29690 - Rational Mechanics T (A-K)

Course contents

Course contents

Vectors and Linear Algebra
Vectors - Cartesian components of a vector- Product of a scalar and a vector – Vector sum – Scalar, vectorial and mixed products – 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 – Symmetric and antisymmetric matrices –  Rotation matrix and similarity transformation– Eigenvalues and eigenvectors –   Positive definite matrices, negative definite matrices, semidefinite matrices.                                                                                                                               

Outlines of differential geometry of curves- Vector functions – Tangent, normal and binormal vectors – Curvature, Frenet's frame.

Kinematics of a point
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 – Angular velocity addition 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.

Geometry of masses                                                                                                                      

Mass – Centre of mass for a discrete or a continuous system – Location theorems for centre of mass – Definition of inertial momentum – Huygens- Steiner theorem – Inertial momentum with respect to concurrent axes – Inertial matrix and ellipsoid of inertia – Gyroscopes.  

Kinematics of masses                                                                                                                      

Momentum – Angular momentum – Kinetic energy – Theorem of the centre of mass and Koenig's theorems.     

Forces, Work and Energy
Modeling and classification of forces – 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 – Virtual work for rigid bodies and for holonomic systems.
  
Principles of mechanics                                                                                                              

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

Statics of the rigid body                                                                                                               

Cardinal equations of statics – Problem of the heavy rigid body on a horizontal plane – Equilibrium of beams and strings.

Statics of holonomic systems                                                                                               

Ideal constraints  – Virtual work principle – Equilibrium stability – Bifurcation diagram – Equilibrium of a holonomic system.
 
Dynamics of points
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 – Central motions – Dynamics with respect to a non-inertial frame - Two-body problem – Eastwards deviation of heavy bodies.

Rigid body dynamics
Cardinal equations of dynamics – Euler equations  - Gyroscopic effects – Poinsot's motion, Motion of a rigid body with a fixed axis and dynamical balancing.

Elements  of analytical mechanics
D'Alembert principle – Genesis of Lagrange equations – Lagrange equations for conservative systems - Small oscillations in the neighborhood of a stable equilibrium position.

Readings/Bibliography

P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale per l'Ingegneria, Ed. Monduzzi, Bologna.

• T. Ruggeri, Appunti di Meccanica Razionale: Richiami di Calcolo Vettoriale e Matriciale, Ed. Pitagora, Bologna.
•  A. 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

Written and oral examination

Links to further information

http://www.ciram.unibo.it/ruggeri

Office hours

See the website of Tommaso Ruggeri