Elements of vector calculus and theory of moments. Elements of rigid body static. Kinematics of rigid systems. Rigid motions and plane rigid motions. Composition of rigid motions. Dynamical systems. General theorems on systems of mass points. Cardinal equations of dynamics.Motion and conservation laws. Geometry and kinematics of masses. Lagrangian formalism. Dynamics of rigid systems. Equilibrium and small oscillations.
Course Content - Last name/s M-Z
1) General theorems on systems of point particles
2) Kinematics of rigid bodies
3) Geometry and kinematics of masses
4) Screw theory and applications to mechanical moments and velocity fields of rigid bodies
5) Lagrangian formalism and virtual work principle
6) Small oscillations
7) Elements of continuum mechanics
1) G. Frosali, E. Minguzzi, Meccanica Razionale per l'Ingegneria. Ed. Esculapio 2017
2) G. Frosali, F. Ricci, Esercizi di Meccanica Razionale. Ed. Esculapio, 2013
3) R. Ricci, Lezioni di Sistemi Dinamici. Firenze University Press, 2016.
4) H. Goldstein, Classical Mechanics. Pearson, 2014.
5) V. I. Arnold, Ordinary Differential Equations. Springer, 1992.
1) Frosali, Minguzzi, Meccanica Razionale per l'Ingegneria, Esculapio 2011
2) A. Fasano, V. de Rienzo e A. Messina, Corso di Meccanica Razionale, Laterza 1989
3) Goldstein, Meccanica Classica, Zanichelli 1991.
Learning Objectives - Last name/s A-L
The lectures aim to provide the mathematical tools necessary for a deep understanding of mechanics. Specularly, the use of mathematical tools in an application context provides positive feedback on the understanding of the mathematical concepts themselves.
Learning Objectives - Last name/s M-Z
cc1: Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems. Knowledge of the principles of computer science and the algorithmic and numerical approach to problems.
cc2: Knowledge and understanding of the relevant laws of physics (mechanics, electromagnetism, thermodynamics) and chemistry in the field of industrial engineering and understanding of the role of these laws in the formulation of representative models of tangible reality.
ca2: Applying knowledge and understanding related to the physical and chemical field to solve mono-disciplinary problems of chemistry, applied chemistry, mechanics, electromagnetism and theoretical thermodynamics as a basis for mechanical engineering problems.
Prerequisites - Last name/s A-L
Mathematical analysis (first-year program) and linear algebra. Mechanics of the first-year program in physics.
Prerequisites - Last name/s M-Z
Analysis I and linear algebra
Teaching Methods - Last name/s A-L
The teaching methods are traditional: lessons on blackboard, with the possible aid of slides projections.
The theory is exposed and accompanied by exercises that clarify its application.
Teaching Methods - Last name/s M-Z
The theory is presented and accompanied by exercises which clarify the applications. A textbook based on the lectures assists the student in the study.
Further information - Last name/s A-L
For further informations:
The final exam consists of a written test followed by an oral exam.
Note: previous exams of Mathematical analysis and Geometry are mandatory.
Type of Assessment - Last name/s M-Z
Written exam followed by an oral exam. For more information
Course program - Last name/s A-L
1) GEOMETRY OF SPACE AND VECTORS
Affine space and vector space. Orthonormal bases and reference systems (RS). Linear transformations. Change of RS and orthogonal matrices. Applied vector systems and polar moment. Funicular polygon.
Trajectory, velocity and acceleration in a given RS. Relative kinematics and angular velocity vector. Addition of angular velocities. Velocity and acceleration fields of a rigid motion. Instantaneous axis of rotation. Ruled surfaces of a rigid motion. Planar rigid motions.
3) DYNAMICS OF THE POINT MASS AND GENERALITIES ON DYNAMICAL SYSTEMS
Inertial RS and Newton laws. Mathematical structure of Newton's law F = ma. Systems of differential equations of the first order (dynamical systems). Cauchy theorem for a dynamical system. Equilibrium of a dynamical system and stability. Examples: logistic equation, predator-prey model, damped harmonic oscillator. Linear stability criterion. Kinetic energy of the point mass and work of a force. Kinetic energy Theorem for the point mass. Conservative forces and conservation of mechanical energy. Characterization of conservative force fields. Poincaré lemma. Potential of a central force. Dirichlet criterion for the stability of the equilibrium of a conservative system.
4) DYNAMICS OF THE SYSTEMS OF POINT MASSES AND RIGID BODIES
Kinetic energy, work, potential energy, kinetic energy Theorem and conservation of mechanical energy for systems of point masses. Cardinal equations of dynamics. Angular momentum and kinetic energy of a rigid body. Geometry of the masses: polar, axial and centrifugal moments of inertia. Matrix of inertia and ellipsoid of inertia. Principal axes of inertia. Rigid bodies rotating around a fixed axis. Precessions. Geometric characterization of the precessions by inertia. Characterization of the permanent rotations and of their stability. Outline of the dynamics of gyroscopes. Outline of graphical methods for the statics of the rigid body.
5) DYNAMICS OF CONSTRAINED SYSTEMS
Ideal holonomic constraints for a system of point masses. Degrees of freedom, Lagrangian coordinates and tangent space. Principle of virtual works and its geometric interpretation. Symbolic equation of the dynamics and Lagrange equations of second kind. Lagrangian function. Quadratic structure of the kinetic energy. Hamiltonian formulation of the Lagrange equations (outline). Generalized Dirichlet criterion. Quadratic approximation of the Lagrangian around a stable equilibrium and linearized equations of motion. Normal modes and frequency of the small oscillations around the stable equilibrium configuration.
Course program - Last name/s M-Z
ELEMENTS OF LINEAR ALGEBRA, DEFINITION OF SPACE AND TIME
Dimensional analysis, Buckingham theorem, construction of dimensionless constants and dimension matrix. Arguments involving scalings. Definition of vector (linear) spaces. Span, linear independence, bases. Dimension of the vector space, isomorphism on R^ n. Change of basis, rule of the transposed inverse. Orientation of a vector space. Scalar product, definition of module. Positively oriented orthonormal bases. special orthogonal matrix. Gram-Schmidt orthogonalization. Cauchy-Schwarz inequality, definition of angle between two vectors. Vector product, formula with the determinant, independent basis, the vector product. Double vector product, the identity of Jacobi. Mixed product and its symmetries, oriented volume. Definition of affine space, systems of reference. Definition of physical space and time.
Reference frames in relative motion. Poisson's theorem and the definition of angular velocity. The case of plane motion. The fundamental formula of rigid motions. Change in mechanical moment and in angular momentum under changes of reference point. Motivations for screw theory. Screw definition. Resultant of the screw and its uniqueness. Examples of screws. The scalar invariant and the vector invariant. The screw axis. The screw pitch, degenerate cases. The screws form a vector space. Composition of rigid motions, additivity of angular velocities. Scalar product of screws: kinetic energy and power. Equivalent systems of forces, balanced systems. Varignon's theorem. Cases in which the resultant vanishes or not. Special cases where the vector invariant is zero: coplanar vectors, parallel and concurrent vectors. The center of parallel forces. Screw of a straight line in space. Scalar product between two lines. Dual numbers and screw calculus, dual angle.
KINEMATICS OF RIGID SYSTEMS.
Definition of rigid system. Degrees of freedom. Inertial and body reference systems. Euler angles. Rigid transformations. Special orthogonal linear transformations. Elements on orthogonal matrices. Transformation of the plane into itself. Rotation and rotation matrix. Relationship between the vector product and antisymmetric matrices. Locus spanned by instantaneous axis of rotation in inertial and body frames. Reconstruction of motion. Special rigid motions: translation, rotation, precession. Plane motion and instantaneous center of motion. Locus spanned by the centers of motions. Chasles theorem. Determination of the instantaneous center of rotation knowing the speed of a point and the angular velocity. Free rigid systems. Euler's theorem and the tennis racket theorem. Poinsot's description of of free motion with the ellipsoid of inertia. Reference systems in relative motion: relative speed, relative acceleration, centripetal and Coriolis. Dragging speed ??and dragging acceleration.
GENERAL THEOREMS ON SYSTEMS OF MATERIAL POINTS.
The cross product. Newton's laws. Action and reaction, internal and external forces. First cardinal equation. Center of mass and its behavior in combination of more bodies. Motion of center of mass theorem. Work. Kinetic energy theorem (forze vive) in three versions: the material point; the system with only the external forces applied to the center of mass; the system of all points considering all forces. Null work of the internal forces in the rigid body, the friction forces. Conservative forces. Gradient, curl and divergence. Theorem of the closed circuit, Stokes and divergence theorems. Irrotational fields, singularity and gradient. Conservation of mechanical energy. Kinetic and potential energy. Examples of conservative forces: gravity and spring. Koenig's theorem for the kinetic energy. Angular momentum, and the second cardinal equation with respect to a point in motion. The center of mass case. Independence of relative angular momentum of the reference point. Koenig's theorem for the angular momentum. Rolling with sliding and conservation of angular momentum with respect to the contact point. The pure rolling condition.
GEOMETRY AND KINEMATICS OF THE MASSES.
Introduction to the geometry of the masses. The matrix of the moments of inertia and its interpretation as a linear application. Link between angular velocity and angular momentum. The kinetic energy for rigid bodies. Parallel axis theorem (or transport) in the matrix formulation. Expression of the moment of inertia with respect to an axis. Principal axes of inertia, principal moments of inertia and diagonalization of the inertia matrix (spectral theorem). Invariants of the inertia tensor. Planar systems, remarkable property. Construction of the ellipsoid of inertia. Graphical calculation of axial moments through the ellipsoid of inertia. Use of symmetries for the determination of the main axes and the inertia matrix. Property of stationarity of the main axes. Exercises with negative masses.
The cardinal equations of statics. The funicular polygon and the meaning of its closure. Solving problems with the funicular polygon. The two and three forces theorems. free system, isostatic and hyperstatic. Examples of constraints. three-hinge system with one or both of arches loaded (superposition principle). Section method and analysis of nodes. Principle of virtual work and its use for the determination of the forces. Examples.
Space of configurations and generalized coordinates. Holonomic and nonholonomic constraints. The principle of virtual work, and the principle of d'Alembert. The Lagrange equations, with or without non-conservative generalized forces.
One-dimensional case. Stationary points for the potential.Stability. The mass matrix and the quadratic approximation of potential and kinetic energies. Simultaneous diagonalization of two matrices. Pulsations and eigenvalues. Eigenvectors and principal modes. Small oscillations.