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, and the Lagrangian formulation.
• To show how to apply these methods in a straightforward way to a wide range of problems in mechanics.

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 concept of generalised coordinates, and their associated generalised forces.
• Express the kinetic and potential energies of simple systems in term of their generalised coordinates, and to use these to obtain Lagrange' equations of motion for the system.
• Explain the link between conservation laws and the form of the energy
• Explain the concept of a constraint, and be able to calculate the force of constraint from Lagrange's equations.
• Approximate the kinetic and potential energies by quadratic forms, and hence deduce the mass and stiffness matrices for small vibration of system about its equilibrium position.

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 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; calculation of forces of constraint.

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

Toggle display of UK-SPEC areas.

 
Last modified: 13/09/2018 14:36