COMPUTATIONAL SIMULATION OF AXIALLY TENSIONED MOVING

COMPUTATIONAL SIMULATION OF AXIALLY TENSIONED MOVING PAPER WEBS Xiaodong Wang y and George Renner

z

ABSTRACT A computational procedure is developed to analyze the out-of-plane motion of pretensioned moving paper webs. The standard Galerkin nite element method is employed along with the Newmark direct time integration scheme for the solution of the governing equation. It is demonstrated that when the tension is the dominant restoring force, this displacement-based nite element formulation will not introduce the well-known shear locking. In addition, results obtained with both the proposed formulation and the mixed nite element formulation with MITC (mixed interpolated tensorial components) elements based on the Mindlin/Reissner plate theory are compared.

1 Introduction

As paper machine speeds continue to increase, eects of the air being drawn along with the moving paper web, cross-machine direction (CD) tension nonuniformity, moisture pro le, basis weight variation, and roll eccentricity can introduce signi cant out-of-plane vibrations. Various mathematical models have been proposed to study such dynamical behaviors [6]. Based on the assumption that the axial tension is the dominant restoring force, the moving paper sheet is often modeled as a moving pre-tensioned string or web. In order to accurately analyze the moving paper web, as discussed by Pramila [7] and Chang and Moretti [4], we may have to include the surrounding air added mass eects. In accord with similar approaches in the vibration analysis of high-speed magnetic tapes and exible disk drives [5] [8], the nite element procedure has been developed to model an axially moving pre-tensioned web controlled through self-acting air bearings [9]. More recently, a mixed nite element formulation based on the Mindlin/Reissner plate theory for a moving orthotropic thin plate has been proposed by Wang [10]. In this work, we present the C 0 displacement-based nite element formulation for the moving paper web with a dominant axial tension. In addition, by comparing the forced vibration results obtained with both formulations, we address issues of bending stiness eects.

Assistant Professor of Engineering, Institute of Paper Science and Technology, 500 10th St., N.W., Atlanta, GA 30318 z Graduate Student, Institute of Paper Science and Technology

y

1

Free boundary y=B

Fixed boundary x=L

V

Fixed boundary x=0 y

x Free boundary y=0

Figure 1: Mathematical model of a typical moving paper web. In the following section, we brie y discuss the governing equation of the moving web, and the corresponding C 0 nite element formulation. A few numerical examples with dierent moving velocities, bending stinesses, and web tensions are presented in Section 3.

2 Governing Equations and Numerical Analysis

The governing equation for the moving web shown in Fig. 1 is constructed as follows: 2 @ 2 w + (mV 2 ? T ) @ 2 w + Dr4 w = f m @@tw2 + 2mV @x@t (1) @x2

where the bending stiness D is given as Ed3 =12(1 ?  2 ), and E , d,  , w, V , m, T , and f stand for the Young's modulus, web thickness, Poisson's ratio, out-of-plane displacement, web transport velocity, web mass density, web tension, and external force distribution, respectively. The xed boundary conditions of the web are based on the contact lines the free edges between the web and the supporting rolls, where w = 0 and @w @x = 0 . Along 2 3 3 2 of the web, both the bending moment @@yw2 +  @@xw2 and the shear force @@yw3 + (2 ?  ) @x@ 2w@y equal to zero. The variation form of Eq. (1) can be written as ) ZB ZL ( ZB ZL 2w 2w @ @ @w @w 4 2 w(m @t2 + 2mV @x@t + Dr w) + @x (T ? mV ) @x dxdy = wfdxdy 0 0

0 0

(2)

where B stands for the web width. Furthermore, by applying the Gauss' theorem along with the boundary conditions, we derive the following bending contributions ZB ZL @ 2 w @ 2 w @ 2w @ 2w @ 2 w @ 2 w ZB ZL 4 wDr wdxdy = D ( @x2 @x2 + @y2 @y2 + 2 @x@y @x@y )dxdy 0 0 0 0 2

+ D

ZB ZL 0 0

(?2

@ 2 w @ 2 w + @ 2 w @ 2 w + @ 2 w @ 2 w )dxdy @x@y @x@y @y2 @x2 @x2 @y2

(3)

Employing the standard Galerkin discretization procedure, we have for a typical nite element @w = B W @w = B W c; c; w = HW x yc @x @y @ 2 w = B W @ 2 w = B W @ 2 w = B W c c ; ; x y xy c @x2 @y2 @x@y c is the vector of the solution variable w. where H is the interpolation matrix, and W Denote the elasticity matrix D as 2 3 1  0 75 0 D = D 64  1 0 0 (1 ?  )=2 we obtain the matrix equations for this C 0 displacement formulation

 + CW c c_ + (K1 + K2)W c =R MW

where

M

=

K1

=

R

=

ZB ZL 0 0

ZB ZL 0 0 ZB ZL 0 0

mH Hdxdy;

C

T

(T

? mV 2)BTx Bxdxdy;

=

K2

=

ZB ZL 0 0

2mV

ZB ZL h 0 0

(4)

HT Bxdxdy

2  3 Bx 7 i T T T 6   Bx By ? 2Bxy D 4 B y 5 dxdy

?2B xy

HT fdxdy

2 We note that the Coriolis term 2mV @ w introduces nonsymmetric terms in the matrix @x@t C, and the existing commercial nite element analysis softwares can no longer be used. In the implementation of the proposed computational procedure, we employ an eective nonsymmetric column solver discussed in Ref. [1] and the Newmark method with  = 0:5 and = 0:25 for the direct time integration t+dt W _ = tW _ + [(1 ? )t W  + t+dt W  ]dt

t+dt W = t W + t W  + t+dt W  ]dt2 _ dt + [( 1 ? )t W 2

(5) (6)

We recognize that although the bending eects are included in Eq. (2) based on the Kirchho assumptions for thin plates, the proposed C 0 nite element discretization will 3

not represent appropriately the bending term K2 governed by second order derivatives. In fact, as pointed out in Ref. [11], simple polynomial expressions for shape functions ensuring full compatibility cannot be obtained when only w and its slopes are prescribed at nodes. Therefore, if the axial tension is not predominant or as the axial moving speed increases, i.e., K1  K2 , reliable numerical procedures for moving plates are needed. It has been widely recognized that the mixed plate formulation based on the Mindlin/Reissner theory can eectively eliminate shear locking and predict accurately and reliably stress levels of thin plates [2] [3]. The extension of the mixed plate formulation with MITC elements to the analyses of axially moving orthotropic thin plates has been presented in Ref. [10].

3 Numerical Results

We analyze the moving tensioned paper web with the following two sets of physical parameters: m = 1:014 kg/m2 (including added mass approximation), d = 0:07 mm, L = 1 m, B = 10 m, E = 1:54 GPa,  = 0:3; or m = 0:49 kg/m2, d = 0:7 mm, L = 2 m, B = 8 m, E = 5 GPa,  = 0:25. The dynamic responses, illustrated in Figs. 2 and 3, clearly show that when bending contributions are signi cant, spurious displacement oscillations may occur, which is typical for the proposed C 0 displacement-based formulation. It is then anticipated that if we have tension slacking near the free edges or the axial velocity is suciently high, the bending stiness of the paper web will play an important role in the web dynamics, and in such cases, the mixed plate formulation is needed. The out-of-plane displacement distributions of the moving plate subjected to a impulse loading (depicted in Fig. 4) at the center of the plate are computed with both the mixed and C 0 formulations. As shown in Fig. 5, the dierence is signi cant. We notice that, by using the Mindlin/Reissner mixed formulation, we can also obtain x-rotation, y-rotation, stress-xx, stress-xy, stress-yy, stress-xz, and stress-yz distributions [10]. To compare two formulations further, we plot the MD and CD out-of-displacement pro les with and without bending stiness eects in Figs. 6 to 9. As expected, the results show that bending eects governed by second order derivatives cannot be captured by regular 4-node element meshes. In the cases without bending stiness, the C 0 formulation results after 60 time steps still correlate with the mixed formulation results, and the dierences can be explained with two rotational inertia terms included in the mixed formulation [10].

4 Conclusion

In this paper, we demonstrate that the proposed C 0 displacement-based nite element formulation can be successfully used to simulate moving paper webs with the dominant axial tension. The spurious oscillatory displacements observed in our numerical tests indicate that the Mindlin/Reissner mixed nite element formulation is needed if the tension is low or the axial velocity is suciently high.

Acknowledgment

We would like to thank the Institute of Paper Science and Technology and its Member Companies for their support.

4

0

−0.1

−0.2

Out−of−plane displacement

−0.3

−0.4

−0.5 "−": Initial position "−.": Time step 100

−0.6

"..": Time step 200 −0.7

−0.8

−0.9

−1

0

1

2

3

4 5 6 y−coordinate (CD), m

7

8

9

10

Figure 2: CD out-of-plane displacement pro le (1  10 plate). Initial displacement / ((y ? 5)2 + y) sin 11x=L, V = 0, dt = 0:001 s, and T = 0. Mesh of 5  50 9-node elements (Kirchho). 0.6

0.4

Out−of−plane displacement

0.2

0

−0.2

−0.4 "−": Initial position "−.": Time step 100

−0.6

"..": Time step 200 −0.8

−1

0

1

2

3

4 5 6 y−coordinate (CD), m

7

8

9

10

Figure 3: CD out-of-plane displacement pro le (1  10 plate). Initial displacement / ((y ? 5)2 + y) sin 11x=L, V = 0, dt = 0:001 s, and T = 10 N/m. Mesh of 5  50 9-node elements (Kirchho). 5

1

Timefunction

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6 Time, s

0.8

1

Figure 4: Impulse load timefunction f (t).

V = 8 m/s T = 50 N/m

Z-DISP 0.02 0.0171429 0.0142857 0.0114286 0.00857143 0.00571429 0.00285714 0 -0.00285714 -0.00571429 -0.00857143 -0.0114286 -0.0142857 -0.0171429 -0.02

MITC4 elements (Mindlin/Reissner)

V = 8 m/s T = 50 N/m

Z-DISP 0.02 0.0171429 0.0142857 0.0114286 0.00857143 0.00571429 0.00285714 0 -0.00285714 -0.00571429 -0.00857143 -0.0114286 -0.0142857 -0.0171429 -0.02

9-node elements (Kirchhoff)

Figure 5: Typical out-of-plane displacement band plot (2  8 plate). t = 1:2 s and dt = 0:02 s. 6

0.04

0.02

Out−of−plane displacement

0

−0.02

"−": MITC4 (Mindlin/Reissner) "o": 9−node (Kirchhoff) "+": 4−node (Kirchhoff)

−0.04

−0.06

−0.08

−0.1

0

1

2

3

4 5 y−coordinate (CD), m

6

4 5 y−coordinate (CD), m

6

7

8

Figure 6: CD out-of-plane displacement pro le without bending stiness (2  8 plate). t = 1:2 s, dt = 0:02 s, V = 8 m/s, E = 0, and T = 50 N/m.

0.04

0.03

Out−of−plane displacement

0.02

0.01

0

−0.01

−0.02

"−": MITC4 (Mindlin/Reissner) "o": 9−node (Kirchhoff) "+": 4−node (Kirchhoff)

−0.03

−0.04

−0.05

0

1

2

3

7

8

Figure 7: CD out-of-plane displacement pro le with bending stiness (2  8 plate). t = 1:2 s, dt = 0:02 s, V = 8 m/s, E = 5 GPa, and T = 50 N/m. 7

0.15

0.1

Out−of−plane displacement

0.05

"−": MITC4 (Mindlin/Reissner)

0

"o": 9−node (Kirchhoff) "+": 4−node (Kirchhoff) −0.05

−0.1

−0.15

−0.2

−0.25

0

0.2

0.4

0.6

0.8 1 1.2 x−coordinate (MD), m

1.4

1.6

1.8

2

Figure 8: MD out-of-plane displacement pro le without bending stiness (2  8 plate). t = 1:2 s, dt = 0:02 s, V = 8 m/s, E = 0, and T = 50 N/m.

0.08

0.06

0.04

Out−of−plane displacement

0.02

0

−0.02 "−": MITC4 (Mindlin/Reissner)

−0.04

"o": 9−node (Kirchhoff) "+": 4−node (Kirchhoff)

−0.06

−0.08

−0.1

−0.12

0

0.2

0.4

0.6

0.8 1 1.2 x−coordinate (MD), m

1.4

1.6

1.8

2

Figure 9: MD out-of-plane displacement pro le with bending stiness (28 plate). t = 1:2 s, dt = 0:02 s, V = 8 m/s, E = 5 GPa, and T = 50 N/m. 8

References [1] K.J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Clis, N.J., 1996. [2] K.J. Bathe and E.N. Dvorkin. A formulation of general shell elements-The use of mixed interpolation of tensorial components. International Journal for Numerical Methods in Engineering, 22:697{722, 1986. [3] F. Brezzi, K.J. Bathe, and M. Fortin. Mixed-interpolated elements for ReissnerMindlin plates. International Journal for Numerical Methods in Engineering, 28:1787{ 1801, 1989. [4] Y.B. Chang and P.M. Moretti. Interaction of uttering webs with surrounding air. Tappi Journal, pages 231{236, March 1991. [5] J.C. Heinrich and D. Connolly. Three-dimensional nite element analysis of self-acting foil bearings. Computer Methods in Applied Mechanics and Engineering, 100:31{43, 1992. [6] A.S. Mujumdar and W.J.M. Douglas. Analytical modeling of sheet utter. Svensk Papperstidning, 6:187{192, 1976. [7] A. Pramila. Natural frequencies of a submerged axially moving band. Journal of Sound and Vibration, 113(1):198{203, 1987. [8] C.A. Tan and C.D. Mote, Jr. Analysis of a hydrodynamic bearing under transverse vibration of an axially moving band. Journal of Tribology, 112:514{523, 1990. [9] X. Wang. Finite element analysis of air-sheet interactions and utter suppression devices. Computers & Structures, 64(5/6):983{994, 1997. [10] X. Wang. Numerical analysis of moving orthotropic thin plates. Computers & Structures, 1998. In press. [11] O.C. Zienkiewicz. The Finite Element Method. McGraw-Hill Publishing Company, third edition, 1977.

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