Università Roma Tre

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.

A. Chopra, Dynamics of Structures - a primer, Earthquake Engineering Research Institute
R.W. Clough and J. Penzien, Dynamics of Structures, Computers & Structures Inc.
L. Meirovitch, Fundamentals of Vibrations, McGraw-Hill.