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ES386 Dynamics of Vibrating Systems

year.jpg15_cats.jpgModule Leader: Dr. P.Brommer

Co-lecturer: Dr. J.R.Kermode

Module Ambassadors: TBD (Systems) TBD (Mechanical)

Module Information


This 15 CATS module is one of the third year modules for:

Core: Optional:
  • Mechanical Engineering
  • Systems Engineering


  • Engineering

This module is a pre-requisite for ES480 Dynamic Analysis of Mechanical Systems and is recommended for students on the optional courses who wish to take the Robotics Elective in Year 4.



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 third-year 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 multi-degrees of freedom (DOF) systems to both periodic and aperiodic excitation. They should also be familiar with engineering solutions for measuring and influencing vibrational behaviour.

Learning Outcomes

By the end of the module the student should be able to...

  • Model planar mechanical systems using Newton’s or Lagrange’s equations: Determine appropriate co-ordinate systems, analyse vibrations.
  • Abstract more complex engineering mechanisms: analyse using lumped system models or simple distributed mass and stiffness models.
  • Evaluate the natural frequencies and modes of vibration of a multi-degree of freedom damped or undamped linear system.
  • Use initiative to apply sensible approximations to develop practical models of complex vibrating systems, to find key parameters such as the natural frequency.
  • Evaluate complex (single- and multi-degree of freedom) systems and via a systematic approach show an understanding of the response of the system to periodic excitations.
  • Demonstrate a sound understanding of the application of vibration analysis to key engineering systems.


  • Generalised co-ordinates, Lagrange's equation (including preliminary study of other classical methods)
  • General application of the Lagrange equation to vibrating systems
  • Multi-degree of freedom systems: lumped system models, continuous system models; geared and branched systems; reduction of an n-DOF system to a set of n single-DOF systems; principal co-ordinates
  • Matrix methods of analysis: conservative and non-conservative (damped) systems; determination of principal co-ordinates
  • Modelling of damping: hysteretic, Coulomb, viscous; measurement of damping factor
  • Forced vibration: harmonic excitation of multi-DOF systems; shaft whirling; transmissibility; vibration isolation; non-harmonic and arbitrary excitation (convolution integral)
  • Approximate methods e.g. Rayleigh's method, Dunkerley's method

Teaching Methods

This module includes 30 hours of lectures and 2 hours of revision classes.

Required self-study: 118 hours


A 15 CATS module: 80% examined via a 3 hour paper:

Exam rubric information

  • 4 Compulsory Questions

and 20% assessed consisting of a computational assignment.