Teacher

prof. Marco GIOVAGNONI

Credits

12 CFU

Language

Italian

Objectives

The course aims to highlight the fundamental concepts which are used in the analysis of vibrations of mechanical systems, both from a theoretical and an experimental point of view. The mathematical models are considered focusing on their experimental validation and their limits of applicability. Particular emphasis is placed on: frequency response, modal analysis, finite element method.

Acquired skills

- Knowledge of the basic techniques for the analysis of vibration problems: frequency response, modal analysis, stability analysis, finite element method.
- Knowledge of the limits of application of the damping models and of the finite element models.
- Knowledge of the basic techniques for the vibration measurement.

Lectures and exercises (topics and specific content)

Introduction: practical examples of vibration problems, modal parameters, unstable systems (8 hours).
Single d.o.f. - frequency response: use of complex algebra to represent sinusoidal functions, force-input displacement-output response, transmissibility, response to unbalance, seismic instruments response, energy balance (14 hours).
Multi d.o.f. - stiffness and mass matrices: small displacements and small strains, symmetry property and elastic energy, system matrix assembly, boundary conditions, mass matrices and kinetic energy (8 hours).
Multi d.o.f. - free response: eigenproblem with two symmetric matrices, eigenvalues and eigenvectors of the dynamic matrix, free undamped response, examples with 2 d.o.f. systems (8 hours).
Multi d.o.f. - forced response: modal loads and modal superposition, multi-input multi-output frequency response, 2 d.o.f. examples, antiresonances, Frahm absorber, torsional vibrations (18 hours).
Fourier analysis of  periodic phenomena: real form of the Fourier series, amplitude and phase distorsion of measuring instruments, response of a slider-crank mechanism on flexible supports, complex form of the series (6 hours).
Digital data processing: fast Fourier transform, aliasing and leakage (6 hours).
Response to general input: response to finite-length impulse and to ideal impulse, convolution integral, convolution theorem, transform of the impulse response and frequency response (10 hours).
Damping models: Kimball and Lovell experiment, complex stiffness method, modal damping and Rayleigh damping, causality requirements, relaxation functions (6 hours).
Introduction to the finite element method: minimum of the potential energy for elastic systems at equilibrium, Ritz analysis of a one-dimensional problem, interpolation functions, convergence of the finite element method (14 hours).
Finite element method in dynamics: stiffness matrices, equivalent nodal loads and consistent mass matrices for truss elements and for beam elements (12 hours).
Self-excited vibrations: response of a 2 d.o.f. unstable system, examples of self-excited vibrations in machine-tools and in screw jacks (8 hours).
Exercises (42 ore).

References

- Marco Giovagnoni - Analisi delle vibrazioni nei sistemi meccanici - Edizioni Libreria Cortina - Padova

Type of exam

Oral