# Structural Dynamics

Institution:
IUSS Pavia
Specialization:
EE - Earthquake Engineering
Term:
Fall 2006
Teacher(s):
EDUARDO MIRANDA
Credits:
6
Date (from - to):
20/11/2006 – 15/12/2006

Definition of SDOF system characteristics. External force and earthquake excitations. Classical solution of second order linear ODE’s. Undamped and damped free vibration, energy in free vibration. Undamped and damped systems, resonance, energy dissipated in viscous damping, equivalent viscous damping. Response to unit impulse, arbitrary force and step force, response spectrum. Newmark’s method, stability and accuracy. Rigid body assemblages, distributed parameter systems, Rayleigh method. Simple MDOF systems, dynamic forces, reduction of DOF’s, static condensation. Natural vibration modes and frequencies, orthogonality and normalization of modes, modal expansion, free vibration response of MDOF systems, eigenvalue problem, vector iteration methods. Construction of damping matrix, Rayleigh damping. Modal response analysis of undamped and damped systems, element forces, modal contribution factors.

PART I – SINGLE DEGREE OF FREEDOM SYSTEMS

• Problem Statement - Definition of SDOF system characteristics.
• Equation of Motion - External force and earthquake excitations.
• Solution of the Equation of Motion - Classical solution of second order linear ODE’s.
• Free Vibration Response - Undamped free vibration, damped free vibration, energy in free vibration.
• Response to Harmonic Excitation - Undamped systems, damped systems, resonance, half power bandwith, energy dissipated in viscous damping, equivalent viscous damping.
• Response to General excitation - Response to unit impulse, arbitrary force and step force, response spectrum.
• Numerical Evaluation of Dynamic Response - Newmark’s method, stability and accuracy, Newmark’s method for nonlinear systems.
• Generalized SDOF Systems - Rigid body assemblages, distributed parameter systems, Rayleigh method.

PART II – MULTI DEGREE OF FREEDOM SYSTEMS

• Equations of Motion - Simple MDOF systems, dynamic forces, reduction of DOF’s, ststic condensation.
• Free Vibration Analysis Natural vibration modes and frequencies, orthogonality and normalization of modes,modal expansion, free vibration response of MDOF systems, eigenvalue problem, vector iteration methods.
• Damping in Structures - Construction of damping matrix, Rayleigh damping.
• Dynamic Analysis of Linear MDOF Systems - Modal response analysis of undamped and damped systems, element forces, modal contribution factors

Chopra, A. K., (2001), Dynamics of Structures: Theory and Applications to Earthquake Engineering. 2nd edition, Prentice-Hall. Clough, R. W., and Penzien, J. (1993):Dynamics of Structures, 2nd Edition, McGraw-Hill.