Dynamic analysis of an axially moving sandwich beam with viscoelastic core

Composite Structures 94 (2012) 2931–2936

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Dynamic analysis of an axially moving sandwich beam with viscoelastic core Krzysztof Marynowski Department of Machine Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland

a r t i c l e

i n f o

Article history: Available online 7 April 2012 Keywords: Sandwich structure Axially moving beam Dynamic stability

a b s t r a c t A development of the beam model of the axially moving sandwich continua with elastic faces and the core characterized by viscoelastic properties is presented in this paper. Two-parameter Kelvin–Voigt rheological model is used to describe material properties of the core. The Galerkin method is used to solve the governing partial differential equation. Dynamic analysis of the composite with two aluminum facings and a polyurethane core is carried out. The effect of the transport speed, the core thickness and the internal damping of the core material on the dynamic behavior of the system is investigated in undercrtitical and supercritical range of transport speed. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Axially moving flat composite objects at high speeds, can be found in many different technical applications. These include paper webs during production, processing and printing, textile webs during production and processing, and flat objects moving at high speeds in space. In all of these various technical applications typically one tends to maximize the transport speed, in order to increase productivity and optimize investment and operating costs of these sometimes very expensive and complicated equipment. An obstacle in the realization of these aspirations are very common dynamic behavior of axially moving systems. Many factors influence the dynamic behavior of the axially moving objects. The most important are: transport speed, tension, an influence of external environment and composite material properties. The effect of composite material properties is particularly apparent in the paper production process. Paper is a very specific material, whose physical properties depend on its structure, raw materials composition, production technology, finishing, processing and hydrothermal state. Large variety of these factors makes complicated nature of the mathematical description is not convenient in engineering applications. The literature especially related to transverse vibrations of viscoelastic running plates is rather limited. A two-dimensional rheological element was applied in dynamic investigations of axially moving systems in [1,2]. Three studies by Hatami et al. [3,4] and Marynowski [5] devoted to free vibrations of axially moving multi-span composite plates, viscoelastic Navier-type plate, and viscoelastic Levy-type plate, respectively were published. In the literature one can find also works in which one-dimensional rheo-

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logical models were used to describe properties of the axially moving material. The level of complexity of the plate model and its size are especially important in calculations of this kind, where computation time plays a significant role. On the other hand, the results of analyses obtained so far indicate that an application of the beam model of the moving wide continua in dynamical calculations can sometimes result in satisfactory results (e.g. [6–8]). One can suppose that in some dynamical calculations aiding designing and building devices that rewind a broad web, its beam model can be useful as less complex in comparison to the plate model. The paper by Fung et al. [9] was the first one where transverse vibrations of a viscoelastic moving belt represented by the spring model were investigated. Nonlinear dynamics of an axially accelerating viscoelastic beam was investigated by Chen et al. [10] and Yang and Chen [11]. In those papers, the beam material was described with the Kelvin–Voigt model. Regular and chaotic vibrations of an axially moving viscoelastic beam subjected to tension variations were studied in Marynowski and Kapitaniak [12]. In this work two different rheological models, namely: a two-parameter Kelvin–Voigt model and a three-parameter Zener model, were applied. This paper presents a development of the beam model of the axially moving sandwich continua with the core characterized by viscoelastic properties. Two-parameter Kelvin–Voigt rheological model has been used to describe material properties of the core. The Galerkin method is used to solve the governing partial differential equation. Dynamic analysis of the composite with two aluminum facings and a polyurethane core is carried out. The effect of the transport speed, the core thickness and the internal damping of the core material on the dynamic behavior of the system is investigated in undercritical and supercritical range of transport speed.

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Substituting Eq. (5) into the strain–displacement relation:

2. Deformation of the beam in equilibrium A sandwich beam of the width b and the thickness (h + 2d) is considered. The geometry and configuration of the layers are shown in Fig. 1. To derive mathematical model of the moving sandwich beam the following assumptions have been made: (1) the core is of soft viscoelastic material, only shear deformation is considered in this layer, two-parameter Kelvin–Voigt rheological model describes material properties of the core, (2) the top and bottom elastic layers have the same properties, they are thin enough and are subjected only to longitudinal stress rx, (3) no slipping occurs at the interfaces between the three layers of the beam, (4) the materials of the beam layers are homogeneous and isotropic, (5) all points which there are on the normal to the axis of the beam undergo the same transverse deflection, (6) u, w denote the displacements of an arbitrary point of the beam respectively along x and z direction, u+, u denote the displacements of the top and bottom face respectively, w denote the rational angle of the core (Fig. 1).

cxz ¼

@ rx @ sxz þ ¼0 @x @z

ð1Þ

  @w @u Q Q þ ¼ þ c0  eat : @x @z Gðh þ dÞb Gðh þ dÞb

sxz ¼

Q ¼ const:; ðh þ dÞb

u ¼ zw;

sxz ¼ g

wðtÞ ¼

ð3Þ

where g is the damping coefficient and G is the Kirchhoff’s modulus. One receives:

@c Q g xz þ Gcxz ¼ ðh þ dÞb @t

ð4Þ

The general solution of Eq. (4) is the following:

cðtÞ ¼

  @w Q Q eat :   c0  @x Gðh þ dÞb Gðh þ dÞb

  Q Q þ c0  eat ; Gðh þ dÞb Gðh þ dÞb

ð5Þ

where c0 is the initial transverse strain and a is the reciprocal of the strain relaxation time (a = G/g).

d

h u ¼  w: 2

ð10Þ

Substituting the first derivative of Eq. (10) into the strain– displacement relation

@u ; @x

ð11Þ

and the result into the constitutive equation of the elastic face

r ¼ Ee;

ð12Þ

yields

h þ d @w : 2 @x

core

d z

bottom layer

ð13Þ

The bending moment in the beam with respect to the neutral axis due to the face layers tension is

ðh þ dÞ dbðrþ  r Þ: 2

ð14Þ

Substituting Eq. (13) into Eq. (14) one receives the overall moment of the sandwich beam in the following form:

M ¼ E

ðh þ dÞ2 @w : db @x 2

ð15Þ

Eq. (15) shows the relationship between bending moment due to the face layers tension and the rotational angle of the core layer. 3. Mathematical model of the moving sandwich beam Axially moving sandwich beam of the finite length l is considered. The beam is moving along the longitudinal direction x at the constant velocity c (Fig. 2). To derive mathematical model of the system the dynamic equilibrium of the moving sandwich beam is considered. The equilibrium of the bending moment, the transverse force and the transverse load, according to Newton’s second law can be expressed as

y

top layer

h

ð9Þ

Furthermore, the displacements of the face layers accounting for the rotational angle of the core layer is calculated as

M¼

@ cxz þ Gcxz ; @t

ð8Þ

where the rational angle of the overall sandwich beam w is

ð2Þ

where Q is the shear force in the core layer. Substituting Eq. (2) into the constitutive equation of the core material:

ð7Þ

Integrating both sides of Eq. (7) leads to the following relationship:

r ¼ E

After integrating both sides of Eq. (1) and based on the assumption (1) the shear stress sxz in the core layer is expressed as

ð6Þ

yields

e¼ At first the relation of the shear force and deformation of the core layer is derived. The equilibrium of the core layer in the longitudinal direction describes the following relation:

@w @u þ ; @x @z

c b

ψ

x

x

h+2d b

Fig. 1. Geometry and configuration of layers.

l z Fig. 2. The moving sandwich beam.

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@M  Q ¼ 0; @x

ð16Þ

@Q þ q ¼ 0; @x

ð17Þ

where q is the transverse load of the moving beam (N/m). Substituting Eq. (9) into Eq. (15) and taking into account Eq. (17) one receives

ðh þ dÞ2 @ 2 w hþd dqð1  eat Þ; M ¼ E db 2  E @x 2G 2

ð18Þ

Differentiate Eq. (18) twice with respect to x and substitute the result into Eq. (17) one obtains

E

ðh þ dÞ2 @ 4 w hþd @2q dð1  eat Þ 2 þ q ¼ 0: db 4  E @x 2G @x 2

ð19Þ

In this analysis the axial tension and inertia forced due to vibration as the transverse loading are considered. Thus, the transverse load q has the following form:

q ¼ q

! 2 @2w @2w @2w 2@ w þP 2 ; þ c þ 2c 2 2 @x@t @x @x @t

ð20Þ

where q is the unity mass of the beam (kg/m) and P is the axial tension (N). Substituting Eq. (20) into Eq. (19) yields the governing equations of the axially moving sandwich beam with viscoelastic core

E

ðh þ dÞ2 @ 4 w hþd dqð1  eat Þ db 4  E @x 2G 2 ! 4 @4w @4w @4w 2@ w þ 2c 3 þ c P 4  @x @t @x4 @x @x2 @t 2 ! @2w @2w @2w @2w þ c2 2  P 2 ¼ 0 þ 2c þq 2 @x@t @x @x @t

ð21Þ

The governing Eq. (21) can be transposed to a dimensionless form using the following terms:

w t¼ ; l

e¼

x n¼ ; l

Eðh þ dÞ2 db 2Pl

2

t k¼ l ;

e1 ¼

sffiffiffiffi P

rffiffiffiffi s¼c

;

q

Eðh þ dÞd 2Gl

2

q P

; ð22Þ

:

Then the dimensionless governing equation has the form:

@2t @2t @4t þ 2s þ ðs2  1Þ 2 þ e 4  e1 ð1  eak Þ @n@k @k @n @n " # @4t @4t @4t 2 þ 2s 3 þ ðs  1Þ 4 ¼ 0  @n2 @k2 @n @k @n

@

2

t 2

ð23Þ

The dimensionless boundary conditions referring to simple supports at n = 0 and n = 1:

tjn¼0 ¼ tjn¼1

 @ 2 t ¼ 2 @n 

n¼0

 @ 2 t ¼ 2 @n 

¼ 0:

ð24Þ

n¼1

Partial differential Eq. (23) and boundary conditions (24) constitute dimensionless mathematical model of the axially moving sandwich beam with two simple supports.

Due to complexity of the problem, the solution to Eq. (23) cannot be obtained analytically. The problem has been solved using the Galerkin method. The following finite series representation of the dimensionless transverse displacement has been assumed:

tðn; kÞ ¼

n X

sinðipnÞqi ðkÞ;

ð25Þ

i¼1

where qi(k) is the generalized displacement. The 4-term finite series representation of the dimensionless transverse displacement of the beam has been taken in numerical investigations. In the Galerkin method the series (25) is substituted in Eq. (23), all terms of the equation are multiplied by sin(ipn), and then the result is integrated in the domain [0, 1]. For n = 4 the set of ordinary differential equations is shown in the Appendix A (Eq. (A.1)). Eq. (A.1) can be written in the state space form:

_ ¼ A;

ð26Þ

where  ¼ ½q_ 1 ; q_ 2 ; q_ 3 ; q_ 4 ; q1 ; q2 ; q3 ; q4 T is the state vector and A is the state matrix describing the dynamics of the system. The spectral analysis of the state matrix permits the analysis of the stability and the mode shapes of the axially moving sandwich beam. 4. Investigation results To test the computational model of the axially moving sandwich beam with viscoelastic core, a comparison between the results of the method and the results obtained by the author in [2] is presented. Results of dynamic analysis of the beam model of axially moving paper web are presented in [2]. The properties of the paper obtained in the experimental way are as follows: basis weight: 220 g/m2; thickness: 0.35 mm, length: 1 m and width: 0.1 m. The Kelvin–Voigt rheological element in the beam model of the web was used to analyze the free vibration frequency and stability of the web motion. Thickness of the elastic face equals to half of the paper thickness was adopted in comparative studies. The core thickness, much smaller than the face thickness, has been used in comparative studies. A comparison of two lowest critical transport speeds of the beam model of the paper web and the present model of sandwich beam for three various axial tensions is presented in Table 1. Table 1 shows that the critical transport speeds generated in this study agree well with the critical speeds obtained from the beam model of axially moving paper web. For the second mode, the greatest discrepancy is equal to 0.25%. Numerical investigations were carried out for the beam model of the axially moving sandwich continua with the core characterized by viscoelastic properties. The composite consists of two aluminum facings and a polyurethane core was analyzed. The following parameters of the face were assumed: E = 70 GPa, d = 0.75 mm, b = 0.2 m, l = 1 m. The Kelvin–Voigt rheological model has been used to describe material properties of the core. Numerical analysis has been carried out for two different values of the longitudinal forces: P = 100 N and P = 500 N and two different values of. the Kirchhoff’s modulus of the polyurethane core: G = 1 MPa and G = 10 MPa. The corresponding values of the

Table 1 Results of comparative calculations. Critical transport speed [m/s] P=5N

Paper model Present model

P = 25 N

P = 50 N

1st mode

2nd mode

1st mode

2nd mode

1st mode

2nd mode

15.035 15.032

15.170 15.131

33.575 33.575

33.637 33.622

47.478 47.475

47.518 47.502

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K. Marynowski / Composite Structures 94 (2012) 2931–2936

(a)

(b)

ω

9

1,8

8

1,6

7

1,4

6

divergence inst. region

1

ω1

4

flutter inst. region

ω2

1,2

ω2

5

ω

ω1

0,8

3

0,6

2

0,4

1

0,2

s

0 0

0,15

0,3

0,45

0,6

0,75

0,9

1,05 scr 1,2

1,35

scr=1.15

0

1,05 1,1 1,15 1,2

1

1,5

s

1,25 1,3

1,35 1,4

1,45 1,5

Fig. 3. Two lowest eigenvalues of the state matrix versus the transport speed (h/d = 0.133; P = 100 N; G = 1 MPa, a = 1; (a) subcritical range of transport speeds and (b) neighborhood of the critical transport speed).

Im(β1) 4 3,5 3

scr

2,5

P =100 N

2 1,5

P = 500 N

1 0,5

Re(β1)

h/d

0 0

ξ

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

5,5

6

Fig. 6. Critical speed versus h/d (a = 1; (__) G = 10 MPa; (- -) G = 1 MPa).

Fig. 4. The first mode shapes (h/d = 1.333; (__) s = 0.4; (. . .) s = 0.8).

7

Im(β2)

ω2

6 5

ω

flutter inst. region

4 3

divergence inst. region

ω1

2 1

Re(β2)

s

scr=1.32

0 1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

Fig. 7. Two lowest eigenvalues in the neighborhood of the critical speed (h/ d = 2.666; P = 500 N; G = 10 MPa, a = 1).

ξ Fig. 5. The second mode shapes (h/d = 1.333; (__) s = 0.4; (. . ..) s = 0.8).

dimensionless coefficients e and e1 for these two forces and two modules have been designated. Dimensionless Eq. (A.1) written in the form of the state Eq. (26) have been analyzed. The state matrix A in the state Eq. (26) is the linear operator describing the dynamics of the investigated system in the neighborhood of the trivial equilibrium position of the beam. The spectral analysis of this operator permits the analysis of the stability and the mode shapes. The appointment of the two lowest eigenvalues of the state matrix versus the transport speed of the composite beam initiated the dynamic analysis of the system under consideration. The investigation results for P = 100 N and G = 1 MPa are shown in Fig. 3. The eigenvalues of the state matrix have the complex form. In subcritical region of transport speeds (s < scr) the imaginary eigenvalues decrease when the transport speed of the beam increases. At the critical transport speed scr the first eigenvalue is equal zero

and the divergence instability occurs. In supercritical transport speeds (s > scr) the first eigenvalue has the real form and the beam experiences divergent instability and next flutter instability (Fig. 3b). In order to determine the mode shapes of the beam in subcritical range of transport speeds the eigenvectors of the state matrix are taken into account. The eigenvector matrix / of the state matrix was created in the form [/1, /2, . . . , /8], where /i is the ith eigenvector of the state matrix. The eigenfunctions bn(n) of the investigated system can be approximated with the eigenvectors of the discrete model in the following way [8]:

bn ðnÞ ¼

4 X /kþ4;n sinðkpnÞ;

ð27Þ

k¼1

where /k+4,n is the (k + 4,n) element of the eigenvector matrix /. The mode shapes of the axially moving composite beam have been determined for two various transport speeds in the subcritical

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K. Marynowski / Composite Structures 94 (2012) 2931–2936

(a)

(b) s

3

3

s sfl100

2,5

2,5

sfl100

2

scr100

1,5

sfl500 scr100 scr500

2 1,5

sfl500

scr500

1

1

0,5

0,5

α

0 0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

α

0 0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

Fig. 8. Critical speeds versus a (h/d = 1.333; (__) scr; (. . .) sfl ; (a) G = 1 MPa; (b) G = 10 MPa).

range: s = 0.4 and s = 0.8. Figs. 4 and 5 show the first mode shapes and the second mode shapes, respectively. Next the effect of core thickness on the critical speed of axially moving sandwich beam has been studied. The values of the transport speed, at which the divergent loss of stability of the beam motion occurs, have been appointed. The values of the core thickness parameter h/d from 0.133 to 6.0 was taken into consideration. Fig. 6 shows the numerical results. Due to the increase of core thickness, the bending moment and flexural stiffness of the beam enlarge, and the critical transport speed increases as well. In the tested range of changes in the thickness of the core, the biggest increase in critical velocity occurs at the lower longitudinal load and at the higher Kirchhoff module of the core material. In addition to the growth of the critical velocity, also increase the range of divergence in the region of supercritical speeds, causes enlarging in the thickness of the core. This is illustrated in Fig. 7, which shows the calculation results of the two lowest eigenvalues of the state matrix in the supercritical region for h/d = 2.66 and P = 500 N. Finally, the effect of damping properties of the core material on the critical transport speed and the width of the divergence region was investigated. Fig. 8 shows the results of calculations of the critical speeds and the transport speeds at which the flutter type instability occurs, for various values of the dimensionless damping parameter a. A small change in the value of the critical velocity can be seen only in Fig. 8a for small values of a at the lower value of the Kirchhoff modulus. Also the biggest changes in the width of the divergence scope in the supercritical region is observed for small values of a. Small values a < 1 correspond to the long relaxation time of strain and that has an influence on the dynamic behavior of the system. Damping of the core material has a significant influence on the width of the divergence instability region in the supercritical range of transport speed. At the lower value of the Kirchhoff modulus, for both longitudinal loads, the range of divergence decreases with increasing of the a value. Different dynamic behavior can be seen in Fig. 8b at the higher Kirchhoff modulus and bigger value of the longitudinal load. Then the increase of the a value also expands the range of divergence.

5. Conclusions Dynamic investigations of beam-like model of the axially moving sandwich continua with elastic faces and viscoelastic core are carried out in this paper. Only shear deformation is considered in the core. Two-parameter Kelvin–Voigt rheological model describes material properties of the core. The Galerkin method is applied to simplify the governing partial-differential equation into fourth-or-

der truncated system defined by the set of ordinary differential equations. Dynamic behavior of the composite with two aluminum facings and a polyurethane core was investigated. Numerical analysis was carried out for two different values of the longitudinal load and two different values of the Kirchhoff’s modulus of the polyurethane core. The effect of the transport speed, the core thickness, and the internal damping of the core material on the critical transport speed ware analyzed. The eigenvalues of the system have the complex form. In subcritical region of transport speeds the imaginary eigenvalues decrease when the transport speed of the beam increases. At the critical transport speed the first eigenvalue is equal zero and the divergence instability occurs. In the supercritical region of transport speeds at first the beam experiences divergent instability and next flutter instability. When the thickness of the core increases, the bending moment and flexural stiffness of the beam enlarge, thus the critical transport speed increases as well. In the tested range of changes in the thickness of the core, the biggest increasing in critical velocity occurs at the lower longitudinal load and at the higher Kirchhoff module of the core material. Damping properties of the core material have a small effect on critical speeds, but have a significant influence on the width of the divergence instability region in the supercritical range of transport speeds. Appendix A The set of ordinary differential equations of the model of the axially moving sandwich beam with two simple supports (n = 4): 1 þ e1 p2 ð1  eak Þ p2 ðs2  1Þ  ep4 þ e1 p4 ðs2  1Þð1  eak Þ €1 ¼ q q1 2 2 8 þ s½1 þ e1 p3 ð1  eak Þq_ 2 3 16 þ s½1 þ e1 p3 ð1  eak Þq_ 4 ; 15 1 þ e1 ð2pÞ2 ð1  eak Þ ð2pÞ2 ðs2  1Þ  eð2pÞ4 þ e1 ð2pÞ4 ðs2  1Þð1  eak Þ €2 ¼ q q2 2 2 8 3  s½1 þ e1 ð2pÞ ð1  eak Þq_ 1 3 24 þ s½1 þ e1 ð2pÞ3 ð1  eak Þq_ 3 ; 5 1 þ e1 ð3pÞ2 ð1  eak Þ ð3pÞ2 ðs2  1Þ  eð3pÞ4 þ e1 ð3pÞ4 ðs2  1Þð1  eak Þ €3 ¼ q q3 2 2 24 3 ak _ s½1 þ e1 ð3pÞ ð1  e Þq2  5 48 þ s½1 þ e1 ð3pÞ3 ð1  eak Þq_ 4 ; 7 1 þ e1 ð4pÞ2 ð1  eak Þ ð4pÞ2 ðs2  1Þ  eð4pÞ4 þ e1 ð4pÞ4 ðs2  1Þð1  eak Þ €4 ¼ q q4 2 2 16 48 s½1 þ e1 ð4pÞ3 ð1  eak Þq_ 1  s½1 þ e1 ð4pÞ3 ð1  eak Þq_ 3 :  ðA:1Þ 15 7

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