• Provide a general introduction to analytical and numerical techniques in structural vibration.

• To give the fundamental concepts of the approximated method by Rayleigh and Ritz for the calculation of natural frequencies and modes of distributed simple structures such as beams and plates.
• To provide a general introduction to the theory and use of finite element techniques for vibration analysis of practical structures.
• To give the student direct experience of the use of computer software which employs numerical methods for the solution of vibration problems.

• Analytical derivation of the wave equations for vibrations in beams and plates.
• Analytical solution of the wave equation for free vibrations in beams with reference to different types of boundary conditions.
• Formulation of the equation of motion for free vibrations in beams and plates using the Rayleigh-Ritz approximate method.
• Formulation of the equation of motion for axial, torsional and bending free vibrations in beams using the Finite Element Method (FEM).
• Formulation of the equation of motion for in-plane longitudinal, shear and out of plane flexural vibrations using the Finite Element Method.
• Numerical reduction of a FEM by means of geometrical considerations (symmetry, periodicity, constraints) and using the Guyan reduction technique.
• Eigenvalue-eigenvector analysis of the matrix equation of motion derived with the Raylegh-Ritz or FEM methods in order to determine the natural frequencies and natural modes of the structure modelled.
• Eigenvalue-eigenvector intrinsic (orthogonality) and normalisation properties (construction of the modal matrix for the forced vibration analysis).
• Forced vibration analysis using the direct and modal methods with reference to both harmonic and periodic excitations.
• Forced vibration analysis using a finite difference time domain approach with reference to a transient excitation.
• General purpose concepts for the use of commercial FEM codes.

• Read, understand and interpret the literature relating to analytical and numerical methods in structural dynamics.
• Recognise and select appropriate techniques for the solution of analytical and numerical problems in structural dynamics.

• Build with MatLab a FEM code for vibrations in beams.
• Use the commercial code ANSYS to solve vibration problems on simplified model systems.

• Acquire a working knowledge of new software packages.
• Acquire a critical thinking about ways of modelling systems with FEM packages and about the results obtained from a FEM analysis with a commercial package. 

4 lectures per week.

Computing laboratories using proprietary engineering software packages to solve vibration problems. The typical lab class size is 20. Two lecturers assist the students to work through the computing sessions. Feedback is given by advice and assistance in the laboratory session.

Students join the course with widely varying experience of using such packages and this is dealt with by proportionate assistance during the computing laboratory sessions.

Students need to work in their own time to complete the laboratory work and are able to go to the lecturers for assistance. 

• Tutorial assistance to cover issues raised through example sheets.
• Computing laboratories provide informal assessment through individual interaction.
• Interactive web notice-board that enables students to write and share comments and observations about the subject.

The first assignment tests students’ ability to apply the knowledge of the mathematical models to carry out analytical modelling correctly and test their ability to interpret the results of models.

The second assignment tests students’ ability to carry out numerical modelling and to build up a FEM code.

The third assignment tests students’ ability to use the ANSYS FEM package to model and carry out free and forced vibration analysis.

The comparison of the results obtained in the three assignments will provide an indication of the capability of critical analysis of the results obtained with the analytical and numerical (Rayleigh-Ritz and FEM) approaches.

Waves in beams

• Derivation of Euler-Bernoulli wave equation for bending vibration of slender beams.
• Derivation of Timoshenko wave equation for bending vibration of deep beams.
• Analytical derivation of natural frequencies and natural modes for bending vibration of slender and deep beams.

The Rayleigh-Ritz approximate method

• The general principle.
• Spatial given functions and convergence criteria.
• Matrix formulation and eigenvalue-eigenvector analysis for the calculus of natural frequencies and natural modes.
• Example: bending vibration of a clamped–simply supported beam.

Introduction to Finite Element Method (FEM)

• The methodology.
• Spatial given functions and convergence criteria.
• Matrix formulation and eigenvalue-eigenvector analysis for the calculus of natural frequencies and natural modes.
• Examples: axial and bending vibration of a clamped–simply supported beam.

FEM for in-plane and out-of-plane vibrations of plates

• The linear rectangular element.
• The linear quadrilateral element.
• Eight nodes elements.
• Eight nodes elements with curved sides.

Free vibration

• Eigenvalue-eigenvector analysis for the calculus of natural frequencies and natural modes.
• Eigenvalue-eigenvector properties and normalisations.
• Methods to solve large eigenvalue–eigenvector problems.
• Reduction of degrees of freedom base on geometrical considerations.
• Guyan reduction of degrees of freedom approach.

Forced vibration

• Modal formulation, modal coordinates.
• Structural and viscous damping.
• Steady state response to harmonic excitation.
• Steady state response to periodic excitation.
• Response to transient excitation.

FEM computer packages

• Commercial computer packages structure.
• DYNAS survey (DYNamic Analysis of Strucures).
• FEM–experimental analysis of DYNAS tapered beam.

As well as the lecturer assigned to this course, there are other two lectures that organise the computer laboratories.

A lecture room with 30 seats is required for four hours a week. The room should be equipped with overhead projection facilities, and blackboard and/or whiteboard. The occasional use of a data projector is required.