J.Norad Makes Paper Ball Joints

This post is more for personal reference than of possible use to any of you. Today, I'll be covering some parameters for designing a ball joint. A ball and socket joint provides excellent movement capabilities but isn't an easy element to build with paper.

The key to building a robust ball joint with paper is to have the appropriate materials needed. A good ball joint should provide adequate joint stiffness, but it may vary on your application.

Designing the Ball and Socket
You'll need to know a few things first:

• Figure out how big your joint is going to be.
• Figure out how much mobility you want.

Here's some calculations to help assist you in figuring out the dimensions of your ball and socket joint.

Figure 1: The Ball Joint Schematic

There's four design parameters that dictate the performance of your ball joint:

1. Ball radius (r1 in Figure 1) (You'll hopefully know this value first)
   
2. Socket depth (Hl in Figure 1) (somewhat adjustable, has a lower limit)
   
3. Support rod diameter (2P in Figure 1) (adjustable, has a lower limit)
   
4. Range of motion (phi in Figure 1) (defined by values 1-3)
   

For a given set of construction variables, r1, Hl and 2p, you can figure out how much motion your ball joint will provide. We'll use some trigonometry to solve for the angle phi, and use this to determine the total angle your joint will provide.

First, we must acknowledge that the support rod diameter 2P and the lower socket height Hl limit our angle. This is because the rod hits the edge of the socket, defined by how deep (Hl) the socket is. Second, we'll define the angle the joint provides as the angle that the center of the support rod makes with the line parallel to the bottom of the socket. This angle will be phi.

From Figure 1:

1. The red angle Beta formed from the point of contact with the socket edge (Hl) and the support rod walls is less than our desired angle, phi. Since it's a right triangle, we know from the pythagorean theorem that the length of from the center to the tip of Hl is the square root of r1^2 + Hl^2.
2. The angle formed from the line OP to the support rod centerline is the difference between angles phi and Beta. We know the thickness is 2P, and that OP forms the hypotenuse, and half the rod thickness P forms the opposite wall. Therefore, the angle Phi- Beta = arcsin(P/sqrt(r1^2 +Hl^2)).
3. From trigoneometry, the angle Beta =arctan(Hl/r1)

Solving for the angle phi, we get Phi = arctan(Hl/r1) + arcsin(p/sqrt(r1^2 +Hl^2)).

The angle we actually want is the complimentary angle to Phi, since that determines the angle relative to the null position. So we take twice the compliment to Phi (two directions) to find the angle of our joint.
Total angle range for the ball and socket joint = 2(90° - Phi)

Where Phi = arctan(Hl/r1) + arcsin(p/sqrt(r1^2 +Hl^2))

This equation tells us some obvious relations, which help support the validity of the result:

1. If Hl is longer than r1, the angle decreases
   
2. if p increases, the angle decreases
3. increasing r1 increases the angle.
   

Now for the actual construction!

The Socket
The socket should ideally consist of a durable material. 110lb cardstock will not work, as it wears out fast and easily over a few cycles. The smooth varnished surface of a Magic: the Gathering card is an excellent material. It will withstand more cycles and is fairly strong. Since paper (and Magic cards) does not have "negligible" thickness anymore (you're now working with a system that will requires a few thousandths of an inch in terms of tolerance to work well), you need to account for the overlap of paper. Magic cards have a thickness of 0.30988 mm (experimentally measured), which translates to 0.0122 in. This is enough to make your cylinder a slight oval if there's overlap. (If you consider that the accuracy of hand building has a tolerance in the range of half a millimeter anyways, it might not matter in the long run. And you can always correct for it later...)

Figure 2: The socket.

By acknowledging paper overlap, we form our socket by cutting out a strip of Magic card of a length equal to our projected ball diameter so that when we curl it up, both ends sit flush with each other, thereby eliminating the overlap. Once you have your cylinder, you can freely complete the cylinder with additional layer of Magic card without worrying as much about overlap. Removing overlap helps reduce wear of the ball and socket over time, since the raised edge is most likely to wear first.

Remember: paper is not incompressible. You'll lose a few thousandths over time. Adjust accordingly.

The Ball
The ball part is perhaps the most difficult part to build. Not also do you need to have the dimensions as close as possible for a tight fit, it needs to be built well and uniform.

First, you'll need a decent support rod to use. 1/8" (3.175 mm) diameter bamboo sticks are a good choice. They're usually $2 for a pack of 100. Pick one with low eccentricity if possible, and look for non-slivering/splintering ones. Those will snap first over time.

Next, you'll need to use the excel sheet for making cylinders. I suggest using 110lb cardstock for the ball, since it's easier to work with and tears less than printer paper. However, printer paper glued together with superglue will provide a nice solid sphere. I like to sand the ball after it's made, so I use 110lb.

Figure 3: Tapering the strip. Note the 5cm allowance before tapering.

To make the spherical shape, you need to taper the strip to a triangular shape. If using the 1/8" rod method, start the taper from 5cm from the starting edge and taper it linearly to 3mm to the other side.

Roll the paper around the rod as tight as possible. Any gaps or loosely bonded sections will result in failure in the rod axial direction, meaning it will start to deform and separate as you push it in the socket.


After you've finished, your sphere will be somewhat octagonal in cross section. Break out those calipers and sand that sphere down to as best as you can to a uniform diameter throughout. Irregularities will result in uneven performance, where certain positions are looser than others. You ideally want the ball to be a few thousandths (0.003-0.010 in) larger than the socket for a nice snug fit.

Adjusting the Fit
Your ball and socket joint may be loose or come loose over time due to thermal expansion, humidity, wear or other factors. You can easily adjust the joint to regain stiffness. Options include:

1. Adding some additional material to pad out the socket to reduce the inner diameter. I suggest using a small section of paper (printer or 110lb works, depending on the looseness) inserted into the joint
2. Thickening the ball with superglue. Make sure the ball is dried completely before re-inserting.

Miscellaneous
If there are any other adjustments or updates, I'll add them as necessary to this page.