FULL-SCALE DYNAMIC CHARACTERISTICS OF TALL · PDF fileThis thesis focuses on the factors influencing the habitability limit state of tall building design

FULL-SCALE DYNAMIC CHARACTERISTICS OF TALL BUILDINGS AND

IMPACTS ON OCCUPANT COMFORT

A Thesis

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science in Civil Engineering

by

James David Pirnia

______________________________ Tracy Kijewski-Correa, Director

Graduate Program in Civil Engineering and Geological Sciences

Notre Dame, Indiana

July 2009

FULL-SCALE DYNAMIC CHARACTERISTICS OF TALL BUILDINGS AND

IMPACTS ON OCCUPANT COMFORT

Abstract

by

James David Pirnia

With every decade, the average height of tall buildings has increased and with

the near completion of Burj Dubai, tall buildings are now reaching unprecedented

heights where the governing limit state has transitioned to occupant comfort. This thesis

focuses on the factors influencing the habitability limit state of tall building design

through an investigation of data from the Chicago Full-Scale Monitoring Program. After

developing and validating a framework for estimating the amplitude-dependent dynamic

properties of tall buildings, frequency and damping are extracted from ambient vibration

data using both spectral and time-domain approaches. By doing so, not only is much

needed information on in-situ damping values made available to aid in habitability

design of tall buildings, but structural attributes facilitating amplitude dependence are

hypothesized, and the shortcomings of a traditional spectral approach are emphasized.

Finally a framework is developed to investigate occupant comfort directly from full-

scale accelerations using existing motion simulator studies to project the likely number

of occupants adversely affected by tall building motion when considering the effects of

waveform, duration, and frequency of oscillation. This framework provides perhaps the

most faithful mechanism, outside of tenant interviewing, to evaluate the performance of

tall buildings from an occupant comfort perspective.

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CONTENTS

FIGURES ......................................................................................................................... iv TABLES ........................................................................................................................... ix ABBREVIATIONS .......................................................................................................... xi SYMBOLS AND VARIABLES ..................................................................................... xii ACKNOWLEDGEMENT ................................................................................................ xv CHAPTER 1: INTRODUCTION ...................................................................................... 1 CHAPTER 2: INTRODUCTION TO THE MONITORED BUILDINGS ........................ 6

2.0 Introduction .............................................................................................................. 6 2.1 Phase 2: Korean Tower Monitoring Program .......................................................... 6

2.1.1 Instrumentation ............................................................................................................. 7 2.1.2 Data Acquisition Configuration .................................................................................... 8 2.1.3 Pre-Processing ............................................................................................................ 11

2.2 The Chicago Full-Scale Monitoring Program ........................................................ 14 2.2.1 Data Acquisition and Pre-Processing ......................................................................... 14

2.3 Summary ................................................................................................................. 15 CHAPTER 3: AMPLITUDE-DEPENDENT DYNAMIC PROPERTIES:SYSTEM IDENTIFICATION METHODS ...................................................................................... 16

3.0 Introduction ............................................................................................................ 16 3.1 Simulations ............................................................................................................. 17

3.1.1 Linear Simulation ....................................................................................................... 17 3.1.2 Nonlinear Simulation .................................................................................................. 17

3.2 Tools and Methods ................................................................................................. 26 3.2.1 Power Spectral Density Method ................................................................................. 26 3.2.2 Half-Power Bandwidth Method .................................................................................. 29 3.2.3 Random Decrement Technique .................................................................................. 30 3.2.4 Analytic Signal Theory ............................................................................................... 35

3.3 Validations .............................................................................................................. 36 3.3.1 Validation of HPBW in Isolation ............................................................................... 36 3.3.2 Validation of Analytic Signal Theory in Isolation ...................................................... 39 3.3.3 Validation of Logarithmic Decrement in Isolation ..................................................... 43 3.3.4 Validation of HPBW in Context ................................................................................. 46 3.3.5 Validation of Analytic Signal Theory in Context ....................................................... 50

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3.3.6 Validation of LD in Context ....................................................................................... 54 3.3.7 Verification of Amplitude-Dependent System Identification Methods ...................... 56

3.4 Summary ................................................................................................................. 71 CHAPTER 4: AMPLITUDE-DEPENDENT DYNAMIC PROPERTIES: APPLICATION TO FULL-SCALE DATA .................................................................... 73

4.0 Introduction ............................................................................................................ 73 4.1 Description of Selected Data .................................................................................. 73 4.2 Spectral Approach .................................................................................................. 74

4.2.1 Results and Discussion ............................................................................................... 74 4.2.2 Modal Isolation/Filter Selection ................................................................................. 85

4.3 Sorted Power Spectral Approach ............................................................................ 89 4.3.1 Korean Tower SSA Results ........................................................................................ 89 4.3.2 Chicago Building 1 SSA Results ................................................................................ 93 4.3.3 Chicago Building 2 SSA Results ................................................................................ 94 4.3.4 Chicago Building 3 SSA Results ................................................................................ 95 4.3.5 Summary ..................................................................................................................... 96

4.4 Time Domain Approach ......................................................................................... 96 4.4.1 Korean Tower RDT Results ....................................................................................... 97 4.4.2 Chicago Buildings .................................................................................................... 103

4.5 Summary ............................................................................................................... 116 CHAPTER 5: PSEUDO-FULL-SCALE EVALUATION OF OCCUPANT COMFORT ........................................................................................................................................ 117

5.0 Introduction .......................................................................................................... 117 5.1 Purpose ................................................................................................................. 119

5.2 Analysis Procedure for Peak Factors .................................................................... 119 5.3 Evaluation of 2007 Korean Tower Response against Perception Criteria Used in Current Practice .......................................................................................................... 121 5.4 Summary of Motion Simulator Observations Regarding Role of Waveform ...... 123 5.5 Classification of Full-Scale Waveforms ............................................................... 124 5.6 Further Discussion ................................................................................................ 128

CHAPTER 6 CONCLUSIONS AND FUTURE WORK .............................................. 132

6.1 Improved Monitoring of the Korean Tower ......................................................... 133

6.2 Framework for Extracting Amplitude-Dependent Dynamic Properties from Tall Building Ambient Responses ..................................................................................... 133 6.3 Extracted Dynamic Properties of Four Tall Buildings ......................................... 134

6.4 Pseudo-Full-Scale Evaluation of Occupant Comfort ........................................... 134 6.5 Future Work .......................................................................................................... 135

APPENDIX A: MODIFIED NONLINEAR NEWMARK‘S METHOD USING A MODIFIED NEWTON-RAPHSON ITERATION PROCEDURE ............................... 137 APPENDIX B: DETAILED DESCRIPTION OF RDT IMPLEMENTATION ............ 140 WORKS CITED ............................................................................................................. 143

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FIGURES

Figure 1.1: Effect of System Properties on Response Levels (Credit: RWDI) ................. 3

Figure 2.1: (A) Finite-Element-Model of the Korean Tower; and (B) Typical Floor Plan with Decoupled Instrumentation Locations ....................................................................... 7

Figure 2.2: Block diagram of Korean Tower Monitoring Program .................................. 8

Figure 2.3 Comparison of Initial and New Configuration of Korean Tower Monitoring Program ............................................................................................................................. 9

Figure 2.4: Schematic of Downsampling Routine of Korean Tower Monitoring Program ......................................................................................................................................... 10

Figure 2.5: Korean Tower In-situ Orientation of Sensors ............................................... 12

Figure 3.1: Sample Time History generated by State-Space Linear Simulation: White Noise Input (top); Response of SDOF System (bottom) ................................................. 18

Figure 3.2: (A) Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation ................................................................................................................. 19

Figure 3.3: Schematic of System Identification using the Logarithmic Decrement ....... 20

Figure 3.4: Frequency and Damping Estimated from MNMwIT Linear Free Decay using the Logarithmic Decrement ............................................................................................. 21

Figure 3.5: (A) AD-F Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation ................................................................................................... 22

Figure 3.6: Frequency and Damping Estimated from MNMwIT AD-F Free Decay using the Logarithmic Decrement ............................................................................................. 23

Figure 3.7: (A) AD-D Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation ................................................................................................... 24

Figure 3.8: Frequency and Damping Estimated from MNMwIT AD-D Free Decay using the Logarithmic Decrement ............................................................................................. 25

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Figure 3.9: (A) AD-F&D Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation ................................................................................................... 25

Figure 3.10: Frequency and Damping Estimated from MNMwIT AD-F&D Free Decay using the Logarithmic Decrement ................................................................................... 26

Figure 3.11: Schematic Representation of the Power Spectral Density Method (Kijewski-Correa, 2003) ................................................................................................................... 27

Figure 3.12: Schematic of Application of the Half Power Bandwidth Method Applied to a Power Spectrum ............................................................................................................ 30

Figure 3.13: Simplified Schematic of the Random Decrement Technique (Kijewski-Correa, 2003) ................................................................................................................... 32

Figure 3.14: Schematic of Local Averaging Technique .................................................. 34

Figure 3.15: Schematic of Analytic Signal Theory for System Identification ................ 36

Figure 3.16: Sample Frequency Response Function ( 2.0nf Hz & 005.0 ) .......... 37

Figure 3.17: Quality of HPBW Application to Theoretical Response Spectra ............... 38

Figure 3.18: Application of Analytic Signal via Hilbert Transform Routine to Noise-Corrupted FDR: (A) Sample of Random Noise; (B) Noise-Corrupted FDR; (C) Amplitude of Analytical Signal; (D) Phase of Analytic Signal ...................................... 40

Figure 3.19: Mean Error and CoV of Hilbert Transform Estimate of Natural Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise ................................................................................................................................ 42

Figure 3.20: Quality of Logarithmic Decrement Estimate of Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise ............... 44

Figure 3.21: Variance in Logarithmic Decrement Estimate of Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise ............... 45

Figure 3.22: Quality of HPBW Estimate of Frequency in a Simulated Linear System ...... ......................................................................................................................................... 48

Figure 3.23: Quality of HPBW Estimate of Damping in a Simulated Linear System ........ ......................................................................................................................................... 49

Figure 3.24: Application of Analytic Signal Theory via the Hilbert Transform to a RDS Obtained from a Linear Simulation: (A) Sample of Linear Simulation Time History; (B) Random Decrement Signature obtained from Time History; (C) Amplitude of Analytic Signal; (D) Phase of Analytic Signal ............................................................................... 52

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Figure 3.25: Bias and Variance in Analytic Signal Frequency and Damping Estimates from a RDS Obtained from a Linear Simulation, as a Function of Overlap ................... 53

Figure 3.26: Bias and Variance of Logarithmic Decrement Frequency and Damping Estimates from a RDS Obtained from a Linear Simulation, as a Function of Overlap ...... ......................................................................................................................................... 55

Figure 3.27: Linear and Non-Linear Simulations to the Same Random Noise Input using the NLS Method Described in Section 3.1.2 ................................................................... 57

Figure 3.28: Comparison of Frequency Estimates using Time and Frequency Domain Approaches ...................................................................................................................... 60

Figure 3.29: Comparison of Damping Estimates using Time and Frequency Domain Approaches ...................................................................................................................... 62

Figure 3.30: Sorted and Full (inset) Power Spectra for a Simulated Linear System ........... ......................................................................................................................................... 67

Figure 3.31: Sorted and Full (inset) Power Spectra for a Simulated Non-Linear System .. ......................................................................................................................................... 68

Figure 3.32: Schematic of Fourier Representation of System with Varying Frequencies ......................................................................................................................................... 70

Figure 4.1: Power Spectral Density Matrix for Korean Tower (rows = instrument locations, columns = primary lateral directions) ............................................................. 78

Figure 4.2: Floor Plan of Korean Tower at 64F with Power Spectra at each Location .. 79

Figure 4.3: Plan view of Building 1 with observed Power Spectra ................................. 82



Figure 4.6: Verification of Korean Tower 1st and 2nd mode filter selection .................... 86

Figure 4.7: Verification of Korean Tower 3rd mode filter selection ................................ 87

Figure 4.8: Verification of Building 1 filter selection ..................................................... 88

Figure 4.9: Verification of Building 2 filter selection ..................................................... 88

Figure 4.10: Verification of Building 3 filter selection ................................................... 89

Figure 4.11: SSA: Spectral Suite for 1st and 2nd modes of Korean Tower ...................... 91

Figure 4.12: SSA: Spectral Suite for 3rd mode of Korean Tower ................................... 92

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Figure 4.13: SSA: Spectral Suite for Building 1 ............................................................. 93



Figure 4.16: Amplitude-Dependent Frequency and Damping Ratio: Mode 1 of Korean Tower ............................................................................................................................... 99

Figure 4.17: Amplitude-Dependent Frequency and Damping Ratio: Mode 2 of Korean Tower ............................................................................................................................. 100

Figure 4.18: Amplitude-Dependent Frequency and Damping Ratio: Mode 3 of Korean Tower ............................................................................................................................. 101

Figure 4.19: Modal Frequency and Damping Ratio Interaction: Building 1 ................ 104



Figure 5.1: Example of Peak Factor Calculation for a Given Sample Response Window ....................................................................................................................................... 120

Figure 5.2: Physical Effects of Acceleration on Occupants as Summarized (Credit: ASCE Tall Buildings Committee) ................................................................................. 121

Figure 5.3: Peak Acceleration as a Function of Frequency for Different Return Periods (Credit: ASCE Tall Buildings Committee) ................................................................... 122

Figure 5.4: Peak Acceleration as a Function of Annual Recurrence Rate (Credit: ASCE Tall Buildings Committee) ............................................................................................ 122

Figure 5.5: Peak Accelerations by Month for the Korean Tower in 2007 (First Mode Isolated, Location 2) ...................................................................................................... 123

Figure 5.6: Waveform Examples (12 Minute Analysis Windows) for Sample Data File Recorded on August 8, 2008 for the Korean Tower (X-Sway, Location 2) ................. 124

Figure 5.7: Response Classification by Waveform Type for X-sway and Y-sway (top and bottom, respectively) (12 Minute Analysis Windows): Korean Tower, 2007 .............. 125

Figure 5.8: Gaussian Long (50 Minute Analysis Window) and Short (12 Minute Analysis Window) Duration Events along X and Y-Axes (top and bottom, respectively): Korean Tower, June 2007 .............................................................................................. 126

Figure 5.9: Comparison of Results of Burton et al. (2005) with Other Occupant Comfort Studies ........................................................................................................................... 130

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Figure 5.10: Peak Accelerations by Month for the Korean Tower in 2007 (Total Response, Location 2) ................................................................................................... 131

Figure 5.11: Peak Accelerations by Month for the Korean Tower in 2007 (Location 2) ....................................................................................................................................... 131

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TABLES

Table 3.1: Frequency and Damping Observed in Simulated Free Decays ...................... 20

Table 3.2: List of Proposed Trigger conditions for the RDT (Ibrahim, 2001) ................ 31

Table 3.3: Application Procedure of the RDT ................................................................. 34

Table 3.4: Comparison of Simulated and Observed Amplitude-Dependent Relationships by Analytic Signal Theory ............................................................................................... 59

Table 3.5: Comparison of Simulated and Observed Amplitude-Dependent Relationships by Logarithmic Decrement .............................................................................................. 59

Table 3.6: Comparison of Frequency estimates from Time and Frequency Domain Approaches ...................................................................................................................... 64

Table 3.7: Comparison of Damping estimates from Time and Frequency Domain Approaches ...................................................................................................................... 64

Table 3.8: Sorted Spectral Approach Results for a Simulated Linear System ................ 67

Table 3.9: Sorted Spectral Approach Results for a Simulated Non-Linear System ........ 68

Table 3.10: Comparison of Damping from Spectral Approach and Gross Damping...... 71

Table 3.11: Evaluation of Gross Damping Using Time Domain Analysis Results ........ 72

Table 4.1: Calculated and Selected Spectral Frequency Resolutions .............................. 75

Table 4.2: Comparison of Design Predictions (Kijewski-Correa et al., 2006) and Spectral Approach Estimates of in-situ Frequency and Damping Ratio ....................................... 77

Table 4.3: Indicators of Variance and bias in Power Spectra .......................................... 79

Table 4.4: Korean Tower Spectral Approach Estimates of In-Situ Frequency and Damping ratio by Month ................................................................................................. 81

Table 4.5: SSA Results: Korean Tower .......................................................................... 90

Table 4.6: SSA Results: Chicago Building 1 .................................................................. 93


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Table 4.9: Summary of Records and Time Domain Approach Results: Korean Tower ..... ......................................................................................................................................... 97

Table 4.10: Amplitude-dependent relationships of Frequency and Damping predicted by the Time Domain Approach ............................................................................................ 98

Table 4.11 Summary of Records and Time Domain Approach Results: Chicago Buildings ........................................................................................................................ 107

Table 4.12: Comparison of Spectral and Time Domain Approach Frequency Results ...... ....................................................................................................................................... 112

Table 4.13: Comparison of Spectral and Time Domain Approach Damping Results ........ ....................................................................................................................................... 113

Table 4.14: Calculation of Gross Damping ................................................................... 114

Table 4.15: Comparison of Spectral Approach Results with Gross Frequency ............ 114

Table 4.16: Comparison of Spectral Approach Results with Gross Damping .............. 115

Table 5.1: Summary of Challenges implementing Full-Scale Monitoring Programs ......... ....................................................................................................................................... 118

Table 5.2: Task Disruption Summary of Gaussian-type Events for 2007 (12 Minute Analysis Window) ......................................................................................................... 127

Table 5.3: Onset of Nausea Summary of Gaussian-Type Events for 2007 (12 Minute Analysis Window) ......................................................................................................... 128

Table AA.1: Modified Newton-Raphson Iteration Procedure (Chopra, 2001) ............. 138

Table AA.2: Modified Nonlinear Newmark‘s Method with Iteration Procedure (Chopra, 2001) .............................................................................................................................. 139

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ABBREVIATIONS

AA Anti-Aliasing AD Amplitude-Dependent CoV Coefficient of Variation FDR Free Decay Response FEM Finite Element Model FFT Fast Fourier Transform HPBW Half Power Bandwidth HT Hilbert Transform NLS Non-Linear Simulation MNMwIT Modified Newmark‘s Method with Iteration PSD Power spectral density RDT Random Decrement Technique RDS Random Decrement Signature SA Spectral Approach SDOF Single Degree of Freedom SI System Identification SSA Sorted Spectral Approach TDA Time Domain Approach

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SYMBOLS AND VARIABLES

Super/Subscripts 0 initial 0t h designates a specific signal within a larger group i denotes a particular time step j denotes an iteration step q vector of FFT indices v denotes a particular trigger Symbols/Operations

CoV coefficient of variation ][be normalized bias error

E expected value/mean ][ Fourier Transform

][1 Inverse Fourier Transform H Hilbert Transform Im imaginary component ln natural logarithm

pdfVV 21, joint probability density function of two random variables Re real component Var variance

convolution given

intersect/and magnitude

phase sum Variables acceleration time history a time stepping and dynamic variable

oA initial amplitude of a free decay b time stepping and dynamic variable c damping constant

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D random decrement signature f frequency

1f , 2f bounding half-power frequencies

Af , Bf bounding net damping frequencies

NETf net (mean) frequency f frequency resolution

dreqf ' required frequency resolution

Df damped natural frequency

nf resonant natural frequency

inf resonant natural frequency at time step i

inf incremental resonant natural frequency at time step i

qf discrete FFT frequency q

Sf resisting force iSf resisting force at time step i )(

1j

iSf resisting force at time step 1i and iteration j

)( jSf incremental resisting force at iteration j

F force )(tF time history of F

HPBW half-power bandwidth k stiffness

ik stiffness at time step i

ik apparent stiffness at time step i

Tk target apparent stiffness m mass M multiplicative factor for local averaging triggers

LAN number of local averaging points

PN number of segment averages in creating a PSD

RN

number of segment averages in creating a RDS NFFT number of FFT points

dreqNFFT ' required number of FFT points p external load

ip incremental external load at time step i

ip incremental apparent external load at time step i )( jP residual force at iteration j VVR auto-correlation function for tV at arbitrary lag

FS sampling frequency

qqS expected spectral energy

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t time vector fpt time of first peak t time step interval

tt sample number nT natural resonant period nf/1

PT time series segment length u ,u , u simulated displacement, velocity, acceleration

iu , iu , iu simulated displacement, velocity, acceleration at time step i

iu , iu , iu incremental displacement, velocity, acceleration at time step i )( ju incremental displacement at iteration j )(1j

iu incremental displacement at time step 1i and iteration j

V ,V ,V displacement, velocity, acceleration 1V 2V random variables of tV at two times ( 1t & 2t ) tV response time history of V ttV sample number tt of time history of tV

hX FFT of sample signal h ty free decay time history tY free decay time history

00uuY free decay time history with initial displacement and velocity )(tz analytic signal

vZ vZ v th displacement or amplitude trigger

vZ local averaging vector for the v th amplitude trigger time stepping constant

A , B half-power bandwidths of net damping limiting frequencies

NET net bandwidth incremental change angular frequency

D damped natural angular frequency 212 nDf

n resonant natural angular frequency nf2 number of cycles between peaks standard deviation damping ratio

i incremental damping ratio at time step i time stepping constant time lag variable, 12 tt

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ACKNOWLEDGEMENT

The order of these acknowledgements is only due to the limitations of this 2D

page; in reality all those mentioned herein equally deserve praise as for the absence of

any of their influences this accomplishment would not be possible.

I would like to thank my whole family (Gilpin and Pirnia) for their patience,

encouragement, and support in all my pursuits: of special note, my first teacher, my

wonderful mother, and my loving wife, Judy.

I would like to thank my teachers for their patience and knowledge; their

examples inspired and provided me the tools to reach this point: of special note, Mr.

Diaz (TJMS), Ms. Genevieve Demos (MHS), Dr. Kevin Sutterer (RHIT), Dr. James

Hanson (RHIT), Dr. Tracy Kijewski-Correa (ND), Dr. Ahsan Kareem (ND), Dr. Yahya

Kurama (ND), and Dr. David Kirkner (ND).

I would like to thank Dr. Tracy Kijewski-Correa for her patience, advisement,

example, and knowledge. You always saw potential in me, and that motivated me to

keep going.

I would like to thank Dr. Tracy Kijewski-Correa, Dr. Ahsan Kareem, and Dr.

Alexandros Taflanidis for their time serving on my committee. I would also like to thank

Dr. Yahya Kurama for his efforts in guiding my graduate program.

Finally, I would like to thank those who collaborated to make this research

possible: the ND NatHaz Lab (Dr. Ahsan Kareem and Dr. Dae Kun Kwon), the Chicago

Full-Scale Monitoring Program [Samsung Corporation (Mr. Ahmad Abdelrazaq and Dr.

Jaeyong Chung), SOM, BLWT Lab at the University of Western Ontario, and Notre

Dame], and the financial support of NSF Grant CMS 06-01143.

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CHAPTER 1:

INTRODUCTION

Advancements in material strengths and structural systems have driven modern

buildings to be taller, lighter, and more flexible in increasingly complex wind

environments. While these advancements have been supported and enabled by solid

research employing a range of computational and scaled experimental techniques, these

settings are generally not effective for addressing some of the community‘s most

pressing research questions. These questions are tied to the fact that tall buildings, unlike

most other structures, must be designed for three performance limit states: survivability

(strength), serviceability (deflections) and habitability (accelerations). These response

quantities depend significantly on dynamic properties such as frequency and damping. In

the case of most tall buildings, these properties are delivered by designers to wind tunnel

consultants who then determine equivalent static wind loads (survivability),

displacements (serviceability) and accelerations (habitability) at various return periods

based on scaled model testing in boundary layer flows. Thus, a structure‘s ability to meet

these limit states greatly depends on the accuracy with which frequency and damping are

estimated during design.

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Although frequency is reasonably estimated by modern finite element models

grounded in theories of mechanics, damping is far more elusive1. Unlike frequency,

which clearly correlates to structural properties like mass and stiffness, damping is a

quantity representing the total energy dissipation intrinsic to a system based on its

construction materials, member connections/interactions, structural system, foundation

type, occupancy and even aerodynamic shape (Kareem and Gurley, 1996; Kareem et al.,

1999; Fang, 1999). Because of its complexity and lack of predictive model, researchers

began conducting full-scale investigations on generally low to mid-rise buildings to

better inform designer‘s guesses of inherent damping (Jeary, 1986; Lagomarsino, 1993;

Suda et al., 1996). Unfortunately, these databases were characterized by significant

scatter, making it difficult to find any clear correlation between damping and other

structural attributes (Satake et al., 2003). This scatter was due not only to the uncertainty

in estimating very low levels of damping (under 2% critical) from ambient vibration

data, but also from the fact that damping and even frequency have demonstrated a

measurable level of amplitude-dependence that results from imposing a linear model on

phenomena that are inherently nonlinear (Jeary, 1996). While some models for

amplitude-dependence have been proposed in the literature, their appropriateness for tall

buildings has received limited attention (Satake et al., 2003; Jeary, 1986). In fact, there

have been only a handful of tall buildings whose dynamic properties have been

thoroughly documented in full-scale: the Central Plaza Tower in Hong Kong (Li et al.,

2005); Di Wang Tower in Shenzhen, China (Li et al., 2005; Li et al., 2004; Li et al.,

1 While frequency certainly is estimated with greater reliability in design, the author‘s research group has also demonstrated in full-scale, inaccuracies in natural frequencies of both concrete and steel buildings (Kijewski-Correa et al. 2006). Still, these are not due to a lack of correlation of this property to geometric and material properties, but rather due to errant assumptions about the in-situ material characteristics or about the most appropriate means to model specific elements of the lateral system. Thus while full-scale observations in general help to better inform the process of estimating all dynamic properties, damping still has the far greater need.

3

2002); Guangdong International Building, China (Li and Wu, 2004); the Jin Mao

Building in Shanghai, China (Li et al., 2006); and an anonymous tall building in Hong

Kong, China (Li et al., 2003; Li et al., 1998). Even when amplitude-dependence has

been detected, there has been no systematic effort to determine the types of structures

most susceptible to this amplitude-dependence, e.g., concrete vs. steel buildings, tubes

vs. frames. While an increase of damping from 0.5% to 1% critical may not seem

significant to those outside the field, when one considers that even modest increases in

damping can dramatically reduce accelerations, more so than any other structural

property (Figure 1.1), this phenomenon clearly becomes worth investigating.

Figure 1.1: Effect of System Properties on Response Levels (Credit: RWDI).

Thus far, our discussion has focused on the predicted structural response

quantities used for assessing the performance of tall buildings and the influence of

dynamic properties on them. In most cases, these predicted responses have clear

performance benchmarks that must be satisfied. For example, prescriptive codes like

ACI 318 and the Manual of Steel Construction ensure that elements are designed with

sufficient capacity for the demands derived from equivalent static wind loads

4

(survivability). And while not code mandated, structural deflections are often restricted

to tolerable levels with respect to non-structural elements and finishes (serviceability).

Meanwhile, highly uncertain and complex human-structure interactions engulf the

habitability limit state in controversy. Just as with dynamic properties, attempts to fully

understand these complex interactions using simplified experiments with artificial

home/office environments hosting human subjects are riddled with limitations (Kareem

et al., 1999). For example, experiments forming the basis of current motion perception

guidelines generally employed only uniaxial, sinusoidal motions, thus neglecting the

effects of other response types such as narrowband Gaussian responses most commonly

associated with tall buildings under wind, or even the more non-Gaussian responses

observed under transient wind events. And thus while full-scale investigations would

certainly help to improve occupant perception criteria, systematic full-scale validations

of accelerations affecting occupant comfort has not been possible due to a number of

practical and even legal barriers. In light of these barriers, motion simulator studies

continue with little correlation to full-scale observations.

In response to these issues surrounding the design practice for tall buildings, this

thesis will employ full-scale data from the Chicago Full-Scale Monitoring Program

(Kijewski-Correa et al., 2006), including a newly acquired tower in Seoul, Korea, to

realize the following objectives:

1. Develop and validate a system identification framework capable of extracting

amplitude-dependent dynamic properties of tall buildings from ambient

vibration response

2. Apply this system identification framework to full-scale data from the

Chicago Full-Scale Monitoring Program to document in-situ dynamic

properties and their level of amplitude-dependence

3. Develop a means to correlate recent motion simulator studies with full-scale

accelerations, considering the effect of waveform, duration and frequency.

5

4. Apply this approach to determine the frequency at which potentially

disruptive motions occur in actual buildings.

This thesis is organized as follows: Chapter 2 overviews the buildings of the

Chicago Full-Scale Monitoring Program and their instrumentation. Chapter 3 will then

introduce and validate the system identification framework for tracking amplitude-

dependent dynamic properties (Objective 1). This framework will then be applied in

Chapter 4 to responses from the instrumented buildings (Objective 2). Chapter 5 will

then discuss occupant comfort criteria and attempt to correlate recent motion simulator

studies with full-scale responses (Objectives 3 and 4). Conclusions and future work are

addressed in Chapter 6, thereby completing this thesis.

6

CHAPTER 2:

INTRODUCTION TO THE MONITORED BUILDINGS

2.0 Introduction

Full-scale response data is necessary to validate and expand our knowledge of

tall building behavior. One of the most comprehensive monitoring programs for tall

buildings, the Chicago Full-Scale Monitoring Program, will serve as the database for this

research. While the initial phase of this program instrumented three tall buildings in

Chicago (Kijewski-Correa et al., 2006), a second phase extended these efforts to Korea

(Abdelrazaq et al., 2005). Details of the monitoring systems of these buildings are

provided in the following sections, while honoring the confidentiality of the

instrumented buildings. More details of the recent Korean addition are offered, as this

instrumentation effort represents a new contribution by this thesis.

2.1 Phase 2: Korean Tower Monitoring Program

As described by Abdelrazaq et al. (2004), the instrumented tower is an 865 ft tall

composite residential building consisting of a concrete core bound to perimeter columns

through exterior belt walls at floors 16 and 55. The belt walls are indicated on the finite

element model provided in Figure 2.1(A). Very stiff floor slabs connect the reinforced

concrete core walls to the exterior belt wall forcing deformation compatibility. The

tower rests upon a 3500 mm thick high-performance reinforced concrete mat over

concrete slab and prepared rock. In April 2005, a monitoring program for this building

7

was initiated through a joint collaboration between the University of Notre Dame and

Samsung Corporation.

Figure 2.1: (A) Finite-Element-Model of the Korean Tower; and (B) Typical Floor Plan with Decoupled Instrumentation Locations.

2.1.1 Instrumentation

Three pairs of Wilcoxon 731A/P31 accelerometers are attached to girders in

orthogonal pairs on the 64th floor of the building at the locations shown in Figure 2.1(B).

These sensors possess a sensitivity of 10 V/g over an amplitude range of 0.5 g and

frequency range down to 0.1 Hz (Wilcoxon Research, 2005). In addition, a single FT

Technologies FT702 Ultrasonic Anemometer attached to a 6.5 ft mast above the roof of

the building provides wind speed and directional data. The sensors are connected to an

IOtech Wavebook/516E data acquisition unit with 16-bit resolution. A WBK13A Low-

Pass Filter card is installed to provide a configurable, hardware-based anti-aliasing (AA)

filter. The data acquisition unit communicates with an on-site computer that is accessible

via FTP.

8

2.1.2 Data Acquisition Configuration

DASYLab by IOtech was selected to configure the sensors and direct the

monitoring program. A schematic of this process is provided in Figure 2.2. The initial

configuration of the monitoring system by Samsung personnel suffered from high noise

levels, as demonstrated in Figure 2.3. Note here that the building‘s modes are scarcely

discernable from the noise. The system was reconfigured through this thesis to reduce

noise levels and improve data quality (see comparison in Figure 2.3). Data acquired

under this new configuration show better resolution of spectral peaks. Details of the data

acquisition routine as well as necessary modifications are now provided.

Figure 2.2: Block diagram of Korean Tower Monitoring Program.

2.1.2.1 Sampling Rate

In the original configuration, the default anti-aliasing filter (20 kHz) was used

with high sample rates (>1000 Hz), thus producing excessive amounts of data that over-

taxed computational resources on-site. A sampling rate of 10 Hz would yield a more

reasonable bandwidth, as the fundamental modes of the building are predicted to be well

below 1 Hz, and thereby reduce the subsequent file size. Unfortunately, the AA filter

Sensors

Wavebook Filter &

Decimate

Scale Vmilli-g

Data Storage Trigger

PC & DASYLab

9

frequencies could not be set low enough to facilitate this sampling rate without the risk

of aliasing. Therefore, a downsampling routine was developed.

Figure 2.3 Comparison of Initial and New Configuration of Korean Tower Monitoring Program.

The standard AA filtering in the Wavebook/516E is fixed at 20 kHz and with the

WBK13A Lowpass Filter Card active, the lowest anti-aliasing frequency setting is 400

Hz. Therefore, an initial sampling rate of 2000 Hz was selected to minimize any aliasing

due to roll-off from this 400 Hz filter2. Following analog to digital conversion, the 2000

Hz data is run through an ―on-the-fly‖ downsampling routine consisting first of

2 Roll-off of this elliptic filter spans approximately 400-512 Hz, which is safely within the Nyquist frequency for a 2000 Hz sampling rate.

0 1 2 3 4 510

-20

10-15

10-10

10-5

100

105

Initial Configuration

April 3, 2005

Spectr

al M

agnitude

Frequency [Hz]

0 1 2 3 4 510

-20

10-15

10-10

10-5

100

105

New Configuration

November 2, 2006

Frequency [Hz]

10

Butterworth filtering at 1 Hz to prevent additional aliasing after decimation, followed by

decimation to obtain the final data at 10 Hz. A schematic of this process is provided in

Figure 2.4.

Figure 2.4: Schematic of Downsampling Routine of Korean Tower Monitoring Program.

11

2.1.2.2 Triggering Conditions

Instead of a peak trigger, which may result in false positives due to noise, sensor

drifts or erroneous spikes, a standard deviation trigger is implemented in the updated

monitoring scheme using a relay switch to regulate data storage. After a fixed length of

response surpasses the trigger threshold, a relay switch permits data storage for at least

an hour or until the trigger condition ceases, whichever occurs last. The level of the

trigger and data length may be adjusted but are currently set at 0.470 milli-g standard

deviation over an 819.2 second interval, corresponding to a peak response amplitude of

approximately 1.5 milli-g assuming a normal distribution.

2.1.2.3 Data Storage

Files are referenced by the date and time their acquisition was initiated. Within

each file are nine columns of data: column 1 refers to time (sec), columns 2-3

correspond to the outputs of orthogonal accelerometers at location 1 (milli-g), columns

4-5 correspond to the outputs of orthogonal accelerometers at location 2 (milli-g),

columns 6-7 correspond to the outputs of orthogonal accelerometers at location 3 (milli-

g), and columns 8-9 respectively correspond to wind speed (m/s) and direction (degrees).

Figure 2.5 provides in-situ locations of each accelerometer on the building floor plan.

2.1.3 Pre-Processing

The triggered files are downloaded via FTP and prepared for detailed analysis by

removing electrical noise and drift in the sensors, followed by projection of the recorded

accelerations onto the building‘s primary lateral axes with the application of filtering to

remove any high frequency noise. The former operation is achieved by identifying

electrical spikes, i.e., seeking response amplitudes more than nine standard deviations

from the mean, and replacing them with a linear interpolation between adjacent points.

Sensor drift is then corrected by de-meaning each response component.

12

Figure 2.5: Korean Tower In-situ Orientation of Sensors.

The projection of response data onto the two lateral axes of the building (X, Y) is

then accomplished by trigonometric operations assuming rigid body motion in the floor

plate. First, let each accelerometer be referenced by location and orientation. For

example, ―α10‖ is the acceleration from the sensor at location 1 orientation 0 (parallel to

the girder). Further let each sensor‘s transformed acceleration posses an ―x‖ or ―y‖

suffix: indicating that they measure X or Y response on the building‘s primary lateral

axes. For example, ―α10x‖ is the transformed acceleration of the sensor at location 1

orientation 0 measuring the X response of the building. Again, the in-situ sensor

locations with respect to the building‘s geometric center are provided in Figure 2.5.

Thus, the following transformations can be used to decouple the accelerations from each

sensor pair to obtain the response of the tower along its X and Y axes. First, the total X

13

and Y responses at location 1 are given by Equations (2.1) and (2.2), where the

components of the response are given by Equations (2.3) to (2.6).

xxX 11101 (2.1)

yyY 11101 (2.2)

)180/5.37cos(1010 x (2.3)

)180/5.37sin(1010 y (2.4)

)180/5.37sin(1111 x (2.5)

)180/5.37cos(1111 y (2.6)

Similarly, the total X and Y responses at location 2 are given by Equations (2.7) and

(2.8), where the components of the response are given by Equations (2.9) to (2.12).

xaxaXa 21202 (2.7)

yyY 21202 (2.8)

)180/5.37cos(2121 x (2.9)

)180/5.37sin(2121 aya (2.10)

)180/5.37sin(2020 x (2.11)

)180/5.37cos(2020 y (2.12)

Finally, at location 3, the sensors are oriented consistent with the building‘s X and Y

axes, therefore only a minor adjustment for sign is required as provided by Equations

(2.13) and (2.14).

303 X (2.13)

313 Y (2.14)

After decoupling, the data (α1X, α1Y, α2X, α2Y, α3X, α3Y) is passed through a

Butterworth filter to isolate only the useable data with frequency content less than 1 Hz.

14

2.2 The Chicago Full-Scale Monitoring Program

The Chicago Full-Scale Monitoring Program has instrumented three buildings

with structural systems common to high-rise design (Kijewski-Correa et al., 2006). This

monitoring program is a partnership between the University of Notre Dame, the

Boundary Layer Wind Tunnel Laboratory at the University of Western Ontario, and

Skidmore, Owings & Merrill LLP in Chicago. In accordance with the wishes of the

owners, each building is referred to by numbers to preserve its anonymity. As this

instrumentation program predates this thesis, only brief details of the instrumentation are

offered here for completeness. Building 1 resists lateral loads primarily through

cantilever action of its exterior columns acting as a stiffened steel tube. Building 2‘s

lateral load-resisting system consists of a reinforced concrete outrigger connecting the

perimeter columns to a shear wall core at two different levels. Building 3 resists lateral

loads primarily through cantilever action of its steel moment-connected, framed tubular

system. Additional discussion of the building systems and dynamic properties can be

found in Kijewski-Correa et al. (2006).

2.2.1 Data Acquisition and Pre-Processing

Each building is instrumented with four accelerometers in orthogonal pairs at

opposite corners of the building floor plan and at the highest possible floor.

Accelerometers were installed on the upper floors of Building 1 and 2 in June 2002 and

later in Building 3 on May 2003. Each accelerometer is a Columbia SA-107LN servo-

force balance accelerometer that is capable of tracking low amplitude motions down to 0

Hz with a relatively low noise floor at 15 V/g sensitivity. Each sensor is sampled at

approximately 8 Hz by a Campbell CR23X Datalogger with 20 MB of memory. Wind

speed and direction data are collected from a nearby NOAA meteorological station in

Lake Michigan, approximately 3 miles offshore from the Chicago Loop. Additional

information on the data acquisition configuration and pre-processing are provided in

15

Kijewski-Correa (2003) and Kijewski-Correa et al. (2006). The acceleration and wind

velocity data have been collected since 2002 and are accessible through a secured online

archive (windycity.ce.nd.edu). This web portal offers similar pre-processing capabilities

through automated removal of spikes and drifts and calculation of global responses

along the primary lateral axes and twist about the building centerline. By addition

operations, the building X-sway responses at the two corners of the building are

condensed to a single averaged X-sway response. The same operation is conducted on

the Y-sway response. Differencing operations between the two sensor locations then

result in two estimates of the torsional response, where an average torsional response is

output. As a result, the analysis of the three Chicago Buildings shall directly utilize the

condensed X, Y and torsional responses from this web-interface, instead of individually

analyzing the raw sensor feeds at the two measurement locations, whereas the analysis

of the Korean Tower data shall include some investigation of the responses at each of the

measurement locations.

2.3 Summary

The instrumentation and buildings of the Chicago Full-Scale Monitoring

Program were introduced in this chapter. The range of building materials and lateral

systems within this tall building database are principal in obtaining a true to life

understanding of amplitude-dependent dynamic properties in tall buildings. In the next

chapter, each of the analysis techniques used in this research to observe and investigate

amplitude-dependent dynamic properties in tall buildings are introduced.

16

CHAPTER 3:

AMPLITUDE-DEPENDENT DYNAMIC PROPERTIES:

SYSTEM IDENTIFICATION METHODS

3.0 Introduction

This chapter details the development of amplitude-dependent system

identification methods used in this research, beginning first with linear and nonlinear

simulation methods that are later used to verify the effectiveness of the system

identification tools. Both time and frequency-domain system identification tools include

two layers of analyses: generation of the response artifact and identification of frequency

and damping from that artifact. As there are potential sources of error at each step,

validations in this chapter first focus on the system identification approaches themselves:

half-power bandwidth, logarithmic decrement and analytic signal theory using the

Hilbert Transform applied to idealized response artifacts. Then validations are performed

on the comprehensive identification scheme including artifact generation and the

aforementioned approaches. The results of these analyses are used to identify parameters

in each identification scheme producing the best performance. These validations are

conducted for linear systems with constant dynamic properties and nonlinear systems

with amplitude-dependent dynamic properties.

17

3.1 Simulations

In this section, two methods of simulation are discussed: linear and nonlinear,

which will later be used to verify the effectiveness of each system identification method

for the frequency and damping range of most relevance to this research.

3.1.1 Linear Simulation

A state-space method was used to simulate response data for a SDOF linear

mechanical oscillator, based upon the linear equation of motion in Equation (3.1), whose

dynamic properties were modified to generate responses with a range of frequencies

and/or damping levels.

m

tFtVtVtV nn)()()(2)( 2 (3.1)

Independent, standard Gaussian white noise served as the input, )(tF , of each simulation

yielding the output, )(tV . Other variables involved in the equation of motion include:

critical damping ratio, ; angular natural frequency, n , related to the natural frequency

fn by n=2fn; and mass; m . A sample time history of input and response is provided in

Figure 3.1.

3.1.2 Nonlinear Simulation

Newmark‘s method is used for nonlinear response simulations in this thesis. By

supplementing the basic time-stepping linear response method with an additional energy

balance equation and iteration, time varying stiffness and damping properties may be

simulated (Chopra, 2001). This additional energy balance equation, provided in Equation

(3.2), is derived from the equation of motion.

iiSii pfvcvm (3.2)

18

Figure 3.1: Sample Time History generated by State-Space Linear Simulation: White Noise Input (top); Response of SDOF System (bottom).

The modified Newton-Raphson iteration method, provided in Table AA.1 of Appendix

A, is used to curtail propagating errors resulting from insufficient resolution of

displacements and the use of tangential stiffness. An iterative solution to Equation (3.2)

permits frequency and damping to be adjusted with respect to amplitude.

Certain adjustments were made to the method described in Chopra (2001) to

achieve the desired amplitude-dependence in frequency and damping. The modified

nonlinear Newmark method with iteration (MNMwIT) is provided in Table AA.2: the

left side assumes average acceleration and includes modifications for amplitude-

dependent dynamic properties; and the right side includes references to the steps

described in Chopra (2001). Changes in frequency and damping in Step 3.1 are

incorporated by updating system dynamic properties ( c , k , and a ) in Step 3.3 of Table

AA.2. Dynamic properties are updated based on peak accelerations and applied over the

subsequent time step.

19

Figure 3.2: (A) Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation.

This method will initially be verified for linear free vibration by comparing its

results with the theoretical free decay equation. The following constant dynamic

properties were chosen for the simulation to be consistent with the properties of the

Korean building analyzed in this research: 2078.0nf Hz and 0061.0 . A time step

of 0.1 sec was used in these simulations, after a more refined time step did not yield

significantly different results, perhaps owing to the energy balance at each step and the

relatively long period dynamics of the system. A comparison of time history generated

by the MNMwIT and theoretical free decay equation and their spectral representations

for a linear system with constant frequency and damping (C-F&D) are presented in

Figure 3.2. In both domains, the results of the MNMwIT are identical to the theoretical

result. Changes in natural frequency and damping in each cycle of oscillation are

extracted using the logarithmic decrement (LD) (Chopra, 2001), shown schematically in

Figure 3.3. Cyclic estimates of frequency and damping in Figure 3.4, referenced to the

(A)

(C) (B)

Am

plitu

de [i

n]

Am

plitu

de [i

n]

Spec

tral M

agni

tude

[in

2 /Hz]

Time [s]

Time [s] Frequency [Hz]

20

amplitude at the initiation of the cycle, compare well with the assumed dynamic

properties. The best-fit line to the observed frequency and damping produced intercepts

within 1% of the values used in the simulation as summarized in the first row of Table

3.1. (Throughout this chapter, dynamic properties will be described by a linear model as

a function of amplitude. Thus a constant parameter system will have an intercept and

zero slope.) Note that oscillations in predicted frequency are commonly observed with

applications of LD and are rectified by simulating with a reduced time step or averaging

over several cycles.

Figure 3.3: Schematic of System Identification using the Logarithmic Decrement.

TABLE 3.1: FREQUENCY AND DAMPING OBSERVED IN SIMULATED FREE DECAYS.

Simulation Type Slope Intercept Slope Intercept Slope Intercept Slope InterceptC-F&D 0.0000 0.2078 0.0000 0.0061 0.0000 0.2075 0.0000 0.0061AD-F -0.0034 0.2078 0.0000 0.0061 -0.0032 0.2075 0.0000 0.0061AD-D 0.0000 0.2078 0.0025 0.0061 0.0000 0.2075 0.0023 0.0061AD-F&D -0.0034 0.2078 0.0025 0.0061 -0.0031 0.2075 0.0023 0.0061

Input Dynamics Observed DynamicsDamping, ζFrequency [Hz] Frequency [Hz] Damping, ζ

21

Figure 3.4: Frequency and Damping Estimated from MNMwIT Linear Free Decay using the Logarithmic Decrement.

Next, the nonlinear simulation (NLS) capabilities will be verified by comparing

the simulation results for a nonlinear free decay with its theoretical linear counterpart.

The amplitude-dependent models for frequency and critical damping ratio, shown in

Equations (3.3) and (3.4), were selected to reflect trends previously observed in

buildings similar to the Korean building studied in this thesis.

2078.00034.0 Vfn (3.3)

0061.00025.0 V (3.4)

Note that the initial natural frequency and damping (y-intercepts) are the same as those

used in the linear simulation in Section 3.1.1. Frequency is assumed to soften due to

increased slippage or gap widening between components resulting in a loss of contact

surface and diminished stiffness. An increase in damping was assumed with amplitude

for the same reason; increased slippage between components dissipates more energy in

22

friction, though eventually with all potential friction surfaces mobilized, this capability

will plateau, as suggested by Jeary (1986).

Figure 3.5: (A) AD-F Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation.

In the first comparison shown in Figure 3.5, a free decay was generated by the

MNMwIT with amplitude-dependent frequency (AD-F) and constant damping. A

softening of frequency is expected and noticeable in Figure 3.5(B) from the longer

periods between peaks for the nonlinear system (the period will approach that of the

linear system as amplitude approaches 0 in). In addition, a frequency domain analysis of

the nonlinear system reveals an asymmetric peak skewed toward the lower frequencies

in Figure 3.5(C) indicative of nonlinearity. The results of a logarithmic decrement

analysis on the free decay are provided in Figure 3.6. As shown in Table 3.1, the

amplitude-dependence in the observed frequency is 6% less than expected.

(A)

(C) (B)

Am

plitu

de [i

n]

Am

plitu

de [i

n]

Spec

tral M

agni

tude

[in

2 /Hz]

Time [s]


23

Figure 3.6: Frequency and Damping Estimated from MNMwIT AD-F Free Decay using the Logarithmic Decrement.

Next, a nonlinear free decay was generated for a system with constant frequency

and amplitude-dependent damping (AD-D), shown in Figure 3.7. As expected, the

nonlinear system decayed faster in the initial cycles due to larger damping with

amplitude. The effect on phase is negligible due to the relatively minor role of damping

in shaping the damped natural frequency. A lower peak magnitude in the frequency

domain for the nonlinear system infers that response amplitudes were diminished due to

the added energy dissipation. Given similar bandwidths in the peaks of the theoretical

and NLS method, a lower peak magnitude in the NLS method would result in a greater

half-power bandwidth. Figure 3.8 presents the amplitude-dependence of frequency and

damping, identified using the logarithmic decrement, for the constant frequency and

amplitude-dependent damping system. As shown in Table 3.1, amplitude-dependence of

damping is observed to be 9% less than expected.

24

The last verification involves a comparison between the constant parameter free

decay and a fully nonlinear, amplitude-dependent frequency and damping (AD-F&D),

free decay. A comparison of the time histories is provided in Figure 3.9 with the results

of a logarithmic decrement analysis in Figure 3.10. As expected, the fully nonlinear

system is a combination of the independent analyses of amplitude-dependent frequency

and damping. In the time domain, the nonlinear system undergoes a greater decay

coupled with an increase in period. The peak in the frequency domain is again

asymmetric and skewed toward the lower frequencies and its magnitude reduced. As

shown by Table 3.1, the identified amplitude-dependence of the frequency and damping

are 9-10% less than the assumed relationships.

Figure 3.7: (A) AD-D Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation.

(A)

(C) (B)

Am

plitu

de [i

n]

Am

plitu

de [i

n]

Spec

tral M

agni

tude

[in

2 /Hz]

Time [s]


25

Figure 3.8: Frequency and Damping Estimated from MNMwIT AD-D Free Decay using the Logarithmic Decrement.

Figure 3.9: (A) AD-F&D Free Decay by the MNMwIT and Theoretical Free Decay Equation for a Linear System (fn = 0.2078 Hz & = 0.0061) with (B) Zoom and (C) Spectral Representation.

(A)

(C) (B)

Am

plitu

de [i

n]

Am

plitu

de [i

n]

Spec

tral M

agni

tude

[in

2 /Hz]

Time [s]


26

Figure 3.10: Frequency and Damping Estimated from MNMwIT AD-F&D Free Decay using the Logarithmic Decrement.

3.2 Tools and Methods

In this section, the techniques used for system identification in this research are

introduced. The first part is devoted to the spectral technique: power spectral density and

half-power bandwidth, while the second half is devoted to the time domain technique:

random decrement and analytic signal theory.

3.2.1 Power Spectral Density Method

The power spectral density (PSD) is perhaps the most fundamental representation

of pseudo-periodic data. While there are several methods to obtain the power spectrum:

the Fourier transform of the autocorrelation function, Filtering-Squaring-Averaging, and

via the Fast Fourier Transform (FFT) (Bendat and Piersol, 2000), the last method is by

far the most popular. Since the ensemble averaging necessary to generate the power

spectrum is not realistic in most practical applications, the assumptions of a stationary

27

and ergodic signal are generally invoked to allow the expectation operator to be replaced

with time averaging. The procedure required to generate the PSD from this FFT

averaging process is summarized below and visually depicted in Figure 3.11.

Figure 3.11: Schematic Representation of the Power Spectral Density Method (Kijewski-Correa, 2003).

Step 1: Break the time series into PN segments of length PT .

Step 2: Apply the FFT to each segment to generate a raw Fourier spectrum.

Step 3: Average the PN raw spectra using Equation (3.5) to obtain the PSD.

PN

hqh

PPqqq fX

TNfS

1

2)(1)(ˆ (3.5)

The reliability of PSDs generated by this method is assessed by bias and variance

errors. In general, only a fixed amount of data is available; therefore, determining PN

and PT in Step 1 must consider the tradeoff between these errors. Generally, bias is

minimized first since it is a systematic error having the tendency to increase the spectral

bandwidth and damping. Since this thesis focuses on the issues of frequency and

damping for such systems, the normalized bias error of a SDOF oscillator‘s power

28

spectrum at its natural frequency, nqqb fSe ˆ , will be used to quantify bias, as presented

in Equation (3.6) (Bendat and Piersol, 2000):

2

231ˆ

nnqqb f

ffSe

(3.6)

This introduces a critical paradox: the bias error to be minimized is a function of the

parameters to be ultimately identified. Therefore, some preliminary estimates of the

natural frequency and damping are required to determine the spectral bias. In general,

this bias error is limited to -2%. This level of bias error ensures that at least four spectral

lines fall within the spectral peak‘s half-power bandwidth. Bearing this in mind, the

required frequency resolution, dreqf ' , can be determined from Equation (3.7).

4

2'

ndreq

ff (3.7)

After selecting the required frequency resolution, the required number of FFT

points, dreqNFFT ' , is calculated based on the sampling frequency, sf , using Equation

(3.8), and rounded up to the nearest power of two (traditional FFT algorithms are more

efficiently employed when NFFT is a multiple of 2):

Sdreq ffNFFT /1' (3.8)

The required time length of the segments being formed in Step 1 is then given by

Equation (3.9):

SP fNFFTT / (3.9)

Finally, after selection of a suitable NFFT and corresponding PT , the discrete

frequencies at which the Fourier transform is calculated, Step 2, can be determined by

Equation (3.10):

P

q Txf (3.10)

)12/(,1,0 NFFTx

29

Since PT is strictly fixed by the normalized bias error minimization, PN is simply the

number of blocks of length PT that can be extracted from the data. The variance error of

the power spectrum is dictated by (Bendat and Piersol, 2000):

P

qqr NfSe 1ˆ (3.11)

Thus, the tradeoff between desired frequency resolution and available number of

averages of a PSD estimate becomes apparent: the need to maximize the number of

segments to reduce the noise in a PSD is in direct opposition to the need to resolve the

spectra as finely as possible to avoid overestimates of damping associated with a

broadened spectral peak. The use of the PSD in this research was conducted according to

the methods previously mentioned, with a normalized spectral bias of -2%.

3.2.2 Half-Power Bandwidth Method

The half-power bandwidth (HPBW) method is a frequency-domain system

identification technique generally applied to transfer/frequency response/mechanical

admittance functions (TF/FRF/MAF); however, in instances where the input to the

system is white noise, the HPBW method can be applied directly to the response or

output PSD, which is proportional to the squared TF/FRF/MAF. A white noise input

assumption is widely invoked in the analysis of ambient vibration and wind-induced

response, as these broadband spectra are generally constant across the resonant

bandwidth of lightly damped systems. ―Lightly damped‖ or underdamped systems are

those with damping ratios much less than critical 1.0 , insuring that the natural

and damped natural frequencies are approximately equal.

A schematic of the HPBW system identification process is provided in Figure

3.12. It initiates with the identification of the spectral ordinates associated with half the

peak amplitude of a given mode (so-called half-power points). The frequencies

30

associated with these half-power points are termed 1f and 2f and their difference

constitutes the half-power bandwidth itself (Equation (3.12)):

12 ffHPBW (3.12)

Figure 3.12: Schematic of Application of the Half Power Bandwidth Method Applied to a Power Spectrum.

The natural frequency and damping ratio are then identified from using the relationships

in Equations (3.13) and (3.14), assuming a symmetric spectral peak (Bendat and Piersol,

2000):

2/)( 12 fff n (3.13)

nfHPBW 2 (3.14)

3.2.3 Random Decrement Technique

The random decrement technique (RDT) is a popular time-domain system

identification tool that does not require explicit knowledge of the system input. The RDT

process yields a decay curve known as a random decrement signature ( D , RDS),

31

which is proportional to the autocorrelation function, VVR , as shown in Equation

(3.15) (Vandiver et al., 1982):

0

001 0

100

VV

VVN

tttttttt

R RRZZVZVV

ND

R

(3.15)

21212

12121 ,,1 2

dVdVVVpdfVVtVtVEttRV VVV (3.16)

This autocorrelation function is proportional to the free vibration response of a SDOF

(linear) system only when the input process is zero mean, stationary Gaussian, white

noise (Kareem and Gurley, 1996; Spanos and Zeldin, 1998). This approach is attractive

since it has been shown to be more resistant to mild nonstationarities than a direct

calculation of the autocorrelation function. The RDS is generated by averaging lengths

of data identified within a response time history, tV , as having specific starting points,

known as triggers. Each time the trigger condition, specified by an amplitude vZ

and/or slope vZ , is satisfied, a specified segment of data is captured, ttV . A wide

variety of trigger conditions have been proposed in the literature; Table 3.2 summarizes

some of the most commonly used triggers (Ibrahim, 2001).

TABLE 3.2: LIST OF PROPOSED TRIGGER CONDITIONS FOR THE RDT

(IBRAHIM, 2001).

The captured segments are then averaged together to obtain a single RDS with

initial conditions equal to the trigger conditions, as shown by Equation 3.15. A

conceptual representation of this process is provided in Figure 3.13 by recognizing that

Level crossing:

Positive point:

Zero up-crossing:

Local extrema:

00 ZVtt

highttlow ZVZ 0

0000 tttt VV

000 tthighttlow VZVZ

32

responses are characterized by a forced and homogenous component. Assuming the

forcing component to be a zero mean random process, progressive averaging will

eventually bring it to zero, leaving only the homogeneous component. Given the

required input conditions are met, dynamic properties (natural frequency and damping)

can be extracted from the free decay response using a variety of methods, including

analytic signal theory via the Hilbert transform or logarithmic decrement.

Figure 3.13: Simplified Schematic of the Random Decrement Technique (Kijewski-Correa, 2003).

The RDT has been shown to have considerable flexibility in a number of its

fundamental assumptions. For example, mild nonlinearities (amplitude-dependence) in

natural frequency and damping may be characterized using this method by varying the

amplitude of the trigger to generate a suite of RDSs and then identifying the dynamic

properties from the first few cycles of each RDS (Tamura and Suganuma, 1996). In fact,

33

identification within the first few cycles is essential to the RDT even in cases where the

system is assumed to be linear, since the variance in the RDS increases with each cycle

of oscillation, as shown in Equation (3.17) (Vandiver et al., 1982):

0101

2

222

VV

VVVV

R RRR

NDEDEDVar (3.17)

Another condition initially assumed by Vandiver et al. (1982) in their derivations

required the segments being captured to be uncorrelated. This mandated a sufficient

temporal separation between two captured segments so they would essentially be

independent. Strict enforcement of this condition reduces the number of segments RN

that can be generated from a fixed amount of data and increases variance. Since the latter

is actually of greater detriment to the RDT, Kijewski and Kareem (2000) showed that

allowing some correlation between captured segments does not significantly hinder

performance and will provide additional segments necessary for reducing variance in the

generated RDS.

The Random Decrement Technique has, however, shown some sensitivity to the

trigger amplitude selected. To eliminate this problem, Kijewski-Correa (2003) proposed

the use of local averaging on the trigger condition. This concept requires the user to

specify an array of trigger amplitudes within a few percent of the desired amplitude

trigger such that the average of this array of trigger amplitudes is the desired amplitude

trigger. The user then executes RDT using each of these triggers in the array and

identifies frequency and damping values from the resulting RDSs. The mean dynamic

properties are then reported, accompanied by their coefficients of variation (CoV). This

process is shown schematically in Figure 3.14.

The use of the RDT in this research, as summarized in Table 3.3, was conducted

according to the methods previously mentioned and consistent with those outlined by

Kijewski-Correa (2003) and Tamura and Suganuma (1996), with an RDS length of 10

34

cycles. A local averaging array of five amplitudes within +/-3% of the target amplitude

was also specified. Additional details of the procedure are provided in Appendix B.

Figure 3.14: Schematic of Local Averaging Technique.

TABLE 3.3: APPLICATION PROCEDURE OF THE RDT.

Step 1 Estimate the dominant frequency of the data.

Step 2 Filter the data to include only one mode.

Step 3 Verify that the input is stationary, Gaussian white noise.

Step 4 Select the length of the captured segments.

Step 5 Select trigger conditions.

Step 6 Select a correlation level.

Step 7 Identify potential starting points by trigger conditions.

Step 8 Identify segments by starting points and correlation level.

Step 9 Average the N R blocks to obtain a RDS.

Step 10 Perform system identification on the RDS.

Step 11 (OPTIONAL) Incorporate local averaging.

35

3.2.4 Analytic Signal Theory

Analytic signal theory has commonly been invoked to extract natural frequency

and damping values from the free decay responses of lightly damped, SDOF systems,

which take the form of:

teAtY Dt

on cos

(3.18)

In order to generate the complex-valued, analytic signal, a quadrature-shifted version of

the original signal must be created. This is conventionally achieved using the Hilbert

transform (HT) (Bendat and Piersol, 2000), a time-domain convolution that shifts the

signal π/2 out of phase. The Hilbert transform is applied to a free decay time history,

tY , in Equation (3.19).

1)()()]([ 1 tYdtYtYH (3.19)

The analytic signal, )(tz , is composed of a real component obtained by Equation (3.20),

and imaginary component calculated by Equation (3.21):

)()(Re tYtz (3.20)

tYHtz )(Im (3.21)

By comparing Equation (3.18) with Equations (3.20) and (3.21), it becomes evident that

Equations (3.22) and (3.23) may be obtained:

)(tzdtd

D (3.22)

|)(|ln tzdtd

n (3.23)

Since the damped natural angular frequency is approximately equal to the resonant

natural angular frequency ( Dn ) in lightly damped systems, natural frequency can

be directly obtained from the slope of the analytic signal‘s phase, and damping can be

identified from the slope of the natural log of the analytic signal‘s magnitude. A

36

schematic, provided in Figure 3.15, may assist in visualizing the procedure.

Furthermore, by using regression procedures to estimate the requisite slopes, reliability

of system identification is considerably improved over techniques reliant on only point-

estimates of decay envelope properties, e.g., logarithmic decrement.

Figure 3.15: Schematic of Analytic Signal Theory for System Identification.

3.3 Validations

In this section the system identification methods and tools are validated through

the use of theoretical formulations and simulations. The first parts of this section are

devoted to tests involving the theoretical frequency response function and equation for a

free decay response. Next, a linear simulation is used to assess the performance of both

the spectral and time domain methods introduced earlier. Finally, the methods are tested

against a nonlinear simulation, which is the main focus of this research.

3.3.1 Validation of HPBW in Isolation

In this experiment, the accuracy of the HPBW method is assessed through its

application to a suite of response power spectra, the product of the input and the square

of the system‘s FRF. If the input is assumed random white noise, the response power

37

spectrum is essentially proportional to the FRF squared. An example FRF is provided in

Figure 3.16, while its calculation is based on Equation (3.24):

nn

kH

/21/1

/1)( 2

(3.24)

Figure 3.16: Sample Frequency Response Function ( 2.0nf Hz & 005.0 ).

An overview of the procedure utilized in this test is now provided. First, response

PSDs are created based on the FRF formula for a range of dynamic properties: a constant

natural frequency of 0.2 Hz, but varying damping ratio: 0.5% to 2.5% in increments of

0.25% of critical with a normalized bias error of at most -2% to specify the discretization

of the frequency variable. Next, the HPBW method is applied to each response PSD to

estimate natural frequency and damping. Last, a percent difference is calculated from the

simulated and estimated dynamic properties.

The results of this analysis are provided in Figure 3.17. Low errors indicate that

the HPBW performs well when applied to the simulated response spectra. Frequencies

are generally underestimated within 0.2% across the damping levels simulated. Damping

38

Figure 3.17: Quality of HPBW Application to Theoretical Response Spectra.

errors are more variable but are less than 4% across the damping levels simulated. Note

that although damping is always overestimated this is expected due to spectral bias. A

slight reduction in error is observed for higher simulated damping levels, as one would

39

expect. Small jumps in damping error are also evident, which coincide with reductions in

the required segment length to maintain at most a -2% bias. Although the damping errors

increase slightly, the required length of data is cut in half. Thus, while an hour of data

was needed to generate a single response spectrum for 0.5% damping, only an eighth of

that length was required for a simulated damping level of 2.5%. In summary, the error

associated with the HPBW itself is on the order of 3% for damping and negligible for

frequency, provided frequency resolutions maintain a -2% bias in the power spectrum.

3.3.2 Validation of Analytic Signal Theory in Isolation

Before applying the analytic signal theory to Random Decrement Signatures, its

performance is first assessed using simulations. To perform this test, the method is

applied to a suite of noise-corrupted free decay responses (FDRs). A sample noise-

corrupted FDR is shown in Figure 3.18(B); it was created by superimposing random

white noise onto an FDR generated by Equation (3.18). Performance of the analytic

signal via Hilbert transform routine was assessed by evaluating the deviation between

the simulated and experimentally identified system properties. This information was then

used to determine the portions of the amplitude and phase plots that should be fit to

minimize frequency and damping errors.

As the analytic signal via Hilbert transform routine is implemented via Fast

Fourier transform, its performance degrades when signals are not periodic in the window

and end effects associated with Gibbs phenomena are often observed. Figure 3.18(C &

D) are sample graphs of the amplitude and phase of the Hilbert transformed analytic

signal for a noise-corrupted FDR. From these graphs, it is apparent that the amplitude is

more affected by Gibbs phenomena, as the first 5 seconds of the amplitude component

demonstrate a shallower slope than the rest of the signal. To reduce this effect, aperiodic

signals can be treated by extending the time series at its ends through reflective padding

(Kijewski and Kareem, 2002) or by characteristic waves (Huang et al., 1998). These

40

techniques can help to improve the integrity of Hilbert-transformed data near the

beginning of a Random Decrement Signature, where it most closely emulates the

autocorrelation function (Kijewski and Kareem, 2000).

Figure 3.18: Application of Analytic Signal via Hilbert Transform Routine to Noise-Corrupted FDR: (A) Sample of Random Noise; (B) Noise-Corrupted FDR; (C) Amplitude of Analytical Signal; (D) Phase of Analytic Signal.

(C)

(A)

(B)

(D)

41

To determine the extent of end effects, FDRs are simulated for systems with a

natural frequency of 0.2 Hz and varying damping ratios of 0.5, 1.5, and 2.5%. Each

simulation contains 10 cycles (10 sec) of decay with superimposed white noise yielding

a signal to noise ratio of approximately 10:1. Next, the analytic signal via Hilbert

transform routine was evaluated over a fixed interval: initiating at the 0.5, 1.0, and 1.5

cycle of oscillation. These points take into account the need to exclude the part of the

segment susceptible to end effects, while restraining the identification to the first few

cycles to ensure amplitude-dependent features are extracted (Tamura and Suganuma,

1996). The ending point of the fixed interval varied between cycle 2.5 and 4, with this

upper limit based on the recommendations in Kijewski and Kareem (2000), to offset the

increasing variance observed in RDSs. The analysis was repeated for 10,000 trials, with

independent additive white noise generated for each trial.

The observed mean error and coefficient of variation for both the estimated

frequency and damping ratio are provided in Figure 3.19. Several important trends can

be observed from the performance of the analytic signal via Hilbert transform routine in

its application to noise-corrupted FDRs. First, increasing the curve-fitting interval

decreases the mean error and CoV in both frequency and damping. Second, a curve fit

that begins at half the first cycle generally underestimates system parameters. Third,

errors in estimated frequencies appear random and do not group together by simulated

damping level. However, damping errors, in both mean and CoV, appear slightly greater

for low simulated damping levels, as one may expect. Thus, it was determined that

system parameters are more accurately and reliably estimated when curve-fitting the

analytic signal amplitude and phase over the first 0.5 to 3.5 cycles of oscillation. This

range provides for largely conservative estimates of frequency and damping and with

lower variance, while remaining close to the trigger condition to allow tracking of

amplitude-dependent features (Tamura and Suganuma, 1996). Damping errors are

42

expected be on the order of a few percent and conservative using this approach, while

frequency errors will be negligible.

Figure 3.19: Mean Error and CoV of Hilbert Transform Estimate of Natural Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise.

43

3.3.3 Validation of Logarithmic Decrement in Isolation

The previous simulation was repeated to validate the performance of the

logarithmic decrement on RDSs. In this test, more critical damping ratios were

investigated: 0.5% to 2.5% in increments of 0.25%. Again only the first four cycles were

investigated to avoid high variance levels as noted by Kijewski and Kareem (2000),

determining the error in the estimated parameters when 1, 2, 3 or all 4 cycles of

oscillation were included in the logarithmic decrement estimate. Several important

trends can be observed from the mean error plots provided in Figures 3.20(A) and

3.21(A). First, increasing the interval between peaks used in the logarithmic decrement

reduces the mean error of estimates of frequency and damping. As noted earlier,

frequency and damping estimates improve when a wider interval is used for

identification. Second, the mean error in estimating frequency using the logarithmic

decrement is not significantly affected by the simulated damping level of the system.

Third, the damping mean error decreases with higher simulated damping levels, again as

expected. Note that the frequency and damping values both converge to a non-zero mean

error, indicating a bias that overestimates frequency and underestimates damping.

The CoV plots in Figures 3.20(B) and 3.21(B) further support the findings that

system identification improves as the identification interval lengthens and that

identification is more reliable as damping increases. The results also verify that CoV is

insensitive to the number of cycles used in the system identification once the damping

level is sufficiently large. Based on the results of this investigation, an averaging interval

of three cycles is suggested for system parameter estimates by the logarithmic

decrement.

44

Figure 3.20: Quality of Logarithmic Decrement Estimate of Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise.

45

Figure 3.21: Variance in Logarithmic Decrement Estimate of Frequency and Damping as a Function of Segment Selection in a Simulated Decay Response with Noise.

46

Comparing the results in Figure 3.19 with those of Figures 3.20 and 3.21, using

the intervals selected earlier, one can deduce the following maximum mean error and

CoV for frequency: -0.03% and 0.2% by the analytic signal approach, and 0.4% and 1%

by the logarithmic decrement approach, respectively. Similarly for damping: -5% and

30% by the analytic signal approach, and -11% and 30% by the logarithmic decrement

approach. While these results indicate that the logarithmic decrement provides more

conservative estimates of damping based on its mean error, the analytic signal approach

provides several key advantages in terms of reliability. First, over the range of simulated

damping levels, the Hilbert transform was found to exhibit the least variability in

estimates of damping. Next, the range of bias by the analytic signal approach was

observed to be less variable over the observed damping range. Finally, frequency

estimates by the analytic signal approach were found to be less variable and 20 times

more accurate. This underscores the challenges point estimators like LD can face when

applied to noise-corrupted signal envelopes.

3.3.4 Validation of HPBW in Context

In this experiment, linear simulations are conducted to assess the error associated

with estimating a power spectral density and then using the HPBW to identify system

dynamic properties. An overview of the procedure is as follows: (1) generate random

noise input; (2) simulate system response to input according to methods in Section 3.1.1;

(3) obtain the PSD of the response with a normalized bias error of -2%; (4) extract

dynamic properties from response PSD using HPBW; (5) repeat steps 1-4 100 times and

perform statistical analyses of the results. For this analysis, the system was selected to

have constant natural frequency (0.2 Hz) and damping ratios from 0.5% to 2.5% in

increments of 0.25% of critical. In addition, the effects of limited data (variance error)

were investigated by evaluating PSDs generated with 10, 100, and 200 averages drawn

from the same time history.

47

The results of this test are presented in Figures 3.22 and 3.23. In general,

estimates of frequency are quite accurate and do not differ significantly with segment

averages. As expected the mean error in estimating frequency is low or within 0.1%

across the range of damping values, though it does increase modestly with increases in

damping ratio. Frequency does not vary with the number of segment averages because it

is more directly related to the resolution of the PSD, which is held constant in this test

through the normalized bias.

On the other hand, mean damping errors were within 10% and became less

biased and variable with 100 or more segment averages. Mean damping errors are

generally controlled by bias, once sufficient averages are obtained. When only 10

segments are used, damping tends to be underestimated and accompanied by a high

CoV, largely due to the jaggedness of the spectral peak. Increasing the number of

segment averages reduces that jaggedness producing smoother peaks that are then

characterized by the bias associated with fixed spectral resolution. This bias tends to

produce overestimates of damping, which as expected does not improve with the

addition of further averages. The CoV understandably decreases with increasing

averages, though again, showing little improvement beyond 100 averages. This then

implies that the number of averages need not be greater than 100 to achieve adequate

performance. Note that the errors in damping are greater in comparison to those

associated with the HPBW applied independent of the PSD generation process (see Fig.

3.17), indicating some additional error is introduced by the PSD generation process.

48

Figure 3.22: Quality of HPBW Estimate of Frequency in a Simulated Linear System.

49

Figure 3.23: Quality of HPBW Estimate of Damping in a Simulated Linear System.

50

3.3.5 Validation of Analytic Signal Theory in Context

In this experiment, the analytic signal via Hilbert transform routine is used in

conjunction with the Random Decrement Technique on simulated linear systems in order

to validate performance. Once again, only a narrow set of dynamic properties were

investigated: constant natural frequency of 0.2 Hz and critical damping ratios between

0.5% and 2.5% in increments of 1.0%. A single run consisted of obtaining the responses

of each system to the same random noise input, a sample of which is shown in Figure

3.24(A), and then using the RDT and analytic signal via Hilbert transform routine to

identify system parameters. Each simulation spanned 100 hr and each run included an

independent random noise input. This process was repeated 100 times to obtain a mean

error and CoV for the estimated parameters.

Several considerations are made regarding implementation of the RDT and

analytic signal via Hilbert transform routine in this linear analysis. First, a single positive

peak trigger was selected for this analysis. The performance of multiple peak trigger

levels for tracking amplitude-dependence has previously been investigated by Jeary

(1992) and Tamura and Suganuma (1996), however only a single peak trigger is

necessary in this analysis because it is a linear system. The number of segment averages

was set to 1000 yielding RDSs such as that provided in Figure 3.24(B). All other

parameters, including the criteria for local averaging and segment length remained

consistent with those mentioned previously in Section 3.2.3. The amount of overlap

permitted between consecutive segments captured in the RDT algorithm was iterated

from full to none in increments of 10%. The curve-fitting interval used with the Hilbert

transform routine was selected based on the recommendations in Section 3.3.2. Again, a

plot of the analytic signal amplitude and phase in Figure 3.24(C & D) reveal the

amplitude‘s greater sensitivity to end effects (in reference to Figure 3.18), as similarly

observed in the case of wavelet transforms (Kijewski and Kareem, 2003).

51

The results of the experiment are provided in Figure 3.25 and are consistent with

the findings of Kijewski-Correa (2003), in that performance is not significantly affected

by correlation levels. Estimates of frequency were on average within 0.05% of the

simulated value, with a CoV of less than 0.2%. Meanwhile, damping was estimated

within +/-4% on average. Recall previously from Figure 3.19 that these errors are

comparable to that observed for the analytic signal theory in isolation, indicating that the

RDT as implemented here does not introduce additional errors. The variability and bias

in damping estimates increases as the damping level decreases and as overlap increases,

likely due to the fact that segments are more likely correlated under these conditions.

In conclusion, segment correlation is advantageous because it generates more

segments for averaging, which tends to be the governing parameter in reducing variance.

Its impacts are largely insignificant here because system identification is performed on

the first few cycles of a RDS, so that the actual interval over which the useful

components of captured segments overlap is very small. Based on the findings of this

investigation, a level of 50% overlap is suggested.

52

Figure 3.24: Application of Analytic Signal Theory via the Hilbert Transform to a RDS Obtained from a Linear Simulation: (A) Sample of Linear Simulation Time History; (B) Random Decrement Signature obtained from Time History; (C) Amplitude of Analytic Signal; (D) Phase of Analytic Signal.

53

Figure 3.25: Bias and Variance in Analytic Signal Frequency and Damping Estimates from a RDS Obtained from a Linear Simulation, as a Function of Overlap.

54

3.3.6 Validation of LD in Context

The following analysis repeats the previous exercise involving the RDT in

Section 3.3.5, but replaces the analytic signal via Hilbert transform routine with the

logarithmic decrement to extract dynamic properties from each random decrement

signature. The logarithmic decrement was applied over a three-cycle interval consisting

of the first and fourth peak as recommended previously in Section 3.3.3.

The results of this analysis are provided in Figure 3.26. Frequency was estimated

with a mean error of less than 0.25% and remained relatively constant despite the level

of overlap. In addition, frequency CoV remained consistently below 0.25%. Damping

was overestimated by less than 20% on average for simulated damping levels of 1.5%

and 2.5%. The low simulated damping level of 0.5% had mean errors in damping

estimates just below 50%. Damping CoV at full correlation was 17% and experienced

reductions to 15% at nearly 40% correlation. The high bias in damping values by the LD

in comparison with the analytic signal theory results in the previous section confirms the

limitation of methods that rely on point estimates of amplitude to estimate damping.

Much of this sensitivity to noise in the RDS envelope is alleviated using the analytic

signal‘s best fit methodology.

Similar to the previous study involving the RDT applying analytic signal theory,

this experiment found that minimal frequency and damping errors and variance occur

with correlation levels below 50% in the RDT. With that correlation level, the mean

error in frequency and damping is expected to be below 0.25% and 50%, respectively. In

addition, the CoV among frequency and damping are expected to be below 0.25% and

17%, respectively.

55

Figure 3.26: Bias and Variance of Logarithmic Decrement Frequency and Damping Estimates from a RDS Obtained from a Linear Simulation, as a Function of Overlap.

56

3.3.7 Verification of Amplitude-Dependent System Identification Methods

Amplitude-dependent RDSs were generated for several linear and nonlinear

simulations and analyzed by time domain approaches to verify their ability to accurately

estimate the amplitude-dependent dynamic properties. This experiment consisted of

three parts: random noise input generation, response simulation by methods in Section

3.1.2, and system identification. Several simulations were performed with a mixture of

amplitude-dependent (AD) and constant dynamic property assumptions:

linear system – constant frequency and damping (C-F&D);

nonlinear system – AD frequency and constant damping (AD-F);

nonlinear system – constant frequency and AD damping (AD-D); and

nonlinear system – AD frequency and AD damping (AD-F&D).

A set of zero mean, Gaussian white noise records were generated to act as input for all

the simulations. After each response was generated, the simulations were individually

analyzed by time domain and spectral approaches. The time domain approach (TDA)

consisted of the RDT and system identification by two routines based on analytic signal

theory using Hilbert transform (TDA-HT) and logarithmic decrement (TDA-LD). The

spectral approach (SA) consisted of obtaining a PSD for each set of responses and then

extracting dynamic properties using the HPBW method.

3.3.7.1 Simulations

The simulations were based on constant and amplitude-dependent dynamic

parameters. Where amplitude-dependency was required, the relationships in Equations

(3.3) and (3.4) were used. Note that the zero-amplitude values (y-intercepts) are the

same as those used in the linear simulation. To obtain enough data for an amplitude-

dependent analysis in the time domain, 200 records of 1-hour length were simulated.

Figure 3.27 illustrates the varying responses of each simulation to the same white noise

input.

57

Figure 3.27: Linear and Non-Linear Simulations to the Same Random Noise Input using the NLS Method Described in Section 3.1.2.

As expected, an amplitude-dependent damping diminishes the response of the

system considerably. To further test the effectiveness of the TDA, an additional

nonlinear system (AD-F&D-II) with amplitude-dependent frequency and damping

relationships as given in Equations (3.25) and (3.26) was also simulated.

1438.00019.0 Vf n (3.25)

58

0061.00035.0 V (3.26)

3.3.7.2 Analysis

Several considerations are made regarding implementation of the RDT in these

validations. First, an array of equally-spaced positive peak triggers was used in each

RDT analysis to obtain amplitude-dependent trends as demonstrated previously by Jeary

(1992) and Tamura and Suganuma (1996). Local averaging and segment length

remained consistent with those mentioned previously in Section 3.2.3. A 50%

correlation level was implemented per Sections 3.3.5 and 3.3.6. Only results meeting the

following criteria were considered: RDS of sufficient averages 1000RN and a local

averaging CoV less than 20%. Finally, settings for the analytic signal routine and the

logarithmic decrement were selected per Section 3.3.2 and 3.3.3, respectively.

In the spectral analysis, each simulation was analyzed to obtain a single PSD

( 400PN , typically). A ceiling on normalized bias was set at -2% as recommended in

Section 3.2.1. The inclusion of spectral analysis in this section is essential to

understanding the effect that amplitude-dependency has on estimates of dynamic

properties in the frequency domain.

3.3.7.3 Results

The results of the time and frequency domain approaches for estimating

frequency are provided in Figure 3.28. System identification by the Hilbert transform

and logarithmic decrement performed very similarly. A summary of the TDA-HT and

TDA-LD results are provided in Tables 3.4 and 3.5, respectively, containing slopes and

intercepts of a best-fit line to the estimated frequency and damping as a function of

trigger amplitude. These are then compared to the actual expression describing the

simulated frequency and damping. A best-fit line of observed frequency yielded errors in

slopes ranging from 0% to -22% for TDA-HT and 0% to -12% for TDA-LD. Errors in

intercepts were in the range of 1% to -2% for both TDA-HT and TDA-LD.

59

TABLE 3.4: COMPARISON OF SIMULATED AND OBSERVED AMPLITUDE-

DEPENDENT RELATIONSHIPS BY ANALYTIC SIGNAL THEORY.

TABLE 3.5: COMPARISON OF SIMULATED AND OBSERVED AMPLITUDE-

DEPENDENT RELATIONSHIPS BY LOGARITHMIC DECREMENT.

Source slope intercept slope intercept

C-F&D Simulated1 0.0000 0.2078 0.0000 0.0061

TDA-HT 0.0000 0.2078 -0.0001 0.0058

Difference2 0.0% 0.0% N/A -4.9%

AD-F Simulated1 -0.0034 0.2078 0.0000 0.0061

TDA-HT -0.0032 0.2048 0.0006 0.0062

Difference2 -5.9% -1.4% N/A 1.6%

AD-D Simulated1 0.0000 0.2078 0.0025 0.0061

TDA-HT 0.0000 0.2081 0.0018 0.0083

Difference2 0.0% 0.1% -28.0% 36.1%

AD-F&D Simulated1 -0.0034 0.2078 0.0025 0.0061

TDA-HT -0.0030 0.2041 0.0025 0.0102

Difference2 -11.8% -1.8% 0.0% 67.2%

AD-F&D-II Simulated1 -0.0019 0.1438 0.0035 0.0061

TDA-HT -0.0015 0.1412 0.0028 0.0116

Difference2 -21.1% -1.8% -20.0% 90.2%

Notes: 1) Based on simulated frequency and damping relationship

2) Percent difference of "Estimate" compared to "Expected"

System Type

Natural Frequency [Hz] Damping, ζ [% ]

Source slope intercept slope intercept

C-F&D Simulated1 0.0000 0.2078 0.0000 0.0061

TDA-LD 0.0000 0.2083 -0.0001 0.0053

Difference2 0.0% 0.2% N/A -13.1%

AD-F Simulated1 -0.0034 0.2078 0.0000 0.0061

TDA-LD -0.0032 0.2051 0.0006 0.0054

Difference2 -5.9% -1.3% N/A -11.5%

AD-D Simulated1 0.0000 0.2078 0.0025 0.0061

TDA-LD 0.0000 0.2084 0.0016 0.0073

Difference2 0.0% 0.3% -36.0% 19.7%

AD-F&D Simulated1 -0.0034 0.2078 0.0025 0.0061

TDA-LD -0.0030 0.2046 0.0024 0.0088

Difference2 -11.8% -1.5% -4.0% 44.3%

AD-F&D-II Simulated1 -0.0019 0.1438 0.0035 0.0061

TDA-LD -0.0017 0.1419 0.0022 0.0141

Difference2 -10.5% -1.3% -37.1% 131.1%

Notes: 1) Based on simulated frequency and damping relationship

2) Percent difference of "Estimate" compared to "Expected"

System Type

Damping, ζ [% ]Natural Frequency [Hz]

60

Figure 3.28: Comparison of Frequency Estimates using Time and Frequency Domain Approaches.

Underestimates in frequency would require an overestimate in damping to

compensate, given the relationship in Equation (3.23). This can be observed in the

estimates of damping by the time and frequency domain approaches, provided in Figure

3.29. Again, both the analytic signal theory using Hilbert transform and logarithmic

decrement performed similarly. Both methods were able to estimate the degree of

amplitude-dependence (slope) of damping for most cases, as summarized in Tables 3.4

and 3.5. In the constant frequency and damping system, C-F&D, the TDA-HT estimated

61

damping to be approximately constant and 5% lower than the assumed ratio of 0.61%.

The TDA-LD similarly identified the system as having constant properties, but

underestimated damping by 14%. Next, the amplitude-dependent frequency and constant

damping system was predicted to contain an amplitude-dependent damping term,

although the slope of damping was simulated to be zero. Both TDA-HT and TDA-LD

under-predicted the slope and overestimated the intercept of damping with the constant

frequency and amplitude-dependent damping system, AD-D. The last simulation is for a

fully nonlinear system; AD F&D. Estimates of slope for this simulation were more

accurately obtained by TDA-HT, whereas the intercept was more accurately obtained by

TDA-LD.

Thus, although both time domain results for the AD-F&D simulation estimated

the assumed slope of damping fairly accurately, the results possessed a considerable bias

with respect to the initial damping (intercept) that is in part due to the inherent errors in

the frequency estimation. Interestingly, the performance of the two methods is nearly

identical for all but the AD-F&D II case, which features a higher initial frequency but

more amplitude-dependence in damping. In fact, it appears the bias in the damping

estimates is more substantial for this lower frequency system, suggesting that natural

frequency has an influence on the accuracy of damping estimation. The degree of

amplitude-dependence in damping was underestimated by 20% by the TDA-HT and by

almost 40% by the TDA-LD in AD-F&D-II.

62

Figure 3.29: Comparison of Damping Estimates using Time and Frequency Domain Approaches.

However, to demonstrate the challenges in general with estimating amplitude-

dependent dynamic properties, spectral results are provided in Tables 3.6 and 3.7 and

compared to the known simulated dynamic properties and RMS statistics of the time

domain approaches (since they are amplitude-dependent). First off, since the spectral

analysis cannot even track amplitude-dependence explicitly, its results are always in

error as they depict a nonlinear system inherently as one that is linear. However, one

may still assume that a spectral analysis could still accurately capture the properties in a

63

mean sense. This is not the case. As amplitude-dependence in frequency is introduced,

the errors in frequency estimates by the spectral approach increased to over -8%. Thus

for even a relatively simple parameter to estimate (frequency), spectral analysis is

wholly ill-equipped, even in a root-mean sense, to accurately estimate its properties. The

time domain methods performed appreciably better.

For the constant frequency and damping system, C- F&D, estimates of damping

by HPBW were slightly greater, by 17%, than the simulated value (Table 3.7), likely due

to inherent spectral bias. However, damping estimated for the AD-F&D system differed

by more than 200% from the expected RMS value for the SA, whereas damping

estimated by the TDA-HT was in error by only 40%. In general, the error in the SA

results for damping ranged from 17% to over 1000%. Note that the errors were greater

for systems with amplitude-dependent frequency as opposed to those with amplitude-

dependent damping. Amplitude-dependence in frequency is essentially interpreted as

multiple frequency components by the stationary spectral analysis, producing larger

spectral bandwidths, as will be discussed later in this chapter. These findings help

reinforce our understanding of the limitations of spectral analyses and the danger of their

application to systems with amplitude-dependent frequencies: they lead to a perception

of damping that is much higher than the inherent damping in the system due to the

blurring of the spectral peak by variations in the natural frequency. Thus it is clear that

an analysis that preserves time-domain information is necessary to capture amplitude-

dependence in any reasonable way, making the time domain methods, although not

errorless, one of the only viable options for the problem at hand.

64

TABLE 3.6: COMPARISON OF FREQUENCY ESTIMATES FROM TIME AND

FREQUENCY DOMAIN APPROACHES.

TABLE 3.7: COMPARISON OF DAMPING ESTIMATES FROM TIME AND

FREQUENCY DOMAIN APPROACHES.

3.3.7.4 Discussion

In summary, it appears that time domain analyses are more effective, particularly

in a root mean sense, in extracting amplitude-dependent relationships from a database of

time history records. As expected, frequency was more accurately estimated than

damping. However, a systemic bias was noted in the results, affecting most significantly

the intercepts. Damping in particular faces a dual challenge, being itself difficult to

identify and by virtue of the expression describing the decay envelope and the half-

power bandwidth, will necessarily reflect any errors in the frequency estimate as well as

its own. Admittedly some of the errors in the amplitude-dependent analyses presented in

this chapter may also be a result of how it was originally simulated, which would come

Expected1

RMS1, fn [Hz] Full DB, fn [Hz] Difference2 RMS1, fn [Hz] Difference3 RMS1, fn [Hz] Difference3

0.2078 0.2076 -0.1% 0.2078 0.0% 0.2083 0.2%

0.1982 0.1817 -8.3% 0.1958 -1.2% 0.1961 -1.1%

0.2078 0.2075 -0.1% 0.2081 0.1% 0.2084 0.3%

0.2019 0.1925 -4.7% 0.1989 -1.5% 0.1994 -1.2%

0.1411 0.1366 -3.2% 0.1391 -1.4% 0.1395 -1.1%

Notes: 1) Based on estimated/predicted AD relationship using average RMS value of simulated time history

2) Percent difference of fn of "Spectral Approach" compared to fn of "Theory"

3) Percent difference of fn of "TDA-X" compared to fn of "Theory"

TDA-LD

AD-F&D-II

TDA-HTSpectral Approach

System Type

C-F&D

AD-F&D

AD-F

AD-D

Expected1

RMS1, ζ [%] Full DB, ζ [%] Difference2 RMS1, ζ [%] Difference3 RMS1, ζ [%] Difference3

0.61 0.71 16.9% 0.55 -9.6% 0.50 -17.8%

0.61 7.33 1102.0% 0.79 29.3% 0.71 16.2%

1.04 1.85 77.9% 1.14 9.6% 1.01 -3.3%

1.04 3.40 226.9% 1.45 39.4% 1.29 24.3%

1.10 2.66 141.0% 1.55 40.9% 1.72 55.9%

Notes: 1) Based on estimated/predicted AD relationship using average RMS value of simulated time history

2) Percent difference of ζ of "Spectral Approach" compared to ζ of "Theory"

3) Percent difference of ζ of "TDA-X" compared to ζ of "Theory"

System Type

C-F&D

AD-F&D

AD-F

TDA-LDTDA-HT

AD-F&D-II

AD-D

Spectral Approach

65

to light only when driven by random input. Still the performance of the time domain

approaches are encouraging, considering the relative minor role that damping plays in

overall response.

3.3.7.5 Additional Analysis: Effect of AD Frequency on Damping

The significant difference the between time and frequency domain results was

further investigated by looking at the effect of amplitude-dependent frequency on

damping. The two simulations utilized in this analysis were the linear (C-F&D) and fully

nonlinear (AD-F&D) simulations. A sorted spectral approach (SSA) was utilized; each

simulation was ranked by energy level, sorted into eight groups of 25 hours each, and

each of those groups analyzed separately by a power spectral analysis. The energy level

was calculated from the RMS of the particular record. In theory, sorting by energy level

will partially isolate the amplitude-dependency because both are related to peak

amplitude. Group 1 contained the highest energy records, while Group 8 contained the

lowest energy records. In addition, a spectral analysis of the full database was created to

provide a baseline for comparison.

The results of the SSA for both simulations are provided in Figures 3.30 and

3.31. The effect of amplitude-dependent frequency is very apparent when comparing the

SSA results for both simulations. First, multiple peaks are present in the nonlinear

system. These multiple peaks indicate that bandwidth inflation due to the presence of

multiple frequencies of oscillation. The bandwidth inflation in Group 1 of the nonlinear

simulation is such that it even appears to have a coupled mode. Note the progressive

softening of frequency as the peaks downshift in each group because of the amplitude-

dependence. Table 3.8 provides the SSA results in tabular form for the linear system.

The estimated frequency and damping do not vary greatly between groups for the linear

system; this is expected, given there is no amplitude-dependence, and thus the errors are

part of the normal variance in the process. However, the results of the sorted spectral

66

approach for the nonlinear system, provided in Table 3.9, present significant variability

which may result from the amplitude-dependence of frequency. Errors in the frequency

estimates are more pronounced in the system with amplitude dependent frequency.

Even when grouping by amplitude, the presence of nonlinearity leads to an

asymmetry of the spectral peak that widens toward the low frequency range. Thus even

when performing a sorted spectral analysis, the influence of amplitude-dependent

frequency cannot be mitigated, except for the lower energy groupings, which

understandably contain less amplitude-dependent behaviors and display spectral peaks

similar to those of the linear system.

67

Figure 3.30: Sorted and Full (inset) Power Spectra for a Simulated Linear System.

TABLE 3.8: SORTED SPECTRAL APPROACH RESULTS FOR A SIMULATED

LINEAR SYSTEM.

Simulated1 Observed Difference [%]2 Simulated1 Observed Difference [%]3

GROUP 1 5.09 0.2078 0.2077 -0.03 0.62 0.64 3.01

GROUP 2 4.37 0.2078 0.2075 -0.13 0.62 0.79 21.50

GROUP 3 3.88 0.2078 0.2075 -0.15 0.62 0.75 17.71

GROUP 4 3.32 0.2078 0.2076 -0.08 0.62 0.71 12.27

GROUP 5 2.63 0.2078 0.2076 -0.09 0.62 0.66 5.36

GROUP 6 1.85 0.2078 0.2075 -0.13 0.62 0.56 -10.56

GROUP 7 1.10 0.2078 0.2075 -0.13 0.62 0.70 11.85

GROUP 8 0.56 0.2078 0.2074 -0.19 0.62 0.68 9.01

Notes: 1) Based on a peak factor of 3.5

2) Percent difference of observed to simulated frequency

3) Percent difference of observed to simulated damping

RMS [milli-g]Frequency, fn [Hz] Damping, ζ [% ]

Spec

tral M

agni

tude

[(in

/s2 )2 /H

z]

Frequency [Hz]

68

Figure 3.31: Sorted and Full (inset) Power Spectra for a Simulated Non-Linear System.

TABLE 3.9: SORTED SPECTRAL APPROACH RESULTS FOR A SIMULATED

NON-LINEAR SYSTEM.

Simulated1 Observed Difference [%]2 Simulated1 Observed Difference [%]3

GROUP 1 2.76 0.1750 0.1887 7.3 3.0 3.4 10.4

GROUP 2 2.50 0.1781 0.1906 6.6 2.8 2.5 -13.8

GROUP 3 2.30 0.1804 0.1918 5.9 2.6 2.3 -12.7

GROUP 4 1.99 0.1841 0.1937 5.0 2.4 2.7 11.8

GROUP 5 1.68 0.1878 0.1968 4.5 2.1 1.3 -61.5

GROUP 6 1.28 0.1926 0.1992 3.3 1.7 1.8 5.9

GROUP 7 0.85 0.1977 0.2020 2.2 1.4 1.2 -17.8

GROUP 8 0.44 0.2026 0.2037 0.6 1.0 1.1 7.9

Notes: 1) Based on a peak factor of 3.5

2) Percent difference of observed to simulated frequency

3) Percent difference of observed to simulated damping

RMS [milli-g]Frequency, fn [Hz] Damping, ζ [% ]

Spec

tral M

agni

tude

[(in

/s2 )2 /H

z]

Frequency [Hz]

69

3.3.7.6 Gross Damping

Although it is understood that spectral bias will lead to overestimates of

damping, considering the resolution used in the spectral analysis, bias alone is not

sufficient to explain some of the errors observed. Instead, it is likely traced to the

amplitude-dependence of the natural frequencies. A simple demonstration of this

concept begins with the HPBW definition (Equation 3.14). As discussed in Kijewski-

Correa and Kareem (2006), harmonic-type analyses will produce spectral representations

that peak at the mean frequency representative of the average frequency of oscillation,

while any variation in the frequency of oscillation will be carried in the system‘s

bandwidth, thus affecting the damping value. If a system‘s frequency varies between two

limiting values ( Af to Bf ), then the effective HPBW about each of these limiting

frequencies will be: AAA f 2 and BBB f 2 , respectively, as shown in Figure

3.32. Since the Fourier Transform has no ability to detect frequency variations in time, it

will present this phenomenon as a widened spectral peak encompassing both limiting

frequency values and their respective bandwidths, centered at a gross (mean) frequency:

2/BAgross fff . Thus, the total bandwidth of this combined system is given by

Equation (3.27):

BBBAAAB

BA

Agross ffffff

22

BBAA ff 11 (3.27)

The unsuspecting analyst will extract an upper bound damping value given by Equation

(3.28) for a variable frequency and variable damping system.

)(11

BA

BBAAgross ff

ff

(3.28)

70

Figure 3.32: Schematic of Fourier Representation of System with Varying Frequencies.

Equation 3.28 may be simplified to approximate gross damping for other system types:

variable frequency and constant damping ( BA , Equation 3.29), and constant

frequency and variable damping ( BA ff , Equation 3.30):

)( BA

BABAgross ff

ffff

(3.29)

2

)( BAgross

(3.30)

Table 3.10 provides a comparison of gross damping levels for each system type

with those estimated by the power spectral analysis. The Equations 3.28-3.30 are

evaluated using frequency and damping levels projected from the frequency and

damping best fit lines obtained from the time domain analyses, as summarized in Table

3.11. The amplitude range evaluated was from 0 to the RMS of the accelerations in the

simulation. The gross damping levels in Table 3.10 are between 10% and 60% of those

predicted by the SA and become increasingly more accurate with the degree of

amplitude-dependence. As higher energy levels may influence bandwidth to a higher

degree (Figure 3.31), differences between gross damping and the SA may result from the

simplification that the range of amplitude dependent dynamic properties would only be

derived from responses up to the RMS value. Still, these results underscore the

importance of time-domain techniques in identifying system parameters whenever

71

amplitude-dependence in frequency, damping or both is suspected, as Fourier analyses

are unable to handle variations in dynamic properties with time.

TABLE 3.10:

COMPARISON OF DAMPING FROM SPECTRAL APPROACH AND GROSS DAMPING.

3.4 Summary

This chapter served to introduce and validate the amplitude-dependent system

identification methods used in this research. These tools were initially validated with

idealized response artifacts: frequency response functions and free vibration decays.

Next, a linear and nonlinear simulation method was introduced to assess the

effectiveness of these tools in identifying amplitude-dependent dynamic properties and

found the performance of the analytic signal theory system identification applied to

random decrement signatures to be satisfactory for tracking amplitude-dependent

features. In the following chapter, the system identification methods introduced in this

chapter are used to investigate the amplitude-dependence of the dynamic properties of

the Chicago Full-Scale Monitoring Program buildings introduced in Chapter 2.

Spectral Approach

Full DB, ζ [%] ζGROSS [%] Difference1 ζGROSS [%] Difference1

7.33 2.95 -60% 2.87 -61%

1.85 0.98 -47% 0.87 -53%

3.40 2.52 -26% 2.36 -31%

2.66 2.11 -21% 2.41 -9%

Notes: 1) Percent difference of ζGROSS compared to ζ of "Spectral Approach"

AD-D

AD-F&D-II

via TDA-HT via TDA-LD

AD-F

AD-F&D

System Type

72

TABLE 3.11: EVALUATION OF GROSS DAMPING USING TIME DOMAIN

ANALYSIS RESULTS.

fnB fnA Difference2 ζB ζA Difference2 ζGROSS [%] Difference4

0.1958 0.2048 -4.4% 0.70 0.70 0.0% 2.95 319%

0.2081 0.2081 0.0% 1.14 0.83 37.3% 0.98 19%

0.1989 0.2041 -2.5% 1.45 1.02 42.3% 2.52 147%

0.1391 0.1412 -1.5% 1.55 1.16 34.0% 2.11 82%

Gross Damping

System Type

AD-F

AD-D

AD-F&D-II

Damping [%]3

TDA-HT1

AD-F&D

Natural Frequency [Hz]

fnB fnA Difference2 ζB ζA Difference2 ζGROSS [%] Difference4

0.1961 0.2051 -4.4% 0.62 0.62 0.0% 2.87 359%

0.2084 0.2084 0.0% 1.01 0.73 37.7% 0.87 19%

0.1994 0.2046 -2.5% 1.29 0.88 47.0% 2.36 169%

0.1395 0.1419 -1.7% 1.72 1.41 22.0% 2.41 71%

Notes:

1) Frequency range obtained from AD relationships using mean + σ (or 0 to RMS)

2) Percent difference calculated as follows: (fnB - fnA) / fnA or (ζB - ζA) / ζA

3) Damping range obtained from AD relationships using mean + σ (RMS)

4) Percent difference of ζGROSS compared to RMS ζ

AD-F&D-II

System Type

Natural Frequency [Hz]Gross Damping

Damping [%]3

AD-F

AD-D

TDA-LD1

AD-F&D

73

CHAPTER 4:

AMPLITUDE-DEPENDENT DYNAMIC PROPERTIES:

APPLICATION TO FULL-SCALE DATA

4.0 Introduction

This chapter will focus on the second thesis objective: the application of the

system identification framework introduced in Chapter 3 to full-scale data from the

buildings described in Chapter 2. First, dynamic properties were estimated using a

spectral approach (introduced in Section 3.2.1) via the half-power bandwidth (introduced

in Section 3.2.2). Next, a sorted spectral approach (introduced in Section 3.3.7.5) was

utilized to investigate the frequency content over a range of energy levels. Finally, a time

domain approach consisting of the RDT (introduced in Section 3.2.3) and the analytic

signal theory (introduced in Section 3.2.4) was used to extract amplitude-dependent

dynamic properties from each of the buildings.

4.1 Description of Selected Data

Spectral system identification techniques require significant amounts of data, as

previously demonstrated in Section 3.3.1 for the HPBW method. These requirements are

substantially increased when amplitude-dependent trends are desired. All analyses

conducted herein incorporate stationary response data identified through a two-part

process. Candidate records were first identified with a specified wind speed and

direction range, and then their stationarity was formally validated using the reverse

74

arrangements test at a 5% level of significance (Bendat and Piersol, 2000). For the

Korean Tower, 128 hours of stationary response collected between October 2006 and

April 2008 were identified resulting from winds primarily out of the west with mean

speeds between 5 and 10 m/s. For Building 1, 24 hours of stationary response collected

between January and December 2003 were identified resulting from winds primarily out

of the SW with mean speeds between 8 and 13 m/s. For Building 2, 12 hours of

stationary response collected between December 2002 and January 2004 were identified

resulting from winds primarily out of the WSW with mean speeds between 10 and 15

m/s. Finally, for Building 3, 56 hours of stationary response collected between January

and December 2004 were identified resulting from winds primarily out of the SW with

mean wind speeds between 7 and 12 m/s.

4.2 Spectral Approach

Dynamic parameters were estimated using a spectral approach that consisted of

applying the half-power bandwidth technique to each of the fundamental modes of the

building response, i.e., lateral sways along the X and Y axis and torsion. The results for

the torsional modes of the three buildings in Chicago will not be considered due to their

comparatively low amplitudes, making the extraction of damping particularly

challenging. To ensure a normalized bias of -2% NFFT was calculated per Section

3.2.1 using the designer‘s estimates of frequency and damping. A summary of the

calculations for NFFT and the resulting frequency resolution for each building are

provided in Table 4.1.

4.2.1 Results and Discussion

The results of the spectral analyses using the selected frequency resolution

calculated in Table 4.1 are provided in Table 4.2. In addition, the normalized bias and

number of spectra included in the average (indicator of variance error) are provided in

Table 4.3. In no case did the normalized bias exceed the desired limit. These in-situ

75

dynamic properties are compared to design predictions from Abdelrazaq et al. (2005), in

the case of the Korean Tower, and from Kijewski-Correa et al. (2006) for the Chicago

buildings. A more detailed assessment of each building now follows.

TABLE 4.1:

CALCULATED AND SELECTED SPECTRAL FREQUENCY RESOLUTIONS.

4.2.1.1 Korean Tower

The spectra for the Korean Tower are provided in Figure 4.1 by location and

superimposed on the building floor plan in Figure 4.2. Based on the repeatability of the

modes at each measurement location, it was determined that the first mode at each

location was the fundamental translational mode in that direction. The average first

mode frequency along the X-axis (averaging over all three measurement locations) was

observed to be 0.197 Hz with a CoV of approximately 0.01%. Similarly, the average

fundamental frequency along the Y-axis was observed to be 0.206 Hz with a CoV of

about 0.01% over the three measurement locations. The second mode was consistently

observed at all but one of the locations in both the X and Y directions, suggesting it is

the fundamental torsional mode. Again averaging over all the locations, this mode‘s

frequency was observed to be 0.239 Hz with a CoV of approximately 0.02%. A torsional

Δt [s] f n [Hz] ζ T p [s] NFFT Power of 2 Power of 2 NFFT T p [min] Δf [Hz]

Mode 1 0.1 0.147 0.015 907 9067 13.1 14 16384 27.31 0.00061

Mode 2 0.1 0.152 0.015 880 8800 13.1 14 16384 27.31 0.00061

Mode 3 0.1 0.182 0.015 733 7333 12.8 13 8192 13.65 0.00122

Mode 1 0.12 0.143 0.01 1400 11667 13.5 14 16384 32.77 0.00051

Mode 2 0.12 0.204 0.01 980 8167 13.0 13 8192 16.38 0.00102

Mode 3 - - - - - - - - - -

Mode 1 0.12 0.149 0.01 1340 11167 13.4 14 16384 32.77 0.00051

Mode 2 0.12 0.156 0.01 1280 10667 13.4 14 16384 32.77 0.00051

Mode 3 - - - - - - - - - -

Mode 1 0.12 0.130 0.01 1540 12833 13.6 14 16384 32.77 0.00051

Mode 2 0.12 0.132 0.01 1520 12667 13.6 14 16384 32.77 0.00051

Mode 3 - - - - - - - - - -

RequiredDesign Predictions

Chi

cago

B

uild

ing

3C

hica

go

Bui

ldin

g 2

Chi

cago

B

uild

ing

1K

orea

n T

ower

Selected

76

mode is absent in one of the spectra shown in Figures 4.1 and 4.2 because of its

orientation with respect to the building‘s center of rotation.

In serviceability design, composite structures are generally assumed to have a

critical damping ratio around 1.5%; therefore, the observed critical damping ratios in

Table 4.2 are consistent with that design assumption. These results may even suggest

that the amount of damping during the 10-year wind event could be even larger, owing

to potential amplitude-dependence and the fact that the data analyzed herein is

essentially ambient vibration. In fact, this potential amplitude-dependence may account

for the variability in the monthly spectral analyses provided in Table 4.4. Certainly, in

instances where limited spectral averages are available, estimated damping and

frequency should be questioned, still noteworthy variability can be observed even when

the number of spectral averages is sufficient.

77

TABLE 4.2: COMPARISON OF DESIGN PREDICTIONS (KIJEWSKI-CORREA ET AL., 2006) AND SPECTRAL APPROACH ESTIMATES OF IN-SITU FREQUENCY

AND DAMPING RATIOS.

f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%]

Characteristics

Design Values 0.147 1.5 0.152 1.5 0.182 1.5

(Mode 1/Mode #) (1.00) (0.97) (0.81)

In-Situ SA 0.197 1.25 0.206 1.11 0.239 1.50

<CoV> <0.01%> <1.88%> <0.01%> <1.35%> <0.02%> <0.91%>

(Mode 1/Mode #) (1.00) (0.96) (0.82)

Difference 33.8% -16.6% 35.7% -25.7% 31.4% -0.2%

Characteristics

Design Values 0.143 1.0 0.204 1.0 - -

(Mode 1/Mode #) (1.00) (0.70) -

In-Situ SA 0.143 1.29 0.206 1.07 - -

(Mode 1/Mode #) (1.00) (0.69) -

Difference 0.0% 29.1% 0.9% 7.3% - -

Characteristics

Design Values 0.149 1.0 0.156 1.0 - -

(Mode 1/Mode #) (1.00) (0.96) -

In-Situ SA 0.182 1.35 0.183 2.35 - -

(Mode 1/Mode #) (1.00) (1.00) -

Difference 22.1% 34.5% 17.1% 134.5% - -

Characteristics

Design Values 0.130 1.0 0.132 1.0 - -

(Mode 1/Mode #) (1.00) (0.99) -

In-Situ SA 0.118 2.02 0.120 1.68 - -

(Mode 1/Mode #) (1.00) (0.99) -

Difference -9.0% 102.3% -8.9% 67.8% - -

Chi

cago

Bui

ldin

g 3

Chi

cago

Bui

ldin

g 2

Chi

cago

Bui

ldin

g 1

Kor

ean

Tow

erMode 2

Full coupled Y-sway

Mode 3

X-sway, slight Y-sway and torsion

Y-sway, slight X-sway and torsion

Torsion, slight X- and Y- sway

Mode 1

-

-Y-sway X-Sway

X-sway, slight torsion Y-sway, slight torsion -

Fully coupled X-sway

78

Figure 4.1: Power Spectral Density Matrix for Korean Tower (rows = instrument locations, columns = primary lateral directions).

79

Figure 4.2: Floor Plan of Korean Tower at 64F with Power Spectra at each Location.

TABLE 4.3: INDICATORS OF VARIANCE AND BIAS IN POWER SPECTRA.

Mode 1 Mode 2 Mode 3

Characteristics X-sway, slight Y-sway and torsion

Y-sway, slight X-sway and torsion

Torsion, slight X- and Y- sway

Avg Spectral Bias -0.5% -0.6% -1.0%

Np [#] 276 276 562

Characteristics Y-sway X-Sway -

Spectral Bias -0.6% -1.8% -

Np [#] 88 31 -

Characteristics X-sway, slight torsion Y-sway, slight torsion -


Np [#] 15 15 -

Characteristics Fully coupled X-sway Full coupled Y-sway -


Np [#] 61 61 -

Bui

ldin

g 3

Bui

ldin

g 1

Bui

ldin

g 2

Kor

ean

Tow

er

Chi

cago

Chi

cago

Chi

cago

80

Design frequencies in Table 4.2 were obtained from a finite element model

(FEM) of the building (Abdelrazaq et al., 2005). In general, the dynamic properties were

extracted from the acceleration data repeatedly across all three measurement locations

(CoV < 0.01%), further supporting not only the reliability of the estimates but also the

validity of the rigid body assumption at each floor, likely aided by this building‘s use of

a belt wall system. The in-situ frequency was observed to be about 30% greater than the

design predictions. There are two possible explanations for differing in-situ stiffness in

reinforced concrete structures. The first, as discussed in Erwin et al. (2007), is associated

with the inclusion of various ―gravity‖ elements assumed to participate in the overall

lateral resistance. Their erroneous inclusion in a finite element representation will result

in differing frequencies in one or both of the lateral directions or possibly in torsion. The

discrepancies between in-situ and FEM predictions will undoubtedly be of varying

degrees depending on the significance of this structural element to that response

component. The other potential explanation is a difference in the in-situ material

properties. This type of global inconsistency tends to equally impact all the response

components and is either attributed to differences in the in-situ concrete strength or

differences in the degree of cracking assumed vs. realized to date. In the development of

FEMs, a level of cracking is assumed and inertial properties are reduced accordingly

leading to a softer structure in the FEM than the current in-situ condition may be,

particularly for a new structure like the Korean Tower.

81

TABLE 4.4: KOREAN TOWER SPECTRAL APPROACH ESTIMATES OF IN-SITU

FREQUENCY AND DAMPING RATIO BY MONTH.

To determine the likely cause in this building, note that the percentage difference

between the design prediction and in-situ observation is similar in all three modes: on

average, in-situ frequency was 30% greater than the design predictions. Secondly, note

that the ratio between adjacent modal frequencies is the same in the FEM and the full-

scale observations: between the 1st and 2nd modes, the design frequencies are within

3.0% of one another, and they were similarly observed to be within 4% of one another in

full-scale. Between the 1st and 3rd modes, the design frequencies differ by 19% and the

observed differed by 18%. This consistency suggests that the modeling of the overall

system behavior was accurate, i.e., the decisions surrounding which elements to include

in the lateral resistance was appropriate, and it is likely that the stiffer features in-situ

Month Mode 1 & 2 Mode 3 Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3

2006-10 - - - - - - - -

2006-11 32 64 0.1980 0.2069 0.2414 0.50 0.73 1.15

2006-12 56 112 0.1971 0.2057 0.2402 1.22 1.12 1.16

2007-01 22 45 0.1961 0.2055 0.2394 1.39 1.26 1.27

2007-02 21 42 0.1967 0.2060 0.2393 1.26 0.63 1.45

2007-03 26 52 0.1965 0.2047 0.2379 1.06 0.88 1.11

2007-04 4 8 0.1975 0.2061 0.2397 0.92 0.73 0.95

2007-05 - - - - - - - -

2007-06 - - - - - - - -

2007-07 - - - - - - - -

2007-08 - - - - - - - -

2007-09 - - - - - - - -

2007-10 7 15 0.1977 0.2063 0.2401 0.75 0.74 1.01

2007-11 22 46 0.1968 0.2062 0.2390 1.52 1.01 1.51

2007-12 36 74 0.1970 0.2052 0.2384 1.12 1.13 1.22

2008-01 16 33 0.1975 0.2057 0.2386 1.44 0.82 1.11

2008-02 30 62 0.1967 0.2055 0.2377 1.27 0.83 1.28

2008-03 - - - - - - - -

2008-04 4 9 0.1965 0.2046 0.2386 0.61 0.47 0.74

Natural Frequency [Hz] Damping Ratio [% ]NP ,[#]

82

are the result of variability in the as-built concrete strength or the level of cracking to

date.

4.2.1.2 Chicago Building 1

The spectra for Chicago Building 1 are provided superimposed on the building

floor plan in Figure 4.3. As demonstrated in Table 4.2, remarkable consistency was

observed between the predicted and in-situ frequencies for Building 1. Frequency

estimates differ from the design predictions by less than 1%, while damping ratios in

both the 1st and 2nd modes exceed design predictions. Note that the frequency estimates

compare well with those published in Kijewski-Correa et al. (2006), while the damping

values observed here are larger than those in the previous study. As a larger range of

events is encompassed by the present analysis, this may suggest that amplitude-

dependence may be playing a role in artificially inflating spectral bandwidth, as

discussed previously in Chapter 3.

Figure 4.3: Plan view of Building 1 with observed Power Spectra.

83


The translational spectra for Chicago Building 2 are provided superimposed on

the building floor plan in Figure 4.4. Observed dynamic parameters for Building 2 in

Table 4.2 are inconsistent with design predictions. Design estimates of frequency are

between 15% and 25% less than the in-situ observations. These findings are consistent

with the earlier observations by Kijewski-Correa et al. (2006). In addition, the observed

ratio between modal frequencies differ in-situ, which may suggest that the discrepancies

in this building depend more upon the assumptions made regarding participation of

various gravity elements in the lateral system modeling. As noted in Erwin et al. (2007),

the effect of gravity components in lateral system modeling can be very pronounced: by

varying the out-of-plane stiffness of concrete floor slabs, fundamental frequencies were

found to vary by up to 25%; however, the need to model the slab stiffness also depended

on the structural system. For systems reliant on tall slender shear walls, the slab played

very little role and could be modeled with 0% effective stiffness. However, for systems

where the slab was integral in linking frame elements, particularly for elongated floor

plate ratios, its stiffness required explicit modeling. Interestingly, Building 2 obtains its

primary resistance from slender shear walls, most notably in its X-direction. Given its

relatively compact floor plate and reliance on slender shear walls, the slab may play a

small role in lateral resistance and its stiffness may be reduced to further calibrate the

model. This may bring the ratio between the fundamental modes closer to 1:1, consistent

with the full-scale observations. With this being said, in-situ material properties are also

likely stiffer than assumed in design.

Damping ratios were 30-140% greater than predicted, although this is not

entirely surprising considering the fact that the designer assumed values were relatively

conservative for concrete construction. Variability of damping values along the

building‘s primary axes may be directly a result of the building‘s varying structural

system. As its x-axis is heavily reliant on slender shear walls, while its y-axis has

84

potentially greater shear racking. This underscores the importance of considering

structural system type and not merely classifiers like material and height when

determining appropriate damping values (Bentz and Kijewski-Correa, 2008). These

observations are consistent with the previous results in Kijewski-Correa et al. (2006).



The spectra for Chicago Building 3 are provided superimposed on the building

floor plan in Figure 4.5. Design predictions and estimates are in better agreement for

Building 3. Both the 1st and 2nd modes differ by approximately 9%, though in this case

indicating a structure that is softer in-situ. Stiffness ratios between the 1st and 2nd modes

differ by 1% in both predicted and observed frequencies. It was speculated by Kijewski-

Correa et al. (2007) that lower in-situ translational frequencies in Building 3 may result

from overestimating panel zone stiffness by utilizing centerline FEM models. The

incorporation of panel zone flexibility did help to further soften the FEM frequencies,

though not entirely resolving the noted discrepancies. Interestingly, damping ratios are

65-105% greater than design predictions over both translational modes, and the observed

85

damping values are greater than those in Kijewski-Correa et al. (2006). As the previous

study considered only an isolated event and the present study considers a wider range of

events, amplitude-dependence of dynamic properties may again be playing some role in

artificially inflating the spectral bandwidth, as demonstrated previously in Chapter 3.


4.2.2 Modal Isolation/Filter Selection

This comprehensive spectral analysis was conducted to obtain a general

understanding about the dynamic properties of each building. Although amplitude-

dependent trends are not observable under this approach, estimates of frequency and

damping are commonly obtained using this approach because of its ease of application.

Therefore it is interesting to now consider the potential errors amplitude-dependence can

introduce to a spectral analysis. To do so, the same data will next be analyzed by a sorted

spectral approach and later by RDT to document amplitude-dependent dynamic features.

This requires us to first isolate each vibration mode by filtering. A Butterworth filter was

selected for its balanced behavior in both the frequency and time domains. Filter settings

were selected to ensure that the peak‘s magnitude and bandwidth were captured

86

sufficiently. A comparison of the unfiltered and filtered modes is provided for the

Korean Tower in Figures 4.6 and 4.7; Building 1 in Figure 4.8; Building 2 in Figure 4.9;

and Building 3 in Figure 4.10. The detail images of Building 3 in particular and to some

extent Building 2 underscore the difficulty in using filtering to isolate modes in buildings

known to have coupling. It is likely in such cases that the HPBW estimates in Table 4.2

were affected by this as well.

Figure 4.6: Verification of Korean Tower 1st and 2nd mode filter selection.

87

Figure 4.7: Verification of Korean Tower 3rd mode filter selection.

88

Figure 4.8: Verification of Building 1 filter selection.


89


4.3 Sorted Power Spectral Approach

In this investigation, the Sorted Spectral Approach (SSA), first introduced in

Section 3.3.7.5, was repeated separately for each building. Records were sorted into

groupings of 25 hours for the Korean Tower, 5 hours for Chicago Buildings 1 and 2, and

10 hours for Chicago Building 3, according to the amplitude of the filtered responses.

Each group is arranged from highest to lowest response amplitude by energy level. This

then permits the use of power spectral estimates to make crude approximations of

amplitude-dependency of the dynamic properties. The results are now presented for each

building.

4.3.1 Korean Tower SSA Results

The results of the SSA applied to the Korean Tower are provided in Figure 4.11

for the 1st and 2nd modes and in Figure 4.12 for the 3rd mode. All the spectra show a

90

tendency to soften (shift of spectral peak toward lower frequencies) as amplitude

increases, though the softening effect is more pronounced in the 3rd mode. In addition,

the complementary asymmetry of the peaks in the 1st and 2nd mode and a tendency to

divide into two closely-spaced peaks in the highest amplitude grouping further suggests

the structure is responding at multiple frequencies and even exchanging energy between

these modes. The trends mentioned above are supported by the HPBW estimates of

frequency and damping, which are provided in Table 4.5. As commonly speculated, the

frequencies in Table 4.5 reduce with increasing amplitude, while the damping increases.

The implications for this level of amplitude-dependence on the power spectra generated

in Section 4.2 will be addressed later in this chapter.

TABLE 4.5: SSA RESULTS: KOREAN TOWER.

MEAN STD [milli-g]

fn [Hz] ζ [%] NP [#] MEAN STD [milli-g]


fn [Hz] ζ [%] NP [#]

GROUP 1 0.52 0.1963 1.24 56 0.52 0.2052 1.07 56 0.18 0.2385 1.38 112

GROUP 2 0.35 0.1968 1.36 55 0.35 0.2057 0.98 55 0.13 0.2391 1.43 111

GROUP 3 0.26 0.1976 1.17 53 0.27 0.2062 0.99 53 0.10 0.2399 1.26 111

GROUP 4 0.20 0.1978 0.76 54 0.21 0.2063 0.83 54 0.08 0.2409 1.21 112

GROUP 5 0.14 0.1981 0.92 56 0.15 0.2068 0.69 56 0.07 0.2406 1.26 111

Mode 1 Mode 2 Mode 3

91

Figure 4.11: SSA: Spectral Suite for 1st and 2nd modes of Korean Tower.

92

Figure 4.12: SSA: Spectral Suite for 3rd mode of Korean Tower.

93

4.3.2 Chicago Building 1 SSA Results

The results of the SSA on Building 1 are provided in Figure 4.13 and the

estimates of dynamic properties are in Table 4.6. Again, decreasing frequency with

amplitude is noted; however, clear trends in damping cannot be discerned, largely due to

the limited number of spectra averaged in each grouping. This results in high variance in

the power spectra, with a jagged spectral peak that is capable of both under or over

estimating damping by HPBW. Note this is particularly pronounced in Mode 1, which

has the fewest number of ensembles, see Table 4.6.

Figure 4.13: SSA: Spectral Suite for Building 1.

TABLE 4.6: SSA RESULTS: CHICAGO BUILDING 1.

MEAN STD [milli-g]


fn [Hz] ζ [%] NP [#]

GROUP 1 0.57 0.1426 1.07 7 0.39 0.2048 0.57 16

GROUP 2 0.34 0.1438 0.59 7 0.20 0.2066 0.48 18

GROUP 3 0.27 0.1426 1.03 6 0.14 0.2068 0.80 17

GROUP 4 0.20 0.1441 0.55 5 0.12 0.2073 0.86 15

GROUP 5 0.11 0.1443 0.72 6 0.08 0.2081 1.02 16

Mode 1 [Y] Mode 2 [X]

94


The results of the SSA on Building 2 are provided in Figure 4.14 and estimates

of dynamic properties in Table 4.7. Again given the high variance of this analysis due to

the minimal amount of available ensembles, damping estimates by HPBW would not be

deemed reliable. As Building 2 is the least dynamically sensitive of the buildings

studied, only two amplitude groupings could be considered. From this limited snapshot

of behavior, frequency does tend to soften with amplitude in both translational modes;

however mode 2 (Y-direction) appears more sensitive to amplitude. This behavior may

again be the result of its different lateral system in that direction, one that relies on

comparatively greater frame action. As the lateral system relies on the weak axis of the

primary shear walls and their link beams in this direction, tied to the exterior columns

through the slab; friction, cracking, and/or out-of-plane deformations in the slab may

explain its greater sensitivity to amplitude. Interestingly, the 1st and 2nd modes appear

well-separated at low energy levels, whereas coupling is apparent at high energy levels

(Fig. 4.14).


95



The results of the SSA on Building 3 are provided in Figure 4.15 and estimates

of dynamic properties in Table 4.8. Similar evidence of amplitude-dependence is

apparent in the frequency spectra, though again the limited number of spectral averages

renders any damping estimate highly unreliable. What is more striking is the

complementary nature of the asymmetry in the spectral peaks, indicative of a residual

coupling of the two modes that will clearly inflate any damping estimates by HPBW.


MEAN STD [milli-g]


fn [Hz] ζ [%] NP [#]

GROUP 1 0.13 0.1821 0.79 6 0.10 0.1805 2.82 6

GROUP 2 0.10 0.1824 1.48 6 0.07 0.1839 1.89 6

Mode 1 [X] Mode 2 [Y]

96


4.3.5 Summary

The SSA results in this section provide some evidence of amplitude-dependence

in frequency, though any inferences regarding damping can only be trusted in the case of

the Korean Tower, as the three Chicago buildings had comparatively fewer spectral

averages in each group and likely variance issues. This underscores the primary

limitation of a spectral approach to investigate amplitude-dependence. More striking is

the evidence of complementary asymmetries between coupled modes and stronger

evidence of their energy exchange at higher amplitudes. This renders the HPBW results

in Section 4.2 quite questionable, as damping was likely inflated by this behavior.

Therefore, the following section presents an alternative time-domain approach, Random

Decrement Technique, to further investigate some of the behaviors that surfaced in the

SSA.

4.4 Time Domain Approach

In this section, the RDT and the analytic signal theory are used for a time-domain

analysis to document amplitude-dependent dynamic properties. The approach adopted

herein was modified from that previously introduced in Section 3.3.7 to include at least

100 segment averages ( RN ). Application of the RDT will utilize filtered building

response as documented in Section 4.2.2. In order to identify amplitude-dependent

MEAN STD [milli-g]


fn [Hz] ζ [%] NP [#]

GROUP 1 0.39 0.1176 1.57 11 0.34 0.1191 1.81 11

GROUP 2 0.22 0.1182 1.60 11 0.19 0.1187 2.10 11

GROUP 3 0.14 0.1194 1.53 11 0.13 0.1201 1.85 11

GROUP 4 0.08 0.1193 1.32 12 0.08 0.1212 2.18 11

GROUP 5 0.01 0.1229 1.83 12 0.01 0.1243 4.10 12


97

properties, a trigger vector was populated with 50 amplitude levels specific to each mode

being analyzed.

4.4.1 Korean Tower RDT Results

Each response component was analyzed separately to account for variations in

the accelerations recorded at each of the three sensor locations. The records and

reliability of the results are summarized in Table 4.9 by documenting the coefficient of

variation observed in the local averaging operation on the RDT. In general, each RDS

was generated using over 500 averages, as reflected in Table 4.9. In addition, variability

in frequency and damping results over the suite of triggers in the local average were low:

mean CoV of less than 0.1% and 11% for frequency and damping, respectively. Results

of the RDT on the first three modes of the Korean Tower are provided in Figures 4.16 to

4.18 and summarized in Table 4.10 using a linear regression of the dynamic properties

against the amplitude. The following discussion expands upon these results.

TABLE 4.9: SUMMARY OF RECORDS AND TIME DOMAIN APPROACH

RESULTS: KOREAN TOWER.

Frequency, Damping,

LA CoV [% ] LA CoV [% ]

Total Mean Min Max Mean Mean Mean

Location 1X 128 535 0.06 0.73 0.28 0.07 9.90

Location 2X 128 525 0.07 0.76 0.29 0.07 9.98

Location 3X 128 542 0.07 0.81 0.31 0.07 9.65

Location 1Y 128 538 0.09 0.75 0.31 0.05 10.38

Location 2Y 128 552 0.10 0.82 0.32 0.06 9.01

Location 3Y 128 489 0.08 0.64 0.26 0.05 10.54

Location 1X 128 633 0.05 0.31 0.14 0.08 7.30

Location 1Y 128 650 0.03 0.20 0.09 0.10 5.96

Location 2X 128 631 0.05 0.37 0.16 0.08 7.66

Location 2Y 128 647 0.04 0.25 0.11 0.09 6.89

Location 3X 128 663 0.03 0.21 0.09 0.08 6.77

Notes: 1) Results of a particular trigger are excluded if NR does not exceed the minimum = 100

2) Mode 3 units [1000 x rad/s 2]

3) LA = Local averaging as introduced by Kijewski-Correa (2003)

[milli-g] or [1000x rad/s2]

Mod

e 1

Mod

e 2

Mod

e 3

Length [Hr] NR1 [#]

Energy Level2, STD

98

TABLE 4.10: AMPLITUDE-DEPENDENT RELATIONSHIPS OF FREQUENCY

AND DAMPING PREDICTED BY THE TIME DOMAIN APPROACH.

slope2 interceptnormalized softening3 slope4 intercept

-0.0011 0.1980 -0.56% 0.0028 0.0065

-0.0013 0.2067 -0.63% 0.0014 0.0041

-0.0048 0.2410 - 0.0033 0.0070

-0.0022 0.1441 -1.53% 0.0069 0.0042

-0.0058 0.2083 -2.78% 0.0034 0.0073

- - - -

-0.0025 0.1824 -1.37% 0.0033 0.0107

-0.0221 0.1858 -11.89% 0.0646 0.0141

- - - -

-0.0013 0.1198 -1.09% -0.0003 0.0097

-0.0037 0.1223 -3.03% -0.0090 0.0121

- - - -

Notes: 1) Mode 1 from Location 1X, Mode 2 from Location 1Y, Mode 3 from Location 2X

3) Slope/Intercept

4) Units of slope for damping under translation [%/milli-g] and torsion [%/(1000 x rad/s 2)]

Kor

ean

Tow

er1 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Chi

cago

B

uild

ing

1 Mode 1 [Y]

Mode 2 [X]

Mode 3 [T]

Chi

cago

B

uild

ing

2 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Chi

cago

B

uild

ing

3 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

2) Units of slope for frequency under translation [Hz/milli-g] and torsion [Hz/(1000 x rad/s 2)]

Natural Frequency [Hz] Damping, ζ [% ]

99

Figure 4.16: Amplitude-Dependent Frequency and Damping Ratio: Mode 1 of Korean Tower.

0.1950.1960.1970.1980.1990.2000.2010.2020.2030.2040.205

0.0 0.5 1.0 1.5 2.0 2.5

Nat

ural

Fre

quen

cy [H

z]

Trigger Amplitude [milli-g]

Location 1X Location 2X Location 3X

0.00%

0.50%

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0.0 0.5 1.0 1.5 2.0 2.5

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Location 1X Location 2X Location 3X

100


0.2000.2010.2020.2030.2040.2050.2060.2070.2080.2090.210

0.0 0.5 1.0 1.5 2.0 2.5

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z]


Location 1Y Location 2Y Location 3Y

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0.0 0.5 1.0 1.5 2.0 2.5

Dam

ping

Rat

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Location 1Y Location 2Y Location 3Y

101


0.2300.2320.2340.2360.2380.2400.2420.2440.2460.2480.250

0.00 0.25 0.50 0.75 1.00

Nat

ural

Fre

quen

cy [H

z]

Trigger Amplitude x 1000 [rad/s2]

Location 1X Location 1Y Location 2X Location 2Y Location 3X

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0.00 0.25 0.50 0.75 1.00

Dam

ping

Rat

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Trigger Amplitude x 1000 [rad/s2]

Location 1X Location 1Y Location 2X Location 2Y Location 3X

102

It is widely understood that stiffness (frequency) reduces with amplitude of

motion, which is affirmed for all the responses, even over the limited amplitude range

considered here. As expected, there is consistency between the outputs of all three

sensors in each direction. A linear fit to the outputs at Location 1 indicates a softening of

frequency in the 1st mode from its low amplitude by a rate of 0.0011 Hz/milli-g. For the

2nd mode, the frequency softens at a similar rate of 0.0013 Hz/milli-g. Similar softening

rates under both translational modes should not be surprising in this building, as its

structural system is consistent in all directions. For the 3rd mode, the frequency softens at

a rate of 0.0048 Hz / (1000 x rad/s2 ). It is interesting to note that frequency in this mode,

and others to a lesser extent, is slightly quadratic, indicating that the degree of softening

will eventually plateau at high enough amplitudes. While the variation in frequency is

less than a percent over the amplitude range considered, recall that it was demonstrated

in Chapter 3 that this may have significant influence on spectral damping estimates.

Similarly, observed amplitude-dependence of both translational modes show

good consistency for damping between locations. Those results are now repeated with

mode 3, where amplitude-dependent trends vary considerably. This may be a

consequence of the difficulty in fully capturing and separating the torsional responses

using basic algebraic operations on the various sensor outputs of a building with such an

irregular plan, particularly compounded when the torsional response is low at a given

location. Thus, results may be most reliable at locations where torsional responses take

on the largest amplitudes, e.g., Locations 1 and 2 in the X-direction. It is interesting to

note that damping tends to cluster into two groupings. This may actually result from

inconsistency of trigger amplitudes and the manner in which torsional contributions to

translation were transformed into rotational accelerations. For example, inaccurate

moment arms in obtaining radial accelerations could account for some variations in the

trigger amplitude, as the geometric center was assumed to coincide with the center of

rigidity. However, setbacks in the building‘s wings could also account for the

103

similarities noted between the results. The wing housing location 3 extends to the roof

(Wing A), the wing housing location 1 extends to level 69 (Wing B), and the wing

housing location 2 extends to level 65 (Wing C). Therefore, similar trigger amplitudes

for Wings A and B would be expected given Wing A shares additional connections with

Wing B versus Wing C. Similarly, in the X-direction, Wing B and C share a common

bond resulting from being the primary wings resisting lateral forces in that direction.

Regardless, the RDT analysis of the Korean Building, and even the SSA results,

indicate that amplitude-dependence is clearly apparent in this building and would be

obscured and potentially misrepresented by traditional spectral analyses in Table 4.2, as

previously demonstrated in Section 3.3.7.6.

4.4.2 Chicago Buildings

Amplitude-dependent curves extracted for Buildings 1 to 3 using the time

domain approach are provided in Figures 4.19-4.21. A summary of the RDT results and

their reliability are provided in Tables 4.10 and 4.11, respectively. In general, each RDS

was generated using over 100 averages. In addition, the variability in the results

associated with a given suite of triggers in the local averaging operation was low: a CoV

of less than 0.5% and 21% for frequency and damping, respectively. The following

discussion expands upon these results. Note that the larger CoV on damping in particular

may result from fewer averages in creating the RDSs for this building collection in

comparison with the Korean Tower.

104

Figure 4.19: Modal Frequency and Damping Ratio Interaction: Building 1.

0.100

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0.150

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0.200

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z]



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0.50%

0.75%

1.00%

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1.50%

1.75%

2.00%

0.00 0.25 0.50 0.75 1.00

Dam

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105


0.175

0.178

0.180

0.183

0.185

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0.00%

0.50%

1.00%

1.50%

2.00%

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Dam

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106


0.1150.1160.1170.1180.1190.1200.1210.1220.1230.1240.125

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Nat

ural

Fre

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z]



0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Dam

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107

TABLE 4.11: SUMMARY OF RECORDS AND TIME DOMAIN APPROACH

RESULTS: CHICAGO BUILDINGS.

4.4.2.1Chicago Building 1

Amplitude-dependent plots of frequency and damping ratio in each mode are

provided in Figure 4.19. As expected, frequency is observed to soften with amplitude as

similarly observed in Figure 4.13. The 1st mode is observed to possess the least level of

amplitude-dependence with a softening rate of 0.0022 Hz/milli-g, while the 2nd mode is

observed undergoing softening at a rate of 0.0058 Hz/milli-g, potentially due to the

elongated floor plate aspect ratio in this direction, increasing shear lag (frame action) in

the X-direction.

Several important trends may be observed from Figure 4.19. First, the modes are

well separated. Next, the observed damping values show a slight increase with amplitude

but significant variability and values generally less than that assumed in design. A

comparable damping level along both axes is likely due to the fact that as a nearly pure

tube, this structure has very similar behaviors on both axes. Also note that average

damping values extracted by the RDT analysis are less than those extracted by the

Frequency, Damping,

LA CoV [% ] LA CoV [% ]

Total Mean Min Max Mean Mean Mean

Mode 1 [Y] 24 185 0.03 0.53 0.19 0.16 18.17

Mode 2 [X] 24 137 0.04 0.87 0.32 0.19 20.17

Mode 3 [T] - - - - - - -

Mode 1 [X] 12 126 0.03 0.20 0.11 0.27 14.62

Mode 2 [Y] 12 129 0.02 0.10 0.08 0.38 15.10

Mode 3 [T] - - - - - - -

Mode 1 [X] 56 161 0.00 1.09 0.14 0.16 14.74

Mode 2 [Y] 56 175 0.00 0.58 0.12 0.16 14.51

Mode 3 [T] - - - - - - -

Notes: 1) Results of a particular trigger are excluded if NR does not exceed the minimum = 100

2) LA = Local averaging as introduced by Kijewski-Correa (2003)

[milli-g]C

hica

go

Bui

ldin

g 1

Chi

cago

B

uild

ing

2C

hica

go

Bui

ldin

g 3

Length [Hr] NR1 [#]

Energy Level, STD

108

traditional spectral analysis (Table 4.2), reflecting the influence of even mild amplitude-

dependence on spectral bandwidth.


Several important trends may be observed from Figure 4.20. First, the 1st and 2nd

modes take on near identical frequencies at some amplitude levels. In fact, if the

amplitude-dependent frequency relationships identified here are extrapolated (based on

linear fits to each mode as presented in Table 4.10), it is possible that at higher

amplitudes, the Y-Sway mode may become the lowest mode of vibration. This

demonstrates the role of amplitude-dependence in fundamentally changing the dynamics

of structures with differing lateral systems on their two orthogonal axes Not only does

the Y-direction have greater sensitivity to amplitude-dependence in frequency (clearly

shown by Table 4.10), but also a higher level of damping and stronger tendency for that

damping to increase with height. This is consistent with the observations of Bentz and

Kijewski-Correa (2008) for other frame-dominated structural systems. Lateral resistance

in the X-direction is provided by shear walls and outriggers that engage the exterior

columns in cantilever action and seems to show a lesser degree of amplitude-dependence

and even energy dissipation overall, despite being formed of concrete. This seems to

further support the suggestion by Kijewski-Correa and Pirnia (2007) that predictive

models of damping should account for structural system characteristics.

Within the observed range of amplitudes, damping levels meet or exceed those

assumed in design (Table 4.2), though these design predictions were somewhat

conservative given the higher energy dissipation generally observed in concrete

compared to steel. However, higher damping levels in the Y-direction were not

unexpected given the greater energy dissipation potentials of that structural system.

Finally, when comparing these results to those in Table 4.2, the averaged damping

values from the RDT analysis are only slightly less than the traditional spectral analysis,

109

likely due to the fact that Building 2 manifests the narrowest range of response

amplitudes and thereby fairly limited ranges of frequency and damping values within the

data analyzed.


The amplitude-dependent frequency trends for Building 3 are provided in Figure

4.21. A clear decreasing trend with amplitude is apparent, varying with the first mode

softening at a rate of 0.0013 Hz/milli-g and the second mode at 0.0037 Hz/milli-g (fits to

linear portion only).

The x and y responses of this building are known to be fully coupled, and this

becomes quite apparent in the amplitude-dependent damping analyses, where there is

evidence of energy exchanging between the two modes, with the levels generally

beneath those assumed in design. It is speculated that some of the potential outlier values

at the low frequency range were due to response levels being bedded in the noise floor.

Contrasting this to the results in Table 4.2, it becomes evident that when strong coupling

and energy exchange are present, spectral damping estimates can experience particularly

strong inflation.

4.4.2.4 Comparison of Degree of Amplitude-Dependence

A comparison of normalized rates of softening (slope/intercept in Table 4.10) for

Building 3 with those of Building 1 reveals that among the steel buildings, mode 2 of

Building 3 possesses the most amplitude softening, though none of the other steel modes

surpass a 3% normalized softening rate. In addition, linear fits of the stabilized damping

regions in Figures 4.19-21 reveal more than 10% higher damping in Building 3 than

Building 1. As both these buildings are steel tubes, similar levels of damping and

amplitude-dependence may be expected, though greater degrees of both are not

surprising in Building 3 due to the aforementioned role of panel zones as a shearing

mechanism in its system.

110

4.4.2.4 Discussion

In this chapter, amplitude-dependent dynamic properties were extracted from the

responses of the Korean Tower and each of the Chicago buildings. The findings

generally confirmed the expected trends of frequency softening with amplitude and some

weak tendency of damping to increase with amplitude. These trends further show some

correlation to features of the structural system and construction material.

Cantilever-dominated structural systems were found to possess an increased

sensitivity to floor plate aspect ratio. In the case of the stiffened tube of Building 1, an

increased amplitude-dependence of frequency and damping was observed in the long

direction, X-sway, where the elongated floor plate creates greater potential for shear lag

as the tube engages. Building 2 differs from Building 1, in that it is concrete, however

similar trends may be observed. Along mode 2, the long direction of Building 2,

amplitude-dependency of frequency is almost 10 times that of the short direction and

static damping is approximately 30% greater. The increased amplitude-dependence and

higher damping in this direction are likely due to the enhanced reliance on floor slabs

and link beams to compensate for the weak axis lateral strength of the shear walls,

whereas in the X-direction (mode 1), the building is dominated by cantilever-action with

the core and outriggers resisting overturning. Thus the findings here seem to support

those of Erwin et al. (2007) and Bentz and Kijewski-Correa (2008), which suggest that

the more efficient a system becomes, the less energy it dissipates.

For the two concrete buildings, the Korean Tower showed less relative softening

(0.56-0.63%) than Building 2 in the Chicago dataset (1.4-11.9%). While it can be

hypothesized that given the young age of the Korean Tower, it has likely developed

fewer cracks responsible for amplitude-dependent dynamic characteristics, much can

also be tied to the structural systems as the Korean building features a virtual outrigger

(belt) wall system that similarly attempts to generate a cantilevered behavior quite

comparable to the X-sway of Building 2. Furthermore, the two steel buildings, Buildings

111

1 and 3 of the Chicago dataset, generally show less amplitude-dependence than the older

concrete building (Building 2). The relative softening of 1.53-3.03% in these buildings

may indeed support the idea that steel-framed structures have limited amplitude-

dependence in their dynamic properties at low response levels. But the fact that the

Korean Tower shows a similar tendency should reiterate that structural system plays

perhaps even a more important role than material type.

Perhaps even more noteworthy is the discernable difference between damping

levels estimated by the spectral approach (Table 4.2) and the time domain approach. For

the Korean Tower, observed stabilized damping levels by the time domain approach can

be less than half those obtained by the spectral approach. A similar trend exists in the

results for the Chicago buildings. While one may initially question the validity of the

analyses conducted herein, it should be noted that the time-domain damping levels are

consistent with, and even greater than, the values noted in similar, independent RDT

analyses conducted on other tall buildings:

Di Wang Building, %6.0~ (Li et al., 2005);

Central Plaza, %5.0~ (Li et al., 2005);

Jin Mao, %55.0~ (Li et al., 2006); and

Bank of China, %4.0~ (Li et al., 2003).

However, it should be noted that damping values reported in this thesis do not

show the characteristic increase with amplitude between plateaus, as proposed by Jeary

(1986) and subsequently documented in-situ by Li et al. (2003; 2005). This may result

from relatively small responses, possibly resigning this data to the low-amplitude plateau

of the classic amplitude-dependent model. In addition, it is particularly noteworthy that

the spectral damping estimates are consistently larger than the time domain estimates,

well beyond the levels that would be expected due to inherent spectral bias. This feature

will now be explored in more detail.

112

4.4.2.4.1 Gross Damping

Previously the differences between spectral and time domain approaches were

investigated in Section 3.3.7.6. Using theoretically generated non-linear systems, the

level of damping was observed to be inflated in estimates derived from low-bias spectra

due to frequency variations in an amplitude-dependent system. The previous approach is

repeated now using in-situ building responses from the Korean Tower and the Chicago

buildings to determine whether this factor explains the discrepancies in frequency and

damping observed here as well (Table 4.12 and 4.13). Damping will conservatively be

assumed to be constant to focus on the effect of frequency variation on observed gross

damping levels.

TABLE 4.12:

COMPARISON OF SPECTRAL AND TIME DOMAIN APPROACH FREQUENCY RESULTS.

Spectral Approach

Full DB, fn [%] RMS, fn [%]2 Difference3

0.1968 0.1977 0.5%

0.2056 0.2063 0.3%

0.2389 0.2402 0.6%

0.1430 0.1437 0.5%

0.2060 0.2064 0.2%

- - -

0.1820 0.1821 0.1%

0.1830 0.1841 0.6%

- - -

0.1180 0.1196 1.4%

0.1190 0.1218 2.4%

- - -

Note:

Chi

cago

B

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2 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Chi

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B

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3 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Chi

cago

B

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1 Mode 1 [Y]

Mode 2 [X]

Mode 3 [T]

Kor

ean

Tow

er1

3) Percent difference of fn of "TDA-HT" compared to fn of "Spectral Approach"

2) Based on predicted AD relationship using average RMS value of response time history

Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

TDA-HT

1) Mode 1 from Location 1X, Mode 2 from Location 1Y, Mode 3 from Location 2X

113

TABLE 4.13: COMPARISON OF SPECTRAL AND TIME DOMAIN APPROACH

DAMPING RESULTS.

Gross damping is calculated per Equation 3.27 and reported in Table 4.14.

Frequency variation is assumed to follow the linear trends observed by the time domain

approach and reported previously in Table 4.10. Only amplitudes up to the RMS value

will be considered. This assumption leads to frequency softening of 0.2-1.0%.

Comparisons between spectral and time domain frequencies are provided in

Table 4.15. Differences between spectral estimates and gross frequency estimates range

from 0.1-2.6%. Although these differences are somewhat large for frequency estimates,

frequency modulation can produce non-symmetric peaks that may lead to bias in

frequency estimates, particularly in the case of coupled buildings like Building 3.

Spectral Approach

Full DB, ζ [%] RMS, ζ [%]2 Difference3

1.24 0.73 -41.5%

1.12 0.45 -59.3%

1.48 0.75 -49.2%

1.29 0.55 -57.5%

1.08 0.84 -22.2%

- - -

1.34 1.11 -17.4%

2.34 1.91 -18.2%

- - -

2.01 0.97 -52.0%

1.95 1.10 -43.6%

- - -

Note:

TDA-HT

Kor

ean

Tow

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Chi

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1C

hica

go

Bui

ldin

g 2

Chi

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3


Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Mode 1 [Y]

Mode 2 [X]

Mode 3 [T]

Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

2) Based on predicted AD relationship using average RMS value of response time history

3) Percent difference of ζ of "TDA-HT" compared to ζ of "Spectral Approach"

114

TABLE 4.14: CALCULATION OF GROSS DAMPING.

TABLE 4.15:

COMPARISON OF SPECTRAL APPROACH RESULTS WITH GROSS FREQUENCY.

fn2 fn1 Difference3 fGROSS [Hz] 5 ζGROSS [%]

0.1977 0.1980 -0.2% 0.73 0.1978 0.81

0.2063 0.2067 -0.2% 0.45 0.2065 0.55

0.2402 0.2410 -0.3% 0.75 0.2406 0.92

0.1437 0.1441 -0.3% 0.55 0.1439 0.69

0.2064 0.2083 -0.9% 0.84 0.2074 1.29

- - - - - -

0.1821 0.1824 -0.2% 1.11 0.1823 1.18

0.1841 0.1858 -0.9% 1.91 0.1849 2.38

- - - - - -

0.1196 0.1198 -0.2% 0.97 0.1197 1.04

0.1218 0.1223 -0.4% 1.10 0.1221 1.29

- - - - - -

Notes: 1) Mode 1 from Location 1X, Mode 2 from Location 1Y, Mode 3 from Location 2X

2) Frequency range obtained from AD relationships using mean + σ (or 0 to RMS)

3) Percent difference calculated as follows: (fn2 - fn1) / fn1

4) Damping range obtained from AD relationships using mean + σ (RMS)5) fGROSS calculated as follows: (fn1 + fn2) / 2

Gross Damping

Mode 2 [Y]

Mode 1 [X]

Kor

ean

Tow

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Time Domain Approach2

Natural Frequency [Hz]Damping, ζ [%]4

Mode 1 [Y]

Mode 3 [T]

Mode 1 [X]

Chi

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3 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Mode 3 [T]

Chi

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1

Mode 2 [X]

Chi

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B

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2

Mode 2 [Y]

Mode 3 [T]

Spectral Approach

Full DB, fn [%]

0.1968 0.1978 0.5%

0.2056 0.2065 0.4%

0.2389 0.2406 0.7%

0.1430 0.1439 0.6%

0.2060 0.2074 0.7%

- - -

0.1820 0.1823 0.1%

0.1830 0.1849 1.1%

- - -

0.1180 0.1197 1.4%

0.1190 0.1221 2.6%

- - -

Note:

2) Percent difference of fGROSS compared to fn of "Spectral Approach"

Chi

cago

B

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3 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]


Mode 1 [Y]

Mode 2 [X]

Mode 3 [T]

Chi

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B

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2 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

fGROSS [Hz] Difference2

Kor

ean

Tow

er1 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Chi

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1

115

TABLE 4.16: COMPARISON OF SPECTRAL APPROACH RESULTS WITH GROSS

DAMPING.

Damping comparisons are provided in Table 4.16. In general, damping levels

observed by the time domain approach are between 15 and 60% less than those

estimated by the spectral approach. Note that in two cases where frequency variations

were high (Table 4.14), gross damping exceeded that observed by the spectral approach,

though this may be due to the fact that many of the spectral estimates in the Chicago

Buildings were plagued by variance that can underestimate damping due to jaggedness

of the spectral peak. Furthermore, the gross damping estimate assumed constant

damping, which was not the case, based on a maximum amplitude set to the RMS value,

which may not be entirely appropriate. In addition, many of the spectral estimates were

suspected of suffering from the effects of closely spaced modes. Thus, the proposed

gross damping measure cannot address every potential error source in the spectral

estimates, but it does at minimum provide a means to explore the potential effects of

amplitude-dependence on spectral bandwidth.

Spectral Approach

Full DB, ζ [%]

1.24 0.81 -35.2%

1.12 0.55 -50.4%

1.48 0.92 -38.2%

1.29 0.69 -46.4%

1.08 1.29 19.7%

- - -

1.34 1.18 -11.6%

2.34 2.38 1.7%

- - -

2.01 1.04 -48.1%

1.95 1.29 -34.1%

- - -

Note:

2) Percent difference of ζGROSS compared to ζ of "Spectral Approach"

ζGROSS [% ] Difference2

Chi

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B

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3

Mode 1 [Y]

Mode 2 [X]

Mode 3 [T]

Chi

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B

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2 Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Kor

ean

Tow

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Chi

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1

Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]

Mode 1 [X]

Mode 2 [Y]

Mode 3 [T]


116

4.5 Summary

This chapter investigated amplitude-dependent trends in four tall buildings: two

steel and two concrete. A spectral approach was incorporated to provide a baseline as

researchers typically use this approach to document tall building response. Observed

damping levels by this spectral approach agree with typical assumptions for tall

buildings based on material type: 1% for steel and 2% for concrete. Not only were

observed damping levels by this time domain approach below those typically assumed,

but this chapter further demonstrated the role a structural system itself can play in

inherent damping levels as well as the degree of amplitude dependence. These

amplitude-dependent trends were found to reduce frequencies and increase damping

ratios as amplitude increased, in agreement with trends observed in the literature. An

investigation of gross damping revealed a possible explanation for the differences

between observed damping levels by spectral and time domain approaches, though not

capable of resolving other error sources tied to spectral variance and coupled modes. The

following chapter will now transition from our investigation of structural dynamic

properties like damping and into the responses they are most critical in mitigating:

accelerations adversely affecting occupant comfort.

117

CHAPTER 5:

PSEUDO-FULL-SCALE EVALUATION OF OCCUPANT COMFORT

5.0 Introduction

As discussed in Bentz and Kijewski-Correa (2009), habitability limit states often

govern the design of tall structures, as wind-induced accelerations increase and become

more perceptible to occupants. Even buildings with acceptable serviceability

performance, as quantified by drift criteria, may still have habitability issues under wind

as prolonged accelerations may cause occupants physical discomfort (Kareem, 1988).

Human perception of motion is dependent upon many factors, some of which are more

difficult to quantify than others. In fact, only recently have the effects of motion

frequency, amplitude, duration, and waveform (peak factors) on human comfort been

investigated using motion simulators (Burton et al., 2005, 2006). While this represents a

considerable enhancement over the sinusoidal motions that were the basis of earlier

perception criteria (Chen and Robertson, 1973; Irwin, 1981; Goto, 1983; Denoon et al.,

2001), there are still other contributing factors that are difficult to accurately capture

even within the most faithful motion simulators; for example, visual and audio cues have

been shown in full-scale occupant surveys to generally be the first stimulus (Bentz and

Kijewski-Correa, 2009). In addition, the extent to which each contributing factor triggers

perception or other undesirable responses (fear, nausea, task disruption) varies from

person to person, and the means to best quantify these accelerations (peak vs. RMS and

118

perception vs. tolerance) is still contested (Hansen et al., 1973; McNamara et al., 2002).

Clearly there are wide ranging uncertainties surrounding not only the criteria used to

evaluate accelerations, but also in the prediction of acceleration responses in design

(Bashor and Kareem, 2009), making this an issue worthy of continued investigation. Still

enhancing the designer‘s ability to predict accelerations accurately in the design stage,

and providing faithful criteria for their evaluation is necessary to prevent the need for

later costly retrofits in response to occupant complaints (Kareem et al., 1999).

To avoid the bias and limitations possible in motion simulator studies, a more

faithful evaluation of habitability criteria would interview the occupants of an

instrumented tall building following wind events so that recorded accelerations could be

correlated with occupant feedback. Unfortunately, there are many practical barriers to

studying occupant comfort in-situ, largely due to a lack of accessibility, as listed in

Table 5.1. Thus it seems entirely likely that motion simulators will continue to be the

primary means by which human sensitivity to motion is investigated (Kareem, 1988;

Burton et al., 2006).

TABLE 5.1: SUMMARY OF CHALLENGES IMPLEMENTING FULL-SCALE

MONITORING PROGRAMS.

Cost Complexity• Monitoring System • Characterization of motion (acceleration vs. jerk, RMS vs. peak)

• Installation and Maintenance • Quantifying perception level (e.g. intensity, duration)

• Analysis of Response Data • Evaluating the role of environmental cues (e.g. swinging doors, rolling chairs, wind-induced noise from cladding)

• On-Going Surveying of Occupants (e.g. web form) • Task type (e.g. typing at a computer, sleeping, dancing)

• Statistical errors (e.g. bias, low participation among occupants)

• Human variables (e.g. conditioning/pre-disposition to motion, frequency sensitivity)

• Legal issues (e.g. obtaining permission from building owners, restrictions on data use)

Sources: Isyumov and Kilpatrick (1996), Boggs (2002)

119

5.1 Purpose

In this chapter, given that the ideal interviewing process cannot be achieved, a

―pseudo-full-scale‖ evaluation of occupant comfort is offered: the characteristics of

motion simulator conditions that were identified by Burton et al. (2005) as being task

disruptive or nauseating are identified, and then full-scale acceleration databases are

queried to determine the number of occurrences of these ―nauseating‖ or ―task

disruptive‖ motions. Specifically, these motions are characterized by waveform (using

peak factors), intensity (using standard deviation), and dominant frequency of oscillation

(using spectral estimate of fundamental frequency). This approach will be used to

evaluate the performance of the Korean Tower in 2007. First, a description of the

methodology used to obtain peak factors will be provided. Next, the building‘s responses

will be evaluated against current occupant comfort criteria, which do not consider the

role of waveform. Finally, Korean Tower responses (using instrumentation Location 2

only) will be evaluated by the proposed ―pseudo-full-scale‖ occupant comfort

assessment proposed here.

5.2 Analysis Procedure for Peak Factors

As most occupant habitability criteria are based on sinusoidal motions, which are

not entirely representative of the actual motions of tall buildings, the effects of other

waveforms on occupant comfort are of interest. Since the motion simulator studies of

Burton et al. (2005; 2006) are being used, the peak factor estimation procedure used in

those studies is also adopted here for application to full-scale data. This methodology

was developed to obtain peak factors based on a constant probability of up-crossing

despite response length, which is important considering motions of varying duration are

considered in this study. To do so, data is analyzed over a moving window of length (12

or 50 minutes) corresponding to the durations of tests in Burton et al. (2005; 2006). First,

a bandpass filter is used to isolate the dominant mode over the analysis window. Second,

120

the response envelope (peak values) is extracted, and the peak factor and standard

deviation are estimated. Peak factors are obtained based on a probability of up-crossing

equal to approximately 1/1000 to provide an equal baseline across different duration

analysis windows. Next, the analysis window is shifted by a translational increment of

10 seconds (~2 cycles of oscillation) and the procedure is repeated. Finally, the results

for uncorrelated segments are grouped by peak factor. Response segments grouped by

peak factor were then sorted from highest to lowest standard deviation. The segments

were assumed to be uncorrelated and retained only if they had no more than 10% of their

data points in common. An illustration of this process is provided in Figure 5.1 for a

sample response window.

Figure 5.1: Example of Peak Factor Calculation for a Given Sample Response Window.

121

5.3 Evaluation of 2007 Korean Tower Response against Perception Criteria Used in

Current Practice

Many different criteria exist for evaluation of occupant comfort. An overview of

the most common criteria, as cataloged by the ASCE Tall Buildings Committee, is

reproduced here in Figures 5.2-5.4. Recall from Chapter 2 that the Korean Tower is a

residential building with a frequency determined in Chapter 4 of around 0.2 Hz in both

fundamental sway directions. A summary of the peak accelerations on these two axes for

the Korean Tower is provided in Figure 5.5; note that the maximum acceleration

observed in 2007 approached 7 milli-g for X-sway and 6.5 milli-g for Y-sway. This

response is marginally acceptable considering the 1-year CTBUH guidelines in Figure

5.4 for residences, which recommend limiting motions to 5-7 milli-g. Revisiting Figure

5.2, disruptive/nauseating motions typically begin at peak accelerations in the range of

10-17.5 milli-g (Isyumov, 1993; Hansen et al., 1973). As the worst case peak

accelerations observed in the Korean Tower during 2007 are well beneath this level, one

would not expect reports of negative sensations from the occupants.

Figure 5.2: Physical Effects of Acceleration on Occupants as Summarized (Credit: ASCE Tall Buildings Committee).

122

Figure 5.3: Peak Acceleration as a Function of Frequency for Different Return Periods (Credit: ASCE Tall Buildings Committee).

Figure 5.4: Peak Acceleration as a Function of Annual Recurrence Rate (Credit: ASCE

Tall Buildings Committee).

123

Figure 5.5: Peak Accelerations by Month for the Korean Tower in 2007 (First Mode Isolated, Location 2).

5.4 Summary of Motion Simulator Observations Regarding Role of Waveform

Of the three waveforms investigated by Burton et al. (2005), sinusoidal,

Gaussian, and burst with peak factors of 1.7, 3.3, and 4.8, respectively, the Gaussian

response was found to induce the most nausea in the occupants. It is hypothesized that

the randomness of the motion kept the occupant effectively ―off guard.‖ This effect

becomes more pronounced in longer duration events (50 minutes vs. 12 minutes). Under

short durations (12 minutes), Gaussian waveforms induced negligible reports of nausea,

while task disruption rates decreased slightly from those under the longer duration test.

The researchers did not evaluate sinusoidal and burst waveforms for short duration

events. These findings are consistent with anecdotal reports collected in Chicago, that

indicate that short duration transient events may cause perception issues due to their

sudden onset and high amplitudes; however, due to their short duration, are not capable

of inducing nausea or task disruption in occupants.

124

5.5 Classification of Full-Scale Waveforms

To assess the extent to which the full-scale responses of the Korean Tower were

characterized by Gaussian waveforms, the peak factors observed over 12 minute analysis

windows were cataloged into three categories: sinusoidal (<2.5), Gaussian (2.5-4.05),

and burst (>4.05). Examples of each of these waveforms are provided in Figure 5.6 and

survey results are provided in Figure 5.7. Gaussian and sinusoidal waveforms compose

approximately equal amounts of the triggered response, while burst waveforms typically

compose 10% or less of the triggered response. Note the Y-sway shows a greater

tendency toward burst-like responses, particularly in the spring and summer months.

Figure 5.6: Waveform Examples (12 Minute Analysis Windows) for Sample Data File Recorded on August 8, 2008 for the Korean Tower (X-Sway, Location 2).

125

Figure 5.7: Response Classification by Waveform Type for X-sway and Y-sway (top and bottom, respectively) (12 Minute Analysis Windows): Korean Tower, 2007.

A similar inventory is now performed to evaluate the intensity of these Gaussian

waveforms over analysis windows of 12 and 50 minutes. Task disruption occurrences

will be evaluated with the shorter duration events (12 minute analysis windows) and

nausea will be evaluated with the longer duration events (50 minute analysis windows).

Disruptive responses, as defined by the duration, frequency and intensity observed by

Burton et al. (2005), will be inventoried within the full-scale data for the Korean Tower

in 2007. These will be correlated with the percentage of subjects experiencing rates of

nausea or task disruption, according to Burton et al.‘s (2005) motion simulator study, to

project the likely number of occupants affected in full-scale. Figure 5.8 provides an

example of an inventory of disruptive events, showing the total number of events

occurring with a specified intensity (standard deviation) and duration for June 2007.

Similar inventories were conducted for the other months in 2007. To demonstrate how

these inventories are correlated with motion simulator test results, consider this example

of June 2007. The maximum short duration RMS acceleration observed in the full-scale

126

data that month (0.75 milli-g), corresponded to a task disruption level of 7.5% in Burton

et al.‘s (2005) motion simulator studies. Meanwhile, the maximum long duration RMS

acceleration observed (0.45 milli-g) corresponded to a nausea rate of 2% in those same

motion simulator studies.

Figure 5.8: Gaussian Long (50 Minute Analysis Window) and Short (12 Minute Analysis Window) Duration Events along X and Y-Axes (top and bottom, respectively): Korean Tower, June 2007.

Monthly results are tabularized in Tables 5.2-5.3. Here various acceleration

levels (standard deviations) are reported along with the corresponding rates of subject

disruption from Burton et al.‘s (2005) motion simulator studies (first two columns) .The

127

numbers of events with this intensity level are then reported, month by month. These

results are not mutually exclusive across analyses; meaning an event inciting a particular

energy level along response X may also be counted for response Y. Task disruptive

events exceeding 1.5 milli-g RMS occurred (at a minimum of) 4 times during 2007,

corresponding to a projected rate of disruption among 14% of occupants. Similarly,

nauseating events exceeding 1.0 milli-g RMS occurred (at a minimum of) 5 times during

2007, causing nausea in a projected 5% of occupants. Note that higher occurrence rates

are observed for responses along the X-axis, due to the higher accelerations observed for

that axis (see Figure 5.5), which is slightly softer than its Y counterpart (see Table 4.2,

Figure 4.11).

TABLE 5.2:

TASK DISRUPTION SUMMARY OF GAUSSIAN-TYPE EVENTS FOR 2007 (12 MINUTE ANALYSIS WINDOW).

Minimum STD Task Disruption

[milli-g] Rate1 [%] J F M A M J J A S O N D Total Rate2

0.5 5 44 67 114 28 18 4 0 35 18 11 10 27 376 0.86%

1.0 10 8 5 14 7 0 0 0 3 0 0 0 0 37 0.08%

1.5 14 1 0 2 1 0 0 0 0 0 0 0 0 4 0.01%

2.0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.5 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

Minimum STD Task Disruption


0.5 5 48 72 126 29 16 0 0 32 16 11 12 26 388 0.89%

1.0 10 10 4 10 4 0 0 0 1 0 0 0 0 29 0.07%

1.5 14 1 0 0 2 0 0 0 0 0 0 0 0 3 0.01%

2.0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.5 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

Notes:1 From Burton et al. (2005)2 Quantity of Exceedance / (Minutes in a Year / Window Length)

Res

pons

e A

long

XR

espo

nse

Alo

ng Y

Quantity of Events Exceeding Minimum STD Yearly Summary

Quantity of Events Exceeding Minimum STD Yearly Summary

128

TABLE 5.3: ONSET OF NAUSEA SUMMARY OF GAUSSIAN-TYPE EVENTS FOR

2007 (12 MINUTE ANALYSIS WINDOW).

5.6 Further Discussion

The ―pseudo-full-scale‖ assessment of occupant comfort in the previous section

was able to project the likely number of occupants at the top floor that were disrupted by

motions observed in the Korean Tower in 2007. However this alone is not sufficient to

determine whether this performance is acceptable. Given the level of motion-

susceptibility or tolerance to motion varies widely among human subjects, acceleration

levels are selected based on the percentage of occupants that will be affected. For

example, some have set a stringent requirement of 2% of occupants (Hansen et al.,

1973), while others have relaxed this to 10-20% of occupants in residential buildings

(AIJ Guidelines, 1991). Although ―perception‖ is a measure of tolerance to motion, in

this case zero tolerance, other tolerance criteria based on task disruption or onset of

nausea would permit some motion beyond that which is perceptible, while excluding

responses that may better be addressed by education of occupants. A similar sentiment is

shared by Hansen et al. (1973): ―the paramount issue to the building designer is the level

of motion tolerance [that] building occupants will accept. In this context it is important

Minimum STD Nausea


0.5 2 12 18 27 7 2 0 0 8 1 1 2 5 83 0.19%

1.0 5 1 0 3 1 0 0 0 0 0 0 0 0 5 0.01%

1.5 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.5 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

Minimum STD Nausea


0.5 2 12 19 29 6 2 0 0 8 1 1 2 6 86 0.20%

1.0 5 1 0 1 0 0 0 0 0 0 0 0 0 2 0.00%

1.5 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

2.5 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00%

Notes:1 From Burton et al. (2005)2 Quantity of Exceedance / (Minutes in a Year / Window Length)

Yearly Summary

Quantity of Events Exceeding Minimum STD Yearly SummaryR

espo

nse

Alo

ng X

Res

pons

e A

long

Y

Quantity of Events Exceeding Minimum STD

129

to distinguish between ‗threshold of perception‘ and ‗level of tolerance‘.‖ If a tolerance

level of 2% is applied to the findings by Burton et al. (2005), short duration motions (12

minutes) should be limited to an RMS value of 0.2 milli-g (minimizing task disruption)

and long duration motions (50 minutes) should be limited to an RMS value of 0.5 milli-g

(minimizing nausea). Note that the Korean Tower experienced motions in 2007 in excess

of those limits.

A comparison of the results of Burton et al. (2005) (for 0.2 Hz frequency of

motion) with those of the AIJ Guidelines (2004) and Chen and Robertson (1973) is

provided in Figure 5.9. Because these criteria differ in their measures, for simplicity

―perception rates‖ or ―occupants experiencing [sensation X] rates‖ will collectively be

referred to as ―exposure rates.‖ In this figure a more restrictive criteria would be one that

dictates a lower amplitude for a given exposure rate. This comparison reveals that

existing criteria would indicate greater exposure rates for peak accelerations exceeding

3-4 milli-g than Burton et al.‘s (2005) findings. This difference may result from various

―acceptable motion‖ definitions among criteria developers: i.e., for a given acceleration

level, motion would be perceived by a larger number of occupants, while a smaller

percentage would experience sensations so severe that it disrupted tasks or induced

nausea (Burton‘s criteria). Ultimately, the client/owner may be the best to decide

whether mere perception is a concern or if more relaxed criteria based on motion

tolerance (e.g., task disruption) is acceptable. For example, hotel developers may be

concerned about perception and the impacts it can have on the reputation among guests

paying for a high level of comfort, whereas office developers may tolerate perception

and see task disruption as a more viable concern in work settings. As a result, client

input into selection of occupant habitability criteria is an integral part of the AIJ

Guidelines (2004).

130

Figure 5.9: Comparison of Results of Burton et al. (2005) with Other Occupant Comfort Studies.

Certainly, when evaluating occupant comfort criteria one must assume that the

profile of the human subjects in motion simulators is similar to the occupants of the

actual tall building, which may not be the case, e.g., occupants of actual tall buildings,

through conditioning or even education, may tolerate higher levels of motion, meanwhile

subjects in motion simulator studies may enter the experiment expecting motion and

thereby biasing their responses. Still there are other more significant factors not captured

by motion simulators. This in part stems from the fact that these simulators emulate

lateral responses neglecting torsional effects. Actual building response is complex;

occupants experience motion resulting from a superposition of variable translational and

torsional modes (McNamara et al., 2002). To demonstrate how significant this can be for

some buildings, consider a comparison of the peak X- and Y-sway responses (Figure

5.5) with the total measured responses along the X- and Y-direction, which includes

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

55%

60%

65%

70%

75%

80%

85%

90%

95%

100%

0 5 10 15 20 25 30

Exp

osu

re R

ate

Peak Acceleration [milli-g]

Nausea, T=50 min Task Disruption, T=50 min Task Disruption, T=12 min

Chen and Robertson (1972)R = 1 year

AIJ Guidelines 2004R = None specifed

131

torsional contributions (Figure 5.10). For example, peak accelerations in the Y-direction

in May of 2007 almost double when torsion is included. These contributions are isolated

in Figure 5.11. In particular, torsional responses are known to be particularly disturbing

to occupants (Kareem et al., 1999). As a result, some criteria provide limits on

translational and torsional accelerations (Isyumov, 1993). However, criteria developed

considering the independence of torsional and translational responses may not be able to

adequately characterize the responses of a coupled building like the Korean Tower.

Figure 5.10: Peak Accelerations by Month for the Korean Tower in 2007 (Total Response, Location 2).

Figure 5.11: Peak Accelerations by Month for the Korean Tower in 2007 (Location 2).

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CHAPTER 6

CONCLUSIONS AND FUTURE WORK

Research efforts such as those presented in this thesis provide a necessary link

between current design practice and in-situ behavior of tall buildings, particularly with

respect to their habitability performance. Addressing uncertainties surrounding the in-

situ natural frequencies and damping ratios are critical to ensuring that design predicted

accelerations are an accurate representation of the actual response of the structure.

Unfortunately, the extraction of these parameters from full-scale ambient vibration data

is non-trivial. Therefore this thesis‘ first objective was the development and validation of

a framework to reliably extract amplitude-dependent dynamic properties from ambient

vibration data. In support of objective 2, this framework was then used to study the in-

situ dynamic properties of several tall buildings. In addition, this thesis‘ third objective

was to develop a means to evaluate habitability performance of instrumented tall

buildings. This was achieved by correlating motion simulator studies, tied to waveform,

duration, intensity and frequency of vibration, with the observed response of tall

buildings to project the likely effects of these accelerations on the building‘s occupants.

This was then applied to the 2007 data from the Korean Tower, in satisfaction of our

fourth objective. Some of the major contributions of this thesis are now reviewed

followed by future work.

133

6.1 Improved Monitoring of the Korean Tower

As discussed in Chapter 2, the data acquisition protocol for the Korean Tower

was redesigned to facilitate present as well as future research efforts. Most importantly,

these modifications enhanced the signal to noise ratio and enhanced resistance to

aliasing, thus providing high quality data for use in this thesis.

6.2 Framework for Extracting Amplitude-Dependent Dynamic Properties from

Tall Building Ambient Responses

Chapter 3 details the development of a framework for extracting amplitude-

dependent dynamic properties from ambient vibration responses of tall buildings. The

appropriateness of power spectral and random decrement approaches was considered,

evaluating performance against linear and non-linear simulations. These optimizations

and validations were carried out on each element of this framework (generating response

artifacts, extracting dynamic properties) in isolation and in conjunction. These findings

were then used to determine the analysis parameters best optimizing their performance.

For example, analytical signal theory applied to Random Decrement Signatures was

found to perform best over the first 0.5-3.5 cycles. Similarly, it was found that allowing

mild correlation when generating Random Decrement Signatures minimized errors while

balancing the need to decrease variance. More importantly, the inability of spectral

approaches to characterize amplitude-dependent dynamic properties, even when

grouping data by amplitude was underscored. Inflation in spectral bandwidths resulting

in significant distortions of damping estimates was noted even for mild levels of

amplitude-dependence. A gross damping parameter was proposed in efforts to quantify

the extent of this distortion. Additionally, the superior performance of best-fit

identification approaches (analytic signal theory with Hilbert Transform) was affirmed

over point estimators (Logarithmic Decrement).

134

6.3 Extracted Dynamic Properties of Four Tall Buildings

Chapter 4 presented the application of this framework to the buildings of the

Chicago Full-Scale Monitoring Program. Traditional dynamic investigations based on

spectral techniques (Table 4.2) were found to be inflated by amplitude-dependent

effects. Amplitude-dependent dynamic properties were extracted for each of the

buildings. While concrete would intuitively be expected to demonstrate more amplitude-

dependence than steel, this thesis particularly demonstrated that structural systems with a

higher degree of cantilever action demonstrated less amplitude-dependence, regardless

of material. These efforts constitute one of the first attempts to explain the factors

contributing to amplitude-dependence of tall buildings.

6.4 Pseudo-Full-Scale Evaluation of Occupant Comfort

While full-scale evaluations of occupant comfort are nearly impossible to

implement, correlation between current motion simulator studies and full-scale response

provides the possibility to assess potential occupant discomfort in tall buildings. In this

thesis, the motion simulator study conducted by Burton et al. (2005; 2006) was used to

evaluate the accelerations of a tall building by developing an approach to identify

waveforms of varying duration and catalog them by peak factor sorted by amplitude.

The occurrences of waveforms that matched the selected attributes of task disruptive and

nauseating motions in Burton et al. (2005; 2006) were identified and the percentage of

occupants potentially disrupted was estimated. The nauseating motions were found to

occur at a level affecting 5% of occupants, while the task disruptive motions observed

were projected to affect 14% of occupants. There were five or less occurrences like this

on each axis of the Korean Tower in 2007. More importantly, this provided a framework

with which a tall building‘s ongoing performance could be assessed when tenant

interviewing is not possible, allowing the client to evaluate wide ranging benchmarks

from motion perception to tolerance.

135

6.5 Future Work

This research was meant to enhance our understanding of tall building

habitability performance. As a result of the system identification framework developed

in Chapter 3 and the improved data acquisition system (Chapter 2) for the Korean

Tower, additional dynamic investigations will be possible for all the buildings in the

Chicago Full-Scale Monitoring Program. Future areas of investigation utilizing these

valuable full-scale datasets should include the continued analysis of long-duration

stationary responses to further identify dynamic properties, investigation of responses

associated with transient wind events like thunderstorms to determine the mechanisms

leading to high amplitude responses, as well as correlating wind tunnel predictions with

in-situ building responses. Additionally, the non-linear simulation developed in Chapter

3 may be improved by more faithfully simulating the amplitude-dependence of

frequency and damping. This may be done through the use of a recursive function that

seeks the amplitude-dependent behavior desired.

With respect to the pseudo-full-scale assessment of occupant comfort, the

analysis conducted for a single year on the Korean Tower should be extended to the

remainder of the data collected, including the other buildings in Chicago. Additionally,

the accelerations at the instrumented floor were correlated with the responses in motion

simulators indicating the likely percentage of occupants affected at that floor. This could

be extended to project the accelerations at other levels of the building and correlate those

with observations by Burton et al. (2005, 2006) to get an indication of the percentage of

occupants affected over the height of the building. Recently, Melissa Burton has agreed

to make data available from her studies to enable such a systematic evaluation of

acceleration performance. The extensions of this effort will also include the torsional

responses of this building, which are considerable for this building and were neglected in

Chapter 5 of this thesis. In total, it is hoped that these continuing efforts will help to

136

further enhance the habitability performance of one of the world‘s most complex and

expensive products.

137

APPENDIX A:

MODIFIED NONLINEAR NEWMARK‘S METHOD USING A MODIFIED

NEWTON-RAPHSON ITERATION PROCEDURE

138

TABLE AA.1: MODIFIED NEWTON-RAPHSON ITERATION PROCEDURE

(CHOPRA, 2001).

139

TABLE AA.2: MODIFIED NONLINEAR NEWMARK‘S METHOD WITH

ITERATION PROCEDURE (CHOPRA, 2001).

140

APPENDIX B:

DETAILED DESCRIPTION OF RDT IMPLEMENTATION

141

Although MDOF RDT has been introduced in the literature, the a priori isolation

of a single mode hastens the convergence of the method and simplifies subsequent

system identification. Therefore, a cursory examination of the power spectrum of the

data is required (Step 1) to identify the modes of interest, which are then each isolated by

bandpass filtering in Step 2. Though rarely executed in practice, Step 3 verifies the

major assumptions regarding the input that are necessary to insure that the RDS takes the

form of a free decay. The length of the captured segments is selected in Step 4, which

will then also be the length of the resulting RDS. As such, the length of captured

segments is often dictated by the number of oscillation cycles required for system

identification. Generally a longer length of data improves the estimates of system

properties; however, system identification is often limited to the first 3-4 cycles of the

RDS due to increasing variance, as shown in Equation (3.17) (Vandiver et al., 1982), and

the desire to quantify amplitude-dependent dynamic properties (Tamura and Suganuma,

1996). Triggers are often selected (Step 5) from the list provided in Table 3.2.

Correlation levels in Step 6 refer to the amount of overlap that will be permitted between

each captured segment: allowing no overlap is a practical application of Vandiver et al.‘s

(1982) segment independence assumption, while allowing overlap will introduce some

dependent segments but increase the number of segments for averaging. The main

analysis of the RDT begins with Step 7 where the entire response time history is

searched for candidate segments based on the trigger conditions selected in Step 5.

Segments are identified for inclusion in Step 8 by permitting only those that satisfy the

correlation level selected in Step 6. The RDS can then be obtained by averaging all the

eligible segments in Step 9. Finally, using any number of techniques for system

identification of a free decay, the natural frequency and damping can be obtained from

the RDS. Local averaging in Step 11 is optional, but can be used to improve accuracy

(Kijewski-Correa, 2003). A schematic of this local averaging process is provided in

Figure 3.14. Instead of using the results from a single amplitude trigger, an array of

142

equally spaced triggers within a few percent (+/-) of the desired trigger amplitude are

specified. RDT is executed for each of these triggers, resulting in a suite of RDSs.

Natural frequency and damping are identified from each RDS in the suite and then

averaged to yield the final dynamic property estimates, in a mean sense, and are

accompanied by a coefficient of variation (CoV) to benchmark the variability associated

with a given trigger.

143

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FULL-SCALE DYNAMIC CHARACTERISTICS OF TALL · PDF fileThis thesis focuses on the factors influencing the habitability limit state of tall building design