Lissajous Plant Stand
I previously applied Lissajous figures to realise a clock and this like many other applications the display of which was 2 dimensional.
Having investigated Lissajous in 2D, it occured to me the next step was to consider Lissajous in 3D and it could lend itself to the upcoming Art and Sculpture contest.
But before embarking further I thought I would check if other people had applied this idea.
An internet search* revealed many 3D renditions but these were mainly plots, prints or animations of familiar shapes.
*A non exhaustive image search with a variety of associated keywords were the lack of hits does not exclude the existance of similar works (not published, deleted, demoted, archived etc.), but gives me a guide to the direction I should take for this project to minimise the likelyhood of direct duplication.
My train of thought was to apply 3D Lissajous to create a form being artistic yet with a practical use.
Read on to find out were the journey led.
Supplies
Filament Red PLA (although any colour and filament material that suits personal taste may be used).
Small plant (Cactus/Succulent)
Plant Pot (Size: 46(H) x 60(dia) rim mm, 8 degree) [50(dia) at rim step x 35(H) to rim step, mm], plastic with drainage holes.
Tools
3D Printer
Cura or other slicer.
Needle files
Sanding Paper
Know your tools and follow the recommended operational procedures and be sure to wear the appropriate PPE.
No affiliation to the suppliers feel free to use your preferred supplier that suits the requirements. Cannot guarantee the availability of specifically listed parts due to supply or obsolescence requiring alternatives to be sourced that may vary from those listed.
Links valid at the time of publication.
Background
Just a short recap of the history and examples.
Lissajous figures named after Jules-Antoine Lissajous (Also known as Bowditch Curves after the initial work of Nathaniel Bowditch), who devised a method to study compound vibrations. Originally employing pendulums, tuning forks and optical methods. Reflecting light from mirrors on two tuning forks vibrating at right angles will display Lisssjous figures as will other devices such as Harmonographs and Pantographs.
Lisaajous figures are used in many areas Acoustics, Electronics and Communication to name a few areas.
Different patterns will be produced by the variation in frequency and/or phase between the two input stimuli.
Graphically these can be displayed using the following:
X = A sin(W); Y = B sin(H + P)
Where A & B are the amplitude, W & H are the fundamental angles, P is the phase angle.
The frequency being determined by the number of sinewave cycles in a given period (E.g. 360 degrees)
E.g 1 (See Image 1.1)
With W maintained at a constant frequency and H = W the phase angle the same for both.
This will display a straight line inclined at 45 degrees.
(Example code using OpenSCAD)
for (i = [0 : abs(1) : 360]) {
translate([(7 * sin(i)), (7 * sin((i / 1 + 0))), 0]){
cylinder(r1=1, r2=1, h=3, center=false);
}
}
E.g. 2 (See Image 1.2)
With W maintained at a constant frequency and H at twice the frequency of W.
This will display a figure of eight.
for (i = [0 : abs(1) : 360]) {
translate([(7 * sin(i)), (7 * sin((i / 0.5 + 0))), 0]){
cylinder(r1=1, r2=1, h=3, center=false);
}
}
Although these shapes are 3D due to the thickness (Z), they are essentially the same as the 2D shapes.
Shape to Form
The previous step illustrates the creation of familiar Lissajous shapes.
However, I was looking for a more substantial self supporting form.
Essentially, this was accomplished by rotation of the shape about the origin such that multiple shapes are produced resulting in the desired form. Although, a few other steps are required to finalised the design.
Using one of the earlier shapes as an example.
The 45 degree line. (See Image 2.1)
for (i = [0 : abs(1) : 360]) {
translate([(7 * sin(i)), (7 * sin((i / 1 + 0))), 0]){
cylinder(r1=1, r2=1, h=3, center=false);
}
}
The shape is rotated in a circle (360 degrees) about the origin (x0,y0,z0).
Applying rotation to the Z axis results in a cylinder with the same height as the original shape although not applied in this application.
In this case the 90 degree rotation can be applied to either the X or Y axis resulting in a form similar to a Diabolo / Yoyo.
From a printing perspective the form needs to be rotated from it's side to it's end and is rotated by 90 degree on the X or Y axis.
But the form straddles the Z axis and needs to be raised up on this axis by half the height of the form.
The code is below and form is shown in Image 2.2.
translate([0, 0, 8]){
rotate([0, 90, 0]){
for (j = [0 : abs(1) : 360]) {
rotate([j, 0, 0]){
// chain hull
for (i = [0 : abs(2) : 360 - 2]) {
hull() {
translate([(7 * sin(i)), (7 * sin((i / 1 + 0))), 0]){
cylinder(r1=1, r2=1, h=3, center=false);
}
translate([(7 * sin((i + 2))), (7 * sin(((i + 2) / 1 + 0))), 0]){
cylinder(r1=1, r2=1, h=3, center=false);
}
} // end hull (in loop)
} // end loop
}
}
}
}
The example illustrates the basic process that will be applied to the final form.
Final Form
Regarding the final form, it was decided that this should be an open design such that the Lissajous pattern is an integrat and visible part of the structure.
In this case it was decided to create a Plant Stand for a small plant (cactus/succulent), slow growing so it will take a while before it outgrows the pot. It's actually a cutting from the plant in the Tensegrity planter.
The next consideration is the shape, style and size of the stand to suit the pot.
The base shape has a frequency which is 2.22 times the reference. (See Image 3.1)
In its current state if fully formed it would have a closed top. Verified with a plot with rotations at 0 & 180 degrees. (See Image 3.2)
Therefore, a basic shape requires 800 steps (360/0.45), each step positions a horizontally aligned cylinder, radius 1.25mm and length 4mm.
In order to create the open top 480 steps (216 degrees), are required. (See Image 3.3)
The open areas of the planter are created by separating each placement of the basic shape by 30 degrees.
The fingers are positioned such that the edge of the pot sits on the edge of the fingers.
To add stability during printing and use a circular base is included as part of the build.
Final code below and rendering in Image 3.4
scale([1.55, 1.55, 1.55]){
union(){
translate([0, 0, 51.125]){
rotate([0, 90, 0]){
for (j = [0 : abs(30) : 360]) {
rotate([j, 0, 0]){
for (i = [0 : abs(0.3) : 480]) {
inc = 0.45 * i; inb = -1 * i;
translate([((sin(inc) * 1) * 50), ((sin(inb) * 1) * 20), 0]){
cylinder(r1=1.125, r2=1.125, h=4, center=false);
}
}
}
}
}
}
translate([0, 0, 0]){
cylinder(r1=22, r2=22, h=2, center=false);
}
translate([0, 0, 0]){
cylinder(r1=22, r2=0, h=5, center=false);
}
}
}
Print Details
Print details:
Layer Height: 0.15mm
Infill Density: 100%
Shell, Wall thickness 2mm
Build Adhesion: Skirt
No supports required.
Size: 68.4(dia) x 126.4(h), pot holder cavity depth 45mm.
Weight: 86g
Can be scaled by the user to suit a different sized plant pot subject to the build volume of the 3d printer.
Post Processing
Some post processing may be required to remove aberrations in the cavities and around the edges with sanding paper and needle files.
Usage
Place the plant pot into the opening bounded by the fingers such that the rim of the plant pot sits on the tip of the fingers. The fingers of the pot holder flex slightly which will allow it to accomodate small variations in pot sizes.
The plant pot used in the project has drain holes but if using a cactus or succulent moisten rather than saturate the soil with water to minimise overflow which may discolour the plant stand overtime.
Minor water overflow can be managed by sitting the plant stand in a saucer.
Alternative options are to use a pot without drainage holes, seal the holes in the existing pot or fit the draining pot into a non draining pot.
Situate the Plant Stand in a suitable location out of strong direct sunlight which can discolour the material or in an enclosed space were the temperature exceeds the glass transition temperature of the material which could warp the stand.
Finally
Hope you enjoyed this project.