This repository contains the code and the reports of the assignments of the course of Fundamentals of Vibration Analysis (Module A) and Vibroacoustics (Module B) held in Politecnico di Milano by professors Alfi Stefano and Corradi Roberto.
During exercise classes, students are assigned case studies on the modelling and numerical analisys of vibrating systems (Module A). In addition to traditional lectures, the course includes experimental and computer labs, focused on specific methodologies, case studies or practical applications (Module B). Students are requested to prepare short reports on the assignments to be delivered at the final exam.
This module deals with mechanical vibrations of lumped parameter systems, considering both theory and applications. First, it covers free and forced vibration of linear single-degree-of-freedom systems. It then extends this analysis to two- and multi-degree-of-freedom linear systems. Particular attention is given to frequency domain models and to the principal coordinate formulation based on modal superposition approach. The course is held by professor Alfi Stefano.
The aim of the course is to introduce students to engineering methods for vibration and sound radiation analysis of continuous structures. Both analytical/numerical models and experimental techniques are covered and particular attention is given to examples and applications, especially in the field of string musical instruments. Each topic is treated theoretically and practically, through traditional lectures and experimental/computer labs. The course is held by professor Corradi Roberto.
The topics covered by the first module (Fundamentals of Vibration Analysis) are prerequisites to the second one (Vibroacoustics). Experimental and computer labs are held by professor Cesare Lupo Ferrari (in both modules).
Five case studies were assigned to students: three for module A and two for module B.
Vibration of single d.o.f. linear system:
- Derivation of the equation of motion of single degree of freedom linear systems through dynamic equilibrium equations and Lagrange equations
- Free vibration: response of the system to initial conditions, definition of natural frequency and damping ratio
- Forced vibration: response to constant, harmonic and periodic forces, frequency response function
Vibration of a 3 d.o.f. linear system:
- Definition of the equations of motion of the system through Lagrange equations (scalar and matrix formulations)
- Vibration modes: definition of natural frequencies, damping ratios and mode shapes.
- Analysis of free and forced vibrations
- State space and frequency response models
Modal superposition approach for a 3 d.o.f. system:
- Formulation of the equations of motion in terms of principal coordinates
- Modal parameters of the system
- Analysis of free and forced vibration in principal coordinates
- Representation of the frequency response functions in terms of modal coordinates
In this problem, we're given a small set of measurements we have to extract information from. The virtual experiment consists in a hammer hitting impulsively a structure in one point, and measuring the displacements in four different positions, one of which is where the force is applied (the first one).
Starting from the evaluation of the FRFs between displacements and force, it is requested to:
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Plot the “experimental” FRF diagrams
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Estimate and comment the natural frequencies, damping ratios and mode shapes of the resonating modes in the range 0 − 5 Hz employing simplified methods (e.g. half power point method).
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Set up a modal parameter identification program for estimating natural frequencies, damping ratios and modes in the range 0 − 5 Hz Compare the identified FRF with the “experimental” ones. Comment the results obtained.
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Compare and comment the modal parameters identified with the methods mentioned above.
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Employing a modal approach, reconstruct the FRFs and compare with the original ones.
Vibration and waves in one-dimensional continuous system (axial vibration of bars).
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Natural frequencies of the free-fixed bar in the frequency range (0 - 10) kHz and plot of the corresponding mode shapes.
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Natural frequencies of the same bar in free-free conditions and plot of the corresponding mode shapes.
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For the free-fixed bar excited at the free-end, plot and comment of the FRF for two different output positions in the following cases:
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undamped bar (standing wave solution);
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damped bar (wave propagation solution) w/ a loss factor
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damped bar (modal superposition approach) w/ a loss factor
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For the free-fixed bar excited at the free-end, plot and comment of the driving-point impedance in the following cases:
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damped bar (wave propagation solution) w/ a loss factor
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damped bar (modal superposition approach) w/ a loss factor
Starting from experimental data:
- implemented a Matlab script for identifying the vibration modes;
- tested the software capabilities on the experimental FRFs;
- summarized the results in terms of:
- natural frequencies and damping ratios;
- comparison of experimental and identified FRFs (for a certain reference channel);
- visualization of the identified modes.