Elements of vector calculus and theory of moments. Elements of rigid body static: analytical and graphical methods. Kinematics of rigid systems. Rigid motions and plane rigid motions. Composition of rigid motions. General theorems on systems of mass points. Cardinal equations of dynamics. Motion and conservation laws. Geometry and kinematics of masses. Momentum and kinetic energy of mechanical systems. Lagrangian formalism. Dynamics of rigid systems.
Course Content - Last names 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, Esculapio 2017
2) A. Fasano, V. de Rienzo e A. Messina, Corso di Meccanica Razionale, Laterza 1989
3) H.Goldstein, Meccanica Classica, Zanichelli 1991.
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 names A-L
The course of Rational Mechanics aims to teach the Elements of Theoretical Mechanics to the students of the Degree Course in Mechanical Engineering. In particular, the course teaches how to construct a mathematical model of a mechanical phenomenon, to study it with rigorous methods and to interpret its results.
The student acquires the ability to use mathematical methods and models to study simple problems that arise in the field of mechanics.
At the end of the course, the student will be able to apply the theoretical mechanics to engineering problems, having understood the principles.
Learning Objectives - Last names 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 names A-L
During the course we will use the concepts of Euclidean geometry, learned at high school and elements of the geometry of vectors and matrices (linear algebra).
Moreover, we will make use of both the differential calculus (total and partial derivatives) and the integral calculus.
Prerequisites - Last names M-Z
Analysis I and linear algebra
Teaching Methods - Last names 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 names 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.
The final exam consists of a written test followed by an oral exam.
Type of Assessment - Last names M-Z
Written exam followed by an oral exam. For more information
http://www.dma.unifi.it/~minguzzi/DidatticaMain.html
Course program - Last names A-L
1st PART: STATIC AND KINEMATIC
INTRODUCTION.
What is the Rational Mechanics. Physical phenomena and their models. Examples. Analogy between a mechanical and an electric phenomenon. First example: forced oscillator and LRC circuit. What is a mathematical model? The physical quantities of mechanics: scalars, vectors and tensor.
ELEMENTS OF VECTOR CALCULUS.
Point space, geometric vectors space (vector space). Geometric vectors and representation of vector quantities with geometric vectors. Affine space. Free and applied vectors.
Summary of vector calculation and notations. Cartesian representation. Scalar and vector products. Mixed product and double vector product
THEORY OF MOMENTS.
Polar moment and axial moment. Applied vector systems and applied vector torque. Resulting moment of a system of applied vectors. Change in the moment as the reduction center varies. Couples of vectors. Scalar and vector invariants. Existence of the central axis. Analytical search of the central axis. Equation of the central axis. Equivalent systems, balanced systems. Examples of reduction of vector systems in the plane. Systems of applied vectors: concurrent, parallel, coplanar. Varignon's theorem. Parallel vectors, center of parallel vectors. Vector systems are reducible to the resultant vector applied on the central axis. Representation of the vector field of moments. Exercises on the calculation of central axis. Some graphic exercises
(displacement of a vector, decomposition in three directions, etc.)
RIGID BODY STATICS ELEMENTS. Terminology. Free and constrained rigid systems. Degrees of freedom. Cardinal equations of the statics. Plane materials systems. Examples. Main types of constraints in the plan. Position of the static problem: labile, isostatic and hyperstatic problem. Statically determined systems. Effective constraints. Some examples of planar systems. Static problems for systems consisting of only one rigid body. Static problems for systems of several rigid bodies. Bow with three hinges. Notes on internal analysis of structures, diagrams of normal stress, cutting and bending moment. Overview of graphic statics. Methods of composition and decomposition of vectors. Funicular polygon. Observations on the funicular polygon. Conditioned polygons and graphical conditions for balance. Some examples of graphic resolution: horizontal and oblique beam, jammed beam, three-hinged arc with concentrated and distributed loads. Reticular structures, knot balance, Ritter method.
KINEMATICS OF RIGID SYSTEMS.
Definition of a rigid system. Degrees of freedom. Reference systems. Configuring a rigid system. Rigid transformations. Orthogonal linear transformations. Transformation of the plane in itself. Rotation of the plane and rotation matrix. Examples. The Euler angles. Orthogonal transformations and their representation with angular parameters. Poisson formulas. Expression of a function of an angular coordinate. Characteristics of a rigid motion. Calculation of in some cases, practical rule for planar motions. Fundamental relationship between the simultaneous speeds of two points. Speed and acceleration of the points of a rigid system in motion. Instantaneous axis of motion. Ruled surface of a rigid motion. Existence and analytical research of the instantaneous motion axis. Example of a disk that rolls without sliding. Particular rigid motions: translations, rotations, precessions. Velocity of the instantaneous center of motion. Planar rigid motions. Polar surfaces of a planar rigid motion. Instantaneous center of motion in planar rigid motions. Chasles' theorem. Examples. Motion of a rigid rod with the ends on two orthogonal guides. Search of the instantaneous center of motion center and polar surfaces. Rod on a circular guide, resting on a disk. Oscillograph. Roulette curve for a system consisting of a disk in contact with a moving inclined plane. Other exercises on the roulette curve. Pair of contact disks, one of which on an inclined plane. Recalls of relative kinematics. Fundamental theorem of relative kinematics. Coriolis' theorem. Absolute and relative derivative. Uniform motion of a point on a rotating guide. Examples of Coriolis acceleration. Composition of rigid motions. Composition of rotations. Poinsot cones. Composition of rotations. The differential. Examples of compositions of rigid motions: the technigraph. Epicycloid and hypocycloidal motions. Exercise on a wheel with a second wheel in relative motion, calculation of the instantaneous center of motion and roulette motion.
2nd pRT: DYNAMICS OF SYSTEMS
GENERAL THEOREMS ON MASS POINT SYSTEMS.
Classification of forces: internal and external forces, active forces and binding reactions. Cardinal equations of dynamics and their derivation. Center of mass and its properties. Comments on the cardinal equations of dynamics. Cardinal equations of the statics. The case of rigid systems. Systems composed of several rigid parts. Energy issues. Conservative forces systems. Again on the cardinal equations of dynamics. Rigid systems or systems composed of rigid parts. Potentials of conservative systems of internal and external forces. Mechanical quantities. Kinetic and potential energy. Conservation of mechanical energy.
GEOMETRY AND KINEMATICS OF THE MASSES.
Introduction to the geometry of the masses. Mass center and its properties, static moments. Examples of calculation of the center of mass. Second degree moments (with respect to a point, a line, a plane) and centrifugal moments. Huygens' (or Transport) theorem: proof in a particular case. Direct and inverse application of the transport theorem. Determination of moments of inertia (rod, rectangle, square, parallelepiped). Inertia structure of a system. Expression of the moment of inertia with respect to a generic line. Matrix of inertia. Construction of the inertia ellipsoid. Principal axes of inertia and canonical form of the ellipsoid of inertia. Recalls on the spectral theorem for linear symmetric applications. Inertia tensor. Principal directions of inertia: eigenvectors and eigenvalues of the inertia matrix. Search for the principal reference system. Examples and exercises. Search for the principal axes in a rectangular plate and a square plate. Geometric-material properties of the principal axes of inertia. Determination for some planar systems. Properties of stationarity of the principal axes. Examples: plane systems, square and rectangular plates, disc and half-disc, square and half square. The case of rigid systems. Expression of T, through the inertia matrix. König's theorem, with proof. Kinematics of the masses. Linear and angular momentum. Motion relative to the center of mass and the center of mass theorem. Kinetic energy. Calculation of in the case of rigid systems.
LAGRANGIAN FORMALISM.
The Lagrangian function. Lagrange equations in conservative form. Applications of the Lagrangian formalism. Non-conservative case. The case of the free point in space. Comments on the Lagrange equations of II kind. Planar motion in polar coordinates. Atwood's machine. First integrals of motion.
MECHANICS OF RIGID SYSTEMS.
Free rigid systems. Precession. Euler equations. Precessions by inertia. First integrals of motion. Motion à la Poinsot. Dynamic properties of the principal axes of inertia. Stability of permanent rotation axes. Brief discussion on the rotation around a fixed axis. Analysis of the moment of reaction forces.
Course program - Last names 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.
SCREW THEORY
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.
STATICS
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.
LAGRANGIAN MECHANICS
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.
SMALL OSCILLATIONS.
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.