# Mechanical Systems Dynamics

Synthetic program:

DINAMICA DEI SISTEMI MECCANICI
Il corso fornisce nozioni avanzate relative alle vibrazioni dei sistemi meccanici, con particolare riferimento alle vibrazioni dei sistemi discreti a più gradi di libertà e dei sistemi continui, rendendo disponibili per l'allievo strumenti di analisi e modellazione di sistemi complessi a corpi rigidi (con le moderne tecniche multicorpo) e deformabili (con l'approccio ad elementi finiti). Alcune nozioni essenziali inerenti la dinamica di sistemi non lineari ed esempi applicativi inerenti la dinamica di macchine e strutture soggette a campi di forze non conservativi vengono fornite a completamento del percorso formativo avanzato sulla dinamica dei sistemi.

Vibration analysis of discrete systems
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 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. Modal 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. Detailed formulation of the 6-dof beam element for in-plane analysis: shape functions for axial and bending deformation, mass and stiffness matrices. 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.

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 about a fixed axis.

Rotordynamics
Overview of the dynamic problems in rotating machinery. Balancing conditions and balancing procedures for a rigid rotor. Bending critical speeds and balancing procedures for a flexible rotor.

## Lecture Notes

### Complete course:

Type File name Year
Digital notes Complete notes 2022/2023

### Divided by topic:

Type File name Year
Digital notes M01-Axial vibrations of a bar 2022/2023
Digital notes M01-Tensioned String 2022/2023
Digital notes M01-Transversal vibrations of a beam 2022/2023
Handwritten notes M03-Dynamic of rigid body in space 2022/2023
Digital notes Parte 1 - Sistemi dinamici discreti e continui. Fune (equazione delle onde). Analisi modale dei continui. Travi. 2011/2012
Digital notes Parte 2 - Metodo degli Elementi Finiti (FEM) per le funi e le travi. Identificazione modale dei parametri. Dinamica dei rotori. Bilanciamento dei rotori. Sistema [rotore + cuscinetti + fondazione]. 2011/2012

### Other:

Type File name Year
Digital notes Laboratory notes - Report 1 2022/2023
Digital notes Laboratory notes - Report 2 2022/2023

## Exercises

### Complete course:

Type File name Year
Handwritten notes Complete exercises of the course 2022/2023
Digital notes Exercises of the course 2019/2020

### Divided by topic:

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## Exams

### First partial exam:

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### Full exam:

Type Date
Text and solution 01/02/2024
Text and solution 11/01/2024
Text and solution 04/09/2023
Text and solution 19/07/2023
Text and solution 14/07/2023
Text and solution 06/07/2023
Text and solution 29/06/2023
Text and solution 23/06/2023
Text and solution 23/06/2023
Text and solution 15/06/2023
Text and solution 03/03/2023
Text and solution 13/02/2023
Text 03/02/2023
Text and solution 03/02/2023
Text 13/01/2023
Text and solution 13/01/2023
Text and solution 07/09/2022
Text and solution 30/08/2022
Text 25/08/2022
Text and solution 25/08/2022
Text and solution 25/07/2022
Text and solution 25/07/2022
Text 15/07/2022
Text and solution 15/07/2022
Text and solution 15/07/2022
Text and solution 06/07/2022
Text and solution 06/07/2022
Text and solution 30/06/2022
Text and solution 30/06/2022
Text 24/06/2022
Text and solution 24/06/2022
Text and solution 24/06/2022
Text and solution 16/06/2022
Text 16/06/2022
Text and solution 16/06/2022
Text and solution 04/02/2022
Text and solution 14/01/2022
Text and solution 26/08/2021
Text and solution 21/07/2021
Text and solution 16/07/2021
Text and solution 09/07/2021
Text and solution 02/07/2021
Text and solution 25/06/2021
Text and solution 18/06/2021
Text and solution 05/02/2021
Text and solution 15/01/2021
Text and solution 15/01/2021
Text and solution 16/11/2020
Text and solution 16/11/2020
Text and solution 10/09/2020
Text and solution 03/09/2020
Text and solution 27/07/2020
Text and solution 17/07/2020
Text and solution 10/07/2020
Text and solution 07/07/2020
Text and solution 02/07/2020
Text and solution 26/06/2020
Text and solution 26/06/2020
Text and solution 16/06/2020
Text and solution 05/09/2019
Text and solution 26/07/2019
Text and solution 14/06/2019
Text 12/02/2016
Text 14/09/2015
Text 04/09/2015
Text 21/07/2015
Text 10/07/2015
Text 22/06/2015

### Oral exam:

Type Date
Text and solution 30/06/2023

### Multiple choice test:

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### Other:

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## Other

### Laboratory:

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### Projects:

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### Presentations:

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### Tables:

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## Live

Quick daily notes, exercises and audio recordings. Files will be approved on priority but deleted after 365 days. 2 points will be assigned by default.

### Quick contents:

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