# Analysis of motion non-uniformity of chain drives

chain entrySprocketAfter forming a broken line, the motion of the chain drive is very similar to that of the belt drive wound on a regular polygon wheel, see Figure 9.The side length corresponds to the chain pitch p, and the number of sides corresponds to the number of sprocket teeth z.For each revolution of the sprocket, the distance the chain moves is zp. Let z1 and z2 be the number of teeth of the two sprockets, p is the pitch (mm), and n1 and n2 are the rotational speeds of the two sprockets (r/min). The average speed v (m/s) is

v=z1pn1/60*1000=z2pn2/60*1000             (4)

In fact, both the instantaneous chain speed and the instantaneous ratio of a chain drive vary.The analysis is as follows: The tight side of the chain is in a horizontal position during transmission, see Figure 6.9.Assuming that the driving wheel rotates at an equal angular velocity ω1, its indexed peripheral speed is R1ω1.When the chain link enters the drive wheel, its pin always changes its position as the sprocket rotates.When at the instant of angle β, the instantaneous speed of the horizontal movement of the chain is equal to the horizontal component of the peripheral speed of the pin.i.e. chain speed v

v=cosβR1ω1 (6)

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The variation range of the angle is between ±φ1/2, φ1=360. /z1.When β=0, the chain speed is large, vmax=R1ω1; when β=±φ1/2, the chain speed is small, vmin=R1ω1cos(φ1/2).Therefore, even when the driving sprocket rotates at a constant speed, the chain speed v changes.The cycle changes every time a chain pitch is rotated, see Figure 10.In the same way, the instantaneous speed v`=R1ω1sinβ of the vertical movement of the chain also changes periodically, so that the chain shakes up and down.

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slaveSprocketSince the chain speed v≠constant and the constant change of γ angle (Fig. 9), its angular speed ω2=v/R2cosγ also changes.

Obviously, the instantaneous transmission ratio cannot get a constant value.Therefore, the chain drive works unstable.