Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2021/2022
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Docente Zasso Alberto
Cfu 5.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento

Obiettivi dell'insegnamento

The aim of the course is to enable students to master advanced engineering methods for vibration analysis of mechanical systems, with a specific focus on continuous structures. It covers theoretical models, discretisation techniques and numerical solutions based on the finite element method. Particular attention is given to practical examples and engineering applications, especially in the field of structural dynamics and rotordynamics. Each topic is treated both theoretically and practically, through traditional lectures, class exercises, experimental labs and computer room sessions.

Risultati di apprendimento attesi

By the end of the course, students will learn contents and practices according to what defined in the learning objectives.


In terms of acquired knowledge and understanding, students will develop a knowledge of the modelling approaches to vibration analysis of continuous systems (DD1) and the ability to handle vibration problems for arbitrary combinations of geometrical configurations, structural properties and boundary conditions (DD2); in addition, students will understand the basic physical phenomena involved in two of the most significant issues of rotordynamics (i.e. bending critical speeds and oil film instability) as well as the fundamental engineering approaches to rotor balancing and dynamic modelling of fluid lubricated journal bearings (DD1);


Concerning the ability to apply the acquired knowledge and understanding, students will develop the ability to apply theoretical knowledge to the analysis of vibration problems in complex mechanical systems and make judgements concerning possible alternative choices for modelling a real structure/machinery (DD2, DD3).


Through lab activities will be able to process experimental data from dynamic tests and extract basic information regarding the modal parameters of a vibrating structure, i.e. natural frequencies, damping ratios, mode shapes (DD2);


Finally, students will improve their communication skills, being able to develop skills in summarizing and presenting the results achieved during lab activities (DD4);

Argomenti trattati

The course contents are delivered through lectures, exercises courses and lab activities.


The course covers the following topics divided in modules as per the description below:


Module 1        Vibration analysis of discrete systems (a short summary)

Topic 1.1          Free and forced vibration of single-dof and multi-dof systems. Modal superposition approach and principal coordinates formulation of the equations of motion.


Module 2        Vibration analysis of one-dimensional continuous systems

Topic 2.1          Transverse vibration of strings. Axial vibration of bars. Torsional vibration of circular shafts. Bending vibration of slender beams (Euler-Bernoulli formulation).

                       Wave propagation and standing wave solutions. Free vibration: natural frequencies and vibration modes for different boundary conditions, transient response to assigned initial

                       conditions. Forced vibration: steady-state response to harmonic input.

Topic 2.2          Modal superposition approach and principal coordinates formulation of the equations of motion.


Module 3        Experimental modal analysis

Topic 3.1          Overview of test procedures and Frequency Response Function (FRF) estimation. Mathematical formulation of the FRF based on modal superposition.

Topic 3.2          Algorithms for frequency-domain modal parameters identification (multi-mode curve fitting method and simplified single-mode identification procedure).


Module 4        The finite element method in structural dynamics

Topic 4.1          General introduction to the finite element discretization of continuous systems. Detailed formulation of the 6-dof beam element for in-plane analysis: shape functions for

                       axial and bending deformation, mass and stiffness matrices.

Topic 4.2          Procedure for developing the finite element model of a structure: global and local reference systems, work-equivalent nodal forces, matrix assembling, structural damping

                       model, boundary conditions and matrix partition. Classical dynamic analysis: natural frequencies and vibration modes, frequency response.


Module 5        Three-dimensional dynamics of a rigid body

Topic 5.1          Kinetic energy and inertia tensor. 3D kinematics: rotation matrix and body angular speed. Dynamics of a rigid body rotating about a fixed axis.


Module 6        Rotordynamics

Topic 6.1          Overview of the dynamic problems in rotating machinery. Balancing conditions and balancing procedures for a rigid rotor.

Topic 6.2          Bending critical speeds and balancing procedures for a flexible rotor


Mathematics: fundamentals of matrix algebra and vector analysis, Fourier series, Taylor series, linear ordinary and partial differential equations. Basic Mechanics: planar kinematics of a particle and of a rigid body, forces and moments, in-plane static equilibrium of a rigid body, in-plane kinetics of a particle and of a rigid body, Lagrange’s equations. Solid Mechanics: stress, strain and constitutive equations for a linear elastic material bending of slender beams and axial loading of bars, elastic potential energy. Fundamentals of Vibration Analysis: free and forced vibration of single and multi-dof discrete linear systems.


Tutoring support on prerequisites will be available during the first four weeks of the course.

Modalità di valutazione

The exam consists of:


  • a written test;
  • a discussion on lab reports;
  • an oral examination.


Students are requested to deliver short reports on the two assignments given during the experimental and computer labs. Delivering lab reports and passing the written test is mandatory for having access to the final oral examination.


With reference to expected learning outcomes:


  • the written test is intended to verify module 2;
  • the discussion on lab reports is intended to verify items 3 and 4;
  • the oral examination will cover all topics illustrated in the course, with a specific focus on items 5 and 6.


Final Grading

To pass the course, students must successfully pass the written test, complete the project and being able to discuss it, pass the oral examination.

Risorsa bibliografica facoltativaF. Cheli, G. Diana, Advanced Dynamics of Mechanical Systems, Editore: Springer, Anno edizione: 2015, ISBN: 978-3319181998
Risorsa bibliografica facoltativaL. Meirovitch, Fundamentals of Vibrations, Editore: Waveland, Anno edizione: 2010, ISBN: 978-1577666912
Risorsa bibliografica facoltativaM. Petyt, Introduction to Finite Element Vibration Analysis, Editore: Cambridge University Press, Anno edizione: 2010, ISBN: 978-0521191609
Risorsa bibliografica facoltativaCourse handouts available on BeeP https://beep.metid.polimi.it/

Software utilizzato
Nessun software richiesto

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
Ore di studio autonome
Laboratorio Informatico
Laboratorio Sperimentale
Laboratorio Di Progetto
Totale 50:00 75:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
schedaincarico v. 1.6.8 / 1.6.8
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