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THERMOACTIVE STABILIZATION OF ACOMPOSITE LEIPHOLZ COLUMNWłodzimierz Kurnik a & Piotr M. Przybyłowicz a

a Warsaw University of Technology, Warsaw, PolandPublished online: 21 Jun 2010.

To cite this article: Włodzimierz Kurnik & Piotr M. Przybyłowicz (2003) THERMOACTIVESTABILIZATION OF A COMPOSITE LEIPHOLZ COLUMN, Journal of Thermal Stresses, 26:11-12, 1263-1276,DOI: 10.1080/714050885

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THERMOACTIVE STABILIZATION OF A COMPOSITE

LEIPHOLZ COLUMN

Wl

=

odzimierz Kurnik and Piotr M. Przybyl

=

owicz

Warsaw University of TechnologyWarsaw, Poland

A model of the Leipholz column made of orthotropic layers is analyzed. Apart fromreinforcing fibers, the layers contain thermoactive shape memory alloy fibers. A non-linear equation of motion that takes into account moderate deflections of the column, aswell as Brazier’s cross-section flattening effect, is examined. The effect of the mar-tensite transformation due to external cooling and heating on the system stability ex-pressed in terms of the critical load is investigated. The temperature change is alsoshown to affect the characteristics of the column nonlinear response, because it can leadto conversion of subcritical bifurcation into a supercritical one.

Keywords bifurcation, follower load, Leipholz column, shape memory alloy,stability

The problem of dynamic loss of stability and occurrence of flutter-type self-excitedvibration was formulated and discussed in detail by Leipholz [1,2], although the firstto initiate the investigations was Beck [3] as he showed that an excessive compressiveload applied tangentially to a cantilever results in transverse vibration of the system.The classically formulated Beck or Leipholz problems have undergone variousmodifications. The dynamic loss of stability brought about by distributed followerforces was also found to concern plates in a supersonic gas flow or pipes conveyingfluids, which can respond chaotically if laterally constrained (see [4,5]). A detailedstudy of the geometrically nonlinear damped Leipholz column was presented byKurnik and Pe� kalak [6] who observed near-critical behavior of the system andformulated periodic solutions to the discovered limit cycles. They showed that atensile follower load always entails a supercritical Hopf bifurcation (soft self-excitation), whereas compression can generate either super- or subcritical bifurcation(hard self-excitation also possible) leading to a catastrophic loss of the equilibriumstability. The work by Bogacz et al. [7] indicated the possibility of flutter occurrence

Communicated by Theodore R. Tauchert on June 13, 2003.

Presented at the Symposium dedicated to Richard B. Hetnarski at the Fifth International Congress

on Thermal Stresses, Thermal Stresses 2003, Blacksburg, Virginia, June 8–11, 2003.

Address correspondence to Wl=odzimierz Kurnik, Warsaw University of Technology, Narbutta 84,

Warsaw, 02-524 Poland. E-mail: wku@rekt.pw.edu.pl

Journal of Thermal Stresses, 26: 1263–1276, 2003

Copyright # Taylor & Francis Inc.

ISSN: 0149-5739 print/1521-074X online

DOI: 10.1080/01495730390232002

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via coalescence of the third and fourth eigenvalues, not the first two ones as in theclassical Beck–Reut problem.

The present article is an extension of classical works on the Leipholz column asit takes up the problem of its thermoactive stabilization by making use of shapememory alloys (SMAs). The analogous problem of active stabilization, however,with the help of piezoelectric elements, was examined by the authors in a previousstudy [8].

Recently, much attention has been paid to thermally adaptive properties ofSMAs related to their martensitic transformation (see [9,10]). These materials showunusual phenomena, including pseudoelasticity and shape memory effect, whichhave been successfully applied to vibration control of beams and plates. SMAs seemto be promising also in the stabilization of systems in which they can be applied towhat is called the active modal modification and active strain energy tuning.

Following the first concept, one makes use of considerable changes of Young’smodulus and the loss factor of SMAs in a relatively small temperature interval.(Between 20 and 35�C Nitinol increases its Young’s modulus 3–4 times while its lossfactor decreases even up to 10 times). This growth of temperature can be achieved byeither external heating (open-loop control) or internal dissipation (self-control). Ithas been already shown that such a technique yields very advantageous effects in thecase of flexible high-speed rotating shafts (see [11,12]).

Presumably, the application of SMA fibers to the structure of the Leipholzcolumn might be desirable because such pronounced changes in the elasticitymodulus and the level of internal friction simply must affect the dynamics of thesystem (hopefully its stability boundary) and its resistance to flutter-type vibration.Variable and controllable temperature conditions would prevent the column fromthe catastrophic loss of equilibrium that may result from hard self-excitation. Ap-plication of SMA fibers and appropriate setting of the temperature of, for example,the medium being transported through the column (pipe) would enable one to shiftthe critical value of the external follower load toward greater levels (i.e., notthreatened by self-excitation).

GOVERNING EQUATIONS AND SYSTEM STABILITY

The transverse vibration of a uniform cantilever column subject to tangential dis-tributed follower load, shown in Figure 1, is described by the following equation ofmotion (see [8]):

rA@2w

@t2þ h

@w

@xþ @

@x

@M

@x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ @w

@x

� �2s2

435� @2w

@x2

Z l

x

qðxÞdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ @w

@x

� �2r ¼ 0 ð1Þ

where r is the mass density of the column material, A is the cross-sectional area, w isthe transverse displacement, q is the follower load, h is the external damping, M isthe bending moment, t is time, and x (or x) is the longitudinal coordinate.

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As is known for Leipholz column systems, the key factors responsible for thevalue of the critical load, and in the case of tensile loads for the criticality, is thepresence of internal damping that results from natural properties of the given col-umn material (in this study, of the composite containing SMA fibers). Assuming asimple rheological model of a single lamina (e.g., the Kelvin–Voigt [K–V] model), theYoung and Kirchhoff moduli Y1;Y2;G12 in the main anisotropy directions (1,2) havethe following operational forms:

Y �i ¼ Yi 1þ bii

@

@t

� �i ¼ 1; 2 G�

12 ¼ G12 1þ b12@

@t

� �ð2Þ

Taking into account that, in composites, predominantly, Y2 � Y1, one finds that theelasticity constant corresponding to the entire laminate structure is

Y �B ¼ YB

1þ b1@@t þ b2

@2

@t2

1þ c @@t

ð3Þ

where

YB ¼�Q11

�Q22 � �Q212

�Q22

b1 ¼�Q11

�Qb22 þ �Q22

�Qb11 � 2 �Q12

�Qb12

�Q11�Q22 � �Q2

12

b2 ¼�Qb11

�Qb22 � ð �Qb

12Þ2

�Q11�Q22 � �Q2

12

c ¼�Qb12

�Q12

ð4Þ

Figure 1. Model of the system.

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and where

�Q11 ¼ Q11 cos4 yþQ22 sin

4 yþ 1

2Q12 þQ66

� �sin2 2y

�Q11 ¼ Q11 sin4 yþQ22 cos

4 yþ 1

2Q12 þQ66

� �sin2 2y

�Q12 ¼1

4Q11 þQ22 � 4Q66ð Þ sin2 2yþ 1

43þ cos 4yð ÞQ12

�Qb11 ¼ b11Q11 cos

4 yþ b22Q22 sin4 yþ b12Q66 þ

1

2b22Q12

� �sin2 2y

�Qb22 ¼ b11Q11 sin

4 yþ b22Q22 cos4 yþ b12Q66 þ

1

2b22Q12

� �sin2 2y

�Qb12 ¼

1

4b11Q11 þ b12Q66 þ

1

4b22Q22

� �sin2 2yþQ12

4b22ð3þ cos 4yÞ

Q11 ¼Y1

1� v212ðY2=Y1ÞQ22 ¼

Y2

1� v212ðY2=Y1Þ

Q12 ¼v12Y2

1� v212ðY2=Y1ÞQ66 ¼ 2G12

ð5Þ

Simplifying Eq. (3), that is, finding a local K–V model of the entire structure, theoperator Y �

B is then linearized with respect to the differential operator @=@t. Toensure precise calculation, such linearization should be done in the neighborhood ofthe critical follower load, qcr, responsible for the loss of stability in the columnsystem. This yields the following local and equivalent form of the operational Youngmodulus:

Y�B ¼ Y0 1þ b0

@

@t

� �ð6Þ

where

Y0 ¼ YB1þ 2cO0 þ ðb1c� b2ÞO2

0

ð1þ cO0Þ2

b0 ¼b1 � cþ O0b2ð2þ cO0Þ1þ 2cO0 þ ðb1c� b2ÞO2

0

ð7Þ

where O0 denotes the frequency of self-excited vibration that appears just after thecritical threshold is reached. The value of O0 is to be found iteratively from equationsof motion discussed later in this article. The global dynamic Young’s modulus Y0

and internal damping b0 depend on the viscoelastic properties of the laminated

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column components, including those corresponding to SMA fibers,Y0 ¼ Y0ðYSMAÞ; b0 ¼ b0ðbSMAÞ:

YSMA ¼ YM þ T� TS

Tf � TSYA � YMð Þ

bSMA ¼ bM � T� TS

Tf � TSbM � bAð Þ

ð8Þ

where T is a given temperature during heating from the start of the transformation ofthe martensite phase (TS) into the austenite one which ends at Tf [9,10]. The sub-scripts A and M refer to austenite and martensite. The important parameters of theconsidered system are given in Table 1. Functions of the dynamic elasticity coeffi-cient Y0 and dynamic internal damping b0 versus temperature T and laminationangle y are shown in Figures 2 and 3.

Taking into account the Brazier flattening effect [13], the bending moment M inEq. (1) can be expressed as follows:

M ¼ Y0J 1þ b0@

@t

� �k 1� gk2� �

ð9Þ

where the cross-section flattening coefficient is

g ¼ 3r2Y0

2b2 �Q22

ð10Þ

where r is the mean radius of the column and b is its thickness, and J is the inertiamoment of the cross section. The curvature k is assumed nonlinear (for moderatedeflections):

k ¼ @2w

@x21þ @w

@x

� �2" #�3

2

� @2w

@x21� 3

2

@w

@x

� �2" #

ð11Þ

Substituting Eqs. (9)–(11) into Eq. (1) one obtains an explicit form of the partialdifferential equations of motion. Since the flutter phenomenon is a two-dimensionaldefinite problem, the thus-obtained equation is then discretized via the Galerkin

Table 1 Main structural and operating parameters

l ¼ 1:0m bM, bA ¼ 0:05; 0:005 sr ¼ 0:015m YM, YA ¼ 0:3; 1:3� 1011 Pa

b ¼ 0:002 m Y1, Y2, G12 ¼ 2:1; 0:053; 0:026� 1011 Pa

r ¼ 2500kg=m3 TS ¼ 39�Ch ¼ 0:25 s Tf ¼ 77�C

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orthogonalization method. The orthogonalization itself is based on the firsttwo eigenfunctions corresponding to a cantilever beam, wðx; tÞ ¼ F1ðxÞU1ðtÞþ

Figure 2. Dynamic elasticity coefficient of the SMA Leipholz column.

Figure 3. Dynamic internal damping of the SMA Leipholz column.

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F2ðxÞU2ðtÞ, where F1ðxÞ and F2ðxÞ are eigenforms and U1;2ðtÞ are arbitrary timefunctions to be determined. Incorporating Galerkin’s discretization, one demands

Z l

0

= w F1ðxÞ;U1ðtÞ;F2ðxÞ;U2ðtÞ½ �f gFiðxÞ dx ¼ 0 i ¼ 1; 2 ð12Þ

Figure 4. Trajectory of the first eigenvalue.

Figure 5. Effect of elasticity modulus and internal damping on critical follower load.

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where =f g represents the partial differential equation (see Eq. (1)). After trans-forming and introducing new variables ðu1 ¼ U1; u2 ¼ _U1; u3 ¼ U2; u4 ¼ _U2Þand denoting u ¼ ½u1; u2; u3; u4�T one can rewrite the newly obtained governingequations in the form

_u ¼ Aðq;T; yÞuþNðq; u;T; yÞ ¼ fðq; u;T; yÞ ð13Þ

where A denotes a matrix of the linear part of the equations of motion, N describestheir nonlinear part, and f is a full representation of the right-hand sides. Obviously,

u ¼ ½u1; . . . ; u4�T, N ¼ ½N1; . . . ;N4�T, f ¼ ½ f1; . . . ; f4�T, and A ¼ A4�4:

A ¼ 1

rA

0 1 0 0

�Y0Jk41 þ qC11 �h� Y0b0Jk

41 qC12 0

0 0 0 1

qC21 0 �Y0Jk42 þ qC22 �h� Y0b0Jk

42

26664

37775 ð14Þ

where

Cij ¼R l

0d2FjðxÞdx2

FiðxÞðx� lÞdxR l

0 F2i ðxÞ dx

ð15Þ

Figure 6. Evolution of stability area with temperature.

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and k1 ¼ 1:8751, k2 ¼ 4:6941 are the eigenvalues of the first two eigenmodes cor-responding to a cantilever column.

Solving the characteristic equation corresponding to the eigenproblemfA� rIgv ¼ 0, where r is the eigenvalue and v is the eigenvector to be found, onecan determine the dynamic stability of the system. This can be done by tracing thetrajectory of the eigenvalue r1 that has the greatest real part. This is shown inFigure 4. Intersection of the imaginary axis and passing into the right-hand side ofthe complex plane means loss of stability. This happens for both tensile andcompressive character of the follower load. Zero load places the eigenvalue at apoint of the minimum real part each time. Surprisingly, unlike in the case of ro-tating shafts with SMA, the external heating shifts the trajectories toward the right;that is, it destabilizes the system. In other words, the Leipholz column requirescooling to become more stable. This ensues from the fact that the columns un-dergoing a follower-distributed load exhibit a rapid increase in the criticalthreshold only in a narrow interval of decreasing internal friction. Before, loweringdamping entails a drop in the critical load, and exactly in that range the mar-tensite–austenite transformation takes place. This is shown, in a rather complicatedway, in Figure 5.

For a better understanding of the role of temperature changes on the systemdynamics, regions of stable and unstable operation of the Leipholz column versuslamination angle are shown in Figure 6.

NEAR-CRITICAL BEHAVIOR

Having derived the equations of motion and having found the stability domain, weproceed now with the nonlinear analysis in order to determine near-critical behaviorof the system. The most important problem is to recognize conditions in which self-excited vibration appears—whether it is of soft or hard type. Expressing the problemin mathematical terms, we deal with Hopf’s bifurcation as the trivial static equili-brium position that evolves into a periodic solution. Another problem is to examinethe orbital stability of such a solution.

The near-critical behavior is to be studied on the grounds of a bifurcating so-lution. The method proposed by Iooss and Joseph [14] will be incorporated.Equation of motion (13) is to be investigated in terms of bifurcation analysis. Thefollower load q is the bifurcation parameter; the lamination angle y and temperatureT are passive parameters.

Examining the properties of Eq. (13) in its explicit form one states the following:

1. The point ðq; 0; y;TÞ is a fixed point of Eq. (13); that is, fðq; 0; y;TÞ ¼ 0 for any q.2. The function fðq; u; y;TÞ is differentiable with respect to the bifurcation para-

meter q and components of the vector u.3. Nð0; u; y;TÞ 6¼ 0 holds.4. At the critical point q ¼ qcr there exists a pair of complex conjugate eigenvalues

r1;2 ¼ zðqcrÞ � iWðqcrÞ which satisfy the conditions

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zðqcrÞ ¼ 0

WðqcrÞ ¼ O0 > 0

dzðqcrÞdq

6¼ 0

ð16Þ

5. For the remaining eigenvalues: Refrð:Þg < 0

The preceding properties of the function fðq; u; y;TÞ are the necessary conditionsof the existence of a periodic bifurcating solution predicted by Hopf ’s theorem.

The bifurcating solution is predicted in the form of a series based on 2pn peri-odic functions unðOtÞ of the frequency O, expressed in terms of a series as:

uðOt; eÞ ¼X1n¼1

en

n!unðOtÞ

O ¼ O0 þX1n¼1

en

n!On

ð17Þ

where O0 is the initial frequency of the self-excited vibration. The solution given byEqs. (17) can be completed by introducing a relationship between the unknownparameter e and the bifurcating parameter q:

q ¼ qcr þX1n¼1

en

n!qn ð18Þ

Consider now the first-order approximation of the bifurcating solution:

uðOt; eÞ ¼ e u1 ð19Þ

Furthermore,

O ¼ O0 þ1

2e2O2 q ¼ qcr þ

1

2e2q2 ð20Þ

because it can be proved that for any n the odd coefficients q2n�1 and O2n�1 vanish(see [14]). The bifurcating solution then assumes the form

u OðqÞt½ � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q� qcrq2

ru1 where u1 ¼ 2Re veiOt

� ð21Þ

The vector v is an eigenvector corresponding to the following eigenproblem:

AðqcrÞ � iO0If gv ¼ 0 ð22Þ

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The coefficients O2 and q2 appearing in Eq. (20) are

q2 ¼ � 1

3

ReC2

ðd=dqÞ Re fr1ðqcrÞgf g O2 ¼ q2d

dqRefr1ðqcrÞgf g þ 1

3ImC2 ð23Þ

where

C2 ¼ 3X4i¼1

X4j¼1

X4k¼1

X4l¼1

@3fiðqcr; 0Þ@uj@uk@ul

�v�i vjvk�vl ð24Þ

Occurring in Eq. (24) vector v� is an eigenvector corresponding to the followingadjoint eigenproblem:

ATðqcrÞ þ iO0I�

v� ¼ 0 ð25Þ

The explicit forms of the vectors v and v� are the following:

v ¼

1

iO0

� a21 þ O20 þ i a22O0

a23 þ ia24O0

�O0ða21 þ O20 þ ia22O0Þ

a24O0 � ia23

26666666664

37777777775

ð26Þ

and

v� ¼ ~D

ia41O0

� ia21ða43 þ O20 � ia44O0Þ

O0ða23 � ia24O0Þ

� a43 þ O20 � ia44O0

O0ða23 � ia24O0Þ

� a23ða44 þ iO0Þ � a24a43a23 � ia24O0

1

266666666664

377777777775

ð27Þ

where the coefficients aij are explicitly given in Eq. (14). The complex constant ~D inEq. (27) is chosen in such a way so that the following orthonormalization conditionsare fulfilled:

v; v�h i ¼X4i¼1

vi�v�i ¼ 1 hv; v

�i ¼

X4i¼1

viv�i ¼ 0 ð28Þ

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These conditions force the constant ~D to be

~D ¼ 1

v�1 þ v�2�v2 þ v�3�v3 þ �v4ð29Þ

The thus-obtained approximation of the bifurcating solution is then examinedwith respect to its orbital stability. It is important now whether the observed

Figure 7. Effect of temperature on Floquet’s exponent.

Figure 8. Exemplary bifurcation diagrams for different temperature and structural parameters.

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bifurcation has super- or subcritical character. Stability of the bifurcating solutiondepends on the sign of the Floquet exponent, defined as

sðeÞ ¼ � d

dqRefr1ðqcrÞgf gq2e2 þOðe4Þ ð30Þ

where Oðe4Þ are negligible terms of higher orders of e. When sðeÞ < 0 then the bi-furcating solution is asymptotically stable (supercritical bifurcation); when sðeÞ > 0,it is unstable (subcritical bifurcation).

Investigate now how far the temperature change responsible for the martensite–austenite transformation affects the orbital stability of the limit cycle. Observe theFloquet exponent in Figure 7 for this purpose. The dashed line corresponds to thecase when the column may exhibit sudden jumps of vibration amplitude even belowthe critical threshold, just due to a minor disturbance coming from the surroundings(subcritical bifurcation). The continuous lines present a much safer case of soft self-excitation with slow evolution of the vibration amplitude above the criticality(supercritical bifurcation). It is clearly seen that in some cases it can definitely changethe character of bifurcation in the system, that is, convert dangerous hard self-excitation into a safer one, soft self-excitation. This happens, however, in a narrowrange of the lamination angle of the composite from which the column is made. Afew examples of bifurcation parameters e to which the limit cycle amplitudes areproportional are shown in Figure 8.

CONCLUDING REMARKS

The application of thermosensitive SMA fibers as integral components of compositesproves to be an interesting method for shifting away the threat of destabilization inLeipholz columns made of such composites. It occurs that in certain conditions(SMA volume share, lamination angle) the column exhibits an increased criticalthreshold if subject to a drop in temperature within the range of austenite-tomartensite transformation. In the case of a Leipholz column used as a pipe con-veying fluid it would be enough to cool the medium pumped through the pipe. Theneed for cooling the system instead of heating results from the well-known fact thatthe nonconservative systems exhibit stabilization when the internal friction dropsdown provided the starting level of the friction is sufficiently low. If it is not—as incomposites—decreasing internal damping entails destabilization. This is un-fortunately in opposition to the effect of stiffening of the SMA fibers when exposedto heat. The austenite-to-martensite transformation decreases the elasticity modulus,which always lowers the critical load and which neutralizes the advantageous effectof the change in the internal damping while being cooled. This can be controlled, to acertain extent, by a proper selection of the SMA volume fraction—however, muchcan also be done by choosing a suitable arrangement of the laminate, that is, plyangle.

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REFERENCES

1. H. H. E. Leipholz, On the Sufficiency of the Energy Criterion for the Stability of Certain Non-

conservative Systems of the Follower Type, J. Appl. Mech., vol. 39, pp. 717–722, 1972.

2. H. H. E. Leipholz, Stability of Elastic Systems, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.

3. M. Beck, Die Knicklast des einseitig eingespannten, tangential gedruckten Stabes, Z. Angew. Math.

Phys., vol. 3, p. 225, 1959.

4. M. P. Paıdoussis, Dynamics of Tubular Cantilevers Conveying Fluid, J. Mech. Eng. Sci., vol. 12,

pp. 85–103, 1970.

5. M. P. Paıdoussis and C. Semler, Nonlinear and Chaotic Oscillations of a Constrained Cantilevered

Pipe Conveying Fluid, Nonlinear Dynam., vol. 4, pp. 655–670, 1993.

6. W. Kurnik and M. Pe� kalak, Stability and Bifurcation Analysis of the Non-linear Damped Leipholz

Column, J. Sound Vibration, vol. 152, pp. 285–294, 1992.

7. R. Bogacz, S. Imiel=owski, and O. Mahrenholtz, On the Shape of Characteristic Curves on Non-

conservative Loaded Structures with Jump Phenomenon, Z. Angew. Math. Mech., vol. 71, pp. T182–

T185, 1991.

8. W. Kurnik and P. M. Przybyl=owicz, Nonlinear Behavior of the Leipholz Column Actively Stabilized

by Piezoelements, Proc. Int. Symposium on Active Control of Sound and Vibration ACTIVE 95,

pp. 139–150, 1995.

9. K. Nagaya and S. Takeda, Active Control Method for Passing Through Critical Speeds of Rotating

Shafts by Changing Stiffnesses of the Supports with Use of Memory Metals, J. Sound Vibration,

vol. 113, pp. 307–315, 1987.

10. C. A. Rogers, C. Liang, and D. K. Barker, Dynamic Control Concepts Using Shape Memory Alloy

Reinforced Plates, in C. A. Rogers (ed.), Smart Materials, Structures and Mathematical Issues,

pp. 39–62, Technomic, Lancaster and Basel 1989.

11. W. Kurnik, Antiwhirl Control of Shafts Using Thermoactive SMA Fibers, Proc. Second Int. Sym-

posium on Thermal Stresses and Related Topics, Rochester, NY, pp. 465–468, 1997.

12. W. Kurnik, Self-Stabilization of a Composite Shaft via Thermally Adaptive Plies, Proc. First Int.

Symposium on Thermal Stresses and Related Topics, Hamamatsu, Japan, pp. 541–544, 1995.

13. V. Birman and C. W. Bert, Non-linear Beam-Type Vibrations of Long Cylindrical Shells, Int. J.

Nonlinear Mech., vol. 22, pp. 327–334, 1987.

14. G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory, Springer-Verlag, New York,

1980.

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