Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we'll discuss the cylindrical coordinate system. Can anyone tell me how this system differs from Cartesian coordinates?
In Cartesian coordinates, the basis directions are fixed, while in cylindrical coordinates, the direction of certain basis vectors changes.
Great! Indeed, the radial and angular components are not constant, but the z-axis remains fixed. This affects how we formulate our equations. Remember this principle when we derive the LMB.
So what does this mean for our analysis?
It means you'll need to account for changing directions when using basis vectors in calculations, particularly for forces.
Let's break down the forces acting on a cylindrical element. What are the faces we need to consider?
The +z plane and the -z plane, plus the radial and angular faces.
Exactly! We calculate the forces from the traction vectors. Can anyone express what the traction vector on the +z plane looks like?
It's expressed as t+z = σ zz e z + τ rz e r + τ θz e θ.
Excellent! We then integrate this over the surface to find the total force. Let's take the area element, dA, on the +z plane. Can someone derive it?
The area element is dA = dξ * (r + ξ) dη.
Now, how do we address the variation in stress components?
We can use Taylor’s expansion around the center of the cylindrical element.
Correct! We find that this is crucial because stress changes depend on θ. Therefore, we can ignore higher-order terms. Why is that?
Because they become insignificant as they increase relative to the principal terms?
Exactly! As we utilize Taylor series, let’s remember that this keeps our equations manageable and accurate.
After applying the Taylor series, how will we finalize our net force expression?
By combining all the individual forces from each plane we analyzed.
Yes! When we compare the forces from different planes, we notice the signs indicate their orientation. Can anyone explain that?
The negative signs for the -z plane indicate forces are acting in the opposite direction!
Exactly right! The forces from the +z and -z planes are crucial for understanding the overall balance equation.
A system that uses radius and angle to define points in a plane, along with height.
The vectors that define the coordinate directions; they can change direction in cylindrical coordinates unlike in Cartesian coordinates.
The process of calculating the total forces acting on an object by adding forces from different surfaces.
A technique used to approximate functions which allows simplification of stress components.
To find a point on the cylinder so fine, use r and theta, they intertwine.
Imagine a race car moving in a circle, where its speed changes with the angle. This mirrors how forces change with θ in cylindrical coordinates.
RAT for cylindrical: Radius- Angle- height.
BAT: Basis changes As Theta modifies.
Term
Cylindrical Coordinate System
Click to flip
Term
Torque in Cylindrical Systems
Click to flip
Term
Traction Vector
Click to flip
A three-dimensional coordinate system where points are defined by a radius, angle, and height.
Vectors that define the directions of the coordinates in a specific system.
The force per unit area acting on a surface.
A mathematical series that approximates functions around a specific point.
This section introduces the Linear Momentum Balance (LMB) formulation in cylindrical coordinate systems, explaining how to derive balance laws for cylindrical elements.
The LMB formulation in cylindrical coordinates encompasses the forces acting on cylindrical elements, accounting for tractions and body forces. The section focuses on deriving these equations, exploring basis vector changes, integration over surfaces, and applying Taylor's series for accurate stress components.
In this section, we derive the Linear Momentum Balance (LMB) for cylindrical coordinate systems, which is vital for analyzing deformation in cylindrical objects. Cylindrical Coordinate System is a crucial tool due to its ability to simplify many problems involving round geometries. We start with understanding the basis vectors within the cylindrical system and how they differ from Cartesian coordinates, culminating in assessing forces on cylindrical elements.
Overall, this formulation is fundamental in solid mechanics, as it provides a systematic approach to analyzing forces in rotating and cylindrical bodies.