Velocity Analysis of Simple Mechanisms: Graphical Method

Definition

The graphical method for velocity analysis is a kinematic technique used to determine the linear and angular velocities of various links in a mechanism by constructing a "velocity diagram." Based on the principle that the velocity of a point on a moving link is always perpendicular to the position vector of that point relative to the center of rotation, this method provides a visual representation of velocity relationships in mechanisms like the four-bar chain or slider-crank.

Main Content

1. Relative Velocity Principle

The velocity of any point B with respect to another point A on the same rigid link is always directed perpendicular to the line joining A and B.
Mathematically, the magnitude of this relative velocity is given by $v_{BA} = \omega \times r_{AB}$, where $\omega$ is the angular velocity of the link and $r_{AB}$ is the length of the link.

2. Velocity of a Point on a Link

Every point on a rotating link has a linear velocity. If a link rotates about a fixed center, the velocity of any point on it is perpendicular to the radius connecting the point to the fixed center.
In a sliding link (like a piston), the velocity is always along the line of the guide or the path of motion.

3. Velocity Image Theorem

The velocity image of a link in the velocity diagram is always similar to the shape of the link itself in the configuration diagram.
If we have a link AB, and we draw the velocity vectors for points A and B, the vector 'ab' in the velocity diagram represents the relative velocity of B with respect to A, and it is always perpendicular to the physical link AB.

Physical Link AB:
      A
     /
    / (Link)
   /
  B

Velocity Diagram:
  a (Vector origin)
  |
  | (Vector ab perpendicular to AB)
  v
  b

Working / Process

1. Preparation of Configuration Diagram

Draw the mechanism to a suitable scale (e.g., 1 cm = 10 cm) based on the given dimensions of the links.
Mark all fixed points and identify the known velocities, such as the input angular velocity ($\omega$) of the driving crank.

2. Construction of Velocity Diagram

Select a fixed point (pole) 'o' on a drawing sheet to represent all fixed points of the mechanism.
Draw velocity vectors for known points. For example, if link OA rotates with $\omega$, calculate $v_A = \omega \times OA$. Draw a vector 'oa' perpendicular to OA, starting from 'o'.
Use the intersection of lines to find unknown points. If point B has two constraints (e.g., one from link AB and one from a slider), draw the perpendiculars from 'a' and 'b' until they intersect at the unknown point.

3. Measurement and Calculation

Measure the length of the vectors on the diagram using a scale.
Convert the vector lengths back to actual units using the velocity scale factor ($v = \text{vector length} \times \text{velocity scale}$).
To find angular velocity of any link (e.g., AB), use $\omega_{AB} = v_{BA} / \text{Length of AB}$.

Advantages / Applications

It provides an intuitive and visual understanding of how motion is transferred through a machine or mechanical linkage.
It is highly efficient for analyzing mechanisms like internal combustion engine slider-crank mechanisms and robotic arms where complex trigonometric solutions might be time-consuming.
It serves as a great verification tool for analytical results obtained through complex vector loop equations or software-based simulations.

Summary

The graphical method is a systematic approach to finding velocities in mechanisms by representing them as vectors perpendicular to their respective links. By constructing a velocity diagram, engineers can determine the linear velocity of any point and the angular velocity of any link within a system. Key terms to remember include the Velocity Pole, Velocity Image, and the Relative Velocity Principle.