| Course Unit Code | 330-0512/01 |
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| Number of ECTS Credits Allocated | 5 ECTS credits |
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| Type of Course Unit * | Compulsory |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | First Year |
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| Semester when the Course Unit is delivered | Summer Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| POD10 | doc. Ing. Jiří Podešva, Ph.D. |
| FUS76 | doc. Ing. Martin Fusek, Ph.D. |
| MAW007 | doc. Ing. Pavel Maršálek, Ph.D. |
| Summary |
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1. Dynamics and FEM
2. Mass matrix
3. Equations of motion of elastic systems
4. Natural frequencies and mode shapes
- properties and normalization of mode shapes
- methods for computing eigenvalues and eigenvectors
5. Reduction of the number of DOf in dynamics
6. Response history: modal method
- proportional damping matrix
- vibration caused by initial conditions
- harmonic response
- general excitation
7. Response history: direct integration method (implicit and explicit methods)
8. Principles of solution nonlinear static problems, fundamental numeric solution of contacts in FEM
9. Newton-Raphson method, arc-length method
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| Learning Outcomes of the Course Unit |
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Outline principle of derivation mass matrix in slope-deflection variant of FEM
Identify meaning solution of natural frequencies and mode shapes and classify differences for Bernoulli and Timoshenko beam
Define methods for numerical computing of eigenvalues and eigenvectors for undamped systems
Construct reduction method for easy problem of natural frequency
Solve matrix equation of motion by modal method
Classify direct integration method, compare implicit and explicit method
Clarify solution principle of nonlinear static problems
Relate methods for analysis contacts problem by FEM
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| Course Contents |
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1.0 Comparison of static and dynamic problems solved by FEM
2.0 Finite element mass matrix
2.1 Bar element
2.2 Beam element
2.2.1 Bernoulli‘s beam, diagonal mass matrix
2.2.2 Thimoshenko‘s beam, the second spectrum of natural frequencies
2.3 Plane and space frame finite element
2.4 Selected types of finite elements
3.0 Natural frequencies and mode shapes of un-damped vibration
3.1 Matrix equation for solving natural frequencies and mode shapes (eigenvectors)
3.2 Normalization of the eigenvectors
3.3 Solution methods
3.3.1 Solutions using determinant, disadvantages
3.3.2 Power method (method of inverse iterations)
3.3.3 Subspace method (method of simultaneous iterations)
3.3.4 Lanczos method
4.0 Reduction method in dynamics
5.0 Damping matrix
6.0 Solving response of the linear mechanical system by using mode shapes (Modal analysis method)
6.1 Self-induced oscillations by changing the initial conditions
6.2 Harmonic excitation
6.3 General continuous excitation
7.0 The response solution of nonlinear dynamical systems
7.1 Implicit method
7.2 Explicit method
8.0 Buckling (collapse of structures shape)
8.1 Introduction
8.2 Geometric stiffness matrix of bar FE
8.3 Geometric stiffness matrix of plane frame FE (compression-tension + bending)
8.4 Practical examples
9.0 Solution of nonlinear static tasks
9.1 Introduction
9.2 Newton-Raphson method
10. Contacts in FEM
10.1. Introduction
10.2. Penalty method
10.3. Langrange multiplier method
10.4. Augmented Lagrange method
10.5. Partitioning (Semi-analytical) method
10.6. Example |
| Recommended or Required Reading |
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| Required Reading: |
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Cook R. D., Malkus D.S., Plesha M.E., Witt R.J. CONCEPTS AND APPLICATIONS OF
FINITE ELEMENT ANALYSIS. 4th edition. J. Wiley & Sons, Inc. NY, 2002, p. 719,
ISBN 0-471-35605-0
Examples for ANSYS solutions: http://www.mece.ualberta.ca/tutorials/ansys/
REDDY, J.N., An Introduction Nonlinear Finite Element Analysis, Oxford
University Press, 2004, p. 463, ISBN 0-19-852529-X
BHATTI, M. A., Fundamental Finite Element Analysis and Applications: with
Mathematica and Matlab Computations, Wiley, 2005, p.590, ISBN 0-471-64808-6
HORYL, P. FEM Finite Element Method Introduction,2014,https://www.fs.vsb.cz/330/cs/MKP-II/
|
Cook R. D., MaFElkus D.S., Plesha M.E., Witt R.J. CONCEPTS AND APPLICATIONS OF
FINITE ELEMENT ANALYSIS. 4th edition. J. Wiley & Sons, Inc. NY, 2002, p. 719,
ISBN 0-471-35605-0
Examples for ANSYS solutions: http://www.mece.ualberta.ca/tutorials/ansys/
REDDY, J.N., An Introduction Nonlinear Finite Element Analysis, Oxford
University Press, 2004, p. 463, ISBN 0-19-852529-X
BHATTI, M. A., Fundamental Finite Element Analysis and Applications: with
Mathematica and Matlab Computations, Wiley, 2005, p.590, ISBN 0-471-64808-6
HORYL, P. FEM Finite Element Method Introduction,2014,https://www.fs.vsb.cz/330/cs/MKP-II/
|
| Recommended Reading: |
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BHATTI,M.A., Advanced Topics in Finite Element Analysis of Structures: with
Mathematica and Matlab Computations, Wiley, 2006, p.590, ISBN-13 978-0-471-
64807-9
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BHATTI,M.A., Advanced Topics in Finite Element Analysis of Structures: with
Mathematica and Matlab Computations, Wiley, 2006, p.590, ISBN-13 978-0-471-
64807-9
|
| Planned learning activities and teaching methods |
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| Lectures, Tutorials, Project work |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 35 | 20 |
| Examination | Examination | 65 | 16 |