Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 2, pp. 323-340, Warsaw 2003 DYNAMIC BEHAVIOUR OF AN AXIALLY MOVING MULTI-LAYERED WEB Krzysztof Marynowski Department of Dynamics of Machines, Technical University of Łódź Zbigniew Kołakowski Department of Strenght of Materials and Structures, Technical University of Łódź e-mail: kola@ck-sg.p.lodz.pl A new approach to the analysis of the dynamic behaviour of an axially mo- ving composite web is presented. The mathematical model of the moving web system constitutes non-linear, coupled equations governing the trans- verse displacement and stress function. The calculations are carried out for two paperboard structures. The results of numerical investigations show so- lutions to linearized problems.The effect of paperboardproperties and axial transport velocity on flexural and torsional vibrations are presented. Key words: composite plate, moving web, dynamic stability Notation b – width of the web c – axial transport speed Ex – Young’s modulus along the x-direction Ey – Young’s modulus along the y-direction G – shear modulus of the plate material h – thickness of the web K – stiffness matrix for the laminate l – length of the web N – axial stress S – surface of the web t – time 324 K.Marynowski, Z.Kołakowski u,v,w – components of the plate displacement in the x, y and z axis direction, respectively U – kinetic energy V – internal elastic strain energy W – work of external forces x,y,z – Cartesian co-ordinates εx,εy,εz – strain components κx,κy,κz – curvature modifications and torsion components λ – scalar load parameter ρ – mass density of the web σ – real part of eigenvalue ω – natural frequency. 1. Introduction Axially moving webs in the form of thin, flat rectangular shape materials with small flexural stiffness occur in the industry as band saw blades, power transmission belts, textile tapes or paper and paperboard webs. Excessive vibrations of moving webs increase defects and can lead to failure of the web. In thepaperandtextile industries involvingmotionof thinmaterials, the stress analysis in the moving web is essential for the control of wrinkle, flutter and sheet break. Although the mechanical behaviour of axially moving materials has been studied formany years, little information is available on the dynamic behaviour and stress distribution in an axiallymovingmulti-layered paper and boardmaterials. Paper and paperboard properties derive from raw materials and paper- making processes. Relatively recent technical developments allow high speed formation of the web in a simultaneous or sequential multi-layered structure like paperboard. Paper is generally considered to be an anisotropic fibrous composite material. Theoretical models describing mechanical properties of paperboard include those based on an thin-walled composite plate structure. A lot of earlier works in this field focussed on investigations of stationary orthotropic composite plates. A more comprehensive review of the literature can be found in investigations of Chandra and Raju (1973), Dawe andWang (1994), Jones (1975), Kołakowski and Królak (1995), Loughlan (2001), Mat- sunaga (2001), Shen and Williams (1993), Walker et al. (1996), Wang et al. (1987). On the other hand, one can find in the literature a lot of works on the dynamic investigations of a one-layered axially moving orthotropic web. Re- Dynamic behaviour of an axially moving multi-layered web 325 cent works in this field analysed non-linear vibrations of an axially moving orthotropicweb (Marynowski andKołakowski, 1999;Marynowski, 1999), equ- ilibriumdisplacement and stress distribution in non-linearmodel of an axially moving plate (Lin, 1997), wrinkling phenomenon and stability of the linear model of an axially moving isotropic plate (Lin and Mote, 1996), and stress distribution in an axially moving plate (Wang, 1999). The aim of this paper is to analyse the dynamic behaviour of an axially moving multi-layered web. Numerical investigations are carried out for a pa- perboardstructure.Paperboard is treatedas a thin-walled composite structure in the elastic range, being under axial extension. The differential equations of motion are derived from the Hamilton principle taking into account the La- grangedescription, the strainGreen tensor for thin-walled plates and the stress tensor. Singular perturbation theory is used to obtain the approximate solu- tion of thegoverning equations.Thenumericalmethodof the transitionmatrix using Godunov’s orthogonalization is used to solve the linearized problems. 2. Formulation of the structural problem Paperboard is in general composed of several pulp fiber sheets bonded by starch or an adhesive material, and is usually a multi-layered structure. A schematic of a three-layered paper-board macrostructure is shown in Fig.1, which also depicts the coordinate system. The x-axis refers to the machine direction, the y-axis refers to the cross-section or transverse direction. The machine and cross directions form the plane of the structure, and the z-axis refers to the out-of-plane (or through-thickness) direction. In this paper, the laminatemodel has been used to describe themechanical behaviour of paper- board. Fig. 1. Schematic of paperboardmacrostructure Let us consider amulti-layer plate element of a thin-walled structuremade of orthotropic materials. The classical laminated plate theory (Jones, 1975) is 326 K.Marynowski, Z.Kołakowski used in the theoretical analysis, the effects of shear deformation through the thickness of the laminate are neglected and the results given are those for thin laminated plates. Thematerials they are made of obey Hooke’s law. For each plate component, precise geometrical relationships are assumed in order to enable the consideration of both out-of-plane and in-plane bending of the plate ε1 =u1,1+ 1 2 um,1um,1 ε2 =u2,2+ 1 2 um,2um,2 ε3 =u1,2+u2,1+um,1um,2 ε4 =−hu3,11 ε5 =−hu3,22 ε6 =−2hu3,12 (2.1) where m = 1,2,3, and ε1 = εx, ε2 = εy, ε3 = 2εxy = γxy, ε4 = hκx, ε5 = hκy, ε6 = hκxy; u1 ≡ u, u2 ≡ v, u3 ≡ w are the components of the displacement vector in the x1 ≡x, x2 ≡ y, x3 ≡ z axis direction, respectively. In themajority of publications devoted to stability of structures, the terms (u21,1 + u 2 2,1), (u 2 1,2 + u 2 2,2), (u1,1u1,2 + u2,1u2,2) are neglected for ε1, ε2, ε3, respectively, in the strain tensor components (2.1). Using the classical plate theory (Jones, 1975), the constitutive equation for the laminate is taken as follows N = [ A B B D ] ε=Kε (2.2) where Aij = 1 h N ∑ k=1 (Qij)k(zk −zk−1) Bij = 1 2h2 N ∑ k=1 (Qij)k(z 2 k −z 2 k−1) Dij = 1 3h3 N ∑ k=1 (Qij)k(z 3 k −z 3 k−1) (2.3) K= [ A B B D ] =          A11 A12 A16 B11 B12 B16 A21 A22 A26 B21 B22 B26 A61 A62 A66 B61 B62 B66 B11 B12 B16 D11 D12 D16 B21 B22 B26 D21 D22 D26 B61 B62 B66 D61 D62 D66          = =          K11 K12 K13 K14 K15 K16 K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36 K41 K42 K43 K44 K45 K46 K51 K52 K53 K54 K55 K56 K61 K62 K63 K64 K65 K66          Dynamic behaviour of an axially moving multi-layered web 327 in which Aij =Aji,Bij =Bji Dij =Dji,Kij =Kji, and Qij is the transfor- med reduced stiffness matrix (Dawe andWang, 1994; Jones, 1975). In the above equations the dimensionless sectional forces N1,N2,N3 and the dimensionless sectionalmoments N4,N5,N6 appear in the following forms N1 = Nx h N2 = Ny h N3 = Nxy h N4 = Mx h2 N5 = My h2 N6 = Mxy h2 (2.4) In the constitutive matrix of Eq. (2.2), the submatrix A, detailed in Eq. (2.3)1 and related to the in-plane response of the laminate, is called the exten- sional stiffness. The submatrix D, described by Eq. (2.3)3, is associated with the out-of-plane bending response of the laminate and is called the bending stiffness,whereas the submatrix B, illustrated byEq. (2.3)2, is ameasure of an interaction (coupling) between themembrane and the bendingaction. Thus, it is impossible to pull on a laminate that has Bij termswithout bending and/or twisting the laminate at the same time. This entails that the extensional force results in not only extensional deformations, but also in twisting and/or ben- ding of the laminate (Dawe and Wang, 1994; Jones, 1975). Moreover, such a laminate cannot be subjected to a moment without suffering simultaneously from extension of the middle surface. Fig. 2. Axially moving web Let suppose now that the multi-layered web of the length l is considered. The web moves at the constant velocity c in the x direction. The geometry of the considered model is shown in Fig.2. The equations of the dynamic stability of a moving composite structure have been derived using the Hamilton principle, which, for the web, can be written as 328 K.Marynowski, Z.Kołakowski δ t1 ∫ t0 Ldt= t1 ∫ t0 (δU −δV + δW) dt= (2.5) = t1 ∫ t0 {1 2 ρ ∫ Ω δ[(c+u1,t+ cu1,1) 2+(u2,t+ cu2,1) 2+(u3,t+ cu3,1) 2] dΩ− − 1 2 ∫ Ω (σxδεx+σyδεy + τxyδγxy) dΩ+ b ∫ 0 hp0(x2)δu1 dx2 ∣ ∣ ∣ x1=ℓ x1=0 } dt=0 where L is the Lagrange function, Ω = l× b×h= S×h, and p0(x2) is the pre-critical external load in the plate middle surface. After grouping the components at respective variations, the following sys- tem of equilibrium equations of motion has been obtained t1 ∫ t0 ∫ S {[N1(1+u1,1)+N3u1,2],1+[N2u1,2+N3(1+u1,1)],2+ +ρ(−u1,tt−2cu1,1t− c 2u1,11)}δu1 dSdt=0 t1 ∫ t0 ∫ S {[N1u2,1+N3(1+u2,2)],1+[N2(1+u2,2)+N3u2,1],2+ (2.6) +ρ(−u2,tt−2cu2,1t− c 2u2,11)}δu2 dSdt=0 t1 ∫ t0 ∫ S [(hN4,1+N1u3,1+N3u3,2),1+(hN5,2+2hN6,1+N2u3,2+N3u3,1),2+ +ρ(−u3,tt−2cu3,1t− c 2u3,11)]δu3 dSdt=0 where ρ= 1 h N ∑ k=1 ρk(zk−zk−1) The boundary conditions for x1 = const t1 ∫ t0 b ∫ 0 [ ρ(c2+ cu1,t+ c 2u1,1)− −(N1+N1u1,1+N3u1,2−hp 0(x2)) ] δu1 dx2dt ∣ ∣ ∣ x1 =0 Dynamic behaviour of an axially moving multi-layered web 329 t1 ∫ t0 b ∫ 0 [ ρ(cu2,t+ c 2u2,1)− (N3+N1u2,1+N3u2,2) ] δu2 dx2dt ∣ ∣ ∣ x1 =0 (2.7) t1 ∫ t0 b ∫ 0 N4δu3,1 dx2dt ∣ ∣ ∣ x1 =0 t1 ∫ t0 b ∫ 0 [ ρ(cu3,t+ c 2u3,1)− (hN4,1+2hN6,2+N1u3,1+N3u3,2) ] δu3 dx2dt ∣ ∣ ∣ x1 =0 The boundary conditions for x2 = const t1 ∫ t0 l ∫ 0 (N2+N2u2,2+N3u2,1)δu2 dx1dt ∣ ∣ ∣ x2 =0 t1 ∫ t0 l ∫ 0 (N3+N2u1,2+N3u1,1)δu1 dx1dt ∣ ∣ ∣ x2 =0 (2.8) t1 ∫ t0 l ∫ 0 N5δu3,2 dx1dt ∣ ∣ ∣ x2 =0 t1 ∫ t0 ℓ ∫ 0 (hN5,2+2hN6,1+N2u3,2+N3u3,1)δu3 dx1dt ∣ ∣ ∣ x2 =0 The boundary conditions for the plate corner (x1 = const and x2 = const) t1 ∫ t0 2N6 ∣ ∣ ∣ x1 ∣ ∣ ∣ x2 δu3 dt=0 (2.9) The initial conditions l ∫ 0 b ∫ 0 [ρ(c+u1,t+ cu1,1)]δu1 dx1dx2 ∣ ∣ ∣ t=0 =0 l ∫ 0 b ∫ 0 [ρ(u2,t+ cu2,1)]δu2 dx1dx2 ∣ ∣ ∣ t=0 =0 (2.10) l ∫ 0 b ∫ 0 [ρ(u3,t+ cu3,1)]δu3dx1dx2 ∣ ∣ ∣ t=0 =0 330 K.Marynowski, Z.Kołakowski 3. Solution to the problem The problem of the dynamic stability has been solved by making use of the asymptotic perturbation method. In the solution and in the developed computer program, the following have been employed: Byskov-Hutchinson’s asymptotic expansion (Byskov and Hutchinson, 1977), the numerical transi- tion matrix method using Godunov’s orthogonalization method (Kołakowski and Królak, 1995). As has beenmentioned above, the fields of displacements U and the fields of sectional forces N have been expanded into power series with respect to the dimensionless amplitude of the web deflection ζn (the amplitude of the nth free vibration frequency of the extension system divided by the thickness h1 of the web assumed to be the first one) U =λU (0) k + ζnU (n) k + ... (3.1) N =λN (0) k + ζnN (n) k + ... where U (0) k ,N (0) k – pre-critical static fields U (n) k ,N (n) k – first order fields for the composite kth web. After substitution of expansions (3.1) into equilibriumEqs. (2.6), continu- ity conditions and boundary conditions Eqs. (2.7)-(2.9), the boundary value problems of the zero and first order can been obtained. The zero approximation describes the pre-critical static state, whereas the first order approximation, being the linear problem of the dynamic stability, allows for the determination of the eigenvalues, eigenvectors and the critical speeds of the system. The panels with linearly varying pre-critical loads along their widths are divided into several strips under uniformly distributed tensile stresses. Instead of the finite strip method, the exact transition matrix method is used in this case (Marynowski andKołakowski, 1999). The inertial forces corresponding to the in-plane displacements u and v are neglected. The pre-critical solution of the kth composite web consisting of homogeneous fields is assumed as u (0) 1k =− (ℓ 2 −x1k ) ∆k u (0) 2k =−x2k∆k K12k K22k u (0) 3k =0 (3.2) where ∆k is the actual loading. This loading is specified as the product of a unit loading and a scalar load factor ∆k. Dynamic behaviour of an axially moving multi-layered web 331 The inner sectional forces of the pre-critical static state for the assumed homogeneous field of displacements (3.1) are expressed by the following rela- tionships N (0) 1k =− ( K11− K212 K22 ) ∆ N (0) 4k =− ( K41−K42 K21 K22 ) ∆ N (0) 2k =0 N (0) 5k =− ( K51−K52 K21 K22 ) ∆ N (0) 3k =− ( K31−K32 K21 K22 ) ∆ N (0) 6k =− ( K61−K62 K21 K22 ) ∆ (3.3) In the one before last dynamical component of Eg. (2.6)3 there is a deriva- tive with respect to x1 and t. Because of the incompatibility of trigonometric functions in the x1 ≡x-direction, theGalerkin-Bubnov orthogonalization pro- cedure is used to find an approximating solution to this equation. Numerical aspects of the problem being solved for the first order fields (Marynowski and Kołakowski, 1999) have resulted in an introduction of the following new orthogonal functions with the nth harmonic for kth composite web in the sense of the boundary conditions for two longitudinal edges a (n) k =N (n) 2k ( 1+λ∆k K21k K22k ) +λN (0) 3k d (n) k,ξ bk d (n) k =u (n) 21k b (n) k =N (n) 3k (1−λ∆k)+λN (0) 3k c (n) k,ξ bk e (n) k =u (n) 3k c (n) k =u (n) 1k f (n) k = u (n) 3k,η bk = e (n) k,η bk h (n) k =hk g (n) k,η bk +2hk N (n) 6k,ξ bk +λN (0) 3k e (n) k,ξ bk g (n) k =N (n) 5k (3.4) where ξk = x1k bk ηk = x2k bk The solutions to Eq. (2.6)3 corresponding to the free support at the seg- ment ends can be written in the following form 332 K.Marynowski, Z.Kołakowski ak = N ∑ n=1 Tn(t)A (n) k (ηk)sinβ bk = N ∑ n=1 Tn(t)B (n) k (ηk)cosβ ck = N ∑ n=1 Tn(t)C (n) k (ηk)cosβ dk = N ∑ n=1 Tn(t)D (n) k (ηk)sinβ ek = N ∑ n=1 Tn(t)E (n) k (ηk)sinβ fk = N ∑ n=1 Tn(t)F (n) k (ηk)sinβ gk = N ∑ n=1 Tn(t)G (n) k (ηk)sinβ hk = N ∑ n=1 Tn(t)H (n) k (ηk)sinβ (3.5) where β= nπbkξ ℓ and Tn(t) is an unknown function of time, and the eigenfunctions have been determined for the non-moving tensioned web (for c=0). The eigenfunctions A (n) k ,B (n) k ,C (n) k ,D (n) k ,E (n) k ,F (n) k ,G (n) k ,H (n) k (with the nth harmonic) are initially unknown functions that will be determined by the numerical method of transition matrices. The obtained system of homogene- ous ordinary differential equations has been solved by the transition matrix method, having integrated numerically the equilibrium equations along the circumferential direction in order to obtain the relationships between the sta- te vectors on two longitudinal edges. During the integration of the equations, Godunov’s orthogonalization method is employed (Kołakowski and Królak, 1995; Marynowski, 1999). The presented way of solution allows for carrying out amodal dynamic analysis of complex composite webs. In system of Eq. (2.6)3 for the non-moving, tensioned web, there are two components of the pre-critical loading N (0) 1k and N (0) 3k . The component N (0) 3k has an insignificant effect on the value of the critical load in comparison with N (0) 1k . Let us return to the case of amoving compositeweb (i.e. for c 6=0).Becau- se of the incompatibility of trigonometric functions in the x1-direction, after substituting Eq. (3.4) into Eq. (2.6)3 the Galerkin-Bubnov orthogonalization methodhas beenused. In thisway, the set of N ordinarydifferential equations with respect to the function Tn(t) can be determined in the following form d2Tm dt2 a2m+ N ∑ n=1 dTn dt a1nm+Tma0m =0 m=1,2, ...,N (3.6) Substituting new variables into Eq. (3.5) one can receive an autonomous set of 2N first order differential equations with respect to time. On the basis Dynamic behaviour of an axially moving multi-layered web 333 of Eq. (3.6) one can determine eigenvalues, eigenvectors and critical speeds of the moving system. 4. Numerical results and discussion Numerical investigations were carried out on the basis of the mathema- tical model, which was presented in the previous sections. First, the calcula- tions results are comparedwith the available solutions. The comparison of the dimensionless fundamental natural frequencies (ωb2 √ ρ/(Eyh2)) of a simply supported anti-symmetric angle-ply laminate, obtained in the present study and by Jones (1975) is shown in Fig.3. The following dimensionless numerical data was used: Ex/Ey = 40, Gxy/Ey = 0.5, νxy = 0.25. The greatest di- screpancies in the compared results can be observed for a two-layer laminate. When the number of layers increases there is little and little discrepancy in the compared values. Fig. 3. Comparison of numerical results, θ – play angle, (—–) – present study, ( – – ) – Jones (1975) The numerical investigations were carried out for a moving paperboard web. The numerical results of the paperboard obtained from the experimental investigations are available in the recent literature (Xia et al., 2002). To reco- gnize the paperboard layers properties, which are not available in approacha- 334 K.Marynowski, Z.Kołakowski ble literature, an identification procedure has been employed. Known natural frequencies of a simply supported homogenous paperboard plate model have been compared with analogous values of a three-layered paperboard structure (Fig.1). Additionally, the outer chemical pulp layers stiffer and stronger than the inner mechanical pulp layer (E1M = E2M = E2C/100) was assumed in the identification procedure (Xia et al., 2002). The identification results as the data for the numerical study are shown in Table 1. Table 1.Numerical data System I System II Length of the web (l) 1 m Width of the web (b) 0.2 m Thickness of the chemical pulp layer (hC) 3.89 ·10 −4 m 5.83 ·10−4 m Thickness of the mechanical pulp layer (hM) 2.22 ·10 −4 m 3.34 ·10−4 m Total thickness of the web 1 ·10−3 m 1.5 ·10−3 m Young’s modulus along the x-axis of the chemical pulp layer (E1C) 5.6 GPa Young’s modulus along the y-axis of the chemical pulp layer (E2C) 2 GPa Poisson’s ratio in the machine direction of the chemical pulp layer (ν12C) 0.392 Poisson’s ratio in the cross direction of the chemical pulp layer (ν21C) 0.14 Shear modulus of the chemical pulp layer (GC) 34MPa Young’s modulus along the machine direction and along the cross direction of the mechanical pulp layer (E1M =E2M) 20MPa Poisson’s ratio along the machine direction and along the cross direction of the mechanical pulp layer (ν12M = ν21M) 0.266 Shear modulus of the mechanical pulp layer (GM) 7.782MPa Mass density (ρ) 133.33 kg/m3 Initial stress (N (0) x ) 55 N/m The calculations were carried out for two kinds of paperboard web struc- tures of different thickness. The effect of the paperboard properties and axial transport velocity on the transverse and torsional vibrations have been studied in numerical investigations. Let σ and ω denote the real part and imaginary part of the eigenvalues, respectively. Simultaneously, ω is the natural frequen- cy of the web. The positive value of σ indicates instability of the considered system. Dynamic behaviour of an axially moving multi-layered web 335 At first, eigenvalues of all investigated stationary systems well calculated. Fig.4 shows the values of the three lowest flexural andflexural-torsional eigen- frequencies of the considered paperboard systems for c=0. Fig. 4. Three lowest flexural (ω11, ...,ω31) and flexural-torsional (ω12, ...,ω32) eigenfrequencies for c=0 Thedynamic investigations of the axiallymoving systemswere begun from the vibration modes definition. Fig.5 shows the modes of the two lowest fle- xural (ω11 and ω21) and two lowest flexural-torsional (ω12 and ω22) eigenfre- quencies of the considered web systems. The results of the dynamic stability investigations of web system I for N = 10 in Eq. (3.6) are shown in Fig.6, Fig.7 and Fig.8. The values of the imaginary part (solid line) and the real part (dotted line) of the three lowest flexural natural frequencies versus the transport velocity are shown in Fig.6. Fig.7 andFig.8 show analogical plots of the flexural and flexural-torsional eigenfrequencies. Both plots in Fig.5 and Fig.6 show that in the undercritical region of transport speeds the lowest natural frequencies decrease during the increase in the axial velocity. At the critical transport speed the fundamental eigenfrequency vanishes indicating in divergence instability (the fundamental mode with non-zero σ and zero ω). For supercritical transport speeds (c > ccr), the web experiences at first the divergent instability, and above that there is the second stability area where σ=0. In paperboard web system I this region is very narrow (Fig.8). Its appearance and position is strictly connected with the distribution of the 336 K.Marynowski, Z.Kołakowski Fig. 5. Non-trivial equilibrium positions of axially moving web: (a) ω11, (b) ω21, (c) ω12, (d) ω22 Fig. 6. Flexural natural frequencies (paperboard web system I) Dynamic behaviour of an axially moving multi-layered web 337 Fig. 7. Flexural and flexural-torsional natural frequencies (paperboardweb system I) Fig. 8. Flexural and flexural-torsional natural frequencies in the supercritical region (web system I) 338 K.Marynowski, Z.Kołakowski flexural and flexural-torsional eigenfrequencies. In the case of paperboardweb system II, the second stable area does not appear at all (Fig.9). Fig. 9. Flexural and flexural-torsional natural frequencies in the supercritical region (web system II) Fig.9 shows a plot of the two lowest flexural and flexural-torsional natu- ral frequencies of web system II in the supercritical region of the transport velocity. The critical transport speed of system II is smaller than in system I. In web system II the distance between the critical transport speed of the fle- xural vibrations (ccrf) and flexural-torsional vibrations (ccrt) is greater than in system I, and the second stable area does not appear. Above the diver- gence instability area there appears a flutter instability region of supercritical transport speeds. 5. Conclusions In the paper, a new approach to the analysis of the dynamic behaviour of an axially movingmulti-layered web is presented. Themathematical model of the moving web system is derived on the basis of the asymptotic perturba- tionmethod for the classical laminated thin-walled plate theory. The solution is based on the numerical method of the transition matrix using Godunov’s orthogonalization. Dynamic behaviour of an axially moving multi-layered web 339 Thenumerical calculations are carried out for twokinds of paperboardweb structures. The axially moving paperboard web is treated as a thin-walled composite structure in the elastic range, being under axial extension. The numerical data of the investigated paperboard structure have been received from experimental investigations addressed in literature. For a constant axial tension of the web, the calculation results show that in the subcritical region of transport speed the lowest flexural and flexural- torsional natural frequencies decrease with growth of the axial velocity. At the critical transport speed the fundamental flexural eigenfrequency vanishes indicating the divergent instability. The critical transport speed of the system is dependent on the thickness of the paperboard layers. The critical speed value decreases when the thickness of the layers increases. In the supercritical regionof transport speedabove thedivergent instability area the second stable region may appear. The width and position of the second stable region is strictly connected with the distribution of the flexural andflexural-torsional eigenfrequencies.Above thedivergent instability and the second stability region there occurs a flutter instability area of supercritical transport speeds. References 1. Byskov E., Hutchinson J.W., 1977, Mode interaction in axially stiffened cylindrical shells,AIAA J., 15, 7, 941-948 2. Chandra R., Raju B.B., 1973, Postbuckling analysis of rectangular ortoth- ropic plates, Int. Journal of Mechanical Science, 16, 81-89 3. DaweD.J.,WangS., 1994,Buckling of composite plates and plate structures using the spline finite strip method,Composites Engineering, 4, 11, 1099-1117 4. 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Wang C., Pian T.H.H., Dugundji J., Lagace P.A., 1987, Analitycal and experimental studies on the buckling of laminated thin-walled structures,Proc. AIAA/ASME/ASCE/AHS 28th Structures, Struct. Dyn. and Materials Conf., Part 1, 135-140 16. Wang X., 1999, Numerical analysis of moving orthotropic thin plates, Com- puters and Structures, 70, 467-486 17. Xia Q., Boyce M., Parks D., 2002, A constitutive model for the anisotro- pic elastic-plastic deformation of paper and paperboard, Int. Journal of Solid Structures, 39, 4053-4071 Zachowanie dynamiczne poruszającej się osiowo wielowarstwowej wstęgi Streszczenie Wpracy przedstawiononowąmetodę analizy dynamicznej przesuwającej się osio- wo kompozytowej wstęgi. Model matematyczny badanego układu wyznaczają nieli- niowe, sprzężone równania przemieszczeń poprzecznych oraz funkcji naprężeń. Bada- nia numeryczne przeprowadzono dla dwóch rodzajów wstęgi kartonu. Wyniki tych badań pokazują rozwiązanie problemu liniowego badanego zagadnienia. Przedstawio- no wpływ własności kompozytu oraz prędkości przesuwu wstęgi na drgania giętne i giętno-skrętne układu. Manuscript received October 7, 2002; accepted for print January 17, 2003