First post, using NB!

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Behind the title

Linearization is an essential part of most classical engineering analyses. Say there’s a dynamic system, like a pendulum swinging from a hook. You want to describe the motion of the pendulum. Looking at the weight at the end of the pendulum; there are two forces acting on it (we’ll ignore friction to keep things simple):

  1. Gravity is pulling it down
  2. Tension in the rod is pulling it towards the pivot.

When the pendulum is hanging straight down, these two forces are in balance with each other:

todo: add image of hanging pendulum

But when the pendulum is hanging at an angle, the tension force and gravity are out of line with each other, and the pendulum begins to accelerate. Observe:

todo: add FBD of pendulum at an angle

The amount of acceleration depends on the angle of the pendulum.

$$ \mathrm{Resultant Force} = m\,g,\sin (\theta) = \mathrm{Tangential Acceleration} \times m $$

Since the angular acceleration \(\ddot{\theta}\) is equal to the tangential acceleration divided by the length \(\ell\) of the rod (imagine pushing around the end of a really long pendulum: you might be moving quite quickly, but the angle of the pendulum will be changing slowly!), we end up with an equation to define the motion of the pendulum:

$$ \ddot{\theta} = \frac{g}{\ell} \sin{\theta} $$

So! An equation for how our pendulum will evolve through time. You could plug this into a computer, tell it where the pendulum starts, and the computer will tell you how the system will evolve.

todo: add charts of swinging pendulum

Here’s the thing, though. That equation that we defined above? Its solutions are pretty nasty. You can find an analytic definition but, trust me… you probably don’t want to. However, we notice that when you have a little displacement of the pendulum, it kind of looks like a sine wave:

todo: add chart of pendulum with a little displacement vs a sine wave

So here’s the magic part: we simply linearize our equation of motion. This will make it easier for us to find a closed-form solution. We take the engineer’s magic incatation:

$$ \begin{matrix} \sin(x) &= &x \\ \cos(x) &= &1 \\ \end{matrix} $$

* Approximately true for small values of \(x\).

Then our new equation of motion is

$$ \ddot{theta} = \frac{g}{l}\theta $$

And this can be solved, in closed form, by sines and cosines:

$$ \theta (t) = A \sin\left( \sqrt{\frac{g}{l}} t \right) + B \cos\left( \sqrt{\frac{g}{l}} t \right) $$

As long as the angle of the pendulum stays pretty close to zero, then the swing of the pendulum stays pretty close to a sine wave!

The pendulum won’t stay like a sine wave at large angles. This is why, when you were a kid swinging on a swing, and you got swinging really high, it might feel like you were hovering in the air at the end of the swing. You really were hovering like that.

todo: add chart of high-angle pendulum annotated with a kid going whee

But for small displacements, the linearization holds, and the math is a lot nicer. You have a closed-form solution, and it’s a lot easier to make generalizations and understand how other similar things might behave. It wouldn’t even be a huge leap to set up a controller that holds the pendulum upside down, for example. Do I sense a project in the works?

Anyway, that’s the ethos of linearized.xyz. It’s all about making stuff conceptually simpler, to take it out of the realm of theory and into the realm of practice. When there’s math, it’ll be the interesting parts: the parts relating to real-world problems, and taking us towards real-world solutions. When in doubt:

keep it simple, stupid!