Aerospace engineering

The course treats the description and the prediction of the motion of rigid bodies, in the framework of Lagrangian mathematical formalism. We analyse in some detail the geometry of masses of rigid bodies and its relevance to motion, as well as the structure of the equatons of motion. We also consider moving reference frames. Motions as solutions to Ordinary Differential Equations (25 hours): Examples of motions. Ordinary Differential Equations and Systems. Autonomous systems. Equilibrium and stability. The conservative case. Representations in phase plane. Central motions. Motion of a point constrained to a curve or to a surface, with or without friction. Kinematics and dynamics of holonomic systems (25 hours): Constraints. Non-degeneracy conditions. Independent coordinates and degrees of freedom. Lagrangian coordinates. Mechanical quantities in lagrangian coordinates. Initial conditions. Virtual displacements. Heuristic introduction to constraints without friction. Projection of the equation of motion on the tangent space. The hypothesis of virtual work. Lagrange equations. Moving reference frames. Fictitious forces. Lagrange equations in moving reference frames. The conservative case. Lagrangian function. Equilibrium and stability. Kinematics (10 hours): Change of reference frames. Orthonormal triples. Angular velocity. Relative derivative in a moving triple. Relative kinematics. Coriolis theorem on accelerations. Curves in space. Frenet-Serret frame and formulas. Rigid bodies (30 hours): Local coordinates for rigid bodies. The inertia tensor. Principal axes. Lagrange equations. Lagrange equations in moving reference frames. Motions of a rigid body with a fixed point. Torque-free motion of a rigid body. Rotations. Cardinal equations. Euler's equations. Instantaneous axis of rotation. Ruled surfaces of the motion. A more detailed description of the program can be found in the Diary of the course, available in https://www.sbai.uniroma1.it/~daniele.andreucci/didattica/mmmecc/mmmecc_index.html

Important: Elementary linear algebra. Differential and integral calculus, in one and several real variables. Curves and surfaces. Ordinary differential equations, basic theory and techniques. Indispensable: exams of Analisi Matematica I and Geometria.

Meccanica Razionale. Modelli matematici per l'Ingegneria. D.Andreucci Sistemi di equazioni differenziali. D.Andreucci, E.N.M.Cirillo Esercizi con risoluzioni D.Andreucci available in: https://www.sbai.uniroma1.it/~daniele.andreucci/didattica/mmmecc/materiale_mm/materiale_mm_index.html The correspondence between texts and the chronology of lectures is pointed out in the analytical part of the Program (Italian version).

Standard lectures devoted to teaching the theory. Problem solving devoted to the development of the capability of applying the theory. Some problems are assigned as homework. Learning is supported by the availability of the instructor in his office hours.

Lectures take place according to the notifications of the Dean Office. Attendance is encouraged, but it does not contribute to the grading procedure.

* Instruments and methods of evaluation The written exam measures the student's capability of solving problems in Rational Mechanics, applying the methods and the techniques taught in the Course. It also measures the student's understanding of the logical structure of the theory. The oral exam is aimed at a more comprehensive assessment of the understanding of the fundamental structures of Rational Mechanics. The exam is administered after the end of the course. * Evaluation criteria Minimal knowledge (grade 18-20); average knowledge (grade 21-23); sufficient capability of applying knowledge (grade 24-25); good capability of applying knowledge (grade 27-28); optimal capability of applying knowledge (grade 29-30 cum laude).

E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria. E. DiBenedetto, Classical Mechanics: Theory and Mathematical Modeling.

Standard lectures devoted to teaching the theory. Problem solving devoted to the development of the capability of applying the theory. Some problems are assigned as homework. Learning is supported by the availability of the instructor in his office hours.

System of ordinary differential equations, phase space, equilibrium and stability, application to mechanics. The course treats the description and the prediction of the motion of rigid bodies, in the framework of Lagrangian mathematical formalism. We analyse in some detail the geometry of masses of rigid bodies and its relevance to motion, as well as the structure of the equations of motion. We also consider moving reference frames. Motions as solutions to Ordinary Differential Equations (30 hours): Examples of motions. Ordinary Differential Equations and Systems. Autonomous systems. Equilibrium and stability. The conservative case. Representations in phase plane. Kinematics (10 hours): Change of reference frames. Orthonormal triples. Angular velocity. Relative derivative in a moving triple. Relative kinematics. Coriolis theorem on accelerations. Kinematics and dynamics of holonomic systems (25 hours): Constraints. Non-degeneracy conditions. Independent coordinates and degrees of freedom. Lagrangian coordinates. Mechanical quantities in lagrangian coordinates. Initial conditions. Virtual displacements. Heuristic introduction to constraints without friction. Projection of the equation of motion on the tangent space. The hypothesis of virtual work. Lagrange equations. Moving reference frames. Fictitious forces. Lagrange equations in moving reference frames. The conservative case. Lagrangian function. Equilibrium and stability. Rigid bodies (25 hours): Local coordinates for rigid bodies. The inertia tensor. Principal axes. Lagrange equations. Lagrange equations in moving reference frames. Motions of a rigid body with a fixed point. Torque-free motion of a rigid body. Rotations. Cardinal equations. Euler's equations. Instantaneous axis of rotation. Ruled surfaces of the motion.

Important: Elementary linear algebra. Differential and integral calculus, in one and several real variables. Curves and surfaces. Ordinary differential equations, basic theory and techniques. Indispensable: exams of Analisi Matematica I and Geometria.

E.M.N. Cirillo, Appunti delle Lezioni di Meccanica Razionale per l'Ingegneria, 2018, Compomat, Configni (RI). D. Andreucci, E.N.M. Cirillo, Sistemi di equazioni differenziali ordinarie, 2025, Compomat, Configni (Ri).

Attending classes is voluntary and takes place in the classrooms and according to the schedule published by the Dean office. Attending classes, although warmly encouraged, does not contribute to the final mark.

The written exam consists of one test: the test is technical in nature and measures the student's capability of solving problems on ordinary differential equations and on Rational Mechanics, applying the methods and the techniques taught in the Course. The oral exam is aimed at a more comprehensive assessment of the understanding of the fundamental structures of Rational Mechanics and the theory of ordinary differential equations. The exam is administered after the end of the course. Evaluation criteria: Minimal knowledge (grade 18-20); Average knowledge (grade 21-23); Sufficient capability of applying knowledge (grade 24-25); Good capability of applying knowledge (grade 27-28); Optimal capability of applying knowledge (grade 29-30 cum laude).

Bibliografia di riferimento in Italiano D. Andreucci, Meccanica Razionale. Modelli Matematici per l'Ingegneria, 2022. H. Goldstein, C.P. Poole, J.L. Safko, Meccanica Classica, Zanichelli.

Standard lectures devoted to teaching the theory. Problem solving devoted to the development of the capability of applying the theory. Some problems are assigned as homework. Learning is supported by the availability of the instructor in his office hours.