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 anupams@iitk.ac.in FB 361 Prepared by: Abhishek Sharma Final Year Dual Degree Student Department of Mechanical Engineering Indian Institute of Technology Kanpur abhishr@iitk.ac.in FB369

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur M02- Recap Definitions Kinematic Diagrams Mobility Introduction Special Cases Examples Mechanism vs Manipulator Assignment-1 30-07-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

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 30-07-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

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 30-07-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Aronhold-Kennedy Theorem 1 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 30-07-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur