Home | | **Dynamics of Machines** | Approximate Analytical Method for Velocity and Acceleration of the Piston

Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig. 15.7. Let OC be the crank and PC the connecting rod.

**APPROXIMATE** **ANALYTICAL** **METHOD** **FOR** **VELOCITY AND ****ACCELERATION OF THE PISTON**

Consider the motion of a crank and connecting
rod of a reciprocating steam engine as shown in Fig. 15.7. Let *OC* be the crank and *PC* the connecting rod. Let the crank rotates with angular velocity
of ω rad/s and the crank turns through an angle θ from the inner dead centre (briefly written as I.D.C). Let *x* be the displacement of a reciprocating
body *P* from I.D.C. after time *t* seconds, during which the crank has
turned through an angle θ .

Fig.
15.7. Motion of a crank and connecting rod of a reciprocating steam engine.

Let l = Length
of connecting rod between the centres,

r = Radius
of crank or crank pin circle,

φ
= Inclination of connecting rod to the
line of stroke PO, and

n = Ratio
of length of connecting rod to the radius of crank = l/r.

**Velocity of the piston**

*Acceleration of the piston*

Since the acceleration is the rate of change of
velocity, therefore acceleration of the piston *P*,

*Q.In a slider crank mechanism, the length of the
crank and connecting rod are 150 mm and 600 mm respectively. The crank position
is 60° from inner dead centre. The crank shaft speed is 450 r.p.m. *(*clockwise*).* Using analytical method, determine: 1.
Velocity and acceleration of the slider, and 2. Angular velocity and angular
acceleration of the connecting rod.*

**Solution. **Given :** ***r*** **= 150 mm = 0.15 m ;** ***l*** **= 600 mm = 0.6 m ;** **θ** **= 60°;** ***N***
**= 400 r.p.m or** **ω = π × 450/60 = 47.13 rad/s

**1. Velocity and acceleration of
the slider**

We know that ratio of the length of connecting
rod and crank,

*n* = *l */*
r *= 0.6 / 0.15 = 4

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