ENGINEERING TRIPOS PART IIA - 2012/2013

Module 3C5 - Dynamics

Courses:

1. Introduction and Rigid-body Dynamics: 2 lectures/week, weeks 1-5, Michaelmas term (Dr H.E.M. Hunt)

2. Lagrangian Mechanics: 2 lectures/week, weeks 6-8, Michaelmas term (Professor R.S. Langley)

AIMS

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.

The specific 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 be able to apply  these methods in straightforward way to a wide range of problems in mechanics.

SYLLABUS

1. Introduction and Rigid-body Dynamics (11L)

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

2. Lagrangian Mechanics (5L)

• Lagrange's equations; connection to Newton's laws; generalise coordinates and generalised forces;
• Applications; calculation of forces of constraint;

OBJECTIVES

On completion of the course students should be able to;

Rigid-body Dynamics

• 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.

Lagrangian Mechanics

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.

Further module selection:

Part IIA: 3C6 Vibration

Part IIB: 4C8 Applications of dynamics

REFERENCES

Please see the 2006/07 Booklist for Part IIA Courses for references for this module.