KINEMATIC COMPARISON OF A HEALD FRAME DRIVEN BY A ROTARY DOBBY WITH A CAM-SLIDER, A CAM-LINK AND A NULL MODULATOR

comparison of a heald frame driven by a rotary dobby with cam-slider and cam-link modulators is introduced. The cam-modulated linkage, also known as the cam-integrated linkage or the combined cam-linkage mechanism, is a composite mechanism consisting of at least one cam–follower pair in combination with a linkage [23-27]. Moreover, in this paper, 200 mapping points based on two conjugate cam profi les are measured using a three-coordinate measuring machine, and non-uniform rational B-splines (NURBS) [28,29] are used to form the cam contour curves. Kinematic comparison mathematical models of the heald frame motion generated by rotary dobby with cam-link or cam-slider modulator are established based on the two different curves. In Abstract: The kinematics of the heald frame of a rotary dobby with two different modulator types are analyzed and compared. Kinematic mathematical models of the modulator main shaft, cam unit, and heald frame driven by the rotary dobby with a cam-slider modulator and a cam-link modulator were constructed based on two different cam contours derived from measured points on the conjugate cams of the two modulators. The motion characteristics of the two modulators and a null modulator, the cam unit, and the motion transmission unit are analyzed. The purpose of the present study was to establish the kinematic models, investigate the motion characteristics, and analyze their differences. At the same time, a calculation method for each motion transmission process was established and numerical models were developed. The results demonstrated that the two different modulators produce almost the same heald frame motion characteristics. Despite that both modulator types can be adapted to the requirements of a loom, the cam-link modulator can produce a more stable and reliable motion.


Introduction
Rotary dobby is a commonly used shedding device that can be applied to shuttleless looms such as high-speed rapiers and air-jet and water-jet looms. Today, it is a dominant type of dobby in the industry that can be used on all types of weaving machines [1]. At opening, the warp yarn is lifted and lowered by the heald frame. The characteristics of the heald frame motion have great infl uence on the force of warp yarn in weaving, and this is the fundamental factor affecting the shedding motion.
Dobby constructions are classifi ed according to their shedding principles, control program, structure, and motion transmission mechanism to the heald frame [2]. Eren et al. introduced mechanism models for rotary dobbies and cranks, and cam shedding motions have been introduced and equations governing heald frame motion have been derived. Heald frame motion curves have been obtained and compared with each other, while heald frame motion characteristics have been mainly determined by the design of modulator mechanisms. According to References [1,[3][4][5][6][7], the eccentric mechanism of rotary dobbies also has a signifi cant effect on heald frame motion, and the motion is transmitted to the link of the heald frame and the heald blade to perform the upper and lower motions of the heald frame. Guo and Chen [8] introduced a new type of microprocessor-controlled dobby, which has been proved to be practical, simple, and easy to be manufactured compared to recent positive dobbies. Gao et al. [9,10] analyzed the motion mechanism of the rotary dobby GT241, established the kinematics model for the main transmission component, and carried out kinematics simulation analysis. The description of a mechanism based upon the method of devising motion equations by means of Lagrange's equations was successfully applied in Mrazek [11] and Jin et al. [12]. The relevant mathematical model is described with more details in Bílek and Mrazek [13,14]. Zhang et al. [15] established a variable rotation speed mechanism of dobby and performed kinematics simulation analysis.
At present, the dobby development technology is focused toward higher speed, higher effi ciency, and easier control and operation [16][17][18]. Conjugate cams are also used in most rotation transmission mechanisms of rotary dobby lifting mechanisms. The input uniform rotational motion is transformed into the output motion of the main shaft [19][20][21]. In another study, the redesign of crank connection mechanism ensured that the heald frame moved according to the law of simple harmonic motion [22]. In the present study, a kinematic comparison of a heald frame driven by a rotary dobby with cam-slider and camlink modulators is introduced. The cam-modulated linkage, also known as the cam-integrated linkage or the combined camlinkage mechanism, is a composite mechanism consisting of at least one cam-follower pair in combination with a linkage [23][24][25][26][27]. Moreover, in this paper, 200 mapping points based on two conjugate cam profi les are measured using a three-coordinate measuring machine, and non-uniform rational B-splines (NURBS) [28,29]  [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] addition, theoretical numerical calculations of the mathematical models are performed using Visual Studio 2012. Figure 1(a)-(c) shows the basic structure of the cam-slider, cam-link, and null modulators, respectively. In Figure 1(a), large gear 1; conjugate cams 5 and 10; cam rollers 3, 6, 9, and 12; cam swing arms 2 and 8; modulator sliders 4 and 7; and main shaft 11 constitute the cam-slider modulator. The cam swing arms are fi xed on the large gear. When the loom is running, the slider is embedded in the groove of the slide rack, so that when the large gear continuously rotates, it leads the slider rack and the main shaft to rotate synchronously. The rotational speed of large gear is uniform. The groove direction of the cam swing arms is always toward the center of the main shaft. The groove direction can be changed by conjugate cams. The cam swing arms are equipped with four cam rollers (3, 6, 9, and 12). With the rotation of the large gear, the groove direction of the cam swing arms changes according to certain rules, resulting in the rotation of the main shaft.

Motion principle
The cam-link modulator is similar to the cam-slider modulator.
In Figure 1( Therefore, the modulator transforms the uniform rotary motion of the weaving machine into a non-uniform rotary motion of the dobby main shaft. The use of conjugate cam and linkages or sliders with precise characteristics and the requirements of heald frame motion principle can be met.
The null modulator is similar to the cam-link modulator. The difference is that the conjugate cam is replaced by a circular one. As it can be seen in Figure 1    In order to perform numerical calculations, 200 cam contour points were measured using a coordinate measuring machine. Cubic NURBS curve interpolation was used to construct the conjugate cam contour of the rotary dobby. In Figure 3, the coordinates of the cam profi le mapping points are presented.     The theoretical cam curve is described by From the geometric relationship: From the law of cosines: In addition: B on the large gear is centered at point 1 O . The trajectory formed by point A is the theoretical cam profi le.
As it can be seen in Figure 8(a), the cam-slider modulator and cam-link modulator angular displacement curves are basically consistent. Approximately 10° dobby modulator main shaft double-dwells are formed. Therefore, the modulator transformed the uniform rotary motion of the weaving machine into a non-uniform rotary motion of the dobby main shaft. On the other hand, the null modulator cannot form dwell, and thus, the dobby main shaft performed a uniform rotary motion. Figure 8(b) shows that the maximum angular velocity was higher with the cam-slider modulator than with the cam-link modulator, while the cam-slider modulator angular velocity was about 7% higher than that of the cam-link modulator. The camlink modulator generated a motion with a longer approximately uniform motion period than the cam-slider modulator. From the angular velocity curves, it can be clearly seen that the camlink modulator was more stable than the cam-slider modulator. This can be more clearly observed in the modulator angular acceleration curves in Figure 8(c). The surface irregularity of the cam profi le is one of the primary sources of vibration noise. The angular acceleration of the cam-link modulator had a gradual period, but the cam-slider modulator did not. This means that the motion characteristics produced by the cam-link modulator were more stable and reliable than those produced by the cam-slider modulator.
The cam unit is the main executive mechanism for controlling the shedding motion of the heald frame. The design of the (19) Equation (19) can be written as (20) Then where 2 2

Results and discussion
The parameters of the cam-slider and cam-link modulators are shown in Table 1, and the parameters of the cam unit and the MTU are shown in Table 2.
A program was developed in Visual Studio 2012 using the data in Tables 1 and 2, and Equations (1) to (21). The rotational speed of the large gear of the main motor was set to 100 rpm. One work cycle of the large gear was calculated.     The motion characteristics of the heald frame are similar to those of the lifting arm. The motion transmission mechanism does not affect significantly the motion characteristics of the heald frame.
In Figure 10(a), it can be concluded that the cam-slider modulator and the cam-link modulator generate a heald frame motion with an approximately same dwell period. However, it may be recalled from Figure 6 that the cam unit is a crank-rocker mechanism without snapback characteristics. The motion characteristics of heald frames 21 and 21′ are not exactly the same. The heald frame 21 has a longer approximate dwell than heald frame 21′ at both the lower and upper shed positions. At a displacement of 46 mm, the heald frames 21 and 21′ reach the middle shed position and complete the shedding motion.
Since the null modulator's main shaft performs a uniform circular motion where no heald frame dwell is obtained, the actual working conditions of the loom are not met. This result indicated that the dwell of the heald frame is controlled by the cam-slider or the cam-link modulator of the rotary dobby.
On the other hand, Figure 10(b) and (c) shows that the maximum velocities of the heald frames 21 and 21′ with the cam-slider modulator were higher than those with the cam-link modulator. The velocity of heald frames 21 and 21′ driven by the cam-slider modulator was approximately 7% higher in the upper position and 6% higher in the lower position than that driven by the cam-link modulator. The maximum acceleration of heald frames 21 and 21′ with the cam-slider modulator was higher than that with the cam-link modulator. The acceleration of heald frame driven by the cam-slider modulator was approximately 9% higher in the upper position and 5% higher cam unit determines the output motion characteristics of the lifting arm, thus affecting the shedding motion of the heald frame. Figure 9 demonstrates the motion characteristics of the cam unit's lifting arms 15 and 15′ generated by the three different modulators. In Figure 9(a), the input motion of the cam unit is a nonuniform rotating motion transmitted by the cam-slider and cam-link modulators, and the output motion is the swing motion of the cam unit's lifting arm. The cam-slider and cam-link modulators' angular displacement curves were basically consistent. It can be concluded that the cam-slider and the cam-link modulators can generate a cam motion with an approximately same dwell period. As it was previously mentioned, the null modulator cannot form dwell. This can be more clearly seen in the curves of angular velocity and angular acceleration in Figure 9(b) and (c), respectively.
In Figure 9(b), it can be seen that the maximum velocities of both the cam lifting arms 15 and 15′ are higher with the cam-slider modulator than with the cam-link modulator. The velocity of the cam driven by the cam-slider modulator was approximately 7% higher in the upper position and 6% higher in the lower position than that driven by the cam-link modulator. Similarly, as it can be observed in Figure 9(c), the maximum acceleration of the cam unit was higher with the cam-slider modulator than with the cam-link modulator, while the acceleration of the cam unit driven by the cam-slider modulator was approximately 10% higher than that driven by the cam-link modulator in both the upper and lower positions. A detailed comparison is given in Tables 4 and 5. The results indicated that the motion curves of the cam unit have continuous velocity and acceleration. Therefore, the cam-link modulator should be used to prevent impact stresses and noise in the structure.  in the lower position than that driven by the cam-link modulator. A detailed comparison is given in Tables 6 and 7. The motion characteristics of the heald frame require the velocity of the heald frame to be stable and avoid sudden changes. The warp should move slowly in its tense state and faster in the relaxed state. Therefore, the motion of the heald frame should be slower near the fully open shed position and faster near the middle shed position. The transition of the heald frame from stationary to moving and from moving to stationary should be slow and the acceleration should be stable, in order to avoid causing vibration of the heald frame. Owing to this, the cam contour in the modulator was obtained by mapping; there were measurement and fi tting errors. However, by comparing the curves in Figure 10(b) and (c), it can be clearly seen that the velocity and acceleration of heald frames 21 and 21′ produced by the cam-link modulator were more stable compared to those produced by the cam-slider modulator.
When rotary dobby with this cam-link modulator is applied to different looms or different fabrics, especially with different speeds, further analysis to the characteristics of the dynamics of the heald frame is required.

Conclusions
The displacement, velocity, and acceleration curves of the modulator main shaft, cam unit, and heald frame with a camslider or a cam-link modulator were obtained by numerical calculations based on the established mathematical model. At the same time, the motion characteristics of the cam unit and heald frame with the null modulator were obtained. Based on the above analysis, it can be concluded that the dwell of the heald frame is controlled by the modulator of the rotary dobby. The motion characteristics of the heald frame are similar to those of the cam unit. The MTU enlarges the swing motion of the cam unit and transfers it to the heald frame, producing the up and down motion of the heald frame.
The comparison of the displacement, velocity, and acceleration curves of the modulator main shaft, cam unit, and heald frame with a cam-slider or a cam-link modulator indicated that the motion characteristics of the modulator and heald frame produced by the cam-link modulator are more stable than those produced by the cam-slider modulator. This difference is more apparent when comparing the velocity and acceleration curves. The heald frame driven by the rotary dobby with the cam-link is more stable and reliable than that with the camslider modulator.