ES386 Dynamics of Vibrating Systems

Module Leader: Dr I.K. Liu

Co-lecturer: Dr X. Liu

Module Information

Scope

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

Core:

Optional:

Mechanical Engineering

Automotive Engineering
Engineering
Systems 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.

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. This module therefore 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 immediate usefulness.

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-DOF systems to both periodic and aperiodic excitation.

Learning Outcomes

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

Identify and apply an appropriate co-ordinate system for the modelling of planar mechanical systems by Newton’s or Lagrange’s equations
Use either Newtonian mechanics or Lagrangian mechanics to analyse the vibration of planar systems
Make simplifying approximations to more complex engineering mechanisms to enable analysis using a lumped system model or a simple distributed mass and stiffness model.
Determine the natural frequencies and modes of vibration of a multi-DOF damped or undamped linear system using standard matrix methods.
Determine estimates for the fundamental natural frequency of an undamped or lightly damped vibrating mechanism using approximate methods.
Determine the response of a single DOF system to aperiodic excitation and of a multi-DOF system to periodic excitation.
Appreciate the application of the methods referred to above to important engineering systems.

Syllabus

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