AIMS AND CONTENT

LEARNING OUTCOMES

The course provides the mathematical methods for describing mechanical systems. In particular the motion of systems with many degrees of freedom is studied the rigid body mechanics analyzed in details.

AIMS AND LEARNING OUTCOMES

After the course completion the student is familiar with the statics and the dynamics of mechanical systems with finite degrees of freedom (particles systems and systems composed by rigid bodies).

PREREQUISITES

mathematical calculus, basic linear algebra, mathematical analysis I, Physics I. 

TEACHING METHODS

The course includes lectures at the blackboard in which the topics of the program are presented. Examples and exercises, designed to clarify and illustrate the concepts of the theory, are also carried out.

Students who have a valid certification of physical or learning disability on file at the University and who wish to make use of compensatory and/or dispensatory tools in relation to lectures, courses and examinations, should contact both the lecturer and Prof. Federico Scarpa, the Scuola Politecnica's disability contact person.

SYLLABUS/CONTENT

• Elements of Vector Algebra: Free and applied vectors. Vector quantities. Geometric representation of vector quantities. Vector structure of the space of free vectors. Scalar product of vectors. Orthonormal bases. Vector, triple scalar and triple vector product of vectors and their component representations. Orthogonal matrices. Change of orthonormal bases. Euler angles. Linear operators. Linear symmetric and skew-symmetric operators. Vector functions. Elements of geometric theory of a curve.
• Absolute Kinematics: Observer. Absolute Space and time. Frame of reference. Velocity, acceleration and their Cartesian and intrinsic representations. Rectilinear, uniform and uniformly accelerated motion. Circular motion. Harmonic motion. Ballistics problems. Central motions and Binet’s formula. Polar, cylindrical and spherical coordinates.
• Relative kinematics: Relative motion of frames of reference. Angular velocity. Poisson formulae. Theorem on composition of angular velocities. Transportation motion. Theorems on composition of velocities and accelerations.
• Dynamics: Newton’s first law. Inertial mass. Momentum of a particle. Momentum conservation for isolated systems. Newton’s second and third laws. Kinetic energy. Work and power of a force. Theorem of energy. Conservative forces. Potential of a conservative force. Theorem on conservation of energy.
• Relative Dynamics: Transportation inertial force. Coriolis inertial force. Earth Mechanics.
• Particle mechanics: Motion of a free particle. Friction laws. Motion of a  particle along a curve. Motion of a particle on a surface.
• Mechanics of systems: Systems of applied vectors. Resultant and resultant moment of a system of vectors. Scalar invariant. Central axis. Reducible and irreducible systems of vectors. Centre of parallel vectors and centre of gravity. Mechanical quantities of a system. Konig’s theorem. Momentum and angular momentum theorems. Theorem of energy for systems. Conservation laws for systems.
• Mechanics of a rigid body: The body-fixed reference frame of a rigid body. Rigid motion. Velocities and accelerations of the particles of a rigid body. Translational and rotational motions of a rigid body. Composition of rigid motions. Mechanical quantities of a rigid body. Inertia Tensor and its properties. Moment of a rigid body with respect to an Axis. Moments and products of Inertia. Inertia matrices. Huygens and parallel axes theorems. Momentum and angular momentum theorems for a rigid body. Power of a system of forces acting on a rigid body. Energy theorem for a rigid body. Motion of a free rigid body. Ideal constraints applied to a rigid body. Rotational motion of a rigid body about a fixed axis. Rotational motion of a rigid body about a fixed point. Poinsot motions. Elementary theory of a gyroscope and its application to the gyroscopic compass.
• Elements of Lagrangian mechanics: Principle of stationary potential for the study of equilibrium in a conservative holonomic system (without proof). Lagrange's equations for a conservative holonomic system (without proof).

RECOMMENDED READING/BIBLIOGRAPHY

• Enrico Massa, Elementi di Meccanica Razionale, dispense Università di Genova.

• Bampi, Zordan “Meccanica Razionale”. Con elementi di probabilità e variabili aleatorie” ECIG (2003)

• Goldstein “Classical Mechanics”, Addsion-Wesley; 3 edition (2001)

• Fasano, Marmi, Pelloni “Analytical Mechanics” Oxford Uiversity Press (2006)

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/en/corsi/11949/studenti-orario

Class schedule

The timetable for this course is available here: EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test that typically includes one problem and some theoretical questions.

No materials may be consulted, nor may a calculator be used (except for accommodations provided to students with learning disabilities).

To take the exam, students must register through the appropriate online services

ASSESSMENT METHODS

The exam assesses the ability to formulate and solve problems in kinematics and dynamics of systems of particles, and in particular, the mechanics of constrained rigid bodies. Theoretical questions also evaluate whether the student has acquired the ability to explain and apply knowledge of the mechanics of systems of particles. The final evaluation takes into account the quality of the presentation, the correct use of technical terminology, and the ability for critical reasoning

FURTHER INFORMATION

Although the course provides an introductory part, it is appropriate that the student is familiar with: linear algebra (vectors and linear transformations), derivation and integration, kinematics and dynamics of a material particle.

Lecture streaming is a support tool to be used exceptionally in the event that the students are unable to attend the lectures in person.