Principal Vector Linkage

In this build, I combined my engineering knowledge and passion for art. The linkage is an example of a force-balanced mechanism, which makes it stable in different configurations. Force balance can be achieved by making the center of mass (COM) of the system stationary. If the COM of a mechanism is kept stationary, the mechanism will not exert reaction forces on the supports. In this sculpture, the stationary COM is desired not for removing the reaction forces, but to make it hold its configuration. This linkage is constructed using the method of Principal Vectors developed by Otto Fisher. For those interested, an overview of his work is published in English by Volkert van der Wijk and Just L. Herder (The Work of Otto Fisher and the Historical Development of His Method of Principal Vectors for Mechanism and Machine Science).

For this project, I used a 3D printer to make the parts and connected them with metal hardware. You can still build this project if you do not have a 3D printer available. However, this will require some design effort of your own as it is unlikely the weight of the components will match the printed ones. I will include a basic and simplified explanation of the theory required to balance this sculpture so you can compensate for this.

What did I use?

• 3D printer
• Black PLA filament
• Threaded rod (M4, outer diameter 4 mm)
• Aluminium tube (inner diameter larger than threaded rod)
• 32x M4x10 bolt
• 20x M4 cap nut
• M4 thread cutter
• 4mm and 3.3mm drill bits
• Iron saw
• Hex keys
• Sand

To start this project print the parts listed below, some parts need to be printed more than once. The number between brackets in the filename indicates how many you need. So for example Part1(4x) you need four times.

(Part 3 is currently 3 mm thick, during my build I noticed that it would not hurt to print this a bit thicker. So if you want a stiffer base make sure to change this dimension in your slicer before printing.)

Now you have the parts printed it is time to prepare them for the assembly process. Part 6 requires tapped holes, in order to do this first run the 3.3mm drill through the holes and then use the thread cutter to make the threads. The threads need to be placed on both sides, so make sure to use the thread cutter on both sides. Furthermore, I advise drilling with the 4mm drill through the holes in the other parts. The printed holes are mostly not the most accurate in size and it is really annoying if the thread does not go in easily while assembling.

The only thing that has to be done before the linkage can be assembled is cutting the threaded rods and aluminium to length. Below you can see the lengths you need to cut.

Aluminium Tube:

• 34mm (4x)
• 28mm (2x)
• 40mm (2x)

Threaded Rod:

• 60mm (8x)
• 56mm (2x)

Now the linkage can be assembled. Start with bolting part 1 to part 6, once one side is connected fill the space with sand and close it by bolting part 1 to the other side (you need to do this procedure twice). The subassembly you created should weigh 430 grams, if this is not the case you can throw a bit of sand out or choose to increase the mass of the top link.

Use the pictures as a reference on how to put the parts together. The green circles highlight where the 56mm threaded rods need to be used and the red circles the 60mm threaded rods. The 28 aluminium tube needs to be used where the yellow circles are placed, the 34mm tube where the orange circles are placed and the purple circles indicate where the 40mm tube goes.

If something is unclear make sure to ask a question, otherwise you are finished building your own principal vector linkage. Congrats!

In order to be balanced, the COM of the linkage needs to be in point S for all configurations of the linkage. To find the conditions for which this is the case you can do the following. First, write down the vectors from point S to the masses of the links. I neglected some links in my calculation as the mass of these can be considered insignificant. Feel free to include them in yours to get your linkage balanced even better. Now multiply each vector with the associated mass divided by the total mass and add them up, this results in a vector describing the position of the COM of the system. As we wanted to keep this COM located in S this vector should be zero vector. From this equation, you can derive the balance conditions.

For the version depicted, the balance condition is:

a1*m1 = L*m1 + L*m2