# RATIONAL MECHANICS

**Academic Year 2019/2020**- 2° Year

**Teaching Staff:**

**Orazio MUSCATO**

**Credit Value:**9

**Scientific field:**MAT/07 - Mathematical physics

**Taught classes:**42 hours

**Exercise:**45 hours

**Term / Semester:**2°

## Learning Objectives

To provide basic knowledge of vector calculus, statics and dynamics of material systems and rigid bodies.

## Detailed Course Content

1. Elements of vector calculus. Recalls. Scalar product, vector, mixed, double-vector product. Vector valued functions. 2. Applied Vectors and moments. Polar and axial moment of applied vector systems. Couple. central axis. equivalent and balanced vector systems. Parallel and concurrent Applied vector systems. Center. Parallel vector systems. Funicular polygon. 3. Kinematics of the point. Generality. Space and time. Smooth curves and Frenet reference. Speed and acceleration of a material point. Plane motion, circular motion, harmonic motion, helical motion. Exercises. 4. Kinematics of particle systems. General information about the constraints for particle systems. Holonomic, nonholonomic, fixed, mobile, one-sided, two-sided. Degrees of freedom and Lagrangian parameters. Rigid body. Degrees of freedom of a rigid system. Euler angles. Poisson formula. fundamental formula of the kinematics of the rigid bodies. rigid motions translational, rotational, helical, roto-shifters, polar and precession. Act of motion. Mozzi's Theorem. Mozzi axis. Relative kinematics. Absolute and relative speed. absolute, relative and Coriolis accelerations. Theorem of velocities and accelerations composition. equivalent reference systems. rigid motion plan. the instantaneous center of rotation. Notes on base and rulletta. Pure rolling motion. 5. Dynamics and statics for a material point. Principles of dynamics. Matherial point Statics. Friction. Harmonic oscillator. Material point Dynamics and statics in non-inertial frame of reference. terrestrial mechanics. Weight. Kinetic energy. Work and power. Conservative forces. Energy theorem. Mechanical energy conservation theorem. 6. Geometry of areas. Center of mass. Moment of inertia. Examples and applications. Huygens theorem. Inertia ellipsoide and Principal axis frame. Kinetic energy of particle systems. Kinetic energy in motion around the center of mass. König's theorem. kinetic energy for a rigid system. Linear Momentum. Examples and exercises. 7. Dynamics and statics of particle systems. internal and external forces. Cardinal equations of statics and dynamics. balance equations. Conservation laws. Examples and applications. Energy theorem. Work for a infinitesimal rigid displacement. Conservative forces. Examples. Mechanical energy conservation theorem. Statics of constrained rigid body. Statics of a rigid body with a fixed point and fixed axis. Statics of a rigid body leaning against a floor. Leaning ladder. Statics of articulated systems (notes). 8. Elements of analytical mechanics. Possible, virtual and elementary displacements. Smooth constraints. Principle of reaction forces. Examples. symbolic equation of dynamic. Principle of virtual work. Principle of stationary potential. Principle of Torricelli. Lagrange equations. Stability of equilibrium. Theorems of Dirichlet and Liapunov (notes). Examples and exercises. 9. Continuous systems. Cardinal equations. Deformable continuum mechanics Postulate. constitutive equations. Static of wires and rods (notes).

## Textbook Information

1. G. Frosali, E. Minguzzi, Meccanica Razionale per l'Ingegneria, Esculapio, Bologna

2. G. Frosali, F. Ricci, Esercizi di Meccanica Razionale, Esculapio, Bologna