Engineering and the Environment, University of Southampton

Knowledge and understanding
Having successfully completed the module, you will be able to:

• Demonstrate understanding of the modal behaviour of discrete and continuous systems.
• Demonstrate understanding of the need for modelling techniques dedicated to high frequency problems and of the assumptions inherent in these methods.
• Calculate the response of infinite rods and beams to a point force and demonstrate the relevance to finite structures.
• Derive approximate formulae for the natural frequencies and modal densities of beams and plates using the phase closure principle.
• Explain wave reflection and transmission at junctions between simple structures.
• Construct appropriate models of simple structures using SEA.

Cognitive (thinking) skills
Having successfully completed the module, you will be able to:

• Distinguish and select appropriate techniques for the solution of vibration problems.
• Formulate and solve analytical models for wave propagation in structures, such as rods, beams and plates.

Practical, subject specific skills
None.

Key transferable skills
None.

The aims of this module are to:

• enable you to solve vibration problems using appropriate analytical and energy methods;
• provide the theoretical background to facilitate a knowledge and understanding of these methods and their limitations.

• To develop your understanding of vibration analysis methods, modes of vibration and modal analysis.
• To develop your understanding of vibration analysis methods for high frequency structural vibration.
• To derive wave analysis techniques applied to a variety of structures.
• To introduce the assumptions and theoretical background of statistical energy analysis.

Introduction

• Terminology.
• Review of single degree of freedom systems.
• Difficulties of applying conventional numerical methods at high frequency.
• Alternatives available for high frequencies.

Modal analysis of continuous structures

• Review of multiple degree of freedom systems.
• Free vibration and modes of vibration.
• Forced vibration, modal decomposition.
• Experimental modal analysis.

Modal analysis at high frequencies

• Difficulties at high frequencies, high frequency approximations.
• Mean square response, kinetic energy.
• Frequency average input power and mobility of infinite system.

Elastic wave motion in rods and beams

• Longitudinal, torsional, flexural wave equations.
• Wave solution, dispersion diagrams.
• Energy flow in propagating waves, group velocity, damping.

Forcing, reflection and transmission in beams

• Receptance of infinite or semi-infinite beam excited by force or moment, input power.
• Reflection of wave at different types of boundary.
• Interaction of wave in beam with a discontinuity.

Prediction of natural frequencies by wave approach

• Use of phase closure principle for modes in a finite beam.
• Comparison with exact analysis.
• Mode count and modal density of 1D systems.

Waves in plates

• Bending and in-plane waves.
• Boundary conditions.
• Reflection at an edge, phase closure, modal density.
• Transmission at simple support.

Statistical energy analysis

• Introduction: power and energy, power balance, coupling power proportionality.
• SEA equations, weak and strong coupling.
• Energy equations of a simple oscillator, coupled oscillators and multi-modal systems.
• Wave transmission and coupling loss factors, structural-acoustic coupling.
• SEA modelling.
• Problems and pitfalls with SEA.
• Experimental SEA.

2 lectures a week.

Example sheets are provided so that students can practise their analytical skills and these are backed up with worked answers and tutorial sessions. Students are encouraged to read supporting texts and a booklist is provided.

Secondary text

Sound Transmission through Buildings, using Statistical Energy Analysis, 1996, R J M Craik, Gower
0566075725

Theory and Application of Statistical Energy Analysis
2nd Edition, 1995, R H Lyon
R G DeJong, Butterworth-Heinemann
0750691115

Sound and Structural Vibration, 1985, F J Fahy, Academic Press
0122476700 hbk
0122476719 pbk

Structure-Borne Sound
2nd Edition, 1988
1st Edition, 1973, L Cremer
M Heckl
E E Ungar, Springer-Verlag (Berlin)
0471847389

Principles and Techniques of Vibrations, 1997, L Meirovitch, Prentice-Hall
0132704307

Wave Motion in Elastic Solids, 1991, K F Graff, Dover Publications
0486667456