Instantaneous centres and Kennedy's theorem

Instantaneous centre (IC)

For two bodies in relative plane motion, the instantaneous centre is the point common to both that has no relative velocity at that instant. Each body appears to rotate about it relative to the other. It is found at the intersection of the perpendiculars to the velocity directions of two points.

Number of ICs

For a mechanism of $n$ links, each pair of links has one IC:

$$N = \frac{n(n-1)}{2}$$

A four-bar mechanism ($n = 4$) has $N = 6$ ICs.

Types

• Fixed IC: its location is fixed in space (for example, the pivot of a link on the frame).
• Permanent IC: its location moves, but it is always at the pin joint between two moving links (for example, the pin between crank and coupler).
• Neither fixed nor permanent: it is not at a joint and its position changes as the configuration changes, for example the IC between the coupler and the frame.

Kennedy's theorem (three-centre theorem)

Any three bodies in relative plane motion have three instantaneous centres, and they lie on a straight line.

Proof. Let the three bodies be 1, 2 and 3, and take body 1 as fixed, so that body 2 rotates about $I_{12}$ and body 3 about $I_{13}$. Assume $I_{23}$ is not on the line joining $I_{12}$ and $I_{13}$, and call it point $P$ (a point of both body 2 and body 3). The velocity of $P$ as a point of body 2 is perpendicular to $I_{12}P$, and as a point of body 3 it is perpendicular to $I_{13}P$. These two directions are different (since the lines $I_{12}P$ and $I_{13}P$ are different), but at the IC $I_{23}$ the velocities of the two coincident points must be identical in magnitude and direction. This is possible only if $P$ lies on the line through $I_{12}$ and $I_{13}$, where the two perpendicular directions coincide. Hence all three ICs are collinear.

Four-bar mechanism ABCD (links: 1 = AD frame, 2 = AB, 3 = BC, 4 = CD)

Obvious ICs (pin joints): $I_{12}$ at A, $I_{23}$ at B, $I_{34}$ at C, $I_{14}$ at D.

Remaining two by Kennedy's theorem:

• $I_{13}$: lies on the line $I_{12}I_{23}$ (AB extended) and on $I_{14}I_{34}$ (DC extended). It is the intersection of AB and DC produced.
• $I_{24}$: lies on $I_{23}I_{34}$ (BC extended) and on $I_{12}I_{14}$ (AD extended). It is the intersection of BC and AD produced.

   I13 *
        \        .
         \   B---C
          \ /     \
        A-----------D  ---- * I24

Use: $\omega_2 \cdot I_{12}I_{24} = \omega_4 \cdot I_{14}I_{24}$ (both equal the speed of $I_{24}$), and $\omega_2 \cdot I_{12}I_{23} = \omega_3 \cdot I_{13}I_{23}$ (both equal the speed of the pin B).