Review of linear algebra basics: matrix and algebraic vectors operations, matrix diagonalization and eigenvalue problem, eigenvectors and their normalization, complex numbers and Euler formula, complex eigenvalue problem.
Review of statics: kinematics of rigid bodies, Euler axioms to study the equilibrium of rigid bodies. Force-displacement and moment-rotation relationships and equilibrium equations in terms of kinematic parameters.
Dynamic problem: geometrical and mechanical aspects of the analytical modelling, time-depending parameters, and motion of a structure, linear constitutive behaviors and rheological models. Forces, moments, and energies of a dynamical system.
Formulation of the equations of motion: direct formulation via the balance of forces and moments and the variational approach. Conservative and nonconservative forces.
Single-degree-of-freedom systems (SDoFS): Equation of motion via direct and variational formulation, free undamped oscillations, the effect of initial conditions. The space-state variables and the formulation, and the solution, of the equation of motion of SDoFS in the space state. Complex and real forms of the solution. Free damped oscillations: direct and variational formulations of the equation of motion (second-order, and first-order in time equation of motion) and its solution. Complex and real forms of the solution. Definition of damped frequency and damped period of oscillation. Critically damped systems, overdamped systems, and underdamped systems.
Forced oscillations of undamped and damped SDoFS: Harmonic excitation, frequency-response curve for undamped and damped harmonically excited SDoFS, the dynamic amplification factor and the resonance phenomenon. The Fourier series and the periodic excitation. The Fourier transform and the logarithmic decrement method.
Arbitrary excitations in SDoFS, some case study and analytical solution. Methods for the numerical solution of the equation of motion of SDoFS (pills)
Displacement-induced excitations: the case of the ground motion or earthquake excitation. Equation of motion and its solution. Response spectrum: deformation, pseudo-velocity, and pseudo-acceleration response spectra. Difference between design and response spectra. The effect of the dissipation in response spectra.
The geometric stiffness and the example of system not possessing elastic stiffness: the rigid pendulum. Nonlinear form of the equation of motion of the pendulum (direct and variational approach), solution of the linearized equation of motion and its limitations. Direct and parametric excitations.
Multi-degree-of-freedom systems (MDoFS): direct and variational formulations of the equations of motion. Free undamped oscillations and the modal analysis. Solution in the state-space. Complex and real forms of the solution. Natural frequencies and linear normal modes. Orthogonality of the modes and their normalization. The modal space and the modal coordinates, solution of the free undamped oscillations in the modal space. Solution of the free damped oscillations in the modal space the case of classically damped systems and discussions on non-classically damped systems. Solution of the forced oscillations in undamped and damped MDoFS by means of the modal analysis.
The case of earthquake excitation in MDoFS: excited mass and response spectrum method.
Analysis of a two-degree-of-freedom (TDoFS) system and example of vibration absorbers (Tuned Mass Dampers)
Systems with distributed mass and elasticity: short recap on the kinematics and the equilibrium of the Euler-Bernoulli beam, linear elastic force-strain relationships, and equilibrium in terms of axial and transversal displacement. Equations of motion of the free undamped axial and bending oscillations of the Euler-Bernoulli beam subjected to different boundary conditions. Mode shapes, wave lengths and frequencies.
Through all the course are planned applications on all the topics investigated in the lectures. To this end, the teacher will use a dedicated software, i.e., Mathematica, licensed by Sapienza. Homeworks assignments will be given during the course.

Erasmo Viola
Fondamenti di Dinamica delle Strutture
Volume Primo: Sistemi discreti

Erasmo Viola
Fondamenti di Dinamica delle Strutture
Volume Secondo: Sistemi continui