Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 4, pp. 993-1007, Warsaw 2008 STABILITY OF HYBRID ROTATING SHAFT WITH SIMPLY SUPPORTED AND/OR CLAMPED ENDS IN A WEAK FORMULATION Andrzej Tylikowski Warsaw University of Technology, Warsaw, Poland e-mail: aty@simr.pw.edu.pl In this paper, a technique of dynamic stability analysis proposed for the conventional laminated structures is extended to activated shapememo- ry alloy hybrid rotating structures axially loaded by a time-dependent force. In the stability study, the hybrid shaft is treated as a thin-walled symmetrically laminated beam containing both the conventional fibers, and the activated shape memory alloy fibers parallel to the shaft axis. The stability analysis method is developed for distributed dynamic pro- blemswith relaxed assumptions imposed on solutions. Theweak form of dynamical equations of the rotating shaft is obtained using Hamilton’s principle.We consider the influence of activation through the change of temperature on the stability domains of the shaft in the case when the angular velocity is constant. The force stochastic component is assumed in the form of ergodic stationary processes with continuous realisations. The studyof stability analysis is basedonexaminingproperties ofLiapu- nov’s functional alongaweak solution. Solution to theproblem is presen- ted for an arbitrary combination of simply supported and/or clamped boundary conditions. Formulas defining dynamic stability regions are written explicitly. Key words: weak equation, rotating shaft, thermal activation, almost sure stability analysis 1. Introduction The dynamic stability of isotropic elastic simply supported shafts rotating with a constant speed has been studied for several years (cf., Bishop, 1959; Parks and Pritchard, 1969; Tylikowski, 1981). The increased use of advanced composite materials in various applications has caused a great research effort 994 A. Tylikowski in the structural dynamic and acoustic analysis of composite materials. Com- posite materials find an increased range of applications for high-performance rotating shafts (e.g., see Napershin and Klimov, 1986; Bauchau, 1983; Song and Librescu, 1997). The uniform stability of laminated shafts modelled as composite shells rotating with a constant angular velocity under a combi- ned axisymmetric loading was investigated byTylikowski (1996). Thin-walled standard angle-ply laminated tubes meet relatively easy the requirements of torsional strength and stiffness but aremore flexible to bending and have spe- cific elastic and damping properties which depend on the system geometry, physical properties of plies, and on the laminate arrangement. Such systems are also sensitive to lateral buckling. Using the Liapunovmethod, Pavlović et al. (2008) investigated the effect of rotary inertia of the shaft cross-section on almost sure stability of a rotating viscoelastic shaft. Shape Memory Alloy (SMA) hybrid composites are a class of materials capable of changing both their stiffnesses through the application of in-plane loads and their elastic properties. The stiffnessmodification occurs as a result of the thermally inducedmartensite phase transformation of SMAfiberswhich are embedded in standard laminated composite structures. Young’s modulus of the nitinol (nickel-titanium alloys), which is an example of such amaterial, increases 3 to 4 timeswhen the temperature changes from that below Mf (i.e. in themartensite phase) to that above Af (i.e. in the austenite phase) (Cross et al., 1970). The damping of vibrations in the SMA due to internal friction exhibits also important characteristics. The low-temperature martensic pha- se is characterised by a large damping coefficient while the high-temperature austenic phase shows a low damping coefficient. The decrease ratio is ap- proximately equal to 1 : 10. Comprehensive studies of eigen-frequencies and eigen-functions of SMA hybrid adaptive panels with uniformly and piecewise distributed actuation are presented in papers by Anders and Rogers (1991), Baz et al. (1995), Krawczuk et al. (1997). One of the possible ways towards improving dynamic properties and smo- othing rotary motion of shafts consists in the implementation of control and semiactive control methodology. The ones considered are based on the incor- poration of adaptivematerials such as piezoceramics (Przybyłowicz, 2004) and shapememory alloys (Tylikowski, 2005, 2007) into the structures. Stability of rotating shaftsmade of a functionally gradedmaterial with piezoelectric fibers was examined by Przybyłowicz (2005). The present work investigates dynamic stability of thin-walled shafts ro- tating with a constant angular velocity and subjected to an axial force having fluctuations from the constant average value. The time-dependent force com- Stability of hybrid rotating shaft... 995 ponent introduces new terms to dynamic equations and lead to the parametric excitation. In this paper, a technique of dynamic stability analysis proposed for conventional laminated structures is extended to activated shapememory alloy hybrid rotating structures. The hybrid elements are treated as a thin- walled symmetrically laminated beam containing both the conventional (e.g., aramid, graphite or glass) fibers, arbitrarily oriented to the laminate coordina- te axis, and the activated shapememory alloy fibers parallel to the shaft axis. We will consider the influence of activation through the change of temperatu- re on the stability domains of the shaft in the case when the force stochastic component is an ergodic stationary process with continuous realisations. The structure buckles dynamically when the axial parametric excitation becomes so large that the structure does not oscillate about the unperturbed state de- scribed by w, and a new increasing mode of oscillations occurs. In order to estimate the perturbed solutions of dynamic equations, we introduce a me- asure of distance ‖ · ‖, of the solution to dynamic equations with nontrivial initial conditions from the trivial solution. We say that the trivial solution w=0 of the dynamic equations is almost sure asymptotically stable (cf. Ko- zin, 1972) if the measure of distance between the perturbed solution and the trivial solution, ‖ · ‖, satisfies the condition P( lim t→∞ ‖w(t, ·)‖=0)=1 (1.1) where P denotes the probability. Using the appropriate energy-like Liapunov functional, the sufficient stability conditions for the almost sure asymptotic stability of the shaft equilibrium are derived. Finally, the influence of SMA activation on stability regions is examined. The action of distributed control- lers is reduced, in the first approximation, to bendingmoments and transverse forces distributed on the actuator edges. The fourth order differential opera- tors are present in classical strongdynamic equations.Theuseof theHeaviside generalised functions in an analytical description of loading leads to irregu- larities. In order to avoid the irregular terms resulting from differentiation of the force and moment terms, the dynamic equations are written in a weak form. The weak form of systems contains only the second order derivative of displacements, and there is no need to differentiate the terms describing the loading. The first analysis of the stability of simply supported or clamped rectan- gular plates in aweak formulationwas due toTylikowski (2008). The problem here is focused on the stability analysismethod of the equilibrium state of be- ams and plates with relaxed assumptions imposed on solutions. We consider dynamical systemsdescribedbypartial differential equations that include time 996 A. Tylikowski dependent coefficients implyingparametric vibrations.TheHamilton principle is used to derive the weak form of rotating beam dynamic equations. Assu- ming the Lagrangian as a difference of kinetic and elastic energy, and taking the viscotic force and the destabilising force as external ones, we obtain the weak form of the equations. Due to elimination of the fourth order deriva- tive, the solutions of weak equations are not so smooth as solutions of the strong one. The classical Liapunov technique for stability analysis of continu- ous elements is based on choosing or generating a functional which is positive definite in the class of functions satisfying boundary conditions of a structure. The time-derivative of Liapunov functional has to be negative in some defined sense. If the constant component of the axial force is smaller than the Euler critical force, the square root of energy-like Liapunov functional is assumed as a measure of distance between the disturbed solutions and the trivial one. Substituting functions related with beam displacements and velocities as the test functions, yields indentitiesmaking algebraic transformations of the func- tional time-derivative easy. The transformations are performed without the previous discretisation. Sufficient stability conditions of the compressed beam are derived for commonly applied boundary conditions. 2. Dynamic equations The shaft, treated as a thin-walled symmetrically laminated beam, contains both the conventional (e.g. graphite or glass) fibers oriented at +Θ and −Θ to the shell axis and the activated shape memory alloy fibers parallel to the shell axis. The shaft rotates with a constant angular velocity ω. By forcing the martensite austenite transformation of the SMA layer, we modify the basic mechanical properties such as Young’s modulus and the in- ternal damping coefficient. The shaft of length ℓ is assumed tohave a constant circular cross-section of mean radius a, and thickness h without initial geo- metrical imperfections. The mean density is denoted by ρ and the area and the geometrical moment of inertia of the shaft cross-section are denoted by A=2πAh and J = πa3h, respectively. A viscous model of external damping with a constant proportionality coefficient be is assumed to describe the dis- sipation of the shaft energy. The beam-like approach due to Bauchau (1981) is used in order to derive the shaft bending stiffness EJ = ( A11− A212 A22 ) πa3 (2.1) Stability of hybrid rotating shaft... 997 where Aij, i,j =1,2 denote in-plane stiffnesses of the thin-walled beam.More sophisticated considerations of thin-walled composite beams were performed by Song and Librescu (1990). Displacements of the center shaft line in the movable rotating coordinates are denoted by u and v. Introducing the di- mensionless time with the time scale kt = ℓ2 √ ρA/EJ and the dimensionless coordinate divided by ℓ, we obtain a shaft model with the unit mass density, unit bending stiffness, dimensionless angular velocity Ω =ωkt, and modified damping coefficients of external and internal damping βe = bekt, βi = bikt, respectively. Starting from the rotating shaft without damping and axial lo- ading, wewrite the action integral as a time integral of the difference between kinetic and bending energy A(w)= 1 2 t∫ to 1∫ 0 [(u,t+Ωv) 2+(v,t−Ωu)2− (u2,xx+v2,xx)] dx dt (2.2) where w = [u,v]⊤ ∈ W = [H2b(0,1)]2, the index b denotes the set of func- tions satisfying the essential boundary conditions, the time interval (t,to) is arbitrarily chosen. Consider ŵ=w+ ǫθ= [ u(t,x) v(t,x) ] + ǫ [ η1(t)Φ(x) η2(t)Ψ(x) ] (2.3) where η(t)=η(to), θ∈W. According to Hamilton’s principle, of motion the shaft must have a stationary value to the action integral, therefore d dǫ A(u+ ǫθ) ∣∣∣ ǫ=0 =0 (2.4) Using equation (2.2) in equation (2.3) and integrating the time-derivatives of functions η1 and η2, by parts with respect to time we obtain dynamic equations of the shaft in a weak form ∀Φ 1∫ 0 [(u,tt−Ω2u+2Ωv,t)Φ+u,xxΦ,xx] dx=0 (2.5) ∀Ψ 1∫ 0 [(v,tt−Ω2v−2Ωu,t)Ψ +v,xxΨ,xx] dx=0 Adding the internal viscous damping with the modified coefficient βi, the external viscous damping with the modified coefficient βe and the axial force 998 A. Tylikowski fo+f(t) as external works, the shaft dynamic equations can bewritten in the weak form as follows ∀Φ 1∫ 0 [(u,tt−Ω2u+2Ωv,t)Φ+βe(u,t+Ωv)Φ+βiu,tΦ+ +u,xxΦ,xx+(fo+f(t))u,xxΦ] dx=0 (2.6) ∀Ψ 1∫ 0 [(v,tt−Ω2v−2Ωu,t)Ψ+βe(v,t−Ωu)Ψ+βiv,tΨ+ +v,xxΨ,xx+(fo+f(t))v,xxΨ] dx=0 where Φ,Ψ are arbitrary sufficiently smooth test functions satisfying essential boundaryconditions.There isnodemand for theexistenceofderivatives higher than the secondorder.As reportedbyBanks et al. (1993), theusual integration bypartsof the termscontainingderivatives of the test functionswith respect to the variable x and the assumption of sufficient smoothness of the components of shaft displacements lead to the commonly used strong formulation. The shaft is assumed to be simply supported or clamped on its ends. Therefore, the essential boundary conditions have the following form at its ends u(t,0)=u(t,1)= v(t,0)= v(t,1)= 0 (2.7) It means that the displacements of the shaft in supporting bearings are small as comparedwith thedisplacements of a thin-walled flexible shaft.Weak linear equations (2.6) have the trivial solution (equilibrium state) u= v=0. 3. Stability analysis In order to determine conditions of smooth shaftmotion corresponding to the Liapunov stability of the trivial solution w=0,we choose thepositive-definite functional Liapunov as a sumof themodified kinetic and elastic energy of the shaft (Tylikowski, 2007) V = 1 2 1∫ 0 [(u,t+Ωv+βu) 2+(u,t+Ωv) 2+(v,t−Ωu+βv)2+ (3.1) +(v,t−Ωu)2+2(u2,xx+v2,xx)−2fo(u2,x+v2,x)] dx Stability of hybrid rotating shaft... 999 where β=βi+βe. Functional (3.1) is positive-definite if the constant compo- nent of the axial force fo fulfils the static buckling condition, i.e. is sufficiently small. Therefore, the measure of distance of solutions with nontrivial initial conditions from the trivial required in stability analysis can be chosen as a square root of the functional ‖w‖ = √ V . As trajectories of the solution to equations (2.6) are physically realisable, the classical calculus is applied to calculation of time-derivative of functional (3.1). Its time-derivative is given by dV dt = 1∫ 0 [(u,t+Ωv+βu)(u,tt+Ωv,t+βu,t)+(u,t+Ωv)(u,tt+Ωv,t)+ +(v,t−Ωu+βv)(v,tt−Ωu,t+βv,t)+(v,t−Ωu)(v,tt−Ωu,t)+ (3.2) +2(u,xxu,xxt+v,xxv,xxt)−2fo(u,xu,xt+v,xv,xt)] dx In order to avoid integration by parts in equations (3.2) and generation the third and the fourth partial derivatives of displacements, we substitute 2u,t, 2Ωv, βu as the test functions in equation (2.6)1. Therefore, we have three identities, respectively 1∫ 0 [2(u,tt−Ω2u+2Ωv,t)u,t+2βu2,t+2βeΩvu,t+2u,xxu,xxt+ +2(fo+f(t))u,xxu,t] dx=0 1∫ 0 [2Ω(u,tt−Ω2u+2Ωv,t)v+2Ωβe(u,t+Ωv)v+2Ωβiu,tv+ +2Ωu,xxv,xx+2Ω(fo+f(t))u,xxv] dx=0 (3.3) 1∫ 0 [β(u,tt−Ω2u+2Ωv,t)u+ββe(u,t+Ωv)u+ββiu,tu+ +βu2,xx+β(fo+f(t))u,xxu] dx=0 Ina similarwaywe substitute 2v,t, 2Ωu, βv as the test functions in equation (2.6)2 1∫ 0 [2(v,tt−Ω2v−2Ωu,t)v,t+2βv2,t−2βeΩuv,t+ +2v,xxv,xxt+2(fo+f(t))v,xxv,t] dx=0 1000 A. Tylikowski 1∫ 0 [2Ω(v,tt−Ω2u−2Ωu,t)v+2Ωβe(v,t−Ωu)u+2Ωβiv,tu+ +2Ωv,xxu,xx+2Ω(fo+f(t))v,xxu] dx=0 (3.4) 1∫ 0 [β(v,tt−Ω2v−2Ωu,t)v+ββe(v,t−Ωu)v+ββiv,tv+ +βv2,xx+β(fo+f(t))v,xxv] dx=0 Subtracting identities (3.3), (3.4)1 and (3.4)3 from equation (3.2) and adding identity (3.4)2 we obtain the following form dV dt =− 1∫ 0 {β(u2,t+v2,t)+β(u2,xx+v2,xx)−βfo(u2,x+v2,x)+ +(βe−βi)ω2(u2+v2)+2Ωβe(vu,t−uv,t)+ (3.5) +[(2u,t+βu)u,xx+(2v,t+βv)v,xx]f(t)} dx After rewriting, we receive dV dt =−βeV +U (3.6) where the auxiliary functional U is known. Now, we look for a function χ which satisfies the following inequality U ¬χV (3.7) Substituting inequality (3.7) into equation (3.6) yields the first order differen- tial inequality dV dt ¬ (χ−βe)V (3.8) which implies ‖w(t,x)‖¬‖w(0,x)‖exp [ − ( βe− 1 t t∫ 0 χ(τ) dτ ) t ] (3.9) The ergodicity of the axial loading leads to the following almost sure stochastic stability condition Eχ(t)¬βe (3.10) Stability of hybrid rotating shaft... 1001 where E denotes the mean value operator. It should be noticed that the way to obtain equation (3.5) is purely algebraic contrary to systems described by strong equations, where the derivation of stability conditions is based on integrations by parts andmanipulations with higher order partial derivatives. Usually, the Liapunov stability analysis of shafts is performed for both ends simply supported (cf. Bishop, 1959; Tylikowski, 1996). In order to extend the field of possible applications, let us assume the following combinations of boundaryconditions: a) s-s, b) c-s, c) c-c,where sdenotes the simply supported end, and c denotes the clamped end. In order to find χ effectively, we use the expansions of the shaft displace- ments [ u(t,x) v(t,x) ] = ∞∑ n=1 Wn(x) [ Sn(t) Tn(t) ] (3.11) where Wn are the beam functions (cf. Graff, 1975) depending on the assumed boundary conditions. In a similar way, velocities of transverse shaft motion are given by [ u,t(t,x) v,t(t,x) ] = ∞∑ n=1 Wn(x) [ Ṡn(t) Ṫn(t) ] (3.12) Integrating, we have the following equality (Tylikowski, 2008) 1∫ 0 W2n,xx dx= γnα2n 1∫ 0 W2n,x dx (3.13) 1∫ 0 W2n,x dx= α2n γn 1∫ 0 W2n dx where αn is an eigen-value of the corresponding boundary problem and the sequence {γn} is known. Due to the existence of even-order space derivatives in functional (3.1) and in its time-derivative (3.5), the value of functionals can be calculated as a sum of suitable quadratic terms V = ∞∑ n=1 Vn U = ∞∑ n=1 Un (3.14) where Vn and Un are calculated for a single term of expansions (3.11) and (3.12). If χn, which satisfies a single term inequality, is known dVn dt ¬ (χn−βe)Vn (3.15) 1002 A. Tylikowski then the function χ can be effectively calculated χ= max n=1,2,... χn (3.16) Denoting κn =χn−βe and substituting the n-th terms of expansions (3.11) and (3.12) into inequality (3.15), we obtain the second order quadratic inequ- ality with respect to the four variables Ṡn, Sn, Ṫn, Tn [ Ṡ2n+T 2 nΩ 2+2ṠnTnΩ+ 1 2 β2S2n+ ( β− 2α 2 n γn f(t) ) ṠnSn+ Ṫ 2 n +S 2 nΩ 2+ −2ṪnSn+ 1 2 β2T2n + ( β− 2α2n γn f(t) ) ṪnTn+ ( α2n− fo γn ) α2n(S 2 n+T 2 n) ] κn+ (3.17) +(Ṡ2n+ Ṫ 2 n)β+2βeΩ(ṠnTn−SnṪn)+ + [ (βe−βi)Ω2+ ( α2n− fo γn ) α2n−β 2α2n γn f(t) ] (S2n+T 2 n)­ 0 After some reduction, we obtain an auxiliary matrix of quadratic form (3.17)    a b 0 d b c −d 0 0 −d a b d 0 b c    (3.18) where a=κn+β b= 1 2 βκn− 2α2n γn f(t) d=Ω(βe+κn) c=κn [ α2n ( α2n− fo γn ) +Ω2+ β2 2 ] +βα2n ( α2n− fo γn ) +Ω2(βe−βi)− βα2n γn f(t) We recall Sylvester’s conditions for the positive-definiteness of matrix (3.18) a> 0 ac− b2 > 0 a(ac− b2−d2)> 0 (b2+d2−ac)2 > 0 (3.19) As latest inequality (3.19)4 is satisfied, it is easy to notice that third Sylvester inequality (3.19)3 is essential from the stability point of view. It is equivalent to the elementary second order inequality with respect to κn κ2n+2βκn+4 β2α2n(α 2 n−fo/γn)−β2iΩ2 β2+4α2n(α 2 n−fo/γn) > 0 (3.20) Stability of hybrid rotating shaft... 1003 which leads to the determination of κn and finally to the almost sure stability condition from equations (3.16) and (3.10) Eχ=E max n=1,2,... √ β4+4(β2+α2nf(t)/γn)α 2 nf(t)/γn+4Ω 2β2i β2+4α2n(α 2 n−fo/γn) ¬β (3.21) Estimating the limit behaviour of χn as n tends to ∞, we find the critical value of angular velocity for a constant axial force fo Ω2 < ( 1+ βe βi )2 α21 ( α21− fo γ1 ) (3.22) The function χ in inequality (3.21) is random due to randomness of the axial force f(t). Therefore, the probability distribution must be known in order to calculate the average in equation (3.21). 4. Results Numerical calculations basedon formula (3.21) are performed for s-s boundary conditions with the changing time-dependent component of the axial force and the coefficient βe of external damping. A number of iterative steps are performed in order to determine the value of βe. The dimensions of hybrid shafts are: length ℓ = 1m, radius r = 0.04m, total thickness h = 0.004m. Thematerial data are given in Table 1. Table 1. SMA hybrid shaft specification Nitinol-Epoxy Glass- -Epoxy Graphite- -Epoxy Material NiTi – 40%, Epoxy – 60% activated unactivated Density [kg/m3] 2350 2350 1790 1560 E11 [GPa] 41.93 19.31 53.98 211.0 E22 [GPa] 20.93 17.25 17.93 5.30 G12 [GPa] 7.54 6.43 8.96 2.60 ν12 0.25 0.25 0.25 0.25 βi 0.01 0.012 0.01 0.01 The shaft consists of seven layers of equal thickness: of two external lay- ers with activated SMA fibers parallel to the shaft axis and of five internal conventional layers symmetrically arranged with the lamination angle π/4. 1004 A. Tylikowski Thus, the laminate configuration can be uniquely defined by the following no- tation: [0 / (π/4) / (−π/4) / (π/4) / (−π/4) / (π/4) /0]. We can calculate the in-plane stiffnesses Aij (c.f. Whitney, 1987) and then also from Eq. (2.1) the reduced Youngmodulus E of the beam-like cylindrical shell. The main results are shown in Fig.1. The figure compares the stochastic stability domains in the plane (βe,σ2) for the zero and nearly critical axial force. The critical variance of the time-dependent component of the axial force strongly depends on the external damping coefficient βe. The time scale in the activated and unactivated state is kt = 1.2697 ·10−2 s, kt = 1.0635 · 10−2 s, respectively.An increase of angularvelocity significantlydecreases the stability region. The thermal activation significantly increases the stability regions due to increasing stiffness of the nitinol fibers. The influence of the shaft material is not observed in Fig.1 due to the dimensionless quantities. Fig. 1. Boundaries of stability domains for zero and nearly critical axial force components 5. Conclusions Atechnique has beenpresented for the analysis of dynamic stability of an acti- vated simply supportedhybrid shaft rotatingwith a constant angular velocity. The shaft consists of classical symmetrically angle-plied layers and symmetri- cally laminated active plies with axially oriented SMA fibers. The dynamic stability and the stochastic stability problem is reduced to the problem of po- sitive definiteness of the auxiliary matrix. The explicit criteria derived in the paper define stability regions in terms of geometrical andmaterial properties, lamination angle aswell as constant values andvariances of the axial force. For Stability of hybrid rotating shaft... 1005 a constant axial force, the criterion asssumes a closed formof an algebraic ine- quality. If the axial force is time-dependent, the almost sure stability criterion has a form of a transcendental equation involving the axial force probability distribution. Analytical results are presented to demonstrate how the thermal activation affects critical parameters. The influence of fluctuation class (Gaus- sian or harmonic) is not significant. The influence of boundary conditions on stability domains is negligible when the constant component of the axial force is small as compared with the critical loading. References 1. Anders W.S., Rogers C.A., 1991, Vibration and low frequency acoustic analysis of piecewise-activated adaptive composite panels, In: Proceedings of First Joint U.S./Japan Conference on Adaptive Structures, B.K. Wada, J.L. Fanson, K.Miura (Edit.), Technomic Publishing, Lancaster-Basel, 285-303 2. Banks H.T., Fang W., Silcox R.J., Smith R.C., 1993, Approximation methods for control of structural acousticsmodels with piezoelectric actuators, Journal of Intelligent Material Systems and Structures, 4, 98-116 3. Bauchau O.A., 1983, Optimal design of high speed rotating graphite/epoxy shafts, J. Composite Materials, 17, 170-181 4. BazA.,PohS.,RoJ.,GilheanyJ., 1995,Control of the natural frequencies of nitinol – reinforced composite beams, Journal of Sound and Vibration, 185, 171-185 5. BishopR.E.D., 1959,The vibrations of rotating shafts,Journal ofMechanical Engineering Sciences, 1, 50-63 6. Cross W.B., Kariotis A.H., Stimler F.J., 1970, Nitinol characterization study,Goodyear Aerospace Corporation Report,No.Ger 14188NASACR-1433, Akron, Ohio 7. Graff K.F., 1975, Wave Motion in Elastic Solids, Dover Publications, New York, 155-158 8. Kozin F., 1972, Stability of the linear stochastic system, Lecture Notes in Mathematics, 294, 186-229 9. Krawczuk M., Ostachowicz W., Żak A., 1997, Dynamcs of a composi- te beam with nitinol fibres, Zeszyty Naukowe Katedry Mechaniki Stosowanej, Politechnika Śląska, Gliwice, 3, 111-116 10. NepershinR.I.,KlimovW.W., 1986,Optimal design of composite transmis- sion shaftswith respect to costs andweight,Mechanics of CompositeMaterials, 4, 690-695 [in Russian] 1006 A. Tylikowski 11. ParksP.C.,PritchardA.J., 1969,On the construction anduse ofLiapunov functionals,Proceedings of 4-th IFAC, Techn. Session 20, Stability, Warszawa, 59-73 12. Pavlović R., Rajković P., Pavlović I., 2008, Dynamic stability of visco- leastic rotating shaft subjected to random excitation, J. Mech. Eng. Sci., 50, 359-364 13. PrzybyłowiczP.M., 2004,Asymetric rigid rotor systemsupportedon journal bearings with piezoactive elements,Machine Dynamics Problems, 27, 87-106 14. PrzybyłowiczP.M., 2005,Stability of activelycontrolledrotating shaftmade of functionally gradedmaterial, Journal of Theoretical and Applied Mechanics, 43, 609-630 15. Rogers C.A., Fuller C.R., Liang C., 1990, Active control of sound radia- tion from panels using embedded shapememory alloy fibers, Journal of Sound and Vibration, 136, 164-170 16. SongO., Librescu L., 1990,Anisotropy and structural coupling on vibration and instability of spinning thin-walledbeams,J. SoundVibration,204, 477-494 17. Tylikowski A., 1981, Dynamic stability of rotating shafts, Ingenieur Archiv, 50, 41-48 18. Tylikowski A., 1996, Dynamic stability of rotating composite shafts,Mecha- nics Research Communication, 23, 2, 175-180 19. Tylikowski A., 2005, Stabilization of plate parametric vibrations via distri- buted control, Journal of Theoretical and Applied Mechanics, 43, 695-706 20. Tylikowski A., 2007, Semiactive control of a shape memory alloy composite shaft under a torsional load,Machine Dynamics Problems, 31, 121-132 21. Tylikowski A., 2008, Dynamic stability of weak equations of rectangular plates, Journal of Theoretical and Applied Mechanics, 46, 679-692 22. Whitney J.M., 1987, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing, Lancaster-Basel Słabe sformułowanie stabilności hybrydowych obracających się wałów swobodnie podpartych i sztywnie utwierdzonych Streszczenie W pracy rozszerzono możliwości analizy stabilności układów ciągłych na obra- cający się wał hybrydowy poddany czasowo zmiennej sile osiowej przy osłabionych założeniach spełnianych przez rozwiązania. Kompozytowy wał hybrydowy obracają- cy się ze stałąprędkościąkątową traktowany jest jako cienkościennabelka zawierającą Stability of hybrid rotating shaft... 1007 obok klasycznychwłókien równieżwłóknawykonane zmateriału z pamięcią kształtu. Słabą postać równań ruchu wału wyprowadzono z zasady Hamiltona. Rozpatrzony jest wpływ aktywacji termicznej na obszar stabilości wału przy założeniu nie tylko swobodnego podparcia obu końcówwału, lecz również przy podparciu utwierdzonym i mieszanym. Podczas wyprowadzania warunków stabilności korzysta się z badania właściwości funkcjonału Lapunowa wzdłuż rozwiązania słabych równań ruchu wału. Wyprowadzono jawną postać warunków stabilności. Manuscript received May 19, 2008; accepted for print October 7, 2008