Step 18.3: Acceleration: Find the Radius-of-Curvature of the Cam

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Step 18.3: Acceleration: Find the Radius-of-Curvature of the Cam

Plot the Radius-of-Curvature for the Cam we have found for the Epitrochoid Curve

A plot of a curve's Radius-of-Curvature is a good example because Radius-of-Curvature needs the Velocity and Acceleration Equations for the X and Y coordinates.

We must symbolically differentiate the Velocity Equations, given in Step 18.2, to find the Acceleration Equations.


Remember, MechDesigner does these calculations automatically with Gear-Pair. We are doing this with a Math FB only as an example of how to use the Math FB.


Plot the Radius-of-Curvature

GST-T18-3-EpiCurve-Acc-A

Before we can plot the Radius-of-Curvature for a 2D-Cam, we must add a Cam-Data FB

1.Mechanism-Editor : Click Machine elements toolbar > Add Cam-Data FB , then click the graphic-area

2.Mechanism-Editor : Open the Cam-Data dialog-box , then click the 2D-Cam. Close the Cam-Data dialog-box

The 2D-Cam is now linked with the Cam-Data FB

3.Mechanism-Editor : Click Kinematic FB toolbar > Add a Graph FB, then click the graphic-area.

4.Mechanism-Editor : Drag a wire from the Radius-of-Curvature to the Graph input-connector.

GST-T18-3-EpiCurve-Acc-B

We can see graph for the Radius-of-Curvature is nonsense.

The Parametric-Equations for the X-axis and Y-axis Acceleration Components

Differentiate the X and Y Parametric Velocity Equations for the Epitrochoid-Curve with respect to Θ.

These are the two Parametric Equations for the X and Y Velocity Components:

PXacc = -(a+b) * cos(Θ) + ( (a+b)2 * h * cos((a+b) * Θ / b) ) / b2

PYacc = -(a+b) * sin(Θ) + ( (a+b)2 * h * sin((a+b) * Θ / b) ) / b2

As before we must replace:

a with p(0) ; b with p(1) ; h with p(2) ; Θ with p(3)

In the Math FB they are:

–((p(0)+p(1))*cos(p(3)))+((p(0)+p(1))^2 * p(2) * cos((p(0)+p(1)) * p(3) / p(1)) / p(1)^2)

-((p(0)+p(1)) * sin(p(3))) + (( p(0)+p(1))^2 * p(2) * sin((p(0)+p(1)) * p(3) / p(1)) / p(1)^2)


Enter the Parametric Equations for the Acceleration Components in the Math FB

GST-T18-3-EpiCurve-Acc-C

These are the 6 equations for the two output-connectors

As entered the Math FB:

Equations 1, 2, 3: Position and Velocity, and Acceleration Equations for the X -axis

Equations 4, 5, 6: Position, Velocity, and Acceleration Equations for the Y-axis

 

Look again at the Plot for Radius-of-Curvature

GST-T18-3-EpiCurve-Acc-D

 

We can now see the the plot for the Radius-of-Curvature is as we expect.

How can we check that the result is correct?

We can check the results if we add a Circles to the Base-Part with a radius equal to the values given in the graph.