ES38615 Dynamics of Vibrating Systems
Introductory description
ES38615 Dynamics of Vibrating Systems
Module aims
Vibrations exert a significant influence on the performance of the majority of engineering systems. All engineers should understand the basic concepts and all mechanical engineers should be familiar with the analytical techniques for the modelling and quantitative prediction of behaviour. Thus, this module provides students with fundamental skills necessary for the analysis of the dynamics of mechanical systems, as well as providing opportunities to apply these skills to the modelling and analysis of vibration.
This thirdyear module is mandatory for students pursuing a degree in Mechanical Engineering, building upon competences acquired earlier in the course. This module introduces students to the use of Lagrange’s equations (applied to 1D and 2D systems only for this module) and to techniques for modelling both lumped and continuous vibrating systems. It includes some coverage of approximate methods both as an aid to physical understanding of the principles and because of their continuing usefulness. The module assumes basic understanding of mechanics and linear algebra consistent with the level of Year 2 modules.
At the end of the module students should have a sound understanding of the wide application of vibration theory and of the underlying physical principles. In particular, they should be able to use either Newtonian or Lagrangian mechanics to analyse 2D systems, and to determine the response of simple damped and undamped multidegrees of freedom (DOF) systems to both periodic and aperiodic excitation. They should also be familiar with engineering solutions for measuring and influencing vibrational behaviour.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Generalised coordinates, Lagrange's equation (including preliminary study of other classical methods)
 General application of the Lagrange equation to vibrating systems
 Multidegree of freedom systems: lumped system models, continuous system models; geared and branched systems; reduction of an nDOF system to a set of n singleDOF systems; principal coordinates
 Matrix methods of analysis: conservative and nonconservative (damped) systems; determination of principal coordinates
 Modelling of damping: hysteretic, Coulomb, viscous; measurement of damping factor
 Forced vibration: harmonic excitation of multiDOF systems; shaft whirling; transmissibility; vibration isolation; nonharmonic and arbitrary excitation (convolution integral)
 Approximate methods e.g. Rayleigh's method, Dunkerley's method
Learning outcomes
By the end of the module, students should be able to:
 1. Model planar mechanical systems using Newton’s and Lagrange’s equations: Determine appropriate coordinate systems, analyse vibrations.
 2. Abstract more complex engineering mechanisms: analyse using lumped system models or simple distributed mass and stiffness models. Use and justify standard methods and approximations for extended and continuous vibrating systems.
 3. Evaluate the natural frequencies and modes of vibration of a multidegree of freedom linear system.
 4. Determine and analyse the free and forced response of singledegree of freedom systems to periodic and aperiodic excitation, as well as the effects of linear and nonlinear damping on the system behaviour.
 5. Evaluate complex (multidegree of freedom) undamped or damped systems numerically, using a systematic approach to analyse the natural frequencies and modes, and the response of the system to periodic and aperiodic excitations.
 6. Demonstrate a sound understanding of the application of vibration analysis to key engineering systems.
Indicative reading list
 Theory of Vibration with Applications, by W. T. Thomson and M. D. Dahleh. Publisher: Pearson. Fifth edition, 1998. ISBN10: 013651068X, ISBN13: 9780136510680.
 Principles of Vibration, by B. H. Tongue. Publisher: Oxford University Press. Second edition, 2002. ISBN10: 0195142462.
 Engineering Vibrations, by D. J. Inman. Publisher: Pearson. Fourth international edition, 2013. ISBN10: 0273768441, ISBN13: 9780273768449.
 Mechanical vibrations, by S. S. Rao, Fook Fah Yap. Publisher: Prentice Hall. Fifth edition in SI units, 2011. ISBN10: 9810687125, ISBN13: 9789810687120
 Vibrations, by B. Balachandran, E. B. Magrab. Publisher: Cengage. Second international SI edition, 2009. ISBN10: 0495411256, ISBN13: 9780495411253.
View reading list on Talis Aspire
Subject specific skills
SSS4: Ability to apply relevant practical and laboratory skills.
SSS8: Ability to be pragmatic, taking a systematic approach and the logical and practical steps necessary for, often complex, concepts to become reality.
Transferable skills
TS1: Numeracy: apply mathematical and computational methods to communicate parameters, model and optimize solutions.
TS2: Apply problem solving skills, information retrieval, and the effective use of general IT facilities.
TS3: Communicate (written and oral; to technical and nontechnical audiences) and work with others.
TS7: Overcome difficulties by employing skills, knowledge and understanding in a flexible manner.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (25%) 
Seminars  2 sessions of 1 hour (2%) 
Practical classes  1 session of 2 hours (2%) 
Private study  86 hours (72%) 
Total  120 hours 
Private study description
Guided independent learning, assignment preparation, etc 86 hours.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D4
Weighting  Study time  Eligible for selfcertification  

Vibration computational and analysis assignment  30%  Yes (extension)  
Matlab code submitted on Matlab Grader (10% of module credit) and a brief computational report (1000 words, 20% of module credit) 

Online Examination  70%  No  
QMP test (2x 1h) ~Platforms  AEP,QMP

Feedback on assessment
 Feedback during laboratory sessions
 Feedback on assignments.
 Model solutions to exam type questions.
 Support through advice and feedback hours.
 Cohort level feedback on examinations
Prerequisites
To take this module, you must have passed:
Courses
This module is Core for:
 Year 3 of UESAH310 BEng Mechanical Engineering
 Year 3 of UESAH315 BEng Mechanical Engineering
 Year 4 of UESAH314 BEng Mechanical Engineering with Intercalated Year
 Year 3 of UESAHH35 BEng Systems Engineering
 Year 3 of UESAHH36 BEng Systems Engineering
 Year 4 of UESAHH34 BEng Systems Engineering with Intercalated Year
 Year 3 of UESAH311 MEng Mechanical Engineering

UESAH316 MEng Mechanical Engineering
 Year 3 of H315 Mechanical Engineering BEng
 Year 3 of H316 Mechanical Engineering MEng
 Year 4 of UESAH317 MEng Mechanical Engineering with Intercalated Year

UESAHH31 MEng Systems Engineering
 Year 3 of HH31 Systems Engineering
 Year 3 of HH35 Systems Engineering
 Year 4 of UESAHH32 MEng Systems Engineering with Intercalated Year
This module is Core optional for:
 Year 3 of UESAH115 MEng Engineering with Intercalated Year

UESAH317 MEng Mechanical Engineering with Intercalated Year
 Year 3 of H317 Mechanical Engineering with Intercalated Year
 Year 4 of H317 Mechanical Engineering with Intercalated Year
 Year 4 of UESAHH32 MEng Systems Engineering with Intercalated Year
 Year 3 of UESAH11L Undergradaute Engineering (with Intercalated Year)
This module is Optional for:
 Year 3 of UESAH113 BEng Engineering
 Year 3 of UESAH114 MEng Engineering
 Year 4 of UESAH115 MEng Engineering with Intercalated Year

UESAH11L Undergradaute Engineering (with Intercalated Year)
 Year 3 of H11L Engineering (with Intercalated Year)
 Year 4 of H11L Engineering (with Intercalated Year)
This module is Option list A for:
 Year 4 of UESAH111 BEng Engineering with Intercalated Year
 Year 3 of UESAH112 BSc Engineering