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

Prerequisites — Familiarity with the fundamental concepts of linear algebra and calculus as covered in the courses of Istituzioni di Matematica.

Kinematics of a Point — Kinematic description of the motion of a point, velocity, and acceleration and their various representations, plane motion in polar coordinates. Mention of central motions. Constraints and their classification, holonomic and non-holonomic systems, degrees of freedom, generalized coordinates, configuration space, possible and virtual displacements of a holonomic system, velocity vector of a holonomic system.

Kinematics of Rigid Bodies — Rigid body and rigidity constraint, body-fixed reference, Euler angles, angular velocity and Poisson's formulas, distribution law of velocities, displacements, and accelerations, derivative of a solid vector and relative derivative theorem, classification of rigid motions, Lagrangian and Eulerian perspectives, motion act of a rigid system, classification of motion acts. Statement of Mozzi's theorem. General information on planar rigid motions, instantaneous center of rotation, and related properties, polar trajectories.

Kinematics of Relative Motions — Galileo's theorem on the composition of velocities and Coriolis' theorem on the composition of accelerations. Composition of angular velocities. Rolling of two rigid curves, pure rolling, and the holonomy of this constraint.

Geometry and Kinematics of Masses — Mass and density, center of mass, theorems regarding the properties of the center of mass location. Moments of inertia, inertia matrix. Principal axes of inertia and their determination, inertia ellipsoid. Momentum, momentum theorem, kinetic energy, theorem of the center of mass motion, motion relative to the center of mass, first and second theorem of König. Momentum, momentum theorem, and kinetic energy of a rigid body, structure of kinetic energy for a holonomic system.

Forces, Work and Potential — Concept of force and work, real and virtual elementary work, work of a force along a finite path, conservative force and its potential, work of a system of forces, work of a system of forces applied to a rigid body and a holonomic system. Polar and axial moment of a force, systems of applied forces, resultant and resultant moment of a system of applied forces, law of moment distribution, couples, scalar invariant, central axis, elementary operations, reducible systems, and reducibility theorems, some systems with null invariants, center of a system of parallel forces.

Principles of Mechanics — Principle of inertia, second principle of dynamics, action-reaction principle. Absolute and non-inertial frames, Galileo's principle of relativity, theorem of living forces, principle of conservation of energy, first integrals of motion and their use in mechanical contexts. Reaction forces, Coulombian friction, frictionless constraints, principle of reaction forces, ideal constraints.

Statics and Dynamics of Points and Systems — Cardinal equations of dynamics. Definitions of rest and equilibrium. Cardinal equations of statics and their necessity and sufficiency. Examples of statics and dynamics of a material point (motion of a mass, harmonic oscillator, damped and forced motion, simple pendulum, weight force as a non-inertial force, deviation of masses to the east, etc.) and of a rigid body (rigid body with a fixed point, with a fixed axis, equilibrium of systems consisting of multiple rigid bodies, the technique of release). Theorem of virtual work with a demonstration of its necessity and sufficiency, equilibrium of a holonomic and conservative system even in the presence of unilateral constraints.

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 evaluation consists of a 2-hour written exam. The written exam consists of a problem, the resolution of which is divided into a series of questions with a pre-assigned score. The final score, out of thirty, is the sum of the scores of the questions answered.

In case of a score higher than 15/30, it is possible to take a complementary oral exam to obtain a higher grade.

The additional oral test is mandatory to pass the exam if the written score is less than 18.

Links to further information

https://gsicuro.github.io/lectures/mr/

Office hours

See the website of Gabriele Sicuro