The primary aim of the course is to provide to the students the skills to formalize a problem of rigid-bodies mechanics using the appropriate mathematical tools. Particular attention is paid on the modeling and analysis of simple engineering problems, in order to provide the cultural background required to cope with engineering analysis and design.
teacher profile teaching materials
Identifying the domain and codomain of functions; example: the motion of a point mass in space is a function from $\mathbb{R}^1$ (domain: time) to $\mathbb{R}^3$ (codomain: position).
Derivatives
Dimensional analysis; example: [acceleration] = $L/T^2$
1) Vector Algebra
The three vector operations:
Dot product (scalar product): given two vectors, construct a scalar.
Cross product (vector product): given two vectors, construct a vector (two methods: pseudo-determinant, antisymmetric matrix).
Tensor product: given two vectors, construct a tensor.
Solving vector equations.
2) Point Mass (Particle)
Curve geometry; velocity and acceleration in the intrinsic triplet
Kinematics and dynamics of the material point
Plane motion in polar coordinates; act of rigid motion; angular velocity and Poisson formulas.
Ferret-Frenet formulas
3) Rigid Body
Rigid body motion:
Formulas and diagrams for velocity and acceleration.
Axis of rotation and center of rotation.
Inertia tensor; principal axes and principal moments of inertia.
Cardinal equations (equations of motion/equilibrium).
4) Statics
Calculating the resultant force and resultant moment of a system of forces.
Determining constraint reactions and balancing a system of forces.
Equilibrium equations for a system of rigid bars (links).
Three-hinged arch.
5) Dynamics
A disk rolling on a fixed track (guide).
Pendulum.
Double pendulum.
Grant R. Fowles, Analytical Mechanics, 3rd edition (PDF)
Stefano Turzi, Exercises in Rational Mechanics (PDF)
Reference textbook: Biscari, Ruggeri, Saccomandi, Vianello, “Meccanica Razionale per l’Ingegneria”, Monduzzi (second edition).
Programme
1) Basic ConceptsIdentifying the domain and codomain of functions; example: the motion of a point mass in space is a function from $\mathbb{R}^1$ (domain: time) to $\mathbb{R}^3$ (codomain: position).
Derivatives
Dimensional analysis; example: [acceleration] = $L/T^2$
1) Vector Algebra
The three vector operations:
Dot product (scalar product): given two vectors, construct a scalar.
Cross product (vector product): given two vectors, construct a vector (two methods: pseudo-determinant, antisymmetric matrix).
Tensor product: given two vectors, construct a tensor.
Solving vector equations.
2) Point Mass (Particle)
Curve geometry; velocity and acceleration in the intrinsic triplet
Kinematics and dynamics of the material point
Plane motion in polar coordinates; act of rigid motion; angular velocity and Poisson formulas.
Ferret-Frenet formulas
3) Rigid Body
Rigid body motion:
Formulas and diagrams for velocity and acceleration.
Axis of rotation and center of rotation.
Inertia tensor; principal axes and principal moments of inertia.
Cardinal equations (equations of motion/equilibrium).
4) Statics
Calculating the resultant force and resultant moment of a system of forces.
Determining constraint reactions and balancing a system of forces.
Equilibrium equations for a system of rigid bars (links).
Three-hinged arch.
5) Dynamics
A disk rolling on a fixed track (guide).
Pendulum.
Double pendulum.
Prerequisiti
It is fundamental to have passed the exams of Calculus and Geometry ( Analisi Matematica I and Geometria. ) Are very important the have basic knowledge about: linear algebra, matrices, differentiation and integration; ODECore Documentation
Course notes provided by the instructor (PDF)Grant R. Fowles, Analytical Mechanics, 3rd edition (PDF)
Stefano Turzi, Exercises in Rational Mechanics (PDF)
Reference textbook: Biscari, Ruggeri, Saccomandi, Vianello, “Meccanica Razionale per l’Ingegneria”, Monduzzi (second edition).
Attendance
Attendance is optional but strongly recommended.Type of evaluation
Student performance is evaluated with a written test lasting 1 and ½ hour, and an oral discussion. Only students with sufficient grade are admitted at the oral discussion. During the discussion, the written test will be examined; afterwards, there will be a session with some random questions about the progam.