Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Cylindrical Coordinates
Unlock Audio Lesson
Today, we start with cylindrical coordinates, crucial for studying cylindrical shapes. Can anyone tell me the main coordinates we use here?
Is it r, θ, and z?
Exactly! r measures the distance from the origin, θ is the angle around the z-axis, and z is the height. Remember the acronym **R-Z-Theta** to recall the order!
What about the basis vectors?
Good question! The basis vectors are e_r, e_θ, and e_z. While e_z remains constant, e_r and e_θ change with θ. Let's visualize that with a diagram!
Characteristics of the Cylindrical Element
Unlock Audio Lesson
Next, let's consider the cylindrical element. What are the faces of this element?
There's the top face, bottom face, and the curved faces.
Correct! The top face has a normal along e_z and the bottom along -e_z. Don't forget that the curved faces have normals that point radially. Think of them as the 'wrap-around' faces!
What happens when the element gets very small?
Excellent point! When the element becomes infinitesimally small, those normals appear flat. That's why we use them in calculus!
Linear Momentum Balance
Unlock Audio Lesson
Now, let’s dive into the Linear Momentum Balance for our cylindrical element. How do we start?
We need to sum the forces on the cylindrical element!
Exactly! We sum the total forces from tractions and body forces. Each face's forces must be taken into account. Remember the formula for traction on the +z plane?
It was t+z =σ e_z + τ e_r + τ e_θ!
Brilliant! Now, integrating these tractions over the area gives us the total force on each face. Let’s set that up for calculations.
Using Taylor Series Expansion
Unlock Audio Lesson
We will apply Taylor series for the stress components. Why do you think this is essential here?
To approximate values for stresses that change over the element?
Exactly! By expanding this, we can express varying components at the center. Can anyone recall the importance of differentiating over r and z?
Those derivatives will be zero for e_r and e_θ since they only change with θ!
Well done! It’s pivotal as we simplify our equations.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore the concept of cylindrical elements within a cylindrical coordinate system, focusing on the derivation of Linear Momentum Balance. We analyze different planes of the cylindrical element, considering forces and stress components to understand the importance of cylindrical coordinates in mechanics.
Detailed
Detailed Summary
In this section, we dive deep into the Cylindrical Coordinate System, a mathematical framework particularly useful for analyzing bodies with cylindrical shapes. The section begins with a brief introduction to cylindrical coordinates, where a general point is defined using coordinates r, θ, z. The basis vectors of this system, which define directions in the cylindrical space, are critically discussed. Two basic vectors, e_r and e_θ, change with the θ coordinate, while the third basis vector, e_z, remains fixed throughout.
We then analyze cylindrical elements, focusing on their construction and orientation. Each face of the cylindrical element is defined in relation to the basis vectors, specifically the top face, bottom face, and curved faces.
Key Points Covered:
- The cylindrical element is analyzed in three different planes:
- +z-plane (top face)
- -z-plane (bottom face)
- θ-planes (front faces in the cylindrical coordinate system).
- Linear Momentum Balance (LMB) is derived from the total forces acting on the cylindrical element, taking into account tractions and body forces.
- A Taylor series expansion is utilized for approximating the varying stress components across the integration domain.
Overall, understanding cylindrical elements and their role in mechanics is crucial for solving real-life engineering problems where such geometrical configurations are present.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Cylindrical Elements
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
In Cartesian coordinate system, we analyzed cuboid elements with their face normals directed along the coordinate axis. In cylindrical coordinate system similarly, we need to analyze cylindrical elements with their face normals parallel to er, eθ, and ez. A cylindrical element for the generalized cylinder and its zoomed view is shown in Figure 3.
Detailed Explanation
Cylindrical elements are three-dimensional shapes used to analyze objects that have a circular cross-section when viewed from above. In the cylindrical coordinate system, these elements are defined such that their normals (the lines perpendicular to the surfaces) align with the three basis vectors—er, eθ, and ez. This is similar to how cuboid elements in Cartesian coordinates are analyzed, with normals pointing along the x, y, and z axes.
Examples & Analogies
Think of a soda can. The curved surface of the can represents the cylindrical element, while the top and bottom of the can represent the flat surfaces that align with the ez direction. Each part can be analyzed to understand the forces acting on the can when it is moved or pressed.
Face Normals of Cylindrical Elements
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The top face has its normal along z axis. Thus, the top face is +z plane and the bottom face is −z plane. The front face shown has its normal along −eθ direction. Thus, this is the −θ plane and the plane opposite to it is +θ plane. Finally, the concave curved face has its plane normal along −er direction and can be imagined to become flat in the limit of the cylindrical element becoming infinitesimally small.
Detailed Explanation
Each face of the cylindrical element has a specific orientation determined by its normal vector. The top face faces the positive z direction, and the bottom face faces the negative z direction. Similarly, the front face points in the direction of decreasing θ, while the back points in the direction of increasing θ. The curved surfaces behave differently compared to flat surfaces; as the cylindrical element becomes smaller, the curved surfaces start to resemble flat planes in their interactions.
Examples & Analogies
Imagine a tall cylindrical glass. The top of the glass is the +z plane, while the bottom is the -z plane. If you were to look at the side of the glass, the curve represents the −er direction, which would flatten as you zoom in on it, akin to examining a tiny slice of the glass at its edge.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Cylindrical Coordinate System: A system used to describe points in space with three coordinates r, θ, and z.
-
Basis Vectors: The vectors that form the basis of the cylindrical coordinate system, including e_r, e_θ, and e_z.
-
Cylindrical Element: A volume element analyzed for stress and force in mechanics, defined within cylindrical coordinates.
-
Linear Momentum Balance: A fundamental concept linking the forces acting on an element to its momentum change.
-
Taylor Series Expansion: A method of approximating functions that is crucial for deriving expressions in mechanics.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
A cylindrical tank holding water can be analyzed using cylindrical coordinates to find the pressure at various depths, which exemplifies the utility of cylindrical elements.
-
When analyzing a pipe under internal pressure, cylindrical coordinate analysis simplifies calculating stresses due to bending.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
In cylindrical shape, we hold tight, r, theta, z, hold on so right.
📖 Fascinating Stories
-
Imagine a cylindrical cake; the layers represent the height z, the slices are the angle θ, and how far from the center you cut, that's the radius r.
🧠 Other Memory Gems
-
Remember 'REM' for cylindrical system: R for radius, E for angle (theta), M for height (z).
🎯 Super Acronyms
CBA for remembering the cylindrical coordinates
- C: for Cylindrical
- B: for Bases
- A: for Area.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Cylindrical Coordinates
Definition:
A coordinate system used to describe locations in a three-dimensional space using radial distance (r), angular position (θ), and height (z).
-
Term: Basis Vectors
Definition:
Vectors that define the direction of the coordinate system axes; in cylindrical coordinates, they include e_r, e_θ, and e_z.
-
Term: Cylindrical Element
Definition:
A small volume element defined in cylindrical coordinates, analyzed for mechanics applications.
-
Term: Linear Momentum Balance
Definition:
An equation that relates the sum of forces acting on an element to the change of linear momentum of that element over time.
-
Term: Taylor Series Expansion
Definition:
A mathematical series that approximates a function as a sum of its derivatives at a single point.