In this blog post, I explore the calculation of the velocity of a point on a landing gear wheel using multiple reference frames. I will show how to break down the complex motion into simpler components using intermediate frames. I start by analyzing the motion of the wheel with respect to the landing gear arm, and then I analyze the motion of the arm with respect to the aircraft. This approach will allow me to find the final velocity of the point using vector algebra. I'll go through the entire derivation, clarifying each step.
In this blog post, I explain how to analyze the velocity of a point observed from two different reference frames, one of which is moving and rotating relative to the other. I start by establishing the relationship between the position vectors in the two frames. Then, through differentiation and consideration of the relative rotation, I derive the general velocity transformation equation. This equation expresses the absolute velocity of a point as the sum of the absolute velocity of the moving frame's origin, the relative velocity of the point within the moving frame, and a term accounting for the frame's rotation.
In this blog post, I explore the complexities of 3D rotational motion by analyzing a practical engineering problem, the retraction of an aircraft landing gear. I examine how to determine the angular velocity and acceleration of a spinning wheel relative to the aircraft body, considering the rotation of both the wheel and the landing gear arm. I use the addition theorems for angular velocities and accelerations, applying them to multiple reference frames. My analysis reveals that even with constant angular velocities, a non-zero angular acceleration can arise due to the changing direction of the angular velocity vector. This leads to a gyroscopic moment. I explain the implications of this moment, particularly its potential to induce stress on the landing gear structure.
In this blog post, I will describe my approach to improving text rendering within Mayavi visualizations. I encountered limitations with Mayavi's built-in text rendering capabilities, particularly regarding LaTeX support and font choices. To overcome these limitations, I developed a hybrid solution that leverages matplotlib to generate text as png images, which are then imported as textures into mayavi. I will detail the steps involved in this process, including generating the text with matplotlib, saving it as a png, and applying it as a texture to a plane within the mayavi scene.
In this blog post, I will build upon my previous work with Mayavi and describe how I implemented functions to create prisms and cylinders. These geometric primitives are essential for representing many mechanical components in my engineering visualizations. I found that while Mayavi provides some basic shape functionality, creating precisely defined prisms and cylinders required a more tailored approach. I will explain the methods I used to generate these shapes, including the mathematical underpinnings and the specific Mayavi functions I employed.
In this blog post, I will describe how I integrated Mayavi, a powerful 3D visualization tool, into my existing software toolkit for mechanical engineering tasks. I needed robust 3D rendering capabilities to visualize complex mechanical designs and simulations, and Mayavi provided the perfect solution. I will explain the reasons behind my choice of Mayavi, the steps I took to incorporate it into my workflow, and the benefits I have experienced since the integration. My goal is to share my experience and provide a practical guide for other engineers who might be considering a similar integration for their own projects. I will also touch upon some of the challenges I faced and how I overcame them.
In this blog post, I explore the angular acceleration in rotating frames of reference. I explain why, unlike angular velocities which simply add, angular accelerations involve an additional term, the gyroscopic term. I derive the relationship between angular accelerations in multiple frames using the transport theorem, highlighting the crucial role of the cross product of angular velocities. My explanation clarifies how the rotation of one frame relative to another influences the observed angular acceleration, leading to the emergence of this often-misunderstood gyroscopic effect.
In this blog post, I explore the extension of rotational kinematics from planar to three-dimensional motion. I focus on the concept of angular velocity as a vector quantity and derive several key properties. I present a formal proof of the uniqueness of the angular velocity vector for a given relative motion between two reference frames. I then demonstrate the relationship between the angular velocity of frame 2 with respect to frame 1 and the angular velocity of frame 1 with respect to frame 2, showing they are negatives of each other. Finally, I provide a detailed derivation of the addition theorem for angular velocities, a crucial result for analyzing complex rotational systems.
In this blog post, I explore the concept of the center of percussion, an idea in mechanics with practical applications in sports and engineering. I examine how to determine the optimal impact point on an object, like a bat or stick, to minimize or eliminate reaction forces at its pivot. This point ensures a smooth, efficient transfer of momentum. My analysis uses the impulse-momentum principle, applied to a simplified model of a rigid body impacting another object. I derive the mathematical expression for the center of percussion and discuss its implications for minimizing unwanted vibrations and maximizing impact efficiency.
In this blog post, I explore the impulse-momentum relationship for rigid bodies and its application to angular momentum conservation, illustrated by a skater spinning with changing arm positions. Using the principles of linear and angular impulse, I examine how internal work done by the skater alters rotational kinetic energy while conserving angular momentum. I analyze the role of the mass moment of inertia in determining angular velocity changes and derive the relationship between rotational kinetic energy and internal energy transfer.
In this blog post, I explore the work-energy principle as it applies to rigid body motion, distinguishing it from its application to particles. By focusing on the work done by external forces and moments, I show how translational and rotational kinetic energies combine to describe the total energy change in two-dimensional motion. I also explain the computation of work for constant and variable forces, as well as gravity's role in energy transfer.
In this blog post, I showcase how I built a responsive visualization tool for mermaid.js using Bootstrap and JavaScript. The app allows users to input mermaid.js code in a textarea and instantly see the rendered diagram in a preview area. I explain the HTML structure, including Bootstrap for layout, custom CSS for styling, and a JavaScript script for real-time diagram rendering. By integrating mermaid.js effectively, this app provides an interactive and professional tool for testing visualizations before embedding them in web pages.
In this blog post, I explore the calculation of kinetic energy in systems involving both translational and rotational dynamics. I examine scenarios such as a wheel rolling without slipping, an unbalanced wheel, and a rotating bar, discussing the interplay between translational motion, rotational motion, and moment of inertia. Using detailed derivations, I demonstrate how to account for the effects of offset mass centers and different reference points in the analysis.
I recently integrated mermaid.js into my work to create advanced visualizations for technical concepts. mermaid.js allows me to transform plain text into professional diagrams, such as Gantt charts for project management, sequence diagrams for process flows, and entity-relationship diagrams for system modeling. Its intuitive syntax and customization options make it a reliable tool for illustrating complex ideas effectively. By combining this with customized themes, I can ensure diagrams meet precise aesthetic and functional requirements.
In this blog post, I explore the kinetic energy of a rigid body in planar motion, deriving the equation for total energy as the sum of translational and rotational components. Using a systematic approach, I relate the velocity of any point in the body to the center of mass and angular velocity. The derivation simplifies to the well-known result that the translational kinetic energy, proportional to the square of the center of mass velocity, and the rotational energy, linked to angular velocity and the moment of inertia about the center of mass. This analysis is essential for understanding rigid body dynamics in physics and engineering.