


Politecnico di Torino  
Academic Year 2015/16  
01NMCQD Mechanical system dynamics 

Master of sciencelevel of the Bologna process in Mechanical Engineering  Torino 





Subject fundamentals
The subject is addressed at providing the knowledge and capabilities for the mathematical modelling of mechanical systems within industrial applications, with particular attention to the analysis of dynamic phenomena that can be verified in mechanical components.

Expected learning outcomes
Knowledge related to dynamic behavior of structures, mechanical systems and rotating machines in stationary or periodical regime and in transitory phases. Capacity to model and analyze the dynamic behavior of structures, mechanical systems and rotating machines.

Prerequisites / Assumed knowledge
Basic knowledge within the mechanical theory and applied mechanics, of mechanical components and of fundamental of differential and integral calculations.

Contents
Vibrations of damped sdof systems [6 h]
Vibrations of mdof systems with proportional viscous damping [12 h] Elements of analytical dynamics [4.5 h] Vibrations of continuous systems (distributed parameters) [9 h] Dynamics of rotors [7.5 h] 
Contents (Prof. A. Vigliani)
• vibrations of damped single degree of freedom systems; free response; harmonic response (approach based on complex numbers); transmissibility; nonperiodic input (step, impulse); convolution integral (theory and tutorials: 12 h.)
• vibrations of multidof 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 (EulerBernoulli) (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.) 
Delivery modes
Practical lectures consisting in the solution of the proposed exercises regarding the topics presented during the lectures (sdof and mdof systems, analytical dynamics, distributed parameters and rotordynamics) [21 h]

Delivery modes (Prof. A. Vigliani)
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 email. 
Texts, readings, handouts and other learning resources
Lectures notes available on the course page
Suggested textbooks: Meirovitch L., Fundamentals of Vibrations, Mc Graw Hill Vigliani A., Lectures on Rotordynamics, Clut 
Test, readings, handouts and other learning resources (Prof. A. Vigliani)
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 course 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 
Assessment and grading criteria
Written only test consisting in the solution of two problems and a theoretical question.
Grade is proportional to the correctness and completeness of the answers to the proposed problems and questions. 
Assessment and grading cirteria (Prof. A. Vigliani)
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. 
