Inelastic Period of Multi Degree of Freedom System Estimation using Wavelet Transformation

Supervisor : Dr. K.K. Wijesundara

H.P. Rathnayaka E/10/279R.P.T.N. Sumathipala E/10/349

Content

• Objective

• Scope

• Literature Review

• Methodology

• Work Schedule

• References

Objective

• To estimate elastic and inelastic periods of multi

degree of freedom system using wavelet

transformation.

Scope

• Why we should estimate the elastic and inelastic periods

– Can use as input parameter in designing phase of a structure.

– Knowing inelastic period, damage state of a existing structure can be determined.

Literature Review

• Basics of Structural Dynamics(Periods of SDOF & MDOF)

• Ambient vibration measurements

• Techniques to extract elastic periods from ambient vibration measurements

• Effects of forced vibrations

Periods of structures• Single Degree of Freedom System and its Period

0 kxxm

m

k

2

T

m = lumped load, k = stiffness, x = displacementω = natural circular frequency, T = natural period

///////////////////////////////////////////

k k

m F

Chopra (1995)

Periods of structures• Undamped Free Vibration

Equation for equilibrium of SDOF for free vibration

ω = natural circular frequency, t = time, x = relative displacement

txtx

tx

cos0sin0

)(

0 0.5 1 1.5 2 2.5 3 3.5 4

-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Disp

lace

men

t (m

m)

0 kxxm m = mass, k = stiffness

Chopra (1995)

Periods of structures• Damped Free Vibration

Equation for equilibrium of SDOF for free vibration

ωD = damped natural circular frequency, λ is damping ratio

txt

xxetx DD

D

t

cos)0(sin)0()0(

)(

0 1 2 3 4

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Undamped10% Damping50% Damping100% Damp-ing

Time (Sec)

Dis

plac

emen

t (m

)

0 kxxcxm m = mass, c = damping coefficient, k = stiffness.

Chopra (1995)

Periods of structures• Multi Degree of Freedom System and its Periods

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

a) Actual Structure b) Mass of the building isconcentrated at floor levels

c) Equivalent System

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\

Younis,(2011)

Periods of structures• Two Degree of Freedom System

2

1

2

1

22

221

2

1

22

221

2

1

2

1 )()(

0

0

P

P

x

x

kk

kkk

x

x

cc

ccc

x

x

m

m

0

0)(

0

0

2

1

22

221

2

1

2

1

x

x

kk

kkk

x

x

m

m

Matrix form of governing equation,

Considering undamped free vibration,

0

0

0

0)(

2

1

2

12

22

221

x

x

m

m

kk

kkk

Based on Eigen values,

02 mk

Younis,(2011)

Periods of structures• Two Degree of Freedom System

Modes of deformations and respective frequencies

21

21

21

2

2

2

1

21

2

2

1

2121

4

2

1

mm

kk

m

k

m

kk

m

k

m

kk

21

21

21

2

2

2

1

21

2

2

1

2122

4

2

1

mm

kk

m

k

m

kk

m

k

m

kkPaul (2002)

Methods to extract periods• Fourier Transformation

Transform the view of the signal from time-base to frequency-base.

dtetfwF iwt)()(

The sum over all time of the signal f(t) multiplied by a complex exponential,and the result is the Fourier coefficients .

Mohammad(2011)

Drawbacks of Fourier Transformation

• Fourier transformation provides a signal which is localized only in the frequency domain

• It does not give any information of the signal in the time domain

• Cannot extract periods from a signal where structure vibrates in inelastic region

Sifuzzaman(2009)

• Continuous Wavelet Transformation

and b = Parameter localizing the wavelet function in the time domain, = CWT coefficients, a = Scale

Methods to extract periods

Continuous wavelet transformation (CWT) method is developed to obtain better decomposition of frequency domain response into frequency-time domain as an alternative method to Fourier Transformation.

CWT method can be used to decompose of a function x(t) into frequency-time domain as defined in the following form:

dta

bttx

aW ba

*),( )(

1

t*

),( baWHongnan(2008)

Vetterli(1995)

• Continuous Wavelet TransformationMethods to extract periods

Methodology• Experimental Measurements

Select two reinforced concrete buildings Obtain ambient vibration measurements using

accelerometers Extract elastic periods using

Fourier transformation Wavelet transformation

Compare the resultsAccelerometer

Methodology• Ambient Vibration Measurements

Advantages of ambient vibration measurements,

They are experimentally measurements of a building which are subjected to natural wind loads, traffic loads etc.

Why?

It is very cheap and easy

Methodology• Implementation of Numerical Model

Implement numerical models of selected buildings using OpenSees Software

Apply seismic loads on models Extract the elastic periods Validate the results and

develop the numerical model Estimation of inelastic periods

using Wavelet transformation

MethodologySelection of two structures

(2 storey and 4 storey reinforced concrete frame structures)

Collecting ambient vibration measurements using

accelerometers in both structures

Estimate the inelastic periods of the structure using Wavelet

transformation

Using Fourier transformation and Wavelet transformation

extraction of elastic periods

Validate the results and development of the numerical

model

Induce ground motions (seismic waves) for heavy shaking.

Collet the responses.

Work ScheduleWORK 7th Semester 8th Semester

11 12 13 14 15 1 2 3 4 5 6 7 8 9 10

Study about OpenSees software

Site Visits

Taking Measurements

Designing of Numerical Model

Analyzing Data

Estimating Inelastic Periods

Conclusion

Prepairing Research Paper

Presentation

References• Mohammad N.H. and Mohammad S.U.,(2011), Accelerating Fast

Fourier Transformation for Image Processing using Graphics Processing Unit, Journal of Emerging Trends in Computing and Information Sciences, Volume 2 No.8, Page no. 368 - 375.

• Sifuzzaman M., Islam M.R. and Ali M.Z.,(2009), Application of Wavelet Transform and its Advantages Compared to Fourier Transform, Journal of Physical Sciences, Vol. 13, Page no. 121 - 134

• Chopra A.K.,(1995), Dynamics of Structures, Prentice Hall, Part I, Page no 35 - 61 and Part II, Page no. 385 - 409.

• Younis M.I.,(2011), MEMS Linear and Nonlinear Statics and Dynamics, Springer New York, Chapter 2, Page No. 13 - 48.

• Martin V. and Jelena K.,(1995), Wavelets and Subband Coding, Prentice Hall, Chapter 4, Page no. 209 -304.

References• Hongnan L., Tinghua Y., Ming G., Linsheng H.,(2008), Evaluation of

earthquake-induced structural damages by wavelet transform, Journal of Progress in Natural Science, Volume 19, Issue 4, Pages 461 - 470

• Kumar M., Castro J. M., Stafford P. J. and Elghazouli A. Y.,(2010), Influence of the mean period of ground motion on the inelastic dynamic response of single and multi-degree of freedom systems, Journal of Earthquake Engineering and Structural Dynamics. Volume 40, Issue 3, pages 237 - 256

• Paul A. L.,(2002), Vibration of Multi Degree-Of-Freedom Systems, Unit 22, Page no. 1 - 8.

• Christopher T. and Gilbert P. C.,(1997), A Practical Guide to Wavelet Analysis, Volume 79, No 1, Page no. 61 - 78.

• Lokenath D.,(1998), Wavelet Transforms and Their Applications, Chapter 6, Page no 337 - 370.

THANK YOU !