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17. Cylindrical Coordinate System

The lecture focuses on deriving the Linear Momentum Balance in a cylindrical coordinate system, essential for analyzing deformation in cylindrical bodies. It introduces the concept of basis vectors in cylindrical coordinates, explains the formulation of linear momentum balance, and details the calculation of forces acting on cylindrical elements. Key aspects include using Taylor’s series for stress components and evaluating changes in basis vector orientations, which are crucial for accurate momentum analysis.

Sections

Cylindrical Coordinate System

This section introduces the cylindrical coordinate system, explaining its significance in analyzing cylindrical bodies and deriving linear momentum balance laws.

1 Section Overview

Start current section content and materials

1.1 Basis Vectors

This section discusses the basis vectors in the cylindrical coordinate system and their significance in analyzing cylindrical deformation.

1.2 Cylindrical Element

This section derives the Linear Momentum Balance in cylindrical coordinates by analyzing cylindrical elements' face normals.

LMB Formulation

This section introduces the Linear Momentum Balance (LMB) formulation in cylindrical coordinate systems, explaining how to derive balance laws for cylindrical elements.

2 Section Overview

Start current section content and materials

2.1 Force on +z and −z Plane

This section discusses the calculation of forces acting on the +z and -z planes in a cylindrical coordinate system and introduces the necessary equations for deriving linear momentum balance.

2.2 Taylor’s Series Expansion

This section delves into the application of Taylor's series expansion for stress components in cylindrical coordinates, illustrating its significance in deriving linear momentum balances.

2.3 Partial Derivative of Basis Vectors w.r.t. θ

This section discusses the concept of partial derivatives of basis vectors in cylindrical coordinates with respect to the angular coordinate θ.

2.4 Final Simplification

This section discusses the final simplification of the linear momentum balance in a cylindrical coordinate system.

Learning Objectives

  • The cylindrical coordinate system is vital for studying deformation in cylindrical shapes.

  • Basis vectors in cylindrical coordinates change with the angle θ, affecting the analysis of forces.

  • Using Taylor's series helps to approximate varying stress components in the context of cylindrical elements.

Key Concepts

Cylindrical Coordinate System

A coordinate system defined by the distance from a reference axis and the angle around that axis, commonly used for cylindrical shapes.

Basis Vectors

Vectors that define the coordinate system at a point; in cylindrical coordinates, they vary based on the angle θ.

Linear Momentum Balance

A principle stating that the rate of change of momentum of a system equals the sum of the forces acting on it.

Taylor's Series

An approximation method used to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting

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