# Hand centrifuge, for power generation.
https://chat.google.com/dm/6AFmRgAAAAE

Eli's expectation from the model:

Updated 5/22: Model should be able to predict power output over the course of a cycle, taking into account all significant torques, including counter torque. I'm hoping the model will be useful for picking design parameters such as string stiffness, necessary string strength, and disk geometries with the larger goal of maximizing electrical power output over the course of a cycle.

1. Estimate the angular velocity and acceleration so that it aligns with the measured angular velocity throughout a cycle of the stationary docked prototype (bearing).
* As a function of time.
* Angular velocity is being measured using video, imaging. Image processing. iPhone.

2. I believe we will need to expand the drag torque term from the Paperfuge model to include drag from bearing and cogging (?).
3. In the end, we will want to maximize angular velocity so that we can maximize power output over the course of a cycle.

$y=a x+b$

$y=a ln(x)+b$

Force : $k A$ </br>
Energy: $\frac{1}{2}k A^2$ </br>

\begin{align}
m \ddot{x}(\tau)+k \,x(\tau)&=F \sin(\omega_{\rm A} \tau)\\
\ddot{x}(\tau)+\frac{k}{m} x(\tau)&=\frac{F_{\rm A}}{m} \sin(\omega_{\rm A} \tau)\\
\ddot{x}(\tau)+\omega_n^2 x(\tau)&=\tilde{F}_{\rm A} \sin(\omega_{\rm A} \tau)\\
x(\tau)&=A(\tau)\sin(\omega_n\tau)
\end{align}

Input, force, $F$ </br>
Output, $\omega$

\begin{align}
\ddot{\phi}(\tau)+T(\phi(\tau);F(\tau))
\end{align}

</br>

Paperfuge: Empirical model. Guess something and fit the parameters in the guess to match the data. They are not good beyond the regime of fitting. They are good for consolidating data.

\begin{align}
\ddot{\phi}+\frac{g}{\ell}\sin(\phi)&=0\\
\ddot{\phi}+\frac{g}{\ell}\phi&=0\\
T_{n}&=2\pi \sqrt{\frac{l}{g}}
\end{align}

$$
\frac{1}{\sqrt{1-\phi^2}}
$$