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Zheng P., Ding X., Yang J, Chen R.; Transverse Vibration of
Papermaking Felt.FIBRES & TEXTILES in Eastern Europe 2010, Vol.
18, No. 3 (80) pp. 106-
106
Transverse Vibration of Papermaking FeltPing Zheng 1, 2 ,Xin
Ding 1, 3 ,Jinkui Yang 2,
Renzhe Chen 4
1. Key Laboratory of Textile Science & Technology,
Ministry of Education,Donghua University,Shanghai 201620,
China
Email: [email protected]
2. Shanghai Golden Bear Fabric Co., Ltd.,Shanghai 201502,
China
3. College of Textiles,
4. College of Mechanical Engineering,Donghua University,
Shanghai 201620, China
Abstract
The papermaking felt of high-speed papermaking machine vibrates
seriously in a trans-verse direction. In this study, the transverse
vibration of papermaking felt was analysed mechanically with
respect to exural rigidity, and a model was developed. The
analyti-cal resolution is obtained by the separation variable
method. The result shows that it isa sinusoidal steady-state
response. The natural frequency is calculated by the principle
of
free vibration. The exural rigidity and density of papermaking
felt are the main factorsin uencing its natural frequency. The
validity of the model is veri ed theoretically and experimentally.
The amplitude of the transverse vibration was tested by an optical
non-contact measuring system. The amplitude-frequency
characteristic curve shows that thenatural frequency measured
agrees well with the theoretical one.
Key words: papermaking felt, exural rigidity, transverse
vibration, natural frequency, PSD.
it is similar to the vibration of general
textiles. Gligorijevic [1] studied yarnoscillation in warp
knitting. The freeoscillatory frequencies of yarns are re-lated to
the length of yarn sections andthe tensioning force. Aubry et al
[2] in-vestigated the vibrating characteristics of textile surfaces
under uniaxial tension.However, they did not consider the
prop-erties of yarn or fabric in their analysis.Matsudaira [3]
discussed the bendingvibration characteristics of fabrics andnew
mechanical parameters which repre-sent this character. The study
evaluated
the wearing characteristics of fabrics, notoperational
stability. Papermaking felt isa special kind of fabric, different
fromyarn and general fabric. As regards paper machines and paper
technology, more at-tention should be paid to the design of
papermaking felt [4]. Papermaking felt isactually a mass-spring
system [5]. Reso-nance can be avoided by designing thestructure of
papermaking felt [5 - 7]. Itis also a damp material that reduces
theamplitude of vibration and even attenu-ates the vibration to
zero. However, the
natural frequency of papermaking feltchanges during its service
life. The exu-ral rigidity of the papermaking felt useddecreases
due to severe wear [8], as a re-sult of which the natural frequency
of the
papermaking felt decreases and gradually becomes closer to the
natural frequencyof the papermaking machine. The natu-ral frequency
of papermaking felt is con-sidered in the designing of Tamfelt
[9].However, the designing and manufactur-ing of papermaking felt
mainly dependon experience.
The transverse vibration of papermakingfelt is studied
mechanically in this paper.An identi ed model is used to
simulate
the transverse vibration of papermaking
felt, in which the exural rigidity is con-sidered. The
experiment designed veri-
es that the model is reliable.
Theoretical analysisof the papermaking felt system
Modelling of papermaking felt
The structure of the press section of a papermaking machine is
shown in Fig-ure 1 . In order to analyse the transversevibration of
papermaking felt, the section
n IntroductionPapermaking felt is used for the con-veyor belt
and ltration belt in the presssection of a papermaking machine.
Thetransverse vibration of papermaking feltincreases steeply during
high-speed op-eration, which will interrupt productionand even
damage the machinery. Nowa-days, growing attention is focused on
theoperational stability of papermaking felt.
The transverse vibration of papermakingfelt is closely related
to its characteris-tics and mechanical devices. Moreover,
2
3
papermaking felt
tension roller
1
Figure 1. Diagram of press section.
w
e i gh t
y
x
0
Figure 2. Sketch map of system.
tension roller
papermaking felt
w e i g
h t
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107FIBRES & TEXTILES in Eastern Europe 2010, Vol. 18, No. 3
(80)
designated as a broken circle is simpli-ed as a sketch map in
Figure 2 .
Under uniaxial tension, the transversevibration of papermaking
felt is some-thing like that of string. Papermaking felthas a
unique base cloth, which makes its
exural rigidity greater than general fab-rics. Therefore, the
transverse vibrationof papermaking felt is similar to that of a
Euler beam.
Equation of motion
The governing differential equation of motion is given using
Newtons secondlaw, which is a combination of the string[10] and
beam models [11].
2 4 2
2 4 2 0 y y d y
T EI m x x dt =
(1)
where:m linear density of the sample in kg/m,T tensioning force
in N, EI exural rigidity of the felt sample in
Nm2.
The boundary conditions are as follows:
y(0,t ) = 0, y(0, t ) = 0,
y(l ,t ) = Ae jwt , y(l ,t ) = 0(2)
The separation variable method is used tosolve Equation 1.
Supposing y = e jm x e jwt ,where m is the wave number, and w
theangular frequency, the equation of mo-tion can be obtained as
follows:
y ( )( 2cosh y a p x= +( ) ( )2 1sinh cosb p x c p x+ + +
(3)
( ))1sin j t d p x e w + where
2 24,
2
T T m EI EI
1 p
2 2
24
2T T m EI p
EI
Substituting the boundary condition inEquation 2 into the
equation of motionin Equation 3 to obtain the integrationconstant,
the equation of motion becomes
( )( )( )
1 2
1 2 2
sinh
sinh
p p x A p p l
y p
+=
+
( )
( )
2 1
1
sin
sin
j t p p x
e p l
w
+
(4)
The equation of motion in Equation 4 isof a standard wave form,
which means
that the behaviour of every point on the papermaking felt is
simple harmonicvibration.
Natural frequency
Substituting the free boundary condition
into Equation 3, we get Equation 5.
In order to obtain a non-zero solutionof Equation 5, the
determinant of theequation system should be zero. That is
( )1sin 0 p l = .The natural frequency of the papermak-ing felt
is
2
2n EI n T
f l m l m
= + (6)
where l - the length of the sample in m.
From Equation 6 we can see that thenatural frequency of
papermaking felt isrelated to the tension T , exural rigidity
EI , linear density m and the length l . For the papermaking
machine, the tension T and length l are invariable. Therefore,
the
exural rigidity EI and linear density m of papermaking felt are
the main factorsin uencing its natural frequency. Thestructure of
papermaking felt is a key
parameter determining the exural rigid-
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
2 1
2 2 1 1
2 2 2 2 2 1 1 1 1
0
0
cosh sinh cos sin 0
cosh cosh cos cos 0
a c
a p cp
a p l b p l c p l d p l
a p p l b p p p l cp p l dp p l
+ = =
+ + + =
+ =
Equation 5.
(5)
papermakingfelt
o'o
x=0 x=l
invisible light source
incident light excident light
PSD signal processing
M
w e i gh t
frequencyconverter
x
y
Figure 3. Diagram of experiment (a).and frequency domain curve
(b).
re ected
0 2 4 6 8 10 12 140
4
8
12
16
y ,
m m
f, Hz
experimentaltheoretical
a)
b)
w e i g
h t
