As I discuss in Chapter 6, you can create a mobile in SketchUp and design it in the software so that – when 3D printed or otherwise fabricated – this mobile hangs perfectly in the balance. This process works for all planar designs, independent of how many levels such a mobile may have. In this post, I’ll show you how to run the calculations for the mobile so that its equilibrium is assured.
Tutorial Steps
Step 1: Break the mobile down into individual Free-Body Diagrams (FBDs)
The previous image shows one of my recent students’ mobiles. It has a great theme and while it doesn’t push this method enough (it is very symmetrical), it helps me explain this step nicely.
From a structural perspective, you need to break down the entire mobile into its sub-structures and their respective free-body diagrams (FBDs). You basically need to separate each of its individual levels in this step. The four sketched outlines in the image show how this looks for my student’s mobile.
Once broken down this way, you would analyze this mobile from the bottom to the top, following the flow of its gravity forces – in reverse. Each FBD needs to get its own set of calculations which then allows you to determine the locations of the hanging points on all pieces.
Going back to the example from the book, which is illustrated in Chapter 6, you will find one such FBD whose important dimensions are broken out in Figure 6.14 (shown in Step 2 below). Because the book’s example is a very simple one-level mobile, only one FBD is required there. We will use those dimensions for the next step.
Step 2: For each FBD, set up and solve the equilibrium equations
In this step you will use equilibrium equations from statics to make sure your levels are in perfect balance. Unless you have a fully symmetric system, this also involves an unknown dimension: The location of the hanging point of the piece. Because you will have designed the locations of suspended pieces and the shape of the suspending piece, those hanging locations are typically known for those. However, to be in equilibrium, the point at which a piece is suspended needs to be adjusted until the equilibrium is found. While this could be achieved by trial and error, we will use statics instead to pre-determine the location.
TIP:
A nice side-effect of this approach is that you can check your calculations easily: If the calculations were wrong, the fabricated mobile will not hang in balance.
Let’s use the following equation to calculate the suspension force in this FBD. You are basically summing up all of the vertical forces (positive direction is assumed to be upward), which comprise the weights of the two pieces and an unknown suspension force. You can then easily solve this to get that unknown force.
Given that we are dealing with gravity forces (i.e. weights) here, it is important to get the accurate weights of the individual pieces. This is, however, made easy if all of the pieces are fabricated with the same material and at the same thickness. As you can see in the next set of calculations, the weight unit actually cancels out, which allows us to simply use the shapes’ areas as a representation of weight. Just note that you will need to work with actual weights if you have shapes of varying thicknesses or materials.
The next set of calculations compute the moments in this FBD. For equilibrium, any overturning moments must of course be resisted by a restoring moment. As before, we need to make some assumptions here: clockwise rotation is the positive direction and the point about which I will be taking the moments is the left centroid (belonging to the larger piece).
This then gives you the missing dimension x, at which you will need to place the hanging point so that everything is suspended in equilibrium.
As mentioned earlier, do this for each FBD and you will have computed all of the missing dimensions for your mobile.
Tips
- You can use any dimension in your FBD as the unknown variable x. It could even be one of the weights. Depending on your design, evaluate what works best in your case.
- Include the suspension point holes in your area calculations, especially if the material you are using is quite light. Their effect on the overall calculation diminishes, however, with increasing material weight. Also, remember that you can move those holes up and down on those shapes because they will remain located on the respective force’s line of action.
- Make sure the shapes are sufficiently stiff so that they don’t warp. My suggestion is to use a heavier cardboard but you can pick any material (plywood and acrylic work well, too). Also, don’t locate support points on a shape lower than any supporting points or those pieces can flip over.
- Use a thin, light (yet strong) thread so that you can ignore its weight. Monofilament (fishing line) works well.
Recap: Modeling and Centroid Finding
The video below (which I had published earlier) uses different dimensions, but it illustrates nicely how you can model this mobile in SketchUp and then use my Face Centroid and Area Properties Extension for SketchUp to find all necessary centroids and areas of the shapes.
Examples
The images below show some of my students’ mobiles from the past years. Also, if you need more inspiration for mobiles (or you just want to buy one), check out Flensted Mobiles (unaffiliated, they are just really nice).
Did you use this approach for one of your mobiles? Add a picture in the comments below.