Fundamentals of harmonic analysis.
Main problems in A/D conversion.
Study of linear time-invariant SDOF and MDOF systems.
Isolation and efficiency of mechanical suspensions.
Modeling with transfer matrices methods.
The finite element method.
Fundamentals of dynamics of nonlinear systems.
E. Funaioli ed altri, "Meccanica applicata alle macchine", vol. II, Ed. Patron Bologna
D.J. Ewins, "Modal Testing - Theory, Practice and Application", Second Edition, Research Studies Press LTD.
G. Genta, "Vibration of Structures and Machines - Practical Aspects", Second Edition, Springer-Verlag
Cyril M. Harris, "Shock and Vibration Handbook", Fourth Edition, Mc.GRAW-HILL
Handouts provided by the Teacher available at the e-learning website of the University of Florence:
(section "Corso Magistrale in Ingegneria Meccanica" - A.A. 2018-19)
1) Course's general purpose
The purpose of the course is to provide the knowledge needed to understand the main dynamic modeling techniques and theoretical and experimental modal analysis. Another purpose is to provide the student with the ability to understand and analyze any problem concerning the vibrations reduction techniques, also introducing the issues concerning non-linear systems.
2) Provided knowledge
In-depth knowledge and understanding of the theoretical-scientific aspects of engineering, with a specific reference to mechanical engineering, in which students are able to identify, formulate and solve, even in an innovative way, complex and/or interdisciplinary problems. The ability to understand a multidisciplinary context in the engineering field and to work with a problem solving approach.
In-depth knowledge and understanding of the theoretical-scientific aspects of mathematics and other basic sciences. To be able to use this knowledge to interpret and describe complex and/or interdisciplinary engineering problems.
3) Applying knowledge
Applying knowledge and understanding related to the choice and application of appropriate analytical and modelling methods, based on mathematical and numerical analysis, in order to better simulate the behavior of components and plants in order to predict and improve their performance.
Applying knowledge and understanding related to the definition, design and implementation of researches useful for understanding problems, through the use of both theoretical and experimental models and techniques.
Knowledge of Physics (mechanics), analytical mechanics, geometry (vectors), Linear Algebra (Matrix Calculus)
Lectures in the classroom
Type of Assessment
1) The evaluation of the student includes an oral exam in which, in general, 3 questions are proposed on the entire program of the course.
2) The student must demonstrate at least basic knowledge of the topics covered in the course. It must be able to demonstrate a minimum of awareness of the free and forced dynamic behavior of mechanical systems.
Topics covered in the Course are:
Fundamentals of harmonic analysis: periodic, harmonic and transient analog signals.
Introduction to signal frequency spectrum and spectral analysis concepts. Series and Fourier Transform. Meanings and problems regarding A/D conversion. The Discrete Fourier Transform. Aliasing and Leakage.
Introduction to physical models, mathematical models, modal and FRF models.
Dynamics of linear SDOF systems (Single Degree of Freedom) by using the simplest dynamic model characterized by time-invariant parameters.
Solution of the equation of motion: study of free and forced behavior of SDOF systems. Viscous damping. Logarithmic decrement and half-power method. Frequency response functions (FRFs): calculation and representation through the Bode diagrams and on Nyquist's plan.
Natural frequency, and resonance. Dynamic models of accelerometers and seismometers as SDOF systems. Piezoelectric accelerometer. Vibration isolation through elastic suspension.
Linear MDOF systems (Multi Degrees Of Freedom) with viscous and structural damping. Free and forced behavior. Natural frequencies and vibration modes of the system. Modal matrix decoupling. Principal and normal coordinates. Resonances and anti-resonances.
Overview on experimental modal analysis: transducers and measuring chains. Transducers for structural excitation signals (transients and random). Main setup for modal test analysis. Parametric identification techniques.
Modeling techniques through transfer matrices: Holzer's method for torsional vibrations, Myklestad method for bending vibrations.
Vibrations of distributed parameters systems: vibrating string, rotary, longitudinal and bending oscillations of a beam with constant cross section.
Finite element method (FEM): single finite element equations. Rotation and assembly of the various elements equations in order to obtain the complete dynamical model of the system. Introduction of mechanical constraints.
Nodal and modal reduction techniques.
Vibrations in nonlinear systems: non-linear elastic behavior of a spring and friction. Exact and approximate techniques to solve nonlinear dynamic equations.