In this blog post, I investigate the rotational stability of a satellite in free space. I model the satellite as a cuboid and analyze its motion using Euler's equations. I focus on steady-state rotation about each of the principal axes and examine the effects of small perturbations. I derive the equations of motion for these perturbed states and analyze their stability based on the moments of inertia. My analysis reveals that stability depends critically on the axis of rotation relative to the principal moments of inertia. I demonstrate through mathematical derivation that spin about the axis of the intermediate moment of inertia leads to instability, while spin about the axes of the largest or smallest moments of inertia results in stable motion. I conclude by discussing the implications of these findings for satellite design and control.
In this blog post, I explore the dynamics of an aircraft landing gear under an applied moment. I revisit a previously analyzed landing gear kinematic model and shift my focus to the moments acting on the system. I consider the wheel's angular momentum and use the moment equation to relate applied moments to the wheel's rotational behavior and the arm's motion. I then calculate the reaction moments at the connection point between the wheel and the arm, a critical location for bearing design. Finally, I examine the resulting equations and discuss the counterintuitive presence of a gyroscopic moment, even though the input velocities are confined to two axes.
In this blog post, I present a detailed explanation of Euler's equations of motion. I begin with a review of the basic principles and then derive the equations in a body-fixed reference frame. I show how the equations simplify when considering the mass center and how to account for an arbitrary point within the body. I also discuss the importance of principal axes and how they simplify the equations. Finally, I introduce the concept of an intermediate reference frame, which I find useful for solving certain types of problems.
In this blog post, I walk through the process of finding the principal axes of inertia for a uniform density triangular plate. I start by recalling the moment of inertia tensor and then proceed to solve the eigenvalue problem. I calculate the eigenvalues, which represent the principal moments of inertia, and the corresponding eigenvectors, which define the directions of the principal axes. I explain how the smallest eigenvalue corresponds to the axis of easiest rotation, while the largest eigenvalue indicates the axis of hardest rotation. Finally, I visualize these principal axes and discuss their significance in understanding the rotational behavior of the triangular plate.
In this blog post, I meticulously derive the moment of inertia tensor for a cylinder. I begin by defining the density and setting up the integral for each component of the tensor. I then proceed to evaluate these integrals, outlining each step of the calculation. I leverage the symmetry of the cylinder to simplify the process, showing how I can deduce some components from others.
In this blog post, I walk through the calculations for determining the moment of inertia tensor of a triangular plate. I begin by establishing the fundamental equations and then proceed to compute the centroid of the triangle, a key step in the process. I then derive the moments of inertia and the product of inertia with respect to the origin. I then apply the parallel axis theorem to shift the reference point to the center of mass, obtaining the corresponding moment of inertia tensor at that point.
In this blog post, I provide a detailed explanation of principal axes and moments of inertia. I begin by defining the inertia matrix and showing how it simplifies when expressed in the principal axis frame. I then explain how I can determine these principal axes and moments by solving an eigenvalue problem. I illustrate the transformation of the inertia tensor under rotation and how this relates to finding the principal axes. Finally, I discuss the significance of principal axes in simplifying the analysis of rigid body rotation, particularly when considering angular momentum and the conditions for torque-free rotation.
本稿では、Webサイトのローカライズを簡略化するために開発した再利用可能なJavaScriptライブラリをご紹介します。以前取り組んだHTML5アプリケーションのローカライズの経験を活かし、このライブラリでは翻訳管理をシンプルにし、動的な言語切り替えを実現しています。ライブラリのアーキテクチャについて解説し、JSONファイルから翻訳データを取得してマージし、ページの内容を更新し、メタデータを扱う方法を示します。さらに、i18nextとLodashを活用することでパフォーマンスと柔軟性を高めています。また、ローカライズプロセスをさらに自動化するために作成した補助スクリプト（HTMLから翻訳可能なコンテンツを抽出するBashスクリプトや、LLMsを利用して自動翻訳を行うPythonスクリプト）についてもご紹介します。
In questo post del blog, presento una libreria JavaScript riutilizzabile che ho sviluppato per semplificare la localizzazione dei siti web. Basata sul mio precedente lavoro con la localizzazione di applicazioni HTML5, questa libreria semplifica la gestione delle traduzioni e consente il cambio dinamico della lingua. Spiegherò l'architettura della libreria, mostrando come recupera e unisce i dati di traduzione da file JSON, aggiorna il contenuto delle pagine e gestisce i metadati. Il mio approccio utilizza i18next e Lodash per migliorare le prestazioni e la flessibilità. Fornirò inoltre approfondimenti sugli script di supporto che ho creato per automatizzare ulteriormente il processo di localizzazione, tra cui uno script Bash per estrarre contenuti traducibili dall'HTML e uno script Python che sfrutta LLM per la traduzione automatica.
In this blog post, I present a reusable JavaScript library I developed to simplify website localization. Building on my previous work with HTML5 application localization, this library streamlines the management of translations and enables dynamic language switching. I will explain the library's architecture, demonstrating how it fetches and merges translation data from JSON files, updates page content, and handles metadata. My approach uses i18next and Lodash to enhance performance and flexibility. I also provide insights into helper scripts I created to further automate the localization process, including a Bash script for extracting translatable content from HTML and a Python script leveraging LLMs for automated translation.
This post details a custom bash syntax highlighting function designed to improve code readability by applying distinct CSS classes to various language constructs. This approach covers keywords, strings, variables, commands, and more, significantly aiding in code comprehension and debugging.
In this blog post, I explore the calculation of the mass moment of inertia for a cuboid. I derive the inertia tensor components, detailing each step from the integral definitions to the final expressions for moments and products of inertia. My focus is on a cuboid with uniform mass density, and I provide the mathematical derivations using a step-by-step approach, making the underlying principles clear. I then show how the final matrix simplifies due to the symmetry of the cuboid and how the off-diagonal elements are zero, resulting in a diagonal matrix.
In this blog post, I explore the concept of angular momentum for a rigid body undergoing three-dimensional motion. I derive the formula for angular momentum about an arbitrary point, starting from the basic definition involving the integral of the cross product between the position vector and velocity. I show how to incorporate the relative velocity equation to describe the velocity of mass elements within the rigid body. I then introduce the inertia tensor and show how it simplifies the expression for the angular momentum. I also cover specific cases where the reference point is the center of mass or a fixed point in the inertial frame, highlighting how the equation simplifies under these conditions.