Chapter 6: Velocity Analysis

Chapter 6: Velocity Analysis

Velocity - Its more than speed!

    Velocity analysis is a crucial part in characterizing a mechanism's motion. Velocity analysis can be used to determine any point on a linkage. This is done by differentiating the position vector of a vector that points from an origin of a coordinate system to the point that velocity is desired. 

Figure 1. A point on a crank that velocity analysis is performed on.

Velocity analysis on this crank would begin with position analysis, where the position vector would be written as Rpa = pe^(jθ). Differentiation of the position vector involves pulling out an omega and an imaginary term, in which it would be rewritten as Vpa = pwje^(jθ). To get the x and y components of the velocity vector, the Euler identity is substituted into the velocity vector. For the figure above, this is written as Vpa = pw(-sinθ +j cosθ) . The presence of the j term causes the sine and cosine terms to swap in the velocity equation, which is further proof of the rotation of the velocity vector with respect to the position vector. These velocity vectors can be found at any point on the mechanism using this process. It is often useful to draw these on the mechanism in order to determine the orientation of the angular velocities. Graphically, this looks like Figure 2.

Figure 2. Velocity vectors drawn on a fourbar mechanism.

A velocity difference can be found as the difference in the velocities on two bodies, For example, the Va vector and Vb vector can be drawn in a velocity triangle. The line that connects the third side of the triangle is the Vba vector, the velocity difference vector of Va and Vb.