MECCANICA RAZIONALE

Knowledge and understanding

By the end of the course, students will be familiar with the fundamentals of vector and tensor calculus, the kinematics of a point and of rigid systems, the composition of relative motions, the principles of classical dynamics of a point and of systems of material points, the elements of mass geometry (centres of mass, moments and tensor of inertia), and the variational principles of mechanics, with particular reference to the principle of virtual work and Lagrange's equations for holonomic systems of constrained rigid bodies.

Applying knowledge and understanding

By the end of the course, students will be able to: perform vector and tensor calculus operations in physical space; study the motion of a point and of a rigid system, including in the presence of relative motion between different observers; set up and solve the fundamental equations of dynamics for systems of material points; compute centres of mass, moments of inertia and inertia tensors of material systems; determine the work, power and kinetic energy of a mechanical system; apply the principle of virtual work and Lagrange's equations to determine the equilibrium conditions and equations of motion of holonomic systems of constrained rigid bodies, and discuss their stability.

The course consists of lectures, during which the theoretical content of the course is presented and discussed rigorously, and tutorials, devoted to the guided solution of applied exercises and to problems representative of the types that will be the subject of the examination.

Taught activities (DE – Didattica Erogativa) include the presentation of theoretical content and applied examples; interactive activities (DI – Didattica Interattiva) include classroom tutorials, with the active involvement of students in solving exercises and discussing questions proposed by the lecturer, also for the purpose of ongoing self-assessment.

Teaching activities are organised so as to ensure consistency between the learning objectives and the methods adopted: the frontal presentation of theoretical and analytical content is constantly accompanied by applied tutorials, aimed at developing the modelling and problem-solving skills required by the intended learning outcomes.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

In order to follow the course profitably, students must have a solid command of differential and integral calculus for functions of one and several real variables, and of the basic elements of geometry and linear algebra (vectors, Euclidean vector spaces, matrices, eigenvalues and eigenvectors), normally provided in the first-year courses of Mathematical Analysis and Geometry.

It is also useful, although not essential, to have the elementary notions of classical mechanics (force, momentum, work and energy) typically provided in General Physics courses, so as to facilitate the physical framing of the mathematical models covered in the course.

Attendance is mandatory, in accordance with the rules set out in the CdS (Degree Programme) teaching regulations.

1.    Recap of vector spaces. 

Elementary properties of vector spaces; dimension and bases of a vector space; vector subspaces; Euclidean vector spaces; orthonormal bases; orthogonalisation; affine point spaces; rectilinear coordinates of an affine point space.

2.    Vectors and variable points. 

Vector-valued functions of a real variable; limits; derivative, differential and Taylor formula of a vector-valued function of one variable.

3.    Elements of tensor algebra. 

Tensors; tensor product; affine tensors and the transformation law of their components; algebraic operations on affine tensors; tensoriality criteria; symmetric and antisymmetric tensors; Euclidean tensors; eigenvectors and eigenvalues of a symmetric tensor.

4.    Vector calculus in physical space. 

Axiom of the physical space of an observer; unit vectors and orthogonal projection of a vector; decomposition of a vector; scalar, vector and mixed products, and double vector product; polar moment of a system of vectors; law of variation of the moment as the pole varies; moment field; moment of a couple; axial moment; equivalence conditions and elementary invariant operations; equivalence of a system of vectors to a vector applied at a point plus a couple; systems equivalent to zero; centre of a system of parallel applied vectors and its properties.

5.    Differential properties of curves. 

Arc length and tangent unit vector; osculating plane; curvature and radius of curvature; first Frenet formula and the principal triad.

6.    Gradient, divergence and curl, conservative fields. 

Gradient of a scalar function; circulation of a vector field; divergence and flux of a vector field; Gauss's lemma and the divergence theorem; Stokes' theorem; conservative fields.

7.    Kinematics of a point. 

Axioms of classical kinematics; finite equations of motion; time law and time diagram; composite motions; scalar and vector velocity and acceleration; uniform and uniformly varying motion; tangential and normal acceleration; plane motions (circular motion, harmonic motion and the differential equation of harmonic motion).

8.    Kinematics of rigid systems. 

Lagrangian and Eulerian viewpoints; rigid displacements and motions; degrees of freedom of a rigid body; kinematic rigidity condition; angular velocity; Poisson's formulas; elementary rigid motions (translational, rotational, helical); instantaneous state of motion, tangent motions, helical state of motion; Mozzi's theorem; spherical and plane rigid motions; infinitesimal rigid displacement.

9.    Relative motion. 

Principle of relative motion; Coriolis' theorem; translational and rotational transport motion; relation between the time derivatives of a vector in two moving frames; composition of rigid motions; Euler angles.

10. Classical dynamics of a point. 

Principle of inertia; inertial space-times; Kirchhoff's and Mach's postulates; inertial mass and force; principle of action and reaction; fundamental equation of the mechanics of a free point; first integrals; statics of a point in inertial and non-inertial spaces; the Galilean principle of relativity; motion and statics in rotating spaces; motion in terrestrial space, apparent forces, weight, inertial and gravitational mass.

11. Foundations of the classical dynamics of systems. 

Fundamental equations of mechanics for systems of material points; fundamental equations of statics; momentum and centre of mass of a system of material points.

12. Mass geometry. 

Coordinates and properties of the centre of mass; moment and radius of inertia; inertia tensor, eigenvalues and principal directions; ellipsoid of inertia; principal axes and planes; central inertia tensor; angular momentum in motion about the centre of mass and of a rigid system.

13. Work, power, energy. 

Work of a system of forces, including for infinitesimal rigid displacements; virtual work and Lagrangian components; conservative forces; kinetic energy and König's theorem; kinetic energy of rigid and holonomic systems; the kinetic energy theorem; the energy integral and the principle of conservation of mechanical energy.

14. Dynamics of holonomic systems of rigid bodies. 

Constraint reactions; the fundamental problem of the dynamics of holonomic systems of rigid bodies; frictionless constrained systems; D'Alembert's and Lagrange's equations; the symbolic relation of dynamics; the integral kinetic energy theorem; Lagrangian equilibrium conditions; the principle of virtual work; Torricelli's principle; determination of the constraint reactions exerted by common constraints; an outline of the stability of equilibrium configurations.

•      Salvatore Rionero, “Lezioni di Meccanica Razionale”, ARACNE edizioni.

•      Mauro Fabrizio, “Elementi di Meccanica Classica”, Zanichelli.

•      L. Barletti, G. Frosali, F. Ricci, “Esercizi di Meccanica Razionale per Ingegneria”, Società Editrice Esculapio.

Any lecture notes and supplementary teaching material prepared by the lecturer will be made available during the course.

The exam consists of:

• Written test – prerequisite for the oral exam

• Oral exam

Written Test

The written test is divided into two parts:

1. First part: calculation of centers of mass, moments of inertia, central axes, and principal axes of inertia.

2. Second part: determination of a system’s equilibrium configurations and calculation of constraint reactions.

Oral Exam

The oral exam assesses the theoretical knowledge of the topics covered during the course.

Grading Criteria

Grade 29–30 with honors

• In-depth knowledge of the subject

• Ability to integrate and critically analyze the presented situations

• Independent resolution of complex problems

• Excellent communication skills and command of language

Grade 26–28

• Good knowledge of the subject

• Ability to analyze situations critically and coherently

• Fairly independent resolution of complex problems

• Clear presentation with appropriate language

Grade 22–25

• Satisfactory knowledge, limited to main topics

• Critical analysis not always coherent

• Presentation fairly clear with adequate command of language

Grade 18–21

• Minimal knowledge of the subject

• Limited ability to integrate and critically analyze situations

• Presentation sufficiently clear, but language skills poorly developed

Exam not passed

• Insufficient knowledge of the main course contents

• Very limited or no ability to use specific terminology

• Unable to independently apply acquired knowledge

Compensatory and Dispensatory Measures

To ensure equal opportunities and comply with current regulations:

• Interested students may request a personal meeting to arrange any compensatory and/or dispensatory measures, according to the learning objectives and specific needs.

• Students may also contact the CInAP representative (Centre for Active and Participatory Integration – Services for Disabilities and/or Learning Disorders) of their Department.

•      Determine the instantaneous state of motion of a given rigid system and identify its Mozzi axis.

•      Compose the motion of a point with respect to two observers in relative motion, applying Coriolis' theorem.

•      Compute the centre of mass and the inertia tensor of a given homogeneous rigid system.

•      Write the fundamental equations of dynamics for a system of material points and discuss its first integrals.

•      Determine the Lagrangian components of the active forces for a given holonomic system.

•      Set up Lagrange's equations for a holonomic system of constrained rigid bodies and determine its equilibrium configurations.

•      Apply the principle of virtual work to determine the equilibrium conditions of a constrained system.

•      Compute the work, power and kinetic energy of a rigid system in a given motion.

•      State and prove one of the fundamental theorems of the course (e.g. Mozzi's theorem, König's theorem, the kinetic energy theorem).

•      Discuss the stability of an equilibrium configuration of a conservative holonomic system.