OALibJOpen Access Library Journal2333-9705Scientific Research Publishing10.4236/oalib.1102632OALibJ-69261ArticlesBiomedical&Life Sciences Business&Economics Chemistry&Materials Science Computer Science&Communications Earth&Environmental Sciences Engineering Medicine&Healthcare Physics&Mathematics Social Sciences&Humanities Calculation on Variable Sag in Chain Drives ChangfaRong1*School of Mechanical Engineering, Hebei University of Technology, Tianjin, China* E-mail:rongchangfa@126.com3105201603051622 April 2016accepted 7 May 10 May 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

This paper presents a computer-aided analysis to calculate sag in chain drives exactly, which is commonly used in mechanical power transmission. With sprocket rotating, sag in chain drives is different. Sag is a function of meshing position of chain and sprocket. Because sag in chain drives is a variable, the maximum and minimum sag can be obtained by numerical calculation. Corresponding tensions in slack chain are obtained. A computer programme to calculate sag and tensions in slack chain is programmed.

Chain Drives Variable Sag Computer-Aided Analysis
1. Introduction

Chain drives are widely used in mechanical engineering. It is useful to calculate sag exactly in practice. In the past, calculation of sag in chain drives is approximate calculation under a series of assumable conditions. These assumable conditions are far different from practical condition of chain drives. In some methods sprocket’s polygon is replaced with pitch circle. But sag in chain drives is variable due to polygonal action. These methods to calculate sag in chain drives have bigger error. This paper presents a computer-aided analysis to calculate sag in chain drives exactly. Because sag in chain drives is a variable, the maximum and minimum sag can be obtained by numerical calculations. Corresponding tensions in slack chain are obtained. A computer programme is made.

2. Determination of Geometrical Shape of Slack Chain Drives

Slack chain sink due to chain weight, slack chain can be considered a slick cure that called catenary, catenary’s equation 

Suppose end of slack chain drives are (x1,y1) and (x2,y2), shown on Figure 1.

Suppose

Form references  , we can know

In above equations:

C―center distance

R1, R2―pitch radius of driving or driven sprocket;

Z1, Z2―number of teeth of driving or driven sprocket;

p― chain pitch

―Angle between horizontal line and centerline of chain drive

s―length of slack chain

Lc― chain length

Lt―tight chain length

K1, K2―coefficient determined in reference 

From reference  , we have

From Equations (1), (5) and, we have

To multiply Equation (6) by Equation (7), and order, , we have

Chain drives

Equation (8) is exceeding equation. Equation (8) can be solved with Newton iterative method. Numerical result of a can be gained.

3. Determination of Principle of End Points of Slack Chain and Corresponding Coordinate Values

When slack chain is tight, end point of slack chain can be determined with θt, ψt

In Equation (9), θt, ψt are angle displacement of driving and driven sprocket.

Because polygonal action and fixed center distance, line is not integer number of pitch, whose end determined by θ and f points are (X1,Y1) and (X2,Y2). Only when slack chain has sag, length of slack chain can be integer number of pitch.

F1―Force between chain link on sprocket and roller, N F2―Force between teeth of sprocket and roller, N T―Force between roller chain and roller, N α―pressure angle

When the angle between sprocket and chain link is outside concave, force of roller shown in Figure 2(a). From Figure 2(a), it is known force in roller is not in balance. Resultant force can make roller go along teeth of sprocket or make roller go out of sprocket.

When the angle between sprocket and chain link is outside protruding, force in roller is in balance force of roller shown in Figure 2(b).

To sum up, the angle between sprocket and chain link should be outside protruding and not outside concave.

When he angle between sprocket and chain link is outside protruding,

From equation of slack chain of chain drive, we have

From Figure 1 and Figure 3, we have

Force diagram of roller in chain drives Analysis and calculation of chain drives

When slack chain is tight, L, H and s can be gained by Equation (2)-(4). Points of slack chain (X1,Y1) and (X2, Y2) can be calculated by Newton iterative method with Equations (6), (2) and (1).

A program to calculate coordinate value of end point is programmed based on Figure 4.

4. Calculation of Sag and Tensions in Slack Chain4.1. Calculation of Sag in Chain Drive

Equation of line between two end points of slack chain is shown in Figure 1.

Equation of slack chain:

Sag in Chain Drives: (15)

Flowchart to calculate end points of slack chain in chain drives

f―Sag in Chain Drives, mm

When x belong to area (X1,Y1), X can has a serious of values. So we can have accurate sag in chain drive enough.

4.2. Calculation of Tensions in Slack Chain

As shown in Figure 5, T1, T2 are tensions of end point slack chain. q is unit weight of roller chain (N). s1 is length of slack chain between (X1,Y1) and (0,a). s2 is length of slack chain between (0,a) and (X2,Y2). H0 is tension at point (0,a).

By Equations (1), (5) and Equation (16), we have

5. Example and Conclusions

A Chain Drive is given. Chain pitch p = 50.8 mm. Number of Sprocket Z1 = 20, Z2 = 50. Number of Chain link LP = 116, Unit Weight of Chain q = 0.1 N/mm.

Calculating results are given in Table 1. From Table 1 the following conclusions can be gained.

Tensions of slack chain in chain drives <xref ref-type="table" rid="table">Table </xref>of sag and t tension of end point slack chain
C (mm)fmax (mm)T1 (N)T2 (N)fmin (mm)T1 (N)T2 (N)
2042.96 2042.92 2042.80 2042.60 2042.40 2042.20 2042.00 2041.80 2041.60 2041.40 2041.20 2041.009.777 12.713 18.630 25.427 30.716 35.357 39.452 43.205 46.612 49.747 52.787 55.6065300.73 4078.05 2785.06 2043.70 1694.03 1473.53 1322.37 1208.89 1121.67 1052.00 993.54 943.025276.61 4053.94 2760.92 2019.57 1669.92 1449.42 1298.26 1184.75 1097.55 1027.87 968.43 918.910.000 0.252 15.789 23.425 29.360 34.135 38.409 42.218 45.714 48.929 51.984 54.8606344.76 4006.83 3280.10 2215.90 1771.22 1525.59 1357.67 1236.66 1143.33 1069.29 1007.47 955.576368.87 4530.91 3255.98 2191.82 1747.12 1501.50 1333.59 1212.57 1119.29 1045.21 983.40 931.49

1) When driving sprocket is running and tight chain is straight, sag in chain drives and tensions of end point of slack chain are changing.

2) A little change in centre distance should make bigger change of sag in chain drives. So when centre distance in chain drive is not adjustable, it must be very careful to design centre distance in chain drives.

Cite this paper

Changfa Rong, (2016) Calculation on Variable Sag in Chain Drives. Open Access Library Journal,03,1-6. doi: 10.4236/oalib.1102632

ReferencesRong, C.F., Zhu, J. and Xu, H. (2000) Determination of Center Distance of a Roller Chain Drives. Journal of Xian Jiaotong University, 34, 47-51.Tongji University (1999) Mathematical Analysis. Advanced Education Publishing Company, Beijing, 344-346.