Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Differential and integral calculus of the functions of one and many variables; ordinary differential equations; differential geometry of curves and surfaces. Matrix and vector algebra; orthogonal and hermitean matrices; eigenvalue problems.

The aim of the course is that of providing the fundamental elements of the kinematics, the statics and the dynamics of point-mass systems, with particular regard to rigid bodies and compound systems. The objective is that of modelling, analyzing and solving engineering problems. The basic elements of lagrangian mechanics will be covered.

On completion of the course, the student will be able to apply the acquired knowledge to real mechnical systems, relevant to control engineering and robotics. In particular, the ability to use the mathematical tools in engineering problems will be developed by the following steps: 1. being able to write the equations of motion for poit-mass systems, particularly for rigid bodies, both using the newtonian and the lagrangian approach; 2. being able to solve the equations of motion in some important cases; 3. being able to determine the equilibrium configurations of the most important mechanical systems and of studying their stability properties; 4. being able to calculate the inertia matrix for a general rigid body and to determine the principal axes of inertia.

The expertise acquired in this course are essential in order to develop the capability of analyzing and modelling rigid and compound systems, such as those often met in control engineering and robotics, with the tools of mathematical physics. Individual and collective problem-solving sessions will improve the ability to develop independent thought and learning capabilities. Oral presentations of the topics taught in the course, with the language proper of the basic disciplines of the degree course will help developing communication skills.

Vector calculus. Kinematics of the point mass: kinematic variables, plane motion and; other types of motion. Kinematics of the systems of particles, rigid motion and relative motion. Fundamental principles of dynamics. Motion under gravity and oscillatory motion. Statics and dynamics of the unconstrained point particle. Statics and dynamics of systems of particles with constrains. Material geometry and dynamical variables of the systems of particles. HuygensÂ’ theorem. General theorems of the mechanics of the systems of particles. Balance equations. Analytical mechanics and Lagrangian mechanics. Some elements of equilibrium and stability theory.

The exam consists of a written test and a colloquium: - both tests will concern the topics of the current academic year; possible exceptions will be assessed on a case-by-case basis; - registration to the first written test is mandatory, and has to be done on line on the university platform Esse3 (the link is available on the teacher's web page); - the written test consists of a number of problems and questions concerning all topics treated during the course; this test will last two or three hours, and the student will not be permitted the use of any kind of material, not even a pocket calculator; - a minimum score of at least 18/30 in the written test is required for the admission to the oral test; - the list of the names of the students admitted to the oral test will be published by the teacher on his official web page; - the oral test will contain mainly theoretical questions, some of which may have to be formulated in written form, and may contain problems and exercises concerning course topics not covered in the written test or course topics in which the student may have shown weaknesses in the written test; - questions of general comprehension may be asked both in the written and in the oral test and may concern any of the course topics; - in the case of a successful written test, the student may sit for the oral test either in the same session or in the next available session, but not later; - in the case of a successful written test, but a not passing grade in the oral test, the student may try the oral test again in the next available session; in case of another failure, the student will have to sit for the whole exam again; - all written tests must be presented in readable form, with a negligible amount of corrections, which must anyway not mar the esthetics of the text; the exposition must be clear, fluent, well organized and consistent both in the mathematical and in the linguistic aspects; - honor code: each student pledges that the written tests are entirely his/her own work and that no input from other students or sources has been used; demeanors which are deemed unfair or not in line with these principles entail the failing of the exam.

In order to pass the exam, the student must demonstrate understanding of all the topics covered and concepts introduced during the course and published on line as "Final program" or "Exam program" at the end of the course, and to be able to use them in solving typical theoretical mechanics problems. The student must demonstrate deep understanding of the principles of mechanics and of their applications in the mathematical-physics meaning. In particular, the ability of writing the equations of motion for the system of point masses, with special emphasis on rigid bodies and systems of rigid bodies, both in the lagrangian and in the newtonian approach; the ability of solving the equations of motion in the most relevant cases; the ability of determining the equilibrium positions and of studying thir stability in the most relevant mechanical systems; the ability of calculating the inertia tensor and of determining the principal exes of inertia for any rigid body.

The highest grade of 30/30 will be given to those students which will have shown deep knowledge and perfect mastering of all the course topics and the ability of working with full independence both in solving the assigned problems and in the oral presentation. The lowest passing grade of 18/30 will be given to the students which will have shown sufficient knowledge and good mastering of all the course topics.

the student must obtain a passing grade in both parts of the exam. The final mark will be the average of the marks in the two parts of the exam.

1) G. Frosali, E. Minguzzi, "Meccanica Razionale per l'Ingegneria", Ed.Esculapio; 2) M. FABRIZIO, Elementi di Meccanica Classica, Zanichelli Ed. 2002.

- Ingegneria Meccanica (Corso di Laurea Triennale (DM 270/04))
- Ingegneria Informatica e dell'Automazione (Corso di Laurea Triennale (DM 270/04))

**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

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