29690 - Rational Mechanics T

Course contents

Academic Year 2015/2016

Course contents

Outlines of vector calculus - Cartesian components of a vector - Product of a scalar and a vector – Vector sum – Scalar, vectorial and mixed products – Double vectorial product.

Applied vectors and their properties. Moment of a vector -Vectors systems - Resultant vector and resultant moment-Couples- Scalar invariant - Central axis - Reduction of a vectors system - Plane vectors- Centre of parallel vectors.

Kinematics of a particle. Methods of describing the motion of a particle - Velocity, acceleration and their properties – Plane motions in polar coordinates.

Constrained systems. Constraints and their classification: analytic description – Holonomic and non holonomic systems - Lagrangian coordinates - Possible and virtual displacements.

Kinematics of a rigid systems. Rigid motion – Cartesian equations of a rigid motion – Euler's 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. Relative derivation theorem – Velocity addition theorem (Galileo's theorem) - Accelerations addition theorem (Coriolis theorem) – Mutual rolling of two curves.

Plane motion of rigid bodies. Instantaneous centre - Polar trajectories in plane rigid motions.

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

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 and potential – Force systems and work of a force system – Virtual work for rigid bodies and for holonomic systems.

Kinematics of masses. Angular momentum – Kinetic energy – Theorem of the centre of mass and Koenig's theorems - Angular momentum and kinetic theory for a rigid body.

Basic laws of mechanics. Newton's laws - Analytical problems of point dynamics -Inertial frame of reference and relativity principle of classical dynamics - First integrals of motion -Conservation of mechanical energy -Many examples of a particle motion (harmonic oscillations, damped oscillations, forced oscillations, pendulum, point moving on a fixed curve or on a fixed surface, relative motion of a particle) - Constraint forces and their properties - Friction. Cardinal equations of dynamics.

Equilibrium and statics of a particle and a rigid body. Definition of equilibrium for a point and a system-Friction- Coulomb-Morin law - Cardinal equations of statics- Many examples of equilibrium of a point and a system.

Analytic statics. Ideal constraints and of virtual work principle-Virtual work in the case of holonomic and conservative systems.

Analytic dynamics. D'Alembert principles. Derivation of Lagrange's equations - Lagrange equations in the case of a potential force field - Lagrangian function - Cyclic coordinates and first integrals of motion.

Stability of equilibrium and small oscillations. Stability of equilibrium - Liapounov's stability - Lagrange-Dirichlet theorem - 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. Springer-Milano;

A. MURACCHINI, T. RUGGERI, L. SECCIA, Esercizi e temi d'esame di Meccanica razionale. Ed. Esculapio-Bologna.

Teaching methods

The lessons are given by explanation of the arguments on the blackboard. Occasionally, in the case of more complex subject, the light board is used. References to textbook and/or to websites are also given.

Assessment methods

The exam consists in a written test (time allowed 2 hours) and in an oral exam. You can take the oral test only if the written test is passed. If the student does not pass the written test can repeat it in the same examination session. The written test focuses on a problem concerning an holonomic system with 1 or 2 degrees of freedom and, usually, is asked to examine equilibrium, stability and general properties of the motion.
In the oral test you respond, in writing, to two questions of theory (allowed time 30 minutes). The paper is then discussed, in the presence of the teacher.
Some texts of the written tests can be found in this same site in the "educational materials" in the queue to the course contents.

Teaching tools

Web site addresses are provided where the student can deepen some of the topics taught in the lesson. On the teacher's web page (http://www.dm.unibo.it/~muracchi/) you can find abstracts of the lessons, texts of the written exams and other teaching aids.

Links to further information

http://www.dm.unibo.it/~muracchi/

Office hours

See the website of Augusto Muracchini