1. 1 Instant center - point in the plane about which a link can be thought to rotate relative to another link (this link can be the ground) An instant center is either (a) a pin point or a (b) two points - - one for each body -- whose positions coincide and have same velocities. Instant centers of velocity (Section 3.13) Link 1 (ground) 2 Instant center: I 12 2 1 (ground) Instant center, I 12

2. 2 Finding instant centers By inspection (e.g. a pinned joint is an instant center) Using rules Aronhold-Kennedy rule

3. 3 Sliding body on curved surface 1 I 12 Sliding body on flat surface 2 1 I 12 is at infinity Rules for finding instant centers I 12 (point of contact) Rolling wheel (no slip) Sliding bodies Common tangent (axis of slip) 2 3 I 23 common normal 2

4. 4 3 1 I 13 Link is pivoting about the instant center of this link and the ground link Link 3 rotates about instant center I 13

5. 5 For each pair of links we have an instant center. Number of centers of rotation is the number of all possible combinations of pairs of objects from a pool of n objects,

6. 6 Aronhold-Kennedy rule Any three bodies have three instant centers that are colinear

7. 7 Instant centers of four-bar linkage I 13 2 3 4 1 I 24 I 12 I 14 I 23 I 34

8. 8 Velocity analysis using instant centers (Section 3.16) I 13 2 3 4 1 I 12 I 14 A B 33 44 22 Problem: Know  2 Find   3 and  4

9. 9 Velocity analysis using instant centers (continued) Steps 1.Find V A, normal to O 2 A, magnitude=  2 (O 2 A) 2.Find  3 =length of V A / (I 13 A) 3.Find V B, normal to O 4 B, magnitude=  3 (I 13 B) 4.Find  4 =length of V B / (O 4 B)

10. 10 Velocity ratio (Section 3.17) 3 4 1 O2O2 O4O4 A B 22 44