29690 - Rational Mechanics T

Learning outcomes

At the end of the course, the student will have a theoretical foundation for the analytical treatment of static and dynamic problems. The course covers fundamental concepts in mechanics, including forces and constraints, statics and dynamics of rigid bodies, equivalent stresses, and the problem of equilibrium.

Course contents

The module assumes some familiarity with the fundamental concepts of linear algebra and calculus as covered in the courses of Istituzioni di Matematica.

Kinematics — Kinematic description of the motion of a point, speed and acceleration and their representations, plane motions in polar coordinates. Reference changes: relative kinematics, Euler angles, angular velocity and Poisson’s formulas. Laws of composition of velocities and Galilei, Coriolis acceleration law, and law of composition of angular velocities. Rigid body and rigidity constraint, solid reference, law of distribution of velocities, displacements and accelerations, classification of rigid motions. Velocity field of a rigid system, classification of velocity fields of rigid systems and Mozzi’s theorem. Generalities on planar rigid motions, instantaneous center of rotation and its properties, polar trajectories and pure rolling of two rigid curves.

Laws of mechanics — Mass and density, barycenter and its properties. Momentum. Concepts of force and work of a force along a path: positional and conservative forces, central forces. Moment of a force. Systems of forces: resultant, law of distribution of moments, couples, scalar invariant, central axis, equivalent systems, work and power of systems of forces. Generalized forces. Principles of mechanics. Mechanical determinism. Galilei’s principle and apparent forces. Two-body problem. Constraints and constraint reactions. Classification of constraints: holonomic systems, ideal and dissipative constraints, Coulomb friction. Example: the gravitational force.

Elements of statics — Criteria of equilibrium and cardinal equations of statics. Theorem of virtual works and search for ordinary and frontier equilibrium points. Equilibrium of rigid bodies and technique of unbinding. Statics of holonomic systems and conservative holonomic systems: potential method. Example: statics of the suspended wire and suspended bridge.

Elements of dynamics — Kinetic energy of a system of points and second theorem of König. Kinetic energy of rigid bodies. Inertia tensor. Principal axes of inertia and their determination, inertia ellipsoid. Huygens-Steiner theorem. Kinetic energy theorem and conservation of mechanical energy. Dynamics of the material point: curvilinear abscissa, intrinsic triad, dynamics along a guide. Harmonic motion and one-dimensional damped harmonic motion. Simple pendulum. Dynamics of systems: moment of momentum and first theorem of König. Angular momentum of the rigid body. Cardinal equations of dynamics. Rigid bodies and Euler’s equations. Physical pendulum. First integrals of motion and examples: conservation of momentum and angular momentum.

Readings/Bibliography

The textbook of the module will be

Paolo Biscari, Tommaso Ruggeri, Giuseppe Saccomandi, Maurizio Vianello
Meccanica Razionale
4a Edizione, Springer Verlag, 2022.

Exercises and exam samples can be found in

Francesca Brini, Augusto Muracchini, Tommaso Ruggeri, Leonardo Seccia
Esercizi e temi d'esame di meccanica razionale
Società Editrice Esculapio, 2019.

Teaching methods

Blackboard lectures.

Assessment methods

The regular exam session consists of a two-hour written test and involves solving an exercise divided into three or more questions, typically but not exclusively focused on analyzing the equilibrium positions of a rigid body and calculating its inertial properties.

All students interested in reviewing the correction of their written test can contact the instructor privately.

Registration for each regular exam session opens two months before the exam and closes strictly one week prior to the exam date: late registrations are not allowed under any circumstances.

Teaching tools

Past exam papers (with solutions) and course materials are available on my personal webpage.

Links to further information

https://gsicuro.github.io/docs/teaching/mecrarch/mecrArch/

Office hours

See the website of Gabriele Sicuro