Machine Dynamics and Vibrations

Module aims

The course aims to enable students to evaluate the dynamic response requirements of a proposed machine design and to produce workable proposals for its safe and effective operation. It will build on the second year Mechatronics and Solid Mechanics courses, introducing a greater range of examples where the dynamic response of a machine must be controlled and making the link between vibration and fatigue failure. This will involve some new subject matter in the vibration of continuous systems, rotor dynamics, signal processing and control analysis. A key aspect of the course is to demonstrate practical vibration measurements and to compare them to solutions of an idealised system in MATLAB. They will also deal with realistic problems involving the assessment of the fatigue life of structures subject to vibration and static loading.

Learning outcomes

On successfully completing this module, students will be able to: 

• Describe simple mechanical systems as lumped parameter models and derive their equations of motion (EOM);

• Explain the alternative methods of deriving the EOMs;

• Predict the natural frequencies and mode shapes of multi-degree of freedom (MDOF) vibration systems; 

• Predict the natural frequencies in bending, torsion and axial vibration of simple continuous systems such as beams; 

• Use MATLAB to obtain solutions to MDOF vibration problems;

• Explain the importance of avoiding shaft whirl and be able to calculate critical speeds;

• Explain the significance of vibration loading in generating high cycle fatigue;

• Predict fatigue life of components by linking their vibration response to alternating stress and using this as input to fatigue calculations;

• Explain the use of Fourier analysis to convert both single and dual channel time domain measurements in the frequency domain;

• Use the frequency domain solutions to predict time domain vibration responses and compare those results to numerical solution techniques.

Module syllabus

Multi degree of freedom systems (MDOF):
• Derivation of equations of motion for mechanical lumped parameter systems
• Forced response of MDOF systems; vibration absorber
• Free and forced response of continuous systems (beams, shafts in torsion and bending)
Vibrations in the frequency domain:
• Frequency response function (FRF) and their main features 
• Fourier Series, Analysis and the Fourier Transform; FFT
• Leakage, Windowing and Aliasing
• Time domain response prediction using Fourier transform (DuHamel’s principle)
Rotor dynamics
• Response to out of balance
• Whirling of shafts
• Critical speeds
• Balancing, static balance and dynamic imbalance
Vibration induced fatigue
• Calculation of stress from vibration mode shape and amplitude
• Input to fatigue calculations; Goodman diagrams
Vibration measurements
• Transducers: linear displacement, angular displacement, angular velocity,  pressure
• Data acquisition:  resolution, aliasing


Pre-requisites: ME2-hDYN; ME2-hSAN; ME2-hMTX; ME1-hCPT; ME2-hCPT or equivalent.

Teaching methods

  • Duration: Autumn and Spring terms
  • Lectures: 1hr each week
  • Tutorials: 1hr every other week
  • Laboratory exercises: tutorial is sometimes held in the PC suite when the main activity is MATLAB based.

Summary of student timetabled hours












Expected private study time

3-4hrs per week plus exam revision


Written examinations:

Date (approx.)

Pass mark

Machine Dynamics and Vibrations (3hrs)


A Data and Formulae book is provided.

This is a CLOSED BOOK Examination

April/ May



Coursework (including progress tests, oral presentations etc.)

Submission date

Pass mark



Weekly Quizes

Generic, in lecture following quiz.




Module leaders

Dr Frederic Cegla