Engineering and the Environment, University of Southampton

Knowledge and understanding
On completion you will be able to model and provide appropriate solutions for MDOF systems. Having successfully completed the module, you will be able to demonstrate the formulation of the equations of motion using Lagrange's equations, a matrix approach for the solution and understand how the solutions are produced. You should be able to extend the analysis of continuous systems and understand the solutions for the axial vibration of rods and the flexural vibration of beams.

Cognitive (thinking) skills
You should be able to produce the appropriate mathematical model for more advanced vibrational systems. You should develop and apply the solutions of the equations of motion to problems for free and forced vibration under harmonic excitation.

Practical, subject specific skills
Having sucessfully completed the module, you will be able to:

• Carry out modelling and measurements on both MDOF and continuous vibrating systems.
  
• Understand the application of practical vibration control measures and know their limitations and applicability.

Key transferable skills
Having successfully completed the module, you will be better able to:

• Carry out experimental work and formulate analytical models and solutions.
  
• Write-up laboratory work and provide critical analysis and conclusions.
  
• Produce a formal technical report.

To introduce the analysis and vibrational behaviour of multi-degree of freedom and continuous elastic systems.

To cover the theoretical appraoches for free and forced vibration response. To introduce the effect of boundaries and discontinuities, wave motion and modal orthogonality. The students on completion of the unit should be able to:
· Predict the free and forced response of multi-degree of freedom systems.
· Analyse flexural vibration of beams and longitudinal vibration of rods
· Apply Rayleigh’s method for approximate natural frequency calculations.

1. Vibration of multi-degree of freedom systems
Lagrange’s equations of motion. Free and forced response using direct solutions of the equations of motion. Forced response by Modal analysis. Introduction to orthogonality and generalised co-ordinates.

2. Free and forced vibration of continuous elastic systems
Longitudinal vibration of rods. The effects of boundaries and discontinuities on longitudinal vibration transmission. Phase closure principle and natural frequencies.

3. Rayleigh’s method for the fundamental natural frequency
Applications to beams with discrete masses and springs attached, the effect of rotation and different boundary conditions.

4. Flexural vibration of beams
Derivation of the equation of motion and the procedure to obtain free vibration solutions. Introduction to forced response (harmonic).

Lectures in Powerpoint notes provided. Tutorial sheets and past exam papers supplied to aid personal study, feedback and revision. Blackboard used to allow the lectures and additional material to be disseminated (including past exam paper solutions).

Lectures, laboratory classes and tutorial/revision classes.

Core text

Mechanical Vibrations (3rd edition), S.S. Rao, Addiston Wesley
ISBN 19950-201-59289-4