## Mechanics of Materials and Structures

OBJECTIVES

LEARNING OUTCOMES: The goal of this course is to provide the student with basic knowledge of the mechanics of linearly elastic structures and of the strength of materials. By completing this class successfully, the student will be able to compute simple structural elements and reasonably complex structures.

KNOWLEDGE AND UNDERSTANDING: At the end of this course, the student will be able to:

• compute constraint reactions and internal actions in rigid-body systems and beams subjected to point/distributed forces and couples
• compute centroid position and central principal second-order moments of area distributions
• understand the formal structure of the theory of linear elasticity for both discrete and continuous systems (beams and 3D bodies)
• analyze strain and stress states in 3D bodies
• compute the stress state in beams subjected to uniaxial bending, biaxial bending, eccentric axial force
• understand the behavior of beams subjected to shear with bending and torsion
• understand how to compute displacements/rotations in isostatic beam systems, how to solve statically underdetermined systems, how to apply yield criteria, and how to design beams against buckling

APPLYING KNOWLEDGE AND UNDERSTANDING: The student will apply the knowledge and understanding skills developed during the course to the analysis of practical problems. This includes the analysis of linearly elastic structures and structural members in terms of strength and stiffness.

MAKING JUDGEMENTS: The student will have to demonstrate his awareness of the modeling assumptions useful to describe and calculate structural elements, as well as his critical judgment on the static response of elastic structures under loads, in terms of stresses, strains, and displacements.

COMMUNICATION SKILLS: The student will demonstrate, mostly during the oral test, his capacity of analyzing and computing the static response of linearly elastic structures, as well as his knowledge of the underlying theoretical models.

LEARNING SKILLS: The student will get familiar with the modeling of structures and structural elements in practical problems, mostly during the development of his skills for the written test. This mainly concerns discrete systems, beams, and three-dimensional bodies.

COURSE SYLLABUS

• Review of basic notions of vector and tensor algebra and calculus.
• Kinematics and statics of rigid-body systems.
• Geometry of area distributions.
• Discrete linearly elastic systems, static-kinematic duality, solution methods.
• Strain and stress in 3D continuous bodies and beam-like bodies.
• Virtual power and virtual work equation for discrete systems, beams, and 3D bodies.
• One-dimensional beam models: Bernoulli-Navier model, Timoshenko model, constitutive equations, governing differential equations.
• Constitutive equation for linearly elastic and isotropic bodies, material moduli.
• Hypothesis in linear elasticity, equilibrium problem for linearly elastic discrete systems, beams, and 3D bodies.
• Three-dimensional beam model: the Saint-Venant problem, uniaxial and biaxial bending, eccentric axial force, shear and bending, torsion.
• Elastic energy of beams and 3D bodies, work-energy theorem, Betti’s reciprocal theorem, Castigliano’s theorem.
• Yield criteria (maximum normal stress, maximum tangential stress, maximum elastic energy, maximum distortion energy).
• Buckling instability, bifurcation diagrams, load and geometry imperfections, Euler buckling load, design against buckling.
• Basic notions on the finite element method and structural analysis software.

## Feedback Control Systems

OBJECTIVES

LEARNING OUTCOMES:
The theory of differential equations is successfully used to gain profound insight into the fundamental mathematical control design techniques for linear and nonlinear dynamical systems.

KNOWLEDGE AND UNDERSTANDING:
Students should be able to understand and use the theory of differential equations and systems theory, along with related mathematical control techniques.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Students should be able to design feedback controllers for linear (and even nonlinear) dynamical systems.

MAKING JUDGEMENTS:
Students should be able to identify the specific design scenario and to apply the most suitable techniques. Students should be able to compare the effectiveness of different controls, while analyzing theoretical/experimental advantages and drawbacks.

COMMUNICATION SKILLS: Students are expected to be able to read and capture the main results of a technical paper concerning the topics of the course, as well as to effectively communicate in a precise and clear way the content of the course. Tutor-guided individual projects (including Maple and Matlab-Simulink computer simulations as well as visits to labs) invite intensive participation and ideas exchange.

LEARNING SKILLS:
Being skilled enough in the specific field to undertake following studies characterized by a high degree of autonomy.

COURSE SYLLABUS

Linear systems
The matrix exponential; the variation of constants formula. Computation of the matrix exponential via eigenvalues and eigenvectors and via residual matrices. Necessary and sufficient conditions for exponential stability: Routh-Hurwitz criterion. Invariant subspaces. Impulse responses, step responses and steady state responses to sinusoidal inputs. Transient behaviors. Modal analysis: mode excitation by initial conditions and by impulsive inputs; modal observability from output measurements; modes which are both excitable and observable. Popov conditions for modal excitability and observability. Autoregressive moving average (ARMA) models and transfer functions.
Kalman reachability conditions, gramian reachability matrices and the computation of input signals to drive the system between two given states. Kalman observability conditions, gramian observability matrices and the computation of initial conditions given input and output signals. Equivalence between Kalman and Popov conditions. Kalman decomposition for non-reachable and non-observable systems.
Eigenvalues assignment by state feedback for reachable systems. Design of asymptotic observers and Kalman filters for state estimation of observable systems. Design of dynamic compensators to stabilize any reachable and observable system. Design of regulators to reject disturbances generated by linear exosystems.
Introduction to adaptive control. Introduction to tracking control. Minimum phase systems and Proportional Integral Derivative (PID) control.
Bode plots. Static gain, system gain and high-frequency gain. Zero-pole cancellation. Nyquist plot and Nyquist criterion. Root locus analysis. Stability margins. Frequency domain design. Realization theory.

Introduction to nonlinear systems Nonlinear models and nonlinear phenomena. Fundamental properties. Lyapunov stability. Linear systems and linearization. Center manifold theorem. Stabilization by linearization.

## Analogue Electronics

OBJECTIVES

COURSE SYLLABUS

• Classification of electrical systems and requirements.
• Analysis of transitory and frequency behavior.
• Distortion in electronic systems and Bode diagrams.
• Diode semiconductor devices and circuit applications: clipper, clamper, peak detector, etc. Bipolar Junction and Field-Effect Transistors.
• Biasing techniques for Transistors. Amplifiers classification, analysis and circuit design.
• Frequency response of single and cascaded amplifiers.
• Differential amplifiers and Cascode.
• Current mirrors.
• Feedback amplifiers and stability issues. Power amplifiers.
• Operational amplifiers and related applications.
• Oscillator circuits. Integrated circuits and voltage waveform generators.

## Physics I

OBJECTIVES

LEARNING OUTCOMES:
Students will improve their knowledge under three main aspects: a) the comprehension and the related capability to articulate the arguments discussed throughout the course, using the proper scientific vocabulary and style; b) the methodological attitude toward complex problems, more specifically, to decompose a physical phenomenon into simpler elements to reach efficaciously a correct explanation; c) an elastic, fluid and reactive way of thinking, based on an inquisitive and hungry developed attitude toward new challenges.

KNOWLEDGE AND UNDERSTANDING:
Knowledge of the main arguments related to classical mechanics, gravitation, mechanical waves, elastic properties of solids and thermodynamics. Identification of the physical quantities and of the conservation principles needed to explain the main physical phenomena behind the above reported topics. Understanding scientific texts related to the course, and the acquisition of proper vocabulary and style.

APPLYING KNOWLEDGE AND UNDERSTANDING:
Capability to achieve the correct solution of complex classical mechanical, gravitation, mechanical waves, elastic properties of solids and thermodynamics problems, through an effective and methodologically correct understanding and application of the principles governing the physical phenomena described. Capability to apply the acquired scientific way of thinking to new problems by comprehending, analysing and also modelling autonomously.

MAKING JUDGEMENTS:
Students are supported to improve their critical way of thinking and to develop an independent judgment able to pose, refine and elaborate scientific questions. The purpose is to pave the way to a free and active research spirit based on a questioning elastic and hungry mind.

COMMUNICATION SKILLS:
The students are expected to master, through familiarity with the arguments studied and through a developed practice refined in the framework of the course, a specific focus, style and proper scientific vocabulary.

LEARNING SKILLS:
Capability to understand, analyse, and model a physical phenomenon crossing different principles learnt during the course. Considering problems related to the topics of the course, students are expected to solve them and articulate the reasons behind the main assumptions followed. An inquisitive attitude, a proper scientific way of reasoning and communicative skills are therefore expected to be acquired during the course.

SYLLABUS

Mechanics (main arguments considered):
• Measurements and fundamental quantities; coordinate systems; elements of vector calculus;
• Relative motion and Galilean transformation; Inertial and non-inertial frames of reference; Fictitious forces;
• The point mass concept; kinematics; from rectilinear to general curvilinear motion on a plane; dynamics; the principle of conservation of momentum; Newton’s laws; the concept of force as interaction; angular momentum; torque; impulse; energy and work: conservative and dissipative forces; conservation principles; discussion of potential energy curves.
• From the point-mass to systems of particles: external and internal forces; Center of Mass (CM): definition, calculus and dynamics; angular momentum and torque for systems; the CM and laboratory frame of reference; König theorem part I and part II; energy of a system; reduced mass;
• Collisions: generality, impact parameter; elastic, partially inelastic and perfectly inelastic collision;
• From systems to the rigid body; moment of inertia: main properties and relationships; statics and dynamics; work on a rigid body; kinetic energy of a rigid body; pure rolling and sliding; rolling friction; systems with variable mass;
• Oscillatory motion: simple harmonic motion; energy of simple harmonic motion; superposition of simple harmonic motions; coupled oscillators; damped oscillations; forced oscillations;

Elastic properties of solids (main arguments considered):
• Young, Shear and Bulk moduli; elastic and plastic deformations; elastic hysteresis.

Gravitation (main arguments considered):
• Kepler’ s laws; Cavendish experiment; Newton’s law of universal gravitation; gravitational potential; a reduced mass approach; bounded and unbounded orbits; escape velocity; geostationary orbit;

Mechanical waves (main arguments considered):
• generality; transverse and longitudinal waves: velocity and equation of the waves; sound waves; intensity of waves; interference; Doppler effect; supersonic waves;

Fluids (main arguments considered):
• Introduction to fluids; mechanical actions on fluids; statics of fluids; Torricelli’s experiment; Archimede’ s principle; Pascal’s principle; fluid and conservative forces; fluids in non-inertial frames of reference; fluids dynamics: Lagrangian and Eulerian approaches; stationary motions; volumetric flow rate; Bernoulli’ s equation; Venturi’ s effect; considerations on real fluids: laminar and turbulent flows; viscosity; Reynolds’ number; considerations on drag forces in fluids;

Thermodynamics (main arguments considered):
• Main concepts of thermodynamics; zeroth principle of thermodynamics; the first principle of thermodynamics; internal energy, heat and work; Ideal gases and ideal gas law; P-V diagram; molar specific heats; heat capacity; main processes in thermodynamics; real gases; Wan der Waals equation; phase transitions; second principle of thermodynamics; Carnot cycle; Carnot theorem; Clausius theorem; entropy; thermodynamic Universe; Enthalpy, Helmholtz free energy, Gibbs free energy; Dalton’ s principle.

Texts

Students can use the book that best suits them. However, the references cited below are the ones considered during the course:
1. “Fundamental University Physics Volume 1: Mechanics”, by Alonso & Finn
2. “Fundamentals of Physics”, by Halliday & Resnick
3. “Physics for Scientists and Engineers”, by Serway & Jewett