Plotting Lissajous Curves with Parametric Equations.

We only need the position equations to plot the Lissajous Curve.

These are simple parametric curves.

This topic here is only because they look nice to me!

Parameters Equations

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

One equation will calculate the X-axis coordinates. Another equation will calculate the Y-axis coordinates.

The parametric equations for the epitrochoid family of curves are:

•x = cos(a*t)
•y = sin(b*t)

Parameters and Variables

a = indicate the frequency of the Slider along the X-axis

b = identifies the frequency of the Slider along the Y-axis

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

Inputs: three parametric-constants: a, b, and the independent variable, t

Outputs: two: x and y.

The parametric equations, above, give the family of Lissajous Curves.

This means if you change a 'parametric-constants' (a, b), you will plot a different Lissajous Curve.

Inputs and Outputs

MathsFB-7MathsFB-6a

We will use Option 2.

Input-Connectors

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

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

MathsFB-8MathsFB-9a

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

Equations for the Position (Displacement) for the two outputs.

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

The parametric equations for the epitrochoid family of curves are:

•x = cos(a×t)
•y = sin(b×t)

We must re-write these for the Maths FB.

Each parameter or variable at the input is represented by:

p(0) = a

p(1) = b

p(2) = t
Parametric Equations for Lissajous Curves.

Parametric Equations for Lissajous Curves.
1.Enter the equations very carefully - see the image to the left.

Add the right-hand side of these equations.

X = cos((p(0)× 1000)× p(2))

Y = sin((p(0)× 1000)× p(2))

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

Note: The parametric constants (p(0) and p(1)) are multiplied by 1000. This is because the distances are in SI units in the Maths FB, which means 5 (outside) becomes 0.005 (inside). A more typical value is 5.

The Piggyback Slider Model for the Lissajous Figures

We give the X and Y position coordinates to two output-connectors.

We can then use a wire to:

•connect the output for the X equation to move a horizontal slider
•connect the output for the Y equation to move a vertical Piggyback Slider,

Then, a Trace-Point in the Piggyback Slider will plot the Lissajous Figures - 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-2a

The 'Variable' Inputs to the Maths FB

We will use Motion-Dimensions for Slider as the Parametric Constant

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

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

MathsFB-3a

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

Video of Lissajous Curves with changing parameters.

The Trace-Point of the Piggyback Slider

When the outputs from the Maths FB are connected to the X and Y Piggyback Sliders, and we add a Trace-Point to to show the Path, then you can see the Lissajous Curve.

The video shows a number of Lissajous Curves.

I have put the parameters 'a' and 'b' in a Design-Set to make them faster and easier to edit.

Tutorials and Reference Help Files for MechDesigner and MotionDesigner 14.2 + © Machine, Cam, Mechanism, and Motion Design Software by PSMotion Ltd