Investigation of vibrations of a multilayered polymeric film/Daugiasluoksnes polimerines pleveles virpesiu tyrimas.
Ragulskis, K. ; Dabkevicius, A. ; Kibirkstis, E. 等
1. Introduction
The quality of printing production and packaging depends not only
on appropriate application of printing and packaging technologies, but
also on the properties of the materials used, their mechanical and other
characteristics [1]. Every year the application of polymeric materials
in packaging industry is growing, with increasing requirements for their
mechanical resistance. Polymeric materials are being adjusted to higher
requirements for print quality and increased productivity, and the
properties of these materials are being upgraded [2].
During the operation of the printing machine, vibrations of the
polymeric film band, of the transportation, printing and other devices
occur [3], as well as deformations of printing materials. They have a
negative effect on the quality of polymeric films and the graphic images
on them, cause noise, reduce precision, durability and other
characteristics of the printing machine.
In order to increase the quality of protective functions of the
packaging materials from external mechanical, chemical and other effects
and to be able to protect the packaged material the surface of the
packaging material is often coated by other materials or by layers of
several materials [4]. The most common combined material is paper coated
by synthetic films. In such multi-layered materials the paper is located
in internal layer, while the synthetic film protects the paper from
humidity, decreases the transmissibility of gasses and vapours and also
enables to perform thermal welding.
For the investigation of surface deformations of multilayered
polymeric films the method of digital speckle photography can be used
[5, 6]. This method is of very high precision.
After performing the analytical investigation of available research
papers it was found that there are some investigations in which the
application of various speckle and holographic methods for the
investigation of deformations of various surfaces is performed [7, 8],
also the application of the speckle methods for the investigation of
displacements of materials and their stresses is performed [9]. In the
research papers [10, 11] the effects of the types of polymeric films or
of their constituent parts (if the films are multilayered) to
mechanical, optical and barrier properties are analyzed. The authors of
the paper consider that there are insufficient investigations in which
mechanical characteristics and vibrations of multilayered polymeric
materials are analyzed. Because of the mentioned reasons this research
is considered important.
Currently market requirements among the producers of polymeric
packages are increasing. Thus the requirements for mechanical,
thermomechanical, qualitative and other parameters become higher. The
quality of production processes of packages to a large amount depends on
qualitative parameters of the polygraphic materials (polymeric films).
So the purpose of this paper is to investigate vibrations of the
multilayered polymeric films used for production of packages in food
industry and to determine their mechanical parameters.
2. Model for the analysis of multilayered polymeric film vibrations
The multilayered polymeric film is assumed to have equal plate-type
top and bottom layers (in such a layer, the displacements of the top and
bottom planes in the direction of the z axis of the orthogonal Cartesian
system of coordinates are equal), while in between them there is an
elastic inner layer (its displacements of the top and bottom planes in
the direction of the z axis are not necessarily equal) (Fig. 1, a). The
parameters of the outer and inner layers-thickness, density,
Young's modulus, Poisson's ratios--are essentially different.
In the outer layer of a multilayered polymeric film, the subelement
has five nodal degrees of freedom [12, 13]: the displacement of the
layer w1 in the direction of z axis, the displacements of the lower
plane of the layer [u.sub.1] and [v.sub.1] in the directions of x and y
axes, the displacements of the upper plane of the layer [u.sub.2] and
[v.sub.2] in the directions of x and y axes (Fig. 1, b). The length and
width of the polymeric film are chosen to be equal to 0.2 m.
Poisson's ratio of the outer layer [v.sub.1] = 0.3, Young's
modulus [E.sub.1] = 8 MPa, density [[rho].sub.1] = 800 kg/[m.sup.3],
thickness [h.sub.1] = 10 [micro]m.
Further [[THETA].sub.x] and [[THETA].sub.y] denote the rotations
about the axes of coordinates x and y. The displacements due to bending
u and v in the directions of axes x and y are expressed as u = z
[[THETA].sub.y] and v = - z [[THETA].sub.x].
Further u and v denote the displacements of the middle plane of the
outer layer in the directions of x and y axes. Thus [u.sub.1 = u -
[h.sub.1]/2 [[THETA].sub.y], [u.sub.2] = u + [h.sub.1]/2
[[THETA].sub.y], [v.sub.1] = v + [h.sub.1]/2 [[THETA].sub.x], [v.sub.2]
= [h.sub.1]/2 [[THETA].sub.x], where [h.sub.1] is thickness of the outer
layer.
This gives the following expressions:
u = [u.sub.1]+[u.sub.2]/2, v = [v.sub.1]+[v.sub.2]/2,
[[THETA].sub.y] = [u.sub.2]-[u.sub.1]/[h.sub.1], [[THETA].sub.x] =
[v.sub.1]-[v.sub.2]/[h.sub.1].
[FIGURE 1 OMITTED]
The mass matrix has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [[rho].sub.1] is the density of the material of the outer
layer, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [N.sub.i] are shape functions of the finite element.
The stiffness matrix has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where [E.sub.1] is the Young's modulus and [v.sub.1] is the
Poisson's ratio of the outer layer.
In the inner layer of the multilayered polymeric (3) film, the
subelement has six nodal degrees of freedom: the displacements of the
lower plane of the layer [u.sub.1], [v.sub.1], [w.sub.1] and the
displacements of the upper plane of the layer [u.sub.2], [v.sub.2],
[w.sub.2] in the directions of the axes of coordinates (Fig. 1, c).
Poisson's ratio of the inner layer [v.sub.2] = 0.3, Young's
modulus [E.sub.2] = 0.8 MPa, density [[rho].sub.2] = 80 kg/[m.sup.3],
thickness h2 = 100 um. Thickness of the inner layer of the polymeric
film is 10 times larger, while density and Young modulus are 10 times
smaller in comparison with the parameters of the outer layer of the
polymeric film.
The displacements u, v, w in the directions of the axes of
coordinates are represented in the following way
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [h.sub.2] is thickness of the inner layer, {[delta]} is the
vector of nodal displacements, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
The mass matrix has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [[rho].sub.2] is density of the material of the inner layer,
and the following integrals have been taken into account
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
The strains are expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
The stiffness matrix has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
where the following integrals have been taken into account
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
where K = [E.sub.2]/3(1-2[v.sub.2]) is the bulk modulus, G =
[E.sub.2]/2(1+[v.sub.2]) is the shear modulus, [E.sub.2] is Young's
modulus and [v.sub.2] is Poisson's ratio of the inner layer.
3. Results of the analysis of vibrations of the multilayered
polymeric film
The finite element consists of three subelements: the lower and
upper plates (outer layers), and an elastic (inner) layer between them.
The square piece of a polymeric film is analyzed. On the lower and upper
boundaries, all the displacements are assumed as equal to zero.
Contour plots of the transverse displacement for the lower and
upper planes for the first eigenmode are presented in Fig. 2, for the
second eigenmode in Fig. 3, for the fourth eigenmode in Fig. 5.
In the first eigenmode both surfaces move in the same direction,
while in the second eigenmode they move in opposite directions. In the
third eigenmode both surfaces move in the same direction, while in the
fourth eigenmode they move in the opposite directions.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
From the presented contour plots of the eigenmodes of vibrations
obtained during the numerical investigation one can note that the open
horizontal lines take place (Figs. 2 and 3) and lines of the shape of
the part of a circle take place (Figs. 4 and 5). In the first eigenmode
of vibrations (Fig. 2) in the transverse direction to the side
boundaries of the analyzed multilayered polymeric film open horizontal
continuous lines are seen, which bend when approaching the central part
of the film. In the second eigenmode of vibrations (Fig. 3) in the
transverse direction to the side boundaries of the analyzed multilayered
polymeric film open horizontal continuous lines are seen. In the third
(Fig. 4) and fourth (Fig. 5) eigenmodes of vibrations the lines are of
the same shape: along the side boundaries the lines of the shape of the
part of a circle take place. For each of the eigenmodes the contour
plots of the upper and lower surfaces look similar, though in some
eigenmodes both surfaces move in the same direction, while in other
eigenmodes they move in the opposite directions.
4. Method of experimental investigations
In order to investigate the effect of vibrations of multilayered
polymeric film when the band of the polymeric film is transported in
printing machine, the experimental setup for the digital speckle
photography investigations was designed and produced (Fig. 6).
The surface of the multi-layered polymeric film was illuminated
with coherent monohromatic light using the Helium-Neon laser (HN-40,
corresponding to IEC 825-1:1993). This laser generates the light beam
with the wavelength [lambda] = 632.8 nm (part of the spectrum of the red
colour seen by an eye). The main parameters and characteristics of the
laser and of the power source are presented in Table 1.
The laser beam was expanded by using expansion lens and directed to
the side of the investigated tape of polymeric material by using the
system of mirrors. By changing frequency and amplitude the shapes of the
obtained speckle images varied. At definite values of frequencies and
amplitudes the images of the eigenmodes occurred.
[FIGURE 6 OMITTED]
Speckle images were registered with high resolution colour digital
image camera Edmund Optics EO-1312C USB Camera. Each photo was
registered with the time interval of 40 ms. Quality of the photo did not
change when changing the time interval between the registrations. The
main characteristics of the image camera are presented in Table 2.
Focusing lens Edmund Industrial Optics 55326 Double Gauss Focus 25
mm was used with the video-camera, which has the lens of wide viewing
field and variable aperture. The main characteristics of the focusing
lens are presented in Table 3.
The camera transferred the obtained images to personal computer
through USB 2.0 connection where the obtained images were processed with
the specialised software uc480viewer (version 2.40.0005) and their
correlation analysis was performed.
In the experimental tests, samples of multilayer polymeric film of
square geometrical shape (0.2 x 0.2 m) were produced.
The tests were carried out at the ambient temperature of 20 [+ or
-] 2[degrees]C and air humidity 65 [+ or -] 2%.
5. Results of experimental investigations and their analysis
The obtained results of the experimental investigations are
presented in Fig. 7: in Fig. 7, a the image of the first eigenmode, in
Fig. 7, b the image of the second eigenmode, in Fig. 7, d the image of
the fourth eigenmode of the multilayered polymeric film PET+PAP+LDPE are
presented.
[FIGURE 7 OMITTED]
The analysis of the obtained results, generated by vibrations of
multilayered polymeric film PET+PAP+ +LDPE, shows that overall four
shapes of eigenmodes were obtained (Figs. 7, a-d), which differ in their
geometrical shape, generating frequency and amplitude: at a higher
number of the eigenmode the generating frequency is higher, and the
amplitude decreases with increasing frequency. After the four eigenmodes
are formed, higher modes are not formed at increased excitation
amplitude and frequency, because the polymeric film vibrates not evenly,
speckle images are located non-uniformly and it is difficult to identify
the eigenmodes.
When comparing the results obtained by experimental method (Figs.
7, a-d) with the results of digital study (Figs. 2-5), it may be noted
that the results according to the sequence of eigenmode shapes correlate
with the digital model of the multilayered polymeric film. Obvious
correspondence can be noted in terms of the geometric configuration of
the shapes and the shape sequence.
Comparing the obtained experimental and numerical results of
contour plots of the field of transverse displacements of the
first-fourth eigenmodes of vibrations one can see that the experimental
results obtained using the method of speckle photography correspond to
the data from numerical investigations.
6. Conclusions
It is assumed that a layered plate has a lower layer and an upper
layer of a plate type and between them there is an elastic layer. The
finite element consists of three subelements: the lower and upper plates
and an elastic layer between them. The square piece of polymeric film is
analyzed. On the lower and upper boundaries all the displacements are
assumed equal to zero.
Contour plots of the transverse displacement for the lower and
upper planes for the first eigenmodes are presented. From the obtained
results it is seen that there are eigenmodes in which both surfaces move
in the same direction and also there are similar eigenmodes in which
both surfaces move in the opposite directions.
The setup for experimental investigations was designed and created
which enabled to determine the images of eigenmodes of the multilayered
polymeric film PET+PAP+LDPE by using the method of digital speckle
photography.
When comparing the configurations of the images of eigenmodes of
the multilayered polymeric film PET+PAP+LDPE obtained by using the
numerical study and the method of digital speckle photography, it may be
noted that the results according to the sequence of eigenmode shapes
correlate with the digital model of the multilayered polymeric film.
The obtained results are used in the process of design of
construction of packaging elements.
Received October 25, 2009 Accepted December 04, 2009
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[12.] Bathe, K.J. Finite Element Procedures in Engineering
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K. Ragulskis *, A. Dabkevicius **, E. Kibirkstis ***, V. Bivainis
****, V. Miliunas *****, L. Ragulskis ******
* Kaunas University of Technology, Kqstucio 27, 44312 Kaunas,
Lithuania, E-mail: kazimieras3@hotmail.com
** Kaunas University of Technology, Studentu 56, 51424 Kaunas,
Lithuania, E-mail: arturas.dabkevicius@ktu.lt
*** Kaunas University of Technology, Studentu 56, 51424 Kaunas,
Lithuania, E-mail: edmundas.kibirkstis@ktu.lt
**** Kaunas University of Technology, Studentu 56, 51424 Kaunas,
Lithuania, E-mail: vaidas.bivainis@ktu.lt
***** Kaunas University of Technology, Studentu 56, 51424 Kaunas,
Lithuania, E-mail: valdas.miliunas@ktu.lt
****** Vytautas Magnus University, Vileikos 8, 44404 Kaunas,
Lithuania, E-mail: l.ragulskis@if.vdu.lt
Table 1
Main parameters of the laser HN-40
Parameter Value
Parameters of Radiated wavelength, nm 632.8
the laser Maximum power, mW 39
Polarisation 1000:1
Diameter of the beam, mm 2.1 ([+ or -]0.1)
Expansion, mrad 2.1
Table 2
Main technical characteristics of the image camera
Edmund Optics EO-1312C USB Camera
Model EO-1312C USB Camera
Sensor Type 1/2" Progressive Scan CMOS
Pixels (H x V) 1280 x 1024
Pixel Size (H x V) 5.2 [micro]m x 5.2 |im
Sensing Area (H x V) 6.6 mm x 5.3 mm
Pixel Depth 8-bitm
Frame Rate 25 fps
Resolution >100 lp/mm image space resolution
50% @ F4, 400 mm WD
(18 mm FL meets 100 lp/mm with
max 2/3" CCD)
Distortion <0.3% @ 400mm WD
<1% @ 400mm WD for 18 mm FL
Dimensions (W x H x L) 34 x 32 x 27.4 mm
Table 3
Main characteristics of the focusing lens Edmund
Industrial Optics 55326 Double Gauss Focus 25 mm
Min. Max.
Primary Magnification 0.106 X [infinity]
FOV (2/3" CCD Hor) 87.5 mm 20.7[degrees]
FOV (1/2" CCD Hor) 63.6 mm 15.1[degrees]
Resolution in Object 11 lp/mm N/A
Space (1/2" CCD)
Working Distance 240 mm CO
Focal Length 25 mm
Aperture (f/#) F4 - closed
Distance to First Lens 21.4~24.1 mm
Filter Thread M 30.5 x 0.5 mm
