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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[70/85]  MECHANICAL ENGINEERING [85/00 - Ord. 2019]  PERCORSO COMUNE 3 30


i. Knowledge and understanding.
The student will acquire basic knowledge of numerical methods for the dynamic analysis of structures. The emphasis is placed on application aspects, especially on the possibility of solving structures in the linear elastic field.

ii. Applying knowledge and understanding
The student will acquire the knowledge that will allow him to choose the types of finite elements and the most suitable algorithms for the solution of the dynamic problem under examination. The student will be able to correctly interpret the results.

iii. Making judgments.
The student will acquire the ability to identify and propose appropriate procedures for calculating the stress and deformation states of mechanical components subjected to dynamic loads.

iv. Communication skills.
The student will develop the ability to communicate requirements and performances to specialized and non-specialized interlocutors, expressed in terms of tensions, deformations, and displacements.

v. Learning skills
The student will acquire the ability to deal with typical problems related to the dynamic study of mechanical structures.


The student should have already attended a course on the finite element method.


1) Dynamic references:
a) Single degree of freedom systems:
a1) Free vibrations not damped and damped;
a2) Forced vibrations with harmonic excitation;
a3) Transient vibrations;
b) Systems with two degrees of freedom.
2) Numerical methods for systems with many degrees of freedom.
3) Energy principles. The Rayleigh-Ritz method.
4) Calculation of stiffness, mass and damping matrices of some types of finite elements:
a) Two and three-dimensional rod and beam elements.
b) Plane and volume isoparametric elements.
5) Calculation of eigenvalues and eigenvectors.
6) Methods of modal superposition
7) Implicit and explicit methods.

Teaching Methods

Duration: 10 weeks.
Lectures: 2h/ week.
Tutorials: 1h/ week.

The course consists of 10 frontal lessons two hours each, presented in a traditional way on the blackboard, sometimes with the aid of projected images and the use of a commercial FEM program.
Students are then required to perform some exercises (with a technical report) to be delivered to the teacher.
The theme of the exercises is as follows:
1) Calculation of a plane frame made up rods;
2) Calculation of a plane frame consisting of beams;
3) Calculation of a plane structure under plane stress;
4) Calculation of a plane shell in bending.

In numerical simulations the previous structures will be subjected to various dynamic loads.
To perform the calculations, students must use an FEM commercial code illustrated in the classroom. In written reports they must compare the results obtained by discretizing the structure with mesh of different density and with finite elements of different type.

Verification of learning

The delivery of the exercises is mandatory to be admitted to the oral exam during which the candidate must illustrate the problems faced and justify his choices.
The student must demonstrate that he can choose the most suitable types of finite elements for solving a specific dynamic problem. He will have to be able to choose the appropriate density of the mesh and decide later on were to refine it. He will have to know how the FEM codes calculate the stiffness, mass, and damping matrices of some elements and the equivalent nodal forces, how the boundary conditions are inserted, calculated the displacements, and therefore the stresses. He will have to know the main algorithms for solving dynamic problems (modal analysis, implicit and explicit methods).

The exam test score is attributed by a note expressed in thirtieths.
The quality of the exercises delivered affects 50% of the final grade.

In assessing the exercises and the examination, the determination of the final mark takes into account the following elements:
1) the logic followed by the student in solving the question;
2) the correctness of the procedure identified for the solution of the question;
3) the adequacy of the proposed solution in light of the theoretical knowledge developed during the course
4) the use of adequate language

To pass the exam, therefore report a mark of not less than 18/30, the student must demonstrate that he knows how to choose the finite elements suitable for given stress distribution, know how to determine the boundary conditions, and must be able to judge the final results.


1] A.Gugliotta: Elementi finiti, OTTO editore, 2002.
[2] R.W. Clough, J. Penzien "Dynamics of Structures", Mc Graw-Hill , 1975.
[3] R.R.Craig, Jr. A.J. Kurdila "Fundamentals of Structural Dynamics", WILEY,
[4] J.He, Z.F. Fu "Modal Analysis", Butterworth Heinemann, 2001.
[5] S.S. Rao "Mechanical Vibrations", Prentice Hall, 2011.
[6] R.D.Cook: Concepts and applications of finite element analysis, John Wiley & Sons. 1974.

More Information

Students will be provided with teaching material (notes, slides, diagrams, graphs, exercises) relating to the topics covered in class. The material can be found at the following address:

Continuous contact is guaranteed (also via e-mail) between the teacher and students throughout the course.
The teacher also provides maximum availability to receive students, even after office hours, for clarifications on the exam topics.

Questionnaire and social

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