27.6 Linearize the Nonlinear ODE for a Simple Pendulum

Types: N/A
Examples: N/A
Constructions: 27.7 Mathematize the Coupled Pendula Problem
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

In 27.5 Motion of a Single Pendulum, we derived the ordinary Differential Equations

\begin{aligned} \ddot{θ} (t) + \frac{g}{ℓ} \sin (θ (t)) & = 0 \\ \frac{d^{2}}{d t^{2}} [θ (t)] + \frac{g}{ℓ} \sin (θ (t)) & = 0 \end{aligned}

whose solution encodes the angle $θ (t)$ made between the cord and the vertical position. Unfortunately, this equation is not easily solved. We cannot write a solution

θ (t) = ?

in terms of elementary functions. However, using Linear Algebra, we can break this problem down into a much easier problem to solve.

Linearize the Nonlinear Ordinary Differential Equation

Previously, we were thinking of determining the position through measurement of Arc Length. However, what if instead of using the arc length, we used the position of the shadow along the ruler? Instead of working in a multidimensional space, we can work in a single dimensional space.

INSERT DIAGRAMS HERE

\begin{proof}[Solution.]

Recall that

\sin (θ (t)) = \frac{u (t)}{ℓ} = \frac{opposite}{hypotenuse}

This implies that we can substitute

\begin{aligned} \ddot{θ} (t) + \frac{g}{ℓ} \sin (θ (t)) & = 0 \\ ⟹ \ddot{θ} (t) + \frac{g}{ℓ} \cdot \frac{u (t)}{ℓ} & = 0 \end{aligned}

However,

\sin (θ (t)) = \frac{u (t)}{ℓ}

doesn't immediately suggest how to relate second order terms

\ddot{θ} (t) and \ddot{u} (t)

But, recall from our work with Taylor Series polynomials:

\begin{aligned} \sin (θ) & = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{2 (k + 1)!} [θ]^{2 k + 1} \\ = θ \underset{as long as θ is small, these terms are close to zero}{\underset{⏟}{- \frac{1}{3!} θ^{3} + \frac{1}{5!} θ^{5} - \dots}} \end{aligned}

In other words, the error between the values of $\sin (θ)$ and $θ$ is negligible for small values of $θ$ .

= θ + \underset{big O notation}{\underset{⏟}{O (θ^{3})}}

Note that it is common practice to assume

θ (t) < 0.1 radian \approx 7^{\circ}

yielding an approximation error $< 0.1 %$ .

Or, the tangent line to $\sin (θ)$ at expansion point $a = 0$ , defined by the linearization

\begin{aligned} L (θ) & = f (a) + f^{'} (a) \cdot (θ - a) \\ = \sin (0) + \cos (0) \cdot (θ - 0) \\ = 0 + 1 \cdot θ \\ = θ \end{aligned}

closely approximates the behavior of sine.

⟹ θ (t) \approx \sin (θ (t)) = \frac{u (t)}{ℓ}

This is known as the small angle assumption.

\begin{aligned} ⟹ \dot{θ} (t) & = \frac{d}{d t} θ (t) \\ \approx \frac{d}{d t} \frac{u (t)}{ℓ} \\ = \frac{1}{ℓ} \cdot \frac{d}{d t} u (t) \\ = \frac{\dot{u} (t)}{ℓ} \\ ⟹ \ddot{θ} (t) & = \frac{d}{d t} \dot{θ} (t) \\ \approx \frac{d}{d t} \frac{\dot{u} (t)}{ℓ} \\ = \frac{1}{ℓ} \frac{d}{d t} \dot{u} (t) \\ = \frac{\ddot{u} (t)}{ℓ} \end{aligned}

Then, using this approximation, we can state the linearized differential equation

\begin{aligned} \ddot{θ} (t) + \frac{g}{ℓ} \sin (θ (t)) & = 0 \\ \frac{\ddot{u} (t)}{ℓ} + \frac{g}{ℓ} \cdot \frac{u (t)}{ℓ} & = 0 \\ ⟹ \ddot{u} (t) + \frac{g}{ℓ} \cdot u (t) & = 0 \end{aligned}

This transformed equation is also a way to state Newton's Second Law. For small angle oscillations with $θ (t) < 0.1$ , we can set

\begin{aligned} u (t) & = \sin (θ (t)) \approx θ (t) \\ ⟹ {\vec{f}}_{g_{t} ⊥} & = - m \cdot g \cdot \sin (θ (t)) \\ = - m \cdot g \cdot \frac{u (t)}{ℓ} \\ = - \frac{m \cdot g}{ℓ} u (t) \\ = {\vec{f}}_{p_{u}} \end{aligned}

\end{proof}

Remark. The transformed 2nd order linear ordinary differential equation describes motion of a simple harmonic oscillator.
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