Aims

The aims of the course are to:

• Introduce the ideas and methods of 3D dynamics: the motion of rigid bodies in three dimensions under given forces and moments.
• Introduce the Lagrange and Hamiltonian formulations of mechanics.
• To show how to apply these methods in a straightforward way to a wide range of problems.

Objectives

As specific objectives, by the end of the course students should be able to:

• Represent the inertia of a rigid body by an inertia matrix, be able to calculate the moments and products of inertia for simple shapes, be able to find the principal axes of inertia.
• Derive Euler's equations for the motion of a rigid body under prescribed moments.
• Apply these equations to the motion of symmetrical rotors, to explain the phenomena of precession, nutation and the rate gyroscope.
• Analyse simple problems involving the rolling of rigid bodies, for example a spinning penny on a table.
• Explain the concepts of generalised coordinates and generalised forces.
• Express the kinetic and potential energies of a system in term of the generalised coordinates, and to use these to obtain Lagrange's equations of motion.
• Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of a system about its equilibrium position.
• Explain the concept of generalized momentum and show how the Hamilton's equations can be used to find the equations of motion.
• Explain the concepts of Poisson brackets, conserved quantities, and canonical transformations.

Content

This module aims to present a systematic approach to the study of dynamics. Once the main techniques have been grasped, a very wide range of problems can be tackled with confidence. The first part of the course presents the tools required to analyse rigid-body motion in three dimensions. These are necessary for a proper understanding of gyroscopic systems, inertial navigation, satellites in space and the stability of high-speed rotating systems such as turbines and compressors. 

The second part of the course deals with Lagrangian and Hamiltonian mechanics, a systematic way to formulate dynamical problems using energy functions.

Introduction and Rigid-body Dynamics (10L)

• Equations of motion of a rigid body in three dimensions.
• The inertia tensor; principal axes.
• Gyroscopes and their application.
• Problems involving rolling bodies.

Lagrangian Mechanics (6L)

• Lagrange's equations; connection to Newton's laws; generalised coordinates and generalised forces.
• Applications to a range of problems.
• Hamilton's equations, Poisson brackets, conserved quantities, canonical transformations.
• Example applications.

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 31/05/2024 07:28