ENGINEERING TRIPOS PART IIA - 2012/2013
Module 3C6 - Vibration
1. Vibration of Continuous Systems: 2
lectures/week, weeks 1-4 Lent term (Professor J. Woodhouse)
2. Vibration of Lumped Systems: Rayleigh's method, 2 lectures/week, weeks 5-8 Lent term (Dr D. Cebon)
course aims to present a systematic approach to the study of small vibration of
engineering components and structures. The course picks up where Part IA Linear
Systems and Vibration left off. Concepts which were barely discussed (e.g.
reciprocity and the orthogonality of vibration modes) are important for
building up qualitative insights into vibration behaviour. Alongside the
mathematical tools for quantitative analysis the course offers vital
ingredients for an engineer's education.
specific aims are:
To introduce the central ideas and tools
for the analysis of small (linear) vibration in mechanical systems:
simple continuous systems which may be combined as components of
larger systems; and the general approach to lumped systems via mass and
stiffness matrices, and the resulting properties of vibration modes
and their frequencies.
Vibration of Continuous Systems (8L)
- Vibration of strings; axial and transverse
vibration of beams, torsional vibration of circular shafts;
- Modal analysis of simple systems;
- Response to impulse and harmonic
- Coupling of systems;
- Rayleigh's method for continuous systems.
Vibration of Lumped Systems (6L)
- Application of Lagrange's equations to
small vibrations; undamped vibration of systems with N degrees of freedom;
- Matrix methods and modal analysis;
- Response functions in frequency and time
domains; properties of frequency-response functions; reciprocal
- Modal damping and modal overlap;
- Rayleigh's method for discrete systems.
completion of the course students should be able to:-
Vibration of Continuous Systems
Vibration of Lumped Systems
- Derive the partial differential equations
governing the forced or free motion of uniform one-dimensional
- Use these equations and appropriate
boundary conditions to obtain vibration modes and natural frequencies.
- Be familiar with musical intervals and
how these are useful in vibration diagnostics.
- Analyse continuous systems using modal
- Compute impulse and harmonic response by
modal and direct methods.
- Calculate the response of a coupled
system from a knowledge of the responses of the separate subsystems.
- Apply Rayleigh's method to continuous
- Take advantage of the link between
Lagrange's equations and small vibration.
- Explain the concept of a vibration mode,
and be able to find the modes and their natural frequencies by an
- Understand orthogonality of modes, modal
damping, modal density and modal overlap factor.
- Express the frequency response functions
or the impulse response functions of a system in terms of the
normal modes, and be familiar with the concepts of resonances and antiresonances.
- Recognise and apply the reciprocal
theorem for responses.
- Use the stationary property of normal
mode frequencies to estimate frequencies given assumed mode shapes,
using minimisation with respect to any free parameters.
selection in Part IIB
year modules that follow on from module 3C7 are:
- 4C6 Advanced linear vibration
- 4C7 Random and non-linear vibration
Please see the Booklist for Part IIA Courses for references for this module.
Last updated: June 2012