TÓM tắt TIẾNG ANH LUẬN án CHUẨN đoán vết nứt của dàm BẰNG PHƯƠNG PHÁP đo DAO ĐỘNG

The structural behaviour used as diagnostic signal for damage detection problem is frequently the vibrational characteristics or dynamic response of structure.. The purpose of present th

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

The thesis has been completed at The Institute of Mechanics, Vietnam Academy of Science and Technology

Supervisors: Professor, Dr.Sc Nguyen Tien Khiem

Assoc Prof Dr Nguyen Viet Cuong

Reviewer 1: Professor, Dr.Sc Dao Huy Bich

Reviewer 2: Prof Dr Hoang Xuan Luong

Reviewer 3: Assoc Prof Dr Tran van Lien

Thesis is defended at The Institute of Mechanics, VAST

on , Date Month Year 2012

Hardcopy of the thesis can be found at

1 Vietnam National Library

2 Library of Institute of Mechanics

INTRODUCTION

Since early prognosis of cracks in structures is most important to keep away from accident, a lot of efforts have been focused to study cracked structures This leads to many papers of the topic published in journals on structural engineering; fracture mechanics; vibrations and control etc

Content of the intensive study consists mainly of two problems The first one is known as the forward problem that investigates structural behaviour with given position and size of cracks The remaider acknowleged as the inverse problem relates to identify the potential cracks from measured behaviour of structure under consideration The latter problem has recently received more interest from practical point of view

There are two approaches to the inverse problem, now could be called generally structural damage detection problem or structural health monitoring The first one is based on “symptoms” of damage occurrence that are directly extracted from the measured data by using different methods of signal processing This approach is termed by direct technique The other one makes use of not only the measured data but also an artificial model of damaged structure for damage detection The damage parameters are identified from fitting the simulated by model behaviour to the measured one The latter approach is called the model-based methods The advantage of the model-based methods in comparison with the symptom-based methods is its ability to encourage the useful knowledge acquired from study of the forward problem

The structural behaviour used as diagnostic signal for damage detection problem is frequently the vibrational characteristics or dynamic response of structure This is because of the dynamical properties of structure contain more information on the structure condition than the static ones The methods that use the vibrational characteristics for damage detection in structure are called the vibration-based method

Using the model-based methods for structural damage detection faces the difficulty associated with measurement and modeling errors and incompleteness of measured data The latter issues may make the problem of structural damage detection be ill-posed The common way to overcome the disturbs is (a) refining the model of damaged structure with aim to decrease modeling error in compution of the structure behaviour; (b) applying the modern mathematical methods that could reduce the effect of measurement error and allow to obtain consistent solution of the ill-posed inverse problem; (c) exposing more information on the structure condition from limited amount of measured data

The purpose of present thesis is to develop an efficient procedure for crack identification in beam from measured mode shape and response to moving load using the well known Tikhonov regularization method and wavelet transform

Outline of thesis is as follows

Chapter 1 Overview on vibration-based method for structural damage detection problem

Chapter 2 Theoretical background for regularization method, wavelet transform and model of cracked beam

Chapter 3 Mode shape-based procedure for multiple crack detection by using the Tikhonov’s regularization method

Chapter 4 Reponse of cracked beam to moving load and crack detection by the wavelet analysis of moving vehicle vibration

Concluding remarks and Referenses

Chapter 1 OVERVIEW

Structural damage is understood as a change in either physical

or geometrical properies of structure in comparison with a baseline configuration of intact structure Damage is often described by its position and degree

The structural damage detection problem was firstly investigated by Adams etc [1] for a bar with single damage modeled

by an axial spring at a position in bar The authors have obtained an equation allowing one to determine the damage position from

measured pair of natural frequencies Latter in [28] Liang and his coworkers extended the result for the case of beam by establishing general form of frequency equation of beam with single crack represented by a torsional spring Morassi [39] proposed a first order approximate frequency equation for cracked beam with variable stiffness Narkis [40] has calculated analytical solution of the problem for crack localization from measured two frequencies of simply supported beam Nguyen Tien Khiem và Dao Nhu Mai [41] investigated in detail change in natural frequencies versus crack position and depth for beam with different cases of boundary conditions Salawu [52] presented an overview on the frequency-based damage detection for structure The problem gets to be more complicated when number of damages increased Ostachowicz và Krawczuk [43] constructed frequency equation for beam with double crack in the form of 12×12 determinant Shifrin và Ruotolo [55], by using the delta function for representation of change in stiffness at crack position obtained the frequency equation for beam with arbitrary

number n of cracks in the form of determinant of order n+4 Khiem và

Lien [21] used the transfer matrix method for deriving the frequency

equation of beam with n cracks of the form of 4x4 determinants

Zhang and coworkers [61] have engaged the equation given by Khiem

và Lien [21] for multi-crack detection from measured natural frequencies Following Liang [28], Patil and Maiti [45] have obtained perturbation equation for natural frequencies of multiple cracked beam based on the energy conception Recently, Lee [24] developed the sensitivity method for crack detection from natural freuencies by using the finite element formulation Fernández-Sáez etc [14] introduced Rayleigh quotient to represent explicitly the first natural frequency of beam with single crack in term crack position and size This generalized Rayleigh quotient gives good approximation only for the fundamental frequency so that it is inadequate for solution of the crack detection

Generally, the frequency-based method of crack detection is limited by that very small number of frequencies could be available and cracks at different positions may produce identical change in a frequency Therefore, solution of the damage detection problem by

using frequencies is often non-unique In such the case in order to have unique solution one has to engage other features of structure that could be axtracted from measurements The mode shape of structure was early used for structural damage detection by Rizos et al [51], Yuen [60], West [65]… At the first time, the mode shape has been taken in use for calculating different damage indices such as Modal Assurance Criteria (MAC) that is unable to be used for damage localization in structure Then, Kim and his coworkers [20] have developed PMAC hay COMAC for the problem but they exposed to

be insensitive to damage Despite that Ho and Ewins [16], Parloo et al [46] proposed different damage indices calculated from given mode shape or its derivatives, Pandey etc [44] demonstrated that the mode shape is less sensitive to damage than mode shape curvature Based on the idea, Ratcliffe [49], Wahab and De Roeck [63] have developed different procedures for damage localization from mode shape and mode shape curvature

Through studying vibration of multiple cracked beam, Li [27] has derived a recurrent relationship between mode shape if the beam

in both sides of a crack However, this is not a closed form solution for the mode shape so that it cannot be used for crack detection from measured mode shape By using the step function Caddemi và Caliò [4] obtained closed form solution for mode shape of beam with arbitrary number of cracks that is an explicit expression of the mode shape through crack parameters The closed form solution of mode shape has not been taken in use for multiple crack detection because it contains Dirac delta function

Although the mode shape of damaged structure could provide more useful information on the damage circumstance of a structure, measurement of mode shape is more difficult The change in mode shape due to damage is usually more difficult to be monitored than its shift caused by measurement error Hence, solving the problem associated with measurement erroneous in using mode shape for structural damage detection presents a great interest

Chapter 2 THEORETICAL BACKGROUND

2.1 Tikhonov’s regularization method

2.1.1 Conception of inverse problem

The essential content of inverse problem is to determine the

“cause” with known “consequent”, Tarantola [56] This problem was

investigated early in Mechanics as determining the force applied to a

body from given its movement trajectory Recently, a novel

formulation of the inverse problem has come from the practical

demands: “establishing model of an existing object from its observed

current behavior” This is very complicated problem that is research

subject in different fields of science and engineering and so far to be

completely solved The problem, called system identification, is also

the substance of the condition monitoring of existing structures

The crucial attributes of the inverse problem are strong

sensitivity of solution to either modeling or measurement inaccuracy

and having non-unique solution because of incompleteness of given

data One of the most effective methods in solving the problem is

proposed by A.N Tikhonov that is called the regularization method

and briefly described below

where A is an arbitrary matrix (might be nonsquare or singular), b is a

vector that is determined only as an approximation to unknown exact

one b.

Tikhonov và Arsenin [57], [58] suggested that solution of

equation (2.1) can be found from solution of the mean square problem

}, {

min arg Ax b2 L(x x0)2

x

with α, L,x0 being regularization factor, regularization operator and a

prior information about solution respectively Leaving outside the

equivalence between equations (2.1) and (2,2) one is going to consider

equation (2,2)

Theorema For αf 0solution of equation (2.2) can be found

uniquely from the equation

)

(ATA+αLTL x=ATb+αLTLx0 (2.3) Obviously, when α→ 0 solution of equation (2.3) becomes the

conventional mean square solution xRLS →xLS =(ATA)−1ATb

The regularization factor α is chosen as solution of the

equation

, RLS α − b =δ

where δ - noise level of vector b

In turn, equation (2.3) can be solved by using the Singular

Value Decomposition (SVD) of matrix A

T

V Σ U

where U, V are square matrices of order m and n respectively and

n T

m

U = = , matrix Σ(m×n) =diag{σ1 , ,σq},q= min(m,n).

Hence, solution of equation (2.3) with x0 =0 is

, ˆ

r

T k

The Fourie transform was the most powerful tool in signal

analysis in frequency domain However, it cannot be used for analysis

of non-stationary processes when frequency is dependent upon time

This gap can be fulfilled by using the newly developed wavelet

transform that is briefly described below

2.2.1 Difinition of wavelet transform

The continuous wavelet transform is defined as

, ) ) ) , ( =+∞∫ ,

∞

−

dt t t f b a

where

), ( ) / 1 (

,

a b t a t

b

−

a is a real number acknowledged as scale or dilation factor, b is

transition, ψ is called mother wavelet function and ( )t ψ*( )t is the

complex conjugate of ψ For every value of a and b, W(a,b) is ( )t

determined as wavelet coefficient of the given signal f(t)

Inverse wavelet transform is

∫ ∫

= +∞

∞

− +∞

∞

−

−

2 ,

)

a

da db b a W C t

where

)

( 2

ξψ

From mathematical point of view, wavelet is convolution of the

signal and wavelet function, allowing compressing a signal

2.2.2 Application of wavelet

Wavelet is widely used in signal processing, especially in

detecting descontineous of a signal It can be utilized also for

detecting similarity of signals, filtering or compressing signals

Recently, wavelet has been used for local damage detection in

structures through wavelet analysis of response of structure

2.3 A model of multiple cracked beam

2.3.1 A crack model

It was approved in Fracture Mechanics that a crack occurred at

a section of beam member introduces local flexibility could be

calculated by the formulae

), ( ) / 6 (

M

where h, b are high and wide of the beam’s rectacular cross section, EI

is the bending stiffness, s = a/h is relative crack depth and F I (s)is an

experimental function In such the case, crack can be represented by a

torsional spring of the stiffness

)).

( 6 /(

2.3.2 Model of beam with crack

The rotational spring model of crack allows one to represent a

crack at a section in beam in a form of compatibility conditions at

crack that should be satisfied by the displacements and forces of beam

in both sides of crack The study of cracked beam with the crack

model can be carried out by dividing the beam into sub-beam

bordered by crack position and beam ends, Rizos và cộng sự [51]

This approach enables to use the governed equations without any

change for solving the problems of cracked structures

2.3.3 Finite element model of beam with crack

Qian et al [48], have shown that stiffness matrix of a cracked

beam can be expressed as

T e e

Where

T e

0 1 1

T and the flexibility matrix ~ ( 0 ) ( 1 )

ij ij e

with

2

2 3

2

2 3 ) 0 (

l EI l

e e

e e

=

1 1

2

1 2 2 1 2 ) 1 (

2

2 2

nR R

nL

R nL mR R nL

e

e e

,

, '

36 ,

'

36

0

2 2 0

2 1 2

In this Chapter, the fundamental of the Tikhonov’s relarization

and Wavelet transform methods has been presented The

regularization method allows obtaining unique solution of the standard

inverse problem with noisy measured data The wavelet transform is

an helpful tool in detecting small change in response of structure due

to damage Additionally, the continuous and finite element models of

cracked beam are briefly described

Chapter 3 CRACK DETECTION BY NATURAL FREQUENCIES AND

MODE SHAPES

3.1 An explicit expressions of frequency and mode shape of

multiple cracked beams

3.1.1 Generalized Rayleigh quotient for multiple cracked beam

Consider a beam free vibration of which is described by

equation

, 2 , 1 , ,

0 ) ( ) (

2 4 4

)

EI

F x

k k

k IV

k

ωρλφ

λ

where 2 ,

k

ω φk (x), k= 1 , 2 , denote the natural frequencies and

associated mode shapes Suppose that the beam length is divided into

N segments(x j−1,x j),j= 1 , ,N , each of them contains a crack (e j,K j)

x kj

N

j x j

k

dx x

B L B x dx

x F

2

) (

) 0 ( ) ( ) (

"

) (

"

φ

φγφ

ρ

Let the eigen-functions be chosen in the form

), ( ) ( ) (x 0 x c x

kj k

with φk0 (x) being mode shape of uncracked beam that is continuous

together with its derivatives φk′0(x),φk′′0(x),φk0′′′ (x) everywhere along

the beam length (0, L) and linear functions

+

=

−

− 1 0

1

), ( ) (

, 0 )

(

j j j j k j

j j

jk kj c

e x x D

x C x

p

p

φγ

,

kj

C D kj are constants and S(x) = [sinhλx+ sinλx] / 2λ,

• For simply supported beam the equation (3.2) is

, sin sin 3

) ( 2 sin

4

1

sin 2 1

1 , 4 1

2

1 2 2

e k e k q k e k

e k

πγπγπ

πγ

πγω

If cracks are small, γj=εηj, the asymptotic approximation of

equation (3,5) with regard to small parameter ε,is

sin 2 1

1 2 2

3 / ) 1 ( [(

2 sin 4 1

sin 2 1

2 4 2 2 2 2

1 2 2

2

j i

i j j

N

j

j j k

e k

π π γ

π γ

π γ ω

ω

− +

N

k k

e e q

e

e

1 , 4 1

2

1 2 2

0 2

) ( ) ( 3

) ( 2 1

) ( 1

γγλ

γ

γω

0 2

) ( 1

Φ +

=

=

j k j k

j k j k

k k

e e

e

γλγ

γω

) ( ) (

2 1 2 1 2 2 2 1 2 1

2 2 2 1 2 1 0

Φ + Φ +

=

=

e e e

e

e e

k k

k k

k

k

γγ

1

k k

3.1.2 Explicit expression for mode shape

Consider the beam with n cracks at e j,j=1, ,n Assume that

the cracks are modeled by spring of stiffness K0j(j= 1 , ,n) that can

be calculated from crack depth a j(j= 1 , ,n).

The governed equation for natural mode is

), 1 , 0 ( , 0 ) ( )

)

ΦIV x λ x x λ=L4 ρFω2/ EI (3.15)

Introducing the function

) sin )(sinh 2 / 1 ( )

e j

j j

0 0

) (

f

x x S

x x

One can express general solution of equation (3.15) in the form

, ) ( )

( ) (

1

Φ

= Φ

=

n

k k K x e k x

where Φ0(x)is general solution of equation (3.15) for uncracked beam

that can be expressed as

) , ( ) , ( ) ,

with constants C, D and functions L1 (x,λ),L2 (x,λ) satisfying

boundary conditions at the left end of beam

Afterward, one gets

, ) , , ( ) ( )]

, ( )

, ( )

,

(

[

1 0

1 2

j j

j

k k j

L e e S x

L D x

L

λ

γ λ μ

λ λ

μμ

=

T n e e

the form

∑

= Φ

=

r l r

l) (x,λ ) ) (x) C α ) (x,λ,e)μ (3.24)