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, 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 session consists of a written test and an oral examination. The written test, which is compulsory, lasts three hours and requires: (i) the solution of one exercise divided into multiple questions, (ii) answering to a theoretical question. The oral examination is compulsory only for students who obtain a score between 15 and 18 in the written test. For students who obtain a score higher than 18, the oral examination is optional and may either increase or decrease the grade obtained in the written test. Students who obtain a score lower than 15 must retake the exam at the next session.

The oral examination may be taken in any examination session within the same exam period.

Students are allowed to refuse a passing grade at most once, in accordance with the resolution of the Academic Senate dated 14/02/2018.

All students wishing to review the correction of their written test may contact the instructor privately.

Registration for each regular examination session strictly closes one week before the exam date; late registrations are not accepted.

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