Dynamic Stability of an Axially Moving Sandwich Composite Web

Mechanics and Mechanical Engineering

Vol. 7, No. 1 (2004) 53-68

c© Technical University of Lodz

Dynamic Stability of an Axially Moving Sandwich Composite Web

Krzysztof MARYNOWSKI

Department of Dynamics of Machines

Zbigniew KOÃLAKOWSKI

Department of Strength of Materials and StructuresTechnical University of ÃLodz, 90-924 ÃLodz, ul Stefanowskiego 1/15, Poland

Received (19 August 2003)

Revised (22 September 2003)

Accepted (1 December 2003)

Dynamic stability of an axially moving corrugated board web has been investigated. Theboard web is modelled as an thin-walled composite plate structure. Mathematical modelof the moving web system has been derived using the classical laminated panel theory.In the solution of the problem the Koiter’s asymptotic expansion and the numericaltransition matrix method have been employed: The results of numerical investigationsshow the solutions to the linearized problem. The effects of the transport velocity andaxial load of the web on its dynamic stability are presented.

Keywords: thin wall structures, moving web, composite materials.

1. Introduction

Since their appearance in the 1960’s, the use of modern composite materials isgrowing rapidly around the world. Composites have been lately commonly usedin aircraft industry, army, automotive industry and marine structures, as well asin packaging. With progress in the technology of composite materials, compositeplates made of solid laminates, sandwich laminates, and laminates reinforced withstiffeners are widely used in packages, cars, wagons, ships, aeroplanes and marinestructures. The positive factors promoting the use of fibrous composites in theconstruction of load carrying elements include good wear (e.g. friction) resistance,fatigue life (under time-variable loads), sound insulation properties or electromag-netic transparency.

Corrugated board is a highly efficient composite material. It is used extensivelyas a packaging material because of its high strength and stiffness properties. Theseproperties allow it to be manufactured into boxes and trays for transporting, storingand distributing a wide range of consumer products.

54 Dynamic Stability of an Axially Moving Sandwich Composite Web

Paper and corrugated board properties derive from the raw materials and pa-permaking processes. Relatively recent technical developments allow high speedformation of the web in simultaneous or sequential multi-layered structure like pa-perboard. Paper is generally considered to be an anisotropic fibrous compositematerial. Theoretical models describing mechanical properties of paperboard in-cluding those based on an thin-walled orthotropic plate structure.

Excessive vibrations of moving board webs in the industry increase the defectsand can lead to failure of the web. In the paper and textile industries involvingmotion of thin materials, stress analysis in the moving web is essential for the controlof wrinkle, flutter and sheet break. Although the mechanical behaviour of axiallymoving materials has been studied for many years, little information is available onthe dynamic behaviour and stress distribution in the axially moving multi-layeredpaper and board materials.

A lot of earlier works in this field focussed on investigations of stationary or-thotropic composite plates. A more comprehensive review of the literature can befound in investigations of Dawe and Wang [3], Jones [4], Kolakowski and Krolak [7],Loughlan [10, 11], Matsunga [12], Walker et al. [17], Wang et al. [18].

On the other hand, in literature one can find a lot of works on dynamic investi-gations of axially moving orthotropic web, i.e. one-layered systems. Recent worksin this field analysed the non-linear vibrations of an axially moving orthotropic web[13, 14], the equilibrium displacement and stress distribution in non-linear modelof an axially moving plate [9], the wrinkling phenomenon and stability of the linearmodel of an axially moving isotropic plate [8], and stress distribution in an axiallymoving plate [19].

The aim of this paper is to analyse the dynamic stability of an axially movingsandwich composite web. Mathematical model of the moving web system has beenderived using the classical laminated panel theory. In the solution of the problemthe Koiter’s asymptotic expansion and the numerical transition matrix methodhave been employed: Numerical investigations are carried out for a corrugatedboard composite structure. The numerical data of the board material received onthe base of experimental investigations have been taken from the recent literature.Paperboard is treated as thin-walled composite structure in the elastic range, beingunder axial extension. The effects of the transport velocity and axial load of theweb on its dynamic stability are presented in this paper.

2. Formulation of the problem

Three-layered composites consist of two thin facings (the skin or sheets) sandwichinga core (Fig. 1). The facings are made of high strength materials having goodproperties under tension while the siding core in the form of corrugated trapezoidalplate is made of lightweight materials. Sandwich composites combine lightness andflexural stiffness. Fig. 1 also depicts the coordinate system. The x-axis refers tothe machine direction, the y-axis refers to the cross or transverse direction. Themachine and cross directions form the plane of the structure, and z-axis refers tothe out-of-plane (or through- thickness) direction.

Let us consider thin-walled web built of plate elements (panels). The web underconsideration is multi-layer plate made of orthotropic materials. The composite

Marynowski, K and KoÃlakowski, Z 55

Figure 1 Schematic of the sandwich composite macrostructure

has been modeled as the thin–walled laminar construction with rigid core. Theclassical laminated panel theory [4] is used in the theoretical analysis, the effectsof shear deformation through the thickness of the laminate are neglected and theresults given are those for thin laminated panels. The materials they are made ofare subjected to Hooke’s law.

For each panel component, precise geometrical relationships are assumed in orderto enable the consideration of both out-of-plane and in-plane bending of the plate:

ε1 = u1,1 + 0.5um,1um,1

ε2 = u2,2 + 0.5um,2um,2

ε3 = u1,2 + u2,1 + um,1um,2

ε4 = −hu3,11; ε5 = −hu3,22; ε6 = −2hu3,12

(1)

where: h is the thickness of the plate, u1 ≡ u, u2 ≡ v, u3 ≡ w – the componentsof the displacement vector in the x1 ≡ x, x2 ≡ y, x3 ≡ z axis direction,respectively, and ε1 = εx, ε2 = εy, ε3 = 2εxy = γxy, ε4 = hκx, ε5 = hκy, ε6 = hκxy.The summation with respect to the factor m is from 1 to 3 (m = 1, 2, 3).

Using the classical plate theory [4, 6], the constitutive equation for the laminateis taken as follows:

N =

[

[A] [B][B] [D]

]

ε = [K] ε (2)

where:

Aij =1

h

N∑

k=1

(Qij)k(zk − zk−1) (3)

Bij =1

2h2

N∑

k=1

(Qij)k(z2k − z2

k−1) (4)

Dij =1

3h3

N∑

k=1

(Qij)k(z3k − z3

k−1) (5)

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

(6)


=

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

in which Aij = Aji, Bij = Bji, Dij = Dji, Kij = Kji and Qij is the transformedreduced stiffness matrix.

In the above equations N1, N2, N3 are the dimensionless sectional forces andN4, N5, N6 – the dimensionless sectional moments:

N1 =Nx

E0h; N2 =

Ny

E0h; N3 =

Nxy

E0h; N4 =

Mx

E0h2; N5 =

My

E0h2; N6 =

Mxy

E0h2, (7)

where E0 is the elastic modulus of reference.

The reverse relation with respect to (2) can be written as:

ε = [K]−1

N =[

K]

N (8)

In the constitutive matrix of Eq. (2), the submatrix [A], detailed in Eq. (3) andrelated to the in-plane response of the laminate, is called extensional stiffness. Thesubmatrix [D], described by Eq. (4), is associated with the out-of-plane bendingresponse of the laminate and is called bending stiffness, whereas the submatrix[B], illustrated by Eq. (3), is a measure of an interaction (coupling) between themembrane and the bending action. Thus, it is impossible to pull on a laminatethat has Bij terms without bending and/or twisting the laminate at the sametime. That is, an extensional force results in not only extensional deformations, butalso twisting and/or bending of the laminate Moreover, such a laminate cannot besubjected to moment without suffering simultaneously from extension of the middlesurface [3, 4, 6].

y ,v

c b

z ,w

x ,u

l

h

Figure 2 Axially moving web

Let suppose now that the multi-layered web of the length l is considered. Theweb moves at the constant velocity c in the x direction. The geometry of the con-sidered model is shown in Fig. 2. The equations of dynamic stability of the movingcomposite structure have been derived using the Hamilton’s principle. Actual tra-jectories differ from the other that satisfy the Hamilton’s principle for the web can


be written as:

δt1∫

t0

Ldt =t1∫

t0

(δU − δV + δW ) dt =

t1∫

t0

0.5∫

Ω

ρδ[

(c + u1,t + cu1,1)2

+ (u2,t + cu2,1)2

+ (u3,t + cu3,1)2]

dΩ

−∫

Ω

(σ1δε1 + σ2δε2 + σ3δε3) dΩ +b∫

0

hp0 (x2) δu1dx2|x1=`x1=0

dt = 0

(9)

where: U – kinetic energy, V – internal elastic strain energy, W – work of externalforces, p0(x2) – external load in the plate middle surface. The expression Ω =` × b × h = S × h has been employed in the above relation.

In order to determine the variation of a single multi-layer panel, the followingidentity has been taken

XδY = δ(XY ) − Y δX . (10)

After grouping the components at respective variations, the following system ofequations of motion has been obtained:

t‘∫

t0

∫

S

[N1(1 + u1,1) + N3u1,2],1 + [N2u1,2 + N3(1 + u1,1)],2 +

ρ(

−u1,tt − 2cu1,1t − c2u1,11

)

δu1dSdt = 0

t‘∫

t0

∫

S

[N1u2,1 + N3(1 + u2,2)],1 + [N2(1 + u2,2) + N3u2,1],2 +

ρ(

−u2,tt − 2cu2,1t − c2u2,11

)

δu2dSdt = 0

t‘∫

t0

∫

S

[

(hN4,1 + N1u3,1 + N3u3,2),1 + (hN5,2 + 2hN6,1 + N2u3,2 + N3u3,1),2 +

ρ(

−u3,tt − 2cu3,1t − c2u3,11

)]

δu3dSdt = 0

(11)

where:

ρ =1

h

N∑

k=1

ρk(zk −zk−1) ;

t∫

t0

∫

S

[

εi − KijNj

]

δNidSdt = 0 , i, j = 1, 2, ...6 . (12)


The boundary conditions for x1=const:

t1∫

t0

b∫

0

ρ(

c2 + cu1,t + c2u1,1

)

−[

N1 + N1u1,1 + N3u1,2 − hp0(x2)]

δu1dx2dt |x1= 0

t1∫

t0

b∫

0

ρ(

c u2,t + c2u2,1

)

− [N3 + N1u2,1 + N3u2,2]

δu2dx2dt |x1= 0

t1∫

t0

b∫

0

N4δu3,1dx2dt |x1= 0

t1∫

t0

b∫

0

ρ(

c u3,t + c2u3,1

)

− (hN4,1 + 2hN6,2 + N1u3,1 + N3u3,2)

δu3dx2dt |x1= 0

(13)The boundary conditions for x2=const:

t1∫

t0

l∫

0

[N2 + N2u2,2 + N3u2,1]δu2dx1dt |x2= 0

t1∫

t0

l∫

0

[N3 + N2u1,2 + N3u1,1]δu1dx1dt |x2= 0

t1∫

t0

l∫

0

N5δu3,2dx1dt |x2= 0

t1∫

t0

∫

0

(hN5,2 + 2hN6,1 + N2u3,2 + N3u3,1)δu3dx1dt |x2= 0

(14)

The boundary conditions for the plate corner (x1= const and x2= const):

t1∫

t0

2N6 |x1|x2

δu3dt = 0 (15)

Initial conditions (t= const):

l∫

0

b∫

0

[ρ (c + u1,t + c u1,1)] δu1dx1dx2 |t = 0

l∫

0

b∫

0

[ρ ( u2,t + c u2,1)] δu2dx1dx2 |t = 0

l∫

0

b∫

0

[ρ ( u3,t + c u3,1)] δu3dx1dx2 |t = 0

(16)

Eq. (9) is a system of equilibrium equations. Eq. (12) are the already employedrelations between strains and external forces, whereas relations (10), (11) and (12)correspond to the boundary conditions and in the plate corner.

3. Solution of the problem

The problem of dynamic stability has been solved with asymptotic perturbationmethod. In the solution of the problem and in the computer program developed,the following have been employed: Koiter’s or Byskov-Hutchinson’s asymptotic


expansion [2, 5], the numerical transition matrix method using Godunov’s orthog-onalization method [6, 7].

As has been mentioned above, the fields of displacements U and the fields ofsectional forces N have been expanded into power series with respect to the dimen-sionless amplitude of the web deflection ζn (the amplitude of the n-th free vibrationfrequency of the extension system divided by the thickness h1 of the web assumedto be the first one):

U = λU(0)k + ζnU

(n)k + . . .

N = λN(0)k + ζnN

(n)k + . . .

(17)

where U(0)k , N

(0)k are the pre-critical static fields, and U

(n)k , N

(n)k – the first order

fields for the composite k-th web.After substitution of expansions (13) into equilibrium equations (9), continuity

conditions and boundary conditions (10) ÷ (12), the boundary value problems of thezero and first order can been obtained. The zero approximation describes the pre-critical static state, whereas the first order approximation, being the linear problemof dynamic stability, allows for determination of the eigenvalues, the eigenvectorand the critical speeds of the system.

In the pre-critical static state the panels are divided along their widths intoseveral strips under uniformly distributed tensile stresses. Instead of the finite stripmethod, the exact transition matrix method is used in this case [6].

The inertial forces from the in-plane displacements u and v are neglected. Thepre-critical solution of the k-th composite web consisting of homogeneous fields isassumed as:

u(0)1k = (x1k − `/2)∆k ; u

(0)2k = −x2k∆kK12k/K22k ;u

(0)3k = 0 , (18)

where ∆k is the actual loading. This loading is specified as the product of a unitloading system and a scalar load factor ∆k.

Inner sectional forces of the pre-critical static state for the assumed homogeneousfield of displacements (14) are expressed by the following relationships:

N(0)1k = −(K11k − K2

12k/K22k)∆k ,

N(0)2k = 0 ,

N(0)3k = −(K31k − K32kK21k/K22k)∆k ,

N(0)4k = −(K41k − K42kK21k/K22k)∆k ,

N(0)5k = −(K51k − K52kK21k/K22k)∆k ,

N(0)6k = −(K61k − K62kK21k/K22k)∆k .

(19)

In the second dynamical component of the last Egs (9) there is derivative withrespect to x1 and t. Because of the trigonometric functions incompatibility in thex1 ≡ x-direction, the Galerkin-Bubnov orthogonalization procedure has beenused to find approximating solution of this equation.

Numerical aspects of the problem being solved for the first order fields haveresulted in an introduction of the following new orthogonal functions with n-thharmonic for k-th composite web in the sense of the boundary conditions for two


longitudinal edges [6]:

a(n)k = N

(n)2k (1 + λ∆kK21k/K22k) + λN

(o)3k d

(n)k,ξ/bk ,

b(n)k = N

(n)3k (1 − λ∆k) + λN

(0)3k c

(n)k,ξ/bk,

c(n)k = u

(n)1k ,

d(n)k = u

(n)21k ,

e(n)k = u

(n)3k ,

f(n)k = u

(n)3k,η/bk = e

(n)k,η/bk ,

g(n)k = N

(n)5k ,

h(n)k = hkg

(n)k,η/bk + 2hkN

(n)6k,ξ/bk + λN

(o)3k e

(n)k,ξ/bk ,

(20)

where ξk = x1k/bk and ηk = x2k/bk.

The solutions of Eq. (9) corresponding to the free support at the segment endscan be written in the following form:

a(n)k =

N∑

n=1Tn(t)A

(n)k (ηk) sin nπbkξ

`,

b(n)k =

N∑

n=1Tn(t)B

(n)k (ηk) cos nπbkξ

`,

c(n)k =

N∑

n=1Tn(t)C

(n)k (ηk) cos nπbkξ

`,

d=k

N∑

n=1Tn(t)D


`,

e(n)k =

N∑

n=1Tn(t)E


`,

f=k

N∑

n=1Tn(t)F


`,

g(n)k =

N∑

n=1Tn(t)G


`,

h=k

N∑

n=1Tn(t)H


`.

(21)

Tn(t) is unknown function of time and the eigenfunctions have been determined forthe unmoved web (for c =0).

The eigenfunctions A(n)k , B

(n)k , C

(n)k , D

(n)k , E

(n)k , F

(n)m , G

(n)k , H

(n)k (with the n-th

harmonic) are initially unknown functions that will be determined by the numericalmethod of transition matrices. The solution assumed in this way allows one todetermine dimensionless sectional forces for the first order approximation in the


form:

N(n)1k =

N∑

n=1Tn(t)N

(n)1k (ηk) sin nπ ,bkξ

`,

N(n)2k =

N∑

n=1Tn(t)N

(n)21k(ηk) sin nπ bkξ

`,

N(n)3k =

N∑

n=1Tn(t)N

(n)3k (ηk) cos nπ bkξ

`,

N(n)4k =

N∑

n=1Tn(t)N

(n)41k(ηk) sin nπ bkξ

`,

N(n)5k =

N∑

n=1Tn(t)N

(n)5k (ηk) sin nπ bkξ

`,

N(n)6k =

N∑

n=1Tn(t)N

(n)6k (ηk) cos nπ bkξ

`

(22)

The obtained system of homogeneous ordinary differential equations has beensolved by the transition matrix method, having integrated numerically the equilib-rium equations along the circumferential direction in order to obtain the relation-ships between the state vectors on two longitudinal edges. During the integrationof the equations, Godunov’s orthogonalization method has been employed [7]. Thepresented way of solution allows for carrying out a modal dynamic analysis of com-plex composite webs.

In system of Eqs. (9), (12) and (14) for the unmoved web, there are two compo-

nents of the zero loading N(0)1k and N

(0)3k . The component N

(0)3k has an insignificant

effect in comparison with N(0)1k .

The developed computer program allows for a division of each web into severalor even more than 50 strips made of different materials and with various thickness.The presented solution method enables a multi-modal dynamic analysis.

Let’s return to the case of moving of composite web (for c 6= 0). Because ofthe trigonometric functions incompatibility in the x1 direction, after substituting(21) into the last Eq. (11) the Galerkin-Bubnov orthogonalization method has beenused. In this way the set of N ordinary differential equations with respect to thefunction Tn(t) can be determined in the following form:

d2Tm

dt2a2m +

N∑

n=1

dTn

dta1nm + Tma0m = 0 , m = 1, 2, . . . N . (23)

Substituting the state variables into Eq. (23) one can receive the autonomous setof 2N first order differential equations with respect to time. On the basis of Eq.(23) one can determine eigenvalues, eigenvectors and critical speeds of the movingsystem.

4. Numerical results and discussion

The numerical investigations has been carried out for the moving microcorrugatedboard with trapezoidal flute profile. The composite has been modeled as the thin–walled laminar construction with rigid core. Under this assumption the classicallaminated plate theory has been satisfied.


The middle layer of the microcorrugated board one can consider as unidirectionalcorrugated trapezoidal plate which is parallel to the x1– direction. The take-upfactor of this profile is small in comparison with overall dimensions of the plate,so in the macro-mechanical analysis one can treat this layer as orthotropic platewith major principal material directions parallel to its edges [1, 6, 10]. Whenone analyses the corrugated plate with large number of trapezoidal segments, onecan take into account an individual segment, assuming the symmetry conditionson the longitudinal edges (Fig. 3). In this case, the boundary conditions on thelongitudinal edges are neglected i.e. an infinitely wide trapezoidal plate is takeninto account [16].

1.0

60o

0.12 1.2

3.25 1.0

Figure 3 Cross-section of the corrugated trapezoidal plate segment

Fig. 3 shows geometric dimensions of the corrugated trapezoidal plate segment[16]. Material properties of this plate are as following: ρ = 750[kg/m3], E1 =E2 = 2 × 109 [Pa], ν12 = ν21 = 0.2, G = 0.833 × 109 [Pa]. The masses of thecorrugated trapezoidal plate and the substitute orthotropic plate are the same. Onthe base of the condition that the stiffness of the corrugated plate and the substituteorthotropic plate are the same one receives substitute material constants,

To check above results natural frequencies of simply supported substitute or-thotropic plate model have been compared with analogous values of corrugatedtrapezoidal plate for two different lengths l = 1000 mm, l = 100 mm for the mvalues: m = 1 ÷ 10. The comparison results are shown in Fig. 4.

757,63024

6780

11990

18620

26610

35900

46430

58100

70860

757,63030

6818

12120

18940

27270

37120

48480

61360

75760

0

10000

20000

30000

40000

50000

60000

70000

80000

1 2 3 4 5 6 7 8 9 10ω ω ω ω ω ω ω ω ω ω

- substitute orthotropic plate

- corrugated trapezoidal plate

a) b)

Figure 4 Natural frequencies comparison a) l= 0.1 m; b) l= 1 m

Fig. 4 shows the values of ten lowest eigenfrequencies of both compared systemsfor c = 0. These eigenvalues of both systems there are near by, but their the greatestdiscrepancy (≈ 7%) one can observe for l = 100 mm and m = 10. Numerical dataof the facing layers and the identification results as the data of the core (Fig. 1) are


Table 1 Numerical data

Length of the web l 1 mWidth of the web b 0.5 mThickness of the facing layer (hF ) 1.2 × 10−4 mThickness of the core layer hC 1.2 × 10−3 mYoung’s modulus along the machine direction and alongthe cross direction of the facing layer E1F = E2F 2.0 GPaPoisson’s ratio of the facing layer ν12F = ν21F 0.2Shear modulus of the facing layer GF 833.3 MPaMass density of the facing layer ρF 750.0 kg/m3

Initial stresses N0 5 and 10 N/mSubstitute orthotropic plate

Young’s modulus along the x axis of the core layer E1C 540.0 MPaYoung’s modulus along the y axis of the core layer E2C 1.36 MPaPoisson’s ratio in the machine direction of the core layer ν12C 0.2Poisson’s ratio in the cross direction of the core layer ν21C 0.0005Shear modulus of the core layer GC 0.76 MPaMass density of the core layer ρC 110.0 kg/m3

shown in Table 1.

The effect of the corrugated board properties and axial transport velocity ontransverse and torsional vibrations have been studied in numerical investigations.Let σ and ω denote the real part and the imaginary part of the eigenvalues, respec-tively. Simultaneously ω is natural frequency of the web. The positive value of σindicates instability of the considered system.

Dynamic investigations of axially moving systems were begun from the vibrationmodes definition. Fig. 5 shows the modes of two lowest flexural (ω11 and ω21)and two lowest flexural-torsional (ω12 and ω22) eigenfrequencies of the moving websystem.

Results of dynamic stability investigations of the corrugated board system areshown in Fig. 6 and 7 for the axial load of the web N0 = 5 N/m and in Fig. 8 forN0 = 10 N/m. Fig. 6 shows the plot of the four lowest flexural and flexural-torsionaleigenfrequencies versus the transport velocity in the pre-critical and supercriticalregions of the transport speeds. The values of the imaginary part (solid line) andthe real part (dotted line) of the four lowest flexural and flexural-torsional naturalfrequencies versus the transport velocity only in supercritical region of the transportspeeds are shown in Fig. 7. The plots in Fig. 6 and 7 show that the lowest naturalfrequencies decrease during the axial velocity increase. At the critical transportspeed the fundamental eigenfrequency vanishes indicating divergence instability (thefundamental mode with non-zero σ and zero ω).

In supercritical transport speeds (c > ccr1), at first the web experiences a di-vergent instability and above that there is the second stability area where σ = 0.The width and position of the second stable region are dependent on the thicknessof the composite layers. The second stable area appearance is strictly connectedwith the flexural and flexural- torsional eigenfrequencies distribution. In the web


a) b)

c) d)

Figure 5 Non-trivial equilibrium positions of axially moving web: a) ω11, b) ω21, c) ω12, d) ω22

0

40

80

120

160

200

0 2 4 6 8 10 12c [m/s]

σ , ω

ω 11

ω 21

ω 22

ω 12

ω 31

ω 32

ω 41

ω 42

c cr= 6 .42 m/s

σ 11

Figure 6 Flexural and flexural-torsional natural frequencies, N0 = 5 N/m


0

5

10

15

20

25

30

35

40

6 7 8 9 10 11 12c [m/s]

σ , ω

ω 11

ω 21

ω 22

ω 12

ω 31

c cr=

σ 11σ 12

σ 11,

σ 21

σ 12,

σ 22

St

ab

il

it

yr

eg

io

n

Divergence instability region S

eco

nd

stab

ility

reg

ion

Flutter instability region

Divergence + flutter instability region

ccr=6.42 m/s

Figure 7 Flexural and flexural-torsional eigenfrequencies in supercritical region of the transportspeed, Nx0 = 5 N/m

systems, where these eigenfrequencies lie near by, the second stable area does notappear at all.

0

40

80

120

160

200

0 3 6 9 12 15

σ , ω

c cr= 8.52 m/s

ω 11

ω 21

ω 22

ω 12

ω 31

ω 32

ω 41

σ 11

c [m/s]

Divergence instability region

S t a b i l i t y r e g i o n

Second stability region

Flutter instability region

Divergence + flutter instability region

σ 21

Figure 8 Flexural and flexural-torsional natural frequencies, Nx0 = 10 N/m

When the axial load of the board web increases the critical transport speed of thesystem increases as well. Above the second stable region at first flutter instabilityoccurs and next flutter and divergence instability of the web motion appear.

5. Conclusions

In the paper, a new approach to the analysis of the dynamic behaviour of an axiallymoving sandwich composite web is presented. Mathematical model of the movingweb system is derived on the basis of asymptotic perturbation method. The math-ematical model solution is based on the numerical method of the transition matrixusing Godunov’s orthogonalization.

The numerical investigations has been carried out for the moving microcorru-gated board with trapezoidal flute profile. The composite has been modeled asthe thin–walled laminar construction with rigid core. Under this assumption the


classical laminated plate theory has been satisfied. The axially moving board webis treated as thin–walled composite structure in the elastic range, being under axialextension.

Calculations results of all investigated systems show in the subcritical region oftransport speed for the constant axial tension of the web, the lowest flexural andflexural-torsional natural frequencies decrease during the axial velocity increase.At the critical transport speed the fundamental flexural eigenfrequency vanishesindicating divergence instability. When the axial load of the board web increasesthe critical transport speed of the system increases as well.

In the supercritical region of transport speed the second stable area may appear.The width and position of the second stable region are dependent on the thicknessof the board layers. In the case of the paperboard webs in which the flexural andflexural-torsional eigenfrequencies lie near by, the second stable area does not appearat all.

References

[1] Brzoska, Z: Statics and stability of beam and thin-walled structures, (1965), PWN,Warszawa, 562.

[2] Byskov, E and Hutchinson JW: Mode interaction in axially stiffened cylindricalshells, AIAA J., (1977), 15, 7, 941-948.

[3] Dawe, DJ, Wang, S: Buckling of composite plates and plate structures using thespline finite strip method, Composites Engineering, (1994), 4, 11, 1099-1117.

[4] Jones, RM: Mechanics of composite materials, (1975), International Student Edi-tion, McGraw-Hill Kogakusha, Ltd., Tokyo, 365.

[5] Koiter, WT: General theory of mode interaction in stiffened plate and shell struc-tures, WTHD, Report 590 (1976), Delft.

[6] KoÃlakowski, Z, Kowal-Michalska, K: Selected problems of instabilities in compos-ite structures, (1999), Technical University of ÃLodz, A series of monographs, 222.

[7] KoÃlakowski, Z, Krolak, M Interactive elastic buckling of thin-walled closed or-thotropic beam-columns, Engineering Transactions, (1995), 43, 4, 571-590.

[8] Lin, CC, Mote, CD Jr: The wrinkling of thin, flat, rectangular webs, Journal ofApplied Mechanics ASME, (1996), 63, 774-779.

[9] Lin, CC: Stability and vibration characteristics of axially moving plates. Int. Journalof Solid Structures, (1997), 34, (24), 3179-3190.

[10] Loughlan, J: The effect of membrane-flexural coupling on the compressive stabilityof antisymmetric angle-ply laminated plate, Proceedings of the Third InternationalConference on Thin-Walled Structures, Elsevier, (2001a), 507-514.

[11] Loughlan, J: The shear buckling behaviour of thin composites plates with particularreference to the effects of bend-twist coupling, Int. J. Mechanical Sciences, (2001),43, 771-792.

[12] Matsunaga, H: Buckling analysis of multilayered angle-ply composite plates, Pro-ceedings of the Third International Conference on Thin-Walled Structures, (2001),Elsevier, 507-514.

[13] Marynowski, K, KoÃlakowski, Z: Dynamic behaviour of an axially moving thinorthotropic plate, Journal of Theoretical and Applied Mechanics, (1999), 37, (1), 109-128.

[14] Marynowski, K: Non-linear vibrations of axially moving orthotropic web, Interna-tional Journal of Mechanics and Mechanical Engineering, (1999), 3, (2), 109-128.


[15] Savolainen, A, (Ed.): Paper and Paperboard Converting. Papermaking Science andTechnology, Book 12, Tappi.

[16] Teter, A, KoÃlakowski, Z: Stability and load carrying capacity of a thin-walledcorrugated trapezoidal plate, Int. J. of Applied Mechanics and Engineering, (2001),6, 2, 311-323.

[17] Walker, M, Adali, S, Verijenko, VE: Optimal design of symmetric angle-plylaminates subject to nonuniform buckling loads and in-plane restraints, Thin-WalledStructures, (1996), 26, 1, 45-60.

[18] Wang, C, Pian, THH, Dugundji, J. and Lagace, PA: Analitycal and experimen-tal studies on the buckling of laminated thin-walled structures, Proc. AIAA/ASME/ASCE/AHS 28th Structures, Struct. Dyn. and Materials Conf., (1987), Part 1, 135-140.

[19] Wang, X: Numerical Analysis of Moving Orthotropic Thin Plates, Computers &Structures, (1999), 70, 467-486.


Documents

Dynamic Stability of an Axially Moving Sandwich Composite Web