29690 - Rational Mechanics T

Academic Year 2015/2016

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Civil Engineering (cod. 0919)

Learning outcomes

After completing the course , and after passing the final exam , the student has basic knowledge about physical and rigorous mathematical modeling of mechanical systems starting from Newtonian mechanics and reaching the Lagrangian - Hamiltonian scheme of analytical mechanics . In particular, it has gained the ability to build, compare and use mathematical models for mechanical systems with few degrees of freedom (such as material point , rigid body, holonomic systems ), he is able to determine the most relevant behavioral properties (static and dynamic and to derive correct possible approximations.


Course contents

PRIORI KNOWLEDGE

A prior knowledge and understanding of differential and integral calculus , the basics of Euclidean geometry , linear algebra and the phenomenology described in Physics is required to attend with profit this course.

In addition, students should master the topics of Analysis, Geometry and Physics (Mechanics).

 

Fluent spoken and written Italian is a necessary pre-requisite: all lectures and tutorials, and all study material will be in Italian

COURSE CONTENTS

Vectors and Linear Algebra
Vectors -
Cartesian components of a vector- Product of a scalar and a vector – Vector sum – Scalar, vectorial and mixed products – 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 Symmetric and antisymmetric matrices – Rotation matrix and similarity transformation– Eigenvalues and eigenvectors – Positive definite matrices, negative definite matrices, semidefinite matrices.

Outlines of differential geometry of curves- Vector functions – Tangent, normal and binormal vectors – Curvature, Frenet's frame.

Kinematics of a point
Velocity, acceleration and their properties – Elementary and effective displacement – Plane motions.

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 – Angular velocity addition 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.

Geometry of masses

Mass – Centre of mass for a discrete or a continuous system – Location theorems for centre of mass – Definition of inertial momentum – Huygens- Steiner theorem – Inertial momentum with respect to concurrent axes – Inertial matrix and ellipsoid of inertia – Gyroscopes.

Kinematics of masses

Momentum – Angular momentum – Kinetic energy – Theorem of the centre of mass and Koenig's theorems.

Forces, Work and Energy
Modeling and classification of forces – 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 – Virtual work for rigid bodies and for holonomic systems.

Principles of mechanics

Inertia principle – Equilibrium of a material point – Equations for a point constrained on a surface – Equilibrium with respect to a non-inertial frame - Terrestrial mechanics: weight .

Statics of the rigid body

Cardinal equations of statics – Problem of the heavy rigid body on a horizontal plane – Equilibrium of beams and strings.

Statics of holonomic systems

Ideal constraints – Virtual work principle – Equilibrium stability – Bifurcation diagram – Equilibrium of a holonomic system.

Dynamics of points
Analytical problems of point dynamics – First integrals of motion equation – Heavy body motion – Harmonic, damped and forced oscillators - Resonance – Simple pendulum – Point moving on a fixed surface or on a fixed curve – Central motions – Dynamics with respect to a non-inertial frame - Two-body problem – Eastwards deviation of heavy bodies.


Rigid body dynamics
Cardinal equations of dynamics – Euler equations - Gyroscopic effects – Poinsot's motion, Motion of a rigid body with a fixed axis and dynamical balancing.

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

Readings/Bibliography


  • P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale, Ed. Springer, Milano.


  • T. Ruggeri, Appunti di Meccanica Razionale: Richiami di Calcolo Vettoriale e Matriciale, Ed. Pitagora, Bologna.


  • A. Muracchini, T. Ruggeri, L. Seccia, Esercizi e Temi d'Esame di Meccanica Razionale, Ed. Esculapio, Bologna.

Teaching methods

Lectures

Assessment methods

The exam is done through a final exam, which ensures the acquisition of knowledge and skills expected by a written exam (2 hours) and one oral exams.

The written exam consists in an exercise of amechanical system with 1 or 2 degrees of freedom and the student have to to find the equilibrium positions and their eventual stability, the calculation of reactions and the equation of motion. If the student pass the written test leads to the oral exam that consists of 2 theory questions that the student answers by writing what he knows in a sheet of paper (20 min). Immediately after the student discusses what is written and integrates with the teacher the questions (about 20 min).

To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject

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


Office hours

See the website of Tommaso Ruggeri