Consider a slider crank mechanism as shown:

> The crank OA is moving with uniform angular velocity ω radians/second in the counter-clockwise direction.

> At point B, the slider moves on the fixed guide G.

> AB is the coupler joining A and B.

We are interested in finding the velocity of the slider at B.

We can write the velocity vector equation for point B as:

$\boldsymbol{v_{bo} = v_{ba} + v_{ao}}$

$\boldsymbol{v_{bo}}$ is the velocity of B relative to O.

$\boldsymbol{v_{ba}}$ is the velocity of B relative to A.

$\boldsymbol{v_{ao}}$ is the velocity of A relative to O.

Since the points and G are on a fixed link with zero relative velocity between them,

$\boldsymbol{v_{bo} = v_{bg}}$

So, in the velocity vector equation for the point B, we can replace the velocity of B relative to O by the velocity of B relative to G.

$\boldsymbol{v_{bg} = v_{ba} + v_{ao}}$

To draw the velocity diagram, we will represent $\boldsymbol{v_{bg}}$ by the vector $\boldsymbol{gb}$, $\boldsymbol{v_{ba}}$ by the vector $\boldsymbol{ab}$ and, $\boldsymbol{v_{ao}}$ by the vector $\boldsymbol{oa}$.

$\boldsymbol{gb = ab + oa}$

$\boldsymbol{v_{ao}}$ is completely known,

$\boldsymbol{v_{ao} = \omega.|OA|}$ ; ⊥ to OA

We draw the velocity vector $\boldsymbol{v_{ao}}$ equal to its magnitude $\boldsymbol{\omega.OA}$ at some convenient scale with its root at point and its direction perpendicular to the link OA.

$\boldsymbol{v_{ba}}$ is not known, but AB being a rigid link, can’t move relative to A along the link AB. Therefore, its velocity direction can only be perpendicular to the link AB. So, in the velocity diagram, we draw a line through point A perpendicular to the link AB.

To locate the point $\boldsymbol{b}$ on this line, we draw a line parallel to the motion of the slider through the point G. This line intersects the previous line and the point of intersection gives us the point $\boldsymbol{b}$.

We indicate the directions on the velocity diagram in accordance with the vector equation, $\boldsymbol{v_{bg} = v_{ba} + v_{ao}}$

The vector $\boldsymbol{gb}$ indicates the velocity of the slider B relative to the guide G.

The slider moves towards left as indicated by the vector $\boldsymbol{gb}$.

> As indicated by the vector diagram, for the given configuration, the coupler AB has an angular velocity in the clockwise direction.

$\boldsymbol{\omega_{ba} = \frac{|v_{ba}|}{|AB|}}$, clockwise direction

We can measure $\boldsymbol{v_{ba}}$ from the velocity diagram.

This completes the velocity analysis of the slider crank mechanism.

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Velocity Analysis – Slider Crank Mechanism (Graphical Method)