The subject is addressed at providing the knowledge and capabilities for mathematical modelling of mechanical systems within industrial applications, focusing on the analysis of dynamic phenomena that may occur in machines.

Knowledge related to the dynamic behaviour of structures, mechanical systems and rotating machines in steady state, periodic and transient motion. Capability of modelling and analysing the dynamic behaviour of structures, mechanical systems and rotating machines with the analytical methods provided in the semester.

Attendance of this module requires fluent spoken and written English as a necessary pre-requisite: all lectures and tutorials, and all study material will be in English.
It is assumed that students taking this subject-module already have knowledge and understanding of analytical and applied mechanics and of fundamental of differential and integral calculus.

• Vibrations of damped single degree of freedom systems; free response; harmonic response (approach based on complex numbers); transmissibility; non-periodic input (step, impulse); convolution integral (theory and tutorials: 12 h.)
• vibrations of multi-dof systems with proportional viscous damping: equation of motion (matrix form), eigenvalues and eigenvectors, orthogonality of modes, modal analysis, free response, frequency response functions, dynamic absorber (theory and tutorials: 14 h.)
• elements of analytical dynamics: principle of virtual work, Hamilton’s principle, Lagrange’s equations (theory and tutorials: 10 h.)
• vibrations of continuous systems (distributed parameters): wave equation (transverse oscillations of strings, axial and torsional oscillations of beams), bending vibrations of beams (Euler-Bernoulli) (theory and tutorials: 9 h.)
• approximate methods: equation of energy, Rayleigh quotient (theory and tutorials: 6 h.)
• dynamics of rotors: Jeffcott’s rotor, stability analysis, Campbell’s diagrams, whirl trajectories, critical speeds, influence of bearing flexibility and damping (theory and tutorials: 9 h.)

Credits 6: 60 classroom hours (39 lecture hours, 21 tutorial hours).
Theoretical lectures are supported by examples and applications. Lectures on a section of the syllabus will be followed by specific tutorials, where students are required to apply knowledge to working context problems. The tutor will provide materials and frames for solutions. However, students are asked to interact with the tutor, especially when setting the solution. The tutor will assist students during the tutorial class hours, supporting students in their learning progression and clarifying their doubts.
Attendance to both lectures and tutorials is strongly recommended, being vital to achieve the expected learning outcomes. The teacher and the tutor are available weekly during the teaching period in order to meet students for consultation; please contact them by e-mail.

Suggested readings:
Meirovitch L., Fundamentals of Vibrations, Mc Graw Hill
Vigliani A., Lectures on Rotordynamics, Clut

Lectures notes on specific topics are available on the subject page.

Tutorials: texts of problems and Matlab codes are provided on the course website before the lectures. Students should either download or print the files before the lecture

Achieved learning outcomes will be assessed by means of a final exam.
This is based on an analytical assessment of student achievement of the "expected learning outcomes" described above.
In order to properly assess such achievement, the examination consists of a written only test, duration 1 h 30 min, closed book, composed of three questions: the first is an exercise similar to those proposed during the tutorial; the second is focused on one of the topics seen during the lectures; the third asks the students to apply the methodologies explained in the course to some new application or machine.
The exam aims at evaluating the ability of the students to model the dynamic behavior of mechanical systems, starting from the model definition and ending with the system analysis.
In particular, the first part of the test assesses the ability to apply knowledge, while the second and third parts aim at assessing knowledge, communication skills and ability to use tools and method taught in the lectures for solving problems not directly proposed in the class hours. During the course, students are given an example of the final test, with discussion of the solution and hints on common errors and evaluation criteria.
A few days after the written test, students are summoned for a review of the written output, in which examiners inform the student on grading criteria, and receive any student appeal supported by appropriate explanations.
Further details on exam rules are given on the official course website.