2 - LMB Formulation
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Interactive Audio Lesson
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Introduction to Cylindrical Coordinates
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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.
Cylindrical Elements
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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.
Applying Taylor Series Expansion
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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.
Final Simplifications
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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.
Introduction & Overview
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Quick Overview
Standard
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.
Detailed
Linear Momentum Balance (LMB) Formulation
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.
Key Steps in the LMB Formulation:
- Understanding Basis Vectors: The basis vectors change with θ, and we have a fixed z-axis, in contrast to the constant direction of Cartesian coordinates.
- Analyzing Cylindrical Elements: We investigate the force components acting on various faces of a cylindrical element, leading to expressions involving stress and tractions.
- Integration and Taylor Series Expansion: Each face's traction is integrated to derive the net force, using Taylor expansion for stress components, focusing on their dependence on radial positions and angular coordinates.
- Final Simplifications: After treating each face of the cylindrical element and integrating appropriately, the overall balance of linear momentum leads to key equations necessary for dynamics in cylindrical systems.
Overall, this formulation is fundamental in solid mechanics, as it provides a systematic approach to analyzing forces in rotating and cylindrical bodies.
Audio Book
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Overview of Linear Momentum Balance
Chapter 1 of 5
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Chapter Content
For the Linear Momentum Balance, we need to sum the total force on the cylindrical element due to tractions and body force and equate it to its rate of change of linear momentum.
Detailed Explanation
This introduction highlights the essential aspects of Linear Momentum Balance (LMB). In classical mechanics, LMB states that the forces acting on an object equal the change in momentum of that object over time. Here, we specifically look at a cylindrical element, which is useful when dealing with bodies shaped like cylinders. We begin by considering all forces acting on the element (due to surface tractions and internal body forces) and relate this to how the momentum of the system changes.
Examples & Analogies
Imagine a cylindrical water bottle. If you push down on it (applying force), the water inside will respond by moving. This is similar to how we analyze forces on the cylindrical element and their effect on momentum.
Forces on the +z and -z Planes
Chapter 2 of 5
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Chapter Content
On +z plane, the traction vector (t+z) is given by t+z = σ e + τ e + τ e. The total force due to this traction can be obtained by integrating it over +z plane...
Detailed Explanation
We start by considering the forces acting on the top surface of the cylindrical element (the +z plane). The traction vector describes how forces are distributed across this surface. The force must be calculated by integrating the traction vector over the surface area of this plane, accounting for components of stress in different directions. This analysis allows us to quantify the total force acting upwards on the cylindrical element.
Examples & Analogies
Think of pushing down on a flat surface like a table with varying pressure across your palm. Each part of your hand exerts a different amount of force. Integrating the pressure over your palm gives the total force applied and how it is distributed, similar to the analysis of forces on the cylindrical surface.
Taylor Series Expansion
Chapter 3 of 5
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Chapter Content
As the stress components are varying in the domain of integration, we can use Taylor’s expansion about the center of the cylindrical element...
Detailed Explanation
Since the stress components change within the cylindrical element, we use Taylor series expansion to express these components at points close to the center of the element. This mathematical technique helps us approximate the values of these components at nearby points, facilitating simpler integrations and calculations for LMB.
Examples & Analogies
Imagine you're observing a moving car from different vantage points along a street. The car's speed seems different depending on your perspective. By using the Taylor series, it's like approximating the car's speed at various points, helping you understand how it moves without needing to know its exact speed at every point.
Partial Derivative of Basis Vectors with Respect to θ
Chapter 4 of 5
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Chapter Content
We show how to obtain the partial derivatives of basis vectors. The θ coordinate of two points differs by ∆θ...
Detailed Explanation
In cylindrical coordinates, as you change the angular coordinate (θ), the direction of basis vectors also changes. By calculating the partial derivatives of these basis vectors with respect to θ, we understand how the basis vectors adjust when moving around the circular path of the cylindrical element. This becomes critical when integrating forces and releases the necessity to deal with constant basis vectors.
Examples & Analogies
Consider how your finger moves around a circular pizza. The angle of your finger changes as you curve around the pizza's edge. Similarly, the basis vector direction changes as you vary the angle θ in cylindrical coordinates, and calculating these shifts helps in understanding related calculations.
Final Simplification of Forces
Chapter 5 of 5
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Chapter Content
We can now plug in the derivatives of basis vectors into equation (6) to get various expressions for the total force...
Detailed Explanation
After obtaining the derivatives of basis vectors, these expressions are substituted back into the equations governing the total force acting on the cylindrical element. This leads to simplified expressions that can be handled mathematically and illustrate how different stress components interact with the coordinates of the cylindrical system, providing clear insights into LMB.
Examples & Analogies
Think of this process like assembling pieces of a puzzle. As you fit each piece together (in this case, the mathematical expressions), you begin to see the complete picture of how forces and stresses distribute over the cylindrical element.
Key Concepts
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Cylindrical Coordinate System: A system that uses radius and angle to define points in a plane, along with height.
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Basis Vectors: The vectors that define the coordinate directions; they can change direction in cylindrical coordinates unlike in Cartesian coordinates.
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Integration of Forces: The process of calculating the total forces acting on an object by adding forces from different surfaces.
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Taylor Series Expansion: A technique used to approximate functions which allows simplification of stress components.
Examples & Applications
The position of a point on a cylindrical body can be defined using coordinates (r, θ, z), where r is the radius, θ the angular position, and z the height.
Using the traction vector formula t+z = σ zz e z + τ rz e r + τ θz e θ, we can calculate the forces acting on cylindrical surfaces.
Memory Aids
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Rhymes
To find a point on the cylinder so fine, use r and theta, they intertwine.
Stories
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.
Memory Tools
RAT for cylindrical: Radius- Angle- height.
Acronyms
BAT
Basis changes As Theta modifies.
Flash Cards
Glossary
- Cylindrical Coordinate System
A three-dimensional coordinate system where points are defined by a radius, angle, and height.
- Basis Vectors
Vectors that define the directions of the coordinates in a specific system.
- Traction Vector
The force per unit area acting on a surface.
- Taylor Series
A mathematical series that approximates functions around a specific point.
Reference links
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