29690 - Rational Mechanics T

Learning outcomes

Notions of kinematics, statics and dynamics of material systems. Motions of rigid bodies. An introduction to analytical mechanics.

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 – Plane motions.

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.

Principles of mechanics
Inertia principle – Proportionality principle between force and acceleration –Action and reaction principle – Principle of force parallelogram – Constraining reaction postulate – Galilean relativity principle – Kepler's laws and the universal gravitation principle.

Static of the point
Equilibrium of a material point – Equations for a point constrained on a surface – Equilibrium with respect to a non-inertial frame - Terrestrial mechanics: weight .

Static of the rigid body
Cardinal equations of static – Problem of the heavy rigid body on a frictionless horizontal plane.

Static of holonomic systems
Constraining reaction principle – Virtual work principle – Equilibrium stability – Bifurcation diagram – Equilibrium of a holonomic system.

Point dynamics
Analytical problems of point dynamics – First integrals of motion equation – Heavy body motion – Harmonic, damped and forced oscillators - Resonance – Simple pendulum – Point moving on a fixed surface or on a fixed curve.

Rigid body dynamics
Cardinal equations of dynamics - Gyroscopic effects – Points motions – 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 per l'Ingegneria, Ed. Monduzzi, Bologna.

Exercises: Muracchini, T. Ruggeri, L. Seccia, Esercizi e Temi d'Esame di Meccanica Razionale per i Corsi di Laurea Triennale in Ingegneria, Ed. Esculapio - Progetto Leonardo, Bologna.

Assessment methods

The exam consists in a written and an oral proof.

Office hours

See the website of Francesca Brini