell, of which
�max¼ 3�min, other parameters are the same as thoseof Figure 5. In this case, the maximum value �max islower than that for the previous PFGM shell, accord-ingly, the rigidity of shell sections B is now more soft inthis system; therefore, as expected, the critical velocitiesof divergent instability are all decreased, as can be seenfrom Figures 5 and 6. Moreover, the so-called ‘modeexchange’ dynamic phenomenon is seen in Figures 5and 6, that is, the coupled-mode flutter is switchedfrom the first and second coupled mode to secondand third coupled mode.
Besides the enhancement of divergence by the PFGMshell, the critical velocity for flutter stability is alsoimproved, as compared to the uniform shell system ofFigure 4. Moreover, one can find that the dynamics ofthe PFGM shell system of Figure 5 has slight differencefrom those of Figures 4 and 6, that is, after the coupled-mode flutter of the first and secondmodes, the thirdmodedevelops divergence at ucr& 15.04, without coupling withthe secondmode; the firstmodenever regain stability afterthe onset of Hamiltonian coupled-mode flutter.Nevertheless, the phenomenon of destabilization-restabi-lization-destabilization for the first mode does also occurhere before the Hamiltonian flutter. It is important tostress that both the restabilization of the system afterdivergence and the coupled-mode flutter are due to gyro-scopic natureof the system, i.e., to theCoriolis terms in theequation of motion, purely conservative systems cannotbe re-stabilized after divergence ‘on their own’, but gyro-scopic force can restabilize an otherwise conservativesystem. However, the point is, as highlighted byPaıdoussis, how some energy of the flowing fluid is chan-neled to generate the oscillatory state (flutter), as thesystem is conservative, remains the question.
The typical plots of stability, as functions of thedimensionless fluid mass �, for the PFGM shells arepresented in Figure 7. In these plots, the conclusionsfor the clamped-clamped PFGM shell systems analyzedabove are validated once more: (i) the divergence-criti-cal velocities are independent of �, and they do beimproved by increase the stiffness of the shell sectionsB of the PFGM shell system; (ii) the characteristics ofdynamical stability (flutter) are enhanced too, again byreinforcing the stiffness of the PFGM shell. It is alsoseen that the two types of coupled-mode flutter areneatly separated in both of these two PFGM shellsystem: Paıdoussis flutter for �<�c1 (�c1¼ 0.37 forthe first PFGM shell and 0.25 for the latter PFGMshell), and flutter via a Hamiltonian Hopf bifurcationfor higher �, for the clamped-clamped constraints; thisvalue is raised as comparing that of 0.139, for the uni-form shell system. Another ‘critical’ � may be of inter-est is �c2, as shown in the plot, near this point, thefrequently occurring feature of ‘mode exchange’ couldbe observed, to be detailedly, i.e., through which, the
0
20
40
60
80
100
fluid velocity u
Re(
ω )
0 5 10 15 20 25–100
–50
0
50
Im(ω
)
1st mode
2nd mode
3rd mode
1sr and 2ndmodes combined
1st and 2nd coupled–mode flutter
1st–mode divergence
3rd–modedivergence
7.75 15.04
11.3
14.5
(a)
(b)
0 20 40 60 80
–80
–60
–40
–20
0
20
40
60
80
11.3
7.54
10
10
12.5
0
12.5
15.04
15
15
17.5
17.5
17.5
17.5
20
20
22.5
22.5
25
25
25
25
0
0
10.7
7.75
3rd mode
2nd mode1st mode
1stand 2nd modecombined
Im( ω
)
(c)
Re(ω)
Figure 5. The eigenfrequencies of the clamped-clamped PFGM
shell, !, as functions of the dimensionless flow velocity, u, for the
lowest three modes; �¼ 0.604, other parameters are the same
with Figure 3. (a) and (b) are the real and imaginary components
of the eigenfrequencies, respectively; (c) is the complex fre-
quency diagram.
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coupled-mode flutter will experience a mode switchfrom the 1st- and 2nd-coupled flutter to the 2nd- and3rd-coupled flutter. Moreover, what is particularlyinteresting is the small stable region that locatednear �c2. In this region, the shell system regains stabilityfrom the unstable state of flutter, and then further suf-fers instability of divergence or flutter, as the flow vel-ocity increased.
Further more, as mentioned in section 1, althoughthe introduction of periodic design in the shell systemcan improve the stability, the serious shortcoming inthe periodic structures is the stress concentration inthe interfaces between the material/geometric disconti-nuities. Hence, naturally, the FGM is introduced to theperiodic system, aiming to reduce the stress concentra-tion for the PS shell system. However, a sagacious
reader may be thinking about the impact of the ‘func-tional grading’ and the changes of, e.g., lengths of seg-ments and/or ‘grading profiles’ on the stability of theshell system. Therefore, at this stage, these aspects aretouched upon.
Firstly, consider the impact of the material param-eters of FGM on the stability for the shell system; typ-ical results are addressed in Figure 8. From this figure,it is clear that the introduction of such a FGM thatemployed in the previous analysis into the shellsystem does influence its stability characteristics.Nevertheless, its effects are not so as strong as theaforementioned PFGM shell models. This is becauseeach FGM shell section is short as compared to theshell section B; accordingly, the contribution of rigidityof these FGM shell sections to overall stiffness of the
0.2 0.4 0.6 0.80
5
10
15
20
25
30
35
βc1
Fluidm ass β
Flu
id v
eloc
ityu
0.2 0.4 0.6 0.80
5
10
15
20
25
30
35
Fluidm ass β
βc2
βc1
βc2
stable stable
divergence divergence
flutte
divergence divergence
stablestable
(a)
divergencedivergence
0.2 0.4 0.6 0.8
11
12
13
14
15
Fluidm ass β
Flu
id v
eloc
ityu
0.2 0.4 0.6 0.810
11
12
13
14
Fluidm ass β
stable stable
flutte
flutterr
fr lutter
stable stable
divergence divergence
divergencedivergence(c)
(b)
(d)
Figure 7. Maps of stability boundaries for the two PFGM shells as studied before. (a) and (b) correspond to the PFGM shells in
Figures 5 and 6, respectively; (c) and (d) are respectively the amplified views for (a) and (b). �c1 indicates such a critical value that
below it, the Paıdoussis-type flutter is observed; while above it, the Hamiltonian flutter appears. �c2 reveals another critical point,
whereupon the coupled flutter will experience a mode switch from the 1st- and 2nd-coupled flutter to the 2nd- and 3rd-coupled
flutter. The dashed line suggests the critical velocity thereafter the 3rd-mode divergence is developed.
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shell system is much smaller than that of shellsections B. If the length of FGM section is even shorterand sufficiently short, then the effects of FGM segmentson the stability for the shell system could be neglected,as illustrated in Figure 9. On the other hand, if eachFGM segment is lengthened, the stability of the shellsystem will be enhanced. By comparing the solid-dottedline with the square-dotted line, one can conclude thatthe mass effect of the FGM segments is to lower theeigenfrequencies of the shell system when the flowingvelocity is not so high, but the mass does not deterior-ate the stability for the system.
Second, take into account the effects of lengths ofshell segments B on the stability for the system, asshown in Figure 10. Also, two types of FGM are uti-lized involving in the stability analysis for the periodicshell system: (i) Emax¼ 3Emin, �max¼ �min; and (ii)Emax¼ 3Emin, �max¼ 3�min; that is the same withthose for Figure 9. Obviously, lengthening shell seg-ment B will result in an appreciable improvement inthe stability for the fluid-conveying shell system, asthe whole rigidity of the system is dramaticallyincreased by such a change.
Finally, the effect of stress-concentration reductionby the PFGM shell system is highlighted in this step.The stress distribution snapshots are presented to illu-minate such an effect, as shown in Figure 11. The finiteelement models involved in the simulation for thePFGM shell are the same with that of Figure 3. Theparameters of the PS shell are the same with the PFGMshell in Figure 8, except that the material of FGM sec-tion is chosen to be the same with shell section A.Apparently, in the PS shell, the stresses for either thefirst, second or the third-mode oscillation, are concen-trated at or near the discontinuous interfaces; whereas,in the PFGM shell, such stress concentration distribu-tion is eliminated, and the stresses are distributed moresmoothly. Also, the maximum value of the stress iseffectively reduced. Hence, the advanced characteristicsof this PFGM shell, in both the stability improvementand the stress-concentration reduction, are clearly seen.
Up to now, the stability characteristics of the fluid-conveying periodic shell are studied, as well as an ana-lysis of some key parameters on the stability of thesystem. However, it is well known that the initial attrac-tion of the periodic structure is due to the usual phys-ical characteristics (Sorokin and Ershova, 2006;Sorokin et al., 2008; Shen et al., 2009, 2012), i.e., thewave bands, within which the propagation of elasticwave is forbidden over some selected frequencyranges (stop bands). Thus, one may be curious aboutthe role of periodicity on the stability for system here.To clarify such a problem, a generally recognized fea-ture, the frequency response function (FRF), is per-formed for the fluid-conveying shell. Figure 12
Re(
ω)
0 2 4 6 80
20
40
60
80
100
120
Fluid velocity u
Uniform shell
FGM shell: Φmax = 3Φmin
FGM shell: Εmax = 3Εmin
10 12
Figure 8. Real parts of eigenfrequencies for the uniform shell
and the shells with FGM embedded periodically in the shell wall
along the axial direction; �¼ 0.1. Geometric parameters for
these shell system are the same as those for Figure 6. The
square-dotted line corresponds to such a PFGM shell that the
FGM (i.e., Emax¼ 3Emin and �max¼ 3�min as described in equa-
tion. (5)) shell sections are the same with those for Figure 6,
while other shell sections (shell sections A and B) take on the
same material with the uniform shell; the solid-dotted line cor-
responds to a PFGM shell with the same material and geometric
parameters as the former shell, but here the material parameters
for FGM are Emax¼ 3Emin and �max¼ �min; the circle-dotted line
is eigenfrequencies for the uniform shell.
0 2 4 6 8 10 120
20
40
60
80
100
120
Fluid velocity u
FGM shell: Εmax= 3Εmin
FGM shell: Φmax= 3Φmin
Uniform she ll
Re
(ω)
Figure 9. Real parts of eigenfrequencies for various fluid-con-
veying shell system; �¼ 0.1. Material parameters for these shell
system are the same as those for Figure 8. The square-dotted,
solid-dotted and circle-dotted lines correspond to the corres-
ponding shells system as studied in Figure 9, except that here the
length for FGM shell sections is reduced to be 1/3 length of those
FGM in Figure 8.
Shen et al. 9
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addresses the real part of eigenfrequencies of thePFGM shell for the lowest five axial modes, andFigure 13 presents the FRFs for the clamped-clampedshell conveying fluid. Close inspection of these plotsreveals that the PFGM shell system produces a stopband in the frequency range 0-500Hz; whereas, nonestop band is found in the uniform shell system. Such
a stop band is moved toward the low frequency rangeas the flowing velocity is increased. This can be easilyunderstood by observing Figure 13. If the flowing vel-ocity is sufficiently high, the real parts of the lowestnatural frequencies may decrease to zero or loci coa-lesce, whereupon the instability form of divergence orflutter is generated. Embodiment of such phenomenon
(a)
(b)
(c)
(d)
PS shell
PFGM shell
PFGM shell
PS shell
PFGM shell
PS shell
PFGM shell
PS shell
Figure 11. Stress distributions for the PS and PFGM shells for the lowest three modes: (a) finite element models; (b), (c) and (d) are
corresponding to the stress distributions for the first, second and third modes, respectively. The dashed lines in the plot denote the
positions of the discontinuous interfaces for the PS shell.
0 5 10 150
50
100
150
Fluid velocity u
Re
(ω)
Segment B1Segment B2Segment B3
(a)
Fluid velocity u
Re
(ω)
Segment B1Segment B2Segment B3
(b)
0 5 10 150
20
40
60
80
100
120
Figure 10. Real parts of eigenfrequencies for various PFGM shell system;�¼ 0.1. The solid-, circle- and square-dotted lines cor-
respond to the cases when lb¼ 0.15lc, lb¼ 0.4lc and lb¼ 0.5lc, respectively; lf and lc are kept the same as Figure 6. (a) and (b)
correspond to two sets of material parameter for the FGM and shell segment B, i.e., (Emax¼ 3Emin, �max¼ �min) and (Emax¼ 3Emin,
�max¼ 3�min), respectively.
10 Journal of Vibration and Control
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in the FRF is the lowest resonant peaks disappear orcombined to be one peak, as illustrated in Figure 14.On closer examination of Figure 13, it can find thatwithin the stop-band frequency range, no resonantpeaks existed; whereas, in the uniform shell system,one resonant peak is located in this stop band. Hence,the role of a periodic design relating to stability char-acteristics of the fluid-conveying shell system may beconcluded to (i) increase the overall stiffness for theshell system, and hence enhancing the system’s stability;(ii) produce a broad band (or several bands) wherenone natural frequency existed such that the instabilityof subsequent modes (whose natural frequencies arelocated above the stop band) are all enhanced.
Before closing the current work, it should be pointedout that all of the results of this paper are obtainedbased on linear theory, hence the coupled-mode flutteris possible in this fluid-structure interaction system.Nevertheless, whether the coupled-mode flutter stilloccurs in such a PFGM shell system under certain con-dition if a nonlinear shell theory is utilized remainsunknown to us. We admit that the nonlinear dynamicsof fluid-conveying shells is of more interest to dynami-cists, because some key questions (e.g., where are thenew fixed points for the static stability and the predic-tion of an unstable ‘inner’ limit cycle in addition to astable outer one for the dynamic stability), as pointedout by Paıdoussis (1998), cannot be answered except bynonlinear theory. Nevertheless, the linear theory is cap-able of predicting the post-divergence dynamics reason-ably well; hence, it is not pointless to examine thestability dynamics via linear theory. A natural
extension of this work may be the nonlinear dynamicsand an experimental design.
4. Conclusions
In summary, the characteristics of the beam-mode sta-bility of the PFGM shells are investigated in this paper,under the clamped-clamped constraints. A FEM algo-rithm is developed to conduct the investigation. Thecorrectness of the FEM algorithm is validated by com-paring with the previously reported results for the uni-form shell system.
For such a clamped-clamped fluid-conveying shellsystem, which is served as the gyroscopic conservativesystem, the system may firstly lose stability by diver-gence, then be further suffered from Paıdoussis-type(coupled-mode) flutter for small �, as u is increased;or be further suffered from the Hamiltonian flutter
10–1
100
101
uniformperiodic
100 200 300 400 50010–3
10–2
10–1
100
101
wm
Frequency Hz
uniformperiodic
wm
(a)
(b)
Figure 13. The FRF curves for the clamped-clamped PFGM
(Emax¼ 3Emin) and uniform shells; (a) and (b) correspond to the
dimensionless flow velocities u¼ 0 and u¼ 7.5, respectively; the
solid and dashed lines correspond to the uniform and PFGM
shells.
0
50
100
150
200
250
300
350R
ealf
req
uenc
y
0 5 10 15Fluid velocity
Figure 12. The eigenfrequencies of the clamped-clamped
PFGM shell (Emax¼ 3Emin), !, as functions of the dimensionless
flow velocity, u, for the lowest five modes; geometric parameters
are the same with Figure 6.
Shen et al. 11
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after a temporary period of restabilization, if � is large.However, the critical velocities ucr for the divergentform of instability is independent of �, since divergenceis a static rather than dynamic form of instability.Generally, the stability characteristics for the PFGMshell with clamped-clamped ends, either the divergenceor the flutter stability, are improved by the PFGMshell, as compared with the uniform shell system.Moreover, the stability of the PFGM shell system canbe further improved by changing some key parameters,e.g., lengths of segments and/or ‘grading profiles’, andappreciable results can thereby obtained. Besides, as theapplication of FGM in the connection of various shellsegments, the serious shortcomings of stress concentra-tion produced by the material/geometric discontinuitiesof periodic structures is overcome, as demonstrated bythe stress distribution images.
It is hoped that this research will be of some help andinterest to the research being conducted on the dynam-ics of shell structures conveying fluid.
Acknowledgment
Special thanks go to Professor Michael P Paıdoussis for hissupervision to the first author at McGill.
Funding
This work was funded by the National Natural Science
Foundation of China (grant numbers 51275519, 51175501,11372346). The first author is grateful for the financial aidprovided by the Hunan Provincial Innovation Foundationfor Postgraduate.
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