Relative Velocity Method

Definition

The Relative Velocity Method is a graphical technique used in the kinematic analysis of mechanisms to determine the velocity of various points on a moving link. It is based on the principle that the velocity of any point on a rigid link with respect to another point on the same link is always perpendicular to the line joining the two points.

Main Content

1. Velocity of a Point on a Rigid Link

• When a link rotates about a fixed center, any point on it moves in a circular path.
• The absolute velocity of a point is always directed perpendicular to the radius of rotation of that point.

2. Relative Velocity Principle

• If we consider two points A and B on a rigid link, the velocity of B with respect to A ($V_{BA}$) is the velocity of B as seen by an observer sitting on A.
• Mathematically, $V_B = V_A + V_{BA}$. The vector $V_{BA}$ must be perpendicular to the line joining points A and B.

3. Velocity Image Principle

• If a mechanism has a series of links, their velocities can be represented in a "velocity polygon."
• The shape formed by the velocity vectors of the points in the mechanism is similar to the configuration of the mechanism itself, rotated by 90 degrees.

       Link AB
      /
     / 
    A-------B

    Velocity Representation (Perpendicular):

       v_ba (Perpendicular to AB)
       ^
       |
    a--+---b

Working / Process

1. Drawing the Configuration Diagram

• Draw the mechanism to a suitable scale in its given position.
• Clearly label all fixed points and the points for which velocity needs to be calculated.

2. Establishing Velocity Vectors

• Identify fixed points (points with zero velocity), which act as the pole of the velocity polygon.
• Calculate the known velocity of the input link using $v = \omega \times r$. Draw this as a vector starting from the pole.

3. Constructing the Velocity Polygon

• Draw lines perpendicular to the links for points with unknown velocity directions.
• Where these lines intersect defines the position of the point in the velocity diagram.
• Measure the length of the vector from the pole and multiply by the scale factor to find the actual magnitude.

Advantages / Applications

• It provides a visual and intuitive way to understand the motion of complex mechanisms like four-bar chains and slider-crank mechanisms.
• It is highly effective for solving kinematic problems where analytical (calculus-based) methods become mathematically cumbersome.
• Used extensively in the design of internal combustion engines, robotic arms, and industrial automated machinery to ensure correct timing and speed.

Summary

The Relative Velocity Method is a geometric approach to analyzing the motion of mechanisms by representing velocities as vectors perpendicular to link positions. It simplifies the study of complex machinery by transforming link geometry into a velocity polygon. Important terms to remember include the Velocity Polygon, Fixed Pole, and Relative Velocity vector.