Acceleration Analysis: Coincident Points and Coriolis Component of Acceleration

Definition

In the kinematic analysis of mechanisms, coincident points refer to two points—one on a moving link and another on a sliding member—that occupy the same physical position at a specific instant. When one link rotates while a second link slides along it, the sliding point experiences an additional acceleration known as the Coriolis component of acceleration ($a_c$). This component arises because the velocity vector of the sliding point is constantly changing its direction due to the rotation of the link upon which it slides.

Main Content

1. The Concept of Coincident Points

Coincident points are used to analyze mechanisms like the Quick Return Mechanism. We consider a point 'A' on a rotating link and a point 'B' on a sliding block that are momentarily at the same location.
Because these points overlap, their radial positions relative to a fixed pivot are identical at that specific instant.

2. The Coriolis Component ($a_c$)

Coriolis acceleration occurs when a point moves along a path that is itself rotating.
Mathematically, it is defined as $a_c = 2 \omega v$, where $\omega$ is the angular velocity of the rotating link and $v$ is the sliding velocity of the point along that link.

3. Direction and Vector Representation

The Coriolis component is always perpendicular to the rotating link.
To determine the direction, rotate the sliding velocity vector $v$ by 90 degrees in the direction of the angular velocity $\omega$ of the link.

       Path of sliding (v)
            |
            |
      (Coincident Point)
      /     |     \
     /      |      \
    /       |       \
   /________|________\  <-- Rotating Link (omega)
            |
            |
      Direction of Coriolis (ac = 2*omega*v)

Working / Process

1. Identify Coincident Points

Locate the point on the rotating link (let's call it $A$) that is coincident with the slider point (let's call it $B$).
Ensure the slider is actually moving relative to the rotating link. If there is no relative motion ($v=0$), there is no Coriolis acceleration.

2. Calculate Magnitudes

Determine the angular velocity ($\omega$) of the link from the velocity polygon or given data.
Determine the relative sliding velocity ($v$) of the block along the link.
Calculate $a_c = 2 \omega v$.

3. Construct the Acceleration Polygon

Plot the fixed points and draw the acceleration vectors known (centripetal and tangential).
Incorporate the Coriolis vector as a separate line in your acceleration polygon, starting from the point of coincidence.
Close the polygon to solve for unknown accelerations in the mechanism.

Advantages / Applications

Quick Return Mechanisms: Essential for analyzing shaper machines where the cutting stroke is slow and the return stroke is fast.
Slider-Crank Variations: Used in complex engine linkages where the piston pin slides within a pivoting guide.
Precision Engineering: Allows engineers to predict vibration and stress levels in high-speed reciprocating machinery.

Summary

Coriolis acceleration is the acceleration experienced by a slider moving along a rotating guide. It is calculated as $2\omega v$ and acts perpendicular to the path of rotation.

Coincident Points: Two distinct points occupying the same space momentarily.
Coriolis Component ($a_c$): Additional acceleration caused by the combination of rotation and translation.
Key Terms: Angular velocity ($\omega$), Sliding velocity ($v$), Centripetal acceleration, Tangential acceleration.