Calculation and analysis of polygon-like effect of non-circular synchronous belt transmission

Xian-bing BIAN, Zhi-wei CHEN, Xin-cheng SUN, Jian-neng CHEN, 2, Da-du XIAO, Jia-you CHEN

(1Zhejiang Sci-Tech University, Hangzhou 310018, China)(2Key Laboratory of Transplanting Equipment and Technology of Zhejiang Province, Hangzhou 310018, China)(3Fujian Jiayou Tea Machinery Intelligent Technologies Inc., Quanzhou 362400, China)

Abstract: Non-circular synchronous belt could be used in the non-uniform speed transmission with large center distance. The closed figure extracted from the synchronous belt wound around the non-circular synchronous belt pulley is composed of curves and straight lines alternately connected. Due to the differences between the closed figure and the pitch curves of the non-circular synchronous belt, the synchronous belt is not always tangent with synchronous belt pulley during the transmission, which will influence the accuracy of the transmission. Consider that, a mathematical model was proposed to calculate this phenomenon above which is called “polygon-like effect” in this paper. By calculating and analyzing some samples on the basis of this model, three key factors that influence the non-uniform transmission ratio of non-circular belt pulley in the transmission were concluded, which include the initial position, tooth number and non-uniform transmission parameter. The more teeth of the non-circular belt pulley have and the smaller the non-uniform transmission parameter is, the smoother the real transmission ratio curve is. And the real transmission ratio curve of master-slave wheel in phase is smoother than the curve of master-slave wheel out of phase. This study could provide the important reference for the tooth number design and manufacture of non-circular synchronous pulley as well as calculating the initial position.

Key words: Non-circular synchronous belt transmission, Polygon-like effect, Law of transmission ratio

Non-circular belt (chain) transmission has the both advantages of non-circular gear transmission and belt (chain) transmission. It can realize the non-uniform transmission with large center distance, and has been successfully applied in various occasions. For example, Zheng Enlai [2] used a non-circular belt transmission on a punching machine to obtain the better quick-return characteristics. But the non-circular belt (chain) transmission has the polygonal-effect. The closed figure extracted from the chain during the non-circular chain transmission is all connected by straight lines. While the closed figure extracted from the synchronous belt during the non-circular synchronous belt transmission is alternately connected by the curves and straight lines, that is called “polygon-like” in this paper. And the dynamic effect caused by the closed figure in the non-circular synchronous belt transmission is called “polygon-like effect”. This phenomenon affects the accuracy of the transmission ratio of the non-circular belt (chain) transmission, so it should be fully considered in the transmission design.

Guo Jianhua et al. [3] proposed the relationship between pulley angle and synchronous belt speed by piecewise function and established a accuracy calculation model of synchronous belt transmission to get the accuracy of the transmission which influenced by the tight side length of the synchronous belt, the tooth profile of the belt teeth and the tooth profile of pulley teeth. But this calculation model is just developed for the new-type special synchronous belt transmission and only analyzed the error in linear distance of transmission influenced by the polygon-effect, didn’t analyze the law of the transmission ratio. Yang Yuping et al. [4] established the mathematical model to calculate the transmission error in linear distance of circular synchronous belt transmission influenced by polygon-effect during one rotation period. However, the quantitative relationship between polygonal-effect and transmission error in linear distance was not cleared in this study. Wang Yanhua et al. [5] developed the speed and acceleration model of the chord motion and the circle motion of the synchronous belt influenced by the polygon-effect in the gasoline engine, and found that the major reason of the transverse and longitudinal vibration of the synchronous belt was the changes in speed and acceleration. But the transmission ratio law of synchronous belt influenced by the polygonal-effect was not analyzed. Cao Lixin[6] established the discrete mapping model of chain transmission ratio law on the static coordinate system by using differential geometry and considering the polygon-effect. They not only analyzed the feature of the transmission ratio law of non-circular chain transmission, but also studied the transmission ratio law of continuous mapping model in chain drive without polygon-effect. However, this model is only fit for the non-circular chain transmission, but not suitable for the non-circular belt transmission.

In summary, until now, most researchers analyzed the influence of polygon-effect on both the kinematic characteristics (speed, acceleration, vibration) of chain (belt) [7-12] and the transmission error in linear distance. Nevertheless, the impact of polygon-effect on the transmission ratio law of the non-circular synchronous belt transmission was not studied.

This paper analyzed the factors of the transmission ratio law of the non-circular synchronous belt influenced by the polygon-like effect, which could provide the important reference for the tooth number design and manufacture of non-circular synchronous pulley as well as calculating the initial position, and further improve the transmission accuracy of non-circular synchronous belt.

Fig.1 Schematic diagram of synchronous belt and belt pulley meshing

1 The cause of “polygon-like effect” in non-circular synchronous belt transmission

In order to quantitatively analyze the influence of “polygon-like effect” on the transmission ratio law of non-circular synchronous belt transmission, a numerical method was used in this paper to calculate the transmission ratio of non-circular synchronous belt influenced by the “polygon-like effect” on the real transmission situation. As shown in Fig.2, the non-circular synchronous belt wound on the non-circular belt pulley in the form of alternate connection of tooth thickness (straight line) and slot width (curve), which transmit in the form of alternate connection of straight line and curve on the meshing pitch curve of belt pulley and belt. It is assumed that the belt pulley pitch isP, the tooth number isz, the center distance isa, the rotation centers of driving pulley 1 and driven pulley 2 areO1andO2respectively, the tooth thickness of belt pulley ism, and the slot width of the belt pulley isn. The intersection of tooth thickness and slot width on the pitch curve is called “node”. Then, at the initial position, the angleα0between the connecting line from the node of driving pulley 1 to the rotation centerO1and the perpendicular is the initial angle of the driving pulley, and theC10is the initial tangent point. Similarly, the angleβ0between the connecting line from the node of driven pulley 2 to the rotation centerO2and the perpendicular is the initial angle of the driven pulley, and theC20is the initial tangent point. AssumeL0is the tangent length ofC10C20. And the tangent points becameC1andC2respectively when the driving pulley 1 turns at an angle anticlockwise, meanwhile, the rotation angles of the driving and driven pulleys areα1andβ1respectively. It is assumed that the initial tangent length ofC1C2isL1, the length ofC10C1isS0, and the length ofC20C2isS1. At arbitrary times, the perpendiculars are constructed from rotation centersO1andO2to the tangent lines, and the vertical distances ared1andd2respectively. The driven pulley rotates synchronously with the driving pulley at a certain transmission ratio. On the basis of the tangent polar coordinate theory of curve [1], the vertical distanced1andd2can be expressed as the functions ofd1(α) andd2(β) respectively, whereαandβare the rotation angles of the driving and driven pulleys at any time.

Fig.2 Schematic diagram of calculation method of non-circular synchronous belt transmission considering “polygon-like effect”

At the initial time, the tangent lengthL0between the two belt pulleys could be expressed as follows:

(1)

When the driving pulley 1 turns one pitch, the rotation angle isα1, and the driven pulley turnsβ1angle according to the transmission ratio. In the meantime, the tangent points of two pulleys and synchronous belt correspondingly becameC1andC2, and the vertical distance from the rotation center to the tangent line ared1(α1) andd2(β1) respectively, whereα1andβ1are the angles from the initial pitch line to the vertical axis. The tangent lengthL1between the two belt pulleys could be expressed as：

(2)

The tangent lengthLrbetween the two belt pulleys at any time is：

(3)

whereαris the angle between the initial pitch line of the driving pulley and the vertical axis at any time;βris the angle between the initial pitch line of the driven pulley and the vertical axis at any time;Lris the length of the tangent line between the driving and driven pulleys at any time;C1ris the tangent point of driving pulley and synchronous belt at any time, andC2ris the tangent point of driven pulley and synchronous belt at any time.

When the driving pulley rotates over one pitch, theS0=p. Meanwhile, the length of driven pulley turns by is the sum ofktooth thicknesses of the driven pulley andhslot widths of the pulley, which isS1, can be written as：

S1=k×m+h×n

(4)

wherek=0, 1, 2, …,z,h=0, 1, 2, 3，…，z,mis the tooth thickness of the belt pulley, andnis the slot width of the belt pulley.

According to the invariance of the total band length, the following equation can be obtained as：

L0-S0+S1-L1=0

(5)

(6)

(7)

The specific steps are as follows: As the driving pulley turnsαrcounterclockwise from the initial position, looking for the tangent pointC2which satisfy the formula (1) from the initial tangent pointC20of the non-circular driven pulley 2, then the new tangent lineC1C2can be determined. And the transmission ratio can be obtained as：

(8)

2 Calculation and analysis example of “polygon-like effect” in non-circular synchronous belt transmission

It is assumed that the transmission ratio law of the non-circular synchronous belt driving pulley and the driven pulley isβ=α+m×sin(α), wheremis the non-uniform transmission parameter. Considering the accuracy of non-circular synchronous belt transmission ratio influenced by the “polygon-like effect” during the transmission, in this section, the transmission ratio of the non-circular synchronous belt transmission is analyzed at the initial position (α=β=0). Due to the tooth thickness and slot width, the initial position can be divided into the four types according to the next corresponding position of the driving and driven pulleys at the initial position is the tooth thickness or tooth slot, as shown in Fig.3.

Fig.3 Four types of the initial positions

(1)When the tooth numberz=12, the curve of transmission ratio on different initial positions and different transmission ratios of driving and driven pulleys is shown in Fig.4 and 5.

Fig.4 Transmission ratio contrast diagram of non-circular belt transmission influenced by “polygon-like effect”(z=12, m=-0.1)

Fig.5 Transmission ratio contrast diagram of non-circular belt transmission influenced by “polygon-like effect”(z=12, m=0.2)

It can be proven by the figures 4 and 5 that when the tooth numberzis 12 andmis invariant, the curves of transmission ratio with different initial positions are quite different, but the number of inflection points of the transmission ratio curves both on the two figures are equal to the number of teethz, and the inflection point exactly corresponding to the vertex of polygon-like at that moment. According to the shape of the transmission ratio curves, the transmission ratio curve of the initial positions corresponding to figures 3 (a) and 3(c), which is tooth thickness (driving)-tooth thickness (driven) and tooth slot (driving)-tooth slot (driven), is “serrated”, so the vibration and shock are large in the real transmission; the transmission ratio curve of the initial positions corresponding to figures 3 (b) and 3(d), which is tooth thickness (driving)-tooth slot (driven) and tooth slot (driving)-tooth thickness (driven), is “wavy”, so the vibration and shock are small in the real transmission. Non-circular synchronous belt driving and driven pulleys at different initial positions cause the alternate and cyclic change of tooth thickness and tooth slot of the driving and driven pulleys in the rotation cycle, which brings about the great changes on the transmission ratio curve. In terms of the fluctuation amplitude of transmission ratio curve, according to the tooth thickness (driving)-tooth thickness (driven) and tooth slot (driving)-tooth slot (driven) as shown in figures 4 and 5, the fluctuation amplitudes of these transmission ratio curves are basically the same and much smaller than the fluctuation amplitudes of curves of the tooth thickness (driving)-tooth slot (driven) and tooth slot (driving)-tooth thickness (driven).

(2)Fig.6 and 7 show the transmission ratio curves of the different initial positions and different transmission ratios for driving and driven pulleys, when the tooth numberz=16.

Fig.6 Transmission ratio contrast diagram of non-circular belt transmission influenced by “polygon-like effect”(z=16,m=-0.1)

Fig.7 Transmission ratio contrast diagram of non-circular belt transmission influenced by “polygon-like effect”(z=16,m=0.2)

Fig. 6 and 7 show that although the transmission ratio curves at different initial positions are quite different when the tooth numberzis 16 andmis constant, the number of inflection points in the transmission ratio curves both on the two figures is still equal to thez. In terms of the fluctuation amplitude of transmission ratio curve, as show in figures 6 and 7, the fluctuation amplitudes of transmission ratio curve of the tooth thickness (driving)-tooth thickness (driven) and tooth slot (driving)-tooth slot (driven) are approximately equal and much smaller than the curves of the tooth thickness (driving)-tooth slot (driven) and tooth slot (driving)-tooth thickness (driven). According to the figures 6 and 4 as well as the figures 7 and 5, if the initial positions of the driving pully and driven pully were same, but the tooth numbers were different, the transmission ratio curves would be different. Then, compare thez=16 withz=12, if the tooth number increased, the number of the inflection points would increase and the amplitude of these points would be small. Simultaneously, these inflection points also correspond to the vertexes of the polygon-like.

(3)When the tooth numberz=20, the transmission ratio curves with different initial positions and different transmission ratios are shown in Fig.8 and 9.

Fig.8 and 9 show that the transmission ratio curves of the non-circular belt transmission with the same tooth numberz=20 and the same transmission ratio law but different initial positions are still different. However, the number of inflection points which appear in the transmission ratio curves in both figures is still equal to the tooth numberz. In terms of the transmission ratio curve, the fluctuation amplitudes of the tooth thickness (driving)-tooth thickness (driven) and tooth slot (driving)-tooth slot (driven) transmission ratio curve in figures 8 and 9 are basically the same and much smaller than the fluctuation amplitudes of the curves of the tooth thickness (driving)-tooth slot (driven) and tooth slot (driving)-tooth thickness (driven). According to the figures 8, 6 and 4 as well as the figures 9, 7 and 5, if the initial positions were same and the tooth numbers were different, the transmission ratio curves are also different. Compare thez=20,z=16 andz=12, if the tooth number increased, the number of the inflection points of the curve would increase and the amplitude of these inflection points would be small. Simultaneously, these inflection points also correspond to the vertexes of the polygon-like.

Fig.9 Transmission ratio contrast diagram of non-circular belt transmission influenced by “polygon-like effect”(z=20,m=0.2)

As shown in figures 4 to 9, the real transmission ratio curve fluctuates on the theoretical transmission ratio curve with differentm, and if themis nearer to 0, the amplitude of curve fluctuation would be smaller.

(9)

The average errors of the four types are calculated by equation (9), as shown in Tables 1 and 2.

Table 1 Average error of transmission ratio with different initial position and different tooth number (m=-0.1)

Table 2 Average error of transmission ratio with different initial position and different tooth number (m=0.2)

According to the four initial positions, if the initial positions of driving pulley and driven pulley are in the same phase state (which is the tooth thickness (driving)-tooth thickness (driven) and tooth slot (driving)-tooth slot (driven)), withm=-0.1 orm=0.2, the average error (fluctuation amplitude) is much smaller than that of the driving pulley and the driven pulley when their initial positions are out of phase (which is the tooth thickness (driving) -tooth slot (driven) and tooth slot (driving)-tooth thickness (driven)), but the transmission ratio curve is much rougher. If the initial positions were in the same phase, the average error would be basically the same; and if the initial positions were out of phase, the average error would be also basically the same. Both under in-phase and out of phase state, the fluctuation times of transmission ratio curve are equal to thez, or equal to the number of the sides of polygon-like. Meanwhile, the inflection points of the curve correspond to the boundary point of the tooth slot and the tooth thickness, the more teeth the belt pulleys have, the smaller the fluctuation amplitude is, and the closer the transmission ratio curve is to the theoretical transmission ratio curve. Therefore, the transmission ratio of the non-circular belt transmission influenced by polygon-like effect is related to the initial position, and the same phase state should be selected as the initial installation position of the non-circular synchronous belt transmission mechanism.

3 Conclusion

(1)A mathematical model was proposed to calculate the non-circular synchronous belt transmission ratio influenced by the “polygon-like effect”, which was build and tested on the MATLAB. And it also analyzed the influence of polygon-like effect on transmission ratio law of the non-circular synchronous belt transmission under the different initial positions, different tooth numbers and different transmission ratios.

(2)According to the analysis of the calculation results, considering the non-circular synchronous belt pulley with different transmission ratios, the more teeth the non-circular belt pulley have and the smaller the non-uniform transmission parameter is, the smoother the real transmission ratio curve is, and the closer the transmission ratio curve is to the theoretical transmission ratio curve. Meanwhile, the average error of master-slave belt pulley in phase is obviously smaller than that of master-slave belt pulley out of phase.