Approximate Method for Slider Crank Mechanism:

The slider crank mechanism OAB is illustrated in Figure. Assume l and r is lengths of the connecting rod AB & crank OA respectively. Assume x be the distance of the piston pin from its following position B, (for this point OA & AB are aligned).

x = (r + l ) - (OC + CB) = (r + l ) - (r cos θ + l cos φ)

Also     AC = r sin θ = l sin φ

∴          sin φ= r/l sin θ

Also

By putting (1 - cos φ) in expression of x you will have the following

In this mechanism, r /l negligible value is about 0.25 and higher powers of (r /l) will have negligible value.

Thus,

Velocity of slider,

d θ /dt=ω        (angular speed of crank)

Thus,

V =ω  dx /d θ

Assume crank rotate at the constant angular velocity.

 Acceleration of slider,