Step 18.1: Position: Plot an Epitrochoid Curve

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Step 18.1: Position: Plot an Epitrochoid Curve

Plotting 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 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.

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 Xaxis coordinates. A different equation will calculate the Yaxis 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×pi)

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.

There are two procedures to enter different the parametric-constants in the Math FB.

Procedure 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-box to edit the parameters.

Procedure 2:

Specify 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.


Piggyback Sliders and Parametric Equations.

GST-T18-1-EpiCurve-4

The Mechanical model in MechDesigner

Because we will use Parametric Equations, it is very convenient to use the standard model of Piggyback Sliders.

1.Edit the Base-Part, to add a horizontal Line; Exit the Part-Editor

2.Add the X-SliderRed-14-1b to move parallel with the X-axis, Do: Add a Part; Add a Slide-Joint between the Part and the horizontal Line in the Base-Part; Add a Motion-Dimension FB to the Sliding-Part to make it the X-Slider.

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

4.Add the Y-SliderRed-14-2 to move parallel with the Y-axis of of the Base-Part. Do Add a Part; Add a Slide-Joint between the Part and the vertical Line in the X-Slider; Add a Motion-Dimension FB to the Sliding-Part to make it the Y-Slider.

5.Rename the two Motion-Dimensions to X-axis and Y-axis, as appropriate.

The Piggyback Sliders are now in the graphic-area.


Add a Math FB and Open the Math FB dialog-box

GST-18-1-addmathfb

Add a Math FB

1.Click Modeling FB toolbar > Add  Math FB

2.Click the graphic-area

The Math FB is now in the graphic-area.

Open the Math FB dialog-box

1.Double-click the Math FB

- or -

1.See How to open a dialog-box

The Math FB dialog-box is now open.


How many Input-Connectors and Output-Connectors?

GST-T18-1-EpiCurve-0

GST-T18-1-AddMathFB-Connectors

Input-Connectors

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

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

1.Click the Add InputRed-14-1 button four(4) times to add four(4) input-connectors

2.Click the UpdateRed-14-2 button


Output-Connectors

We need a total of two(2) output-connectors as we will calculate parametrically the motion of the X-axis Slider and the Y-axis Slider

There is one output-connector when we add the Math FB to the graphic-area.

STEP 2: Add one(1) output-connector (a total of two(2) output-connectors):

1.Click the Add OutputRed-14-3 button one(1)

2.Click the UpdateRed-14-2 button


Output Data Type

The Units at the output-connectors from the Math FB are set with the Output Data Type.

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

1.Click the Output Data Type drop-down handle


Drag the right of the Math FB dialog-box to see the Output Data Type drop-down handle.


2.Select Linear CoordinatesRed-14-4 - at the top of the list.

3.Click the Update buttonRed-14-2


Default Equations and Data-Channels and Output-Connectors

Note - See image above for Red-14-5Red-14-6Red-14-7Red-14-8


Red-14-8 # of Equations = 3 × # of Output-Connectors

3 Equations × Red-14-6 2 Output-connectors = Equations: 0 - 5

Red-14-7 Data-Channels = 3 for each Output-Connector

Dis - Linear Displacement, m

Vel - Linear Velocity - m/s

Acc - Linear Acceleration, m/s/s

The units are always SI units

Red-14-5 The default equations for the output-connectors 0 and 1 are:

p(0),v(0),a(0) - three(3) Data-Channels connected to input-connector 0

POSITION Parametric Equations in the Math FB

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 four(4) parameter 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:

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

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 to confirm the changes

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 to confirm changes.

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-box: 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-box : 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 and it is .

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

See schematic image at the very 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 Y-Slider of the Piggyback Slider

3.Mechanism-Editor: Click OK-tiny-13-17 in the Command-Manager.

Gearing FB: Add after Gearing Ratio parameter

Gearing FB: Add after Gearing Ratio parameter

GST-T18-1-EpiCurve-6

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

GST-Icon-FB-DesignSet

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 graphic-area.

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

3.Design-Set dialog: Add three Element Rows, select one Element-Row

4.Design-Set dialog: Click a Gearing FB - for Parametric-Constant a

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

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

6.Select Property for Design-Set: Click OK-tiny-13-17 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

 

Epitrcohoid Curve - Position Equations Only

 

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 loop in the Epitrochoid-Curve.