Velocity Analysis using Instantaneous Centres and Kennedy’s Theorem

Definition

An Instantaneous Centre of Rotation (ICR) is a point common to two bodies in relative motion that possesses the same linear velocity in both magnitude and direction. At any given instant, the motion of a link relative to another can be considered as pure rotation about this centre. Kennedy’s Theorem (or the Three-Centre Theorem) states that if three bodies move relative to each other, their three instantaneous centres must lie on a single straight line.

Main Content

1. Types of Instantaneous Centres

• Fixed Centres: These are centres where the velocity is zero relative to the frame of reference (e.g., the pivot point of a pendulum).
• Permanent Centres: These are centres that remain the same throughout the motion of the mechanism (e.g., the pin joint connecting two links).
• Neither Fixed nor Permanent: These centres change their position as the mechanism moves (e.g., the contact point between a rolling wheel and the ground).

2. Kennedy’s Theorem (The Three-Centre Theorem)

• If three bodies (A, B, and C) move relative to each other, there are three instantaneous centres: $I_{AB}$, $I_{BC}$, and $I_{AC}$.
• The theorem asserts that these three points must be collinear. If two centres are known, the third must lie on the line connecting them.

3. Locating Instantaneous Centres

• The total number of instantaneous centres ($N$) in a mechanism with $n$ links is calculated by the formula: $N = \frac{n(n-1)}{2}$.
• For a four-bar mechanism ($n=4$), there are $N = \frac{4(4-1)}{2} = 6$ centres.

Visual representation of a four-bar mechanism:
   Link 2 (AB)       Link 3 (BC)
      /-------------/
     /             /
  Link 1 (AD)----Link 4 (CD)
  (Fixed)

Working / Process

1. Identification of Links and Centres

• List all the links in the mechanism, including the fixed frame.
• Calculate the total number of centres ($N$) using the formula $N = \frac{n(n-1)}{2}$.
• Sketch a circle and mark $n$ points representing the links to help identify all pairs.

2. Locating Simple Centres

• Identify centres that are visible by inspection, such as pin joints or pivots.
• Mark these on the diagram. For example, in a four-bar linkage, the joints where links connect are the "simple" centres.

3. Applying Kennedy's Theorem

• Identify the remaining centres by drawing lines through the already located centres.
• If two links have a common third link, their instantaneous centre must lie on the intersection of the lines connecting the other known centres.
• Use the relation $V = \omega \times r$ (where $V$ is velocity, $\omega$ is angular velocity, and $r$ is the distance to the ICR) to solve for unknown velocities.

Advantages / Applications

• Efficiency: It provides a graphical, intuitive method to determine the velocity of any point on a link without complex vector calculus.
• Versatility: It is widely used in the design of automotive steering linkages (like the Ackermann steering gear) to ensure proper turning geometry.
• Kinematic Synthesis: Engineers use these centres to optimize the motion of robotic arms and complex machinery like presses and engines.

Summary

Velocity analysis using instantaneous centres is a powerful kinematic technique that treats complex mechanism motion as a series of simple rotations. By applying Kennedy’s Theorem, engineers can pinpoint the rotational axes of moving parts, allowing for the straightforward calculation of linear and angular velocities. Key terms to remember include Instantaneous Centre (ICR), Three-Centre Theorem, and Linkage Geometry.