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
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Before we can plot the Radius-of-Curvature for a 2D-Cam, we must add a Cam-Data FB
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We can see graph for the Radius-of-Curvature is nonsense. |
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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
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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
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Look again at the Plot for Radius-of-Curvature
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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. |
