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.

Slack chain sink due to chain weight, slack chain can be considered a slick cure that called catenary, catenary’s equation [1]

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

Suppose

Form references [2] , 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

L_c― chain length

L_t―tight chain length

K₁, K₂―coefficient determined in reference [2]

From reference [2] , we have

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

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

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

When slack chain is tight, end point of slack chain can be determined with θ_t, ψ_t

In Equation (9), θ_t, ψ_t 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

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.

A Chain Drive is given. Chain pitch p = 50.8 mm. Number of Sprocket Z₁ = 20, Z₂ = 50. Number of Chain link L_P = 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.

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.

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