The aim of the course is to introduce students to the fundamentals of vibration theory and to the vibroacoustic 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 pratically, through traditional lectures and experimental/computer labs. 

The course will provide students with:

1. a knowledge of the modelling approaches to vibration analysis of both lumped-parameter and continuous systems (DD1);

2. an understanding of the basic physical principles of sound radiation from a vibrating structure (DD1);

3. the ability to apply theoretical knowledge to the handling of complex vibroacoustic problems in real structures (DD2, DD3);

4. the ability to perform standard vibration and sound measurements and to critically apply data processing techniques for modal parameters extraction or sound analysis (DD2, DD3);

5. an opportunity to develop skills in summarizing and presenting the results achieved during the lab activities (DD4).

Module 1: Fundamentals of vibration analysis

Vibration of single d.o.f. linear systems
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 single d.o.f. non-linear systems
Analysis of static equilibrium positions. Linearization of the equation of motion about a static equilibrium position: effect of constant forces.
Vibration of two- and multi-d.o.f. linear systems
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 multi d.o.f. systems
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.

Module 2: Vibroacoustics
Vibration and waves in one-dimensional continuous systems
The partial differential equation governing transverse vibration of stretched strings and axial vibration of bars (one-dimensional wave equation). Traveling waves and standing waves. Wave propagation speed, wavenumber, mechanical impedance. Transmission and reflection at an impedance change. Natural frequencies and vibration modes for different boundary conditions. The partial differential equation governing bending vibration of slender beams (Euler-Bernoulli beam). Dispersion and evanescent waves. Vibration modes, principal coordinates and modal superposition approach.
Vibration and waves in two-dimensional continuous systems
The partial differential equation governing bending vibration of thin plates (Kirchhoff plate). Bending waves propagation, the wavenumber vector, point mobility. Bending vibration of finite plates. Vibration modes of plates with different geometry and boundary conditions. Theoretical soundboard models for stringed instruments. Introduction to finite element method and examples of application to musical instruments.
Sound radiation from vibrating structures
General formulation of the problem. Sound radiation from an infinite plate. Wave/boundary matching. Critical frequency. Sound radiation from finite plates. Radiation efficiency. Introduction to boundary element method and examples of application to the simulation of the vibroacoustic behaviour of musical instruments.
Experimental techniques
Fundamentals of experimental modal analysis: test equipment and test procedure, data processing and identification algorithms. Sound intensity and sound power measurement techniques. Practical examples relevant to stringed musical instruments.

Mathematics: fundamentals of matrix algebra and vector analysis, Fourier series, Taylor series, linear ordinary and partial differential equations.

Basic Mechanics: planar kinematics of a particle and of a rigid body, forces and moments, in-plane static equilibrium of a rigid body, in-plane kinetics of a particle and of a rigid body.

In addition to traditional lectures, the course includes labs activities, focused on specific methodologies, case studies or practical applications. Students are requested to prepare short reports on the assignments given during the labs, to be delivered at the final exam. The exam consists of two oral test sessions, one for each module, which include a discussion on the corresponding lab reports.