Page 175 - Analysis and Design of Machine Elements

p Chordal rise R-r Chain Drives 153
p
B A A
φ B
R
2 R
r r
ω
ω

(a) (b)
Figure 7.6 Chordal action of roller chain. Source: Adapted from Juvinall & Marshek 2001, Figure 19.7,
p. 790. Reproduced with permission of John Wiley & Sons, Inc.

7.2.2.3 Chordal Action
As introduced before, when a chain sequentially meshes with sprocket teeth, the
sprocket resembles a polygon. Figure 7.6a,b illustrates two positions where the centre-
lines of chain are at chordal radius r and sprocket pitch radius R,respectively. As the
sprocket rotates, the amount of chain rise and the fall of the chain centreline is
( ∘ )
180
Δr = R − r = R 1 − cos (7.10)
z
Similarly, the velocity reaches the minimum and maximum value at these two posi-
tions. The minimum velocity occurs at a chordal radius r is
∘
2 rn n p 180
v = = × × cos (7.11)
min ∘
60 × 1000 60 × 1000 180 z
sin
z
The maximum velocity occurs at pitch radius R,expressed as
2 Rn n p
v max = = × (7.12)
60 × 1000 60 × 1000 180 ∘
sin
z
Employing Eq. (7.4), we have the chordal speed variation as
Δv v max − v min ⎡ 1 1 ⎤
= = ⎢ − ⎥ (7.13)
v v 180 ∘ ∘
avg avg z ⎢ tan 180 ⎥
⎣ sin
z z ⎦
The rise and fall of chain becomes harmful when resonance occurs, which is known
as chain whip [5]. The rise and fall of chain, as well as the variation of instantaneous
velocity, are caused by the cyclic fluctuation between the sprocket pitch radius R and
chordal radius r, or by the polygon, as the chain engages the sprocket. This is called
chordal action, or polygonal action. Chordal action is a kinematic consequence of the
polygon due to the pitch length in chains.
Chordal action affects operating smoothness of a roller chain drive, particularly in
high speed applications. Both chordal speed variation and chordal action decrease as