Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2019/2020
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Docente Pizzigoni Bruno
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 the 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 class exercises, experimental labs and computer room sessions.


Risultati di apprendimento attesi

The course will provide students with:

  1. a knowledge of the modelling approaches to the 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);
  2. the ability to implement finite element models of plane frames (with arbitrary geometries (?), loads and boundary conditions) and perform structural dynamics analysis (DD2);
  3. the ability 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);
  4. an opportunity to develop skills in summarizing and presenting the results achieved during lab activities (DD4);
  5. an understanding of both 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 fundamental engineering approaches to the rotor balancing and dynamic modelling of fluid lubricated journal bearings (DD1);
  6. an ability to apply theoretical knowledge to the analysis of vibration problems in complex mechanical systems (DD2) and make judgements concerning possible alternative choices to modelling a real structure/machinery (DD3).

Argomenti trattati

Vibration analysis of discrete systems (a short summary)

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

Vibration analysis of one-dimensional continuous systems

Transverse vibration of taut 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. Forced vibration: steady-state response to harmonic input. Modes superposition approach and principal coordinates formulation of the equations of motion.

Experimental modal analysis

Overview of test procedures and Frequency Response Function (FRF) estimation. Mathematical formulation of the FRF based on modal superposition. Algorithms for frequency-domain modal parameters identification (multi-mode curve fitting method and simplified single-mode identification procedure).

The finite element method in structural dynamics

General introduction to the finite element discretization of continuous systems. A detailed formulation of the 6-dof beam element for in-plane analysis: shape functions for axial and bending deformation, mass and stiffness matrices. Procedures 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.

Three-dimensional dynamics of a rigid body

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


Overview of dynamic problems in rotating machinery. Bending critical speeds and balancing procedures for rigid and flexible rotors. Numerical model of the fluid lubricated journal bearing, steady-state journal position, equivalent stiffness and damping matrices. Oil film instability.


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 rigid body, forces and moments, in-plane static equilibrium of a rigid body, in-plane kinetics of a particle and 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 access to the final oral examination.

With reference to expected learning outcomes:

  • the written test is intended to verify item 1;
  • the discussion on lab reports is intended to verify items 2, 3 and 4;
  • the oral examination will cover all topics illustrated during the course, with a specific focus on items 5 and 6.

Risorsa bibliografica facoltativaCourse handouts www.polimi.it/BeeP

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. 1.6.5 / 1.6.5
Area Servizi ICT