# RATIONAL MECHANICS

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- Versione italiana
- Academic year
- 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