Abstract
In this paper, the transmission process of the heald frame driven by the dobby is analyzed. The equivalent motion model of the dobby modulator, the eccentric mechanism, and the motion transmission unit are constructed. Then, based on the given movement characteristics of the heald frame, the mathematical model is built to achieve the cam pitch curve and the cam profile of the modulator. The numerical solution method for this is developed. The preparation of a mathematical model for the new concept of the solving cam profile based on the motion characteristics of heald frames is explained in this study. By setting a 11th polynomial motion law of the heald frame, due to the inconsistency between the outward and return motion laws of the crank-rocker mechanism, an asymmetrical cam profile is obtained under the premise of ensuring that the heald frame’s ascending and descending motions are consistent. Through the kinematics simulation analysis, the correctness of the reverse process is verified.
1. Introduction
At present, the weaving technology is focused towards higher speeds, higher efficiency, and easier control and operation [1, 2]. Eren [3–6] introduced kinematic models for the motion of rotary dobbies, cranks, and cam shedding mechanisms, and equations governing heald frame motion were derived. In addition, the obtained heald frame motion curves were compared with each other, while the heald frame motion characteristics were mainly determined the design of the modulator mechanism. Nowadays, in modern high-speed weaving machines, rotary dobbies have been developed for the shedding operations. Rotary dobbies convert the rotational motion of the main shaft of the weaving machine into an up-down frame motion by means of various gears, arms, and eccentrics (cams) [7, 8].
In optimal cam profile design, particularly focused on reducing vibration, Wiederrich [9–11] has been the subject of numerous investigations. Stoddart [12–15] continued Dudley’s work on polydyne cams for valve trains and developed a characteristic relationship between vibration amplitude and cam speed. Mermelstein and Acar [16] used piecewise polynomials, and the complete cam profile can be designed as a combined linear system. Qiu, et al. [17] generated optimized motion curves using B-splines and simultaneously handled many design objectives, including the control of residual vibrations. Mosier [18–21] started with a candidate acceleration function that was adjusted to satisfy the necessary constraints. In that method, no candidate profile was needed and all of the desired acceleration properties were defined from the outset with no adjustments needed to produce the final form. The result was better control over all important acceleration events during the cam cycle. Once the acceleration was defined, the velocity was obtained by integrating the acceleration while enforcing boundary conditions and continuity between segments [22].
However, no other publications found the research of relationship between heald frame movement and cam profile in the literature. In particular, the cam profile is calculated based on the motion characteristics of the heald frame. This paper conducts a research study on the construction of the cam profile of the modulator through mathematical models based on the heald frame motion characteristics.
2. Motion Principle
The rotary dobby and heald frame shedding control mechanism consist of a modulator, an eccentric mechanism, a motion transmission unit (MTU), and a heald frame. The motion principle is presented in Figure 1. The modulator is used to convert the continuous rotation of the loom main shaft into an intermittent movement of the shaft. The schematic view of the modulator is shown in Figure 1(a). The conjugate cam (5 and 10) is fixed to the dobby body. The cam follower arms (2 and 8) that are used as an engagement element are pivoted on gear 1 and can rotate around its pivot axis , transmitting the motion of the loom to the cam rollers (3, 6, 9, and 12). The rotation of the follower rollers makes links 4 and 7 to move and drive the main shaft 11. Therefore, the modulator transforms the uniform rotary motion of the weaving machine into a nonuniform rotary motion of the main shaft.
(a)
(b)
Figure 1(b) illustrates the schematic views of the eccentric mechanism, MTU, and heald frame. The link 13 is an eccentric disc cam, as its center of rotation is different from its geometrical center . Due to this characteristic, when the gear of the modulator rotates, for example, clockwise, the motion of the main shaft is transmitted to the eccentric disc cam and its motion is transmitted to the lifting arm 15 by the ring link 14. The motion of link 15 is transmitted to the heald frame by the MTU (Figure 1(b)). The MTU consists of a four-bar linkage mechanism (, MTU-I) and a crank slider mechanism (, MTU-II). The forwardmost and rearmost positions of the lifting arm 15 correspond to the lower and higher positions of the heald frame 21, respectively. Due to this construction, the eccentric mechanism generates a heald frame motion only for plain weaving. In order to convert it to a rotary dobby mechanism, it is necessary to include the necessary means to get the lifting arm 15 (hence, the heald frame) dwelt at its forwardmost and rearmost positions as many loom revolutions as required by the weave [3].
3. Mathematical Model of the Shedding Motion
3.1. MTU and Heald Frame Modeling
The schematic diagram of MTU is shown in Figure 2. MTU-II is equivalent to a slider-crank linkage that changes at each position, and the kinematic characteristics of the heald frame are transformed to those of the crank 14 and the link 13. The link 11 is rigidly coupled to the link 13. The angle between them is around the fixed point . The positive angle between the link 13 and the axis is .
Stoddart [12] demonstrated how to develop double-dwell polynomials suitable for polydyne applications. Define as the kinematic characteristic function of the heald frame (15), where is the offset of the slider 15, is the distance between the initial point and the axis. , and . , , , and are known parameters in the slider-crank mechanism . is an 11th-degree polynomial [23] motion law and is described as follows:
The straight-line motion of the heald frame is transformed from the rotational motion of the link 13. The displacement equation, relating the parameters , , and to the input and output variables and , may be written as [24]
Equation (3) takes a derivative of time and obtains angular velocity of :
MTU-I is also equivalent to a four-bar linkage mechanism that changes its geometry at each position (Figure 2). Angle of the link 13 can be obtained from equations (1) to (4). Angle of the link 11 can be obtained by the geometric relationship , where and are the known parameters.
In this study, represents the distance from the point to the pivot point , represents the input link length of , represents the link length of , and represents the output link length of . If a Cartesian coordinate system is centered on the pivot point , then angle can be expressed in terms of .
The complex vector form of the closed vector equation is defined as
The Euler formula is applied:
The real part is separated from the imaginary part of equation (5), yielding
Angle is eliminated to get
After some manipulation and substitution, the final result iswhere , , and .
Equation (7) is differentiated by time and gives
Equation (12) is written as a matrix:
From equation (13), the and angular velocity functions can be derived:
Equation (13) is differentiated by time and gives
The and theoretical anger acceleration functions of the follower, respectively, are
3.2. Eccentric Mechanism Modeling
Figure 3 illustrates the eccentric mechanism, which is equivalent to a crank-rocker mechanism. From equations (5) to (18), the angular motion characteristic functions can be obtained. is considered to affect the output of the . According to Figure 3, , where is a known parameter.
Moreover, represents the distance from the pivot point to the pivot point , represents the lifting arm length of , represents the link length of , and represents the output link length of . During one revolution of the link 5, the lifting arm 7 swings between its limit positions and the swing angle is . When points , , and are on the same line and and are extended, the lifting arm 7 reaches its forwardmost position. When and are folded on top of each other, the link 7 reaches its rearmost position. The eccentric mechanism has no quick-return characteristics.
If a Cartesian coordinate system is centered on the pivot point , the output angle can be expressed in terms of :
Then,
Then,
After some manipulation and substitution, the final result iswhere , , and .
Equation (19) is differentiated by time and gives
Equation (24) is written as a matrix:
From equation (25), the and angular velocity functions can be derived:
Equation (25) is differentiated by time and gives
The and angular acceleration functions are
3.3. Modulator Modeling
Figure 4 shows a schematic diagram of the cam-link modulator. Angle of the link can be obtained from equations (19) to (30). Angle is equal to angle , , where is the main shaft angle of the modulator.
(a)
(b)
In Figure 4(a), is the angular velocity of the gear. The starting position of the cam swing arm coincides with the hinge point of the gear. A Cartesian coordinate system is centered on the cam point . The rotation angle of the gear in the Cartesian coordinate system is defined as , and it is expressed by (Figure 4(b)).
From the geometric relationship,
Then,where , , , , , , , , , and .
The distance between the cam actual profile and the cam pitch curve is equal to the roller radius . The cam profile along a direction normal to any point is taken as a distance in order to obtain the coordinates of the corresponding point on the actual cam profile.
The slope of the normal line at the cam pitch curve iswhere is the angle between the normal line and the axis and are the coordinates of any point on the cam pitch curve.
Equation (33) can be written as
According to equations (31) to (34), the coordinates of the point of the actual cam profile can be worked out as follows:
From equations (31) to (35), the analytical solution of the cam profile equation can be determined computationally.
4. Results and Discussion
4.1. Calculation
A mathematical model was used to ascertain the relationship between heald frame motion characteristics and cam profile. In the calculation process, the program was written in Visual Studio 2012 and ran on a personal computer.
Figure 5 shows the kinematic curves of the heald frame for displacement, velocity, acceleration, and jerk [20] for the 11th-degree polynomial functions. These polynomial functions are taken as the motion characteristics (Displac-11th, Velocity-11th, Accel-11th, and Jerk-11th) of the heald frame in combination with equation (1). The program sets as the rotation step of the gear and calculates from 0° to 180°.
According to the actual size of the test bench as shown in Figure 6, the parameter is 106.8 mm; the length of is 375 mm, is 200 mm, the fixed angle is 97.65°, the length of is 200 mm, and the distance is 325 mm.
This program employed equations (1) to (4) and used the above parameter values as input data, and the motion characteristics (Angular displac-, Angular velocity-, Angular accel-, and Jerk-) can be obtained and are presented in Figure 7, normalized with respect to the calculated maximum angular displacement value of 31°. Results are shown for the half gear cycle only, after which a steady state is achieved.
Based on the test bench, the length of link is 150 mm, the length of is 695 mm, the length of is 550 mm, the fixed angle is 100°, and the length of link is 185 mm. According to equations (5) to (18) and the above parameter values, the motion characteristics (Angular displac-, Angular velocity-, Angular accel-, and Jerk-) can be obtained and the results are presented in Figure 8, normalized with respect to the maximum angular displacement value of 36.4°.
The length of link is 30 mm, the length of is 170 mm, and the length of link is 96 mm. In order to ensure no quick-return motion of eccentric mechanism, the length of is calculated to be 192.94 mm. According to equations (19) to (30) and the above parameter values, when moves counterclockwise and clockwise to , the motion characteristics (Angular displac-, Angular velocity-, Angular accel-, and Angular jerk-) and (Angular displac-, Angular velocity-, Angular accel-, and Angular jerk-) are obtained, respectively, which are presented in Figure 9, normalized with respect to the maximum angular displacement value of 180°.
The length of is 42.5 mm, the length of is 111 mm, the length of is 81 mm, and the length of is 56 mm. In Figure 4, the rotation angle of the gear is , and the step is , . According to equations (31) to (35) and the above parameter values, the 2000 points are obtained through the program. The final cam profiles 1 and 2 and cam pitch curves 1 and 2 fitted through nonuniform rational B-splines (NURBS) can be established. They are presented in Figure 10.
4.2. Simulation
In order to verify the correctness of the proposed mathematical model, a virtual prototype of the rotary dobby and the heald frame shedding control mechanism is developed using SolidWorks 2016. The noncentral symmetrical cam defined by the cam profile curve 1 and 2 is employed on the modulator. The motion simulation is performed on the heald frame driven by the rotary dobby with this modulator. The obtained simulation characteristics (Displac-SHF, Velocity-SHF, Accel-SHF, and Jerk-SHF) of the heald frame are compared with the characteristics (Displac-11th, Velocity-11th, Accel-11th, and Jerk-11th) (Figure 11).
As it can be seen in Figure 12, the period of the gear is 180° and the modulator shedding motion characteristics (Displac-SHF, Velocity-SHF, Accel-SHF, and Jerk-SHF) of the heald frame are very close to the 11th polynomial curves, which proves the correctness of the proposed mathematical model. At the same time, the main shaft of the rotary dobby is rotated for one cycle and the same heald motion characteristics as motion characteristics (Displac-SHF, Velocity-SHF, Accel-SHF, and Jerk-SHF) can be formed, meaning that regardless the forward or reverse rotation of the gear, the heald frame rise laws are consistent with the fall laws.
The cam profile 3 and 4 are formed by copying and rotating the cam profile 1 and 2 by 180° and combining them (Figure 12), whereas the cam profile 3 is shown as the dash dot line and the cam profile 4 is shown as the solid line. The cam pitch curve 3 and 4 are shown as the dash double-dot line and the dashed line, respectively.
The two cams formed by cam profile 3 and 4 are employed in the modulator. The two cams are mounted on the rotary dobby transmission mechanism in order to obtain the movement law of the heald frame. Motion characteristics (Displac-CP3, Velocity-CP3, Accel-CP3, and Jerk-CP3; Displac-CP4-1, Velocity-CP4-1, Accel-CP4-1, and Jerk-CP4-1) are obtained from the simulation of the heald frame motion driven by the rotary dobby with the aforementioned modulators, . The results are compared with each other and presented in Figure 13.
It can be seen that the motion characteristics (Displac-CP3, Velocity-CP3, Accel-CP3, and Jerk-CP3; Displac-CP4-1, Velocity-CP4-1, Accel-CP4-1, and Jerk-CP4-1) of the heald frame formed by cam profiles 3 and 4, respectively, are very different. Figure 14 shows the deviation curves. The motion of the heald frame by the modulator with cam profile 3 or 4 resulted in different characteristics in the upward and downward sections. More specifically, the forward and reverse rotation of the dobby modulator main shaft resulted in different heald frame motion characteristics.
However, in Figure 15, the heald frame motion characteristics (Displac-CP3, Velocity-CP3, Accel-CP3, and Jerk-CP3) generated by the dobby modulator when it rotates forward with cam profile 3 and the movement characteristics (Displac-CP4-2, Velocity-CP4-2, Accel-CP4-2, and Jerk-CP4-2) generated by the dobby modulator when it rotates reversely with cam profile 4 are compared. As it can be observed, the values of motion characteristics (Displac-CP3, Velocity-CP3, Accel-CP3, and Jerk-CP3; Displac-CP4-2, Velocity-CP4-2, Accel-CP4-2, and Jerk-CP4-2) have the same variation trend. This implies that modulators with different cam profiles can produce exactly the same heald frame motion characteristics.
The above findings form the foundation for the modulator design and further theoretical analysis of the rotary dobby structure.
5. Conclusions
When the geometric dimensions of the modulator, eccentric mechanism, and MTU, as well as the motion curve, of the heald frame are given, the analytical mathematical model of the cam profile and the cam pitch curve can be determined. In this mathematical model, it is possible to ascertain the relationship between heald frame motion characteristics and cam profile. By means of this model, 2000 points of cam profile is obtained, and the error of this model can be reduced by increasing the number of calculation points. The model proposed in this article enables the analysis of the heald frame motion characteristics and the cam profile design. Given different heald frame displacement curves and parameter values, the cam profile and motion characteristics of each motion transfer process can be obtained based on the proposed mathematical model.
Based on the particularity of the rotary dobby structure, a cam profile is obtained, and the motion law of the heald frame is solved in the forward direction to verify the correctness of the inverse model. At the same time, regardless of whether the dobby is rotating forward or backward, the motion of the heald frame is the same, while the cam is asymmetrical.
Two cam profiles are obtained, the heald frame motion characteristics are solved in the forward direction, and the asymmetric motion characteristics of the heald frame are obtained. The asymmetric deviation revealed the cam profiles, the eccentric mechanism, and the motion transmission mechanism. When the roller moves clockwise along one of the cam profiles and counterclockwise along the other cam profile, the exact same heald frame motion characteristics are produced.
A good correlation is found between the simulation and calculation results of the heald frame displacement, velocity, acceleration, and jerk. Future work will include the development of a virtual prototype, which will verify the mathematical model and may give a wealth of dynamic information about the system, as well as similar systems yet to be built.
Data Availability
All data's used to support the findings of this study are included within the article.
Disclosure
Honghuan Yin and Hongbin Yu are co-first authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Key R&D Program of China (2017YFB1104202).