Università Roma Tre

Brief compendium of Lagrangian mechanics. Force balance, constitutive equations, inertia principle. Virtual variations and the Principle of Virtual Work. Constraints, constraint reactions and their characterization. Lagrangian coordinates. Lagrangian forces. Lagrange equations. Conservative forces. Examples: the simple pendulum, the double pendulum. Dissipative forces and dissipation potential. Autonomous systems. The phase space. Equilibrium configurations. Small oscillations about equilibria. Mass, stiffness, and viscous damping matrices.

Lumped systems: springs and dampers. Series and parallel arrangements, rheological models. System of one mass and one spring, system of two masses and two springs, system of two masses and three springs.

The simple oscillator. Undamped free oscillations. Representation of the solution through the complex notation. Assignment of the initial conditions. Energy conservation.

Free damped oscillations. Damping factor. Subcritical, critical, supercritical case. Methods for estimating the damping factor.

Harmonic excitation of systems with one degree of freedom. Forced oscillations in the absence of damping. Resonance curves. Forced oscillations in the presence of damping. Amplification factor,
quality factor, polygon of forces, dissipated power. Stationary solutions.

Distributed mass systems. Rigid bodies and linear elastic beams. Rotational springs. Deduction of the equations of motion for mechanical systems which combine devices with concentrated elasticity, rigid bodies, material points and beams without mass. Static condensation.

Analysis in the frequency domain. Periodic functions, Fourier series, fundamental frequency and Fourier coefficients, fundamental interval and prolongation of a periodic function, odd and even functions, Dirichlet's Theorem. Fourier series in complex form. Spectrum of a periodic function. Determination of the steady-state response of linear systems subject to periodic forcing. Spectrum of the amplitudes and spectrum of the phases. Fourier transform. Autocorrelation function. Spectral density function. Parseval theorem.

Analysis in the time domain. Response to the unit impulse, relationship with the Fourier transform. Arbitrary excitation. Duhamel's integral.

Linear systems with more degrees of freedom. Modal analysis. Natural frequencies and vibration modes. Rayleigh quotient. Modal matrix. Principal coordinates. Modal mass- and stiffness-matrices. Proportional dissipation matrix.

Vibrations of frames. Methods for the construction of the stiffness matrix: the displacement method and the finite-element method. Consistent mass matrices.

Introduction to vibration analysis of continuous systems. Beams, frames and plates.

Main textbooks.
E. Viola, "Fondamenti di Dinamica e Vibrazione delle Strutture", Voll. 1 e 2. Pitagora Editrice, Bologna.
Clough & Penzien, "Dynamics of Structures", Third Edition. Computers & Structures, Inc. 2003.
G. Muscolino, "Dinamica delle strutture". McGraw-Hill.

Other textbooks.
L. Meirovitch, "Analytical Dynamics", McGraw-Hill.
L. Meirovitch, "Fundamentals of Vibrations", McGraw-Hill.
P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, "Meccanica Razionale".