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 Scheda Riassuntiva
 Anno Accademico 2019/2020 Scuola Scuola di Ingegneria Industriale e dell'Informazione Insegnamento 088775 - DYNAMICS OF MECHANICAL SYSTEMS Docente Bruni Stefano Cfu 10.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - MI (473) AUTOMATION AND CONTROL ENGINEERING - INGEGNERIA DELL'AUTOMAZIONE*AZZZZ088775 - DYNAMICS OF MECHANICAL SYSTEMS

 Obiettivi dell'insegnamento
 The course is an advanced module on dynamics and vibrations of mechanical systems, with emphasis on multi degree of freedom systems and distributed parameter systems. First, the techniques for writing the equations of motion of a multi-d.o.f. system are introduced and the solutions for free undamped, free damped and forced motion of the system are derived. Then, beam-type distributed parameter systems are introduced ant their motion is determined under the assumption of small displacements from the un-deformed state. Techniques for the discretisation of distributed parameter systems are also covered by the course with special reference emphasis on the mode superposition approach and on the finite element method. The final part of the course treats the analysis of stability for mechanical systems subjected to field forces, also introducing some examples from real applications.

 Risultati di apprendimento attesi
 Lectures and other didactic activities will allow students to: Understand the dynamics and vibrations of complex mechanical systems Define mathematical models to describe the motion of mechanical systems formed by rigid and / or flexible bodies Analyse the stability of mechanical systems, focussing on some significant case studies

 Argomenti trattati
 1. Dynamics and vibration of single, two and multi-degree-of-freedom systems 1.1 Free and forced vibrations of single-degree-of-freedom linear systems. Natural frequency and damping factor. The Frequency Response Function. The superposition principle and the response to different kinds of input (step, harmonic, periodic, non-periodic). 1.2 Non-linear vibration of single-degree-of-freedom systems. Non-linear motion equation and linearization about a certain equilibrium position. Free non-linear vibration: dependency of the natural frequency on the oscillation amplitude. Forced non-linear vibration: jump phenomenon, subharmonics and combination harmonics. 1.3 Two and multi-degree-of-freedom systems. Non-linear motion equations and linearization about a certain equilibrium position. Free and forced vibrations of two and multi-degree-of-freedom linear systems. Natural frequencies, vibration modes, Frequency Response Function. 1.4 Principal coordinates and modal superposition approach. Modal parameters identification.   2. Vibrations in continuous systems 2.1 Transversal vibrations in stretched strings. Wave equation. Wave propagation solution and standing wave solution. Natural frequencies, vibration modes, forced response. 2.2 Introduction to simple elastic beam models. Euler-Bernoulli beam theory. 2.3 Axial vibrations in beams. Natural frequencies, vibration modes, forced response. 2.4 Torsional vibrations in circular shafts. Natural frequencies, vibration modes, forced response. 2.5 Bending vibrations in slender beams. Natural frequencies, vibration modes, forced response. 2.6 Principal coordinates and modal superposition approach.   3. Introduction to finite element method 3.1 Discretization of the continuous domain into a set of sub-domains: mesh and degrees-of-freedom. 3.2 Euler-Bernoulli beam element: reference systems, degrees-of-freedom, shape functions, inertia and stiffness matrices. 3.3 Modelling of planar beam structures. Coordinate transformation from the local to the global reference system. Assemblage procedure to obtain the matrices of a complete structure. Expression of the nodal forces corresponding to concentrated and distributed loads. Structural damping model. 3.4 Natural frequencies, vibration modes, forced response, motion imposed to the supports. 3.5 Examples and applications.   4. Stability of mechanical systems subjected to non-linear force fields 4.1 Definition of force field. Conservative and non-conservative force fields. Mathematical models of aerodynamic forces and contact forces. 4.2 Stability analysis of single-degree-of-freedom systems, about a certain equilibrium position. Static and dynamic instability. 4.3 Stability analysis of two-degrees-of-freedom systems, about a certain equilibrium position. Flutter instability. 4.4 Examples and applications.   Prerequisites     Handouts  Lecture handouts, training material and proposed exercises will be made available on the course web page.

 Prerequisiti
 Fundamentals of matrix algebra and of vector analysis. Taylor Series Expansion. Fourier Series. Linear ordinary differential equations. Planar kinematics of a point mass. Planar kinematics of a rigid body: translation, rotation, roto-translation, angular speed, centre of rotation. Holonomic constraints, degreeds of freedom for a system of rigid bodies moving in a plane. Constraint forces. Planar kinematics of systems of rigid bodies.Statics and dynamics of a point mass: Newton’s laws, D’Alembert principle Statics and dynamics of a point mass: Newton’s laws, D’Alembert principle.Statics of a rigid body, moment of a force. Inertia properties of a rigid body: centre of mass, moment of inertia.Planar dynamics of a rigid body: Newtonian approach, generalization of the D’Alembert principle. Newtonian dynamics for a system of rigid bodies.The principle of virtual work in statics.The principle of virtual work in dynamics. Kinetic energy for a system of rigid bodies.Conservative force fields, potential energy.Lagrange Equations for a system of rigid bodies.

 Modalità di valutazione
 The examination consists of a written exam, followed by an oral exam. In the written exam the student will be asked to demonstrate her/his ability regarding the following skills: - Define a mathematical model for a system of rigid bodies and derive the motiion of the system caused by assigned excitation effects In the oral exam the student will be asked to demonstrate her/his ability regarding the following skills: - Describe the motion of systems formed by flexible bodies, with special reference to planar systems of beams - Understand the principles underlying the dynamics and stability of mechanical systems.

 Bibliografia
 Lecture notesNote:Lecture notes, training material and proposed exercises will be made available on BeeP. F. Cheli and G. Diana, Advanced Dynamics of Mechanical Systems, Editore: Springer, Anno edizione: 2015, ISBN: 978-3-319-18199-8 L. Meirovitch, Fundamentals of Vibrations, Editore: McGraw-Hill, Anno edizione: 2001 M. Petyt, Introduction to finite element vibration analysis, Editore: Cambridge University Press, Anno edizione: 2010

 Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
60:00
90:00
Esercitazione
24:00
36:00
Laboratorio Informatico
14:00
21:00
Laboratorio Sperimentale
2:00
3:00
Laboratorio Di Progetto
0:00
0:00
Totale 100:00 150:00

 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 25/11/2020