# Mathematical Model of Cam Profile Based on Heald Frame Motion Characteristics

Authors: Honghuan Yin; Hongbin Yu; Junqiang Peng; Hongyu Shao
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 Figure1. 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 O1, 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.Figure 1
Heald frame motion principle. (a) Modulator. (1) gear; (2, 8), cam swing arms; (3, 6, 9, 12) cam rollers; (4, 7) modulator links; (5, 10) conjugate cams; and (11) main shaft. (b) Eccentric mechanism, MTU, and heald frame. (13) Eccentric disc cam; (14) ring link; (15) lifting arm; (16) lifting arm link; (17) swivel arm; (18) rotor link; (19) heald frame link; (20) support link; and (21) heald frame.
(a)
(b)Figure1(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 O1 is different from its geometrical center D. 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 (O2FGO3, MTU-I) and a crank slider mechanism (O3HI, 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 Figure2. 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 φ9 around the fixed point O3. The positive angle between the link 13 and the x axis is φ10.Figure 2
Schematic diagram of the MTU.Stoddart [12] demonstrated how to develop double-dwell polynomials suitable for polydyne applications. Define Lu=Su+S0 as the kinematic characteristic function of the heald frame (15), where e is the offset of the slider 15, S0 is the distance between the initial point I and the x axis. l13=O3H, and l14=HI. e, S0, l13, and l14 are known parameters in the slider-crank mechanism O3HI. Su is an 11th-degree polynomial [23] motion law and is described as follows:(1)Su=h336u5−1890u6−4740u7−6615u8+5320u9−2310u10+420u11.The straight-line motion of the heald frame is transformed from the rotational motion of the link 13. The displacement equation, relating the parameterse, l14, and l13 to the input and output variables Lu and φ10, may be written as [24](2)2l13Lusinφ10−ecosφ10+l142−l132+L2u+e2=0,(3)φ10=arc sinl132+L2u+e2−l1422l13L2u+e2+arc taneLu,forφ10∈−90°,90°.Equation (3) takes a derivative of time and obtains angular velocity of φ10:(4)ω10=−2Lu⋅dLu/du+2l15Lusinφ102l13Lucosφ10+esinφ10.MTU-I is also equivalent to a four-bar linkage mechanism that changes its geometry at each position (Figure2). Angle φ10 of the link 13 can be obtained from equations (1) to (4). Angle φ8 of the link 11 can be obtained by the geometric relationship φ8=φ9+φ10, where φ5 and φ9 are the known parameters.In this study,l12 represents the distance from the point O3 to the pivot point O2, l11 represents the input link length of O3G, l10 represents the link length of FG, and l9 represents the output link length of O2F. If a Cartesian coordinate system X,Y is centered on the pivot point O3, then angle φ6 can be expressed in terms of φ8.The complex vector form of the closed vector equation is defined as(5)l9eiφ6+l10eiφ7=l11eiφ8+l12.The Euler formula is applied:(6)eiφ=cosφ+isinφ.The real part is separated from the imaginary part of equation (5), yielding(7)l9cosφ6+l10cosφ7=l11cosφ8+l12,l9sinφ6+l10sinφ7=l11sinφ8.Angleφ7 is eliminated to get(8)l102=l112+l122+l92+2l11l12cosφ8−2l9l12cosφ6−2l9l11cosφ8−φ6,(9)2l12l11cosφ8−l9cosφ6−l11l9φ8−φ6+l122+l112−l102+l92=0.After some manipulation and substitution, the final result is(10)tanφ62=B−A2+B2−C2A−C,where A=l12+l11cosφ8, B=l11sinφ8, and C=A2+B2+l92−l102/2l9.(11)φ6=2arc tanB−A2+B2−C2A−C,forφ6∈0°,90°.Equation (7) is differentiated by time and gives(12)l9ω6sinφ6+l10ω7sinφ7=l11ω8sinφ8,l9ω6cosφ6+l10ω7cosφ7=l11ω8cosφ8.Equation (12) is written as a matrix:(13)l9sinφ6l10sinφ7l9cosφ6l10cosφ7ω6ω7=ω8l11sinφ8l11cosφ8.From equation (13), the ω6 and ω7 angular velocity functions can be derived:(14)ω6=ω8l11sinφ7−φ8l9sinφ7−φ6,(15)ω7=ω8l11sinφ8−φ6l10sinφ6−φ7.Equation (13) is differentiated by time and gives(16)l9sinφ6l10sinφ7l9cosφ6l10cosφ7α6α7=−l9cosφ6l10cosφ7−l9sinφ6−l10sinφ7ω6ω7+α8l11sinφ8l11cosφ8+ω8l11cosφ8−l11sinφ8.Theα6 and α7 theoretical anger acceleration functions of the follower, respectively, are(17)α6=ω62l9cosφ6−φ7+ω72l10−ω82l11cosφ8−φ7l11sinφ8−φ7,(18)α7=−ω62l9cosφ6−φ8−ω72l10cosφ7−φ8−ω82l11l11sinφ7−φ8.


### 3.2. Eccentric Mechanism Modeling

Figure3 illustrates the eccentric mechanism, which is equivalent to a crank-rocker mechanism. From equations (5) to (18), the φ6 angular motion characteristic functions can be obtained. φ4 is considered to affect the output of the φ2. According to Figure 3, φ4=φ6+φ5, where φ5 is a known parameter.Figure 3
Schematic diagram of the eccentric mechanism.Moreover,l8 represents the distance from the pivot point O2 to the pivot point O1, l7 represents the lifting arm length of O2E, l6 represents the link length of ED, and l5 represents the output link length of O1D. During one revolution of the link 5, the lifting arm 7 swings between its limit positions and the swing angle is ∅. When points O1, D0, and E are on the same line and O1D0 and D0E are extended, the lifting arm 7 reaches its forwardmost position. When O1D0 and D0E 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 systemX,Y is centered on the pivot point O2, the output angle φ2 can be expressed in terms of φ4:(19)l5cosφ2+l6cosφ3=l7cosφ4+l8,l5sinφ2+l6sinφ3=l7sinφ4.Then,(20)l62=l72+l82+l52+2l7l8cosφ4−2l5l8cosφ2−2l5l7cosφ4−φ2.Then,(21)2l8l7cosφ4−l5cosφ2−l7l5cosφ4−φ2+l82+l72−l62+l52=0.After some manipulation and substitution, the final result is(22)tanφ22=B±A2+B2−C2A−C,where A=l8+l7cosφ4, B=l7sinφ4, and C=A2+B2+l52−l62/2l5.(23)φ2=2arc tanB±A2+B2−C2A−C,forφ2∈−180°,180°.Equation (19) is differentiated by time and gives(24)l5ω2sinφ2+l6ω3sinφ3=l7ω4sinφ4,l5ω2cosφ2+l6ω3cosφ3=l7ω4cosφ4.Equation (24) is written as a matrix:(25)l5sinφ2l6sinφ3l5cosφ2l6cosφ3ω2ω3=ω4l7sinφ4l7cosφ4.From equation (25), the ω2 and ω3 angular velocity functions can be derived:(26)ω2=ω4l7sinφ3−φ4l5sinφ3−φ2,(27)ω3=−ω4l7sinφ4−φ2l6sinφ2−φ3.Equation (25) is differentiated by time and gives(28)l5sinφ2l6sinφ3l5cosφ2l6cosφ3α2α3=α4l7sinφ4l7cosφ4+ω4l7cosφ4−l7sinφ4−l5cosφ2l6cosφ3−l5sinφ2−l6sinφ3ω2ω3.Theα2 and α3 angular acceleration functions are(29)α2=ω22l5cosφ2−φ3+ω32l6−ω42l7cosφ4−φ3l7sinφ4−φ3,(30)α3=−ω22l5cosφ2−φ4−ω32l6cosφ3−φ4−ω42l7l7sinφ3−φ4.


### 3.3. Modulator Modeling

Figure4 shows a schematic diagram of the cam-link modulator. Angle φ2 of the link O1D can be obtained from equations (19) to (30). Angle φ2 is equal to angle φ1, φ1=φ2, where φ1 is the main shaft angle of the modulator.Figure 4
Cam-link modulator.
(a)
(b)In Figure4(a), ω is the angular velocity of the gear. The starting position of the cam swing arm coincides with the hinge point C of the gear. A Cartesian coordinate system X,Y is centered on the cam point O1. The rotation angle of the gear in the Cartesian coordinate system is defined as θ, and it is expressed by θ=ωt (Figure 4(b)).From the geometric relationship,(31)xA−xB2+yA−yB2=l22,xA−xC2+yA−yC2=l12.Then,(32)xA=−B1±B12−4A1C12A1,yA=C−AxB,where A=xC−xB, B=yC−yB, C=l4−l3−xB2−yB2+xC2+yC2/2, A1=A2+B2, B1=2AByB−2AC−2B2xB, C1=−2BCyB+C2−B2l42−yB2−xB2, xB=l3cosφ1, yB=l3sinφ1, xc=l4cosθ, and yc=l4sinθ.The distance between the cam actual profile and the cam pitch curve is equal to the roller radiusr. The cam profile along a direction normal to any point A is taken as a distance r in order to obtain the coordinates of the corresponding point TxT,yT on the actual cam profile.The slope of the normal linen−n at the cam pitch curve A is(33)tanγ=dxAdyA=−dxA/dθdyA/dθ=sinγcosγ,where γ is the angle between the normal line n−n and the x axis and xA,yA are the coordinates of any point A on the cam pitch curve.Equation (33) can be written as(34)sinγ=dxA/dθdxA/dθ2+dyA/dθ2,cosγ=−dyA/dθdxA/dθ2+dyA/dθ2.According to equations (31) to (34), the coordinates of the point TxT,yT of the actual cam profile can be worked out as follows:(35)xT=xA±rcosγ,yT=yA±rcosγ.From equations (31) to (35), the analytical solution of the cam profile equation can be determined computationally.


## 3.1. MTU and Heald Frame Modeling

The schematic diagram of MTU is shown in Figure2. 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 φ9 around the fixed point O3. The positive angle between the link 13 and the x axis is φ10.Figure 2
Schematic diagram of the MTU.Stoddart [12] demonstrated how to develop double-dwell polynomials suitable for polydyne applications. Define Lu=Su+S0 as the kinematic characteristic function of the heald frame (15), where e is the offset of the slider 15, S0 is the distance between the initial point I and the x axis. l13=O3H, and l14=HI. e, S0, l13, and l14 are known parameters in the slider-crank mechanism O3HI. Su is an 11th-degree polynomial [23] motion law and is described as follows:(1)Su=h336u5−1890u6−4740u7−6615u8+5320u9−2310u10+420u11.The straight-line motion of the heald frame is transformed from the rotational motion of the link 13. The displacement equation, relating the parameterse, l14, and l13 to the input and output variables Lu and φ10, may be written as [24](2)2l13Lusinφ10−ecosφ10+l142−l132+L2u+e2=0,(3)φ10=arc sinl132+L2u+e2−l1422l13L2u+e2+arc taneLu,forφ10∈−90°,90°.Equation (3) takes a derivative of time and obtains angular velocity of φ10:(4)ω10=−2Lu⋅dLu/du+2l15Lusinφ102l13Lucosφ10+esinφ10.MTU-I is also equivalent to a four-bar linkage mechanism that changes its geometry at each position (Figure2). Angle φ10 of the link 13 can be obtained from equations (1) to (4). Angle φ8 of the link 11 can be obtained by the geometric relationship φ8=φ9+φ10, where φ5 and φ9 are the known parameters.In this study,l12 represents the distance from the point O3 to the pivot point O2, l11 represents the input link length of O3G, l10 represents the link length of FG, and l9 represents the output link length of O2F. If a Cartesian coordinate system X,Y is centered on the pivot point O3, then angle φ6 can be expressed in terms of φ8.The complex vector form of the closed vector equation is defined as(5)l9eiφ6+l10eiφ7=l11eiφ8+l12.The Euler formula is applied:(6)eiφ=cosφ+isinφ.The real part is separated from the imaginary part of equation (5), yielding(7)l9cosφ6+l10cosφ7=l11cosφ8+l12,l9sinφ6+l10sinφ7=l11sinφ8.Angleφ7 is eliminated to get(8)l102=l112+l122+l92+2l11l12cosφ8−2l9l12cosφ6−2l9l11cosφ8−φ6,(9)2l12l11cosφ8−l9cosφ6−l11l9φ8−φ6+l122+l112−l102+l92=0.After some manipulation and substitution, the final result is(10)tanφ62=B−A2+B2−C2A−C,where A=l12+l11cosφ8, B=l11sinφ8, and C=A2+B2+l92−l102/2l9.(11)φ6=2arc tanB−A2+B2−C2A−C,forφ6∈0°,90°.Equation (7) is differentiated by time and gives(12)l9ω6sinφ6+l10ω7sinφ7=l11ω8sinφ8,l9ω6cosφ6+l10ω7cosφ7=l11ω8cosφ8.Equation (12) is written as a matrix:(13)l9sinφ6l10sinφ7l9cosφ6l10cosφ7ω6ω7=ω8l11sinφ8l11cosφ8.From equation (13), the ω6 and ω7 angular velocity functions can be derived:(14)ω6=ω8l11sinφ7−φ8l9sinφ7−φ6,(15)ω7=ω8l11sinφ8−φ6l10sinφ6−φ7.Equation (13) is differentiated by time and gives(16)l9sinφ6l10sinφ7l9cosφ6l10cosφ7α6α7=−l9cosφ6l10cosφ7−l9sinφ6−l10sinφ7ω6ω7+α8l11sinφ8l11cosφ8+ω8l11cosφ8−l11sinφ8.Theα6 and α7 theoretical anger acceleration functions of the follower, respectively, are(17)α6=ω62l9cosφ6−φ7+ω72l10−ω82l11cosφ8−φ7l11sinφ8−φ7,(18)α7=−ω62l9cosφ6−φ8−ω72l10cosφ7−φ8−ω82l11l11sinφ7−φ8.


## 3.2. Eccentric Mechanism Modeling

Figure3 illustrates the eccentric mechanism, which is equivalent to a crank-rocker mechanism. From equations (5) to (18), the φ6 angular motion characteristic functions can be obtained. φ4 is considered to affect the output of the φ2. According to Figure 3, φ4=φ6+φ5, where φ5 is a known parameter.Figure 3
Schematic diagram of the eccentric mechanism.Moreover,l8 represents the distance from the pivot point O2 to the pivot point O1, l7 represents the lifting arm length of O2E, l6 represents the link length of ED, and l5 represents the output link length of O1D. During one revolution of the link 5, the lifting arm 7 swings between its limit positions and the swing angle is ∅. When points O1, D0, and E are on the same line and O1D0 and D0E are extended, the lifting arm 7 reaches its forwardmost position. When O1D0 and D0E 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 systemX,Y is centered on the pivot point O2, the output angle φ2 can be expressed in terms of φ4:(19)l5cosφ2+l6cosφ3=l7cosφ4+l8,l5sinφ2+l6sinφ3=l7sinφ4.Then,(20)l62=l72+l82+l52+2l7l8cosφ4−2l5l8cosφ2−2l5l7cosφ4−φ2.Then,(21)2l8l7cosφ4−l5cosφ2−l7l5cosφ4−φ2+l82+l72−l62+l52=0.After some manipulation and substitution, the final result is(22)tanφ22=B±A2+B2−C2A−C,where A=l8+l7cosφ4, B=l7sinφ4, and C=A2+B2+l52−l62/2l5.(23)φ2=2arc tanB±A2+B2−C2A−C,forφ2∈−180°,180°.Equation (19) is differentiated by time and gives(24)l5ω2sinφ2+l6ω3sinφ3=l7ω4sinφ4,l5ω2cosφ2+l6ω3cosφ3=l7ω4cosφ4.Equation (24) is written as a matrix:(25)l5sinφ2l6sinφ3l5cosφ2l6cosφ3ω2ω3=ω4l7sinφ4l7cosφ4.From equation (25), the ω2 and ω3 angular velocity functions can be derived:(26)ω2=ω4l7sinφ3−φ4l5sinφ3−φ2,(27)ω3=−ω4l7sinφ4−φ2l6sinφ2−φ3.Equation (25) is differentiated by time and gives(28)l5sinφ2l6sinφ3l5cosφ2l6cosφ3α2α3=α4l7sinφ4l7cosφ4+ω4l7cosφ4−l7sinφ4−l5cosφ2l6cosφ3−l5sinφ2−l6sinφ3ω2ω3.Theα2 and α3 angular acceleration functions are(29)α2=ω22l5cosφ2−φ3+ω32l6−ω42l7cosφ4−φ3l7sinφ4−φ3,(30)α3=−ω22l5cosφ2−φ4−ω32l6cosφ3−φ4−ω42l7l7sinφ3−φ4.


## 3.3. Modulator Modeling

Figure4 shows a schematic diagram of the cam-link modulator. Angle φ2 of the link O1D can be obtained from equations (19) to (30). Angle φ2 is equal to angle φ1, φ1=φ2, where φ1 is the main shaft angle of the modulator.Figure 4
Cam-link modulator.
(a)
(b)In Figure4(a), ω is the angular velocity of the gear. The starting position of the cam swing arm coincides with the hinge point C of the gear. A Cartesian coordinate system X,Y is centered on the cam point O1. The rotation angle of the gear in the Cartesian coordinate system is defined as θ, and it is expressed by θ=ωt (Figure 4(b)).From the geometric relationship,(31)xA−xB2+yA−yB2=l22,xA−xC2+yA−yC2=l12.Then,(32)xA=−B1±B12−4A1C12A1,yA=C−AxB,where A=xC−xB, B=yC−yB, C=l4−l3−xB2−yB2+xC2+yC2/2, A1=A2+B2, B1=2AByB−2AC−2B2xB, C1=−2BCyB+C2−B2l42−yB2−xB2, xB=l3cosφ1, yB=l3sinφ1, xc=l4cosθ, and yc=l4sinθ.The distance between the cam actual profile and the cam pitch curve is equal to the roller radiusr. The cam profile along a direction normal to any point A is taken as a distance r in order to obtain the coordinates of the corresponding point TxT,yT on the actual cam profile.The slope of the normal linen−n at the cam pitch curve A is(33)tanγ=dxAdyA=−dxA/dθdyA/dθ=sinγcosγ,where γ is the angle between the normal line n−n and the x axis and xA,yA are the coordinates of any point A on the cam pitch curve.Equation (33) can be written as(34)sinγ=dxA/dθdxA/dθ2+dyA/dθ2,cosγ=−dyA/dθdxA/dθ2+dyA/dθ2.According to equations (31) to (34), the coordinates of the point TxT,yT of the actual cam profile can be worked out as follows:(35)xT=xA±rcosγ,yT=yA±rcosγ.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.Figure5 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 180°/2000 as the rotation step of the gear and calculates from 0° to 180°.Figure 5
11th-degree polynomial curves.According to the actual size of the test bench as shown in Figure6, the parameter h is 106.8 mm; the length of l14 is 375 mm, e is 200 mm, the fixed angle φ9 is 97.65°, the length of l13 is 200 mm, and the distance S0 is 325 mm.Figure 6
The test bench.This program employed equations (1) to (4) and used the above parameter values as input data, and the φ10 motion characteristics (Angular displac-φ10, Angular velocity-φ10, Angular accel-φ10, and Jerk-φ10) 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.Figure 7
Motion characteristics of angularφ10.Based on the test bench, the length of linkl11 is 150 mm, the length of l12 is 695 mm, the length of l10 is 550 mm, the fixed angle φ5 is 100°, and the length of link l9 is 185 mm. According to equations (5) to (18) and the above parameter values, the φ6 motion characteristics (Angular displac-φ6, Angular velocity-φ6, Angular accel-φ6, and Jerk-φ6) can be obtained and the results are presented in Figure 8, normalized with respect to the maximum angular displacement value of 36.4°.Figure 8
Motion characteristics of angularφ6.The length of linkl5 is 30 mm, the length of l6 is 170 mm, and the length of link l7 is 96 mm. In order to ensure no quick-return motion of eccentric mechanism, the length of l8 is calculated to be 192.94 mm. According to equations (19) to (30) and the above parameter values, when D0 moves counterclockwise and clockwise to D1, the φ2 motion characteristics (Angular displac-φ2−1, Angular velocity-φ2−1, Angular accel-φ2−1, and Angular jerk-φ2−1) and (Angular displac-φ2−2, Angular velocity-φ2−2, Angular accel-φ2−2, and Angular jerk-φ2−2) are obtained, respectively, which are presented in Figure 9, normalized with respect to the maximum angular displacement value of 180°.Figure 9
Motion characteristics of angularφ2.The length ofl3 is 42.5 mm, the length of l4 is 111 mm, the length of l2 is 81 mm, and the length of l1 is 56 mm. In Figure 4, the rotation angle θ of the gear is ωt, and the step is 180°/2000, θ∈0°,180°. 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.Figure 10
Cam profile and cam pitch curve 1 and 2.


### 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).Figure 11
11th-degree polynomial curves and simulation characteristics of the heald frame.As it can be seen in Figure12, 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.Figure 12
Cam profile and cam pitch curve 3 and 4.The cam profile 3 and 4 are formed by copying and rotating the cam profile 1 and 2 by 180° and combining them (Figure12), 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,θ∈0°,360°. The results are compared with each other and presented in Figure 13.Figure 13
The motion characteristics of the heald frame with the same rotation direction.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. Figure14 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.Figure 14
The deviations of motion characteristics.However, in Figure15, 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.Figure 15
The motion characteristics of the heald frame with different rotation directions.The above findings form the foundation for the modulator design and further theoretical analysis of the rotary dobby structure.


## 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.Figure5 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 180°/2000 as the rotation step of the gear and calculates from 0° to 180°.Figure 5
11th-degree polynomial curves.According to the actual size of the test bench as shown in Figure6, the parameter h is 106.8 mm; the length of l14 is 375 mm, e is 200 mm, the fixed angle φ9 is 97.65°, the length of l13 is 200 mm, and the distance S0 is 325 mm.Figure 6
The test bench.This program employed equations (1) to (4) and used the above parameter values as input data, and the φ10 motion characteristics (Angular displac-φ10, Angular velocity-φ10, Angular accel-φ10, and Jerk-φ10) 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.Figure 7
Motion characteristics of angularφ10.Based on the test bench, the length of linkl11 is 150 mm, the length of l12 is 695 mm, the length of l10 is 550 mm, the fixed angle φ5 is 100°, and the length of link l9 is 185 mm. According to equations (5) to (18) and the above parameter values, the φ6 motion characteristics (Angular displac-φ6, Angular velocity-φ6, Angular accel-φ6, and Jerk-φ6) can be obtained and the results are presented in Figure 8, normalized with respect to the maximum angular displacement value of 36.4°.Figure 8
Motion characteristics of angularφ6.The length of linkl5 is 30 mm, the length of l6 is 170 mm, and the length of link l7 is 96 mm. In order to ensure no quick-return motion of eccentric mechanism, the length of l8 is calculated to be 192.94 mm. According to equations (19) to (30) and the above parameter values, when D0 moves counterclockwise and clockwise to D1, the φ2 motion characteristics (Angular displac-φ2−1, Angular velocity-φ2−1, Angular accel-φ2−1, and Angular jerk-φ2−1) and (Angular displac-φ2−2, Angular velocity-φ2−2, Angular accel-φ2−2, and Angular jerk-φ2−2) are obtained, respectively, which are presented in Figure 9, normalized with respect to the maximum angular displacement value of 180°.Figure 9
Motion characteristics of angularφ2.The length ofl3 is 42.5 mm, the length of l4 is 111 mm, the length of l2 is 81 mm, and the length of l1 is 56 mm. In Figure 4, the rotation angle θ of the gear is ωt, and the step is 180°/2000, θ∈0°,180°. 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.Figure 10
Cam profile and cam pitch curve 1 and 2.


## 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).Figure 11
11th-degree polynomial curves and simulation characteristics of the heald frame.As it can be seen in Figure12, 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.Figure 12
Cam profile and cam pitch curve 3 and 4.The cam profile 3 and 4 are formed by copying and rotating the cam profile 1 and 2 by 180° and combining them (Figure12), 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,θ∈0°,360°. The results are compared with each other and presented in Figure 13.Figure 13
The motion characteristics of the heald frame with the same rotation direction.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. Figure14 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.Figure 14
The deviations of motion characteristics.However, in Figure15, 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.Figure 15
The motion characteristics of the heald frame with different rotation directions.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.



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*Source: 2106373-2020-02-19.xml*