1.4 - Final balance
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Derivation of the Angular Momentum Balance (AMB)
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Today we focus on the Angular Momentum Balance, or AMB, a crucial equation that allows us to analyze the angular effects of both internal and external forces acting on a cuboid.
Why is it important to consider both traction and body forces?
Great question! Both types of forces influence how the momentum of the cuboid changes. The traction forces affect it externally while body forces act internally.
Can you explain what the first integral means in the equation?
The first integral sums the momentum from a fixed mass, while the second encompasses changes in the mass distribution over time. This distinction is vital!
Understanding Control Mass vs. Control Volume
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When deriving the AMB, we prefer a control mass approach. Why do you think that is?
I assume it's because it simplifies the integration process?
Exactly! Moving the time derivative within the integral is much easier when we work with identifiable masses.
What happens during the transition to the limit when we shrink the volume to its centroid?
As we shrink the volume, many terms disappear, and we get the ultimate AMB equation reflecting the total contributions from the significant forces.
Tensor Representation of AMB
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Let’s explore how to represent AMB in coordinate terms. This helps make the equations applicable in real-world problems.
How does the symmetry of the stress matrix come into play here?
The symmetry shows that certain stress components are equal, which simplifies our calculations and is vital for theoretical applications.
So does this mean AMB holds true under all conditions, even when the body is accelerating?
Correct! The derivation confirms that the AMB remains valid regardless of external accelerations or forces applied.
Introduction & Overview
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Quick Overview
Standard
This section details the derivation of the Angular Momentum Balance equation by integrating the contributions from traction forces and body forces acting on a cuboid. It emphasizes the importance of considering both fixed/identifiable mass and control mass settings to simplify the analysis.
Detailed
Final Balance
In this section, we derive the Angular Momentum Balance (AMB) equation that holds for a cuboid of any size, which is crucial for analyzing dynamics in Solid Mechanics. We start by substituting previously derived equations for traction forces and body forces into the dynamics of the cuboid's motion.
The derivation notes that the first integral in the equations considers a fixed mass while the second involves a changing volume, highlighting the need for careful integration techniques.
Ultimately, as we shrink the cuboid to a point at its centroid, we arrive at the final AMB equation, which shows that the total angular momentum is conserved even under the influence of external forces or acceleration of the body. The section concludes by offering a tensorial representation of the AMB equation that further elaborates on its implications, including the symmetry of stress matrices irrespective of the presence of external body forces. This provides a foundational understanding essential for advanced studies in mechanics of materials.
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Substituting Previous Equations
Chapter 1 of 4
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Chapter Content
So, now we can substitute equations (1), (2), and (7) in (8)
Detailed Explanation
In this step, we take equations from previous sections (1, 2, and 7) and plug them into equation (8). This is a crucial step in our calculation, as it allows us to combine previous findings into a single expression which takes into account all the effects on angular momentum balance.
Examples & Analogies
Think of it like making a recipe. You have several ingredients (equations) from previous steps, and now you are combining them to create a final dish (the final balance equation). Just as you need the right amounts of each ingredient to make a successful meal, you need the right equations to derive the correct balance.
Applying Limit to the Equation
Chapter 2 of 4
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As this equation is valid for a cuboid of any size, we can shrink it to its centroid (x). So, first we divide by ∆V on both sides and then take the limit: ∆V→0
Detailed Explanation
This chunk describes how to generalize our findings. By stating the equation is valid for a cuboid of any size, we can simplify it further. We focus on the centroid of the cuboid, which represents an average center point. Dividing by ∆V and taking the limit approaches a situation where the volume is infinitesimally small, which helps in refining our calculations to find specific behavior of the system.
Examples & Analogies
Consider it like zooming in on a map. Initially, the map shows a whole city, but when you zoom in, you focus on a specific neighborhood (the centroid). As you continue to zoom in, details become clearer, just like how we refine our equations by focusing on smaller scales.
Final Angular Momentum Balance
Chapter 3 of 4
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We finally get → Angular Momentum Balance (AMB)
Detailed Explanation
After substituting the equations and taking the limit, we arrive at the final expression known as Angular Momentum Balance (AMB). This equation summarizes all the previous analyses and gives us a formula to evaluate the angular momentum in relation to the forces acting on the body. Its formulation captures the essence of dynamic interactions within the system.
Examples & Analogies
Imagine constructing a model. At first, you have many pieces (equations and concepts). But as you assemble them thoughtfully, you create a final model (the AMB equation) that represents a complete picture of how things work together. This model allows you to analyze how the system behaves under different conditions.
Applicability of the AMB
Chapter 4 of 4
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This equation holds even if a body force is acting or if the body is accelerating as these terms vanished in the derivation itself.
Detailed Explanation
This statement emphasizes that the Angular Momentum Balance equation is robust; it remains valid regardless of additional forces acting on the body or whether the body itself is in motion. This illustrates the power of the balance equations in mechanics where they can simplify complex scenarios and help us analyze the motion without additional complications.
Examples & Analogies
Consider a car moving on a road. Whether the car is going uphill (experiencing a body force) or changing speed (accelerating), the principles governing its motion still apply. In the same way, the AMB holds true under different conditions, providing a reliable framework for understanding angular momentum.
Key Concepts
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Angular Momentum Balance (AMB): Important for analyzing dynamics in solid mechanics.
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Body and Traction Forces: Critical to understanding the effects of forces on a material.
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Control Mass vs. Control Volume: Simplified analysis using identifiable mass leads to easier integration.
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Tensor Representation: Symmetrical properties of stress tensors simplify derived equations.
Examples & Applications
Consider a cuboidal block experiencing rotational motion under varying traction forces. The AMB helps compute the angular momentum accurately.
When analyzing forces acting on a spinning top, applying the principles of AMB can elucidate the effect of body and traction forces on its motion.
Memory Aids
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Rhymes
If body forces push and traction flows, angular momentum balance surely grows!
Stories
Imagine a smooth spinning top on a table. If someone gently pushes it from the side, the balance of forces dictates how fast it wobbles. Each push affects its momentum, illustrating the AMB.
Memory Tools
Use 'BAT' to remember: 'B' for Body forces, 'A' for Angular momentum, and 'T' for Traction forces.
Acronyms
Remember AMB as 'Always Manage Balance' for all forces acting in dynamics!
Flash Cards
Glossary
- Angular Momentum Balance (AMB)
An equation that expresses the balance of angular momentum considering both traction and body forces acting on a body.
- Body Force
A force that acts throughout the volume of a material, such as gravitational force.
- Traction Force
An external force applied per unit area on the surface of a body, affecting its motion.
- Control Mass Setting
An approach in mechanics where a fixed mass is monitored to derive dynamics.
- Finite Element Method
A numerical technique for finding approximate solutions to boundary value problems for partial differential equations.
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