Advanced Rigid Body Mechanics in Three Dimensions

Mathematical intuition behind the fundamental equations in rigid body mechanics

5.00 (3 reviews)
Udemy
platform
English
language
Engineering
category
99
students
4 hours
content
May 2021
last update
$44.99
regular price

What you will learn

How to derive the fundamental equations on rigid body kinematics and dynamics

How to derive energy conservation from Newton's laws (Virtual Work Theorem)

Fundamental properties of the angular velocity of a rigid body

Description

This is a course on the fundamental equations and concepts which revolve around rigid bodies. All the equations are derived with detailed explanations, but the following mathematical prerequisites are needed: vectors, dot and cross products, some linear algebra (matrices, determinants, eingenvectors, eigenvalues), some calculus (especially: derivatives, volume integrals). As regards the physics of the course, the only prerequisite is the knowledge of Newton's equations. In fact, these equations constitute the physical foundation of the course, since the rigid body mechanics are constructed from point-particle dynamics (i.e. the law: F=ma, where F is the total force acting on a point-particle, a is the acceleration, m is the mass, is postulated to be true for point-particles).

In the course, the inertia matrix is derived, which will appear in the equation of moments, as well as in the expression of the kinetic energy of a rigid body. The concept of angular velocity is also derived, and it will be shown that it is unique. Other important formulae regarding kinematics are derived, which will relate velocities and accelerations of generic points of a rigid body.

In kinematics, we will derive Chasles' theorem, or Mozzi–Chasles' theorem, which says that the most general rigid body displacement can be produced by a translation along a line (called Mozzi axis), in conjunction with a rotation about the same line.

Content

Derivation of the angular velocity

Relation between velocities of points belonging to a rigid body
Derivation of the angular velocity
Angular velocity: application of the theory
Angular velocity as a function of three angles

Formulae for: Velocities, Accelerations, and Mozzi's theorem

Relative velocities and formula for accelerations
Introduction to Mozzi (or Chasles) theorem
Proof of Mozzi theorem
Special cases of Mozzi theorem

Derivation of the equations of motion

First equation of motion
Highlighting the need for the equation of moments
Derivation of the equation of moments part 1
Derivation of the equation of moments part 2
Euler rotation equations derivation part 1
Euler rotation equations derivation part 2

Derivation of the energy conservation (Virtual work theorem)

Derivation of the conservation of energy part 1
Derivation of the conservation of energy part 2
Derivation of the conservation of energy part 3
Kinetic energy of a rigid body

Some important properties of the inertia matrix

Properties of the inertia matrix

Stability analysis of the motion of a torque-free and force-free rigid body

Writing out the components of the equations of motion
Stability analysis of the torque-free motion part 1
stability analysis of the torque-free motion part 2

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3636252
udemy ID
11/15/2020
course created date
11/21/2020
course indexed date
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