29690 - Rational Mechanics T

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.

Kinematics of constrained systems
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.

Rudiments of analytical mechanics
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