First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Engineering
Course unit
IN05105686, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2018/19

Information on the course unit
Degree course Second cycle degree in
IN0529, Degree course structure A.Y. 2011/12, A.Y. 2018/19
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Number of ECTS credits allocated 9.0
Type of assessment Mark
Course unit English denomination MECHANICAL VIBRATIONS
Website of the academic structure
Department of reference Department of Management and Engineering
E-Learning website
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge ALBERTO TREVISANI ING-IND/13

Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses ING-IND/13 Applied Mechanics for Machinery 9.0

Course unit organization
Period Second semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Lecture 9.0 72 153.0 No turn

Start of activities 25/02/2019
End of activities 14/06/2019
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Board From To Members of the board
10 2019 01/10/2019 15/03/2021 TREVISANI ALBERTO (Presidente)
RICHIEDEI DARIO (Membro Effettivo)
9 2018 01/10/2018 15/03/2020 TREVISANI ALBERTO (Presidente)
RICHIEDEI DARIO (Membro Effettivo)
8 2017 01/10/2017 15/03/2019 TREVISANI ALBERTO (Presidente)
RICHIEDEI DARIO (Membro Effettivo)

Prerequisites: Basics of Mechanics of Machines (kinematics and dynamics of rigid bodies and mechanisms)
Target skills and knowledge: Providing students with technical and practical knowledge in the mechanics of vibrating systems.
Providing students with knowledge in discrete and continuous dynamics modeling of mechanical systems.
Developing abilities in the understanding of the dynamic behavior of a mechanical system through the autonomous development of single or multi dofs models reproducing the chief vibratory phenomena of a mechanical system and the excitation mechanisms.
Developing knowledge and abilities in the measurement and analysis of the vibrations in mechanical systems.
Examination methods: The assessment of knowledge and abilities is carried out through an exam divided into three parts distributed over two separate days.
Two parts are written assessments done in the same day. The third part consists of an oral test that can be optional or mandatory depending on the overall grade obtained in the two written assessments.
The written test is divided into:
- a first part in which, through three separate open-ended questions, the student's knowledge of the course topics is ascertained,
- a second part in which, through the solution of a practical problem (exercise), the abilities related to the analysis of a vibratory phenomenon, the autonomous writing of a dynamic model, and the evaluation, also numerical, of the vibratory phenomena are ascertained.
All students who have passed the written test can, on a voluntary basis, take an oral test to discuss in more detail the topics of the course, also in order to obtain a better evaluation. The oral test is mandatory for those students who aspire to record grades higher or equal to 25/30.
Assessment criteria: The evaluation criteria by which the knowledge and abilities acquired will be verified are:
- the completeness of the theoretical knowledge acquired on the course topics;
- the level of autonomy acquired in the interpretation and solution of mechanical vibrations problems.
- the proved ability to apply theoretical knowledge to the development of dynamic models and to the evaluation, through calculations, of vibratory phenomena (natural and anti-resonance frequencies, vibrating modes, responses to external excitations, etc.);
- the presentation capabilities and the rigorousness in the discussion of the issues discussed.
Course unit contents: VIBRATIONS OF SINGLE-DEGREE-OF-FREEDOM MECHANICAL SYSTEMS: the harmonic oscillator, natural frequency and damping ratio. Free vibrations, transient response, damping ratio estimation. Instability, self-excited vibrations. Forced vibrations under harmonic excitation, complex vector notation, frequency response of a damped harmonic oscillator, transmissibility, vibration excited by an unbalanced mass. Vibration isolation, choice of antivibration mountings. Impulse response. Response to an arbitrary force, convolution integral, convolution theorem, correspondence between Fourier transform of the impulse response and frequency response. Harmonic-oscillator-like mechanical systems, linearization of dynamic models. Torsional vibrasions. Vibrations induced by inertial forces in slider crank mechanisms. Balancing of multi-cylinder engines. Examples and exercises.

VIBRATIONS OF MULTI-DEGREE-OF-FREEDOM MECHANICAL SYSTEMS: equations of motion in linear matrix form. Mass and stiffness matrices: definitions and properties. Stiffness matrices and elastic energy definition. Properties of the eigenvalues of stiffness matrices. Positive definite and positive semidefinite systems. Mass matrices and kinetic energy definition. Modal analysis, eigenvalue problem, natural frequencies and modes of vibration, modal matrix and equation of motion decoupling. Free undamped vibrations, examples. Beat. Modal and Rayleigh damping. Modal load and time response by modal superposition. Resonance and antiresonance, Frahm’s active vibration absorber, dynamic vibration absorbers (DVA, TMD). Examples and exercises.

VIBRATIONS OF CONTINUOUS SYSTEMS: continuous models for slander beams, frequency equations, modes of vibration of pinned-pinned beams, clamped-clamped beams, free beams and cantilever beams. Free and forced vibration. Exercise: experimental evaluation of the natural frequencies of a free beam.

VIBRATION MEASUREMENT AND CONTROL: vibration measurement equipments, contact and non-contact devices. Focus on piezo accelerometers. Vibration measurement methods, spectral analyzers. Experimental data conditioning techniques. Methods and instruments for the experimental evaluation of the modal parameters of structures and mechanisms: electrodynamic shakers and impact test. Optimal estimate of the frequency response by means of averages of auto and cross spectra. Example of active vibration control: sky-hook damping. Laboratory experiences.
Planned learning activities and teaching methods: - Lectures also with the support of computer material (power point and pdf files prepared by the lecturer, plots and elaborations in Matlab, videos and images, industrial catalogs)
- Exercises developed on the blackboard
- Experimental laboratories
- Seminars and workshop held by experts in the field.
Additional notes about suggested reading: All teaching material (lecture notes, exercises, exam topics, catalogs, software code, etc.) is made available by the "moodle" platform (
Textbooks (and optional supplementary readings)
  • Giovagnoni, Marco, Analisi delle vibrazioni nei sistemi meccanici. Padova: Libreria Internazionale Cortina, 2009. Cerca nel catalogo
  • Hartog, Jacob Pieter Den, Mechanical vibrations. New York: Dover, 1985. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Laboratory
  • Problem based learning
  • Case study
  • Problem solving
  • Video shooting made by the teacher/the students
  • Use of online videos
  • Loading of files and pages (web pages, Moodle, ...)
  • Seminars held by experts in the field.

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • Matlab
  • Possible software used in the laboratories and workshops (e.g. Test.Lab, Adams)

Sustainable Development Goals (SDGs)
Industry, Innovation and Infrastructure