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01DRRTR

A.A. 2024/25

Course Language

Inglese

Degree programme(s)

1st degree and Bachelor-level of the Bologna process in Civil And Environmental Engineering - Torino

Course structure

Teaching | Hours |
---|---|

Lezioni | 28 |

Esercitazioni in aula | 12 |

Lecturers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
---|---|---|---|---|---|---|---|

Loy Nadia | Ricercatore L240/10 | MATH-04/A | 28 | 12 | 0 | 0 | 2 |

Context

SSD | CFU | Activities | Area context |
---|---|---|---|

MAT/07 | 4 | C - Affini o integrative | Attività formative affini o integrative |

2024/25

Aim of this course is to teach the students the fundamental laws of Classical Mechanics: the equations of motion of a mechanical system with a finite number of degrees of freedom, with an emphasis on statics. The teaching addresses kinematics, statics, and dynamics of a rigid body and of systems of rigid bodies. Equations of balance of momentum and moment of momentum are derived for a rigid body from a system of constrained particles. Emphasis is given to the notions of equilibrium and stability thanks to the principle of virtual work. A large number of exercises and examples favours a deeper understanding of the subject.

Aim of this course is to teach the students the fundamental laws of Classical Mechanics: the equations of motion of a mechanical system with a finite number of degrees of freedom, with an emphasis on statics. The teaching addresses kinematics, statics, and dynamics of a rigid body and of systems of rigid bodies. Equations of balance of momentum and moment of momentum are derived for a rigid body from a system of constrained particles. Emphasis is given to the notions of equilibrium and stability thanks to the principle of virtual work. A large number of exercises and examples favours a deeper understanding of the subject.

At the end of the course, the student is expected to be able:
- to correctly describe the kinematics of a mechanical system, to identify the degrees of freedom and to choose the most suitable generalized coordinates,
- to introduce formally the forces acting on a mechanical system,
- to compute the mechanical quantities of a system,
- to formulate the proper mathematical equations underlying the mechanical problem, choosing the most appropriate solution method.
The overall target is to possess a solid knowledge of the theoretical mechanics of systems with a finite number of degrees of freedom.

At the end of the course, the student is expected to be able:
- to correctly describe the kinematics of a mechanical system, to identify the degrees of freedom and to choose the most suitable generalized coordinates,
- to introduce formally the forces acting on a mechanical system,
- to compute the mechanical quantities of a system,
- to formulate the proper mathematical equations underlying the mechanical problem, choosing the most appropriate solution method.
The overall target is to possess a solid knowledge of the theoretical mechanics of systems with a finite number of degrees of freedom.

Elementary notions of Newtonian physics, linear algebra and matrix calculus. Differential and integral calculus. Basic knowledge of curves in the plane and in the space. All these notions have been addressed in the introductory courses of mathematics and physics.

Elementary notions of Newtonian physics, linear algebra and matrix calculus. Differential and integral calculus. Basic knowledge of curves in the plane and in the space. All these notions have been addressed in the introductory courses of mathematics and physics.

1. Elements of linear algebra. vector calculus in R^3, tensor algebra, second order tensors.
2. Kinematics. Kinematics of a point particle and of a system of point particles. Holonomic and non-holonomic constraints, generalized coordinates and degrees of freedom. The constraint of rigidity. Generalized coordinates of a rigid body. Rigid velocity distribution. Poisson’s formulas. Analysis of some rigid motions. Characterization of rigid velocity distributions. Infinitesimal rigid displacements and virtual displacements. Plane rigid motions.
3. Statics. Statics of a point mass and of a system of point particles, reaction forces, friction. Equivalent systems of forces and their reduction. Center of mass. Balance equations of statics. Statics of articulated systems. Internal stresses. Principle of virtual work, equilibrium of holonomic systems, conservative forces. Equilibrium and stability using the potential energy.
4. Dynamics. Fundamental law of dynamics. Dynamics of a point mass. Mechanical quantities of mechanical systems: mass, center of mass, moment of inertia, linear and angular momentum, kinetic energy, work, power. Balance equations of dynamics. Dynamics of a rigid body. Lagrangian mechanics. Dynamics of holonomic systems; Lagrange’s equations, first integrals.

1. Elements of linear algebra. vector calculus in R^3.
2. Kinematics. Kinematics of a point particle and of a system of point particles. Holonomic and non-holonomic constraints, generalized coordinates and degrees of freedom. The constraint of rigidity. Generalized coordinates of a rigid body. Rigid velocity distribution. Poisson’s formulas. Analysis of some rigid motions. Characterization of rigid velocity distributions. Infinitesimal rigid displacements and virtual displacements. Plane rigid motions.
3. Statics. Statics of a point mass and of a system of point particles, reaction forces, friction. Equivalent systems of forces and their reduction. Center of mass. Balance equations of statics. Statics of articulated systems. Principle of virtual work, equilibrium of holonomic systems, conservative forces. Equilibrium and stability using the potential energy.
4. Dynamics. Fundamental law of dynamics. Dynamics of a point mass. Mechanical quantities of mechanical systems: mass, center of mass, moment of inertia, linear and angular momentum, kinetic energy, work, power. Balance equations of dynamics. Dynamics of a rigid body. Lagrangian mechanics. Dynamics of holonomic systems; Lagrange’s equations, first integrals.

1. Elementary notions of material point mechanics, Newton laws, curves (3 hours lesson + 2 hours exercises)
2. Kinematics of rigid body (7 hours lesson + 5 hours exercises)
3. Fundamental laws of the mechanics of a rigid body: statics and dynamics (10 hours lessons + 5 hours exercises)
4. The principle of virtual works (3 hours lessons + 2 hours exercises)
5. Stability and equilibrium using the potential energy (2 hours lessons + 1 hours exercises)

1. Elementary notions of material point mechanics, Newton laws, curves (3 hours lesson + 2 hours exercises)
2. Kinematics of rigid body (8 hours lesson + 4 hours exercises)
3. Fundamental laws of the mechanics of a rigid body: statics and dynamics (11 hours lessons + 4 hours exercises)
4. The principle of virtual works (4 hours lessons + 1 hours exercises)
5. Stability and equilibrium using the potential energy (2 hours lessons + 1 hours exercises)

A. Romano, Classical mechanics with Mathematica, Springer

Irving Shames, Engineering Mechanics: Statics and Dynamics
D. Gross et al, Engineering mechanics 1: Statistcs

...
The final exam consists of a written test and an oral exam. In the written test, the student will be asked to solve problems related to the statics and dynamics of systems of rigid bodies, together with a question on theoretical aspects. The written test is followed by a mandatory oral discussion, where the understanding of the theory will be assessed. Only students who pass the written test will be admitted to the oral discussion.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.

The final exam consists of a written exam and an oral exam.
Written exam: it will be a 2 hour long written exam. There will be one exercise on the statics and one on the dynamics of the rigid body.
The minimum mark for being admitted to the oral exam is 12.
The written exam is followed by a mandatory oral exam.
The oral exam consists of 4 open questions on the theory of the course
The relative weight of the written and oral exam is 2:3.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.