velocity analysis procedure by instantaneous centre method

VELOCITY ANALYSIS PROCEDURE BY INSTANTANEOUS CENTRE METHOD

In this method, the velocity of links in a mechanism are determined using the following steps.

Step 1: Construction of configuration diagram.

Step 2: Calculation of velocity of input link (see Section 2.5)

Step 3: Locate the required instantaneous centres (see Section 2.21)

Step 4: Determination of velocity of various links

The velocity and angular velocity of various links can be determined by measuring the distance between various instantaneous centres and using the following relations.

Example 2.12

Locate all the instantaneous centres for a four-bar mechanism shown in Fig.2.34. The lengths of various links are: AD = 125 mm; AB = 62.5 mm; BC = CD = 75 mm. If the link AB rotates at a uniform speed of 10 rpm in clockwise direction, find the angular velocity of the links BC and CD.

Given data:

AD = 125 mm = 0.125 m; AB = 62.5 mm = 0.0625 m; BC = CD = 75 mm = 0.075 m; N_AB = 10 rpm (CW)

Solution: Instantaneous centre method.

Procedure:

Step 1: Construction of configuration diagram: First of all, draw the configuration diagram, to some suitable scale (say, 1 cm = 25 mm), as shown in Fig.2.35(a).

Step 2: Velocity of input link:

Step 3: Locate the required instantaneous centres:

(ii) Write down a list of all instantaneous centres in a book keeping table, as shown below.

(iii) Locate the fixed instantaneous centres (I₁₂ and I₁₄) and permanent instantaneous centres (I₂₃ and I₃₄) by inspection and tick mark it on the book keeping table.

(iv) Locate the remaining, neither fixed nor permanent centres (I₁₃ and I₂₃), by drawing a circle diagram. Mark points 1, 2, 3 and 4 on the circle equal to the number of links in the mechanism, as shown in Fig.2.35(b).

(v) Join these points by solid lines such as 1-2, 2-3, 3-4 and 4-1 to indicate the centres I₁₂, I₂₃, I₃₄ and I₁₄ respectively.

(vi) The required other two instantaneous centres I₁₃ and I₂₄ are located by joining points 1 to 3 and 2 to 4 by dotted lines. Now all the instantaneous centres are located, as shown in Fig.2.35(a).

Step 4: Determination of velocity of various links:

Angular velocity of link BC:

Angular velocity of link CD:

Angular velocity of link CD can be obtained by using the angular velocity ratio theorem and is given by

Example 2.13

Locate all the instantaneous centres of the slider-crank mechanism shown in Fig.2.36. Find:

(i) velocity of piston, and

(ii) angular velocity of connecting rod.

Given data:

OA = 0.5 m; AB = 2 m; BOA = 45°; ω_AO = 12 rad/s (CW)

Solution: Instantaneous centre method.

Procedure:

Step 1: Construction of configuration diagram: First of all, draw the configuration diagram, to some suitable scale (say, 1 cm = 0.2 m), as shown in Fig.2.36(a).

Step 2: Velocity of input link:

Velocity of input link i.e., crank OA, v_AO = ω_AO × OA

= 12 × 0.5 = 6 m/s

Step 3: Locate the required instantaneous centres:

(ii) Write down a list of all instantaneous centres in a book keeping table, as shown below.

(iii) Locate the fixed instantaneous centres (I₁₂ and I₁₄) and permanent instantaneous centres (I₂₃ and I₃₄) by inspection and tick mark it on the book keeping table. Since the slider (link 4) moves on straight line relative to link 1, therefore its instantaneous centre I₁₄ will lie at infinity.

(iv) Locate the remaining, neither fixed nor permanent centres (I₁₃ and I₂₄), by drawing a circle diagram. Mark points 1, 2, 3 and 4 on the circle equal to the number of links in the mechanism, as shown in Fig.2.37(b).

(v) Join these points by solid lines 1-2, 2-3, 3-4 and 4-1 to indicate the centres I₁₂, I₂₃, and I₁₄ respectively.

(vi) The required other two instantaneous centres I₁₃ and I₂₄ are located by joining points 1 to 3 and 2 to 4 by dotted lines. Now all the instantaneous centres are located, as shown in Fig.2.37(a).

Step 4: Determination of velocity of various links:

(i) Velocity of piston:

(ii) Angular velocity of connecting rod AB: