Abstract

The prediction and measurement of vibrations of the low-frequency transverse modes of tensioned webs are of increasing interest for process monitoring, quality control, and process stability in roll-to-roll flexible hybrid and stretchable electronics manufacturing, nanomanufacturing, coated layer patterning, and other continuous manufacturing technologies. Acting as distributed added mass, the surrounding air significantly affects the frequency responses of taut thin webs in ambient roll-to-roll processes in comparison with those in vacuo. In this paper, we present closed-form, semi-analytical, universal hydrodynamic functions used to accurately predict the lowest symmetric and anti-symmetric transverse frequency response for any uniaxially tensioned web of arbitrary material and aspect ratio used in roll-to-roll processes. Experimental validation is carried out by using pointwise laser measurements of acoustically excited webs with different pre-tensions, web materials, and aspect ratios. These closed-form hydrodynamic functions provide roll-to-roll process designers a convenient way to predict the lowest frequencies of such web systems without the need to resort to computationally intensive methods; alternately, these functions allow for the quick identification of conditions when air-coupling is important to determine the web’s vibration response. The results presented herein are expected to help ongoing efforts to improve process monitoring and control in a variety of roll-to-roll continuous manufacturing technologies.

References

References
1.
Shivakumar
,
S.
,
2014
,
The Flexible Electronics Opportunity
,
The National Academies Press
,
Washington, DC
.
2.
Cao
,
Q.
,
Kim
,
H.
,
Pimparkar
,
N.
,
Kulkarni
,
J. P.
,
Wang
,
C.
,
Shim
,
M.
,
Roy
,
K.
,
Alam
,
M. A.
, and
Rogers
,
J. A.
,
2008
, “
Medium-Scale Carbon Nanotube Thin-Film Integrated Circuits on Flexible Plastic Substrates
,”
Nature
,
454
(
7203
), pp.
495
500
. 10.1038/nature07110
3.
Krebs
,
F. C.
,
Tromholt
,
T.
, and
Jørgensen
,
M.
,
2010
, “
Upscaling of Polymer Solar Cell Fabrication Using Full Roll-to-Roll Processing
,”
Nanoscale
,
2
(
6
), pp.
873
886
. 10.1039/b9nr00430k
4.
Sekitani
,
T.
,
Zschieschang
,
U.
,
Klauk
,
H.
, and
Someya
,
T.
,
2010
, “
Flexible Organic Transistors and Circuits With Extreme Bending Stability
,”
Nat. Mater.
,
9
(
12
), pp.
1015
1022
. 10.1038/nmat2896
5.
Webb
,
R. C.
,
Bonifas
,
A. P.
,
Behnaz
,
A.
,
Zhang
,
Y.
,
Yu
,
K. J.
,
Cheng
,
H.
,
Shi
,
M.
,
Bian
,
Z.
,
Liu
,
Z.
,
Kim
,
Y. S.
,
Yeo
,
W. H.
,
Park
,
J. S.
,
Song
,
J.
,
Li
,
Y.
,
Huang
,
Y.
,
Gorbach
,
A. M.
, and
Roger
,
J. A.
,
2013
, “
Ultrathin Conformal Devices for Precise and Continuous Thermal Characterization of Human Skin
,”
Nat. Mater.
,
12
(
10
), pp.
938
944
. 10.1038/nmat3755
6.
Palavesam
,
N.
,
Marin
,
S.
,
Hemmetzberger
,
D.
,
Landesberger
,
C.
,
Bock
,
K.
, and
Kutter
,
C.
,
2018
, “
Roll-to-Roll Processing of Film Substrates for Hybrid Integrated Flexible Electronics
,”
Flexible Printed Electron.
,
3
(
1
), p.
014002
. 10.1088/2058-8585/aaaa04
7.
Cooper
,
K. P.
,
2019
, “
Advanced and Nano Manufacturing Research at NSF
,”
Micro- and Nanotechnology Sensors, Systems, and Applications XI
, Conference 1098213 (May), 38.
8.
Lu
,
Y.
, and
Pagilla
,
P. R.
,
2014
, “
Modeling of Temperature Distribution in Moving Webs in Roll-to-Roll Manufacturing
,”
ASME J. Therm. Sci. Eng. Appl.
,
6
(
4
), p.
041012
. 10.1115/1.4028048
9.
Feng
,
D.
, and
Raman
,
A.
,
2019
, “
Thermomechanics of Axially Moving Webs in Roll-to-Roll Manufacturing Processes
,”
Int. J. Heat Mass Transfer
,
129
, pp.
1317
1327
. 10.1016/j.ijheatmasstransfer.2018.10.038
10.
Raman
,
A.
,
Wolf
,
K. D.
, and
Hagedorn
,
P.
,
2001
, “
Observations on the Vibrations of Paper Webs
,”
Proceedings of 11th Annual Symposium on Information Storage and Processing Systems
,
Santa Clara, CA
,
June 10–13
, pp.
415
429
.
11.
Vaughan
,
M.
, and
Raman
,
A.
,
2010
, “
Aeroelastic Stability of Axially Moving Webs Coupled to Incompressible Flows
,”
ASME J. Appl. Mech.
,
77
(
2
), p.
021001
. 10.1115/1.2910902
12.
Nguyen
,
Q. C.
, and
Hong
,
K. S.
,
2012
, “
Transverse Vibration Control of Axially Moving Membranes by Regulation of Axial Velocity
,”
IEEE Trans. Control Syst. Technol.
,
20
(
4
), pp.
1124
1131
. 10.1109/TCST.2011.2159384
13.
Subbaraman
,
H.
,
Lin
,
X.
,
Xu
,
X.
,
Dodabalapur
,
A.
,
Guo
,
L. J.
, and
Chen
,
R. T.
,
2012
, “
Metrology and Instrumentation Challenges With High-Rate, Roll-to-Roll Manufacturing of Flexible Electronic Systems
,”
Instrumentation, Metrology, and Standards for Nanomanufacturing, Optics, and Semiconductors VI
,
San Diego, CA
,
Oct. 11
, Vol.
8466
, p.
846603
.
14.
Kimbrough
,
B.
, and
Novak
,
E.
,
2015
, “
In-line Roll-to-Roll Metrology for Flexible Electronics
,”
Appl. Adv. Opt. Metrol. Solutions
,
9576
, p.
957603
. 10.1117/12.2189905
15.
Eum
,
S.
,
Lee
,
J.
, and
Nam
,
K.
,
2016
, “
Robust Tension Control of Roll to Roll Winding Equipment Based on a Disturbance Observer
,”
IECON Proceedings (Industrial Electronics Conference)
,
Florence, Italy
,
Oct. 23–26
, pp.
625
630
.
16.
Ma
,
L.
,
Chen
,
J.
,
Tang
,
W.
, and
Yin
,
Z.
,
2017
, “
Transverse Vibration and Instability of Axially Travelling Web Subjected to Non-Homogeneous Tension
,”
Int. J. Mech. Sci.
,
133
, pp.
752
758
. 10.1016/j.ijmecsci.2017.09.047
17.
Ma
,
L.
,
Chen
,
J.
,
Tang
,
W.
, and
Yin
,
Z.
,
2018
, “
Vibration-Based Estimation of Tension for an Axially Travelling web in Roll-to-Roll Manufacturing
,”
Meas. Sci. Technol.
,
29
(
1
), pp.
1
8
. 10.1088/1361-6501/aa9046
18.
Ali
,
S.
, and
Hawwa
,
M. A.
,
2019
, “
A Parametric Study on the Dynamics of Two-Span Roll-to-Roll Microcontact Printing System
,”
Sadhana—Acad. Proc. Eng. Sci.
,
44
(
5
), pp.
1
11
. 10.1007/s12046-019-1113-1
19.
Lee
,
J.
,
Shin
,
K. H.
, and
Kang
,
H.
,
2019
, “
Design of a Register Controller Considering Inherent Characteristics of a Roll-to-Roll Continuous Manufacturing System
,”
Int. J. Adv. Manuf. Technol.
,
102
, pp.
3725
3737
. 10.1007/s00170-019-03428-4
20.
Dorrestijn
,
M.
,
Bietsch
,
A.
,
Açikalin
,
T.
,
Raman
,
A.
,
Hegner
,
M.
,
Meyer
,
E.
, and
Gerber
,
C.
,
2007
, “
Chladni Figures Revisited Based on Nanomechanics
,”
Phys. Rev. Lett.
,
98
(
2
), pp.
1
4
. 10.1103/PhysRevLett.98.026102
21.
Vuillermet
,
G.
,
Gires
,
P. Y.
,
Casset
,
F.
, and
Poulain
,
C.
,
2016
, “
Chladni Patterns in a Liquid at Microscale
,”
Phys. Rev. Lett.
,
116
(
18
), pp.
2
6
. 10.1103/PhysRevLett.116.184501
22.
Shabaniverki
,
S.
,
Thorud
,
S.
, and
Juárez
,
J. J.
,
2018
, “
Protocol for Assembling Micro- and Nanoparticles in a Viscous Liquid Above a Vibrating Plate
,”
MethodsX
,
5
, pp.
1156
1165
. 10.1016/j.mex.2018.09.008
23.
Shabaniverki
,
S.
,
Thorud
,
S.
, and
Juárez
,
J. J.
,
2018
, “
Vibrationally Directed Assembly of Micro- and Nanoparticle-Polymer Composites
,”
Chem. Eng. Sci.
,
192
, pp.
1209
1217
. 10.1016/j.ces.2018.06.068
24.
Finn III
,
M.
,
Martens
,
C. J.
,
Zaretski
,
A. V.
,
Roth
,
B.
,
Søndergaard
,
R. R.
,
Krebs
,
F. C.
, and
Lipomi
,
D. J.
,
2017
, “
Mechanical Stability of Roll-to-Roll Printed Solar Cells Under Cyclic Bending and Torsion
,”
Sol. Energy Mater. Sol. Cells
,
174
, pp.
7
15
. 10.1016/j.solmat.2017.08.015
25.
Niemi
,
J.
, and
Pramila
,
A.
,
1987
, “
FEM-Analysis of Transverse Vibrations of an Axially Moving Membrane Immersed in Ideal Fluid
,”
Int. J. Numer. Methods Eng.
,
24
(
12
), pp.
2301
2313
. 10.1002/nme.1620241205
26.
Koivurova
,
H.
, and
Pramila
,
A.
,
1997
, “
Nonlinear Vibration of Axially Moving Membrane by Finite Element Method
,”
Comput. Mech.
,
20
(
6
), pp.
573
581
. 10.1007/s004660050277
27.
Chang
,
Y. B.
, and
Moretti
,
P. M.
,
2002
, “
Flow-Induced Vibration of Free Edges of Thin Films
,”
J. Fluids Struct.
,
16
(
7
), pp.
989
1008
. 10.1006/jfls.2002.0456
28.
Hara
,
K.
, and
Watanabe
,
M.
,
2013
, “
Stability Analysis of Rectangular Plates in Incompressible Flow With Fourier Multiplier Operators
,”
ASME 2013 Pressure Vessels and Piping Conference
,
Paris, France
,
July 14–18
, pp. V004T04A044–V004T04A044.
29.
Hara
,
K.
, and
Watanabe
,
M.
,
2014
, “
Formulation of the Aeroelastic Instability Problem of Rectangular Plates in Uniform Flow Based on the Hamiltonian Mechanics for the Constrained System
,”
ASME 2014 Pressure Vessels and Piping Conference
,
Anaheim, CA
,
July 20–24
, pp. V004T04A060–V004T04A060.
30.
Yao
,
G.
, and
Zhang
,
Y. M.
,
2016
, “
Dynamics and Stability of an Axially Moving Plate Interacting With Surrounding Airflow
,”
Meccanica
,
51
(
9
), pp.
2111
2119
. 10.1007/s11012-016-0365-7
31.
Kulachenko
,
A.
,
Gradin
,
P.
, and
Koivurova
,
H.
,
2007
, “
Modelling the Dynamical Behaviour of a Paper Web. Part II
,”
Comput. Struct.
,
85
(
3–4
), pp.
148
157
. 10.1016/j.compstruc.2006.09.007
32.
Bidkar
,
R. A.
,
Raman
,
A.
, and
Bajaj
,
A. K.
,
2008
, “
Aeroelastic Stability of Wide Webs and Narrow Ribbons in Cross Flow
,”
ASME J. Appl. Mech.
,
75
(
4
), p.
041023
. 10.1115/1.2871192
33.
Sader
,
J. E.
,
1998
, “
Frequency Response of Cantilever Beams Immersed in Viscous Fluids With Applications to the Atomic Force Microscope
,”
J. Appl. Phys.
,
84
(
1
), pp.
64
76
. 10.1063/1.368002
34.
Green
,
C. P.
, and
Sader
,
J. E.
,
2002
, “
Torsional Frequency Response of Cantilever Beams Immersed in Viscous Fluids With Applications to the Atomic Force Microscope
,”
J. Appl. Phys.
,
92
(
10
), pp.
6262
6274
. 10.1063/1.1512318
35.
Chu
,
W. H.
,
1963
, “
Vibration of Fully Submerged Cantilever Plates in Water
,”
South-West Research Institute
, Technical Report 2.
36.
Meyerhoff
,
W. K.
,
1970
, “
Added Mass of Thin Rectangular Plates Calculated From Potential Theory
,”
J. Ship Res.
,
14
(
1
), pp.
100
111
10.1300/J087v40n03_08
37.
Yadykin
,
Y.
,
Tenetov
,
V.
, and
Levin
,
D.
,
2003
, “
The Added Mass of a Flexible Plate Oscillating in a Fluid
,”
J. Fluids Struct.
,
17
(
1
), pp.
115
123
. 10.1016/S0889-9746(02)00100-7
38.
Ulsoy
,
A. G.
, and
Mote
,
C. D.
,
1982
, “
Vibration of Wide Band Saw Blades
,”
ASME J. Eng. Ind.
,
104
(
1
), pp.
71
78
. 10.1115/1.3185801
39.
Turnbull
,
P. F.
,
Perkins
,
N. C.
, and
Schultz
,
W. W.
,
1995
, “
Contact-Induced Nonlinearity in Oscillating Belts and Webs
,”
J. Vib. Control
,
1
(
4
), pp.
459
479
. 10.1177/107754639500100404
40.
ABAQUS
6.13 Analysis User's Guide.
Dassault Systèmes Simulia Corp
,
Providence, RI
.
41.
Applied Materials, Inc.
,
SMARTWEB®‌ Roll-to-Roll Coating System for State-of-the art Flexible Electronics, https://3.imimg.com/data3/LQ/YL/MY-956144/applied-smart-web.pdf, Accessed June 11, 2019
.
42.
TECNAU, Inc.
,
Roll 20 Roll-to-Roll machine, https://www.tecnau.com/product/roll-20/, Accessed June 11, 2019
.
43.
Davis-Standard, LLC
, .
You do not currently have access to this content.