Design and Experiment of Non-circular Combined Gear Train Beating-up Mechanism

CHEN Jianneng ()， YU Chennan()，TONG Lin ( )，WANG Ying ( )， XIA Xudong ()， ZHAO Xiong ( )

1 College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018， China 2 Faculty of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China

Abstract： The most important performance of a beating-up mechanism is that the dwelling time of the sley must ensure the completion of the weft insertion. To meet this requirement, a new non-circular combined gear train beating-up mechanism which is composed of two-stage planetary gear trains is proposed. The first-stage is a Fourier planetary gear train and the second-stage is a non-circular planetary gear train. For designing of this new mechanism, the ideal kinematic equations of the sley are constructed first. Then the kinematic model of the first-stage Fourier planetary gear train is established and the reverse solution for the pitch curves of the second-stage non-circular gears is deduced. With a computer-aided design program, the influences of several important parameters on the pitch curves of the second-stage non-circular gears are analyzed, and a set of preferable structural parameters are obtained. Finally, a test bed of this mechanism is developed and the experimental results show that this new beating-up mechanism can achieve the designed dwelling time, namely it can meet the requirements of beating-up process.

Key words： beating-up mechanism; Fourier gear; non-circular gear; planetary gear train; kinematic

Introduction

The function of the beating-up mechanism is to transform the constant rotation of the main shaft into the non-uniform swaying of the sley, by which the weft yarn weaved with the warp yarn to be fabric[1]. In order to ensure enough weft insertion time, the beating-up mechanism should be designed to keep the sley with enough dwelling time (dwelling time means in a rotation cycle of the main shaft, the sley should keep unmoved in the backmost position when the main shaft rotates some angle)[2]. At present, the conjugate cam beating-up mechanism is most widely used. It can meet any reasonable kinematic characteristics requirements of the sley by altering the profile curve of the conjugate cam[2-4]. However, the high requirement of the profile curve makes the design and manufacture of the conjugate cam very difficult. Meanwhile, with the wear of the conjugate cam, the accuracy of the beating-up mechanism will degrade[5]. Another widely used beating-up mechanism is the linkage beating-up mechanism. It is easy to be designed by optimum design method which can obtain optimal structural parameters[6]. However, the linkage beating-up mechanism can only achieve an approximate dwelling time. The four-linkage beating-up mechanism can only achieve about 70° approximate dwelling time[7]and the six-linkage beating-up mechanism achieves about 140°[8]. Meanwhile, the large rotational inertia of the linkage mechanisms will cause the loom vibrate greatly[9]. Guetal. used elliptical gears to enhance the dwelling time of the linkage beating-up mechanisms to 200°[10-11], but the dwelling time is still not enough.

To overcome the shortages of the existing beating-up mechanisms, a non-circular combined gear train beating-up mechanism is proposed in this paper. The non-circular gear drive can achieve a non-uniform transmission requirement. The Fourier gear is a special non-circular gear with Fourier function pitch curve, which can provide more adjustable parameters to easily optimize the mechanical properties[12-13]. Thus, the non-circular gears including the Fourier gears are used to improve the transmission characteristics of the beating-up mechanism to achieve enough dwelling time. The ideal kinematic equations of the sley are constructed and the kinematic model of the beating-up mechanism is established. Subsequently, the influences of several important parameters on the pitch curves of the second-stage non-circular gears are analyzed. Moreover, a test bed of this mechanism is developed to test the working performance of this new mechanism.

1 Non-circular Combined Gear Train Beating-up Mechanisms

The structure of the non-circular combined gear train beating-up mechanism is shown in Fig. 1. The Fourier sun gear 3 is fixed on the rack. The Fourier planetary gear 6 and the non-circular planetary gear 7 are fixed on the planetary shaft 5 which is hinged with planetary carrier 1. The non-circular sun gear 8 and the cylindrical driving gear 9 are both fixed on the rockshaft 4. The cylindrical driven gear 10 and the sley 12 are both fixed on the rockshaft 11. The Fourier sun gear 3, the Fourier planetary gear 6 and the planetary carrier 1 compose the first-stage Fourier planetary gear train. The non-circular planetary gear 7, the non-circular sun gear 8 and the planetary carrier 1 compose the second-stage non-circular planetary gear train. The cylindrical driving gear 9 and the cylindrical driven gear 10 compose the amplification gear train. When the loom is working, the motor drives the main shaft 2 and the planetary carrier 1 to rotate at a uniform speed. With the rotation of the planetary carrier 1, the Fourier planetary gear 6 rolls around the Fourier sun gear 3 with a non-uniform auto-rotational speed by achieving the first-stage non-uniform transmission. With the rotation of the planetary carrier 1 and the non-circular sun gear 7, the non-circular sun gear 8 rotates at a non-uniform speed by achieving the second-stage non-uniform transmission. Then the motion of the non-circular sun gear 8 is transmitted to the sley by amplification gear train. Thus, the beating-up mechanism can transform the constant rotation of the main shaft into the intermittent non-uniform swing of the sley.

Fig.1 Structure of non-circular combined gear train beating-up mechanism

2 Kinematic Equations of Sley

For a beating-up mechanism, the sley must swing to the backmost position before the rapier head enters the shed. Therefore, the sley should keep unmoved when the main shaft rotates from 60° to 300°. In order to achieve about 240° absolute beating-up dwelling time, the ideal acceleration equation of the sley in interval [0,π] is established based on Hermite interpolation polynomials with 3 sections. It can be expressed as

(1)

wherehis the maximum positive beating-up acceleration,hxis the maximum negative beating-up acceleration which is calculated by Eq. (2),αis the rotary angle of main shaft,α1is the turning point of beating-up acceleration which is calculated by Eq.(3), andα2is the starting point of beating-up.

(2)

wheresmaxis the total beating-up displacement, andωis the angle velocity of the main shaft.

(3)

The velocity equation can be obtained by integrating the acceleration equation and the displacement equation can be obtained by integrating the velocity equation.

Fig.2 Acceleration, velocity and displacement curves of the sley

3 Kinematic Model of Non-circular Combined Gear Train Beating-Up Mechanisms

The combined gear train in this new beating-up mechanism includes two-stage planetary gear trains. The first-stage is a Fourier planetary gear train and the second-stage is a non-circular planetary gear train. So the kinematic model is divided into two parts according to the structure.

3.1 Kinematic model of first-stage Fourier planetary gear train

As shown in Fig. 3, the Fourier gear 3 works as the sun gear and the conjugated Fourier non-circular gear 6 works as the planetary gear. With the rotation of the planetary carrier 1, the conjugated non-circular gear 6

Fig.3 Conjugation of the first-stage Fourier gear train

The pitch curve of the Fourier sun gear 3 is represented by

(4)

(5)

According to the basic requirement of non-circular gear engagement, the following equations can be constructed

r6=a-r3,

(6)

(7)

(8)

Equation (8) can be integrated for timetand it can then be written as

(9)

So the pitch curve of the conjugated non-circular planetary gear 6 can be given as

(10)

In order to ensure the pitch curve of the non-circular planetary gear 6 closed, the following equation must be observed

(11)

From Eq. (11), it can be seen that, the center distanceacan be solved by the numerical analysis method[14].

3.2 Reverse solution for pitch curves of second-stage non-circular gears

Fig.4 Conjugation of non-circular combined gear train

According to the equationφ3=0, the following transmission ratio is calculated as

1-f′(φ8)=1-i81,

(12)

(13)

The distance between the main shaft 2 and the planetary shaft 5 is

r3+r6=r7+r8=a.

(14)

From Eqs. (9),(12)and(14), the contact radius of non-circular planetary gear 7r7can be constructed

(15)

The pitch curve of the non-circular planetary gear 7 is constructed as

(16)

And the pitch curve of the conjugated non-circular sun gear 8 can be given as

(17)

whereφ8can be calculated based on the ideal kinematic equation of the sley. The kinematic model of the small cylindrical gear 10 is obtained by kinematic equations of sley. It can be written as

(18)

whereφ10is the rotary angle of small cylindrical gear 10 andris the height of sley. The kinematic model of the large cylindrical gear 9 and the non-circular sun gear 8 is obtained by the kinematic model of the gear 10. It can be written as

(19)

whereifis the transmission ratio of the amplification gear train.

4 Computer-Aided Design Program and Influence Analysis of Parameters on the Pitch Curves of the Second-Stage Non-circular Gears

4.1 Computer-aided design programs

In order to facilitate the design of this new mechanism, a computer-aided design program with a user-friendly graphical user interface was compiled based on MATLAB, which is shown in Fig. 5.

After inputting the parameters of movement rule of the given sley such as the beating-up starting angle, the beating-up total displacement, the maximum positive beating-up acceleration and the Fourier parameters,etc., the program can automatically calculate the displacement, velocity, and acceleration of the sley, the center distance, the maximum swing angle of sley,etc. The program can also draw the corresponding non-circular pitch curves and conduct the movement simulation of the beating-up mechanism.

Fig.5 Computer-aided design program for the beating-up mechanism

4.2 Influence analysis of Fourier parameters

The influences of the first-stage Fourier parameters on the pitch curves of the second-stage non-circular gears are shown in Fig. 6. When the parametersa0,a2,b1,b2andm31are the same, the pitch curves of the second-stage non-circular gears corresponding to differenta1are shown in Fig. 6(a). It can be seen that with the increase ofa1, the concave extent of the pitch curves decreases, and meanwhile the center distance increases. However, the maximuma1should not be too large because the non-circular gear with excessively flat pitch curve has a large autobiographical speed change, which will affect the stability of the beating-up mechanism. When the parametersa0,a1,b1,b2andm31are the same, the pitch curves of the second-stage non-circular gears corresponding to differenta2are shown in Fig. 6(b) .With the increase ofa2, the concave extent of the pitch curves is also reduced and the center distance becomes slightly larger. It can also improve the curve flattening caused by the increase of parametera1. By adjusting other Fourier parameters, the decrease of the parameterm31also reduces the concave extent of pitch curve, butb1andb2have little influence on the pitch curve.

(a)

(b)

Fig.6 Influences of the first-stage Fourier parameters on the shape of pitch curves of second-stage non-circular gears:(a)a1; (b)a2

4.3 Influence analysis of amplification gear train ratio

The maximum rotation angle of the non-circular gear 8 is determined by the amplification gear ratio trainif1. The larger the amplification gear train ratio, the smaller the rotary angle of the non-circular gear 8 is; conversely, the rotary angle of the non-circular gear 8 is greater. The variation of the pitch curve of the second-stage non-circular gears corresponding to differentif1are shown in Fig. 8. It can be seen that with the decrease ofif1, the convex angle of the pitch curve of the non-circular gear 8 becomes sharp, and the pitch curve of the non-circular gear 7 is enlarged. Whenif1=1.5, the concave of the pitch curve of the non-circular gear 7 is obvious. Because the pitch curve of the non-circular gear with excessive concave and convex cannot generate ideal tooth profiles, theif1must not be too small. However, the amplification gear train ratio should not be too large, because a large amplification gear ratio will cause larger transmission fluctuations.

Fig.7 Influences of the amplification gear train ratio if1 on the shape of pitch curves of second-stage non-circular

5 Kinematic Experiments

With the computer-aided design program, a set of preferable structural parameters are obtained by adjusting the parameters and analyzing the pitch curves：a0=47 mm,a1=14 mm,a2=4 mm,b1=0,b2=0,m31=0.9 andif1=2. And the pitch curves of second-stage non-circular gears are shown in Fig. 8.

Fig.8 Pitch curves of the preferable second-stage non-circular gears

In order to further verify the working performance of the new beating-up mechanism, the test bed of this mechanism shown in Fig. 9 is established based on the preferable structural parameters.

Fig.9 Beating-up mechanism test bed

5.1 Kinematic experiment methods

Fig.10 Experiment result processing interface

Two marking points in the beating-up mechanism are selected. The beating-up motion of this new mechanism is recorded by the high-speed camera, as shown in Fig. 10. The recorded video results can be processed by using video analysis software Blaster’s MAS. The two marking points are connected in astraight line, and the position of the line is tracked until the main shaft finishes a rotation cycle. The data of position change of the line are imported into MATLAB, and then the obtained experimental displacement and velocity curves are compared with theoretical curves.

5.2 Analysis of kinematic experiment results

It can be seen from Fig. 11 that the experimental curves and theoretical curves of the sley are basically consistent, although some fluctuations exist in the experimental beating-up curve. The fluctuation is caused by errors in the manufacture of the non-circular gears, the video analysis and the transmission error of the test bed. Hence, it can prove that the reverse kinematic model of this beating-up mechanism deduced is correct, and the new mechanism can be used for actual working.

(a) Velocity experimental curve

(b) Velocity theoretical curve

(c) displacement experimental curve

(d) Displacement theoretical curve

Fig.11 Experimental and theoretical curves of the sley

6 Conclusions

(1) A new non-circular combined gear train beating-up mechanism is proposed to transform the constant rotation of the main shaft into the intermittent non-uniform swing of the sley. The non-circular combined gear train is composed by the first-stage Fourier planetary gear train and the second-stage non-circular planetary gear train. The ideal kinematic equations of the sley are established based on Hermite interpolation polynomials that can ensure the mechanism achieve enough dwelling beating-up time.

(2) The influences of the Fourier parameters and the amplification gear train ratio on the pitch curves of the second-stage non-circular gears are analyzed based on the kinematic model of this beating-up mechanism. The results showed that the Fourier parametersa1,a2and the amplification gear ratioif1are too small or too large which will cause the pitch curve in a poor shape. Based on the influence analysis, a set of preferable structural parameters are obtained:a0=47 mm,a1=14 mm,a2=4 mm,b1=0,b2=0,m31=0.9 andif1=2.

(3) A test bed based on the preferable structural parameters was developed. The experimental results are basically identical with the theoretical analysis, which indicates the reverse solution model for the pitch curves of the second-stage non-circular gears is correct. And it also indicates that the new mechanism can be used for actual working．