Posts
Today I got the news that my self-proposed undergraduate thesis project was approved by the school. This means that I’ll spend the next semester working nearly full-time on the optimal control of fixed-wing aircraft, using an approach pioneered in the last two years for drone racing control. If successful, this project will have major implications for autonomous navigation and control in disaster recovery, fire fighting, and package delivery with fixed-wing aircraft.
The typical derivation for the equations of motion of a beam uses a free-body diagram with shear and moment forces to derive the equations of motion. This derivation suffers from the usual lack of clarity in differential diagrams – it’s not immediately clear, for example, what conditions of smoothness need to be satisfied for the equation to hold. It’s also more difficult to generalize to beams with odd shapes or varying properties.
Question Is \((\ell^p, \|\cdot\|_{\ell^q})\) Banach for \( 1 \lt p \lt q \) ?
Note This proof is wrong. There is a simple counterexample. If anyone knows where I messed up, please email me at (jromanoday) at gmail.com.
Terminology \(\ell^p\) is the space of sequences \((x_n)_{n \in \mathbb{N}}, x_n \in \mathbb{F}\), where \(\mathbb{F}\) is some field, and
$$ \|(x_n)\|_{\ell^p} := \left( \sum_{n=1}^\infty |x_n|^p \right)^{1 \over p} \lt \infty $$ \( (\ell_p, \| \cdot \|_{\ell^q}) \) is the space of \( \ell^p \) sequences, equipped with the \( \ell^q \) norm.
Many years ago, my dad bought a Raspberry Pi. A Raspberry Pi can be many things: it can be your computer, it can be a server, it can store your files, it can listen to your kids. This particular model was a Raspberry Pi Model 3, with 1 GB of RAM and a 3-core CPU. I wound up connecting it to my 3D printer and using it to remotely monitor my prints using the (excellent) open-source software OctoPrint, which worked very well.
Welcome! This is the more professional side of my public output. If you would like less professional stuff, check out my blog here.
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):
\[
C_{M_{ CG }} = C_{M_{ wing }} \bar{c} + C_{L} x_{CG}
\] So for a \(C_M\) of 0 at a at a particular \(\alpha\),
\[
x_{CG} = -\frac{ C_{M_{ wing }}(\alpha) \bar{c}}{ C_{L}(\alpha)}
\] Static margin:
The neutral point is located at
\[
x_{NP} = - \bar{c} \frac{\partial C_M / \partial \alpha}
{\partial C_L / \partial \alpha}
\] which gives a static margin of
\[
\mathrm{Static Margin} := \frac{x_{CG} - x_{NP}}{\bar{c}} = \frac{\partial C_M / \partial \alpha}
{\partial C_L / \partial \alpha}
\] with moments calculated about the center of gravity, as they are in XFLR5.