- Docente: Francesca Brini
- Credits: 6
- SSD: MAT/07
- Language: Italian
- Moduli: Francesca Brini (Modulo 1) Andrea Mentrelli (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mechanical Engineering (cod. 0927)
Learning outcomes
Notions of kinematics, statics and dynamics of material systems. Motions of rigid bodies.
Course contents
Course contents
Recalls of vector and matrix calculus
Free vectors - Cartesian components of a
vector- Product of a scalar and a vector – Vector sum – Scalar,
vectorial and mixed product – Double vectorial product.
Applied vectors- Resultant of a
vector system – Polar moment, axial moment – Central axis – Couple -
Elementary operations – Reduction of an applied vector system – Plane
vector system – Parallel vector system.
Linear operator – Rotation matrix –
Eigenvalues and eigenvectors - Symmetric and antisymmetric matrices –
Positive definite matrices, negative definite matrices, semidefinite
matrices.
Outlines of differential geometry of curves - Vector functions – Tangent, normal and binormal vector – Frenet's frame.
Point kinematics
Velocity, acceleration and their properties – Elementary and effective displacement.
Kinematics of rigid systems
Rigid motion – Cartesian equations of a rigid motion – Euler angles –
Poisson's formulas – Angular velocity –Law of velocity, acceleration and
elementary displacement distributions –Classification and Properties of
rigid motions – Motion acts – Mozzi's theorem.
Relative kinematics
Velocity addition theorem – Relative derivation theorem – Coriolis
theorem – Mutual rolling of two surfaces – Polar trajectories in rigid
motions.
Constraints and their classification – Analytic description – Holonomic systems - Possible and virtual displacements.
Mass geometry
Mass – Barycentre of a discrete or continuous system – Theorem of barycentre location – Definition of inertial momentum – Huygens- Steiner theorem – Inertial momentum with respect to concurrent axes – Inertial matrix and ellipsoid of inertia – Gyroscope.
Mass kinematics
Momentum – Angular momentum – Kinetic energy – Barycentre theorem and Koenig's theorems.
Work
Definition of elementary and effective work – Work along a finite path
for a general force and for positional non-conservative forces–
Conservative forces – Force systems and work of a force system – Work
for rigid bodies and for holonomic systems.
Recalls of principles of mechanics
Inertia principle – Proportionality principle between force and
acceleration –Action and reaction principle – Principle of force
parallelogram – Constraining reaction postulate.
Static of the point
Equilibrium of a material point – Equations for a point constrained on a surface.
Static of the rigid body
Cardinal equations of static.
Static of holonomic systems
Ideal constraints – Virtual work principle – Equilibrium stability – Equilibrium of a holonomic system.
Point dynamics
Analytical problems of point dynamics – First integrals of motion
equation – Harmonic, damped and forced oscillators - Resonance – Simple
pendulum.
Rigid body dynamics
Cardinal equations of dynamics – Poinsot motions – Gyroscopic effect --
Motion of a rigid body with a fixed axis and dynamical balancing.
D'Alembert principle – Genesis of Lagrange equations – Lagrange equations for conservative systems - Small oscillations in the neighbourhood of stable equilibrium position.
Readings/Bibliography
Theory: P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale, Ed. Springer.
Exercises: F. Brini, A. Muracchini, T. Ruggeri, L. Seccia, Esercizi e Temi d'esame di Meccanica Razionale, Ed. Esculapio, Bologna (2019).Teaching methods
The basic theory is explained in standard lessons and illustrated with several examples and exercises. Homework is also proposed every week and will be corrected during the lessons to help students in the exam preparation.
Assessment methods
Achievements will be assessed by the means of a final exam. This is based on an analytical assessment of the "expected learning outcomes" described above.
Teaching tools
Blackboard, projector.
Office hours
See the website of Francesca Brini
See the website of Andrea Mentrelli