29690 - Rational Mechanics T

Learning outcomes

Good operative knowledge of mathematical models especially in terms of energy balance for mechanical system with a finite number of state variables, with an eye to extensions to electrical and electro-mechanical systems.

Course contents

Elements of vector calculus.

Kinematics

Point kinematics. Constraints and holonomic systems. Rigid body kinematics.  Mozzi's theorem. Relative kinematics. Theorems of addition of velocities and accelerations. relative motion for rigid bodies. Rigid plane motion, instantaneous center of motion, polhode and hyperpolhode.

Geometry of mass distribution and dynamical quantities.

Center of mass and its properties. Moments of inertia. Huygens's theorem. Inertia matrix. Momentum. Angular momentum. Kinetic energy. Koenig's theorems. Kinetic energy for rigid bodies and for holonomic systems.

Forces

Point vectors. Newton's principles. Force law and examples. Weight. Work of a force system. Conservative force systems.

Statics and dynamics

Cardinal equations. Virtual displacements and virtual work. Principle of constraints forces. Theorem of kinetic energy. Theorem of conservation of mechanical energy.

Analytical mechanics

Symbolic relation of dynamics. Lagrange's equations. First integrals for lagrangian systems. Principle of virtual work. Equilibrium for a holonomic system. Stability of equilibrium. Qualitative analysis of motion: the method of Weierstrass. Small oscillation around a stable equilibrium point.

Readings/Bibliography

main textbook:

M. FABRIZIO, Elementi di Meccanica Classica, Zanichelli. Bologna

Solved problems:

A. MURACCHINI, T. RUGGERI, L. SECCIA, Esercizi e temi d'Esame di Meccanica Razionale, Progetto Leonardo, Bologna

F.BAMPI, M.BENATI, A.MORRO, Problemi di Meccanica Razionale, Ecig, Genova

Suggested reading:

P.BISCARI, T. RUGGERI, G. SACCOMANDI, M. VIANELLO, Meccanica razionale per l'ingegneria. Ed. Springer-Milano;

Teaching methods

Class lessons and exercise sessions. Homework problems are also assigned

Assessment methods

The exam takes the form a written test consisting in twelve multiple choice questions, one for each of the following topics:

degrees of freedom- angular velocity - other topic in kinematics - centers of mass - moments of inertia and/or inertia operator - kinetic energy and/or inertia operator - qualitative study of motion - work, generalized forces and conservative systems of forces - equations of motion, conservation of energy and first integrals - ordinary equilibrium points - boundary equilibrium points -stability and small oscillations.

Part of the questions are theoretical while part require tackling an exercise (usually concerning a system with 1 or 2 degrees of freedom about which the student has to determine equilibria, stability and general properties of motion).

Teaching tools

Blackboard, video projector, online video and audio connection

Office hours

See the website of Carlo Alberto Bosello