 Original Article
 Open Access
 Published:
Research and Test of ThreePulley Noncircular Synchronous Pulley Transmission Mechanism with Minimum Slack
Chinese Journal of Mechanical Engineering volume 34, Article number: 29 (2021)
Abstract
The noncircular synchronous belt drive mechanism has demonstrated certain achievements and has been used in special fields. Research regarding noncircular synchronous belt drive mechanisms has focused on optimization design and kinematic analysis in China, whereas two pulley noncircular synchronous belt transmissions have been developed overseas. However, owing to the noncircular characteristics of the belt pulley, the realtime variation in the belt length slack during the transmission of the noncircular synchronous belt is significant, resulting in high probabilities of skipping and vibration. In this study, a noncircular tensioning pulley is added to create a stable threepulley noncircular synchronous belt driving mechanism and a good synchronous belt tensioning, with no skipping; hence, the nonuniform output characteristic of the driven pulley is consistent with the theoretical value. In the circular noncircular noncircular threepulley noncircular synchronous belt mechanism, the pitch curve of the driving synchronous belt pulley is circular, whereas those of the driven synchronous belt and tensioning pulleys are noncircular. To minimize the slack of the belt length of the synchronous belt and the constraint of the concavity and circumference of the tensioning pulley, an automatic optimization model of the tensioning pulley pitch curve is established. The motion simulation, analysis, and optimization code for a threebeltpulley noncircular synchronous belt drive mechanism is written, and the variation in belt length slack under different speed ratios is analyzed based on several examples. The testbed for a circular–noncircular–noncircular threepulley noncircular synchronous belt transmission mechanism is developed. The test shows that the threepulley noncircular synchronous belt drives well. This study proposes an automatic optimization algorithm for the tensioning pulley pitch curve of a noncircular synchronous belt transmission mechanism; it yields a stable transmission of the noncircular synchronous belt transmission mechanism as well as nonuniform output characteristics.
Introduction
During one rotation period of a noncircular synchronous belt transmission mechanism, the required theoretical belt length changes in real time owing to the realtime changes of the noncircular synchronous belt pulley diameters in a transmission cycle. The amount of slack is the difference between the maximum and minimum values of the required belt length. An excessive amount of slack can cause tooth skipping, rendering it impossible to achieve accurate nonuniform output characteristics.
Satchel [1] and Wu et al. [2, 3] presented models for calculating the noncircular belt pulley pitch curve required for the transmission ratio based on different methods, but no discussion regarding tensioning was provided. Konishi et al. [4] and Guo et al. [5] used eccentric circular gears to design a rice transplanting mechanism that can satisfy the specific motion of planting. Liu et al. [6], Ottaviano et al. [7], and Li [8] investigated many types of noncircular gears, such as elliptical gears, eccentric circular gears, noncircular helical gears, and gears with lobes. Finally, eccentric circular gears were used to design a rice transplanting mechanism that can satisfy the specific motion of planting. Karpov et al. [9] and Freudenstein et al. [10] proposed a method to calculate slack in the noncircular twopulley synchronous belt transmission process. It was discovered that the twopulley synchronous belt transmission could not maintain realtime tension during a drive cycle. Zheng et al. [11] used a noncircular belt drive on a punching machine to obtain better quickreturn characteristics, satisfying the requirements of a slow operating speed. However, only two noncircular pulleys were discussed. The calculation showed that the amount of slack changed during a drive cycle, and the maximum slack was 114.4 mm. Feng et al. [12] attempted to obtain the minimum theoretical slack of the belt (chain) length when the center distance was 100 mm, and the transmission law involved the driving and driven pulley angles. The parameters of the eccentric circle belt (chain) pulley mechanism were optimized. When the eccentricity was 0.16, a minimum theoretical slack of 2.62 mm was obtained without considering the polygon effect. Yang et al. [13] established a threepulley noncircular synchronous belt transmission model and regarded the driving pulley as a circle. Using an objective function with the smallest change in the theoretical belt length at any time during the rotation period, the objective function was optimized to obtain a set of optimal parameters. After the optimization was performed, the variance of the length change of the band was 46.91 in one cycle. Cao et al. [14] established a continuous mapping model of twobelt (chain) pulley transmissions, and the speed ratio function was calculated while considering the polygon effect, and the law of belt (chain) transmission was obtained.
In summary, whether it is a twopulley noncircular synchronous belt drive without a tensioner or a threepulley noncircular synchronous belt drive with a noncircular tensioner, the slack of the belt length is large when a slight change occurs in the transmission ratio. Furthermore, the amount of slack will not cause realtime tensioning or skipping in the synchronous belt drive, thereby nullifying the predetermined precise nonuniform speed transmission rule.
The uneven variation in the belt length slack during a noncircular synchronous belt transmission renders it difficult to achieve accurate nonuniform speed transmission in engineering applications. He [15] proposed a circular–eccentric circular–noncircular belt pulley drive mechanism, in which the noncircular belt pulley was the tensioning pulley and the belt length slack was the objective function. The slack value was 10.36 mm when the center distance of the belt pulley was 110 mm after manual optimization. The method involved fitting and backward generation, which necessitated a significant amount of calculation; hence, the results obtained were low in precision.
Following the results of He [15], the author studied the variation in slack in a threepulley noncircular synchronous belt transmission with a noncircular tensioning pulley, analyzed the characteristics of the threepulley noncircular synchronous belt transmission, and conducted experimental research.
ThreePulley Noncircular Synchronous Belt Drive Belt Length Calculation Model
As shown in Figure 1, a circular–noncircular threepulley noncircular synchronous belt transmission mechanism that included driving pulley 1 with a circular pitch curve, driven pulley 2 with a noncircular pitch curve, and tension pulley 3 with a noncircular pitch curve was considered. The circumferences of the three pulleys were identical. The pitch curve of synchronous pulley 2 was determined based on the transmission relationship between the driving and driven pulleys, and the pitch curve of the synchronous belt tension pulley 3 was determined based on the theoretical optimization of the minimum slack of the synchronous belt in the transmission.
Tangent Polar Coordinate Equation of Plane Curve
The plane coordinate system oxy comprised an arbitrary curve c. As shown in Figure 2, the tangent at point \(C\left( {x,y} \right)\) on the curve is t, ON is the vertical distance from the origin O to the tangent t, the foot point is N, the vertical distance ON recorded is denoted as p (hereinafter referred to as the tangential diameter), the length of the line segment CN is denoted as q, and the length of vector OC is denoted as r. The angle between the oxaxis and vector OC is φ, and the angle between the oxaxis and the perpendicular ON is θ, (hereinafter referred to as the tangential angle). Hence, the tangent polar coordinate equation of the curve is [16, 17]
Based on Eq. (1), the first derivative of p with respect to θ is expressed as
Based on the geometric relationship shown in Figure 2 [18], we obtain
Solving Eqs. (1) and (2), the relationship between the Cartesian coordinates of curve c and the tangent polar coordinates is as follows:
Pitch Curve Calculation Model of Synchronous Belt Pulley in ThreePulley Noncircular Synchronous Belt Drive
As shown in Figure 3, the rotation centers of driving pulley 1, driven pulley 2, and tension pulley 3 are denoted as O_{1}, O_{2}, and O_{3}, respectively. We define O_{1}XY as the fixed reference frame; and \(o_{1}  x_{1} y_{1}\), \(o_{2}  x_{2} y_{2}\), and \(o_{3}  x_{3} y_{3}\) as the moving reference frames attached to pulleys 1, 2, and 3, respectively. The center distance between the two pulleys is \(o_{1} o_{2} = a,\) the transmission ratio between the driving and driven pulleys is i_{12}, and the initial rotation angle of driving pulley 1 and driven pulley 2 is \(\varphi_{1} = \varphi_{2} = 0\). The driving pulley rotates counterclockwise, and the rotation angle between the driving and driven pulley exhibits the following relationship:
Figure 4 shows that, in moving coordinate system \(o_{1}  x_{1} y_{1}\), the angle between axis x_{1} and the tangential diameter p_{1} of the driving pulley is θ_{1}. In moving coordinate system \(o_{2}  x_{2} y_{2}\), the angle between axis x_{2} and the tangential diameter p_{2} of the driving pulley is θ_{2}. The relationship between the tangential and rotation angles can be written as
The driving pulley pitch curve is a circle with a fixed radius, and the tangential polar coordinate equation is expressed as follows:
Based on the relationships shown in Figure 4, Eqs. (7) and (8) yield the following:
Based on Eqs. (5) and (8), the rectangular coordinates of the pitch curve of the driving pulley in coordinate system \(o_{1}  x_{1} y_{1}\) are expressed as follows:
According to tangent polar coordinate theory, the relationship between the driving and driven pulley tangential diameters is expressed as
For a rotation angle φ_{1}, Eqs. (8) and (9) can be solved to obtain p_{1} and θ_{1}, whereas Eqs. (6), (7), and (10) can be solved to obtain p_{2} and θ_{2}. Hence, we obtain a tangent polar coordinate equation for the driven pulley curve as follows:
Based on Eqs. (5) and (12), the driving pulley rotation angle is φ_{1}, and the driven pulley rotation angle is φ_{2}. Moving coordinate system \(o_{2}  x_{2} y_{2}\) can be expressed in rectangular coordinates as follows:
The initial tension pulley pitch curve is a circle of radius the same as that of the driving pulley. The driving and tension pulley rotation angles are denoted as φ_{1} and φ_{3}, respectively. Hence, the tangent polar coordinate equation of the tension pulley pitch curve is as follows:
Based on Eqs. (5) and (14), moving coordinate system \(o_{3}  x_{3} y_{3}\) can be expressed in rectangular coordinates as follows:
ThreePulley Noncircular Synchronous Belt Drive Belt Length Calculation Model
The driving, driven, and tension pulleys rotate by \(\varphi_{1}\), \(\varphi_{2}\), and φ_{3}, respectively. The tangent points between the synchronous belt and driving pulley 1 and the between driven pulley 2 and tension pulley 3 are shown in Figure 3. The tangent polar coordinate parameters of tangent points \(C_{1} {,}\;C_{2} {,}\;C_{3} {,}\;C_{4} {,}C_{5} {\text{, and}} C_{6}\) are p_{1} and θ_{12}, p_{2} and θ_{21}, p_{2} and θ_{23}, p_{3} and θ_{34}, p_{3} and θ_{35}, and p_{1} and θ_{16}, respectively. The total belt length of the noncircular threepulley synchronous belt transmission mechanism can be written as
In Eq. (16), \(\overline{{C_{1} C_{2} }}\), \(\overline{{C_{3} C_{4} }}\), and \(\overline{{C_{5} C_{6} }}\) are the lengths of the common tangents between the two pulleys, and \(\mathop {C_{2} C_{3} }\limits^{\frown }\), \(\mathop {C_{4} C_{5} }\limits^{\frown }\), and \(\mathop {C_{6} C_{1} }\limits^{\frown }\) are the arc lengths corresponding to the wrapping angles formed between the two tangential points on each pulley and the synchronous belt. As shown in Figure 3, the driving, driven, and tension pulleys rotate by \(\varphi_{1}\), \(\varphi_{{2}}\), and \(\varphi_{{3}}\), respectively. Inserting Eqs. (9) and (13) into Eq. (15), the pitch curves of driving pulley 1, driven pulley 2, and tension pulley 3 in fixed coordinate system \(O_{1}  XY\) are expressed as follows:
The coordinates of tangent points \(C_{1} ,C_{2} ,C_{3} ,C_{4} ,C_{5} {\text{ and}} \; C_{6}\) in fixed coordinate system \(O_{1}  XY\) are \(X_{12}\) and \(Y_{12}\), \(X_{21}\) and \(Y_{21}\), \(X_{23}\) and \(Y_{23}\), \(X_{34}\) and \(Y_{34}\), \(X_{35}\) and \(Y_{35}\), \(X_{16}\) and \(Y_{16}\), respectively. The lengths of common tangent lines \(\overline{{C_{1} C_{2} }}\), \(\overline{{C_{3} C_{4} }}\), and \(\overline{{C_{5} C_{6} }}\) between the driving and driven pulleys, driven and tension pulleys, and driving and tension pulleys are expressed as follows, respectively:
The tangential angles of tangent points \(C_{6} {,}\;C_{2} {,}\;C_{3} {,}\;C_{4} {,}\;C_{5} {\text{, and}} \; C_{1}\) in moving coordinate systems \(o_{1}  x_{1} y_{1}\), \(o_{2}  x_{2} y_{2}\) and \(o_{3}  x_{3} y_{3}\) are \(\theta_{12}\) and \(\theta_{16}\), \(\theta_{21}\) and \(\theta_{23}\), \(\theta_{34}\) and \(\theta_{35}\), respectively. The synchronous belt forms wrapping arc lengths with the driving, driven, and tension pulleys in areas \(\mathop {C_{2} C_{3} }\limits^{\frown }\), \(\mathop {C_{4} C_{5} }\limits^{\frown }\), and \(\mathop {C_{6} C_{1} }\limits^{\frown }\), respectively, which can be calculated as follows:
Substituting Eqs. (20) and (21) into Eq. (16), the belt length of the threepulley noncircular synchronous belt transmission mechanism is expressed as follows:
Tension Pulley Pitch Curve Calculation of ThreePulley Noncircular Synchronous Belt Drive with Minimum Slack
The circular–noncircular–noncircular threepulley synchronous belt transmission mechanism comprises driving pulley 1 with a circular pitch curve, driven pulley 2 with a noncircular pitch curve, and tension pulley 3 with a noncircular pitch curve. The three pulleys had equal circumferences. The rotational relationship between the driving and driven pulleys is denoted as \(i_{12}\), and the center distance of the three pulleys is \(l_{1} = l_{2} = l_{3}\). An automatic optimization algorithm was developed, and the optimal solution was used to obtain the noncircular tension pulley pitch curve with the minimum slack.
Noncircular Belt Pulley Pitch Curve Concave and Convex Judgment Based on Vector Ipsilateral
As shown in Figure 5, n discrete points on the noncircular pulley pitch curve were used as n vertices of the constituent polygon, denoted counterclockwise as \(P_{i} (i = 1,2 \cdots ,n)\). The coordinates of the vertices \(P_{i}\) were \((x_{i} ,y_{i} )(i = 1,2 \cdots ,n)\). To determine the positional relationship between the edge of the polygon and the discrete point to be measured, i.e., to determine whether the discrete point to be measured is on the left or right of the edge (where the left and right sides are relative to the forward direction, which refers to the starting point to the end point of the line segment), one side of the polygon \(P_{1} P_{2}\) was assigned to vector \(\overrightarrow {{P_{1} P_{2} }}\), and the discrete point to be tested was \(P_{3}\). Based mathematical geometry, the area of the triangle formed by three points \(P_{1}\), \(P_{2}\), and \(P_{3}\) on the plane is expressed as follows [19]:
If \(S(P_{1} ,P_{2} ,P_{3} ) > 0\), then \(P_{3}\) is on the left side of the vector \(\overrightarrow {{P_{1} P_{2} }}\). If \(S(P_{1} ,P_{2} ,P_{3} ) < 0\), then \(P_{3}\) is on the right side of the vector \(\overrightarrow {{P_{1} P_{2} }}\). If \(S(P_{1} ,P_{2} ,P_{3} ) = 0\), then \(P_{3}\) is on line segment \(\overrightarrow {{P_{1} P_{2} }}\). Similarly, if \(S(P_{1} ,P_{2} ,P_{3} ) > 0\) and \(S(P_{1} ,P_{2} ,P_{n} ) > 0\), then the polygon is convex at edge \(P_{1} P_{2}\). Similarly, the positional relationship between each edge of the polygon and its two adjacent discrete points is calculated until all edges become convex [20, 21].
If it is determined that the polygon composed of discrete points is a convex polygon, then a convex noncircular pulley pitch curve can be obtained by fitting a cubic nonuniform Bspline [22, 23].
ThreePulley Noncircular Synchronous Belt Slack Calculation Model
During one rotation period of the driving pulley, at some point j (j = 1, 2, …, n), the circumference of the synchronous belt is S_{j}; max(S_{j}) and min(S_{j}) are the maximum and minimum of the theoretical belt length in the next transmission cycle, respectively. The slack F is calculated as follows:
Figure 6 shows that the initial tension pulley pitch curve is the same as the driving pulley pitch curve, where the radius is R. Coordinate system \(O_{1}  X_{1} Y_{1}\) is established based on the initial tension pulley rotation center \(O_{1}\)_{.} To ensure the accuracy of the tension pulley pitch curve and minimize the number of calculations, based on the principle of initial point selection of the optimization algorithm, points on the initial tension pulley pitch curve were obtained at every 45° and labeled as \(A_{i} (i = 1,2 \ldots ,8)\). \(B_{i} (i = 1,2 \ldots ,8)\) is the point on the extension line toward the diameter \(\overrightarrow {{O_{1} A_{i} }} (1,2, \ldots ,8)\). The length \(x_{i}\) (increment in radius) of vector \(\overrightarrow {{A_{i} B_{i} }}\) was used as the design variable.
To minimize the belt slack of the noncircular threepulley synchronous belt drive mechanism, the minimum value of the belt slack was obtained directly as the objective function [24, 25]:
Optimal Calculation of Tensioned Pulley Pitch Curve with Minimum Slack
As shown in Figure 7, to ensure that the algorithm can quickly search for the optimal solution within the optimization region, the value range of the design variable (the optimization region) must be selected, where X(i)_{max} and X(i)_{min} are the upper and lower limits of the ith variable, respectively. \(D(i) = X(i)_{\max }  X(i)_{\min }\) is the allowable change in diameter. The selection of the upper and lower limits of the design variables should be considered comprehensively based on the objective function, design requirements, and algorithm execution time. The expression for the range of the design variables is as follows:
To ensure that the noncircular tension pulley pitch curve is convex, i.e., to ensure that the polygon composed of discrete points \(B_{i} (i = 1,\;2, \ldots ,\;8)\) on the noncircular tension pulley is convex, the radius of the circular pitch curve of the initial tension pulley is R, the coordinates of \(B_{i} (i = 1,2, \ldots ,8)\) are \((M_{i} ,N_{i} ),\) the abscissa \(M_{i} = (R + x_{i} )\cos ({\uppi }i/4),\) the ordinate \(N_{i} = (R + x_{i} )\sin ({\uppi }i/4),\) and the constraint equations are as follows [26]:
One rotation cycle of the driving pulley in the noncircular synchronous belt drive mechanism is considered as one cycle segmented into 720 moments at the jth time. The belt length of the threepulley noncircular synchronous belt transmission mechanism is \(S(j)\) [27]. The minimum (\(\Delta S_{\min }\)) value of the difference between the maximum (\(S_{\max }\)) pulley length and minimum pulley length (\(S_{\min }\)) in a period is used as a fitness function, as follows:
Based on the objective function and constraint conditions, 720 points around the tension pulley were considered as the optimized individual group. Every 45° point on the circumference of a noncircle was selected, and eight points were used as optimization points. An adaptive evolutionary algorithm was designed to iteratively optimize the initial tension pulley based on the minimum synchronous belt slack. The largest number of iterations was 100, and the convergence criterion was a change of less than 1×10^{−4} in the objective function. The optimized point was interpolated using a Bspline function to fit the curve of the noncircular tension pulley. The algorithm flowchart is shown in Figure 8, and the optimization solution was obtained.
Transmission Characteristics Analysis Software for Threepulley Noncircular Synchronous Belt with Minimum Slack
As shown in Figure 9, Region 1 is the parameter input area. The basic parameters of the noncircular threepulley synchronous belt transmission mechanism were input to the fields in this region. The main parameters of the noncircular threepulley synchronous belt transmission mechanism included the three belt pulley rotation center coordinates, driving pulley radius, driving and driven pulley transmission ratios, maximum allowable synchronous belt slacks, and initial radius of the tension pulley [28, 29]. Region 2 is the basic control and optimization control area, which included three basic control buttons for the starting, ending, and equal perimeter calculations as well as two optimization control buttons for tension pulley fitting and automatic optimization. The display area of the basic parameters was used to display the center distance, realtime belt length, and slacks of two adjacent pulleys during the simulation of the noncircular threepulley synchronous belt transmission mechanism. Region 4 is the pitch curve display area, which was used to display the optimal pitch curve of the tension pulley obtained after an automatic optimization, and the noncircular pitch curve of the tension pulley was calculated using an equal circumference. Region 5 is the motion simulation area, which was used to display the realtime motion simulation process of the noncircular threepulley synchronous belt transmission mechanism. Region 6 is the calculation result area, which was used to display the calculation results of the noncircular threepulley synchronous belt drive mechanism, including the transmission ratio between two adjacent pulleys and the belt length variation curve.
Comparison and Analysis of Noncircular ThreePulley Synchronous Belt Drive with Minimum Slack
Analysis of Minimum Slack in Eccentric Circular ThreePulley Synchronous Belt Transmission Mechanism
Because the pitch curve of the driving pulley was circular, the coordinates of rotation center o_{1} were xo_{1} = 0 and yo_{1} = 0. The radius of the driving pulley was R_{1} = 30. The curve of the driven pulley was an eccentric circle, and the coordinates of rotation center o_{2} were xo_{2} = 110 and yo_{2} = 0. The eccentricity e_{2} = 15 and R_{2} = 30. The rotation center coordinates of the tension pulley were xo_{3} = 62 and yo_{3} = −102. The mechanism was a circular eccentric–noncircular threepulley synchronous belt transmission mechanism. A method reported previously [7] was adopted for manual optimization. The belt length change curve obtained is shown in Figure 10; its slack was approximately 10.36 mm. However, the automatic optimization algorithm presented above was adopted for optimization and adjustments. The variation curve of the band length is shown in Figure 11; its slack was 0.03 mm. The pitch curves of the tension pulley before and after optimization are shown in Figure 12; it is clear that the automatic optimization algorithm used in this study was high in accuracy [30].
Analysis of Minimum Slack under Different Transmission Ratio Laws
Because the pitch curve of the driving pulley was circular, the coordinates of rotating center o_{1} were xo_{1} = 0 and yo_{1} = 0. The radius of the driving pulley, R_{1} = 30. The pitch curve of the driven pulley was noncircular, and the coordinates of the rotating center o_{2} were xo_{2}=110 and yo_{2}=0. The eccentricity e_{2}=15 and R_{2}=30. The rotation center coordinates of the tension pulley were xo_{3} = 62 and yo_{3} = −102. The driving and driven pulleys were rotated by angles \(\varphi_{1}\) and \(\varphi_{2} = \varphi_{1} + m \cdot \sin (\varphi_{1} )\), respectively, where m was set as different values to verify the rationality and validity of the automatic optimization algorithm. The other parameters of the noncircular threepulley synchronous belt transmission mechanism were the same. For m = −0.2, −0.1, 0.1, and 0.2, the belt length variation and tension pulley pitch curves are as shown in Figures 12 and 13, respectively. The corresponding slacks were 0.858, 0.379, 0.166, and 0.725 mm, respectively. By setting different m values, i.e., for different transmission ratios, the automatic optimization algorithm can solve for the discrete data points on the optimal noncircular tension pulley pitch curve; subsequently, the latter can be obtained by nonuniform rational Bsplines fitting.
The specific parameters of the automatic optimization algorithm were compared for different transmission ratios, as shown in Table 1. Table 1 and Figures 13 and 14 show that when other conditions remained unchanged, the execution time and slacks of the automatic optimization algorithm differed for different transmission ratios (different values of m). The “zigzag” fluctuations of the transmission ratio curve were related to the number of data points on the pitch curve and the accuracy in the tangent calculation method. Specifically, the closer m is to 0 (the driven pulley closer to the circle), the smaller is the slack, as calculated by the allowable variation in the same directional diameter, and the shorter is the execution time of the corresponding algorithm. In general, the automatic optimization algorithm converged, and the convergence speed was faster.
Analysis of Minimum Slack Amount under Different Radial Diameter Allowable Changes
To study the effect of the directional diameter (i.e., D values) on the automatic optimization algorithm, for m = −0.1, D was assigned the values shown in Table 2 and substituted into the algorithm. The belt length variation curve obtained after the operation is shown in Figure 15. As shown in Table 2, the allowable variation in the directional diameter D significantly affected the slack and execution time of the algorithm after automatic optimization. In particular, the larger the range of the absolute value of variable D, the larger was the optimization area that allowed for the maximum slack \(\Delta L_{\max }\), which reduced the possibility of obtaining an optimal solution. Furthermore, the possibility of obtaining an exact solution containing a smaller slack became greater, and the smaller the slack, the longer was the execution time of the algorithm. As shown in Figure 14, for different diameter values D, the automatic optimization algorithm converged quickly and obtained discrete data points on the optimal noncircular tension pulley pitch curve.
In summary, it is critical to set a reasonable D value for a noncircular threepulley synchronous belt transmission mechanism using an automatic optimization algorithm to calculate the optimal noncircular tension pulley.
Experimental Research on ThreePulley Noncircular Synchronous Belt Transmission
Development of Testbed
The coordinates of the rotation center O_{1} of the driving pulley of the testbed were xo_{1} = 0 and yo_{1} = 0, and the radius of the driving pulley was R_{1} = 30. The coordinates of the rotation center O_{2} of the driven pulley were xo_{2} = 110 and yo_{2} = 0. The central coordinates of the tension pulley were xo_{3} = 62 and yo_{3} = −102, and the rotation angle of the driven pulley was \(\varphi_{2} = \varphi_{1}  0.1 \times \sin (\varphi_{1} )\). The testbed primarily included a transmission mechanism, testbed frame, driving motor, counting equipment, and tension control equipment, as shown in Figure 16.
A direct current motor deceleration motor was connected to the circular driving pulley (the original part) through plum blossom coupling and rotated at a uniform speed, followed by a noncircular driven pulley and a tension pulley. The other end of the three synchronous pulleys was connected to the encoder, and the encoder was fixed on the testbed frame, which was used to measure the rotational speed of the three synchronous pulleys to obtain the actual transmission ratio of the experiment. The DC deceleration motor was connected to the speed regulation module to adjust the speed of the motor output shaft. To analyze the effect of the load torque on the transmission period of the noncircular threepulley synchronous belt drive mechanism, a micro magnetic powder brake was installed on the driven pulley shaft, and the load torque loaded on the driven pulley through the current was adjusted by the tension controller.
Test Results and Analysis
A load torque was applied to the driven wheel shaft through a magnetic powder brake. The rotational speeds of encoders A and B were obtained using an IPAM7404, and the data were transmitted to a computer through an RS485 converter. The fitting curve was compared to the theoretical curve by fitting the test discrete data points in MATLAB.
Figure 17 shows the discrete data point fitting results of the transmission ratio of the driving pulley to the driven pulley, and Figure 18 shows a comparison of the fitted and theoretical curves of the transmission ratio of the driving pulley to the driven pulley. As shown in Figure 17, the discrete data points of the test transmission ratio generally exhibited a regular trend and fluctuated within the minimum error range, and the data exhibited statistical significance. A smooth transmission ratio fitting curve was obtained using MATLAB. As shown in Figure 18, the trend of the actual transmission ratio curve fitted by the test discrete point data was similar to the theoretical transmission ratio curve. The transmission characteristics of a threepulley noncircular synchronous belt transmission mechanism with a minimum amount of slack in the synchronous belt were verified in this study. The change in the transmission ratio obtained from the test was consistent with the theoretical value; therefore, an accurate nonuniform speed and a significant center distance transmission can be achieved.
Conclusions

(1)
In this study, a model for calculating the slack of a threepulley noncircular synchronous belt transmission mechanism was developed, and an optimal calculation method for the noncircular tensioned pulley pitch curve of a threepulley noncircular synchronous belt transmission mechanism with minimum slack was proposed.

(2)
Based on the calculation and optimization models of the threepulley noncircular synchronous belt transmission mechanism, the motion simulation, analysis, and optimization code of the transmission mechanism was written, and the characteristics of the minimum slack of different nonuniform transmission ratio laws were analyzed based on multiple examples.

(3)
A circular–noncircular–noncircular synchronous belt pulley transmission mechanism test rig was developed, and experimental research was conducted. It was discovered that the output characteristics of the driven pulley were consistent with the theoretical characteristics. The reliability of the analytical calculation and optimization models of the threepulley noncircular synchronous belt transmission mechanism was verified.
References
 [1]
H Satchel. Nonuniform chainwheel drives. 8thWorld Congress on TMM, Pragal, Czechoslovakia, 1991.
 [2]
Xutang Wu, Guihai Wang. Noncircular gear and nonuniform ratio transmission. Beijing: China Machine Press, 1997. (in Chinese)
 [3]
Xutang Wu. Calculation principle of joint curve of noncircular sprocket (belt pulley). Mechanical Transmission, 1993, 17(2): 1317.
 [4]
I Yamakage, T Konishi, M Horio, et al. Studies and development of a high performance rice transplanter. J. Jpn. Soc. Agric. Mach., 1989, 51(6): 604610.
 [5]
L S Guo, W J Zhang. Kinematic analysis of a rice transplanting mechanism with eccentric planetary gear trains. Mech. Mach. Theory, 2001, 36(11): 11751188.
 [6]
D Liu, T Ren. Study on deformed limacon gear and motion iterative of its serial mechanism. J. Mech. Des., 2011, 133(6): 061004.
 [7]
E Ottaviano, D Mundo, G A Danieli, et al. Numerical and experimental analysis of noncircular gears and camfollower systems as function generators. Mech. Mach. Theory, 2008, 43(8): 9961008.
 [8]
F S Li. Noncircular gear. Beijing: Mechanical Industry Press, 1975. (in Chinese)
 [9]
O Karpov, P Nosko, P Fil, et al. Prevention of resonance oscillations in gear mechanisms using noncircular gears. Mech. Mach. Theory, 2017, 114: 110.
 [10]
F Freudenstein, C K Chen. Variableratio chain drives with noncircular sprockets and minimum slacktheory and application. Journal of Mechanical Design, 1991, 113(3): 253262.
 [11]
Enlai Zheng, Fang Jia, Hongwei Sha, et al. Noncircular belt transmission design of mechanical press. Mechanism and Machine Theory, 2012, 57(11): 126138.
 [12]
Liping Feng, Jiahong Zheng. Optimization of noncircular sprocket drive segment curve. Mechanical Design, 2006, 23(1): 2527.
 [13]
Yuping Yang, Binbin Ji. Optimal design of tensioning wheel installation position in synchronous belt transmission. Journal of Nantong University, 2010, 9(1): 6870.
 [14]
Lixin Cao, Jian Liu. Research on the mapping models and designing methods of variableratio chain/belt drives. Mechanism and Machine Theory, 2002, 37: 955970.
 [15]
Xiaojing He. Research on design method and key technology of threepulley noncircular synchronous belt transmission. Hangzhou: Zhejiang University of Science and Technology, 2017.
 [16]
Xutang Wu. Contour design of noncircular steel belt transmission. Mechanical Design, 1999(11): 1012.
 [17]
Wei Wang, Chen Zhao. Proof of calculation formula of radial, vector and tangent positive angle of plane curve. Journal of Tianjin Normal University (Nature Edition), 1999, 19(3): 67–70.
 [18]
Kai Yu. Algorithm of distinguishing inner and outer points of convex polygon based on dichotomy. Electronic Design Engineering, 2016, 24(16): 187190.
 [19]
Yajun Yang, Mingyi Duan. An improved algorithm for determining the relationship between a point and a specified polygon region. Computer Knowledge and Technology, 2014, 10(22): 53625364.
 [20]
Xiaoxiang Zhao. Using iteration method to find common tangent. Science and Education Guide, 2015(13): 2832.
 [21]
M Masal, M Tosun, A Z Pirdal. Euler savary formula for the one parameter motions in the complex plane C. Int. J. Phys. Sci., 2010, 5(1): 610.
 [22]
N R Miller, D Ross. Design of variableratio chain drives for bicycles and ergometersapplication to a maximum power bicycle drive. J. Mech. Des., 1980, 102(4): 711717.
 [23]
L A Piegl, W Tiller. Fitting NURBS spherical patches to measured data. Eng. Comput., 2008, 24(2): 97106.
 [24]
M F Tsay, Z H Fong. Study on the generalized mathematical model of noncircular gears. Math. Comput. Model, 2005, 41(4): 555569.
 [25]
Y M Zhang, Q Liu, B Wen. Practical reliabilitybased design of gear pairs. Mech. Mach. Theory, 2003, 38(12): 13631370.
 [26]
X Liu, A Fahad, Y Kazuo, et al. Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics. Int. J. Mach. Tools Manuf., 2005, 45(4): 433444.
 [27]
H X Ju. Design and kinematics analysis of noncircular belt drive. J. Mech. Transm., 2015, 39(1): 126130.
 [28]
F Zheng, L Hua, X Han, et al. Generation of noncircular bevel gears with freeform tooth profile and tooth tengthwise based on screw theory. J. Mech. Des., 2016, 138(6): 064501.
 [29]
F Y Zheng, H Lin, X H Han, et al. Synthesis of indexing mechanisms with noncircular gears. Mech. Mach. Theory, 2016, 105: 108128.
 [30]
Carlo Innocenti, Davide Paganelli. Designing synchronous belt transmissions with variable velocity ratio. Journal of Mechanical Design, 2008, 130(1): 842849.
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Authors’ Information
Jianneng Chen, born in 1972, is currently a professor at Zhejiang SciTech University, China. He received his PhD degree from Zhejiang University, China, in 2004. His research interests include mechanism analysis and synthesis, agricultural machinery equipment, and technology.
Xincheng Sun, born in 1979, is currently a PhD candidate at Zhejiang SciTech University, China. He received his bachelor’s degree from Northwest A&F University, China, in 2007. His research interests include mechanism design and agricultural planting equipment.
Chuanyu Wu, born in 1976, is currently a professor at Zhejiang SciTech University, China. His research interests include theory and application of mechanism innovative design, robot technology, and application.
Dadu Xiao, born in 1993, is currently an engineer at Zhejiang Dahua Technology Co., Ltd., China.
Jun Ye, born in 1989, is currently a PhD candidate at Zhejiang SciTech University, China. His research interests include mechanism design and robotics.
Funding
Supported by National Natural Science Foundation of China (Grant Nos. 51675486, 51805487).
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XS was responsible for the entire trial; XS wrote the manuscript; XS, JC, CW, DX, and JY assisted with experiments and data analysis. All authors read and approved the final manuscript.
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Chen, J., Sun, X., Wu, C. et al. Research and Test of ThreePulley Noncircular Synchronous Pulley Transmission Mechanism with Minimum Slack. Chin. J. Mech. Eng. 34, 29 (2021). https://doi.org/10.1186/s10033021005485
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Keywords
 Noncircular synchronous belt drive
 Slack calculation model
 Nonuniform speed transmission characteristics