Plotting the Epitrochoid Curve: Position Equations

In this Tutorial, we enter, in the Maths FB, the kinematic equations for the 'Epitrochoid Curve'.

To plot a curve, we need the position equations only.

The Epitrochoid Curve is a good example. We can compare the curve we get with the equations we enter in the Maths FB with the curve we can get with a Gear-Pair, as provided in MechDesigner.

The Epitrochoid Curve given by a Gear-Pair will be correct - it is found by MechDesigner!
The Epitrochoid Curve given by our equations will only be correct when our equations in the Maths FB are correct!

See Tutorial 14 and Epitrochoid Curve given by a Gear-Pair.

Parameters Equations

We will use parametric equations in the Maths FB to calculate the X,Y coordinates of the Epitrochoid Curve.

One equation will calculate the Xaxis coordinates. A different equation will calculate the Yaxis coordinates.

Epitrochoid

The parametric equations for the epitrochoid family of curves are:

x = (a+b)*cos(t) – h*cos((a+b)/b)*t)
y = (a+b)*sin(t) - h*sin((a+b)/b)*t)

Note: Replace 'a+b' with 'a-b' to give a Hypotrochoid Curve.

Parameters and Variables

a = radius of the fixed circle: parameter 1

b = radius of the rolling circle: parameter 2

h = distance to the point, P, from the centre of the rolling circle: parameter 3

t = the independent variable: 0 to 360 (0 or 2*pi)

Inputs: three 'parametric-constants': a, b, h ; and one variable: t

Outputs: two: x and y.

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.

You can use two different procedures to enter different values for the parametric-constants in the Maths FB.

Procedure 1:

Enter the equations, but enter the values for the parametric-constants - a,b and h - explicitly. For example, enter the values 110, 20, 14 for each constant.

In this case, when you want to plot a different Epitrochoid Curve, you must open the Maths FB dialog-box to edit the parameters..

Procedure 2:

Enter the equations, but use a separate wire and input-connector to the Maths FB for each parametric-constant.

In this case, to plot a different Epitrochoid Curve, you edit the input to the Maths FB to change a parametric constant.

We recommend Procedure 2 when you want to experiment with the parametric-constants. You can easily combine this method with a Design-Set FB.

Input and Output-Connectors

MathsFB-7MathsFB-6

We will use Procedure 2.

Input-Connectors

We need four input-connectors: three parameters (a, b, h) and one variable (t)

1.Double-click the Maths FB to open a Maths FB dialog-box
2.Click the Add Input button four times to add four inputs to the Maths FB
3.Click the 'Update' button at the bottom of the Maths FB dialog-box

MathsFB-8MathsFB-9

Output-Connectors

We need two output-connectors to give the data separately to the X and Y-axes

1.Double-click the Maths FB to open the Maths FB dialog-box
2.Click the Add Output button two times to add two outputs to the Maths FB in the graphic-area

The units for the output-channel is dependent on the Output Data Type you specify.

3.Set the Units to Linear Units
4.Click the 'Update' button at the bottom of the Maths FB

Parametric Equations in the Maths FB

Position Equations: X and Y

The Epitrochoid Curve equations for the X and Y coordinates are given here again:

The parametric equations for the family of Epitrochoid Curves are:

x = (a+b)*cos(t) - h*cos((a+b)/b)*t)
y = (a+b)*sin(t) - h*sin((a+b)/b)*t)

We must re-write these in the correct syntax in the Maths FB.

Each parameter or variable at the input is represented by:

p(0) = a

p(1) = b

p(2) = h

p(3) = t

MathsFB-10

1.Enter the equations exactly as follows...

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

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

I frequently enter the parameters without the brackets, then add the brackets last.

2.Select the Output Data Type as Linear Coordinates
3.Click the Update button to confirm changes.

The Piggyback Slider Model and Epitrochoid Curve

We connect the two output-connectors of the Maths FB to the input-connectors two Motion-Dimension FBs.

1.Connect the output that is the X equation to a Motion-Dimension FB to move a horizontal slider
2.Connect the output that is the Y equation to a Motion-Dimension FB to move a vertical Piggyback Slider
3.Add a Trace-Point to the Piggyback Slider.

The Trace-Point plots the Epitrochoid Curve - when the equations are correct!

MathsFB-1

The Mechanical model in MechDesigner

Because we will use parametric equations, it is very convenient to use Piggyback Sliders

MathsFB-2

The 'Variable' Inputs to the Maths FB

We will use Motion-Dimensions of a Slider as the Parametric Constant.

The Base-Value of each Motion-Dimension FBs can define a, b, or h.

The Linear Motion FB can be used as the angle, t, as the independent variable

Note:

You can also use the output-connector of a Gearing FB.

I find it better to use Motion-Dimensions. I can see their values in the graphic-area.

MathsFB-3

The Maths FB - input variable connectors

The Maths FB has four inputs to represent:

a ... the radius of the fixed circle
b ... the radius of the rolling circle
h ... the radius of the point from the centre of the rolling circle
t .... angle of the centre of the rolling circle from 0° to 360º

MathsFB-4

Maths FB - output variable connectors

The Maths FB has two outputs:

x ... the horizontal position of the point, P, at machine angle, t, relative to the centre of the fixed circle
y ... the vertical position of the point, P, at machine angle, t, relative to the centre of the fixed circle

Epitrcohoid Curve with Position Equations Only

The Trace-Point of the Piggyback Slider

When you connect the Maths FB output-connectors to the X and Y Piggyback Sliders, and we show the a Trace-Point, then you can see the Epitrochoid Curve.

You can edit the Base-Values of the Slider Motion-Dimensions to change the radius, the number of 'loops', and distance 'h' from the centre of the rolling circle to the point, P.

Click the video to watch the sliders 'plot' the Epitrochoid Curve and the curve update as I edit the parameter: 'h'.

Tutorial and Reference Help Files for MechDesigner and MotionDesigner 13.2 + © Machine, Mechanism, Motion and Cam Design Software by PSMotion Ltd