1. THOERY OF MECHANISMS AND MACHINES
Module-03
Instantaneous Center
Instructed by:
Dr. Anupam Saxena
Associate Professor
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
FB 361
Prepared by:
Abhishek Sharma
Final Year Dual Degree Student
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
FB369

2. Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur
M02- Recap
Definitions
Kinematic Diagrams
Mobility
Introduction
Special Cases
Examples
Mechanism vs Manipulator
Assignment-1
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

3. Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur
Instantaneous Center B 𝒗𝐵 A 𝒗𝐴 I
(2)
A link/body in motion has a point with zero instantaneous velocity about which the whole link/body seems to be rotating momentarily. This point is called instantaneous center for zero velocity. If I represents the instantaneous center for zero velocity and A & B are two points on the same rigid body with known velocities, then 𝒗𝐴 = 𝒗𝐼 + 𝝎 𝑥 𝒓𝐴/𝐼 and 𝒗𝐵 = 𝒗𝐼 + 𝝎 𝑥 𝒓𝐵/𝐼 (1) Where 𝝎 is angular velocity of the body. Since 𝒗𝐼 = 0, therefore 𝒓𝐴/𝐼 ⊥ 𝒗𝐴 and 𝒓𝐵/𝐼 ⊥ 𝒗𝐵 (2) And if 𝒗𝐴 ∥ 𝒗𝐵 , then |𝒗𝐴| |𝒗𝐵| = |𝒓𝐴/𝐼| |𝒓𝐵/𝐼| (3) A B 𝒗𝐵 I
(3) 𝒗𝐴
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

4. Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur
Instantaneous Center 1 2 3 4
P12
P23
P34
P14
𝑷𝒊𝒋 represents a point on body 𝑖, which is the relative instantaneous center of rotation of body 𝑗 with respect to body 𝑖 Also, 𝑷𝑖𝑗= 𝑷𝑗𝑖 , and 𝒗𝑃𝑖𝑗= 𝒗𝑃𝑗𝑖 Proof (refer to the diagram on bottom right): 𝒗𝐵 = 𝒗𝑃𝑗𝑖 + 𝝎𝑗𝑥 𝒓𝐵/𝑃𝑗𝑖 𝒗𝐵 = 𝒗𝑃𝑖𝑗 +𝝎𝑖𝑥 𝒓𝐵/𝑃𝑖𝑗+𝒗𝑟𝑒𝑙 Where, 𝒗𝑟𝑒𝑙= 𝝎𝑗−𝝎𝑖 𝑥 𝒓𝐵/𝑃𝑖𝑗 Using above equations, 𝒗𝑃𝑖𝑗= 𝒗𝑃𝑗𝑖 But, 𝒂𝑃𝑖𝑗≠ 𝒂𝑃𝑗𝑖
Pij = Pji B j i
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

5. Aronhold-Kennedy Theorem1 2 3
P12
P23
P34
P14 4 ∞
P13
P24
If three bodies (whether or not connected) are in relative motion with respect to one another, then their relative ICs are collinear i.e. if body 1, 2 & 3 are in reltive motion wrt one another, their relative ICs 𝑷13, 𝑷12 & 𝑷23 will be collinear 1 2 3 4
P12
P23
P34
P14
P13
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur