## Vibroacoustics

##### Scientific area of pertinence: ING-INF/13

This course addresses the modeling of vibrations in all kinds of structures from its fundamentals to its applications. This is an essential course for addressing a wide range of problems, from noise modeling and control, to the characterization of vibrations in resonating structures (such as musical instruments).

### Module 1: “Fundamentals of vibration analysis” (5 CFU)

*Free vibration of single-DOF linear systems.*Procedure for deriving the differential equation of motion. The harmonic oscillator. Undamped and damped free vibration.*Forced vibration of single-dof linear systems.*Frequency response. The superposition principle.*Two and multi-dof vibrating systems.*Procedure for deriving the equations of motion. Matrix formulation. Free vibration of undamped systems: natural modes. Free damped vibration. Eigenvalue problem. Frequency response.*Modal superposition.*Orthogonality of modal vectors. Coordinate transformation from physical to principal coordinates.

### Module 2: Vibroacoustics of Musical Instruments (5 CFU)

*Vibration and wave propagation in one-dimensional and two-dimensioanl structural elements.*Transversal vibration of stretched strings, quasi-longitudinal vibration of bars, bending vibration of slender beams and thin plates.*Sound waves.*The homogeneous acoustic wave equation. Sound pressure and particle velocity. Propagation of plane and spherical sound waves. Sound intensity and sound power. Fundamental acoustic source models.*Sound radiation from vibrating structures.*General formulation of the vibroacoustic problem. Sound radiation from an infinite plate. Wave/boundary matching. Critical frequency. Sound radiation from finite plates. Radiation efficiency.*Experimental techniques in vibroacoustics.*Fundamentals of experimental modal analysis. Sound intensity and sound power measurement. Sound source mapping through microphone arrays. Examples of application to stringed musical instruments.*Numerical methods in vibroacoustics.*Introduction to finite-element and boundary-element methods. Examples of application to the simulation of the vibroacoustic behaviour of stringed musical instruments.