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 during 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 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. Neither intermediate formal checks of the learning process nor reports on projects are programmed. 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, exercises and other material are available on the subject page. Tutorials: texts of problems and Matlab codes are provided on the subject website before the lectures. Students should either download or print the files before the lecture

Modalità di esame: Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

The online exam consists of a written test, for a maximum of 33 points. The exam is considered sufficient when the score obtained is at least 18 points. The online exam is based on EXAM platform, with the simultaneous use of Respondus proctoring, and has a duration of 70 minutes, taking into account the additional time for the authentication on the exam platform. The test is composed of both multiple-choice questions, counting and two exercises. All the questions are freely accessible for all the test duration. Pocket calculators are admitted. In analogy to the past traditional sessions, students have to answer each question by producing the equations, demonstrations, schemes and diagrams relevant to develop the answers, writing on white A4 paper sheets. When explicitly required by the question, students should upload one or more pictures (taken using the PC webcam). The photos have to clearly framing what is required by the text of the question only. If the required pictures are not uploaded, or if their quality is insufficient, the relative answer is marked with a score of zero. For this reason, students are requested to take great care in answering to the parts requiring a direct upload, in a tidy and precise manner. The required parts are expressly requested to be written on dedicated A5 paper sheets and to be photographed via the webcam during the examination. Only photos and numerical values inserted in the appropriate fields have value for marking purposes. Within 15 minutes from the end of the exam, students must upload (mandatory) a complete scan in PDF format of all the material regarding the two exercises, including A5 papers (clearly readable, taken by any means, i.e. cell phones) on the section “Elaborati” of the course page of the “Portale della didattica”. Any other means of sending copies of the exam is forbidden (NO emails!). In case no material is uploaded within the required deadline, the exam will be considered null and registered as fail (“respinto”). The uploading time of the file on the teaching portal will be checked. The last page of the written exam is used to write Withdrawn, in case of this choice. In this case, no uploading phase of the pdf file is required on the section “Elaborati” of the course page of the “Portale della didattica”. At the end of the exam evaluations, the marks will be published on the teaching portal. The methods to view the online exam will be indicated together with the publication of the mark.

Modalità di esame: Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

The online exam consists of a written test, for a maximum of 33 points. The exam is considered sufficient when the score obtained is at least 18 points. The online exam is based on EXAM platform, with the simultaneous use of Respondus proctoring, and has a duration of 70 minutes, taking into account the additional time for the authentication on the exam platform. The test is composed of both multiple-choice questions, counting and two exercises. All the questions are freely accessible for all the test duration. Pocket calculators are admitted. In analogy to the past traditional sessions, students have to answer each question by producing the equations, demonstrations, schemes and diagrams relevant to develop the answers, writing on white A4 paper sheets. When explicitly required by the question, students should upload one or more pictures (taken using the PC webcam). The photos have to clearly framing what is required by the text of the question only. If the required pictures are not uploaded, or if their quality is insufficient, the relative answer is marked with a score of zero. For this reason, students are requested to take great care in answering to the parts requiring a direct upload, in a tidy and precise manner. The required parts are expressly requested to be written on dedicated A5 paper sheets and to be photographed via the webcam during the examination. Only photos and numerical values inserted in the appropriate fields have value for marking purposes. Within 15 minutes from the end of the exam, students must upload (mandatory) a complete scan in PDF format of all the material regarding the two exercises, including A5 papers (clearly readable, taken by any means, i.e. cell phones) on the section “Elaborati” of the course page of the “Portale della didattica”. Any other means of sending copies of the exam is forbidden (NO emails!). In case no material is uploaded within the required deadline, the exam will be considered null and registered as fail (“respinto”). The uploading time of the file on the teaching portal will be checked. The last page of the written exam is used to write Withdrawn, in case of this choice. In this case, no uploading phase of the pdf file is required on the section “Elaborati” of the course page of the “Portale della didattica”. For onsite exam, students can provide exam sheets at the end of the exam directly to the commission, avoiding to upload the PDF file on the portal. At the end of the exam evaluations, the marks will be published on the teaching portal. The methods to view the online exam will be indicated together with the publication of the mark.