4.3 INSTANTANEOUS CENTERS OF ROTATION

---

À  p

If we look at any body in a mechanism it will be rotating and translating.

And, if we pick the right point to watch, it will appear to be rotating, we will call this the center of rotation.

And, because the translation and rotation often changes as the mechanism moves through different positions we say that it is instantaneous.

For a binary link the instant center is defined by the velocities at the joints.

The easiest way to find an instantaneous center is to find the intersection (P) of vectors perpendicular to the velocity vectors at the joints (A and B).

Try the example below,

We can also find an instantaneous center for a chain of links.

If we consider an entire mechanism with `n' links, it will generally have `N' instantaneous centers.

We can help keep track of which centers are to be found using points mapped out on a circle. In this technique we lay out a circle, and then put on a point for each of the links (including the ground). Draw on lines to connect each point where links exist. Also draw on lines for the Kennedy theorem.

Try the example below,

The instantaneous center has no velocity (VP=0), but there is also a rotational velocity about this point. This can be found using the relationship,

4.3.1 Aronhold-Kennedy Theorem of Three Centers

(|

The theorem states that for three rigid links in a mechanisms, the instantaneous centers for each of the three mechanisms lie on the same straight line.

We can consider a linkage with three members as pictured below,

4.3.2 Some Examples of Instantaneous Centers

(|

Consider the following mechanism - use the Aronhold-Kennedy theorem,

Consider the following example. Find all of the instant centers.

Find the instant centers for the mechanism below,

4.3.3 The Angular-Velocity-Ratio Theorem

(|

"The angular velocity ratio of any two bodies in planar motion relative to a third body is inversely proportional to the segments into which the common instant center cuts the line of centers"

In general, after finding the instant centers, we can find rations between angular velocities using,

4.3.4 Mechanical Advantage

(|

Keeping in mind that energy is conserved we can develop equations for mechanical advantage.

These two relationships can then be used with the results of earlier calculations to find the mechanical advantage.

4.3.5 Freudenstein's Theorem

(|

If in a 4 bar linkage we find instant centers for adjacent links (i.e. 1 & 3 and 2 & 4). These form a collinear axis if there is a perpendicular to the connecting rod then that position has the minimum mechanical advantage.

4.3.6 Centrodes

(|

Instant Centers are valid for only a single point in time. If we watch the locations of these centers evolve, we will see them move through space. Below the centrode for the four bar linkage shows how P13 sweeps through space as the mechanism moves.

In the example above the centrode is called `fixed'. If we had used the kinematic inversion where 3 was the frame, then the instant center would trace out another called a `moving' centrode.