|SCIENTIFIC DISCIPLINARY SECTOR||MAT/07|
|MODULES||This unit is a module of:|
Course attendance and of the class for will allow the students to:
- understand the mathematical foundation of Newtonian mechanics, in particular the kinematics and dynamics of rigid bodies
- be able to calculate the center of mass and the elements of the inertia matrix for systems with the rigidity constraint
- being able to deduce and use the most suitable reference and pole for the setting of the force-torque equations for rigid bodies.
- being able to identify and apply the most suitable useful form of the force-torque equations needed to describe an idealized rigid mechanical system
-being able to discern the cases in which an aspect of the motion of a rigid mechanical system can be deduced using the work energy theorem
Classroom teaching, exercises in class. Attendance (and active participation) to the course is recommended.
Elements of vector algebra and of the theory of geometric curves:
Free and applied vectors. Vector quantities. Geometrical representation of free and applied vectors. Orthogonal projections. Scalar product. Orthonormal bases. Vector product. Mixed product, double vector product. Parallel and normal component of a vector with respect to another. Matrix algebra. Polar, spherical an cylindrical coordinates.
The concept of observer. Absolute space and absolute time axioms. Rectification formula. Arc length parameter. Frenet frame. Curvature and Torsion.Velocity, acceleration and their Cartesian representation. Examples of elementary motions (linear, uniform, harmonic, circular, in cartesian poral and spherical coordinates) and their relation with the arc length parameter.
Relative motion between frames. Transformation of vectors. Orthogonal matrices and change of orthonormal bases. Euler Angles. Angular velocity. Poisson formulas. Composition of angular velocities. Frame dragging motions. Theorem of composition of velocities and accelerations.
First Principle of dynamics. Inertial mass. Momentum. Conservation of momentum for isolated systems. Second and third principle of dynamics. Work and Power of a force (including ). Conservative forces. Potential of conservative forces. Kinetic energy. Work-energy theorem. Conservation of energy. Rheonomic constraints on a point mass. Friction. Extension of the Work-Energy theorem in the case of non-conservative forces.
Mechanics of systems of particles:
Systems of applied vectors. Resultant and total angular momentum of systems of vectors. Scalar invariant. Central axis. Reducible and irreducible systems of vectors. Center of a system of parallel applied vectors. Barycenter. Key mechanical quantities of systems of particles. Koenig theorem. Force-torque equations. Work-Energy theorem for systems of particles. Conservation laws of systems of particles.
Mechanics of the rigid body:
Reference frame comoving with a rigid body. Act of motion of a rigid body. Velocity and acceleration of the points of a rigid body. Examples of rigid motion. Composition of rigid motions. Key mechanical quantities of the rigid motion. Linear operators and their representation in terms of matrices. Symmetric linear operators and their matrix representation. Symmetric and antisymmetric linear operators. Eigenvalues and eigenvectors. Inertia tensor and its properties. Inertia matrices. Huygens’ theorem and parallel-axis theorem. Force-Torque equations for rigid bodies. Power of a system of forces acting on a rigid body. Work-Energy theorem for rigid bodies. Poinsot motion and permanent rotations. Stability of permanent rotations. Poinsot Gyroscope. Motion of a free rigid body. Ideal constraints applied to a rigid body. Rigid body with fixed point. Rigid body with fixed axis. Compound pendulum. Pure rolling. Static and dynamic balancing.
The main topics of the course can be found in Biscari P. et al. “Meccanica razionale”, Monduzzi editore (2007)--third edition. Lectures also contain elements 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.
Reference for 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”, edizionei Pitagora (2011).
SANTE CARLONI (President)
DAVIDE GIGLIO (President)
All class schedules are posted on the EasyAcademy portal.
The final exam is composed by a written test and an oral examination. The written test has a duration of three hours and the oral one a maximum of 30 minutes.
A student is admitted to the oral exam only if s/he has achieved at least 50% of the maximum score (15 marks). The final mark is the average of the written and the oral marks. In order to pass the exam, it is necessary he that student score at least 18 marks in the oral exam and that the final mark is bigger or equal to 18 marks
The written exam consists in the resolution of a problem concerning the dynamics of a rigid body. The exam will allow to evaluate the capability of the students to determine the best resolution strategy of the problem, their ability to perform the calculations necessary, and their knowledge of the theory required to solve the problem.
The oral exam will consist in the presentation and demonstration of some aspects/theorems of Newtonian mechanics for point particles and material systems. The exam will allow to evaluate the capability of organization and presentation of the course material and to verify, via suitable questions, the degree of integration of the course material in the cultural background fo the student.