Calculation and Analysis of Horizontal and Vertical Lapping Angles in Tricot Warp Knitting

Abstract The lapping angle, which affects the style and quality of production, has been studied as a parameter of weft knitting. But the importance of the lapping angle has not been considered during the warp-knitting cycle. This paper shows that the lapping angle exists in the process of warp knitting and can be divided into horizontal and vertical lapping angles. Models for the lapping angles of closed and open loops were devised, and the lapping angles (horizontal and vertical lapping angles) of closed and open loops were calculated and analyzed. Furthermore, the paper seeks to investigate the factors that influence the lapping angle of tricot warp-knitted fabrics and summarize the rules. Moreover, the vertical lapping angle can affect the loop coverage. Results reveal that the decrease in number of underlaps and an increase in take-off density enables loops of the front guide bar to show on the face of the fabric. Moreover, it is also advantageous for an apparent front loop visibility when the front guide bar knits in open loop.


Introduction
A large amount of work has been carried out to study the stages of the warp-knitting cycle and the cooperation among the knitting elements [1][2][3]. Almost none of these studies discusses the lapping angle during the warp-knitting cycle. There are also many reports on the lapping angle, which is divided into the horizontal lapping angle and the vertical angle during the weft-knitting process [4,5] The lapping angle is used as a parameter to adjust the machine so that the machine is at the best condition for production. To some extent, the warp-knitted fabric is similar to the plating stitch. However, the overlap and underlap present during the knitting cycle and the two kinds of loop (open and closed loops) distinguish weft-knitted fabrics from warp-knitted ones [6]. This paper reveals that there is a lapping angle during the warp-knitting cycle, which also can be divided into horizontal and vertical lapping angles. Moreover, it is generally acknowledged that by ensuring a larger horizontal lapping angle for the ground yarn, as opposed to the supply yarn, the front yarn will surely show on the face side of the fabric when weft knitting in plating stitch [7][8][9]. The principle of warp knitting involves the yarn of the front bar covering that of the back bar in a relationship of loops of the two guide bars, usually referred as loop coverage [10]. However, various factors in actual production lead to variations in loop coverage. If the yarns of the two bars are the same, loop coverage will not affect the fabric style; otherwise, it will impact the style and the quality of the product regardless of the linear density, composition, or color. A previous study has shown that a lower lapping height is considered to provide favorable conditions for the yarn to show on the face side of the warp-knitted fabric [1] and that the vertical lapping angle is related to the lapping height.
This paper aims to establish the theoretical models of closed and open loops, and it reveals that there is a lapping angle during the warp-knitting cycle, which also can be divided into horizontal and vertical lapping angles. Through the study of the lapping angle during the warp-knitting cycle, this study can provide the theoretical basis for the study on loop coverage.

Calculation of lapping angle
The trail of the guide needle establishes the models for the lapping angle of closed and open loops [11].

Calculation of lapping angle for closed loop
The top view of lapping movement of closed loop is shown in Figure 1, the vertical lapping angle is shown in Figure 2, and the top view of lapping movement of the open loop is shown in Figure 3, where: t is the needle gauge, r is the radius of needle bar, n is the maximum distance of guide needle swing toward the front of machine, m is the maximum distance of guide needle swing toward the rear of machine, c is the height of each course, k is the number of underlaps (Figures 1 and 3 show the case for k=1), h is the height from the contact point of the old loop and needle to the guide needle role (The value of h varies with the swing, but the amount of change is too small to be ignored), O 1 and O 2 are the centers of the two needles and O 1 is the coordinate origin, A is the origin of the new extension cord, 1 Bi (i=1,2,¼¼.8) is the tangent point between yarn and needle bar, Cj(i=1,2,¼¼.8) is the lapping point, O 1 E= O 2 A= c-r, l 1 =k·t, Fi (i=1,2,¼¼.8) is the location of the guide needle (points F 1 , F 2 ¼¼F 8 are eight locations at different times), and the coordinates of Fi are (x, y), S is the total movement length of guide needle, and D is the origin point of the movement of guide needle; so S = (x -xD) + (yD -y) = x+ n-y (x>0) and S = (xD -x) + (yD -y) = -x+ n -y (x<0).
During the knitting cycle, the tension on the yarn makes the length of segment AFi′ (i=1,2,¼¼.8) the shortest, even if there is the angle of wrap between the yarn and the needle bar. Figure 2 shows the situation when the yarn is straightened. Here, A, Ci′, and Fi′ are the points on the yarn, and Ci, Fi are the projections of C i ′ and Fi′ on the horizontal section.
ai (i=1,2,¼¼.8) is the horizontal lapping angle, meaning the angle between the projection of yarn on the horizontal section and the X-axis. bi (i=1,2,¼¼.8) is the vertical lapping angle, meaning the angle between the yarn and the horizontal section. qi (i=1,2,¼¼.8) is the yarn angle around the needle.
The calculation was made based on the process of D-G-H-I-J for the closed loop. There is no angle of wrap during the process D-G. The guide needle F moves between D and G, as in F 1 There is no angle of wrap during the process G-H. The guide needle F moves between G and H, as in F 2 . Thus, x=0.5t, y~[y D, n], so S=0.5t+n -y.
0.5 arctan arccos arctan The angle of wrap exists during the process H-I. The guide needle F moves between H and I, as in F 3 . Thus, x=0.5t, y~[0, y D ], so S=0.5t+n -y.
There is the angle of wrap during the process I-J. The guide needle F moves between I and J, as in F 4 . Thus, x=0.5t, y~[-m, 0], so S=0.5t+n -y.

The calculation of lapping angle for open loop
In There is no angle of wrap during the process D-P. The guide needle F moves between D and P, as in F 5 . Thus, x~[-0.5t,0], y=m, so S=n -x -m.
There is no angle of wrap during process P-N. The guide needle F moves between P and N, as in F 6 . Thus, x=-0.5t, y~[0,n], so S=0.5t+n -y.              As shown in Figure 7, the vertical lapping angle increases from b 1 to b 2, and the lapping height decreases from H 1 to H 2 . It can be seen that the larger the vertical lapping angle, the lower is the lapping height. Combined with the previous conclusion, the decrease of vertical lapping angle is detrimental to loop visibility on the face of the fabric.
The results show that both the horizontal lapping angle and the lapping height of an open loop increase, which is unfavorable for loop visibility on the fabric face, thereby confirming the findings in previous research. However, it was found that with an open loop, it is easier to cover other loops in practical production. To a large extent, the direction of the extension cords for the open loop make the loop expand, but for a closed loop, the direction of extension cords tends to make the loop contract.
In order to allow consistent loop coverage of the front guide bar over the back guide bar, the vertical lapping angle has an important role to play, followed by the horizontal lapping angle and the take-off density.

Conclusion
The investigations show that there is a lapping angle during the knitting cycle, and the lapping angle can be divided into horizontal and vertical lapping angles.