I am going to make a prediction. As people start to get bored with their fidget spinners, they are going to start playing with these double pendulum fidget spinners. The normal spinner has a bearing in the center of some object such that you can hold it and spin it—moderately cool, I’ll admit. But the double pendulum spinner has two bearings with two moveable arms. Here’s how that might look:
In this case, you hold one of the bearings and then let the two arms move about in a fun and entertaining fashion. Here’s a description of how you could make one of these double pendulum fidget spinners yourself.
Besides just being entertaining, there is some serious physics at play here. Let me go over some of the coolest things about double pendulums.
Modeling the Motion of a Double Pendulum
A double pendulum has two degrees of freedom. That means that with two variables, you could describe the orientation of the whole device. Typically we use two angles—θ1 and θ2 as shown in this diagram (assuming constant length strings).
You might think that with just these two angles to determine the position it might be fairly straightforward to model the motion of this double pendulum—but no. There are really two things that make this problem difficult. First, the two strings exert forces on the two masses, but these string forces are non-constant: They change in both direction and magnitude. You can’t just use some equation to calculate these forces because they are forces of constraint, meaning they exert whatever is needed to keep the object in a particular path. For mass 1, it must stay a certain distance from the top pivot point.
The second problem is with the lower angle (θ2). This angle is measured from a vertical line but this variable by itself does not give the whole motion of the lower mass. Angle θ2 could stay at zero but the lower mass could still be moving due to the motion of mass 1. This means that the time derivatives of θ2 can be rather complicated.
In the end the best method to solve this problem is to use Lagrangian mechanics—a system that uses energy and constraints to obtain an equation of motion. For the double pendulum, Lagrangian mechanics can get an expression for angular acceleration for both angles (the second derivative with respect to time) but these angular accelerations are functions of both the angles and the angular velocities. There is no simple solution for the motion of the two masses. Really, you need to do a numerical calculation using some type of computer code to find the motion of the system.
If you want to go over all the details of getting a double pendulum solution, check out this site—it does a fairly nice job showing how to get expressions for the angular accelerations.
For my model, I am going to use Python (hopefully, you could have guessed that). Here is what I get. Just a note, you can look at and change the code. But first, just run it by pressing “play” to run and “pencil” to edit. If the model stops running, just click the “play” button again to start over.
I put some comments at the top of the code to point out the things that you might want to change. The first thing to try is starting with different initial angles of θ1 and θ2—but you can also change the value of the masses and the lengths of the strings. It’s pretty fun to watch it move around.
The double pendulum is a great example of a chaotic system. What does that even mean? Let me start with an example. Here are two double pendulums right on top of each other (well, almost). For one of the pendulums the starting angle for the lower mass is just 0.01 degrees different than the other pendulum—so they essentially start with the same initial conditions. Watch what happens as the two double pendulums swing back and forth. Again, you can click “play” to run it more than once.
If you take a plain pendulum with just one mass, then small changes to the initial conditions won’t do too much to the long term outcome of the system. However, with this double pendulum just a tiny change at the beginning gives a completely different motion after some amount of time. When any system is highly dependent on the initial conditions it is considered a chaotic system. Of course, in the real world we’re surrounded by such chaotic systems—the most famous being the weather. We can still predict the motion of a chaotic system, but it gets more and more difficult the further in the future you want to make a prediction. You can get a better prediction with more accurate initial conditions—but it’s still chaotic.
Even though a double pendulum is chaotic, we can put it into certain cases where it behaves more orderly. Let me start with one such example. Watch this:
Notice that the two masses oscillate in a predictable way. Although the two masses oscillate with different amplitudes, they have the same frequency such that they get back to the same starting place. In this case the pendulum isn’t exactly chaotic; I could find the location of the two masses at any point in the future. But wait! There’s more! Here is another normal mode for a double pendulum:
There’s a bunch of other stuff I could talk about in regards to normal modes—but for now I just wanted to show you what they looked like because they are cool.
Another Mass System
What if I replaced the strings in the double pendulum with springs? How many degrees of freedom would the system have now? Each mass could still swing back and forth so that would be two angles (and two degrees of freedom) but the springs could also move towards or away from the attachment points (two more degrees of freedom). This gives a total of four degrees of freedom. If the double pendulum is difficult to model, the double spring pendulum must be nearly impossible. Right?
Nope. It’s easier.
Consider the bottom mass (mass 2) in this spring pendulum thingy. There are essentially two forces acting on this mass. There is the gravitational force pulling down, which depends on the mass of the object and the gravitational field, and then there is the force from the spring. Both of these forces are deterministic forces—meaning you can calculate both their magnitude and direction at any instant. The spring force depends on the stiffness of the spring and the location of the two masses. Once I have the total force acting on mass 2, I can use the momentum principle to find how its momentum changes. With the momentum of mass 2, I can find out where it is after some short time interval. This the basic recipe of a numerical calculation—I don’t have to use Lagrangian mechanics to find the motion. It’s perfect for a computer to calculate.
OK, here is my double pendulum spring model. Press “play” to run it.
Now, if you take a look at the code (click the “pencil”) you should be able to see that this program is much simpler than the previous code. It’s more complicated and simpler at the same time.
If you want to play around with the code (and you should), see if you can adjust the spring constant such that this double spring pendulum starts to act like a normal double pendulum. You might have to decrease the time step to get it to behave. But really, this should work. Strings are just really stiff springs. They have to stretch a little bit when the string exerts a force. So, in a way you can take a constraint force and make it a deterministic force to make a super difficult problem just medium difficult.