<< Click to Display Table of Contents >> Navigation: Getting Started Tutorials - MechDesigner > Tutorial 18: Math FB > Step 18.1: Position: Plot an Epitrochoid Curve |
In this step, we enter in the Math FB the parametric equations that define the X and Y coordinates for the shape of the Epitrochoid-Curve.
The Epitrochoid-Curve is a good example. We can compare the Epitrochoid Curve derived from the Math FB with the curve we can get from a Gear-Pair.
•The Epitrochoid-Curve given by a Gear-Pair will be correct - it is found by MechDesigner!
•The Epitrochoid-Curve from the Math FB will only be correct when our equations are correct.
See Tutorial 14 : Epitrochoid Curve given by a Gear-Pair.
We will add parametric equations to a Math FB to calculate separately the X and Y coordinates of the Epitrochoid-Curve. One equation will calculate the X–axis coordinates. A different equation will calculate the Y–axis coordinates. |
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The parametric equations for the family of epitrochoid curves are: Px = (a+b)*cos(Θ) – h*cos((a+b)/b)*Θ) Py = (a+b)*sin(Θ) - h*sin((a+b)/b)*Θ) Parametric-Constants a = radius of the fixed circle b = radius of the rolling circle h = distance to the point, P, from the center of the rolling circle Independent-Variable Θ = the independent variable: 0 to 360 (0 or 2×Π) |
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The parametric equations, above, give the family of Epitrochoid Curves. Thus, if you change a Parametric-Constant - a, b, h - you will plot a different Epitrochoid-Curve. To enter different the parametric-constants in the Math FB, use: Method 1 Enter the actual values for the Parametric-Constant - a, b, h - explicitly in the Math FB. For example, enter the values 110, 20, 14 for each constant. To plot a different Epitrochoid Curve, we open the Math FB dialog to edit the parameters again. Method 2 Enter a constant in a Gearing FB as the input to the Math FB To plot a different Epitrochoid Curve, edit the Gearing FB to change a parametric-constant. Or, even more efficient, add the parameters in the Gearing FB to a Design-Set. |
It is convenient to use Piggyback Sliders to represent the Parametric-Equations
The Piggyback Sliders are now in the graphic-area. |
Add a Math FB
The Math FB is now in the graphic-area. Open the Math FB dialog
The Math FB dialog is now open. |
Input-Connectors There are three(3) Parametric-Constants (a, b, h), plus the Independent Variable (Θ) STEP 1: Add four(4) input-connectors:
Output-Connectors We need two(2) output-connectors to output the motions for the X-axis Slider and the Y-axis Slider. STEP 2: Add one(1) output-connector to give a total of two(2) output-connectors:
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Output Data Type Use the Output Data Type to edit the Units at the output-connectors from the Math FB. STEP 3: Change the Output Data Type to Linear Coordinates:
Default Equations and Data-Channels and Output-Connectors
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STEP 1: Enter the Parametric Equations for the two output-connectors
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The equations in the Math FB are now: |
Define the 3 Parametric-Constants
Note: I have entered a = 120, b = 40, h = 40 <<< Gearing FB dialog : Add after Gearing Ratio for the Parametric-Constant 'a' Add the Independent-Variable
Connect the FBs
Add a Trace-Point
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Gearing FB: Add after Gearing Ratio parameter |
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Add a Design-Set and add to it the three Parametric-Constants.
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Epitrcohoid Curve - Position Equations Only
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When we choose 120 for Parameter a, we can enter 60, 40, 30, 20, 10 as factors for Parameter b. Each will give a continuous, endless, Epitrochoid-Curve. •If Parameter h = b, there is a cusp in the Epitrochoid-Curve. •If Parameter h > b, there is a loop in the Epitrochoid-Curve |