Analysis of yarn tension based on yarn demand variation on a tricot knitting machine

Abstract

To date, the trend in the development of knitting promotes higher requirements for warp tension control, and the fluctuation of knitting warp finally comes down to the imbalance between yarn demand and feed while forming every loop. According to warp knitting machine characteristics, four typical points are chosen to calculate the value of yarn demand, and in this way a model is built to obtain and plot the overall change trend in yarn demand in one knitting cycle. Combined with the theoretical tension curve calculated based on the yarn demand and the measured one, it is verified as coincident. The model is useful for analyzing the factors causing warp dynamic tension variation, and lays the groundwork for accurate active tension compensation.

Keywords

tricot machine yarn tension model knitting cycle knitting element yarn demand

In order to ensure smooth knitting and to obtain high-class fabric, warp tension needs to be quite well controlled in the warp knitting production process. To cater to the market demand for natural fiber clothing products, using spun yarn in fine-gauge warp knitting machines to knit dress fabrics has become increasingly common, which has created a much higher requirement for the in-depth study of warp tension.

In 1996, a model to simulate the warp tension in weaving processes was described in the dissertation of De Weldige,¹ in which the thread was assumed to be a linear elastic, massless, and flexible spring element, and the friction between the thread and solid bodies was neglected.² He presented an algorithm to calculate the warp tension based on an examination of the equilibrium of torque around the axis of a mass–spring backrest system. An additional dissertation concerning this topic was published in 1998 by Chen,³ which made the same assumptions concerning the thread as the previous work.² According to the characteristics of bearded needle warp knitting machine, researchers like Feng⁴ and Zong,⁵ by combining knitting yarn consumption and warp dynamic tension in the spindle in the range of 360°, theoretically analyzed the dynamic tension variation and the main influencing factors,⁴ and discussed the optimization of the machine parameters. This provided a theoretical basis for the study of the positive warp let-off system, and improved tension fluctuation from the angle of the delivery.⁵ In recent years, Cong⁶ and Chen³ have used a high-frequency tension collection device to test and explore the warp tension of the groove needle warp knitting machine.⁶ They interpreted the fluctuations of the measured knitting tension diagram in an overall trend, according to the knitting cycle, and compared the mean, extreme, and several other characteristic quantities, to get an approximate tension fluctuation law.

Metzkes et al. investigated the thread tension in the looping process, and developed a continuum model of the warp thread in an attempt to reduce the influence of tension fluctuation.² Koo used simulated knitting conditions with a newly developed test rig to study yarn tension variation, and found that the selection of gauge and the yarn feed angle should be taken into account to reduce yarn tension during knitting.⁷

Now the warp knitting machine has largely adopted the positive warp let-off system, which can effectively reduce the tension fluctuation. However, the remaining lag indicators make it unable to accurately feed yarn at a rate that precisely matches the warp consumption. The instantaneous difference between demand and feed causes the tension to continue to fluctuate. The tension analysis results for the bearded needle warp knitting machine have certain reference significance for the grooved needle machine, but due to the differences in the loop forming process and movement of the knitting elements, there are still many inaccuracies. Studies have shown that it is difficult for the maximum tension produced in the process of knitting to reach the breaking strength of yarn; the main reason that causes the warp breakage is the repeated stretching of the yarn and sudden changes of the tension.⁸ Thus, it is necessary to conduct in-depth analysis of the tension fluctuation in one knitting cycle, and the difference between the yarn demand and feed is the main factor causing the changes of yarn tension. A tension compensation device and the elasticity of the yarn are two ways to relieve the tension. The focus of this paper is on exploring the basic laws of yarn tension fluctuation in the warp knitting cycle, to provide a basis for further research concerning the impact of mechanical properties of the yarn on warp knitting, and to develop an active tension compensation device.

Theoretical analysis

As shown in Figure 1, on the KS4 type warp knitting machine as an example, the yarn is unwound from the beam, then its direction is adjusted by a slewing roller, and finally it passes onto a tension rail and through the guide needle into the needle hook.⁹
Figure 1.
Schematic diagram of warp knitting yarn path.

In the knitting process, knitting elements, including the guide needle, the knitting needle, and the sinker, cooperate with each other to weave the yarn woven into a ring, and their motion matching curve is shown in Figure 2,¹⁰ in which θ (°) is the main shaft angle and the S (mm) is the displacement quantity of these knitting elements. At the main shaft angle of 0°, the open stem and the closing element are both in the lowest position, the guide needle underlaps in the front of the machine, and the sinker moves forward to assist in casting off. Then the needle rises, the guide needle swings forward to the front of the hook after underlapping, and overlaps at the angle of 180° or so. After overlapping, the guide needle swings to the front of the machine to lap the yarn into the needle hook, and the compound needle and the closing element descend to close up and kink the loop.
Figure 2.
Matching curve of the motion of knitting elements on a grooved needle machine.

In the process of loop formation, the position of the knitting elements is constantly changing in the range of 360°, which means that the yarn demand, the length of yarn between the backing-off point on the beam and the knitting point, constantly changes as well. The warp run-in is set according to conditions such as structure and density, and the yarn is fed evenly while knitting. The difference between yarn demand and feed cause the warp tension to continue to fluctuate.

Studies have shown that the maximum tension value in the knitting process is far less than the breaking strength value of the ordinary warp yarn, and the major causes of yarn breakage are sudden changes of the tension and yarn damage caused by repeated stretching, rather than too much tension. And the sudden changes are caused by the difference between the yarn demand and yarn feed. Further, studies of the yarn damage by repeated stretching can also be attributed to the change of relative positions of the yarn, which is caused by the imbalance between the yarn demand and feed in every knitting cycle. Therefore, it is very important to analyze theoretically the yarn demand change in the process of forming a loop.

Theoretical analysis of yarn demand in the knitting cycle

Model building

As is shown in Figure 1, it is easy to determine the yarn travel diagram on the tricot groove needle warp knitting machine from the unwinding point on the beam (A) to the knitting point (F). The warp run-in is set in accordance with the knitting requirements, and the yarn is fed uniformly. So the change of the yarn length (L (mm)) between point A and point F in the knitting cycle can reflect the yarn demand fluctuation.

The meaning of the labels in Figure 1 are as follows:
A – the unwinding point on the beam;

B – the contact point between the yarn and the slewing roller;

C – the contact point between the yarn and the tension rail;

D – the contact point between the yarn and the guide needle;

E – the lapping point on the pinned bar;

F – the old loop gripping point (stitch-forming point);

X axis – the horizontal line through the first needle head at the lowest and perpendicular to the needle bar direction;

Y axis –the horizontal line through all the needle head at the lowest and parallel to the needle bed;

Z axis – the vertical line with the first needle bar.

The coordinates of the points are represented by the rule: $C$ ( $X C, Y C, Z C$ ) for point C, $D$ ( $X D, Y D, Z D$ ) for point D, $E$ ( $X E, Y E, Z E$ ) for point E, etc.

The projection on the XY plane of the points is represented as: C_XY for point C, D_XY for point D, E_XY for point E, etc.

Total length of yarn travel is
$L = L + L BC + L CD + L DE + L EF$
(1)
where $L AB$ is the segment length between points A and B (mm), $L BC$ is the segment length between points B and C (mm), $L CD$ is the segment length between points C and D (mm), $L DE$ is the segment length between points D and E (mm), and $L EF$ is the segment length between points E and F (mm).

Take the KS4 type knitting machine as an example; its knitting gauge is E28, and it works at the speed of 800 rpm. The second strickle braids the second and first lapping as 1–0/2–3//, and the results of the measured parameters are as follows
$L = 88 mm, L BC = 36 mm, X = 13, Y C = - 0.454, Z C = 57$

At the speed of 800 rpm, it takes 0.075 s for one turn and 0.0002 s for one degree. So the shogging angle and the corresponding time can be estimated, as shown in Table 1.
Table 1.
Shogging angle and corresponding time for the groove needle knitting machine used

Strickle movement process Overlapping Swinging in Underlapping Swinging out

Range of spindle angles 130–190° 190–230° 230–80° 80–130°

Angle used 60° 40° 210° 50°

Time taken 0.013 s 0.008 s 0.044 s 0.010 s

In order to study the fluctuation law of the tension from the angle of looping theory, the influence of the tension rail is excluded. Also, in the loop forming process it is assumed that the tension rail is fixed, the needle is perpendicular to the needle bed, the sinker is at the position of 1/2 needle gauge, and the guide needle swings right at the centre between the knitting needles. So $L$ and $L$ are constant, and the key point is the change of L brought about by the movement of points D, E, and F in the looping cycle.

Calculation of yarn demand at the beginning of underlap shogging

At this stage, the guide bar just swings to the front of the machine and is ready for underlap shogging. The needle and the closing element descend to the position near to the sinker rib and close the needle hook. The sinker moves towards the rear of the machine. On the projection of the XY plane, points E_XY and F_XY are regarded as coincident at this angle, and point A, B, and C are regarded as fixed. At this stage, the relative positions of those points on the XY plane are as shown in Figure 3. D₁ is used to distinguish the D at this stage. D_2, D₃, and D₄ are the same for the following stages
$L = L + L BC + L CD 1 + L DE + L EF$
(2)

$L CD 1 = \sqrt{} (Z - Z D 1) 2 + L_{C XY D 1 XY}^{2}$
(3)

$L + L = L = \sqrt{(Z D 1 - Z) 2 + L_{D 1 XY E XY}^{2}}$
(5)

$L_{D 1 XY E XY}^{2} = X 2 + Y 2$
(6)

$L = L + L BC + \sqrt{(Z - Z D 1) 2 + (X D 1 - X) 2} + \sqrt{X 2 + Y 2 + (Z D 1 - Z) 2}$
(7)

Figure 3.
Relative positions of special points on the XY plane at the beginning of underlap shogging.

Calculation of yarn demand at the end point of underlap shogging

At this time, the guide needle finishes underlap shogging. The needle moves upward, and the closing element is at the lowest point. The needle head is just at the height of sinker belly. Meanwhile, the sinker reaches the front of the machine. The calculation of yarn demand at the position D₂ is as follows
$L = L + L BC + \sqrt{(X - X_{)}^{2} + (Y - Y D 2) 2 + (Z - Z D 2) 2} + \sqrt{X_{D 2}^{2} + Y_{D 2}^{2} + (Z D 2 - Z) 2}$
(8)

In conclusion, the results of the yarn demand calculation in the process of underlap shogging can be obtained as follows
$L = \sqrt{(X - X D) 2 + (Y - Y D) 2}$
(9)

$L = L + L BC + \sqrt{(X - X D) 2 + (Y - Y D) 2 + (Z - Z D) 2} + \sqrt{X_{D}^{2} + Y_{D}^{2} + (Z D - Z) 2}$
(10)

In the above equations, the values of $L, L BC, X, Y, Z C, X D andZ D$ are constant. The change in the value of $Z E$ is small enough to be ignored. The values of $X D 1, X D 2 andZ D$ have been measured, as follows
$X D 1 = X D 2 = 8, Z D = 19$

So we can be obtain
$L = 124 + \sqrt{1469 + 0.454 + Y D} + \sqrt{425 + Y D}$
(11)

In the process of underlap shogging, the guide bar accelerates first and then decelerates. The change regulation of that speed versus time meets the relationship of a quintic polynomial, but in a high-speed motion process it can be simplified into linear acceleration and linear deceleration, as shown in Figure 4.
Figure 4.
Simplified diagram of the guide bar speed while overlap shogging, where t is the time (s) and V_max is the maximum speed during underlap shogging (mm/s).

Using k to represent the change rate of speed versus time (mm/s²), T for the total time of the current movement process, and d for the needle gauge (mm), then the change regulation of the speed versus time (t (s)) of the guide needle can be simplified to the following equations
$V D = {k \cdot t, if 0 \leq t \leq \frac{T}{2} k \cdot (T - t), if \frac{T}{2} \leq t \leq T$
(12)

$2 d = \frac{T \cdot V max}{2}$
(13)

$V max = \frac{k \cdot T}{2}$
(14)
where T $= 0.044 s$ , $d = \frac{25.4}{28} = 0.907 m$ , and $k = \frac{8 d}{T 2} = 3748.524 mm / s 2$ .

About the y coordinate, the motion of the guide needle while underlap shogging is
$Y D = {- \frac{d}{2} + \frac{k \cdot t 2}{2}, if 0 \leq t \leq \frac{T}{2} \frac{3 d}{2} - \frac{k}{2} \cdot (T - t) 2, if \frac{T}{2} \leq t \leq T$
(15)

$t = 0.0002 \cdot (θ - 230)$
(16)

To get the results of the calculation
$Y D = {- 0.454 + 1874.262 t 2, if 0 \leq t \leq \frac{T}{2} 1.361 - 1874.262 \cdot (0.044 - t) 2, if \frac{T}{2} \leq t \leq T$
(17)

Through processing the function relation, the relative curve of $L$ versus $θ$ during the process of underlap shogging can be obtained as follows (Figure 5)
Figure 5.
Relative curve of $L versus θ$ in the process of underlap shogging.

$230 \circ \leq θ \leq 335 \circ L = 124 + \sqrt{1469 + 5.621 \times 10 - 9 \cdot (θ - 230) 4} + \sqrt{425 + [- 0.454 + 7.497 \times 10 - 5 \cdot (θ - 230) 2]}$
(18)

$335 \circ \leq θ \leq 440 \circ L = 124 + \sqrt{1469 + {1.815 - 1874.262 \cdot [0.044 - 0.0002 \cdot (θ - 230)] 2}} + \sqrt{425 + [1.361 - 1874.262 \cdot [0.044 - 0.0002 \cdot (θ - 230)] 2]}$
(19)

Calculation of yarn demand at the beginning of overlap shogging

At this time, the guide bar just swings to the rear of the machine, and gets ready to start overlap shogging. The needle and the closing element continue rising, and are regarded as reaching the highest point. The sinker is withdrawn, thus relaxing the gripped loop. At this stage, the relative positions of those points on the XY plane are as shown in Figure 6, and the relevant calculation is as follows
$L = L + L BC + \sqrt{(X - X_{)}^{2} + (Y - Y D 3) 2 + (Z - Z D 3) 2} + \sqrt{X_{D 3}^{2} + Y_{D 3}^{2} + (Z D 3 - Z) 2}$
(20)

Figure 6.
Relative positions of special points on the XY plane at the beginning of overlap shogging.

The distance of shogging movements is assumed to be in equilibrium with the n gauge, and it can be obtained as follows
$Y D 3 = (n - \frac{1}{2}) \cdot d$
(21)

$L = L + L BC + \sqrt{(X - X_{)}^{2} + [Y - (n - \frac{1}{2}) \cdot d] 2 + (Z - Z D 3) 2} + \sqrt{X_{D 3}^{2} + [(n - \frac{1}{2}) \cdot d] 2 + (Z D 3 - Z_{)}^{2}}$
(22)

Calculation of yarn demand at the end point of overlap shogging

The guide bar finishes overlap shogging and gets ready to swing back towards the rear of the machine. The knitting needle goes to descend, and the sinker comes forward. The relevant calculation for this stage is as follows
$L = L + L BC + \sqrt{(X - X_{)}^{2} + [Y - (n + \frac{1}{2}) \cdot d] 2 + (Z - Z D 3) 2} + \sqrt{X_{D 3}^{2} + [(n + \frac{1}{2}) \cdot d] 2 + (Z D 3 - Z_{)}^{2}}$
(23)

In conclusion, the results of the yarn demand calculation in the process of overlap shogging can be obtained as follows
$L = L + L BC + \sqrt{(X - X_{)}^{2} + (Y - Y D) 2 + (Z - Z D) 2} + \sqrt{X_{D}^{2} + Y_{D}^{2} + (Z D - Z) 2}$
(24)

In the above equations, the values of $L, L BC, X, Y, Z C, X D andZ D$ are constant. The change of the value of $Z E$ is small enough to be ignored. Some values have been measured and calculated
$X D 3 = X D 4 = - 9, Z D = 9$

So we can obtain
$L = 124 + \sqrt{2788 + (0.45 + Y D) 2} + \sqrt{162 + Y D}$
(25)

Similar to underlap shogging, this can be obtained during the overlap movement
$Y D = {\frac{3 d}{2} + \frac{k \cdot t 2}{2}, if 0 \leq t \leq \frac{T}{2} \frac{5 d}{2} - \frac{k}{2} \cdot (T - t) 2, if \frac{T}{2} \leq t \leq T$
(26)

$T = 0.013 s, k = 21470.837 mm / s 2; t = 0.0002 \cdot (θ - 130)$
(27)

So the results are as follows
$Y D = {1.361 + 4.294 \times 10 - 4 \cdot (θ - 130) 2, if 130 \circ \leq θ \leq 160 \circ 2.268 - 10735.419 \cdot [0.013 - 0.0002 \cdot (θ - 130)] 2, if 160 \circ \leq θ \leq 190 \circ$
(28)

$L = 124 + \sqrt{2788 + (0.454 + Y D) 2} + \sqrt{162 + Y D}$
(29)

Through processing the function relation, the relative curve of $L$ versus $θ$ in the process of overlap shogging can be obtained as follows (Figure 7)
$130 \circ \leq θ \leq 160 \circ L = 124 + \sqrt{2788 + [1.815 + 4.294 \times 10 - 4 \cdot (θ - 130) 2]} + \sqrt{162 + [1.361 + 4.294 \times 10 - 4 \cdot (θ - 130) 2]}$
(30)

$160 \circ \leq θ \leq 190 \circ L = 124 + \sqrt{2788 + {2.722 - 10735.419 \cdot [0.013 - 0.0002 \cdot (θ - 130)] 2}} + \sqrt{162 + {2.268 - 10735.419 \cdot [0.013 - 0.0002 \cdot (θ - 130)] 2}}$
(31)

Figure 7.
Relative curve of $L versus θ$ in the process of overlap shogging.

During the time that the guide bar is returning to the front of the knitting needle, the moving track of the guide needle is an arc, and the value of $Y D$ is constant. Assuming that the X and Z coordinates of the center of the arc are, respectively, a and b, and the radius of the arc is R (mm), we can obtain
$(X D - a) 2 + (Z D - b) 2 = R 2$
(32)

The motion of the guide needle swinging to the front of the needle in the direction of X axis is regarded as a linear acceleration and deceleration process. Similarly available
$X D = {8 - \frac{k \cdot t 2}{2}, if 0 \leq t \leq \frac{T}{2} \frac{k}{2} \cdot (T - t) 2 - 9, if \frac{T}{2} \leq t \leq T$
(33)

$\frac{k \cdot T 2}{4} = 17 mm$
(34)

This process takes about 0.010 s (T = 0.010 s); the calculation results can be obtained
$k = 6.8 \times 105 mm / s 2; t = 0.0002 \cdot (θ - 80)$
(35)

$L = 124 + \sqrt{(13 - X D) 2 + 3.294225 + (57 - Z D) 2} + \sqrt{X_{D}^{2} + 1.852321 + Z D}$
(36)

As measured
$a = - 110, b = 215, R = 229 mm$

The L values of five typical points in the process of the swing from the back to the front of the needle are calculated, and the results are shown in Table 2. The relative curve of $L versus θ$ of this stage is shown in Figure 8.
Figure 8.
Relative curve of $L versus θ$ in the process of swinging in.

Table 2.
Calculation results of the yarn demand

$θ$ ( $\circ$ ) Coordinate of point D L (mm)

80 (8,1.36,19) 183.03

92.5 (5.875,1.36,17.480) 182.689

105 (0.5,1.36,14.424) 182.907

117.5 (–6.875,1.36,10.534) 187.223

130 (–9,1.36,9.476) 189.540

190 (–9,2.268,9) 189.800

200 (–6.875,2.268,10.534) 187.3932

210 (–0.5,2.268,14.424) 187.3577

220 (5.875,2.268,17.480) 182.829

230 (8,2.268,19) 183.164

At the stage that the guide bar swings toward the back of the needle, it can similarly be found that
$X D = {- 9 + \frac{k \cdot t 2}{2}, if 0 \leq t \leq \frac{T}{2} 8 - \frac{k}{2} \cdot (T - t) 2, if \frac{T}{2} \leq t \leq T$
(37)

This process takes about 0.008 s (T = 0.008 s); the calculation results are as follows
$L = 124 + \sqrt{(13 - X D) 2 + 7.409284 + (57 - Z D) 2} + \sqrt{X_{D}^{2} + 5.143824 + Z D}$
(39)

The results are shown in Table 2 and the relative curve of $L versus θ$ of this stage is shown in Figure 9.
Figure 9.
Relative curve of $L versus θ$ in the process of swinging out.

Above all, the yarn demand has been calculated at each main shaft angle. In the high speed knitting process the warp yarn mainly reflects acute elastic deformation characteristics, which can be described by the linear model. According to the changes in the yarn demand, theoretical calculation of the warp tension can be conducted.

Theoretical calculation of yarn tension based on yarn demand

Take the commonly used polyester knitting yarn-75D36F DTY as an example, with the specific technical conditions shown in Table 3.
Table 3.
Experiment conditions

Machine Type Raw material Guide bar Knitting construction Warp run-in Machine speed

KS4 75D36f polyester draw texturing yarn GB2 1–0/2–3// 1680 mm/rack 800 rpm

Using a YG063 fully-automatic single yarn strength tester for the tensile testing of the yarn, and a clamping distance of 50 cm, the tensile curve in Figure 10 is obtained.
Figure 10.
The tensile curve of the experimental yarn.

The yarn tension is generally less than 100 cN while knitting, so the yarn can be regarded as approximately linear elastic, and the relationship between its strength and elongation can be expressed as follows
$σ = p \cdot ɛ$
(40)
where σ is the strength of the yarn (cN), ɛ is the corresponding elongation (mm), and p is the coefficient (cN/mm).

According to the test results, the coefficient p is obtained as
$p = 5.88329 cN / mm$

The relationship between yarn tension and yarn demand in the knitting process can be expressed as
$F = F 0 + E \cdot (L - L 0)$
(41)
where F is the yarn tension (cN), L is the yarn demand at the corresponding main shaft angle (mm), F₀ is the initial value of the yarn tension (cN), and L₀ is the initial length of the yarn (mm).

According to the calculation results, yarn demand reaches the minimum value at the angle of 93 $\circ$ , so the angle is selected as the initial position, and can be obtained as
$L 0 = 182.21467 mm$

In the shutdown state, the yarn tension at the angle of 93 $\circ$ was tested
$F 0 = 9.8 cN$

So the theoretical calculation formula of yarn tension in knitting can be obtained
$F = 9.8 + 5.88329 \cdot (L - 182.21467)$
(42)

$130 \circ \leq θ \leq 160 \circ F = 5.88329 \cdot \sqrt{2788 + [1.815 + 4.294 \times 10 - 4 \times (θ - 130) 2]} + 5.88329 \cdot \sqrt{162 + [1.361 + 4.294 \times 10 - 4 \times (θ - 130) 2]} - 332.69379$
(43)

$160 \circ \leq θ \leq 190 \circ F = 5.88329 \cdot \sqrt{2788 + {2.722 - 10735.419 \times [0.013 - 0.0002 (θ - 130)] 2}} + 5.88329 \cdot \sqrt{162 + {2.268 - 10735.419 \times [0.013 - 0.0002 (θ - 130)] 2}} - 332.69379$
(44)

$230 \circ \leq θ \leq 335 \circ F = 5.88329 \cdot \sqrt{1469 + 5.621 \times 10 - 9 (θ - 230) 4} + 5.88329 \cdot \sqrt{425 + [- 0.454 + 7.497 \times 10 - 5 (θ - 230) 2]} - 332.69379$
(45)

$335 \circ \leq θ \leq 440 \circ F = 5.88329 \cdot \sqrt{1469 + {1.815 - 1874.262 \times [0.044 - 0.0002 (θ - 230)] 2}} + 5.88329 \cdot \sqrt{425 + [1.361 - 1874.262 \times [0.044 - 0.0002 (θ - 230)] 2]} - 332.69379$
(46)

According to the theoretical analysis of the yarn demand, the yarn tension of each angle range is calculated (see Table 4).
Table 4.
Calculation results of the yarn tension

$θ$ ( $\circ$ ) L (mm) F (cN)

80 183.03 14.59682

92.5 182.689 12.59062

105 182.907 13.87318

117.5 187.223 39.26546

130 189.540 52.89785

190 189.800 54.4267

200 187.3932 40.26679

210 187.3577 40.05794

220 182.829 13.41428

230 183.164 15.38518

According to the above calculations, we can find the theoretical fitting tension curve, as shown in Figure 11.
Figure 11.
The theoretical fitting tension curve of the whole knitting cycle.

The theoretical tension curve is obtained based on the yarn demand calculation, and in order to verify the correctness of the above model, the practical tension curve should been tested.

Comparison of theoretical tension curve and observed tension curve

A complete dynamic tension test system was built to collect tension data at the sampling frequency of up to 1 kHz. The system consists of a TS1-200-A2-CE1 type tension meter, a SCM01 four channel sound of vibration analyzer for synchronous signals acquisition of tension and the main shaft, and LMS test.lab software supporting signal analysis and processing. Experimental conditions are the same as those shown in Table 3.

The measured yarn tension fluctuation curve without the action of the tension rail is shown in Figure 12, in which the tag F (cN) is the yarn tension and θ (°) is the main shaft angle.
Figure 12.
The observed tension curve.

On the theoretical fitting tension curve (Figure 11), it can be seen that the tension values fluctuate between 13 cN and 57 cN, and there is a peak that occurs at the range of 80–240°, while in other regions the tension values are significantly smaller and vary little. In the process of the whole forming process, the warp tension increases sharply when the yarn is required to swing from the back to the front of the needle, and reaches its highest when the overlap draws to a close.

On the observed tension fluctuation curve without the action of the tension rail, in the range of 360°, the tension fluctuates within the range of 11–58 cN, and there is an obvious peak in the range of 120–260°.

It can be seen that the overall variation trend and the tension value range of the theoretical tension curve are basically consistent with the observed one; there is a large crest near to the overlapping, apart from which the other sections of the curve are low and easy.

In the knitting process, the change of yarn demand is affected by the motion of the guide bar, the needle, and the sinker. From the knitting elements displacement map and the tension fluctuation diagram, it can be seen that the yarn guide needle displacement curve is basically the same as the tension fluctuation curve in trend, where the maximum exists in the vicinity of 180°. So it can be concluded that the maximum tension wave crest is mainly influenced by the movement of the guide.¹¹

On the other hand, unlike the observed tension curve, where the angle range of the wave crest is about 140°, on the theoretical curve the peak occurs at the angle of 160°, which means the actual tension fluctuation near the peak is more intense.

In the process of modeling and simplified calculation, the movements of the needle and the sinker have been ignored, as the displacement is really small, which means that the yarn curve unable to reflect the details of some slight fluctuations. However, relative to the peak near to 180°, the value is too small to be considered in tension research. Also, the process where the guide bar swings toward the back of the needle in the vicinity of 230° is simplified. The calculation is conducted without regard to the fact that the yarn might be bent by the slabbing blade.¹² However, at this stage, the tension is in continuous reduction, and has no effect on the study of the wave crest region. Further, the effect of friction between the yarn and the guiding elements has not been considered in the model, which makes the theoretical curve a little offset to the left.

The difference between the yarn demand and feed is the main factor causing the change of the yarn tension, but at high machining speed and high thread fineness, the properties of the yarn, which is an elastic material with mass and which creates friction, influence the process.² The inertia of the mass affects the fluctuation of yarn speed, causes yarn quality change, and indirectly influences the yarn tension. The inertia of a warp is due to a stick–slip phenomenon, but it is also due of the actual reaction to fast dynamic warp tension change.¹³ In the knitting process, due to the contact of yarn and yarn guiding devices, the warp yarn must overcome some friction in the movement from the warp beam to the knitting point. This makes the tension increase gradually, reaching the maximum near to the knitting needle.¹⁴ Eventually, the yarn must bear the complex stress caused by the overall effect of yarn feed imbalance and friction, both of which can cause cumulative tension in the yarn. Strain in the elastic yarn is produced at the same time. Active tension compensation, in cooperation with the mutative required amount of warp yarn, must consider friction as well, so as to realize accurate compensation.

Conclusions

From the above analysis, rather than too much tension, the major causes of yarn breakage are sudden changes of tension and yarn damage caused by repeated stretching, which both finally come down to the imbalance between yarn demand and feed in one knitting cycle. The change of yarn demand during the looping process can be calculated by studying the required quantity at several specific points, based on which the tension curve can be plotted theoretically. By comparison, the variation trends of the theoretical tension curve, fitted according to the established model, and the warp knitting tension fluctuation curve without the action of the tension rail, measured by the dynamic tension test system, are basically identical. The highest crests of them both appear before and after overlapping, which is mainly affected by the movement of the guide bar. The value of the tension fluctuates between 10 cN and 60 cN, and is small and stable outside the main crest. So it can be concluded that the model can be used to explain the fluctuation of knitting tension from the perspective of the principle of looping, which can also provide a theoretical basis for further tension control and compensation.

According to the above model, it can be seen that the faster the machine speed is, the less time there is for each turn, and the faster the process of acceleration and deceleration, which can speed up the change of yarn demand. Also, the shogging distance increases, bringing similar results. Furthermore, the tension rail is not fixed in practice, and it can play a large role in balancing the tension by changing the values of $X, Y, Z C$ , which can greatly affect the yarn demand. However, the tension rail presently provides passive tension compensation, and the hysteretic nature limits the effect. To achieve a more timely compensation effect, it will be necessary to look at reducing tension fluctuation by active and real-time compensation of greater precision.

According to the results of the analysis, we obtain a rule of theoretical tension fluctuation in the process of one knitting cycle. However, the effect of friction between the yarn and the guiding elements has not been considered in the model, resulting in a theoretical curve a little offset to the left, which is also an important factor for more precise tension compensation. Combined with the elongation and friction properties of yarn, the fluctuation of tension can be accurately analyzed. A positive tension compensation device can be developed to adjust the warp run-in in a timely way using the law of the theoretical tension change, and at the same time, to compensate for the influence of friction, so realizing real-time compensation for tension.

Strickle movement process	Overlapping	Swinging in	Underlapping	Swinging out
Range of spindle angles	130–190°	190–230°	230–80°	80–130°
Angle used	60°	40°	210°	50°
Time taken	0.013 s	0.008 s	0.044 s	0.010 s

$θ$ ( $\circ$ )	Coordinate of point D	L (mm)
80	(8,1.36,19)	183.03
92.5	(5.875,1.36,17.480)	182.689
105	(0.5,1.36,14.424)	182.907
117.5	(–6.875,1.36,10.534)	187.223
130	(–9,1.36,9.476)	189.540
190	(–9,2.268,9)	189.800
200	(–6.875,2.268,10.534)	187.3932
210	(–0.5,2.268,14.424)	187.3577
220	(5.875,2.268,17.480)	182.829
230	(8,2.268,19)	183.164

Machine Type	Raw material	Guide bar	Knitting construction	Warp run-in	Machine speed
KS4	75D36f polyester draw texturing yarn	GB2	1–0/2–3//	1680 mm/rack	800 rpm

$θ$ ( $\circ$ )	L (mm)	F (cN)
80	183.03	14.59682
92.5	182.689	12.59062
105	182.907	13.87318
117.5	187.223	39.26546
130	189.540	52.89785
190	189.800	54.4267
200	187.3932	40.26679
210	187.3577	40.05794
220	182.829	13.41428
230	183.164	15.38518

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors wish to acknowledge the Fundamental Research Funds for the Central Universities (grant number JUSRP51404A), the National Science Foundation of China (grant number 51403080), the Natural Science Foundation of Jiangsu Province (grant number BK20151129), and the Innovation fund project of Cooperation among Industries, Universities and Research Institutes of Jiangsu Province (grant number BY2014023-20).

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