The University of Southampton
Courses

# ISVR6133 Advanced Vibration

## Module Overview

Vibration is an important consideration for many engineered structures and is caused by dynamic loads arising during operation, e.g. in all transportation vehicles, motors/generators and buildings. It is also an invaluable diagnostic tool for monitoring the long-term health of structures. This module aims to provide an in-depth understanding of established methods for modelling vibration. These range from analytical methods for simple components such as beams and plates through to numerical methods for built-up structures. Experimental techniques are also covered and the most commonly used method for vibration testing is studied in detail through a practical laboratory.

### Aims and Objectives

#### Module Aims

The aim of this module is to equip students with the knowledge, understanding and application of vibration principles that underpin prudent use of both modelling software and experimental techniques used in industry.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Have an in-depth knowledge of the theoretical framework for both continuous and discrete structural dynamic models;
• understand the benefits and limitations of characterising the dynamics of a structure in terms of its vibration modes and wave behaviour;
• understand the process and limitations of Finite Element Analysis (FEA) and Statistical Energy Analysis (SEA) for built-up structures;
##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Translate mathematical formulations into computer code such as Python or Matlab;
• Question the validity of modelling assumptions in the light of experimental data.
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• perform a vibration based transfer function measurement using an instrumented hammer and a commercial frequency analyser;
• assess the reliability of measured transfer functions;
• process/interpret measured transfer functions using an experimental modal analysis technique.
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• analyse the free and forced behaviour of simple structures using modal and wave approaches;

### Syllabus

The syllabus with approximate allocation of teaching sessions is as follows: Introduction - 1 Lecture: • Terminology. • Review of single degree of freedom systems. • Difficulties of applying conventional numerical methods at high frequency. • Alternatives available for high frequencies. • Applicability of the methods covered in the module Continuous Systems - 5 Lectures • Equations of motion, characteristic equations, free vibration, forced vibration, modal summation. • Shafts. • Timoshenko beam theory and then reduction to Euler-Bernoulli beam theory. • Love-Kirchhoff plate theory. • Difficulties at high frequencies, high frequency approximations. • Mean square response, kinetic energy. • Frequency/space averaging of input power and mobility of finite and infinite systems. Discrete Multiple Degree of Freedom (MDOF) systems - 6 Lectures • Review of 2 DOF & extension to n DOF systems. • Free response (eigen problem), orthogonality and scaling of modes. • Forced response (direct and modal summation). • Use of finite element analysis to obtain system matrices. • Model reduction techniques (Guyan reduction). • Damping matrices, normal and complex modes, loss factor. Experimental techniques - 6 Lectures - 1 Lab (3hrs) • Vibration testing (instrumentation, shaker & hammer testing). • Experimental modal analysis (quad picking, circle fitting, rational fraction polynomial method). Waves - 8 Lectures • Free wave propagation in shafts, beams and plates. • Non-dispersive vs. dispersive waves, cut-off frequencies. • Dispersion equation and curves. • Phase/group velocity. • Characteristic impedances. • Wave energy and power. • Reflection & transmission coefficients. • Wave excitation. Statistical energy analysis - 4 Lectures • 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

#### Special Features

The module includes a practical laboratory to perform a vibration test on a structure using typical state-of-the-art equipment and techniques that are used in industry.

### Learning and Teaching

#### Teaching and learning methods

Teaching sessions include PowerPoint based lectures, tutorials, an in-class quiz and a supervised hands-on laboratory. The three assignments facilitate independent study in the areas of analytical, numerical and experimental methods.

TypeHours
Follow-up work5
Revision44
Tutorial3
Practical classes and workshops3
Completion of assessment task60
Preparation for scheduled sessions5
Lecture30
Total study time150

#### Resources & Reading list

R J M Craik (1996). Sound Transmission through Buildings, using Statistical Energy Analysis.

C. de Silva (2007). Vibration: Fundamentals and Practice, 2nd edition.

R H Lyon (1995). Theory and Application of Statistical Energy Analysis.

F Fahy and D Thompson (2015). Fundamentals of Sound and Vibration.

S. Rao (various). Mechanical Vibrations.

L Cremer and M Heckl (2005). Structure-Borne Sound.

K F Graff (1991). Wave Motion in Elastic Solids.

F J Fahy and P Gardonio (2007). Sound and Structural Vibration.

M. Petyt (1990). Introduction to finite element vibration analysis.

L Meirovitch (1997). Principles and Techniques of Vibrations.

### Assessment

.

#### Summative

MethodPercentage contribution
Assignment 15%
Assignment 15%
Assignment 10%
Exam  (120 minutes) 60%

#### Referral

MethodPercentage contribution
Exam  (120 minutes) 100%

#### Repeat Information

Repeat type: Internal & External