Extracting Dynamic Characteristics from Strong Motion Data Palle Andersen Structural Vibration...

Extracting Dynamic Characteristics from Strong Motion Data

Palle Andersen

Structural Vibration Solutions A/S

Denmark

www.svibs.com

Contents

• Introduction

• Traditional Modal Analysis and Natural Input Modal Analysis

• Case Study: The 777 Tower in LA, CA

• Conclusions

Introduction

• Dynamic characterization of civil engineering structures is typically performed by use of ambient excitation (weak motion)

• The obtained dynamic characteristic are usually the modal parameters. Natural frequency, damping ratio and mode shape

Modal Behavior

Showing that y the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Displacements

y(t) = q1(t)1 + q2(t)2 + q3(t)3+ + qn(t) n

m

1rrr qqy

= ++ + +

Introduction

• The modal parameters are constant for a linear structure. If the structure is non-linear the parameters will change when response levels are changing.

Why use strong motion data?

• Other modes excited than in the case of ambient excitation.

• To obtain dynamic characteristics describing the strong motion state if a building is non-linear.

• To gain confidence on FE models used to predict the response of buildings due to strong ground shaking

Traditional Modal Technology

Input

Output Time DomainFrequency Domain

Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s²)/N]Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s²)/N]

Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100u

1m

10m

100m

1

[Hz]

[N] Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100u

1m

10m

100m

1

[Hz]

[N]

Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

1m

10m

100m

1

10

[Hz]

[m/s²] Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

1m

10m

100m

1

10

[Hz]

[m/s²]

Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N]

Time(Response) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s²]

FFT

FFT

Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-2k

-1k

0

1k

2k

[s]

[(m/s²)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-2k

-1k

0

1k

2k

[s]

[(m/s²)/N/s]

InverseFFT

ExcitationResponse

ForceMotion

InputOutput

H() = = =

FrequencyResponseFunction

ImpulseResponseFunction

Combined System Model (analysis procedure)

Measured

Responses

Stationary

Zero Mean

Gaussian

White Noise

Model of the combined system is estimated from measured responses

Excitation Filter

(linear, time-invariant)

Structural System

(linear, time-invariant)

Unknown excitation forces

Combined System

Modal Model of Structural System extracted from estimated model of Combined System

Assumptions

Mathematical Stationary input force signals can be approximated by

filtered zero mean Gaussian white noise

– Signals are completely described by their correlation functions or auto- & cross-spectra

– Synthesized correlation functions or auto- & cross-spectra are similar to those obtained from experimental data

Practical Broadband excitation All modes must be excited

Non-Parametric Modal parameters are estimated directly from curves,

functional relationships or tables

Parametric Modal parameters are estimated from a parametric

model fitted to the signal processed data

Identification Techniques

Experimental data Signal ProcessingParameter Estimation

Parametric ModelModal Parameters

Experimental Data Signal Processing Modal Parameters

Identification Techniques

Non-parametric method: » Frequency Domain Decomposition, FDD

Parametric methods: » Enhanced Frequency Domain Decomposition, EFDD

» Stochastic Subspace Identification, SSI

Signal Processing Modal FitMeasured Data fn, ,

FDD(Pure Signal Processing)

EFDD (Simplest fit)

SSI (Advanced fit)

Time Histories

Tkkk

k

k

k

kn

k

Rj

R

j

RjH

,)(1

k

T

kkk

k

Tkkk

N

kyy j

d

j

djG

1

)(

Frequency Domain Decomposition (FDD)

Txxyy jHGjHjG )()()( White noise excitation:

Partial fraction expansion:

Lightly damped structure:

Power Spectral Density (PSD) estimation

k

T

kkk

k

Tkkk

kyy j

d

j

djG

)(

A number of modes can often not be found by simple peak-picking

Modes may be coupled by small frequency difference or by high damping

The number of modes equals the number of terms in the linear decomposition in the modal transformation

The number of terms is the rank of the PSD matrix

The spectra can be used for Operational Deflection Shapes

but do not contain modal information!


Extracting Modal Parameters from PSD response matrix

Modal Behavior

Showing that y the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Displacements

y(t) = q1(t)1 + q2(t)2 + q3(t)3+ + qn(t) n

m

1rrr qqy

= ++ + +


y(t) = []q(t)

[Cyy()] = E{y(t+)y(t)T}

[G yy()] = [][Gqq()] []H

[Cyy()] = E{[]q(t+)q(t)H[]H} = [][Cqq ()] []H

[G yy()] = [V][S] [V]H

Correlation:

Modal behaviour:

=>

i.e. by Fourier Transform:

Same form as Singular Value Decomposition of PSD:

Singular Value Decomposition of Hermitian matrices

[A] = [V] [S] [V]H = s1 v1 v1H+s2 v2 v2H +..

The Singular Value Decomposition

of the response matrices is

performed for each frequency

A real diagonal matrix Number of non-zero elements in

the diagonal equals the rank

[S] =

0......0

.0.

.0.

.0..

...0

...00

...00

0...000

2

1

s

s

s

[V] =

Orthogonal columns Unity length columns Approximates the Mode shapes

nvvvv ...321


Singular values Singular vectors

Singular Value Decomposition of PSD Matrix


[G][G]

Frequ

ency

[G][G]

i

SVD performed for each frequency, response spectra of modes Frequency at peak found from decoupled modes The singular vectors approximates the mode shapes No damping estimated

[dB]

Frequency [Hz]0 5 10 15 20 25

-80

-72

-64

-56

-48

-40

-32

Magnitude of Pow er Spectral DensityTransducer #1,Transducer #1 [R1,C1] - 1

[dB]

Frequency [Hz]0 5 10 15 20 25

-81

-72

-63

-54

-45

-36

-27

Singular Values of Pow er Spectral Density

PSD Mag. SVD of PSD

S1: At least one mode exists S2: At least two modes exist

Decoupled Modes

Mode 1

Mode 2

[A] = [V] [S] [V]H = s1 v1 v1H+s2 v2 v2H +..

Enhanced FDD Method (EFDD)

Estimates Frequency & Damping from each data sets

Lists Average Values and Standard deviation

Mode Freq. [Hz] Std. Freq. [Hz] Damp. [%] Std. Damp. [%]352.3 Hz - Enhanced FDD 352,3 3.935 564.6m 146.3m487.7 Hz - Enhanced FDD 487,7 0,822 635.3m 200.1m716.5 Hz - Enhanced FDD 716,5 3.463 516.6m 74.43m867.6 Hz - Enhanced FDD 867,6 2.398 527.2m 99.7m971,5 Hz - Enhanced FDD 971,0 6.135 369.3m 197.1m

Simple curvefitting is used (linear regression)

Singular Value Spectral Bell Identification

User definable MAC rejection level (default 0.80)» Compare singular vectors i against singular vector 0

Each mode in each data set specified separately [dB | (1 m/s²)² / Hz]

Frequency [Hz]

0 500 1k 1.5k 2k 2.5k-60

-40

-20

0

20

40

60

Singular Value Spectral Bell Identif icationfor Data Set: Measurement 3

Brüel & Kjaer, Operational Modal Analy sis Pro, Release 3.1

Project: Black Plate.axp

0

i

Damping Calculation, 1

Autocorrelation of SDOF Bell using IDFT

Graphical feedback of selected interval » maximum and minimum correlation values

NormalizedCorrelation

Time Lag [s]

0 30m 60m 90m 120m 150m-1.2

-800m

-400m

0

400m

800m

1.2

Normalized Correlation Function ofSingular Value Spectral Bellfor Data Set: Measurement 3



Damping Calculation, 2

Damping from Logarithmic Envelope of correlation function

Logarithmic Decrement methodLog of AbsoluteExtremum Value

Time Lag [s]

0 30m 60m 90m 120m 150m-6

-5

-4

-3

-2

-1

0

Validation of Damping Ratio Estimatefor Data Set: Measurement 3



Frequency Calculation

From how frequent the correlation function crosses zero

Zero Crossing Number

Time Lag [s]

0 30m 60m 90m 120m 150m0

300

600

900

1.2k

Validation of Natural Frequency Estimatefor Data Set: Measurement 3



Improving and enhancing FDD -> EFDD

IFFT performed to calculate Correlation Function of SVD function Frequency and Damping estimated from Correlation Function Mode shape from weighted sum of singular vectors

H = Complex Conjugate transpose (Hermitian) of vector/shape,

s1

s2

••

• • • ••

•••

•

0•

i

MAC = 8,0)()(

)(

iHi0

H0

2i

H0

Improved shape estimation from weighted sum:

i

iiweight s

Select MAC rejection level(default 0,8):

Case Study: The 777 Tower in LA, CA

Left: Figueroa at Wilshire Tower (1990, 52 Storey)

Right: The 777 Tower (1989, 54 Storey)

Data and pictures are provided by Dr. Carlos E. Ventura, University of British Columbia, Vancouver, BC, Canada.

Both building are permanently intrumented by the Califonia Division of Mines and Geology – Strong Motion Instrumentation Program (CSMIP)

On January 17, 1994 the instrumentation recorded valuable data of the Northridge earthquake.

The simulatenous collected data from the two buildings have been used in a number of comparison cases concerning; base shaking experience, response (shock) spectra, modal characteristics based on strong motion data.


Dimensions of footprint: 64.6 x 41.4 m. Overall elevation above ground is 218 m.

Has a 4-storey garage below ground.

Instrumentation is at: P4 (lowest), ground, 20th, 36th, 46th and penthouse levels.


The epicentre of the Northridge earthquake was approx. 32 km from the building.

The 20 accelerometers recorded motions for about 180 seconds. Sampling rate 100 Hz.

Transducer 18 (Penthouse level). Peak value of displacement is 16.7 cm.

The strong ground shaking is only a few seconds, but the response is more than 180 seconds.

During the ground shaking the higher modes dominates. After the shaking the lower (1st) mode dominates.


Animation og part of the event (~1 min)

Unmeasured points animated through linear interpolation.

Transducer 18 (Penthouse level). Peak value of displacement is 16.7 cm.

The strong ground shaking is only a few seconds, but the response is more than 180 seconds.

During the ground shaking the higher modes dominates. After the shaking the lower (1st) mode dominates.


First 6 modes identified with Frequency Domain Decomposition (FDD)

dB | 1.0 / Hz

Frequency [Hz]

0 0.3 0.6 0.9 1.2 1.5-120

-80

-40

0

40

Frequency Domain Decomposition - Peak PickingAverage of the Normalized Singular Values of

Spectral Density Matrices of all Data Sets.


Mode 1. f = 0.1628 Hz, T = 6.143 Sec.

Mode 2. f = 0.1953 Hz, T = 5.12 Sec.


Mode 3. f = 0.3662 Hz, T = 2.731 Sec.

Mode 4. f = 0.5046 Hz, T = 1.982 Sec.


Mode 5. f = 0.5371 Hz, T = 1.862 Sec.

Mode 6. f = 0.8219 Hz, T = 1.2167 Sec.

Conclusions

• Modal analysis has been applied to strong records of the Northridge earthquake.

• 6 lowest modes has been extracted using the Frequency Domain Decomposition (FDD) method.

• The extracted modes characterizes the dynamics of the investigated building at the strong motion level.

• In case of non-linear structures, the characteristics will be different from what would be observed in case of a more traditional modal survey using ambient (weak motion) excitation.

Extracting Dynamic Characteristics from Strong Motion Data Palle Andersen Structural Vibration...

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