# RATIONAL MECHANICS

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Versione italiana
2022/2023
Teacher
VINCENZO COSCIA
Credits
12
Didactic period
Primo Semestre
SSD
MAT/07

#### Training objectives

Aim of the course is to acquire basic notions and methods of Rational Mechanics, especially on those parts that will be largely used in the course of “Scienza delle Costruzioni”.

The main tools the students will learn concern

equivalence of systems of applied vectors
algebra and elementary tensor calculus
kinematics of constrained systems, rigid kinematics
kinematics of deformations
mass geometry
dynamics of material systems (points mass and rigid bodies)
statics of rigid and deformable material systems
Lagrangean mechanics of constrained holonomic systems
Equilibrium and stability
Empirical laws of friction
At the end of the class the student will be able to

Reduce any system of forces
Describe the kinematics of an arbitrary holonomically constrained system and compose rigid motions
Recognize the characters of homogeneous deformations, decompose any homogeneous deformation, find a homogeneous deformation once their elements are prescribed
Describe the motion of any continuous body both in the material and in the spatial representation, and be able to find the kinematic quantities in both of the representations
Find centers of mass and momenta of inertia of material systems, mainly planar
Work on the dynamics of rigid, articulated and deformable (beams and wires) systems, use balance law sas well as first integrals
Determine equilibria of material systems, mainly with holonomic and bilateral constraints, find equilibrium conditions for boundary configurations of unilateral constraints
Study stability of equilibria, find out normal modes and normal frequencies of motions around stable equilibria, compute constraint reaction forces both in equilibrium and in dynamic conditions, work on constraint in presence of friction forces

#### Prerequisites

Working knowledge of analytic geometry, linear and vector algebra, differential and integral calculus.

#### Course programme

Affine spaces. Point vectors. Torque. Equivalence of vector systems. (10 hours).
Constrained material systems. Classifications of constraints. Holonomic constraints. Degree of freedom and lagrangean coordinates (10 hours).
Linear transformations. Tensors. Tensor algebra and calculus (10 hours).
Kinematic of deformable continua. Homogeneous deformations. Motion. Lagrangean and Eulerian representations (12 hours).
Kinematic of rigid bodies. Relative kinematics. Composition of rigid motions (10 hours).
Center of mass. Momentum of inertia. Huygens theorem and applications (10 hours).
Laws of dynamics. Frame indifference and objective functions. Classes of dynamic problems (12 hours).
Dynamics of systems. Balance equations. Rigid body dynamics. Kinematic of mass. D’Alembert-Lagrange equation (8 hours).
Equilibrium of constrained systems. Reaction forces. Statics of rigid, articulated and deformable systems (12 hours).
Mechanics of holonomic systems. Lagrange equations and function. Principle of virtual work (12 hours).
Stability of equilibria. Lagrange-Dirichlet theorem and its partial inversion. Small oscillations and normal coordinates. Phenomenological laws of friction. (12 hours).

#### Didactic methods

Class lectures that will include both theoretical presentations and exercises.

#### Learning assessment procedures

Written/oral examination. Passing the written test is mandatory to access the final (oral) exam.

The written examination consists in the solution of a problem on different point. Specifically:

Lagrange equations, Equilibrium, Stability, Small oscillations, Reaction forces

The final examination has the aim to ascertain an operative knowledge of the whole contents of the course. Failing the oral exam obliges the candidate to go back to the written exam.

#### Reference texts

Lecture notes

Coscia V., Meccanica Razionale, Pitagora, Bologna 1999
For the exercises
Bampi F., Benati M. Morro A., Problemi di Meccanica Razionale, ECIG, Genova 1984