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 parametric 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 Maths FB only as an example of how to use the Maths FB.


Add a Cam-Data FB; Plot the Radius-of-Curvature

MathsFB-RoC1

To plot the Radius-of-Curvature for a Cam, we must

1.Add a Cam-Data FB
2.Edit the Cam-Data FB to associate it with the Cam
3.Add a Graph FB
4.Drag a wire from the Radius-of-Curvature to the Graph input-connector.
5.Open the Graph FB

MathsFB-RoC2

We can see graph is nonsense.

Add the Equations in the Maths FB for the Acceleration for the coordinates of Epitrochoid Curve

You would need to differentiate the Velocity Equations we derived from the equations for the Epitrochoid Curve.

To make your life easier, here are the two parametric equations for the acceleration components:

Xacc = -(a+b) * cos(t) + ( (a+b)2 * h * cos((a+b) * t / b) ) / b2
Yacc = -(a+b) * sin(t) + ( (a+b)2 * h * sin((a+b) * t / b) ) / b2

These are the acceleration equations written in the Maths FB:

–((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)

It is important to click the 'Update' button to check the equations are valid.

MathsFB-RoC3

These are the equations in the Maths FB

Position, velocity and Acceleration for the X-axis
Position, Velocity and Acceleration for the Y-axis.

 

Radius of Curvature of the Epicycloid Curve.

Radius of Curvature of the Epicycloid Curve.

 

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?

 

MathsFB-RoC5

We can check the results by adding sketch-circles in the Base-Parts with a radius equal to the values given in the graph.

In this case I have added two Circles at the External and Internal Radii-of-Curvature.

The radius of the circles agree with the Radius of Curvature graph.

 

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