Unit RATIONAL MECHANICS I
- Course
- Mathematics
- Study-unit Code
- 55031206
- Curriculum
- In all curricula
- Teacher
- Diletta Burini
- Teachers
-
- Diletta Burini
- Hours
- 63 ore - Diletta Burini
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2024/25
- Learning activities
- Caratterizzante
- Area
- Formazione modellistico-applicativa
- Academic discipline
- MAT/07
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Kinematics: kinematics of the point, kinematics of material systems and the rigid body, relative motions. System dynamics: fundamental principles, free point statics and dynamics, statics and dynamics of constrained systems. Dynamics of material systems: geometry of masses and dynamic quantities of material systems, cardinal equations, prime integrals. Analytical mechanics: D'Alembert's principle, Lagrange's equations, Hamilton's equations. Stability and small oscillations. Canonical transformations.
- Reference texts
- N. Bellomo, L. Preziosi, A. Romano. "Mechanics and dynamical systems with Mathematica®". Springer Science & Business Media. 2012
- Educational objectives
- The course aims to provide students with the mathematical tools and methods useful for the study of mechanical systems, in particular kinematics, statics and dynamics of the material point and rigid body, and Lagrangian and Hamiltonian mechanics. Students are expected to acquire knowledge of mathematical modelling of mechanical systems and learn the relevant methods of analysis for the study of motion and equilibrium.
- Prerequisites
- Knowledge of differential calculus, Euclidean geometry, algebra and the rudiments of classical mechanics.
- Teaching methods
- Classroom lectures on all course topics and related exercises.
Students will find the detailed syllabus of the lectures and additional material on unistudium. - Other information
- Class attendance: optional but recommended.
- Learning verification modality
- Written test and oral examination. The written test consists of solving 2 exercises of similar difficulty to the examples given in class and is passed by obtaining a mark of 16/30 or higher. After passing the written exam, the oral exam can be taken at any call of the current academic year. The oral examination consists of two to three questions of a more theoretical nature and lasts approximately 30 minutes.
Two optional intermediate examinations will be organised during the course: students who pass both examinations with a mark of 16/30 or higher will be admitted directly to the oral examination, which will also be held during the same academic year.
For information on support services for students with disabilities and/or DSA visit http://www.unipg.it/disabilita-e-dsa. - Extended program
- Point kinematics: definition of space and time, motion of a point, trajectory, curvilinear abscissa, scalar velocity, velocity of a point, elementary displacement, acceleration of a point, intrinsic triad, radius of curvature, osculating circle.
Plane motions, description of the motion of a point in polar coordinates, areal velocity, central motions, Binet formula, uniformly varied motion, periodic motion, circular and uniform motion.
Harmonic motion, helical motion, exercises on plane motion, definition of constraint, bilateral and unilateral constraint, scleronome and rheonome constraint, holonomic and anolonomic constraint.
Holonomic material system, Lagrangian coordinates, degree of freedom, anholonomic material system, kinematics of rigid systems, Euler angles, translatory motion, rototranslatory motion.
Act of motion, act of motion of translation, of rotation, of rototranslation, Posson's formulae, fundamental formula of the kinematics of rigid systems, Mozzi's Theorem, rigid plane motion, example of a bar whose ends are constrained to move in two parallel planes.
Kinematics of relative motions, absolute, relative and trailing velocity, composition theorem of velocities, absolute, relative, trailing and Coriolis acceleration, composition theorem of accelerations, translatory trailing motion, equivalent observers, relative motions for rigid bodies, absolute, relative and trailing angular acceleration.
Plane rigid motions, instantaneous centre of rotation, examples on instantaneous centre of rotation, acceleration of plane rigid motions, introduction to system dynamics, mass, forces, Newton's laws, examples of forces.
Weight force, theorem of live forces for a free material system, conservative forces, conservation of mechanical energy, central force, momentum, momentum momentum.
Prime integrals, free material point statics, motion of bodies in vacuum, deflection of bodies due to earth's rotation, harmonic oscillator, damped harmonic motion, resonance, introduction to the two-body problem.
Two-body problem and Kepler's laws, constrained reactions, virtual velocity and displacement, virtual work, principle of constrained reactions, friction force.
Dynamic friction, bound material point statics, dynamics of a material point bound to a surface, spontaneous motion of a point on a surface, simple pendulum.
Weierstrass method, phase diagram, equilibrium stability.
Definition of centre of gravity for continuous and discrete systems, properties of centre of gravity, calculation of centre of gravity for plane systems, momentum and momentum for discrete systems, kinetic energy, moments of inertia.
Konig's theorem, inertia matrix, principal axes of inertia, Huygens theorem.
Momentum, general theorems of material system mechanics, cardinal equations, live force theorem for a constrained material system.
Conservation theorem of mechanical energy for a constrained material system, prime integrals, rigid material systems, cardinal equations for rigid systems, cardinal equations of statics and their applications.
Example of the lever, statics of rigid material systems resting on a surface, rigid body systems, symbolic equation of dynamics and D'Alembert's principle.
Symbolic equation of statics and the principle of virtual work, applications of the principle of virtual work, equilibrium conditions for a holonomic system, calculation of virtual reactions using the principle of virtual work, holonomic systems stressed by conservative forces, Torricelli's theorem, Lagrange equations of the second type.
Examples of the use of Lagrange's equations, kinetic energy of a holonomic system, method of Lagrange multipliers for anholonomic systems, Appell's equations, Lagrange's equations for a conservative system.
Generalised potential, Lagrangian systems and prime integrals, Hamilton's equations.
Stability according to Lyapunov, Lyapunov's first method for stability, Lyapunov's second method, small oscillations around an equilibrium position, secular equation.
Principal oscillations, principal frequencies, exercises on small oscillations.
Routh equations, Poisson brackets, properties of Poisson brackets, introduction to canonical transformations.
Canonical transformations, canonical invariants, Liouville's theorem, variational principles.