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81Mikołajczyk Z..; Optimisation of the Knitting Process on Warp-Knitting Machines in the Aspect of the Feeding Zone Geometry.FIBRES & TEXTILES in Eastern Europe 2011, Vol. 19, No. 4 (87) pp. 81-88.
Introduction
The process of feeding of the knitting zone with warp threads on warp-knitting machines is determined by the follow-ing three groups of parameters: The first group characterises the structure of the feeding device: m – point mass of the tension rail reduced to one thread, ks – coefficient of the rigidity of the tension rail, reduced to one thread, l1 – length of the thread from the tension rail to the guide bar, l2 – length of the thread from the beam to the tension rail, a – angle of the horizontal deviation of the thread “running-on” the tension rail, b – angle of the vertical deviation of the thread “running-of” the tension rail, ρ – angle of wrapping the tension rail with thread, the kind of surface of the tension rail plate determined by the coefficient of friction μ between the tension rail and the thread, np, vp – rotational and linear speed of feeding. The second group re-fers to the mechanical properties of the threads, treated as a visco-elastic object and includes: d the coefficient of thread elasticity kp and the coefficient of thread attenuation bp. The third significant group of parameters are the components of the kinematic input function S’(t) which re-sult from the specificity of the process of forming courses of loops on the warp knitted machine. The parameters model-ling the changes in the absolute elonga-tion of threads in the feeding zone are the following: Sp(t) – input function for the process of knitting the isotropic structures of the stitches, determined by the follow-ing: the dislocation of loop-forming ele-
OptimisationoftheKnittingProcessonWarp-KnittingMachinesintheAspectoftheFeedingZoneGeometryAbstractAn analysis of the structure of warp-knitting machines shows a great variety of the parame-ters of the knitting zone geometry, rigidity and attenuation of the tension rail system, a coef-ficient of friction of the threads against guiding elements and the character of the kinematic input function of loop forming elements. In the process of technical design, the constructors concentrate on the kinematics and dynamics of the components of the machine, the kind and properties of the materials used, optimisation of overall constructional dimensions etc. In many cases the aspects concerning the technology of manufacturing knitted fabrics are ignored at the stage of assumptions and design concepts. A significant element in the modelling of the construction design of a warp-knitting machine is the process of optimisa-tion of the machine feeding parameters, considering the minimum load of the threads being fed. This paper presents a simulation of the feeding zone in order to analyse and optimise its working conditions.
Key words: knitting process, optimisation, warp-knitting machine, feeding zone.
Zbigniew Mikołajczyk TechnicalUniversityofLodz,
FacultyofMaterialTechnologiesandTextileDesign,
DepartmentofKnittingTechnologyul. Żeromskiego 116, 90-924 Łodź, Poland,
E-mail: [email protected]
ments in the warp-knitting machine, the process of reversible dislocations of the threads in the knitting zone, the type of stitch, conditions of the take-up of the fabric and the linear speed of the threads unwinding vp; DS(t) – input function de-termined by the variation of the gradient of the knitting-in, characteristic of aniso-tropic warp knitted fabrics only.
A physical model of the constant-length feeding system presents the groups of parameters mentioned above (Figure 1, see page 82). The keynote and aim of this research is to present and analyse the dependencies describing the influence of the first group of geometry parameters of the feeding zone on the character and val-ues of the dynamic loads of the threads.
The practical aim of this research was to optimise the process of knitting on warp knitting machines by outlining the posi-tive and advantageous areas of the input parameters, which will make the techno-logical process a “correct” one with re-gard to the minimum loads of the threads, without breaking them or, in the extreme situations, damaging the loop-forming el-ements of the machine. The investigations were carried out with the use of computer simulation based on a mathematical mod-el of the constant-length feeding process on warp knitting machines, which is also empirically verified [1-4].
Mathematicalmodelofthefeedingsystem
The equation of the motion of the ten-sion rail referring to the mechanics of the vibration of the system, with one degree of freedom can take the following form Equation 1 (see page 82), where:
h – relative coefficient of system attenuation,
m
bh zr
2= (2),
w0 – frequency of the free vibration of the non-attenuated system,
mk zr=0ω (3),
ai – coefficient describing the geometry of the feeding system,
12
sincoslel
eai +⋅⋅−
=±
±
µρ
µρ αβ (4),
bzri – coefficient of system attenuation,
( ) ipzri abb ⋅−= αβ sincos (5),
kzri – coefficient of system elasticity,
( ) piszri kakk ⋅−+= αβ sincos (6)
The solution of equation (1) fulfilling zero initial conditions takes the follow-ing form Equation 7 .
Forces in the threads are determined ac-cording to the following dependence Equation 8 (see page 82).
( ) ( )
+⋅⋅=++
dttdSbtSk
ma
ydtdyh
dtyd
pipii ''2 2
02
2
ω
Equations 1.
(6)
82 FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
The solution of the mathematical model of the feeding process was based on the numerical and analytical method [1]. The input functions S(t)k, given in a discrete way, are developed into a trigonometric Fourier series. The solution, in the form of dislocations of the tension rail in an analytical way, is an integral of the con-volution of the input function and the response to the impulse input function of the system, which was determined by numerical integration using the Gauss method. The results of the calculations are presented in the form of text files of model coefficients ai, bzri, kzri, hi, w0i and wi of the deflection value of the tension rail y(t), the values of forces in the threads P1(t)k, the derivatives of input function SS(t)k and the input function S(t)k.
The most important feature of the com-puter program elaborated, which is used for simulation of the knitting process in the aspect of feeding, is that it supports the process of the technological design of warp-knitted fabrics.
Simulationtestsoftheknittingprocesswithrespecttothediverseparametersofthefeedingzone
Characteristic of the feeding zone
The identification of the geometry of the feeding zone conducted for particular warp-knitting machines of the follow-ing types: R5NF (E=12), RJSC (E=12), MRSEJ 43/1 (E=18) K2MPS (E20), K13 (E12), RL5NF (E12), made by the K. Mayer company, and Kokett 5223 (E28), made by the Textima company, showed significant diversity in the parameters of the geometry of the feeding zone [5, 6].
The overall length of threads between the beam and guide bar was within the range of 916 to 3249 mm, the partial lengths: l1 = 591÷2525 mm & l2 = 220÷1025 mm; the number of elements (barriers) guid-
ing the threads was from 3 to 10, the summary angle of wrapping the barri-ers γc = 96°÷619°, the angle of threads “running-on” the tension rail α = 1°÷62°, the angle of the threads “running-of” the tension rail β = 80°÷-15°.
For each particular type of machine there is a different way of feeding the threads. For example, on an RJSC warp-knitting machine the threads are fed using a constant-tension or constant-length con-troller, which unwinds the warp threads, made on the basis of a mechanical or electronic control device. On an MRSEJ warp-knitting machine the threads are unwound from the beams, or in the case of the designing wefts, from the creel. On warp-knitting machines of the type MRSEJ and K13, used for producing lac-es, the classic tension rail in the feeding zone is replaced with individual spring compensators through which the pattern threads are guided.
Characteristic parameters of the structure of tension rail devices include the fol-lowing: profile, surface finish, the mass of the tension rail plate, and the rigidity of the tension rail springs.
For the most commonly used steel tension rail plates with their surface polished, the coefficient of friction μ depends on the structure of the threads’ surface, which, unwound from the beams, run across the tension rail. Moreover the parameter of the linear mass of the tension rail plate is constant and equals mpl ≈ 85g/m. The in-
dividual mass Num
m pl= , referring to one
thread and depending only on the needle gauge of the warp-knitting machine.
The rigidity coefficients of the spring ksp in the warp-knitting machine are selected depending on the elastic properties of the threads, their thickness, the configuration, and the number of springs in the particular length of the tension rail plate. In warp-
knitting machines the most commonly used are flat tension rail springs of a rigid-ity within the range 12-168 cN/mm.
For the synthetic threads which are most commonly processed on warp-knitting machines, with an average needle gauge of ks = 0,5÷0,6 cN/mm.
In the process of technical (and eco-nomic) design, machine construction designers concentrate on the following: the kinetics and dynamics of working machine assemblies, the type and proper-ties of the materials used, optimisation of the constructional overall dimensions of machines, and the safety and ergonomics of machine operation [7].
In many cases, aspects concerning the technology of manufacturing knitted fab-rics are ignored at the stage of assump-tions and design concepts.
A significant element in the modelling of the construction design of warp-knitting machines is the process of optimisation of the geometry of the feeding zone from the point of view of the minimum loads of the threads being fed.
Identification of the feeding system on warp-knitted machines in the aspect of an analysis of coefficients of the feeding zone geometry and the dynamic loads of the threads.An analysis of the feeding process with regard to the optimisation of the loads of threads was conducted on the basis of the dependencies between the coefficients of the geometry of feeding ai and the input parameters of the system, as well as the influence of changes in particular param-eters on the forces in the threads. In the description of the process of feeding in warp-knitting machines, coefficient ai is described by Equation (4), where coef-ficients a1, a2 and a3 depend on the three conditions of t movement of the threads on the tension rail plate [1, 8].
, (7)
( )( ) ( )
( ) ( ) ( )
( )
⋅−⋅−
−⋅−⋅−⋅+⋅⋅
+⋅=
±
dtdyb
ykdttdS
btSk
lelktP
p
pk
pkp
αβ
αβµρ
sincos
sincos11
1
121 . (8)
Equations 7 and 8.
ai – coefficient describing the geometry of the feeding system, 12
sincos
lel
eai
(4),
bzri – coefficient of system attenuation, ipzri abb sincos (5),
kzri – coefficient of system elasticity, piszri kakk sincos (6).
The solution of equation (1) fulfilling zero initial conditions takes the following form:
dted
dSbSk
m
tyyh
tyety
tht N i
k
ik
p
N i
kikp
ht
ia
sin
sincos
0 11
00
0
˙
(7).
Forces in the threads are determined according to the following dependence:
dt
dyb
ykdt
tdSbtSk
lelktP
p
p
k
pkp
sincos
sincos1
1
1
12
1 (8).
The solution of the mathematical model of the feeding process was based on the
numerical and analytical method [1]. The input functions S(t)k ,given in a discrete way, are
developed into a trigonometric Fourier series. The solution, in the form of dislocations of the
tension rail in an analytical way, is an integral of the convolution of the input function and the
response to the impulse input function of the system, which was determined by numerical
integration using the Gauss method. The results of the calculations are presented in the form
of text files of model coefficients ai, bzri, kzri, hi, 0i and i of the deflection value of the
tension rail y(t), the values of forces in the threads P1(t)k, the derivatives of input function
SS(t)k and the input function S(t)k.
The most important feature of the computer program elaborated, which is used for
simulation of the knitting process in the aspect of feeding, is that it supports the process of the
technological design of warp-knitted fabrics.
2. Simulation tests of the knitting process with respect to the diverse parameters of the
feeding zone
Characteristic of the feeding zone
The identification of the geometry of the feeding zone conducted for particular warp-
knitting machines of the following types: R5NF (E=12), RJSC (E=12), MRSEJ 43/1 (E=18)
K2MPS (E20), K13 (E12), RL5NF (E12), made by the K. Mayer company, and Kokett 5223
(E28), made by the Textima company, showed significant diversity in the parameters of the
geometry of the feeding zone [5, 6].
The overall length of threads between the beam and guide bar was within the range of
916 to 3249 mm, the partial lengths: l1=591÷2525 mm & l2=220÷1025 mm; the number of
elements (barriers) guiding the threads was from 3 to 10, the summary angle of wrapping the
barriers γc=96°÷619°, the angle of threads “running-on” the tension rail α=1°÷62°, the angle
of the threads “running-of” the tension rail β=80°÷-15°.
For each particular type of machine there is a different way of feeding the threads. For
example, on an RJSC warp-knitting machine the threads are fed using a constant-tension or
constant-length controller, which unwinds the warp threads, made on the basis of a
mechanical or electronic control device. On an MRSEJ warp-knitting machine the threads are
unwound from the beams, or in the case of the designing wefts, from the creel. On warp-
knitting machines of the type MRSEJ and K13, used for producing laces, the classic tension
3
ai – coefficient describing the geometry of the feeding system, 12
sincos
lel
eai
(4),
bzri – coefficient of system attenuation, ipzri abb sincos (5),
kzri – coefficient of system elasticity, piszri kakk sincos (6).
The solution of equation (1) fulfilling zero initial conditions takes the following form:
dted
dSbSk
m
tyyh
tyety
tht N i
k
ik
p
N i
kikp
ht
ia
sin
sincos
0 11
00
0
˙
(7).
Forces in the threads are determined according to the following dependence:
dt
dyb
ykdt
tdSbtSk
lelktP
p
p
k
pkp
sincos
sincos1
1
1
12
1 (8).
The solution of the mathematical model of the feeding process was based on the
numerical and analytical method [1]. The input functions S(t)k ,given in a discrete way, are
developed into a trigonometric Fourier series. The solution, in the form of dislocations of the
tension rail in an analytical way, is an integral of the convolution of the input function and the
response to the impulse input function of the system, which was determined by numerical
integration using the Gauss method. The results of the calculations are presented in the form
of text files of model coefficients ai, bzri, kzri, hi, 0i and i of the deflection value of the
tension rail y(t), the values of forces in the threads P1(t)k, the derivatives of input function
SS(t)k and the input function S(t)k.
The most important feature of the computer program elaborated, which is used for
simulation of the knitting process in the aspect of feeding, is that it supports the process of the
technological design of warp-knitted fabrics.
2. Simulation tests of the knitting process with respect to the diverse parameters of the
feeding zone
Characteristic of the feeding zone
The identification of the geometry of the feeding zone conducted for particular warp-
knitting machines of the following types: R5NF (E=12), RJSC (E=12), MRSEJ 43/1 (E=18)
K2MPS (E20), K13 (E12), RL5NF (E12), made by the K. Mayer company, and Kokett 5223
(E28), made by the Textima company, showed significant diversity in the parameters of the
geometry of the feeding zone [5, 6].
The overall length of threads between the beam and guide bar was within the range of
916 to 3249 mm, the partial lengths: l1=591÷2525 mm & l2=220÷1025 mm; the number of
elements (barriers) guiding the threads was from 3 to 10, the summary angle of wrapping the
barriers γc=96°÷619°, the angle of threads “running-on” the tension rail α=1°÷62°, the angle
of the threads “running-of” the tension rail β=80°÷-15°.
For each particular type of machine there is a different way of feeding the threads. For
example, on an RJSC warp-knitting machine the threads are fed using a constant-tension or
constant-length controller, which unwinds the warp threads, made on the basis of a
mechanical or electronic control device. On an MRSEJ warp-knitting machine the threads are
unwound from the beams, or in the case of the designing wefts, from the creel. On warp-
knitting machines of the type MRSEJ and K13, used for producing laces, the classic tension
3
83FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
rys. 1
400
1000
1600
61
0
76
0
91
0
10
60
12
10
13
60
15
10
16
60
18
10
19
60
21
10
0
0,00002
0,00004
0,00006
0,00008
0,0001
0,00012
0,00014
0,00016
a1
,mm
-1
length of the thread
l2,mm
coefficient of the system geometry a1=f(l
1,l
2)
0-0,00002 0,00002-0,00004 0,00004-0,00006 0,00006-0,00008
0,00008-0,0001 0,0001-0,00012 0,00012-0,00014 0,00014-0,00016
length of the thread
l1, mm
rys. 2
coefficient of the system geometry a1=f(l1) for
l2=const.
y = 0,0037x-0,53
R2 = 0,9895
y = 0,0003x-0,2534
R2 = 0,9743
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
400 600 800 1000 1200 1400 1600 1800
length of the thread l1, mm
co
eff
icie
nt
a1,
mm
-1
l2=610mm l2=760mm l2=910mm
l2=1060mm l2=1210mm l2=1360mm
l2=1510mm l2=1660mm l2=1810mm
l2=1960mm l2=2110mm Potęg. (l2=610mm )
Potęg. (l2=2110mm)powerpower
rys. 3
y = 0,0235x-0,7872
R2 = 0,9986
y = 0,0012x-0,4457
R2 = 0,9908
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
610 810 1010 1210 1410 1610 1810 2010
length of the thread l2,mm
co
eff
icie
nt
a1,m
m-1
l1=400mm l1=550mm l1=700mm
l1=850mm l1=1000mm l1=1150mm
l1=1300mm l1=1450mm l1=1600mm
l1=1750mm l1=1900mm Potęg. (l1=400mm)
Potęg. (l1=1900mm)
coefficient of the system geometry a1=f(l2) dla l1=const
power(
power(l1=400mm)
rys. 4
ai=f(l1+l2)
y = 0,1354x-0,9859
R2 = 1
y = 0,3536x-1
R2 = 1
y = 0,6289x-1,0146
R2 = 1
0
0,0001
0,0002
0,0003
0,0004
0,0005
0,0006
1000 1500 2000 2500 3000 3500 4000
total length l1+l2, mm
co
effi
cie
nts
a1,a
2,a
3, m
m-1
a1 a2 a3
Potęg. (a1) Potęg. (a3) Potęg. (a2)power (a1) power (a3) power (a2)
rys. 5
Coefficients ai are functions of the pa-rameters of the feeding zone geometry, including the angles of the threads “run-ning-on” and “running-of” the tension rail plate α and β, the lengths of threads “behind” and “in front” of the tension rail l1 and l2 , and the segment emr determin-ing the friction force, which depends on the coefficient of friction m and the angle of wrapping r. These coefficients are also significant in the modelling of such pa-rameters as the elasticity and attenuation of the feeding system, as well as indica-tors such as the relative attenuation of the
system and the free vibration frequency of the un-attenuated system, in relation to the character of the forces in the warp threads being fed.
In the general form of the differential equation (1) of the dynamics of tension rail movement, coefficients ai model the level of the input function, depending on
the values of S(t) and ( )
dttdS
. Parameters
such as l1, l2, m & r presented in the form
of a fraction 12
1lel +⋅ ±µρ and the values
of angles α and β determining the differ-ence (cosβ – sinα) determine the values of forces in the threads Pi(t), calculated according to dependence (8). Coeffi-cients ai = f(α, β, l1, l2, μ) take the form of a composite function describing the elements of both the trigonometric and power functions.
Influence of the l1 and l2 lengths of threads on the values of coefficients of the feeding system geometry ai .With constant values of α, β & μ, equa-tion (4) ai = f(li) takes the following form:
Figure 3. Dependence of the coefficient of geometry a1 on the length of thread l1. Coefficient of the system geometry a1=f(l1) for l2=const.
Figure 4. Dependence of the coefficient of geometry a1 on the length of thread l2.
0.000160.000150.000140.000130.000120.000110.0001
0.000090.000080.000070.000060.000050.000040.00003
y = 0.0235x-0.7872
R2 = 0.9986
y = 0.0012x-0.4457
R2 = 0.9908
rys. 1
400
1000
1600
61
0
76
0
91
0
10
60
12
10
13
60
15
10
16
60
18
10
19
60
21
10
0
0,00002
0,00004
0,00006
0,00008
0,0001
0,00012
0,00014
0,00016
a1
,mm
-1
length of the thread
l2,mm
coefficient of the system geometry a1=f(l
1,l
2)
0-0,00002 0,00002-0,00004 0,00004-0,00006 0,00006-0,00008
0,00008-0,0001 0,0001-0,00012 0,00012-0,00014 0,00014-0,00016
length of the thread
l1, mm
rys. 2Figure 1. Physical model of a constant-length feeding system for warp knitted machines (P1k and P2k – forces in the threads before and behind the tension rail).
Figure 2. Dependence of the coefficient of geometry a1 on the l1 and l2 lengths of threads (α = 30°, β = 22°, μ = 0,2).
0.00016
0.00014
0.00012
0.0001
0.00008
0.00006
0.00004
0.00002
0
0-0.00002 0.00002-0.00004 0.00004-0.00006 0.00006-0.000080.00008-0.0001 0.0001-0.00012 0.00012-0.00014 0.00014-0.00016
a 1,mm
-1
coefficient of the system geometry a1=f(l1) for
l2=const.
y = 0,0037x-0,53
R2 = 0,9895
y = 0,0003x-0,2534
R2 = 0,9743
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
400 600 800 1000 1200 1400 1600 1800
length of the thread l1, mm
co
eff
icie
nt
a1,
mm
-1
l2=610mm l2=760mm l2=910mm
l2=1060mm l2=1210mm l2=1360mm
l2=1510mm l2=1660mm l2=1810mm
l2=1960mm l2=2110mm Potęg. (l2=610mm )
Potęg. (l2=2110mm)powerpower
rys. 3
y = 0,0235x-0,7872
R2 = 0,9986
y = 0,0012x-0,4457
R2 = 0,9908
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
610 810 1010 1210 1410 1610 1810 2010
length of the thread l2,mm
co
eff
icie
nt
a1,m
m-1
l1=400mm l1=550mm l1=700mm
l1=850mm l1=1000mm l1=1150mm
l1=1300mm l1=1450mm l1=1600mm
l1=1750mm l1=1900mm Potęg. (l1=400mm)
Potęg. (l1=1900mm)
coefficient of the system geometry a1=f(l2) dla l1=const
power(
power(l1=400mm)
rys. 4
ai=f(l1+l2)
y = 0,1354x-0,9859
R2 = 1
y = 0,3536x-1
R2 = 1
y = 0,6289x-1,0146
R2 = 1
0
0,0001
0,0002
0,0003
0,0004
0,0005
0,0006
1000 1500 2000 2500 3000 3500 4000
total length l1+l2, mm
co
effi
cie
nts
a1,a
2,a
3, m
m-1
a1 a2 a3
Potęg. (a1) Potęg. (a3) Potęg. (a2)power (a1) power (a3) power (a2)
rys. 5
0.00016
0.00015
0.00014
0.00013
0.00012
0.00011
0.0001
0.00009
0.00008
0.00007
0.00006
0.00005
0.00004
0.00003
y = 0.0037x-0.53
R2 = 0.9895
y = 0.0003x-0.2534
R2 = 0.9743
power (l2=2110mm)power (l2=610mm)
coefficientofthesystemgeometrya1=f(l1)forl2=const.
power (l2=1900mm)power (l1=400mm)
coefficientofthesystemgeometrya1=f(l2)forl1=const.
84 FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
coefficient of the system geometry a1=f(l1) for
l2=const.
y = 0,0037x-0,53
R2 = 0,9895
y = 0,0003x-0,2534
R2 = 0,9743
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
400 600 800 1000 1200 1400 1600 1800
length of the thread l1, mm
co
eff
icie
nt
a1,
mm
-1
l2=610mm l2=760mm l2=910mm
l2=1060mm l2=1210mm l2=1360mm
l2=1510mm l2=1660mm l2=1810mm
l2=1960mm l2=2110mm Potęg. (l2=610mm )
Potęg. (l2=2110mm)powerpower
rys. 3
y = 0,0235x-0,7872
R2 = 0,9986
y = 0,0012x-0,4457
R2 = 0,9908
0,00003
0,00004
0,00005
0,00006
0,00007
0,00008
0,00009
0,0001
0,00011
0,00012
0,00013
0,00014
0,00015
0,00016
610 810 1010 1210 1410 1610 1810 2010
length of the thread l2,mm
co
eff
icie
nt
a1,m
m-1
l1=400mm l1=550mm l1=700mm
l1=850mm l1=1000mm l1=1150mm
l1=1300mm l1=1450mm l1=1600mm
l1=1750mm l1=1900mm Potęg. (l1=400mm)
Potęg. (l1=1900mm)
coefficient of the system geometry a1=f(l2) dla l1=const
power(
power(l1=400mm)
rys. 4
ai=f(l1+l2)
y = 0,1354x-0,9859
R2 = 1
y = 0,3536x-1
R2 = 1
y = 0,6289x-1,0146
R2 = 1
0
0,0001
0,0002
0,0003
0,0004
0,0005
0,0006
1000 1500 2000 2500 3000 3500 4000
total length l1+l2, mm
co
effi
cie
nts
a1,a
2,a
3, m
m-1
a1 a2 a3
Potęg. (a1) Potęg. (a3) Potęg. (a2)power (a1) power (a3) power (a2)
rys. 5
y = 7689,8x-0 ,6 0 4 1
R2 = 0,9846
y = 8054,7x- 0 ,5 7 9 4
R2 = 0,9904
50
100
150
200
250
380 580 780 980 1180 1380 1580 1780 1980 2180 2380 2580 2780 2980
length of the thread L,mm
forc
e P
,cN
V=1,4m/s (3cN) V=1,4m/s (10cN)
V=2.5m/s (3cN) V=2,5m/s (10cN)
V=4,75m/s (3cN) V=4,75m/s (10cN)
V=6,05m/s (3cN) V=6,05m/s (10cN)
Potęg. (V=1,4m/s (3cN)) Potęg. (V=6,05m/s (10cN))
Dependence of force on the length
of a thread being stretched J PE 84/72 dtex
power power
Rys 6
Dependence of force Pmax on the length of the warp thread (l1+l2).
y = -7E -12x4 + 6E -08x
3 - 0,0002x
2 + 0,201x - 45,206
R2 = 0,9994
y = -3E -09x3 + 2E -05x
2 - 0,0597x + 61,81
R2 = 0,9996
0
5
10
15
20
25
30
35
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200
length of the thread (l1+l2),mm
forc
e P
max.,
cN
P symul.l=2.3mm,Lc1 P symul.l=2,1mm,Lc1
P symul.l=1.85mm,Lc1 P symul.l=2,3mm,Lc2
P symul.l=2,1mm,Lc2 P symul.l=1,85mm,Lc2
Wielom. (P symul.l=1.85mm,Lc1) Wielom. (P symul.l=2,3mm,Lc2) Polyn. Polyn.
rys. 7 (11 cm)
122
12,1 lkl
ka+⋅
= ,
(9)
12
33 ll
ka+
= , (10)
where:αβ µρ sincos1 ⋅−= ±ek , µρ±= ek2 ,
αβ sincos3 −=k .
The dependence of ai on li or ∑li is a hy-perbolic function.
With an increase in the argument of function l1, l2 or Lc = l1 + l2, the val-ue of ai decreases. The character of changes a1 = f(l1 and l2) and dependence ai = f(Lc) are presented on the graphs in Figures 2, 3, 4 & 5. From a physi-cal point of view of coefficients ai of the feeding system geometry, it can be stated that with a decrease in ai and an increase in length of the warp threads, a decrease in the values of forces in the threads can be expected.
Influence of the length of the threads on the character and values of forces in the themThe problem of changes in the loads of forces in the threads, depending on the length of their feeding zone, can be ex-plained by basic research on the process of the dynamic stretching of the threads.The character of the dependencies of the stretching force on the length of the thread can be determined on the basis of particular solutions of rheological equa-tions of the state of the threads from the point of view of elastic and visco-elastic deformations [9-13].
The dependencies of the stretching force and the length of the thread for the mod-els chosen can be presented in the follow-ing form:naccording to Hooke’s model:
1−⋅= lKP HH , where: HH ElK ⋅∆= (11)naccording to the Voigt – Kelvin model:
1−⋅= lKP VV , where: .rozVVV VlEK ⋅+∆⋅= η (12)
naccording to Zener’s model:
1−⋅= lKP ZZ , where:
∆⋅+
⋅−−⋅= lEtEVK Z
Z
ZrozZZ 2
1. exp1
ηη
(13)
where: EH, EV, EZ1,2 – Young modulus of linear elasticity, ηV, ηZ – coefficients of viscosity, Δl – absolute elongation, Vroz. = const – stretching rate.
The dependencies P = f(l) presented are hyperbolic decreasing functions asymp-totically approaching P = 0.
The dependencies resulting from the rheological models were confirmed by experimental research of the dynamic stretching of the threads [14, 15].
The characteristics of stretching P = f(E) for the following PET filament yarns: JPE84/f72 dtex, JV167/f40 dtex, and textured JPA 72/2/f17 dtex [15] were de-termined on a measuring stand specially constructed for the dynamic stretching of the threads with rates Vroz within the
range of 0-6 m/s and absolute elongation with the possibility of changing within the range of Δl = 0÷30mm.
The threads were stretched at the follow-ing rates: V = 1.4, 2.5, 4.75 and 6.0 m/s, corresponding to the input function of the growth in force on the warp-knitting machines nd = 1200, 2000, 3600 & 4800 rows/min, respectively, for the speeds of knitting.
The investigations showed that with an increase in the length of the thread, the value of forces decreased by about 56% for viscose filament yarns, 71% for poly-ester filament yarns and 86% for textured polyamide threads.
A mathematical analysis of the character-istics of Pmax = f(l) showed that the most adequate regression functions describing the dependence between Pmax and l are the square and power functions (Figure 6).
For the length of thread being stretched, limited within the range of l = 400÷1300 mm, the power functions of dependence P = f(l) take a specific form of hyperbolic function P = a·l-1 for R2 = 0,95, which corresponds to the func-tions of rheological models.Simulation of the knitting process in the aspect of changes in P = f(L) were car-ried out taking into account calculations of the real parameters of the process for a Kokett 5223 warp-knitting machine of needle gauge E28. The growth in thread
Figure 5. Dependence of the coefficients of geometry ai on the total length of threads Lc = l1 + l2.
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0
y = 0.3536x-1
R2 = 1
y = 0.1354x-0,9859
R2 = 1
Figure 6. Dependence of force Pmax on the length of thread being stretched.
y = 8054.7x-0.5794
R2 = 0.9904
y = 7689.8x-0.6041
R2 = 0.9846
V=1.4m/s(3cN)V=2.5m/s(3cN)V=4.75m/s(3cN)V=6.05m/s(3cN)power(V=1.4m/s(3cN))
V=1.4m/s(10cN)V=2.5m/s(10cN)V=4.75m/s(10cN)V=6.05m/s(10cN)power(V=6.05m/s(10cN))
y = 0.6289x-1.0146
R2 = 1
power(a1) power(a3) power(a2)
a1
a3
a2
85FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
Dependence of an amplitude of forces on the length Lc2.
y = 3E-12x4 - 2E -08x
3 + 7E -05x
2 - 0,078x + 44,381
R2 = 0,9994
y = 3E -12x4 - 3E-08x
3 + 7E -05x
2 - 0,0809x + 44,157
R2 = 0,9992
2
3
4
5
6
7
8
9
10
11
12
1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100
length of the thread Lc2 ,mm
am
plitu
de s
ił D
P,c
N
P for l=1,85mm,Vo av . P for l=2,1mm,V o av .
P for l=2,3mm,Vo av . W ielom. ( P for l=1,85mm,Vo av .)
W ielom. ( P for l=2,3mm,Vo av .)
D
D
D
Polyn.
Polyn.
( D
( D P
rys. 8
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1-1
-0,4
0,2
0,8
-0,0005
0
0,0005
0,001
0,0015
0,002
co
eff
icie
nt
a1,m
m-1
coefficient a1=f(a ,b)
-0,0005-0 0-0,0005 0,0005-0,001 0,001-0,0015 0,0015-0,002
angle b ,rad.angle a ,rad.
rys. 9
being fed was changed within the range of 0÷288%.
The results showed a decrease in the values of the average forces of 56%, the minimum forces – 47%, and the maxi-mum forces – 72% (Figure 7), and the amplitude of change in force DP was 65% (Figure 8) within the limits of changes (l1 + l2) established. The charac-teristics of the results of P = f(l) obtained were experimentally verified to obtain relative differences between the values calculated and those measured ε Pmax within the range of 9.8÷19.4%.
Influence of the values of the angles of threads „running-on” and „running-of” the tension rail and the coefficients of friction on the values of the coefficients of feeding a1 and the forces in the threadsInfluence of angles α and β on the values of coefficients of the geometry of feeding ai .Dependence (4) describes coefficients a1 and a2:
12
2
2
2,1sincos
lel
ea
+⋅
⋅−=
−+±
−+±
αβπµ
αβπµαβ , (14)
assuming that (l1, l2 & μ) = const takes the form of composite function a1,2 = f(α, β).The characteristics of a1 and a2 are very similar, thus graphs a1 = f(α, β) are pre-sented in Figures 9, 10 and 11. From the graph of a1 = f(α) for β = const., it results that a decrease in the value of coefficient a1 can be well described by a linear func-tion, depending on angle α (Figure 10) (with the correlation coefficient of the boundary curves r2 ≥ 0.995).The character of function a1 = f(β) for α = const is described by polynomial equations of the third degree (r230.9995) (Figure 11). It can be stated with high probability that function a1,2 = f(β) for α > 0 is a symmetric function in relation to the OY axis.The formula for function a3 = f(α, β) can be written as follows:
( ) ka ⋅−= αβ sincos3 , where const
llk =
+=
12
1 (15)
According to the formula above, function a3 = f(α) for β = const takes the following form: kka c ⋅−= )sin(3 α ,
which indicates that the value of a3 de-creases with an increase in angle a on the basis of the character of changes in the sine function.
Figure 8. Dependence of the amplitude of force ΔPmax on the length of thread being fed.
y = 3E-12x4 + 3E-08x3 + 7E-0.5x2 - 0.0809x + 44.157R2 = 0.9992
y = 3E-12x4 + 2E-08x3 + 7E-0.5x2 - 0.078x + 44.381R2 = 0.9994
For a3 = f(β) for α = 0 we obtain ( ) kka s ⋅−= βcos3 , described by the co-
sine function.
The character of the curves of functions a1,2,3 = f(a) analysed is very similar, de-termining the same tendency of changes in function ai of angles a. The analogous similarity of the curves can be seen for the dependencies of a1,2,3=f(b).
Dependence of coefficients ai on the friction coefficient m of the thread against the tension rail plate
The dependence of ai=f(m) concerns co-efficients a1 and a2, which correspond to the conditions of the dislocation of the threads on the tension rail.
Dependence (4) can be written in the fol-lowing form:
y = 7689,8x-0 ,6 0 4 1
R2 = 0,9846
y = 8054,7x- 0 ,5 7 9 4
R2 = 0,9904
50
100
150
200
250
380 580 780 980 1180 1380 1580 1780 1980 2180 2380 2580 2780 2980
length of the thread L,mm
forc
e P
,cN
V=1,4m/s (3cN) V=1,4m/s (10cN)
V=2.5m/s (3cN) V=2,5m/s (10cN)
V=4,75m/s (3cN) V=4,75m/s (10cN)
V=6,05m/s (3cN) V=6,05m/s (10cN)
Potęg. (V=1,4m/s (3cN)) Potęg. (V=6,05m/s (10cN))
Dependence of force on the length
of a thread being stretched J PE 84/72 dtex
power power
Rys 6
Dependence of force Pmax on the length of the warp thread (l1+l2).
y = -7E -12x4 + 6E -08x
3 - 0,0002x
2 + 0,201x - 45,206
R2 = 0,9994
y = -3E -09x3 + 2E -05x
2 - 0,0597x + 61,81
R2 = 0,9996
0
5
10
15
20
25
30
35
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200
length of the thread (l1+l2),mm
forc
e P
max.,
cN
P symul.l=2.3mm,Lc1 P symul.l=2,1mm,Lc1
P symul.l=1.85mm,Lc1 P symul.l=2,3mm,Lc2
P symul.l=2,1mm,Lc2 P symul.l=1,85mm,Lc2
Wielom. (P symul.l=1.85mm,Lc1) Wielom. (P symul.l=2,3mm,Lc2) Polyn. Polyn.
rys. 7 (11 cm)Figure 7. Dependence of force Pmax on the length of warp thread being fed(data for calculations: Nu = 28, nd = 600 rows/min, l1 and l2, α = 5°, β = 15°, ks = 0,6 cN·mm-1, locz. = 2,3 mm (y0 = 4mm); 2,1 mm (y0 = 7mm); 1,85 mm (y0 = 12mm), μ = 0,2, kp = 1250 cN, bp = 100 cN·ms (for JPA 44 dtex), m = 0,185 g, S’(t) dane_t.txt).
y = -7E-12x4 + 6E-08x3 - 0.0002x2 + 0.201x + 45.206R2 = 0.9994
y = -3E-09x3 + 2E-05x2 - 0.0597x + 61.81R2 = 0.9996
Psymul.I=2.3mm,Lc1Psymul.I=1.85mm,Lc1Psymul.I=2.1mm,Lc2Polyn.(Psymul.I=1.85mm,Lc1)
length of the thread (l1+l2), mm
Psymul.I=2.1mm,Lc1Psymul.I=2.3mm,Lc2Psymul.I=1.85mm,Lc2Polyn.(Psymul.I=2.3mm,Lc2)
amplitud
eofforce
,DP,cn
DPforI=1.85mm,Voav.DPforI=2.3mm,Voav.Polyn.(DPforI=2.3mm,Voav.)
DPforI=2.1mm,Voav.Polyn.(DP’forI=1.85mm,Voav.)
86 FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
Dependence of an amplitude of forces on the length Lc2.
y = 3E-12x4 - 2E -08x
3 + 7E -05x
2 - 0,078x + 44,381
R2 = 0,9994
y = 3E -12x4 - 3E-08x
3 + 7E -05x
2 - 0,0809x + 44,157
R2 = 0,9992
2
3
4
5
6
7
8
9
10
11
12
1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100
length of the thread Lc2 ,mm
am
plitu
de s
ił D
P,c
N
P for l=1,85mm,Vo av . P for l=2,1mm,V o av .
P for l=2,3mm,Vo av . W ielom. ( P for l=1,85mm,Vo av .)
W ielom. ( P for l=2,3mm,Vo av .)
D
D
D
Polyn.
Polyn.
( D
( D P
rys. 8
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1-1
-0,4
0,2
0,8
-0,0005
0
0,0005
0,001
0,0015
0,002
co
eff
icie
nt
a1,m
m-1
coefficient a1=f(a ,b)
-0,0005-0 0-0,0005 0,0005-0,001 0,001-0,0015 0,0015-0,002
angle b ,rad.angle a ,rad.
rys. 9Figure 9. Dependence of coefficient a1 on the angles α of the thread „running-on” and β „running-of” the tension rail plate (l1 = 400 mm, l2 = 610 mm, μ = 0,2).
-0.0005-0 0-0.0005 0.0005-0.001 0.001-0.0015 0.0015-0.002
0.002
0.0015
0.001
0.0005
-0.0005
0.8
0.2
-0.4
0.80.60.40.2-0.6-0.4 -0.2
Dependence of an amplitude of forces on the length Lc2.
y = 3E-12x4 - 2E -08x
3 + 7E -05x
2 - 0,078x + 44,381
R2 = 0,9994
y = 3E -12x4 - 3E-08x
3 + 7E -05x
2 - 0,0809x + 44,157
R2 = 0,9992
2
3
4
5
6
7
8
9
10
11
12
1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100
length of the thread Lc2 ,mm
am
plitu
de s
ił D
P,c
N
P for l=1,85mm,Vo av . P for l=2,1mm,V o av .
P for l=2,3mm,Vo av . W ielom. ( P for l=1,85mm,Vo av .)
W ielom. ( P for l=2,3mm,Vo av .)
D
D
D
Polyn.
Polyn.
( D
( D P
rys. 8
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1-1
-0,4
0,2
0,8
-0,0005
0
0,0005
0,001
0,0015
0,002
co
eff
icie
nt
a1,m
m-1
coefficient a1=f(a ,b)
-0,0005-0 0-0,0005 0,0005-0,001 0,001-0,0015 0,0015-0,002
angle b ,rad.angle a ,rad.
rys. 9
-0.8angleα,deg angleβ,deg
coefficienta1=f(α,β) coefficient a1=f(a ) d la b=const.
y = 0,0002x3 + 6E-05x
2 - 0,0011x + 0,0004
R2 = 1
y = 0,0002x3 + 6E-05x
2 - 0,001x + 0,0008
R2 = 1
-0,0006-0,0005-0,0004-0,0003-0,0002-0,0001
00,00010,00020,00030,00040,00050,00060,00070,00080,0009
0,0010,00110,00120,00130,00140,00150,00160,00170,0018
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1
angle a ,rad.
co
eff
icie
nt
a1,m
m-1
b=-1 rad b=-0,8 rad b=-0,6 rad
b=-0,4 rad b=-0,2 rad b=0 rad
b=0,2 rad b=0,4 rad b=0,6 rad
b=0,8 rad b=1 rad Wielom. (b=1 rad)
Wielom. (b=-0,2 rad)polyn. (b = -0,2 rad)polyn. (b = 1 rad)
rys. 10
coefficient a1=f(b) dla a=const.
y = -0,0003x2 - 1E-05x + 0,0017
R2 = 0,9939
y = -0,0004x2 - 0,0001x + 5E-05
R2 = 0,9963
-0,0005
0
0,0005
0,001
0,0015
0,002
-1 -0,5 0 0,5 1
angle b ,rad.
co
eff
icie
nt
a1,m
m-1
a=-1 rad a=-0,8 rad a=-0,6 rad
a=-0,4 rad a=-0,2 rad a=0 rad
a=0,2 rad a=0,4 rad a=0,6 rad
a=0,8 rad a=1 rad W ielom. (a=-1 rad)
W ielom. (a=1 rad)
polyn. (a = -1 rad)
polyn. (a = 1 rad)
rys. 11
rys. 12coefficient ai=f(mi)
y = -0,0009x + 0,0003
R2 = 0,9989
y = 0,0011x + 0,0003
R2 = 1
-0,0002
0
0,0002
0,0004
0,0006
0,0008
0,001
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5
coefficient of friction mi
co
eff
icie
nt
ai,
mm
-1
a1 a2 a3 Liniowy (a1) Liniowy (a2)linear (a2)linear (a1)
Figure 10. Dependence of coefficient a1 on angle α of the thread „running-on” the tension rail plate.
0.00180.00170.00160.00150.00140.00130.00120.00110.001
0.00090.00080.00070.00060.00050.00040.00030.00020.0001
0-0.0001-0.0002-0.0003-0.0004-0.0005-0.0006
y = 0.0002x3 + 6E-0.5x2 - 0.001x + 0.0008R2 = 1
y = 0.0002x3 + 6E-0.5x2 - 0.0011x + 0.0004R2 = 1
coefficienta1=f(α)forβ=const.
angleα,degb=-1 degb=-0.4 degb=0.2 degb=0.8 degpolyn. (b=-0.2 deg)
b=-0,8 degb=-0.2 degb=0.4 degb=1 deg
b=-0.6 degb=0 degb=0.6 degpolyn. (b=1 deg)
coefficient a1=f(a ) d la b=const.
y = 0,0002x3 + 6E-05x
2 - 0,0011x + 0,0004
R2 = 1
y = 0,0002x3 + 6E-05x
2 - 0,001x + 0,0008
R2 = 1
-0,0006-0,0005-0,0004-0,0003-0,0002-0,0001
00,00010,00020,00030,00040,00050,00060,00070,00080,0009
0,0010,00110,00120,00130,00140,00150,00160,00170,0018
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1
angle a ,rad.
co
eff
icie
nt
a1,m
m-1
b=-1 rad b=-0,8 rad b=-0,6 rad
b=-0,4 rad b=-0,2 rad b=0 rad
b=0,2 rad b=0,4 rad b=0,6 rad
b=0,8 rad b=1 rad Wielom. (b=1 rad)
Wielom. (b=-0,2 rad)polyn. (b = -0,2 rad)polyn. (b = 1 rad)
rys. 10
coefficient a1=f(b) dla a=const.
y = -0,0003x2 - 1E-05x + 0,0017
R2 = 0,9939
y = -0,0004x2 - 0,0001x + 5E-05
R2 = 0,9963
-0,0005
0
0,0005
0,001
0,0015
0,002
-1 -0,5 0 0,5 1
angle b ,rad.
co
eff
icie
nt
a1,m
m-1
a=-1 rad a=-0,8 rad a=-0,6 rad
a=-0,4 rad a=-0,2 rad a=0 rad
a=0,2 rad a=0,4 rad a=0,6 rad
a=0,8 rad a=1 rad W ielom. (a=-1 rad)
W ielom. (a=1 rad)
polyn. (a = -1 rad)
polyn. (a = 1 rad)
rys. 11
rys. 12coefficient ai=f(mi)
y = -0,0009x + 0,0003
R2 = 0,9989
y = 0,0011x + 0,0003
R2 = 1
-0,0002
0
0,0002
0,0004
0,0006
0,0008
0,001
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5
coefficient of friction mi
co
eff
icie
nt
ai,
mm
-1
a1 a2 a3 Liniowy (a1) Liniowy (a2)linear (a2)linear (a1)
Figure 11. Dependence of coefficient a1 on angle β of the thread „running-of” the tension rail plate.
a=-1 dega=-0.4 dega=0.2 dega=0.8 degpolyn. (a=1 deg)
a=-0,8 dega=-0.2 dega=0.4 dega=1 deg
a=-0.6 dega=0 dega=0.6 degpolyn. (a=-1 deg)
y = -0.0003x2 - 1E-0.5x + 0.0017R2 = 0,9939
coefficienta1=f(β)forα=const.
angleα,deg
y = -0.0004x2 - 0.0001x + 5E-05R2 = 0,9963
coefficient a1=f(a ) d la b=const.
y = 0,0002x3 + 6E-05x
2 - 0,0011x + 0,0004
R2 = 1
y = 0,0002x3 + 6E-05x
2 - 0,001x + 0,0008
R2 = 1
-0,0006-0,0005-0,0004-0,0003-0,0002-0,0001
00,00010,00020,00030,00040,00050,00060,00070,00080,0009
0,0010,00110,00120,00130,00140,00150,00160,00170,0018
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1
angle a ,rad.
co
eff
icie
nt
a1,m
m-1
b=-1 rad b=-0,8 rad b=-0,6 rad
b=-0,4 rad b=-0,2 rad b=0 rad
b=0,2 rad b=0,4 rad b=0,6 rad
b=0,8 rad b=1 rad Wielom. (b=1 rad)
Wielom. (b=-0,2 rad)polyn. (b = -0,2 rad)polyn. (b = 1 rad)
rys. 10
coefficient a1=f(b) dla a=const.
y = -0,0003x2 - 1E-05x + 0,0017
R2 = 0,9939
y = -0,0004x2 - 0,0001x + 5E-05
R2 = 0,9963
-0,0005
0
0,0005
0,001
0,0015
0,002
-1 -0,5 0 0,5 1
angle b ,rad.
co
eff
icie
nt
a1,m
m-1
a=-1 rad a=-0,8 rad a=-0,6 rad
a=-0,4 rad a=-0,2 rad a=0 rad
a=0,2 rad a=0,4 rad a=0,6 rad
a=0,8 rad a=1 rad W ielom. (a=-1 rad)
W ielom. (a=1 rad)
polyn. (a = -1 rad)
polyn. (a = 1 rad)
rys. 11
rys. 12coefficient ai=f(mi)
y = -0,0009x + 0,0003
R2 = 0,9989
y = 0,0011x + 0,0003
R2 = 1
-0,0002
0
0,0002
0,0004
0,0006
0,0008
0,001
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5
coefficient of friction mi
co
eff
icie
nt
ai,
mm
-1
a1 a2 a3 Liniowy (a1) Liniowy (a2)linear (a2)linear (a1)
Figure 12. Dependence of coefficient ai on the friction coefficient m of the thread against the tension rail plate (l1 = 400 mm, l2 = 610 mm, α = 36°, β = 22°).
coefficientai=f(mi)
y = 0.0011x + 0.0003R2 = 1
y = -0.0009x + 0.0003R2 = 1
8070
6050
4030
2010
0-10
-200
20
40
10
12
14
16
18
20
22
24
forc
e P
ma
x ,
cN
Dependence of force P max on angles a and b
10-12 12-14 14-16 16-18 18-20 20-22 22-24
angle b , deg
angle a , deg
rys. 13 (7,5 cm)
15
20
25
30
35
40
0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
coefficient of friction m
forc
e P
ma
x ,
cN
fi=120 deg fi=70 deg fi=20 deg
Dependence of force Pmax on the friction coefficient m
rys. 14 (7,5 cm)
Figure 13. Dependence of force Pmax on angles α and β.
angleα,degangleβ,deg
DependenceofforcePmaxonanglesαandβ
Figure 14. Dependence of force Pmax on the friction coefficient of a thread against the tension rail plate (ρ = 20° for α = 40° and β = -30°, ρ = 70° for α = 40° and β = 20°, ρ = 120° for α = 40° and β = 70°).
8070
6050
4030
2010
0-10
-200
20
40
10
12
14
16
18
20
22
24fo
rce
Pm
ax
, c
N
Dependence of force P max on angles a and b
10-12 12-14 14-16 16-18 18-20 20-22 22-24
angle b , deg
angle a , deg
rys. 13 (7,5 cm)
15
20
25
30
35
40
0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
coefficient of friction m
forc
e P
ma
x ,
cN
fi=120 deg fi=70 deg fi=20 deg
Dependence of force Pmax on the friction coefficient m
rys. 14 (7,5 cm)
DependenceofforcePmaxonthefrictioncoefficientμ
coefficientoffrictionμ
87FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
52
4
32
12,1
kek
keka k
k
+⋅
⋅−=
±
±
µ
µ
(16)
with constant coefficients: k1 = cosβ, k2 = r, k3 = sinα, k4 = l2 & k5 = l1 = const.
Fraction a1,2 is a quotient of exponential functions. The character of a1,2 for a range of m from 0.005 to 0.5 is presented in Figure 12.
With an increase in the argument, the values of function a1,2 = f(m) increase for coefficient a2 and decrease for coefficient a1. Within the range of m assumed for r230,999 with high probability, depend-ence (16) can be described by a linear function. Straight lines approximating functions a1,2 = f(m) are symmetrical to
the straight line 54
313 kk
kka+−
= .
Results of simulations of the dependencies of the force on angles a and b and the friction coefficient m of the thread against the tension railFor calculations of the forces of the func-tion of angles a of thread „running-on” and b “running-of” the tension rail plate (Figure 13), which depend on the angle of wrapping r, the following input data was selected: E 20, nd = 700 rows/min, l1 = 400 mm, l2 = 610 mm, ks = 0,6 cN·mm-1, locz. = 4 mm, kp = 3150 cN, bp= 1050 cN·ms (for thread JPE 110/24 f dtex), and y0=4 mm.
The results of the investigations are pre-sented in Figure 13, which show the fol-lowing:nWithin the range of angles α = -30÷10°
with an initial constant value of the force of around 17 cN, Pmax decreases by 24% and next within the range of 0°(10°) – 40° the force rapidly in-creases, reaching 21.5 cN (of 65%) and then it stabilises.
nFunction Pmax = f(b) has a parabolic character (Pmax = aβ2 + bβ + c). The minimum of the function occurs for angle b approaching zero and equals Pmax = 13.2 ÷ 21.8 cN.
The dependencies mentioned above can help to optimise the parameters of the feeding process from the point of view of the lowest values of force Pmax.
This way, for α = 30 ÷ 10° and β = -20÷10° the extreme forces Pmax take the mini-mum values.
The dependence of the force in the threads on the friction coefficient of a thread against the tension rail plate is presented in Figure 14. Function Pmax = f(m) has an increasing character, while the value of the angle of wrapping r determines the intensity degree of changes in Pmax on m. For the linear function of regression (y = ax + b) of Pmax = f(m) changes, the slopes of the function equal a = 54.9 cN for r = 20°, a = 16.8 cN for r = 70° and a = 4.4 cN for r = 120°.
Summary1. Simulations of the process of knitting
on warp-knitting machines were con-ducted in order to determine optimal input parameters with regard to the minimum load of threads.
Analysis of the knitting process was conducted in the aspect of the deter-mination of advantageous areas of an important group of parameters characterising the geometry of the structure of the feeding device, in-cluding as follows: the length of warp threads “behind” and “in front of” the tension rail l1 and l2, angles α and β of the thread “running-on” and “running-of” the tension rail plate, friction coefficient m of a thread against the tension rail, the mass of the tension rail plate m, and the rigidity of springs support-ing the tension rail ks. Simulations were carried out in both directions, primarily to identify the character of changes in the coefficient of the feeding system geometry a1. In the mathematical function of the model of the feeding process, the coeffi-cient describes the parameters of the structure of the feeding device men-tioned above. The second direction of the analysis, and at the same time the main goal of this research, was to find optimum solutions with re-gard to limiting the values of forces in the threads, beyond which disad-vantageous disturbances in the knit-ting process can occur, further lead-ing to unfavourable changes in the fabric structure. An algorithm of the mathematical model of the knitting process was elaborated in order to be used as a tool for simulations of the behaviour of the feeding system on warp-knitting machines.
From the simulations of the knitting process conducted in order to analyse
the structural features of warp-knit-ting machines and to identify opti-mum parameters of the feeding proc-ess with regard to the dynamic load of the threads, it results that:– The coefficient of the geometry of
the feeding system describing, in a synthetic way, the parameters of the geometry of the structure of the feeding device shows a ten-dency to decrease with an increase in the length of the threads being fed, which in a physical meaning is characterised by a decrease in the values of forces in the threads. Coefficient ai is characterised by a decreasing function that depends on the angle a of threads “running-on” the tension rail and is a function of the argument of angle β of the thread “running-of” the tension rail, being of a tendency described by a polynomial equation of the third de-gree.
– With an increase in the free length of the threads within a range of 0 – 280%, the forces in the threads decrease by 74%, while the ampli-tude of force ΔP decreases by 65%. An optimal range of changes ΔL significantly decreasing the values of the maximum forces and ampli-tude of forces are the increments in length, starting from a value of 100%. The data presented and char-acter of changes in P = f (L) were experimentally verified. The results obtained confirm the experimen-tal investigations of the dynamic stretching of the threads, where the dependence of P = f (L) is a decreasing hyperbolic function as-ymptotically approaching P = 0.
– The character of changes in the force, depending on angles α of the threads “running-on” and β “running-of” the tension rail, can be described by hyperbolic tan-gent and cosine functions, where optimum values of the angles from the point of view of the lowest values of Pmax equal α = -30°÷10°, β = -20°÷10° and the angle of wrap-ping ρ = 60°÷110°.
– The extreme forces in the threads increase in a ’linear’ way depending on the coefficient of friction of the thread against the tension rail plate, and the intensity of the increase de-pends on the angle of wrapping r.
88 FIBRES & TEXTILES in Eastern Europe 2010, Vol. 19, No. 4 (87)
2. A computer simulation of the knitting process in the aspect of the dynamics of feeding is of great importance for the process of designing technological parameters, as well as for the mechani-cal designer-engineer in the process of designing warp-knitting machines from the point of view of the optimisation of modulus constructional solutions for the machine and also optimisation of the geometry of the feeding zone, and the technical parameters of the machine.
References 1. Mikołajczyk Z.: Identification of the process
of knitting of the warp-knitted anisotropic structures on warp-knitting machines (in Polish), Scientific Letters TUL, 2009, No 1047, Dissertation, Z.381, p. 234.
2. Kopias K.: Technologia dzianin kolum-ienkowych. Ed. WNT, Warszawa 1986, p. 177.
3. Mikołajczyk Z.:Model of Spatial Struc-ture of Anisotropic Warp Knitted Fabrics.
Fibres & Textiles in Eastern Europe, 2001, Vol. 9 No 2 (33), pp. 23-27.
4. Mikołajczyk Z.: Model of Feeding Proc-ess of Anosotropic Warp Knitted Fab-rics. Fibres & Textiles in Eastern Europe, Vol. 11, No. 2 (41), 2003, pp. 58-62.
5. Mikołajczyk Z.: Character of dynamic loads of the threads on warp-knitting machines, depending on the geometry of tension rail device (in Polish), Pro-ceedings of the International IMTEX’95 Conference, TUL, Łódź 22-23 May 1995, p. 9.
6. Piotrowska J. Analysis of geometry of the feeding zone on the warp-knitting machines in the aspect of changes of dynamic forces in the threads (in Polish), MSc Thesis supervised by Z. Mikołajczyk, TUL, 1997, p. 102.
7. Pahl G., Beitz W.: Nauka konstruow-ania, Ed. WNT, Warszawa, 1984, p. 505.
8. Mikołajczyk Z. Modeling of Knitting Process on Warp-Knitting Machines. 44 Congress IFKT “Knitting round the clock”, Saint-Petersburg, 23-27 Septem-ber 2008, p. 9.
9. Derski W., Ziemba S.: Analiza modeli re-ologicznych. Ed. IPPT PAN, Warszawa 1968, p.158.
10. Bland D.R.: The theory of linear viscoe-lasticity. Oxford, Print in Northern Ireland at the Universities Press, Belfast 1960, p.125.
11. Urbańczyk G.W.: Fizyka włókna. Ed. TUL, 2002, p. 420.
12. Reuman R.D.: Investigations of Polya-mide Monofilaments during high draw-ing velocities (in Polish), Przegląd włókienniczy, No. 9, 1979, pp. 494÷498.
13. Mamich A.M., Vssamane M.H., Barella A.: A New contibuation to the viscoe-lastic behavior of polypropylene fibres. World Conference, Advances in fibre and textile science and technologies, Mulhouse (France), 21-24 April, 1997.
14. Mikołajczyk Z.: Identification of the Dynamic Properties of Yarn Used in Rib Knitting. Fibres & Textiles in Eastern Europe, Vol. 3, No 4 (11), 1995, p. 29÷33.
15. Prężyńska A.: Modelling of the Drawing Process of Polyester Fibres. MSc the-sis supervised by Z. Mikołajczyk, TUL, 2002, p.252.
Received 29.11.2010 Reviewed 05.04.2011
XVIISeminarandWorkshopon‘NewAspectsoftheChemistryandApplicationsofChitin
anditsDerivatives’
INVITATION
On behalf of the Board of the Polish Chitin Society I have both a pleasure and an honour to invite you to participate in the XVIISeminarandWorkshopon“NewAspectsoftheChemistryandApplicationsofChitinanditsDerivatives” which will be held in Warsaw,Poland,September21st–23rd,2011.
The aim of the conference is to present the results of recent research, development and applications of chitin and chitosan.It is also our intention to give the conference participants working in different fields an opportunity to meet and exchange their experiences in a relaxing environment.
Best regardsDr Malgorzata M. Jaworska
For more information please contact:
CONFERENCE SECRETARYM. Sklodowskiej-Curie 19/27, 90-570 Łódź, Polandtel. (+48) 42 638 03 339, fax (+ 48) 42 637 62 14,e-mail: [email protected] www.ptchit.lodz.pl
XVII Seminar and Workshop on ‘New Aspects of the Chemistry
and Applications of Chitin and its Derivatives’
INVITATION
On behalf of the Board of the Polish Chitin Society I have both a pleasure and an honour to invite you to participate in the XVII Seminar and Workshop on “New Aspects of the Chemistry and Applications of Chitin and its Derivatives” which will be held in Warsaw, Poland, September 21st – 23rd, 2011.
The aim of the conference is to present the results of recent research, development and applications of chitin and chitosan.It is also our intention to give the conference participants working in different fields an opportunity to meet and exchange their experiences in a relaxing environment.
Best regardsDr Malgorzata M. Jaworska
More information:
CONFERENCE SECRETARYM. Sklodowskiej-Curie 19/27, 90-570 Łódź, Polandtel. (+48) 42 638 03 339, fax (+ 48) 42 637 62 14,e-mail: [email protected] www.ptchit.lodz.pl