Getting Started Tutorials - MechDesigner > Tutorial 18: Math FB > Step 18.1: Position: Plot an Epitrochoid Curve

Plot the Epitrochoid-Curve: Position Equations

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 from the Math FB and a curve 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.

Parameters and Parametric Equations

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.

GST-T18-1-Epitrochoid-A

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)

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 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 dialog. For example, enter the values 100, 20, 14 for each constant.

To plot a different Epitrochoid Curve, we must open the Math FB dialog to edit the parameters again.

Method 2

Enter the three constants in three Gearing FBs which are three inputs to the Math FB.

To plot a different Epitrochoid Curve, edit the Gearing FB to edit a parametric-constant.

Or, even more efficiently, add the parameters in the Gearing FB to a Design-Set.

Piggyback Sliders and Parametric Equations.

It is convenient to use Piggyback Sliders to represent the Parametric-Equations

1.Edit the Base-Part and dd a horizontal Line; Exit the Part-Editor.

Add Slider-X to move parallel with the X-axis:

2.Add Part; Add Slide-Joint between the Part and the Line in the Base-Part; Add Motion-Dimension FB to the Sliding-Part to give Slider-X.

Add a vertical Line to the Slider-X:

3.Edit Slider-X; Add a vertical Line that is parallel to the Y-axis; Exit the Part-Editor.

Add Slider-Y to move parallel with the Y-axis of the Base-Part:

4.Add a Part; Add Slide-Joint between the Part and the vertical Line in the Slider-X; Add Motion-Dimension FB to the Sliding-Part to give Slider-Y.

5.Rename the two Motion-Dimension FBs to X-axis and Y-axis.

The Piggyback Sliders are now in the graphics-area.

Add a Math FB and Open the Math FB dialog

GST-18-102

Add a Math FB

1.Click Modeling FB toolbar > Add Math FB

2.Click the graphics-area

The Math FB is now in the graphics-area.

Open the Math FB dialog

1.Double-click the Math FB

1.See How to open a dialog

The Math FB dialog is now open.

How many Input-Connectors and Output-Connectors?

GST-18-103

GST-18-104

Input-Connectors

There are three(3) Parametric-Constants (a, b, h), plus the Independent Variable ()

STEP 1: Add four(4) input-connectors:

1.Click the Add Input button four(4) times.

2.Click the Update button.

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:

1.Click the Add Output button one time, to give a total of two output-connectors.

2.Click the Update button.

Output Data Type

Use Output Data Type to change the units at the output-connectors from the Math FB. to Linear Coordinates.

STEP 3: Change the Output Data Type to Linear Coordinates:

1.Click the Output Data Type drop-down arrow.

2.Select Linear Coordinates (at the top of the list).

3.Click the Update button

Default Equations, Data-Channels, Input-Connectors and Output-Connectors

Input-Connectors

Numbers

The input-connector numbers start at Zero (0) for the top input-connector.

The input-connectors numbers, from the top, are therefore : 0, 1, 2, ... .

Data-Channels

All wires that connect Function-Blocks that control motion-values have three Data-Channels.

The Data-Channels labels in the equations for the Input-Connectors

•Parameter label of input Data-Channel 1 is “p” for Displacement, m

•Parameter label of input Data-Channel 2 is “v” for Velocity, m/s

•Parameter label of input Data-Channel 3 is “a” for Acceleration, m/s/s

We combine the Input-Connector Number (0,1,2, ...) and the Data-Channel parameter label (p,v,a) as:

p(0), v(0), a(0) - are the three Data-Channels of the top input-connector

p(1), v(1), a(1) - are the three Data-Channels of the next input-connector

Output Equations

Each equation evaluates one Data-Channel for one Output-Connector.

Because each Output-Connector has three Data-Channels, there are three equations that evaluate the top output-connector.

Numbers

The output-connector numbers also start at Zero (0) for the top input-connector.

In the equations, we prepend the output-connector number with the label “Q”.

In the equations, the Output-Connector Numbers, from the top, are therefore : Q0, Q1, Q2, ... .

Data-Channels

All wires that connect Function-Blocks that control motion-values have three Data-Channels.

The Data-Channels labels in the equations for the Output-Connectors

•Parameter label of output Data-Channel 1 is Dis. for Displacement, m

•Parameter label of output Data-Channel 2 is Vel. for Velocity, m/s

•Parameter label of output Data-Channel 3 is Acc. for Acceleration, m/s/s

We combine the Output-Connector Number (Q0, Q1, Q2, ...) and the Data-Channel parameter label (Dis,Vel,Acc) as:

Q0(Dis.), Q0(Vel.), Q0(Acc.) - the three Data-Channels of the top output-connector

Q1(Dis.), Q1(Vel.), Q1(Acc.) - the three Data-Channels of the one down from the top output-connector

POSITION Parametric Equations in the Math FB

STEP 1: Enter the Parametric Equations for the two output-connectors

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)

We must re-write these equation with the correct syntax for the Math FB.

Each of the wires we connect to an input-connector has a wire-number (0, 1, 2, 3) and a data-channel (p, v a).

We use the p data-channel for the parameters. The v data-channel of a constant is equal to zero(0). Therefore, in the parametric equations we:

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

1.Equation #0 is now ...

(p(0)+p(1))*cos(p(3))-(p(2)*cos(((p(0)+p(1))/p(1))*p(3)))

2.Click the Update button

3.Equation #3 is now ...

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

4.Click the Update button

The equations in the Math FB are now:

GST-T18-1-EpiCurve-3

Function-Blocks for the Parametric-Constant and the Independent Variable

GST-T18-1-EpiCurve-5

Define the 3 Parametric-Constants

1.Mechanism-Editor: Click Kinematic FB toolbar > Add Gearing FBs - 3 x

2.Mechanism-Editor: Rename each Gearing FB to a, b, and h - see image to the left.

3.Mechanism-Editor: Edit each Gearing FB 'a'

4.Gearing FB dialog: Enter a value for the Add after Gearing Ratio parameter,

5.Close the Gearing-FB

Do 3-5 again and again to enter values also for the Parametric-Constant b, and h.

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

1.Mechanism-Editor: Click Kinematic FB toolbar >Add Linear Motion FB

The output from a Linear Motion FB increases from 0 to 360.

It will represent the angle, Θ, the Independent-Variable,

See schematic image at the top of this topic.

Connect the FBs

1.Mechanism-Editor: Drag wires from the output of Gearing FB to the respective input-connectors of the Math FB.

2.Mechanism-Editor: Drag a wire from the output of the Linear-Motion FB to the bottom input-connector of the Math FB.

3.Mechanism-Editor: Drag a wire from the Math FB for the X-axis to the input-connector of the Motion-Dimension FB for the Slider that will move parallel with he X-axis.

4.Mechanism-Editor: Drag a wire from the Math FB for the Y-axis to the input-connector of the Motion-Dimension FB for the Slider that will move parallel with he Y-axis.

Add a Trace-Point

1.Mechanism-Editor: Click Kinematic elements toolbar > Add Trace-Point

2.Mechanism-Editor: Click a Point on Slider-Y of the Piggyback Slider

3.Mechanism-Editor: Click in the Command-Manager.

Gearing FB: Add after Gearing Ratio parameter

GST-T18-1-EpiCurve-6

Add a Design-Set - select the 3 Parametric-Constants

GST-T18-1-EpiCurve-8

Add a Design-Set and add to it the three Parametric-Constants.

1.Mechanism-Editor: Click Modeling elements toolbar > Add Design-Set, and click the graphics-area.

2.Mechanism-Editor: Open the Design-Set dialog

3.Design-Set dialog: Add three Element Rows

4.Design-Set dialog: Click one Element-Row ; Click a Gearing FB - for Parametric-Constant a

Select Property for Design-Set interface opens - see image to the left. In the interface:

5.Select Property for Design-Set : Select OutputStartUnits: Add after Gearing Ratio

6.Select Property for Design-Set: Click to close the Select Property for Design-Set interface.

Do 4 -6 again, and again, to add Parametric-Constants b and h

Video - to show the Epicycloid-Curve family

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