Engineering and the Environment, University of Southampton

Knowledge and understanding
Having successfully completed the module, you will be able to demonstrate knowledge and understanding of:

• Reminder of the wave equations for vibrations in beams and plates.
• Introduction to variational calculus and the Lagrangian.
• 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 and bending free vibrations in beams using the Finite Element Method (FEM).
• 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 Rayleigh-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.
• General purpose concepts for the use of commercial FEM codes.
• Finite element model validation.

Cognitive (thinking) skills
Having successfully completed the module, you will be able to:

• Make appropriate modelling decisions (regarding meshing, element type, boundary conditions, solution type etc) which are consistent with the anticipated structural dynamic behaviour and knowledge of the FE formulation.
• Assess, by comparisons with experimental data as directed, the appropriateness of the model.

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

• Operate the commercial code ANSYS to solve vibration problems on simplified model systems.
• Read, understand and interpret the literature relating to numerical methods in structural dynamics.

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

• Program in Matlab to obtain numerical solutions to large order sets of equations.
• Acquire a working knowledge of new software packages.

  Module Details

  Title: Finite Element Vibration Analysis
  Code: ISVR3011
  Year: Acoustical Engineering Acoustics and Music Part 3
  Semester: Semester 2
  

  CATS points: 10 CAT points (= 100 hours) ECTS 5 ECTS points: NaN
  Level: Undergraduate
  Co-ordinator(s): Dr Timothy Waters, Dr Christopher Jones

  Pre-requisites and / or co-requisites

  Vibration II (ISVR2002); Maths (ISVR2025); Design II (ISVR2007) - for Matlab component

The aims of this module are to:

• Provide a general introduction to the use of the finite element method for obtaining numerical solutions to the vibration response of mechanical systems.

• To give the fundamental concepts of variational calculus, Hamilton and Lagrangian formulation and the Rayleigh and Ritz method for the calculation of natural frequencies and modes of distributed simple structures such as beams.
• 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.

The outline algorithm and language of the finite element method using the known example of one, 2 and 3 degrees of freedom mass-spring models.

• Variational calculus with a simple example application for a different equation.
• The Lagrange equation, Hamilton's principle and virtual work.

Vibration of beams: analytical method

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

Vibration of beams: 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.

Vibration of beams: 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 noded elements.

Free vibration

• Eigenvalue–eigenvector analysis for the calculus of natural frequencies and natural modes.
• Eigenvalue–eigenvector properties and normalisations.
• Guyan reduction of degrees of freedom approach.

Forced vibration

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

FEM computer packages

• Commercial computer packages structure.
• DYNAS survey (DYNamic Analysis of Strucures).
• FEM-experimental analysis of DYNAS tapered beam.
• Improvement of FE models with modal analysis and the modal assurance criterion.

Overview of Finite Element Model Validation

Four lectures per week.

Computing laboratories using proprietary engineering software packages to solve vibration problems. The typical lab class size is 20. 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.

Working on a 3-part formal assignment which is based on a real structure in the laboratory, interpret the analysis using the software provided in the light of measurements made on the structure. The assignment includes some development of the formulation used. Students are encouraged to read supporting texts and a booklist is provided.