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: I12

2

1 (ground)

Instant center, I12

2

Finding instant centers

• By inspection (e.g. a pinned joint is an instant center)

• Using rules

• Aronhold-Kennedy rule

3

Sliding body on curved surface

1

I12

Sliding body on flat surface

2

1I12 is at infinity

Rules for finding instant centers

I12 (point of contact)

Rollingwheel (noslip)

Slidingbodies

Commontangent (axis of slip)

2 3

I23

commonnormal

2

4

3

1

I13

Link is pivoting about the instant center of this link and the ground link

Link 3 rotates aboutinstant center I13

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,

number of links number of instant centers

3 3

4 6

5 10

2

1nn )(

6

Aronhold-Kennedy rule

• Any three bodies have three instant centers that are colinear

7

Instant centers of four-bar linkage

I13

2

3

4

1

I24

I12I14

I23

I34

8

Velocity analysis using instant centers (Section 3.16)

AV

I13

2

3

4

1

I12I14

A

B

BV3

4

2

Problem:

Know 2

Find 3 and 4

9

Velocity analysis using instant centers (continued)

Steps

1. Find VA, normal to O2A, magnitude= 2(O2A)

2. Find 3=length of VA/ (I13A)

3. Find VB, normal to O4B, magnitude= 3(I13B)

4. Find 4=length of VB/ (O4B)

10

Velocity ratio (Section 3.17)

BOAO

4

2

2

4

cityinput veloocityoutput vel

ratiovelocity

A3

4

1

O2 O4

A

B

2

4

B