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1.2. Cylindrical Element
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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!
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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!
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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.
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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.
Overview
Short Summary
This section derives the Linear Momentum Balance in cylindrical coordinates by analyzing cylindrical elements' face normals.
Medium Summary
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 Summary
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
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Create a free accountIn 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.
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Create a free accountThe 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.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- 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
Step-by-step examples to apply the section's ideas and test your understanding.
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
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.
Remember 'REM' for cylindrical system: R for radius, E for angle (theta), M for height (z).
Flash Cards
Glossary
Cylindrical Coordinates
A coordinate system used to describe locations in a three-dimensional space using radial distance (r), angular position (θ), and height (z).
Basis Vectors
Vectors that define the direction of the coordinate system axes; in cylindrical coordinates, they include e_r, e_θ, and e_z.
Cylindrical Element
A small volume element defined in cylindrical coordinates, analyzed for mechanics applications.
Linear Momentum Balance
An equation that relates the sum of forces acting on an element to the change of linear momentum of that element over time.
Taylor Series Expansion
A mathematical series that approximates a function as a sum of its derivatives at a single point.