Teacher

prof. aggr. Vittorino TALAMINI

Credits

12 CFU

Objectives

The course aims to develop the fundamental concepts of Newtonian mechanics and of analytical mechanics using deeply vector algebra and calculus. It will be given the tools to be able to treat in a rigourous manner kinematics, static and dynamics of rigid bodies and more complicated systems formed by the union of rigid bodies and material points, with forces and reactions originating from the constraints.

Acquired skills

– Ability to make mathematical models of mechanical systems with few degrees of freedom.
– Ability to study the kinematics of a mechanical system compose by moving points and rigid bodies.
– Ability to determine the equilibrium positons and their stability and the motion of a mechanical system composed by interacting material points and rigid bodies.

Contents

Review of vector algebra and calculus Scalar and vector products, mixed products, double vector products and their properties. Derivatives and integrals of vector functions. The gradients. The rotation matrices. (7 hours)
Torques Torque, couple of forces, field of moments, invariants and central axis, equivalent systems of applied vectors. Center of a system of parallel applied vectors. (9 hours)
Constrained systems and rigid systems Types of constraints, holonomic constraints, degrees of freedom, generalized coordinates. Rigid systems, Euler angles. (3 hours)
Point kinematics Trajectory. Scalar and vector velocity and acceleration. Motions in a plane. Central motions. Intrinsic systems of references. Frénet formulas. (8 hours)
Kinematics of constrained systems Holonomic and non holonomic constaints. Possible and virtual velocities. Fixed and mobile constaints. (1 hour)
Kinematics of rigid bodies Poisson's formulas and angular velocity. Velocities and accelerations in a rigid motions. Different kinds of rigid motions. Plane rigid motions. (9 hours)
Relative kinematics Composition of the velocities, of the accelerations and of the angular velocities. Galileian equivalence. Contact motions. (6 hours)
Principle of dynamics Mass. Inertial and not inertial systems of references. Fundamental equation of dynamics. Fundamental equation of statics. (3 hours)
Dynamics of material points Fundamental dynamical variables. Conservative forces. Constraint reactions. Static, dynamical and viscous friction. Work theorem and energy conservation. Qualitative analysis and phase plane. Equation of motion. Integration of the most simple equations of motion. Forced oscillations and resonance. (16 hours)
Moment of inertia Moments of inertia, inertia matrix, principal axes of inertia and their determination. (8 hours)
Kinetic quantities Linear momentum, angular momentum, kinetic and potential energy for a system of particles. Koenig's theorems. Angular momentum and kinetic energy of a rigid body. Work, power and energy in a system of particles. (14 hours)
Systems mechanics Cardinal equations in static and in dynamics. Work and power, conservation theorems. Dynamics of a rigid body with a fixed axis, a fixed point or a free rigid body. (18 hours)
Analytical mechanics Ideal constraints. Theorem of virtual powers and its applications. Lagrange's equations. Stability of equilibrium positions. Small oscillations around stable equilibrium points. Motion of a symmetric heavy spinning top. (18 hours)

References

Type of exam

Written and oral