212 Ketter Hall, North Campus, Buffalo, NY Fax: Tel: x 2400 Control of Structural Vibrations Lecture #2 Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering

Response of Open Loop

Review of SDOF

Free Vibrations

Efficiency of Damping

Equilibrium in Phase Space - Bode Diagram

Definition of Spectrum Spectrum is a function of frequency Examples: Spectrum of Light Spectrum in Chromatography / Chemistry Spectrum in Acoustics Spectrum in Mechanics

Response Spectrum TnTnTnTn T1T1T1T1 T2T2T2T2 T3T3T3T3dÄ SdSdSdSd T Each pendulum responds differently than the other Each pendulum will have a random response Each Pendulum will have a maximum response at some time The plot of the maximum displacement for each pendulum as a function of its Natural Period, T, is the Displacement Response Spectrum (S d )

Response Computations

Properties of Response Spectra

Pseudo-Spectra

Spectrum characteristics

Uncertainities in Spectrum

Code Spectrum - FEMA / NEHRP 1994

Spectral Amplification - Israeli Standard, IS-413, December 1998 R a = 1.25*S/T 2/3 <2.5 S=R a,max *T o 2/3 /1.25 S = { }

Spectral Amplification - Israeli Standard, IS-413, December 1998 R a = 1.25*S/T 2/3 <2.5 S=R a,max *T o 2/3 /1.25 S = { }

Code Spectrum / NEHRP 1994/ FEMA 303/ NEHRP SPECTRUM FOR 5% DAMPING (FEMA223A/223B) 1997 SPECTRA FOR HIGHER DAMPING (FEMA303/304)

Damping Factors in FEMA 303/304

Example - Uncontrolled

Example w/ Control

Conventional Spectra

Composite Elastic ADRS 22

Composite DARS(pectra)

Composite IDARS

Composite IDARS (2)

Approximation of Actual IDARS Approximation Methods:

Derivation of IDARS form DARS Approximations of factor R d can be obtained form literature

IDARS form DARS (3) From NEHRP 1997

IDARS form DARS (4) From NEHRP 1997

IDARS form DARS (5) From NEHRP 1997 Note: Spectrum for 5% damping

Normalized Composite ADRS T=1.00 T o =0.35 T=5.00 For Soil S 1 only. For other soils need to modify corner period T o

Modal Spectral Analysis

Drift (Displacement) Response