Course code: wb1416

Course name: Numerical Methods in Engineering Dynamics

This concerns a Course

ECTS credit points: 3

Faculty of Mechanical Engineering and Marine Technology

Section of Engineering Mechanics

Lecturer(s): Daniel J. Rixen

Tel.:  015 - 27 81523 /      

Catalog data:

Course year:

MSc 1^st year

Course language:

English

In case of Dutch: Please contact the lecturer about an English alternative, whenever needed.

Semester:

2A / 2B

Hours per week:

Other hours:

16

Assessment:

Oral exam

Assessment period:

2B, August

(see academic calendar)

Prerequisites (course codes):

Statics and Strength of materials, Dynamics (e.g. wb1418, wb1419), Linear Algebra, Numerical Analysis (e.g. wi3097wb), Finite Elements (e.g. wb1212-1214)

Follow up (course codes):

Multibody Dynamics B (wb1413)

Detailed description of topics:

Using engineering tools as black boxes can be dangerous and inefficient. This is especially true when performing dynamic analysis of structures in a finite element package. Choosing the right finite element types and the suitable solution procedure is critical to get accurate results and to compute solutions efficiently. In order to discuss basic principles of numerical methods for dynamics and to explain fundamental concepts related to dynamic analysis, the course will cover the following topics:

-  Linear solvers, storage techniques and singular systems

-  Free vibration modes, mode superposition techniques and eigensolvers for large systems

-  Accuracy of modal superposition, modal acceleration, system excited through support

-  model reduction, including dynamic substructuring

-  time-integration of linear and non-linear systems

-  computing senstitivity of modes and eigenfrequency to design parameters, model updating

-  Parallel computing techniques for fast solvers

 Some topics might be dropped depending on students background. Specific topics might also be discussed if time permits.

In this courses emphasis will be put on understanding fundamental concepts of numerical methods and how they relate to the mechanics of structures. Therefore, the oral (open book) exam will concentrate on the mastering of concepts rather than on formulation details. If time permits, a computational project will be included (using Matlab pre-cooked routines and/or Ansys-Nastran).

Course material:

•  Lecture notes (available through blackboard)

References from literature:

• Mechanical Vibrations, Theory and Application to Structural Dynamics, M. Géradin and D. Rixen, Wiley, 1997.
• The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes Prentice-Hall, 1987.
• Finite Element Procedures, K.J. Bathe, Prentice-Hall, 1996
• Structural Dynamics: an introduction to computer methods, R.R. Craig, Wiley, 1981, ISBN 0-471-04499-7
• Matrix Computation, G.H. Golub and C.F. Van Loan, Johns Hopkins University Press, 1996.

Remarks assessment, entry requirements, etc.:

An assignment will be given in ANSYS/Matlab (topic can be defined by students) time permitting.

Learning goals:

The student must be able to:

1. describe the solutions steps needed to solve linear systems and choose the proper algorithm according to the problem (LU, Cholesky, LDLT) including storage techniques

2. identify singular matrices arising from mechanical systems and compute a generalized inverse of a singular matrix and its nullspace

3. use the concept of eigenmodes to write the dynamic solution as a modal superposition and the system matrices in the form of spectral expansions

4. choose the proper eigensolvers and implement standard techniques  from the family of the power iteration including shifting

5. evaluate the approximations inherent to modal truncation in the mode displacement method and apply the mode acceleration method to correct for the static truncated part

6. solve by mode superposition the dynamics of systems excited by their support and apply the technique of additional mass to replace imposed displacements

7. describe the concept of effective modal mass and explain how it can be used to evaluate the contribution of modes to the approximation by modal series of the response of systems excited by the support

8. describe the concept of model reduction and write the reduced equations and write the reduced dynamic equations according to the static Guyan-Iron reduction

9. outline the idea of substructuring and derive the substructure approximation in the Craig-Bampton method, derive the associated reduced matrices and describe how accurate the Craig-Bampton approximation is in practice

10. solve the normal equations using Laplace transforms and put the solution procedure of the normal equations in a recursive matrix

11. discuss the concepts of consistency, stability and accuracy for simple implicit and explicit direct time-integration schemes

12. derive the time-integration formulas belonging to the Newmark family and discuss the stability limits and the accuracy of the Newmark schemes

13. write the explicit and implicit time-integration algorithms for non-linear systems

14. write the sensitivity of eigenmodes and eigenfrequencies of dynamic systems

15. describe the basic principles of parallel computing and explain the concept of domain decomposition and write the decomposed problem in a dual and primal interface problem suitable for parallel computing

16. write a small program (in Matlab for instance) to perform a dynamic analysis according to the Finite Element method, and implement the proper numerical techniques

Computer use:

Use of ANSYS and/or Matlab for assignment and illustration

Laboratory project(s):

-

Design content:

-

Percentage of design:  -0%