AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the module is to introduce and deepen techniques and methods of Mathematical Physics for the construction of mathematical models and the solution of mechanical problems in Naval Engineering.

AIMS AND LEARNING OUTCOMES

The course aims to provide students with the basic notions of Rigid Body Mechanics, with particular attention to its applications to ship dynamics and hydrodynamics.

At the end of the course, the student will be able to:

- be able to calculate the center of mass and the elements of the inertia matrix in the context of systems that present the rigidity constraint.

- be able to deduce and use the reference system and the most correct pole for the setting of the cardinal equations of the rigid body.

- be able to identify and apply the most correct form of the cardinal equations suitable for describing an idealized rigid mechanical system.

- be able to discriminate the situations in which an aspect of a mechanical problem can be solved using the energy theorem and be able to apply this theorem to solve it.

- use the Lagrangian formalism to set up the Dynamics and Statics of systems with a finite number of degrees of freedom subject to ideal constraints;

- study the stability of the equilibrium and the small oscillations of said systems;

- use the Lagrangian formalism to analytically represent the hydrostatic and hydrodynamic stresses and analyze the dynamics of the ship.

TEACHING METHODS

Frontal lectures. Attendance and active participation in the classwork is recommended.

Students with valid certifications for Specific Learning Disorders (SLDs), disabilities or other educational needs are invited to contact the teacher and the School's contact person for disability at the beginning of teaching to agree on possible teaching arrangements that, while respecting the teaching objectives, take into account individual learning patterns. Contacts of the teacher and the School's disability contact person can be found at the following link Comitato di Ateneo per l’inclusione delle studentesse e degli studenti con disabilità o con DSA | UniGe | Università di Genova

SYLLABUS/CONTENT

Recapitulation of vector algebra and geometric theory of curves:

Vector quantities. Scalar product. Orthonormal bases. Vector product, mixed product, double vector product and their representation in components. Orthogonal matrices and orthonormal change of basis. Rectification formula. Curvilinear abscissa. Intrinsic triplet. Bending and torsional curvature. Frenet formulas.

Recapitulation of kinematics and dynamics of the material point:

Absolute kinematics. Euler and Tait-Bryan angles. The three Newtonian laws of dynamics. Concept of Force and its properties. Work and power of a force. Potential of a conservative force. Kinetic energy. Energy theorem and conservation of energy. Constraints and constraint reactions. Constitutive characterization of constraints. Friction. Motion of the material point in the presence of fixed and mobile constraints.

Relative kinematics:

Relative motion between reference systems. Angular velocity. Poisson formulas. Composition theorem of angular velocities. Dragging motions. Addition theorem of velocities and accelerations.

Dynamics of systems:

Systems of applied vectors. Resultant and resultant moment of systems of vectors. Scalar invariant. Central axis. Systems of reducible and irreducible applied vectors. Center of parallel applied vectors. Barycenter. Koenig's theorem. Cardinal equations. Energy theorem for systems. Conservation laws for systems.

Dynamics of the rigid body:

 Act of rigid motion. Reference system integral with a rigid body. Velocity and acceleration of the points of a rigid body. Composition of rigid motions. Particular rigid motions. Mechanical quantities of the rigid body. Linear operators and their representation by matrices. Symmetric and antisymmetric linear operators. Eigenvalues ​​and eigenvectors. Inertia tensor and its properties. Inertia matrices and moment of inertia. Inertia matrices and moment of inertia. Principal axes of inertia and their determination. Huygens' theorem and the theorem of parallel axes. Cardinal equations for the rigid body. Energy theorem for the rigid body. Ideal constraints applied to a rigid body. Sliding friction for the rigid body and rolling friction.

Lagrangian Mechanics: Space–time of configurations, space of motion acts. Kinetic constraints. Ideal constraints. Lagrange equations. Conservative stresses. Lagrangian. Generalized potential. Lagrange equations in the general case. First integrals. Lagrange equations and Relative Mechanics. Scleronomous systems. Generalized velocities. Kirchhoff equations. First integrals in the Lagrangian formalism. Connection with the dynamics of the rigid body. .Statics of holonomic systems. Stability of equilibrium. Small oscillations.

Applications: 

General scheme for the resolution of rigid body dynamics. Calculation of the center of gravity and the inertia matrix of rigid bodies. Statics of the rigid body with cardinal equations and stationarity of the potential. Pure rolling, sliding and rolling friction. Construction of the Lagrangian. Derivation of the pure equations of motion from the Lagrangian. Calculation of the constraint reactions.

RECOMMENDED READING/BIBLIOGRAPHY

1. Theory: Biscari P. et al. “Meccanica razionale”, Monduzzi editore (2007)--third edition. Lectures also integrate lelements of Massa E., “Appunti di meccanica razionale” (dipense); Grioli G. “Lezioni di meccanica razionale” Edizioni Libreria Cortina." Padua, Italy (1988);  Demeio L. “Elementi di meccanica classica per l’ingegneria”, Città Studi edizioni (2016); Bampi F. , Zordan, C., “Lezioni di meccanica razionale” ECIG 1998; C. Cercignani, “Spazio, Tempo, Movimento”, Zanichelli; M.D. Vivarelli, “Appunti di Meccanica Razionale”, Zanichelli;Goldstein H., Pool C., Safko J. "Meccanica Classica". Zanichelli (2005); Lewandowski E.M. “The dynamics of marine craft”;Newman J.N. ”Marine hydrodynamics”.

   Exercises: G. Frosali, F. Ricci “Esercizi di Meccanica Razionale“, Società editrice Esculapio (2013); Muracchini A. et al. ”Esercizi e temi d’esame di Meccanica Razionale” (2013); Bampi F. et al “Problemi di meccanica razionale” ECIG, (1984); G. Belli, C. Morosi, E. Alberti  “Meccanica razionale. Esercizi”, edizioni Maggioli Editore (2008); S. Bressan, A. Grioli, “Esercizi di meccanica razionale”,  edizioni Libreria Cortina (1981); B. Finzi, P. Udeschini “Esercizi di meccanica razionale”, editore: Massone (1986); V. Franceschini, C. Vernia “Meccanica razionale per l’ingegneria”, edizioni Pitagora (2011). 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/8738/p/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The final exam consists of a written and an oral part integrated into a group project. Students will have to solve a rigid body mechanics problem and deliver a paper containing its solution. Subsequently, this resolution will be exposed through an oral presentation, which will be integrated for each student, with some theoretical questions. The final grade is made up of the grade on the written paper (50%), on the presentation (10%), and on the oral questions (40%).

ASSESSMENT METHODS

The written part of the exam consists of solving a problem on the dynamics of a rigid body with ideal constraints. The written part will allow the evaluation of the student’s ability to determine the best problem-solving strategy, their ability to carry out the necessary calculations, and their knowledge of the theory required to solve the problem. 

The oral part of the exam consists of an exposition of the resolution and demonstration of some aspects/theorems of Newtonian and Lagrangian mechanics of the material point and material systems. It will allow the evaluation of the ability to organize and explain concepts and verify, through transversal questions, the degree of integration of these concepts in the student's global cultural background.

FURTHER INFORMATION

Ask the professor for any additional information not included in this course presentation form.