Course: Vibration Analysis and Vibroacoustics (Module A)
Programme: MSc in Music and Acoustic Engineering, Politecnico di Milano
Work Team: Di Lorenzo Giuliano, Ouali Ernest, Panettieri Francesco
Three MATLAB assignments covering the theoretical and numerical analysis of mechanical vibrating systems: from single degree-of-freedom (SDOF) dynamics to multi-DOF modal analysis and experimental modal identification. All implementations are in MATLAB, with analytical derivations documented in the accompanying PDF reports.
A single degree-of-freedom equivalent mechanical system is derived from a multi-body assembly (masses, disks, springs, dampers connected by ropes and constraints) using the Lagrange energy method.
Topics covered:
- Reduction to equivalent SDOF parameters (
$M_{eq}$ ,$c_{eq}$ ,$k_{eq}$ ) via kinematic constraints and energy formulation - Natural frequency and adimensional damping ratio computation
- Free motion response for underdamped, lightly damped, and overdamped cases
- Forced motion: Frequency Response Function (FRF) derivation in complex form
- Steady-state response to harmonic and multi-harmonic torque inputs
- Superposition principle applied to a three-component harmonic torque
A 3-DOF mechanical system (4 rigid bodies, 9 constraints) is fully characterised using matrix formulations of the Lagrange equations, leading to mass $[M^]$, stiffness $[k^]$, and damping
Topics covered:
- Degrees of freedom count and constraint analysis
- Kinetic, potential, and dissipative energy in matrix form using Jacobian matrices
$[\Lambda_M]$ ,$[\Lambda_k]$ ,$[\Lambda_c]$ - Eigenfrequency and eigenvector computation for both undamped and damped cases
- Rayleigh damping assumption and coefficient fitting (
$\alpha$ ,$\beta$ ) - Free motion time responses with eigenmode isolation
- Frequency Response Matrix
$[H(\Omega)]$ and co-located FRF computation - Harmonic and triangular periodic force responses via superposition
- Modal coordinate transformation and diagonal modal FRF matrix
- Graphical representation of the three vibration mode shapes
Starting from experimentally measured FRFs of a 4-DOF mechanical system (obtained via impulsive force excitation), modal parameters are identified using two independent methods and compared.
Topics covered:
- Experimental FRF interpretation: resonances, nodes of vibration, phase shifts and wraps
-
Simplified method: damping ratio estimation from the phase response derivative at resonance; mode shape extraction from imaginary part of
$H$ at$\omega_{di}$ -
Residual minimisation (MSE): parametric FRF fitting using MATLAB's
fminsearch, modelling each resonance zone as a 1-DOF system with quasi-static and seismographic residual terms - Modal parameter comparison between the two methods (natural frequencies, damping ratios, mode shapes, modal stiffness and damping)
- Full FRF reconstruction via modal superposition and comparison against experimental data