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!
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:
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. |
We will use Option 2. Input-Connectors We need three input-connectors: three parameters (a, b) and one variable (t)
|
|||||||||
Output-Connectors We need two output-connectors to give the data separately to the X and Y-axes
The units for the output-channel is dependent on the Output Data Type you specify.
|
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:
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. |
Add the right-hand side of these equations. X = cos((p(0)× 1000)× p(2)) Y = sin((p(0)× 1000)× p(2))
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. |
We give the X and Y position coordinates to two output-connectors. We can then use a wire to:
Then, a Trace-Point in the Piggyback Slider will plot the Lissajous Figures - when the equations are correct! |
|||||||||
The Mechanical model in MechDesigner Because we will use parametric equations, it is very convenient to use Piggyback Sliders |
|||||||||
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 |
|||||||||
The Maths FB - input variable connectors The Maths FB has four inputs to represent:
|
|||||||||
Maths FB - output variable connectors The Maths FB has two outputs:
|
|||||||||
![]() 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. |