analytical method for velocity and acceleration of slider- crank mechanism

ANALYTICAL METHOD FOR VELOCITY AND ACCELERATION OF SLIDER- CRANK MECHANISM

Consider a reciprocating steam engine mechanism OCP as shown in Fig.2.38. Let crank OC rotates with angular velocity o rad/s and the connecting rod PC makes angle with the line of stroke PO. Let x be the displacement of piston from initial point P' to P, when the crank turns through an angle θ from I.D.C.

Let

r = Crank radius,

l = Length of the connecting rod,

θ = Angle made crank with I.D.C.,

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

n = l/r = Ratio of length of connecting rod to the radius of crank, also known as obliquity ratio.

1. Velocity of the Piston (V_P)

From the geometry of Fig.2.38, displacement of the piston is given by

x = P'P = OP' - OP= (P'C' +C'O) – (PQ + QO)

From ∆CPQ, PQ = l cos ϕ and from ∆COQ, QO = r cos θ

By expanding the above expression by binomial theorem, we get

Substituting equation (iii) in equation (1), we get

Differentiating equation (2.20) with respect to θ, we get

Therefore, velocity of the piston or velocity of P with respect to O,

2 Acceleration of the Piston (a_P)

We know that acceleration is the rate of change of velocity. So, acceleration of the piston P is given by

Differentiating equation (2.21) with respect to θ, we get

Note

1. If the value of n is very large, then a_P = ω²r cos θ, as in case of SHM.

As the direction of motion is reversed at the outer dead centre (ODC) position, therefore, changing the sign of the above expression, we get

3. Angular Velocity of the Connecting Rod (ω_PC)

From the geometry of the Fig.2.38, we find that

Differentiating both sides with respect to time t, we get

Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dϕ/dt, therefore

4. Angular Acceleration of the Connecting Rod (α_PC)

We know that the angular acceleration of P with respect to C,

Differentiating equation (2.23) with respect to θ, we get

Dividing and multiplying by (n² - sin² θ)^1/2, we get

The negative sign indicates that the sense of angular acceleration of the rod is such that it tends to reduce the angle ϕ.

Note

1. Since sin² θ is small as compared to n², therefore it may be neglected. Thus equations (2.23) and (2.24) are reduced to

2. Also in equation (2.25), unity is small as compared to n², hence the term unity can be neglected.

Example 2.14

The lengths of crank and connecting rod of a horizontal reciprocating engine are 125 mm and 500 mm respectively. The crank is rotating at 600. rpm. When the crank has turned 45°from inner dead centre, find analytically,

(i) the velocity and acceleration of the slider,

(ii) the angular velocity, and angular acceleration of the connecting rod.

Given data.

Solution:

(i) Velocity and acceleration of the slider (v_P and a_P):

Velocity of the slider is given by

Acceleration of the slider is given by

(ii) Angular velocity and angular acceleration of the connecting rod (ω_PC and α_PC):

Angular velocity of the connecting rod is given by

Angular acceleration of the connecting rod is given by

Example 2.15

In a reciprocating engine mechanism, the lengths of the crank and connecting rod are 150 mm and 600 mm respectively. The crank position is 45°from inner dead centre. The crank-shaft speed is 300 rpm (clockwise). Using analytical method, determine:

(i) the velocity of the piston;

(ii) the acceleration of the piston; and

(iii) the crank angle for maximum velocity of the piston and the corresponding velocity.

[A.U., Nov/Dec 2012]

Given data:

r = 150 mm 0.15 m; l = 600 mm = 0.6 m; N = 300 rpm; θ = 45°

Solution:

ω = 2πN/60 = 2 π (300)/60 = 31.41 rad/s

Obliquity ratio, = l/r = 0.6/0.15 = 4

(i) Velocity of the piston (v_P):

Velocity of the piston is given by

(ii) Acceleration of the piston (a_P):

Acceleration of the slider is given by

(iii) Crank angle for maximum velocity of the piston the corresponding velocity:

Let

θ  = Crank angle from IDC at which the maximum velocity occurs

We know that the velocity of the piston,

For maximum velocity of the piston,

Substituting the value θ = 77° in the v_P equation, we get

Example 2.16

In a simple steam engine, the lengths of the crank and the connecting rod are 100 mm and 400 mm respectively. The weight of the connecting rod is 50 kg and its centre of mass is 220 mm, from the cross-head centre. The radius of gyration about the centre of mass is 120 mm. If the engine speed is 300 rpm and the crank has turned 45° from IDC, determine:

(i) the angular velocity and angular acceleration of the connecting rod; and

(ii) the kinetic energy of the connecting rod.

Given data:

r = 100 mm = 0.1 m; l = 400 mm = 0.4 m; m = 50 kg; k = 120 mm = 0.12 m; N = 300 rpm; θ = 45°

Solution:

ω = 2πN/60 = 2 π (300)/60 = 31.4 rad/s

Obliquity ratio, n = l/r = 0.4/0.1 = 4

(i) Angular velocity and angular acceleration of the connecting rod (ω_PC and α_PC):

Angular velocity of the connecting rod is given by

Angular acceleration of the connecting rod is given by

(ii) Kinetic energy of the connecting rod:

We know that the kinetic energy of the connecting rod,