# 96360 - MECCANICA RAZIONALE M

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda. ## Learning outcomes

At the end of the course the student is able to model and solve problems related to static and dynamic aspects of a mechanical system autonomously, identifying the main constituents from both the mathematical and the physical standpoint. To this aim the student acquires specific expertise in kinematics, statics and dynamics of mechanical systems, with particular attention to rigid bodies and basic principle of analytic mechanics. In this way the student becomes capable of learning new topics and solving new problems independently.

## Course contents

Course contents

Recalls of vector and matrix calculus
Free vectors - Cartesian components of a vector- Product of a scalar and a vector – Vector sum – Scalar, vectorial and mixed product – Double vectorial product.
Applied vectors
- Resultant of a vector system – Polar moment, axial moment – Central axis – Couple - Elementary operations – Reduction of an applied vector system – Plane vector system – Parallel vector system.
Linear operator – Rotation matrix – Eigenvalues and eigenvectors - Symmetric and antisymmetric matrices – Positive definite matrices, negative definite matrices, semidefinite matrices.
Outlines of differential geometry of curves
- Vector functions – Tangent, normal and binormal vector – Frenet's frame.

Point kinematics
Velocity, acceleration and their properties – Elementary and effective displacement.

Kinematics of rigid systems
Rigid motion – Cartesian equations of a rigid motion – Euler angles – Poisson's formulas – Angular velocity –Law of velocity, acceleration and elementary displacement distributions –Classification and Properties of rigid motions – Motion acts – Mozzi's theorem.

Relative kinematics
Velocity addition theorem – Relative derivation theorem – Coriolis theorem – Mutual rolling of two surfaces – Polar trajectories in rigid motions.

Kinematics of constrained systems
Constraints and their classification – Analytic description – Holonomic systems - Possible and virtual displacements.

Mass geometry

Mass – Barycentre of a discrete or continuous system – Theorem of barycentre location – Definition of inertial momentum – Huygens- Steiner theorem – Inertial momentum with respect to concurrent axes – Inertial matrix and ellipsoid of inertia – Gyroscope.

Mass kinematics

Momentum – Angular momentum – Kinetic energy – Barycentre theorem and Koenig's theorems.

Work
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 and work of a force system – Work for rigid bodies and for holonomic systems.

Recalls of principles of mechanics
Inertia principle – Proportionality principle between force and acceleration –Action and reaction principle – Principle of force parallelogram – Constraining reaction postulate.

Static of the point
Equilibrium of a material point – Equations for a point constrained on a surface.

Static of the rigid body
Cardinal equations of static.

Static of holonomic systems
Ideal constraints – Virtual work principle – Equilibrium stability – Equilibrium of a holonomic system.

Point dynamics
Analytical problems of point dynamics – First integrals of motion equation – Harmonic, damped and forced oscillators - Resonance – Simple pendulum.

Rigid body dynamics
Cardinal equations of dynamics – Poinsot motions – Gyroscopic effect -- Motion of a rigid body with a fixed axis and dynamical balancing.

Rudiments of analytical mechanics
D'Alembert principle – Genesis of Lagrange equations – Lagrange equations for conservative systems - Small oscillations in the neighbourhood of stable equilibrium position.

Theory: P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale, Ed. Springer.

Exercises: F. Brini, A. Muracchini, T. Ruggeri, L. Seccia, Esercizi e Temi d'esame di Meccanica Razionale, Ed. Esculapio, Bologna (2019).

## Teaching methods

The basic theory is explained in standard lessons and illustrated with several examples and exercises. Homework is also proposed every week and will be corrected during the lessons to help students in the exam preparation.

## Assessment methods

Achievements will be assessed by the means of a final exam. This is based on an analytical assessment of the "expected learning outcomes" described above. In order to properly assess such achievement the examination is composed of an oral exam, that consists of a technical conversation.

Higher grades will be awarded to students who demonstrate an organic understanding of the subject, a high ability for critical application, and a clear and concise presentation of the contents.

To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject, some ability for critical application, and a comprehensible use of technical language.

## Office hours

See the website of Francesca Brini