Mechanical 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 equations of motion. We also consider moving reference frames. A more detailed description of the program can be found in the Diary of the course, available in http://www.dmmm.uniroma1.it/~daniele.andreucci/didattica/meccraz/meccraz_index.html Motions as solutions to Ordinary Differential Equations (25 hours): This section of the program introduces some general ideas which are needed in the rest of the course. Some topics have intrinsic relevance, like conservative forces, the phase plane, the global equations of dynamics. This part also serves as a training in basic techniques of analysis of differential equations. Examples of motions. Velocity and acceleration; regular trajectories. Ordinary Differential Equations and Systems. Laws of motion. Kinetic energy and work. Autonomous systems. Equilibrium and stability. The conservative case. Potential. First integrals. One-dimensional systems. Representations in phase plane. Central motions. Representation in polar coordinates. Areal velocity. Systems of points. External and internal forces. Global equations of dynamics. Kinematics and dynamics of holonomic systems (25 hours): This part is an essential component of the course and leads to introducing holonomic constraints, from the point of view of geometry (Dini's theorem), kinematics (space of virtual displacements), dynamics (hypothesis of virtual work). We start with an introductory section on elementary constraints (curves and surfaces). Motion of a point constrained to a curve or to a surface, with or without friction. Length of a curve, arc length. Curvature and torsion of a curve. Intrinsic frame of a curve. Frenet-Serret formulae. Coordinate tangent vectors in a regular surface. Geodesic motion. Coulomb-Morin models of friction. Constraints. Dini's theorem and conditions of non-degeneracy. Independent coordinates and degrees of freedom. Lagrangian coordinates. Mechanical quantities in lagrangian coordinates. Initial conditions. Virtual displacements. Heuristic introduction to constraints without friction and projection of the equation of motion on the tangent space. The hypothesis of virtual work. Solution of the problem of motion. Lagrange equations. Moving reference frames. Fictitious forces. Lagrange equations in moving reference frames. Lagrangian components of Coriolis forces. Equivalent Lagrangian functions. The conservative case. Lagrangian function. Equilibrium and stability. Relative kinematics (10 hours): This part of the program is used in the other ones; essentially in it we introduce a dependence on time in the elementary theory of change of basis in vector spaces. Change of reference frames. Orthonormal triples. Angular velocity. Relative derivative in a moving triple. Relative kinematics. Coriolis theorem on accelerations. Angular velocity of a moving triple with respect to another. Composition of angular velocities. Rigid bodies (30 hours): In this essential section of the course we introduce rigid bodies, also as a continuum (besides the case of bodies comprised of particles). This subject actually falls within the area of systems constrained by holonomic constraints, but it deserves specific attention owing to its importance. It is the first one of the great families of mathematical models for the mechanics of solids and fluids which will be presented to the student in the Laurea Program. Local coordinates for rigid bodies. Reference frames fixed in the body. Density and mass distributions. The inertia tensor. Moments of inertia and products of inertia coefficients. Principal axes. Distributions of forces. Hypothesis of virtual work for systems of rigid bodies. Lagrange equations for systems of rigid bodies. Global (or cardinal) equations for a rigid body. Euler's equations. Motions of a rigid body with a fixed point. Torque-free motion of a rigid body. Rotations. Velocity field of a motion fixed in the moving frame. Instantaneous axis of rotation. Ruled surfaces of the motion. Plane motions of a rigid body.

Important: Students must have an operational knowledge of the following subjects. Elementary linear algebra. Linear systems; rank of a matrix and theorem of Rouché-Capelli. Scalar and vector product. Bases of vector spaces, orthonormal bases, matrices of linear applications and changes of basis. Quadratic forms. Eigenvectors and eigenvalues; diagonalization of symmetric matrices. Orthogonal subspaces. Differential and integral calculus, in one and several real variables. Main results on limits and continuous functions (for example, theorems of intermediate value, boundedness on compact sets, uniform continuity). Ordinary and partial derivatives, theorems of differentiation of inverse and composed functions, in the scalar and in the vector case. Integrals depending on a parameter. Integrals on intervals, curves, multidimensional domains, surfaces. Integration by parts and by substitution. Improper integrals. Taylor's formula and application to the classification of critical points. Finding extrema, both free and constrained. Curves and surfaces. Tangent line, tangent plane. Parametrizations. Conics. Planes, surfaces, curves determined by geometrical characteristics (for example a plane normal to a vector and including a point). Ordinary differential equations, basic theory and techniques. General integral of linear differential equations with constant coefficients, of any order. Separation of variables. First integrals of equations of second order. Theorems of divergence, Stokes, Gauss-Green. Closed and exact vector fields and related criteria. Simply connected open sets. Indispensable: exams of Analisi Matematica I and Geometria.

Meccanica Razionale. Modelli matematici per l'Ingegneria. D.Andreucci 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 lectures is outlined in the analytical part of the Program.

Standard lectures devoted to teaching the theory and the logic of its modeling. Problem solving devoted to the development of the capability of applying the theory, also in view of specific goals. Some problems are assigned as homework. Learning is supported by the availability of the instructor in his office hours, and by the solved problems downloadable from the web site of the course.

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

The written exam consists in two tests. The first one is theoretical in nature and measures the student's understanding of the logical structure of the theory. The second one is technical in nature and measures the student's capability of solving problems in 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. 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.N.M. 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 and the logic of its modeling. Problem solving devoted to the development of the capability of applying the theory, also in view of specific goals. Some problems are assigned as homework. Learning is supported by the availability of the instructor in his office hours, and by the solved problems downloadable from the web site of the course.