Fundamentals of Vibration

Learning Outcomes

Disciplinary Specific Learning Outcomes

Having successfully completed this module you will be able to:

• Select appropriate techniques for the solution of analytical problems in vibrations.
• Describe the behaviour of structures by modal and wave approaches.
• Show understanding of the benefits and limitations of basic vibration control methods
• Undertake measurement of vibration quantities and measure modes of vibration on a simple structure
• Develop the equations of motion for free and forced vibration of simple systems.
• Interpret the behaviour of vibrating systems through an understanding of basic principles and the role of mass, stiffness and damping.

Full CEng Programme Level Learning Outcomes

Having successfully completed this module you will be able to:

• Students take part in two laboratory classes in which they explore the dynamics of two-degree-of-freedom and continuous systems through measurements and analysis.
• As a part of the laboratory coursework, students become familiar with using lab equipment, instrumentation and experimental procedures. The coursework requires the students to understand the limitations of various measurement approaches, interpret the data and are able to implement different preventative vibration measures. The choice of the methodologies and their consequences are subsequently understood from a practical viewpoint to complement the simple theoretical models.
• As a part of the laboratory coursework, students must demonstrate an understanding of the Euler-Bernoulli beam theory, its assumptions and the effect of real boundary conditions, which sometimes differ from the idealised ones.
• As a part of the written exam, the students must demonstrate the ability to use core methods for vibration modelling and analysis, which includes formulating the equations of motion for vibrating systems, solving them, formulating conclusive observations, and suggesting ways for mitigating undesired vibration. Through the tutorial sheets, the students gain practice and develop their skills in these areas before the final assessment.
• As a part of the laboratory coursework, students investigate the effect of different vibration mitigation techniques attempted in the laboratory in the light of uncertainties of the system’s parameters and variability associated with environmental factors, measurement procedure and modifying the system on the fly.

1. Introduction (2 sessions)

Vibration problems in engineering. Terminology. Basic principles.

2. Single degree of freedom system (6 sessions)

Free vibration of mass-spring system: natural frequency. Free vibration with damping: damping factor. Energy methods. Time harmonic forced vibration: resonance. Isolation, base excitation and other applications. Structural damping. Transient vibration; response to transient excitation; Duhamel’s Integral; forced vibration with various loads including earthquake; impulse response.

3. Multiple degree of freedom systems (4 sessions)

Free vibration of two degree of freedom systems: developing stiffness and mass matrices for lumped parameter systems, modes of vibration, natural frequencies and mode shapes. Multiple degree of freedom systems: matrix methods. Time harmonic forced vibration with damping: modal decomposition. Time domain solutions. Multi-storey structure. Introduction to vibration treatment and dynamic vibration absorbers.

4. Modelling methods (3 sessions) Rayleigh’s method. Lagrange’s equations for free undamped vibration. Applications. Finite Element Method (FEM) basis.

5. Continuous systems (6 sessions) Free vibrations of strings, bars and shafts: equation of motion, boundary conditions. Modes of vibration: natural frequencies and mode shapes. Bending vibration of

beams. Forced vibration of continuous systems: modes and resonance. Introduction to structural wave motion in one dimension: propagation, reflection and transmission; coincidence.

6. Fundamentals of Vibration Control (5 sessions) Sources of vibration. Mobility/Impedance Methods. Vibration Isolation: mobility approach, energy considerations. . Vibration Damping (different

types of damping, viscoelastic damping, damping treatments, measurements of damping).

7. Vibration measurement and analysis (Lab sessions) Vibration transducers and measurement of frequency response functions and modal damping.

This is a one-semester module, three lectures per week with two laboratory sessions. The students are provided with copies of the lecture notes. These include problems worked through in class to demonstrate problem formulation and solution, and the thought processes involved. Students are encouraged to discuss difficulties informally with the lecturer. Additional tutorials are provided throughout the duration of the course as preparation for the examination. One-to-one assistance and verbal feedback is facilitated through four tutorial classes. Past exam papers are supplied to aid personal study, feedback and revision. Blackboard is used to allow the lectures and additional material to be disseminated (including solutions to past exam papers). The students have to write-up two laboratory reports. Students are encouraged to read supporting texts and a booklist is provided. Laboratory classes: ‘Multi storey structure’ and ‘Beam Vibration Measurement’.

The laboratory classes will be supervised by a member of academic staff or a suitably qualified postgraduate student. The classes will be restricted to 2-4 students per rig. Students need to work in their own time to complete the laboratory work and are able to go to the lecturer for assistance. Feedback on laboratory reports will be returned to the student normally within three weeks of the submission deadline. A laboratory sheet provides instructions, information and assistance with the experimental procedure.

Feedback to students during module study is obtained by:

• Tutorial assistance to cover issues raised through example sheets
• Informal contact with lecturer encouraged after lectures and at other times.
• Model answers to selected problems are provided.
• Solution notes and comments to selected problems are made available.
• Previous examination papers with model answers are made available.
• Laboratory class provides informal assessment through individual interaction.
• Exam: formal written feedback after exam

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.