Vibrations of Single Degree of Freedom Systems
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2016
This document describes free and forced dynamic responses of single
degree of freedom (
State Variables and Linear Time-Invariant Systems
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
1 Linearity and Time Invariance
A system G that maps an input u(t) to an ouput y(t) i
Structural Element
Stiness, Mass, and Damping Matrices
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
1 Preliminaries
This document describes the formulation of stiness and mass matr
Numerical Integration in Structural Dynamics
CEE 541. Structural Dynamics
Department of Civil & Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
Introduction
A damped structural system subjected to dynamic forces and possibly experiencin
Pulse Response As A Free Response
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2014
1 A pulse wavelet equation
The response of a linear time-invariant (LTI) dynamic system to a pulse w
Real Modes of Vibration of Building Structures
CEE 421L. Matrix Structural Analysis
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2012
1 Mass and Stiness Matrices
Consider a building frame modeled by a set of rigid
Modal Superposition
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2014
1 Solve the eigen-value problem and diagonalize the matrix second order o.d.es
Given a matrix second order ordinar
Fourier Series, Fourier Transforms, and
Periodic Response to Periodic Forcing
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2014
This document describes methods to analyze the steady-st
Inelastic Response Spectra
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall , 2014
Modeling
The following nonlinear ordinary dierential equation describes the behavior of an inelastic singl
Dynamic Models for Yielding and Friction Hysteresis
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
Consider the following nonlinear ordinary dierential equation
z(t) = u(t) |u(t)| z
Mathematical Properties of Stiness Matrices
CEE 421L. Matrix Structural Analysis
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
May 13, 2014
This handout gives an explanation of some of the mathematical properties of stru
Classical Damping, Non-Classical Damping and
Complex Modes
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall, 2014
1 Classical Damping
The equations of motion of an un-forced N degree of fre
Bilinear Hysteresis
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
1 Non degrading bilinear hysteresis
Bilinear hysteresis is a mathematically convenient model for the behavior of yi
Random Processes, Correlation, and
Power Spectral Density
CEE 541. Structural Dynamics
Department of Civil and Environmental Engineering
Duke University
Henri P. Gavin
Fall 2014
1 Random Processes
A random process X(t) is a set (or ensemble) of m random v