A return on the fundamental theorem of engineering

Good evening peeps.

Tonight, I'll be posting some kind of rant(?) about one of the things that bothered me the most when doing physics and control engineering...

The linearization of equations by Taylor expansion.

So, what exactly is the fundamental theorem of engineering?

It's an ongoing joke between mathematicians, physicians and engineers that provides the two following properties:

i)  e = π = √g = 3

ii) sin x = x and cos x = 1 for any x

Here's an application of the fundamental theorem of engineering.

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Simple pendulum problem

A pendulum M of mass m is hanging at the end of a weightless, tight and inextensible rope of length L = 1m and fixed at a horizontal at the point O. Its starting angle is θo >0 and it is released without any initial speed.

System: Pendulum M (of mass m)

Frame of reference: bound to the , assumed to be galilean

Forces applied on the system : The weight P and the tension T, as shown in the picture above. We will neglect air friction in this part.

With Newton's second law, projected on the uθ vector:

Now, let ω0 = √(g/L) (with g = 9.8 m.s-2). By putting both on the same side and dividing by mL, you get:

Now, what happens since this equation is not linear (because of the sinθinstead of θ) ? How do you solve the problem?

Well, engineers will linearize it.

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According to the fundamental theorem:

sinθ = θ

Thus, the equation can be rewritten as:

And here, thanks to League-senpai's posts about linear differential equations, we know that the solution of the equation is under the form:

Now, with the initial conditions: at t = 0, θ(0) = θo = A

Now, as you differentiate this:

Since the pendulum starts without any initial speed, evaluating at t = 0 should give you B = 0.

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So, now...

Okay, this is the only solution to our differential equation that matches the initial conditions.

Now, here's a freshly made graph for that function, with a starting angle of π/2:

Okay, it sounds fairly satisfying. The pendulum oscillates around the position θ = 0 with a sinusoidal movement.

.... Is it good enough?

Let's try with a bigger starting angle.

So, what's wrong with it?

The very fundamental theorem of engineering.

Without simplifying sin θ = θ, a numeric resolution of the non linear differential equation (using an algorithm I will talk about another time) gives this...

... Yeah. The model is pretty much f***ed.

With an even bigger starting angle of 3 rad (for both cases, blue for the simplified solution and orange for the actual solution):

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Partial conclusion

All that stuff, to say what?

Dear physicists, dear engineers, and dear mathematicians:

No. Just. No.