# RATIONAL MECHANICS

## Learning outcomes of the course unit

The Rational Mechanics course is the link between purely mathematical subjects and engineering applications. The purpose of the course is to provide the basics of kinematics and statics necessary for tackling the subsequent Mechanical Engineering degree courses with awareness and criticality.

## Prerequisites

AB analysis, C analysis, Geometry, AB physics are recommended.

## Course contents summary

1. Free Vectors (Definition of vector and of versor - angle between vectors - parallelism and orthogonality - components of a vector - operations with vectors)

2. Outline of geometry (regular curves: main trihedron - plane curves - regular surfaces: tangent plane)

3. Outline of element kinematics (velocity and acceleration - Cartesian, polar, cylindrical coordinates)

4. Rigid kinematics (degree of freedom of a rigid body - equiprojectivity - basic formula of rigid kinematics - acceleration formula - plane rigid motions: angular velocity - spherical motions - rigid motion actions - rigid motion actions in plane motions and in spherical motions)

5. Relative kinematics (absolute reference and relative reference - pulling motion - absolute derivative and relative derivative - relative motion theorem - acceleration theorem - relative motions of a rigid body)

6. Contact motions (contact between regular rigid surfaces - contact between plane rigid curves – sliding velocity - properties of sliding velocity)

7. Spherical motions (non-degenerate spherical motions - Poinsot cones - realization of spherical motions with bevel gears)

8. Plane motions (instantaneous centre of rotation - base and trochoid - realization of plane motions with plane gears - Chasles' theorems - analytical methods for determining base and trochoid)

9. Newton's laws (statement of the three laws - dynamics equation - apparent forces - centrifugal force)

10. Applied vectors (polar moment of an applied vector - systems of applied vectors: polar moment variation formula - pair of applied vectors)

11. Geometry of masses (definition of barycentre and moment of inertia - Huygens' theorem - variation formula of the moment of inertia in a sheaf of lines - principal axes of inertia - application to plane systems)

12. Mass kinematics (momentum and moment of momentum - expression of the moment of momentum of a flat rigid plate moving in its plane)

13. Forces and stresses (description of the main active forces - constraint reactions: Coulomb-Morin law of static friction and dynamic friction - application to plane systems)

14. Cardinal equations (cardinal equations of statics and dynamics - dynamically plane systems - sufficiency of cardinal equations for rigid bodies)

15. Equivalence theory (systems of coplanar forces - the case where the vector sum is null, or else is not null - constraint stress acting on a supported plate - constraint stress of a cylindrical hinge - systems of forces equivalent to zero)

(Free vectors. Kinematics of a point: velocity and acceleration. Kinematics of a rigid body. Absolute, relative and migratory motion of a point. Theorem of composition of velocities. Motion of a body with a fixed point. Plane-parallel motion of a body. The instantaneous centre of velocities. The three basic laws of Galileo and Newton. Applied vectors. Couples of forces. Plane systems of forces. The centre of gravity of a body. The moment of inertia of a body. The theorems on the moments of inertia about parallel or converging axes. Momentum of a system. The theorem on the motion of the centre of mass of a system. Statics and dynamics of plane systems.)

## Recommended readings

lecture notes handed out by the professor.

## Teaching methods

the examination consists of a written test followed by an oral test. In certain cases (which will be explained during the lecture) the written test may be sufficient. About half the course hours are dedicated to practical exercises, consisting of mechanical exercises whose purpose is to illustrate and apply the theoretical knowledge. The same type of exercises will then be given in the examinations