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

Review of mathematical and geometrical tools
Vectors, scalar and vector products. Euclidean spaces, orthogonal frames. Linear maps and matrices (essential review), diagonalization. Second-order multivariable Taylor expansion. Regular curves and arc length. Intrinsic coordinate and the Frenet frame. Point kinematics, planar motion in polar coordinates. Scalar and vector fields. Gradient, brief introduction to divergence and curl. Line integrals of the second kind.

Constrained systems and virtual displacements
Systems of material points. Degrees of freedom and generalized coordinates. Constraints and their classification. Virtual displacements and ideal constraints. Spherical coordinates. Free point particle on the sphere.

Dynamics of systems
Linear momentum and angular momentum. Newton’s laws. Forces and torques. Cardinal equations of dynamics. Brief remarks on the equilibrium of systems. Work and energy. Conservative forces. Work–kinetic energy theorem.

Principle of virtual work and Lagrangian mechanics
Theorem of virtual work and applications to the statics of constrained systems. d’Alembert’s principle. Lagrange’s equations. Metric tensor. First integrals of motion. Harmonic motions and the Lagrangian for small oscillations. Example of an exam problem.

Kinematics and rigid bodies
Angular velocity and Poisson formulas. Change of reference frame, composition of velocities and accelerations, Coriolis acceleration. Definition of a rigid body. Distribution of velocities and accelerations. Mozzi’s theorem. Examples of rigid motions, pure rolling. Euler angles.

Geometry of masses and moments of inertia
Center of mass and its properties. Moments of inertia about an axis. Parallel-axis theorem. Inertia tensor and its derivation from kinetic energy. König’s theorem. Review of eigenvalues and eigenvectors. Principal axes of inertia.

Complements
Elements of the theory of ordinary differential equations. Harmonic oscillators with external driving force and friction. Resonance.

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 examination consists of a written test and an oral examination.

The written test is compulsory, lasts three hours, and consists of:
(i) solving a multi-part exercise;
(ii) answering one theoretical question.

The oral examination is compulsory only for students who obtain a mark between 15 and 18, inclusive, in the written test. For all other students, it is optional. However, without taking the oral examination, the highest final mark that can be awarded is 25, even if the mark obtained in the written test is higher.

The oral examination may either raise or lower the mark obtained in the written test.

Students who obtain a mark below 15 must retake the examination at a subsequent examination sitting.

The oral examination may be taken at any examination sitting within the same examination session. If a student takes the written test more than once during the same session, the best written-test result will be retained. Once the examiner proposes a final mark following an oral examination, the record of previous written-test results is cleared, and the student must retake the entire examination in order to make a new attempt.

In accordance with the resolution of the Academic Senate dated 14 February 2018, students may reject a passing mark no more than once.

Students wishing to inspect the marking of their written test may contact the instructor privately.

Registration for each regular examination sitting closes strictly one week before the examination date. Late registrations will not be accepted under any circumstances.

Teaching tools

Lectures will be complemented by problem-solving sessions. Useful exercises can be found in:

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

Course notes, updated on a weekly basis, will be made available on the Virtuale platform.

At the end of the course, if time permits, a mock examination with self-assessment will be organized.

Links to further information

https://virtuale.unibo.it/course/view.php?id=79184

Office hours

See the website of Francesco Camilli