01379 - Rational Mechanics

Learning outcomes

The main purpose of the course is to furnish the fundamental notions permitting to the students to model, from a mathematical point of wiev,  the physical systems of the real world.

Course contents

Kinematics of a point  - Kinematics description of a particle motion.  Outlines of differential geometry of curves. Vector functions. Tangent, normal and binormal vectors. Curvature.  Frenet's frame. Velocity, acceleration and their properties. Plane motions. Central motions.

Kinematics of constrained systems -  Constraints and their classification. Analytic description. Holonomic and nonholonomic systems. Degrees of freedom.  Possible and virtual displacements.Velocity of holonomic systems.

Kinematics of the rigid systems - Generalities about rigid motions. Poisson's formulae and angular velocity. Law of velocity, acceleration and elementary displacement distributions. Euler's angles. Euler's formulae. 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  curves and surfaces. Polar trajectories in rigid motions.

Rigid plane motions - Generalities. Instantaneous center. Polar trajectories.

Applied vectors -  Outlines of vectorial calculus. Cartesian components of a vector. Product of a scalar and a vector. Vector sum. Scalar, vectorial and mixed products. Double vectorial product. Resultant of a vector system. Polar and  axial moment. Central axis. Couple. Elementary operations. Reduction of  an applied vector system. Plane vectors systems. Parallel vectors systems.

Geometry of masses - Mass. Centre of mass for discrete and continuous systems. Location theorems for centre of mass. Moment of inertia. Huygens- Steiner theorem. Moment of inertia  with respect to concurrent axes.  Generalities on the linear operators  (Symmetric and antisymmetric matrices, rotation matrix and similarity transformation, eigenvalues and eigenvectors, positive definite matrices, negative definite matrices, semidefinite matrices).   Inertia tensor  and ellipsoid of inertia.  Gyroscopes.  

Forces, Work and Energy  - Classification of the 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. Work of a force system. Virtual work for rigid bodies and for holonomic systems.

The principles of mechanics - Frames of reference. Newton's laws.  Equilibrium of a material point . Dynamics with respect to a non-inertial frame. Terrestrial mechanics: weight.  Constraints and their reactions. Coulombian friction. Ideal constraints.

Kinematics of masses - Momentum. Angular momentum.  Kinetic energy. Theorem of the centre of mass and Koenig's theorems.   Momentum, angular momentum and kinetic energy of rigid bodies. Kinetic energy in holonomic systems.

Dynamics and static  of  particles and systems -  Cardinal equations of  dynamics. Analytical problems of point dynamics. First integrals of the motion equations. Harmonic, damped and forced oscillators.  Resonance. Simple pendulum. Point moving on a fixed surface or on a fixed curve. Central motions. Eastwards deviation of heavy bodies.  Dynamics of a rigid body. Euler's equations. Gyroscopic effects. Poinsot's motion. Motion of a rigid body with a fixed axis and dynamical balancing. The static as a  particular case of the dynamics. Cardinal equations of the static. Equilibrium  of rigid bodies.  Problem of the heavy rigid body on a horizontal plane. Equilibrium of beams and strings.

Virtual work's principle  -  Virtual work's principle. Equilibrium of a holonomic and conservative system in the case of unilateral constraints too.

Elements  of analytical mechanics  - D'Alembert's principle. Genesis of the Lagrange equations. Lagrange equations for conservative systems. Ignorable coordinates. The Hamiltonian function.

Stability and small oscillations  -  Liapounov's stability criterion. Liapounov's theorem. Lagrange-Dirichlet theorem. Potential function in equilibrium positions. Small oscillations of a n-degrees of freedom conservative system. Characteristic frequencies.  

Readings/Bibliography

P.BISCARI, T. RUGGERI, G. SACCOMANDI, M. VIANELLO, Meccanica razionale per l'ingegneria. Ed. Monduzzi-Bologna;
C. CERCIGNANI, Spazio tempo movimento.  Ed. Zanichelli-Bologna;
M. FABRIZIO,  La Meccanica Razionale e i suoi metodi matematici. Ed. Zanichelli-Bologna;
G. GRIOLI, Lezioni di Meccanica Razionale.  Ed. Cortina-Padova;
A. STRUMIA, Meccanica Razionale (2 voll.).  Ed. Nautilus-Bologna.
A. MURACCHINI, T. RUGGERI, L. SECCIA,  Esercizi e temi d'esame di Meccanica razionale. Ed. Esculapio-Bologna.               
T. RUGGERI,  Richiami di calcolo vettoriale e matriciale.  Ed. Pitagora-Bologna.

Assessment methods

The exam consists in a written and an oral proof.

More informations are available in my Web page: http://www.ciram.unibo.it/~muracchi

Teaching tools

The lessons are, partly, given by a projector. In the Web-page of the teacher  ( http://www.ciram.unibo.it/~muracchi/ ) a lot of  informations and educational material is found.

Office hours

See the website of Augusto Muracchini