Comparison of the Vibration Response of a Rotary Dobby with Cam-Link and Cam-Slider Modulators


 This article presents the kinematic modeling and analysis of cam profiles of two different types of modulator: a cam-link and a cam-slider modulators. Kinematic and dynamic models of the two different modulators were established based on the motion curves of the main shaft of the rotary dobby. Simulations were carried out in Simulink to analyze the vibration responses under different rotary speeds, and vibration responses of the two mechanisms were compared. The results show that the cam-link modulator vibrates smoothly at speeds of <700 rpm, and theoretically, the speed should not exceed 1,400 rpm. The cam-slider modulator vibrates smoothly at speeds of <500 rpm, and theoretically, the speed should not exceed 1,000 rpm. The cam-slider modulator is more suitable for use at low speeds, whereas the cam-link modulator is more appropriate for high speeds. When both the cam-slider and cam-link modulators operate at high speeds, vibration distortion occurs, leading to bifurcation and chaotic vibration. Further knowledge of the complex behaviors associated with detachment of the follower from the cam can support the design of more sophisticated controllers aimed at avoiding follower detachment.


Introduction
The rotary dobby is currently the most advanced high-speed opening device used in modern high-speed weaving machines. The dobby mechanism divides the warp yarn into two layers according to a certain rule, and a channel is formed, allowing the weft thread to pass through. This is achieved by converting the rotational motion of the main shaft of the weaving machine into the up-and-down motion of the frame. Nowadays, the rotary dobby is widely used in shuttleless looms, such as the rapier loom and air-jet loom, due to its compact structure, high motion accuracy, and the ability to achieve simple control and stable operation [1][2][3][4].
As one of the most important components, the rotary transmission mechanism is crucial to the performance of the rotary dobby. A modulator transforms uniform rotary motion of the weaving machine into non-uniform rotary motion to the dobby main shaft and determines the heald frame motion characteristics of the loom. The requirements of the loom can be met using conjugate cams and linkages or sliders with welldefined characteristics. However, further studies on the dynamic performance of various rotary transmission mechanisms are required [5,6]. Cam-follower mechanisms are widely used in modern machinery due to several attractive features, such as a simple design, low-cost manufacturing, and versatility.
The cam follower represents, in essence, the core of the rotary transmission. In high-speed cam-driven mechanisms, the motion accuracy of the follower affects the overall performance of the machine. Several researchers have investigated the influence of cam profile accuracy and system flexibility on the output motion. Rothbart [7] proposed a method for predicting the follower response to irregularities on the cam surface. In another study, Koster [8] used a four degrees-of-freedom model to simulate the effects of backlash and camshaft deflection on the output of the mechanism. Kim and Newcombe [9] and Grewal and Newcombe [10] investigated the combined effects of geometric inaccuracies, kinematic errors, and dynamic errors. Norton [11][12][13] investigated the effects of manufacturing tolerances on the dynamic performance of eccentric and double dwell cams through experimental analyses. Besides, Wu and Chang [14] proposed an analytical method for analyzing the effects of mechanical errors in disc-cam mechanisms.
Complex behaviors in mechanical devices with impacts, as well as other types of piecewise-smooth dynamical systems, have become important topics of ongoing research [15][16][17][18]. At high rotary speeds, contact loss between the cam and the follower can reduce engine performance [19,20], particularly in terms of efficiency, fuel consumption, and emissions [21]. Gianluca and Domenico [22] presented a preliminary study on the control of follower vibrations in cam-follower mechanisms when acting directly on the follower.
The research described earlier has laid a foundation for further analyses of the kinematic characteristics and dynamics of rotary transmission mechanisms of the rotary dobby. The speed of modern looms is continuously increasing, in pursuit of increased productivity, however, problems associated with higher operating speeds, such as machine vibrations and reduced stability, can shorten the lifespan of machines and must be solved. In-depth studies on the dynamic performance dobby rotary transmission mechanism, a key part of the opening mechanism, could help to solve these problems. However, the complex behaviors of transmission mechanisms have made it difficult to establish an accurate model of the rotary dobby. An accurate dynamics model of the dobby rotary transmission mechanism would provide the basis for studying and optimizing the dynamic performance of the mechanism and improve the efficiency of shuttleless weaving machines and product quality [23].

Motion principle of modulators
Schematic drawings of the cam-link (CL) and cam-slider (CS) modulators are presented in Figure 1(a) and (b), respectively. The CL modulator, shown in Figure 1(a), consists of a gear, conjugate cam (including main cam and auxiliary cam (aux cam)), roller, swing arms, links, and main shaft. The swing arms are fixed to the gear. When the loom is running, the swing arms are hinged to the gear, and motion is transmitted to the cam rollers. The rotational speed of the gear is uniform. The conjugate cam is fixed to the dobby body, such that the four rollers move along the main and auxiliary cam contours of the conjugate cam. Clockwise rotation of the follower rollers causes the links to move, which drives the main shaft of the dobby.
Similarly, the CS modulator, shown in Figure 1(b), consists of a gear, conjugate cam (including main cam and auxiliary cam), rollers, swing arms, sliders, and the main shaft. The swing arms are fixed on the large gear. When the loom is running, the slider is embedded in the groove of the slide rack. As the gear continuously rotates, the slider rack and main shaft rotate synchronously. The groove direction ensures the swing arms always travel toward the center of the main shaft, and the groove direction can be altered by the conjugate cam. The swing arms are equipped with four rollers. The groove direction of the swing arms can be changed by rotating the gear according to certain rules, resulting in rotation of the dobby main shaft.

Acquiring the cam profile
The cam profile is an important aspect of cam-follower systems and is typically designed to move the follower in the desired manner [24][25][26]. Given the wide range of applications, there is a wide variety of possible cam geometries, ranging from cycloidal cams to cam motion designed using spline functions to accommodate discontinuities in the acceleration of the follower. In the present study, first we assume that the main shaft of the dobby will have the same kinematic curves for both modulators. Then, cam profiles of the CL and CS modulator can be obtained. Figure 2 illustrates the kinematic motion characteristics of the main shaft of the modulators.  The slope of the line n n − normal to the main cam pitch curve

The CL modulator
where γ is the angle between the normal line n n − and the x y are coordinates of any point A on the main cam pitch curve.
Eq. (2) can be rewritten as According to Eqs (1) From the geometric relationship: According to the law of cosines: The geometric relationship is where angle β , From Eqs (8)-(13), the point ( ) Similar to the principle of the CL modulator mechanism: The coordinates of point  with the contact points, as shown in Figure 5(b).
The detached bodies of follower 1-CL and follower 2-CL are considered for force analysis. In the vertical direction, follower 1-CL and follower 2-CL are subjected to an elastic restoring force and a damping force. The force analysis is presented in Figure 5(c). The CL and CS modulators have the same dynamic model.

Geometric parameters
The conjugate cam profiles were obtained using geometrical methods and theoretical calculations. Parameter values of the CL and CS modulators are presented in Table 1. Eqs (1)- (18) were used to ascertain the conjugate cam profiles of the CL and CS modulators. All calculations were performed using a computer program written and compiled in Microsoft Visual Studio 2012 on a personal computer.

Dynamic model
The dobby modulator was modeled as two masses, three springs, and three dampers. Figure 1(a) and (b) shows models of the CL and CS modulators, respectively. When the gear rotates at low speeds, the cam surface and the contact points of rollers 1 and 2 never separate. Roller 1 meets the desired displacement characteristic

Dynamic model response
The nonlinear system of equations, presented in Eq. (20), was implemented in Simulink using the parameters listed in Table 2. The Runge-Kutta method was used to calculate 1 y and 2 y of the two mechanisms. Then, the dynamics of the cam-follower

Vibration response of CL modulator
To understand the vibration response of follower 1-CL, a bifurcation diagram was derived for different values of w , as shown in Figure 7. The evolution of gear angle θ with rotational speed is presented in Figure 8(a)-(d).
low. Owing to the different mechanical structures of the two modulators, the cam contours must also differ to achieve the same motion characteristics of the spindle output. Therefore, the ( ) 1 s θ curves produced by each modulator are different.

Results and discussion
Cam-follower systems are particularly sensitive to variations in gear rotational speed w . Here, we sought to uncover the    Figure 9.
According to the above analysis, the maximum value of w cannot theoretically exceed 1,400 rpm in the CL modulator system and should ideally operate under 700 rpm.

Vibration of CS modulator
The bifurcation diagram of follower 1-CS position of CS modulator is presented in Figure 10. As an example, trajectories Figure 7 shows the vibration response of the follower 1-CL at about 700 rpm, and the corresponding vibration response to the gear angle θ is shown in Figure 8( , the vibration of the system is relatively stable. Severe vibration suddenly occurs at about 1,500 rpm, which rapidly evolves into a chaotic attractor, as illustrated in Figure  8  Theoretically, the maximum value of w should not exceed 1,000 rpm, and ideally, the CS modulator system should operate under 500 rpm.

Comparison of vibration results
Bifurcation diagrams of the vibration/displacement of follower 1-CL and follower 1-CS are presented in Figure 13. In Figure  13(a), the vibration/displacement is equal to the vibration of follower 1-CL/ ( ) 1 S θ of the CL modulator. In Figure 13 From Figure 13, it can be observed that, overall, the vibration/ displacement of follower 1-CS is smaller than that of follower 1-CL. The ideal speed of the CL modulator is under 700 rpm, however, theoretically, the modulator can be used at speed of up to 1,400 rpm. On the other hand, the ideal speed of the CS modulator is under 500 rpm, and theoretically, the modulator can be used at speed of up to 1,000 rpm. Based on the analysis of vibration/displacement, it can be concluded that the CS modulator is suitable for gear speeds w under 500 rpm, and the CL modulator is suitable for high speeds.

Conclusions
Based on the kinematic characteristics of the main shaft, this article modeled and analyzed the cam profiles of the CL and CS modulators, which were used to establish physical models of the modulators. The main features of the two proposed modulators were described and the dynamics of each camfollower system were investigated. As the gear speed w changes, the modulator systems exhibit complex vibration behaviors. Periodic impact behavior emerged, characterized by several impacts and chattering. The preliminary results suggest the presence of bifurcations and chaotic vibrations in the system. at different instances in time are shown in Figure 11 for two gear cycles only. Figure 10 shows the vibration response of follower 1-CS at about 500 rpm, and the corresponding vibration response to the gear angle θ is shown in Figure 11 , the vibration response of the CS modulator system is stable. However, severe vibration suddenly occurs at about 1,100 rpm. The behavior is similar to that of the CL modulator, and again, quickly evolves into a chaotic attractor, as shown in Figure 11(c)-(d).
Similar to the analysis of follower 2-CL, vibration characteristics were also calculated for follower 2-CS at different values of w . It can be concluded that the vibration conditions of follower 1-CS and follower 2-CS are approximately the same. Figure 12 shows a comparison of the follower-CS displacements when w = 1,000 rpm.  AUTEX Research Journal, DOI 10.2478/aut-2020-0037 © AUTEX http://www.autexrj.com/ Dynamic models of the two mechanisms were established, and simulations were performed in Simulink to analyze and compare vibration responses at different speeds. The results show that the CL modulator vibrates smoothly when w is <700 rpm, and theoretically, the speed cannot exceed 1,400 rpm; the CS modulator vibrates smoothly when w is <500 rpm, and theoretically, the speed cannot exceed 1,000 rpm. Thus, the CS modulator is more suitable to use at low speeds, whereas the CL modulator is more suitable to use at high speeds. When the speed is high, vibration distortion will occur in the vibration response of both mechanisms, leading to the bifurcation.
We wish to emphasize that the results reported in this paper can be used to (1) establish an inverse model of the cam profile and solve the conjugate cam profiles and (2) determine the vibration characteristics of the CL and CS modulators. A comparative analysis was performed to determine the range of speeds for which complex behaviors, such as vibration stability, sudden vibration changes, and bifurcations, will occur in these systems against the gear rotational speed w. The analysis presents the theoretical results of the shock system vibration and guides future practical work. In addition, we have provided a more in-depth understanding of the complex behaviors associated with detachment of the follower from the cam to support the design of more sophisticated controllers aimed at avoiding follower detachment.