Guided waves in a monopile of an offshore wind turbine

Ultrasonics 51 (2011) 57–64

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Guided waves in a monopile of an offshore wind turbine V. Zernov a,*, L. Fradkin a, P. Mudge b a Waves and Fields Research Group, Department of Electrical, Computer and Communication Engineering, Faculty of Engineering, Science and Built Environment, London South Bank University, London SE1 0AA, UK1 b The Welding Institute, Granta Park, Great Abington, Cambridge CB1 6AL, UK

a r t i c l e

i n f o

Article history: Received 7 January 2010 Accepted 29 May 2010 Available online 4 June 2010 Keywords: Guided waves Wind turbine Composite

a b s t r a c t We study the guided waves in a structure which consists of two overlapping steel plates, with the overlapping section grouted. This geometry is often encountered in support structures of large industrial offshore constructions, such as wind turbine monopiles. It has been recognized for some time that the guided wave technology offers distinctive advantages for the ultrasonic inspections and health monitoring of structures of this extent. It is demonstrated that there exist advantageous operational regimes of ultrasonic transducers guaranteeing a good inspection range, even when the structures are totally submerged in water, which is a consideration when the wind turbines are deployed off shore. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The paper addresses issues arising in ultrasonic inspection of offshore structures, such as wind turbine monopiles. These structures are subject to fatigue loading and corrosion, which can remain unnoticed and undetected until there is a catastrophic failure. It is thus desirable to monitor their structural integrity throughout the entire life cycle. Ultrasonic guided waves are already widely used in nondestructive evaluation [1], in particular for inspection of iron pipes [2]. One of the challenges facing the technology is predicting whether and under what conditions enough wave energy is delivered to the area containing possible damage. Another important issue is the minimization of energy loss, that is increasing an inspection range. The monopiles can be modeled as two-dimensional, because their diameter is much larger than its wall thickness. The relevant two-dimensional structure can be described as two overlapping grouted steel plates. The fundamental behavior of the guided waves known as Lamb waves, in homogeneous plates, has been intensively studied over the past century [3,4]. One of the most interesting problems of practical interest is the conversion of Lamb waves in structures with inhomogeneities (that is, defects, joints, etc.). Rokhlin [5] solved the problem of interaction of Lamb waves with adhesive metal lap joints using the Wiener–Hopf technique. A similar problem of Lamb waves transmission in adhesively bonded lap joints has been addressed in [6] by using a finite element code. The so-called ‘‘projection method” has been employed by Gregory 1 Now at Sound Mathematics Ltd., Cambridge CB4 2AS, UK. * Corresponding author. E-mail address: [email protected] (V. Zernov).

0041-624X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2010.05.008

et al. [7] to study their reflection at the edge of a plate, and by Flores-López et al. [8] to investigate their scattering by a surface breaking crack. The interaction with the crack has been also studied by Castaings et al. [9], who used the modal decomposition to satisfy boundary conditions at a discrete set of points along the crack (a collocation method). Recently, a number of hybrid approaches have been proposed. These are used to solve the problem using a finite element method [10] or boundary element method [11] in a finite region containing inhomogeneities and then utilizing the modal expansion in the regions where the waveguide is homogeneous. The general framework of this approach is described in [12]. The modal expansion can be performed using SAFE (the Semi-Analytical Finite Element) method [13]. It allows the authors to investigate modal conversion in waveguide of complex structure (i.e. three-dimensional, composite, etc.). We present an alternative semi-analytical approach, which is similar to the one described in [14]. We split the construction into several simple waveguides and represent the displacement field inside each of them as a sum of Lamb modes. These are chosen to satisfy the appropriate boundary conditions. The solution of the resulting system of linear equations allows us to calculate the reflection and transmission coefficients on the waveguide boundaries. The technique leads to a straightforward estimation of modal amplitudes. It is applicable when the total displacement can be expanded in Lamb waves only, e.g. when working in two dimensions, it is applicable if all boundary segments are straight. The model allows for the energy losses caused by radiation into water. The paper is organized as follows: in the next section we derive dispersion equations for Lamb waves in the waveguides of interest.

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Section 3 is devoted to the mode conversion coefficients. In Section 4 we discuss the amplitudes of modes present in the loaded overlapping grouted plates. The resulting approach is used to investigate response to loading in symmetric and asymmetric monopiles in Sections 5 and 6, respectively. 2. Dispersion equations in plates and composites immersed in water The monopile structure is schematically shown in Fig. 1. To be specific, let us use realistic dimensions and assume the cylindrical pile to be approximately 30 m long, have a 5 m diameter and be driven approximately 20 m into the seabed. The top end of the pile is typically 2 m above the sea level. Let the tubular turbine superstructure be mounted on top of the pile, with an approximately 6 m overlap, which is filled with grout. Thus, the mid section is a composite consisting of steel/grout/steel layers. Throughout the paper, the parameters H1, H2 and H3 refer to the thicknesses of the bottom pile, grouted layer and top pile, respectively. The waves are excited by a transducer installed on the joint’s butt-end. As mentioned in the Introduction, the structure can be modeled as two overlapping grouted steel plates. The model can be decomposed into three simple waveguides I, II and III as illustrated in Fig. 2. The waveguides I and III are plain steel plates and the composite waveguide II consists of a grout layer sandwiched in between such plates. We assume that both steel and grout layers can be modeled as isotropic and homogeneous elastic solids. Then inside each layer the time harmonic displacement u of the circular frequency x satisfies the elastodynamic equation

r  ðr  uÞ þ v2l r  ðr  uÞ þ K 2l u ¼ 0;

l ¼ 1; 2; 3;

ð1Þ

where the index l = 1, 3 denotes the bottom and top plates, respectively, and the index l = 2 refers to the grouted section. The parameter vl ¼ cLl =cTl stands for the ratio of the shear and compressional wave speeds and K l ¼ x=cTl is the shear wave number. Let rl ¼ frl;ij g2i;j¼1 denote the stress tensor. The displacement field u and the stress tensor rl are related by the following formula:

rl;ij ðuÞ ¼ kl dij r  u þ ll ðui;j þ uj;i Þ; i; j ¼ 1; 2; where kl and ll are the Lamé constants. On the surface C with the outer normal vector n = (n1, n2), the components of the normal stress vector Sl(u) are given by

Sl;i jC ¼ rl;ij ðuÞnj ;

i ¼ 1; 2:

ð2Þ

Fig. 2. Decomposition of the overlapping grouted plates into waveguides I, II and III. The vertical arrows indicate the position of the transducer.

The waveguides II and III can be partially submerged in water, where the motion is described by the potential / satisfying the Helmholtz equation

M/ þ K 2W / ¼ 0;

ð3Þ

with KW = x/cW – the compressional wave number in water and cW – its speed. The corresponding displacement u and pressure p are related to the potential / by

u ¼ r/;

p ¼ qW x2 /;

ð4Þ

where qW is the water density. Let us consider the guided waves in the steel plate, which is submerged in water and has the half width h1 = H1/2. At the plate’s boundaries x2 = ±h1, the normal displacement coincides with the normal displacement in water, normal stress is compensated by the pressure field and plate’s boundary is free of tangent loading. Therefore, the boundary conditions on the plate’s sides x2 = ±h1 can be written as

ðu1;1  uW;1 Þjx2 ¼h1 ¼ 0;

ðr1;11 þ pW Þjx2 ¼h1 ¼ 0;

r1;12 jx2 ¼h1 ¼ 0;

ð5Þ

where ul and uW refer to the displacement in the layer l and water, respectively. Also, as x2 ? ±1 we impose the standard radiation conditions at infinity, that is, request that the waves in water either decay exponentially or carry the energy away from the plate. We look for the guided waves of the form u(x1, x2) = U(x2) exp(iax1). The problem can be decomposed into symmetric and antisymmetric. The resulting symmetric secular equation for a plate, known as dispersion equation, is

 T 2  S1 h1   þ UI ða; xÞ ¼  CT1 a21   T 2 S1 a 1

GL1 CL1 h21 GL1

   2 2  K 1 h1 q  ¼ 0;   2cW 0

ð6Þ

where we have

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ a2  K 2l hl ; T l

Fig. 1. A schematic of a monopile.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 hl ¼ a2  l hl ; 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 c ¼ a2  2l hl ; L l

q ¼

vl

qW ; al ¼ ahl q1

cW ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  K 2W h1 ;

ð7Þ

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V. Zernov et al. / Ultrasonics 51 (2011) 57–64

and for the sake of brevity, we introduce the notation

Sm l ¼

m l Þ

sinhðc

cml

m ¼ L; T;

;

m Cm l ¼ coshðcl Þ;

m m Gm l ¼ cl sinhðcl Þ;

l ¼ 1; 2; 3:

ð8Þ

Eq. (6) specifies implicit functions an(x), n = ±1, ±2, . . ., which are m called the dispersion curves. Note that the expressions Sm l ; C l and Gm do not possess any poles or branch points. The only branch l points in (6) are the ones associated with the parameter cW. The antisymmetric counterpart of (6) is given by

 T 2  C 1 h1    UI ða; xÞ ¼  GT1 a21   T 2 C 1 a1

C L1 SL1 h21 C L1

3. Equations for mode conversion coefficients

   2 2  K 1 h1 q  ¼ 0:   2cW 0

ð9Þ

The well known dispersion equations for the plate with the traction free boundary conditions can be obtained from (6) and (9) by postulating that qW = 0 and cW = 1. The waveguide II consists of steel/grout/steel layers. Let hi = Hi/ 2. On the steel/grout interfaces x2 = ±h2, we impose the condition of continuity of the displacements and normal tractions

  u1  u jx2 ¼h2 ¼ 0; S 1  S 2 jx2 ¼h2 ¼ 0;    3  u  u2 jx2 ¼h2 ¼ 0; S 3  S 2 jx2 ¼h2 ¼ 0: 

is, when H1 – H3 or else when the bounding plates are made of different material the motion cannot be decomposed into symmetric and antisymmetric. In this case the characteristic function UII(a, x) can be represented as a determinant of a 14  14 matrix (if the composite is in air the matrix is 12  12). This determinant is too cumbersome to be reproduced here, though its derivation is straightforward. Below the above dispersion equations are solved using the numerical procedure described in [15].

 2

Let us consider the upper joint of the overlapping grouted plates (Fig. 3). The lower joint can be studied in a similar fashion. There are two types of excitation that can be applied to the joint, a transducer can be placed on the traction free part of the interface BD or else, there can be a guided wave incident on the boundary AD from within. In either case the resulting excitation of the displacement/stress field on the boundary AD can be described as

 T U E ðx2 Þ ¼ uE ðx2 Þ; v E ðx2 Þ; rE ðx2 Þ; sE ðx2 Þ :

ð13Þ

where we use the standard notation

ð10Þ

u ¼ u1 ;

v ¼ u2 ; r ¼ r11 ; s ¼ r12 :

ð14Þ

] nð

On the composite/water interfaces x2 = h2 + H3 and x2 = h2  H1 the boundary conditions are similar to (5). The radiation conditions at x2 = ±1 should also be satisfied. When the bounding steel plates are made of the same material and have the same thickness H1 = H3 the motions in the waveguide II can be decomposed into symmetric and antisymmetric. The symmetric motions are described by the dispersion equation

Let us denote by a xÞ; ] ¼ I; II the wavenumbers in the waveguides I and II. Let the positive indices n = 1, 2, . . . correspond to the waves which either carry energy or decay exponentially in the positive direction. Let them be arranged in ascending order of imaginary parts

UþII ða; xÞ ¼ 0;

Let the wavenumbers for the negative indices be defined by

ð11Þ

Im a]n1 P Im a]n2 ;

n1 P n2 P 1;

] ¼ I; II:

a]n ¼ a]n ; n P 1; ] ¼ I; II:

where

  C T h2  1 1   a2 GT  1 1  2 T  a1 C 1   UþII ¼  a21 C T1   T  G1  T  C h2  1 1   a2 GT 1

1

ST1 h21

C L1

GL1

0

0

a21 C T1 a21 ST1 a21 ST1

SL1 h21

C L1 h21

0

0

C L1 C L1 SL1 C L1 SL1 h21

GL1

0

0

C T1 ST1 h21 2 T 1 C1

a

GL1 C L1

l a22 C T2 l C L2 l GT2 h1 l SL2 h1  

GL1

C T2 h22

C L2

C L1 h21

a22 GT2 h

SL2 h22 h

 0   2 K 21 h1 q   2cW   0 ;   0   0   0 

*

with l = l1/l2 and h* = h1/h2. The corresponding antisymmetric dispersion equation for the three-layer symmetric composite immersed in water is

UII ða; xÞ ¼ 0;

U I ðx1 ; x2 Þ ¼ U II ðx1 ; x2 Þ ¼

1 X

I

C In U In ðx2 Þeian x1 ;

n¼1 1 X

II

C IIn U IIn ðx2 Þeian x1 ;

1

1

C L1

GL1

0

0

a21 C T1 a21 ST1 a21 ST1

SL1 h21

C L1 h21

0

0

C L1 C L1 SL1 CL1 SL1 h21

GL1

0

0

CT1 ST1 h21 2 T 1 C1

a

GL1 CL1

a22 ST2 l GL2 l 1 1 CT2 h l CL2 h l

GL1

ST2 h22

CL1 h21

2 T 2 C2 h 

a

GL2 L 2 C2 h2 h

ð18Þ

where C In and C IIn are unknown coefficients and the vectors U denote the displacements/stresses of the associated Lamb mode

    2 2  K 1 h1 q   2cW   0 :   0   0   0  0

As with a simple steel plate, when the structure is in air, the secular equations are obtained from (11) and (12) by postulating that qW = 0 and cW = 1. When the waveguide II is not symmetric, that

ð17Þ

n¼1

ð12Þ

ST1 h21

ð16Þ

They are then the wave numbers of the guided waves that carry energy or decay exponentially in the negative direction. Clearly, the scattered field contains in the plate I, only the modes associated with the set of dispersion curves faIn ðxÞg1 n¼1 and in the composite II, only set of dispersion curves faIIn ðxÞgþ1 n¼1 . It follows that, the scattered field has the form

where

  C T h2  1 1   a2 GT  1 1  2 T  a1 C 1    UII ¼  a21 C T1   T  G1   CT h2  1 1   a2 GT

ð15Þ

Fig. 3. The upper joint of the overlapping grouted plates.

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V. Zernov et al. / Ultrasonics 51 (2011) 57–64

U ]n ðx2 Þ ¼ ½u]n ðx2 Þ; v ]n ðx2 Þ; r]n ðx2 Þ; s]n ðx2 Þ;

] ¼ I; II:

ð19Þ

The fields (17) and (18) satisfy the equations of motion and radiation conditions whatever the choice of the constants C In and C IIn . The boundary conditions define these constants uniquely. Using (17) and (18), the traction free boundary conditions on BD can be written as 1 X

!

C IIn

n¼1





rIIn ðx2 Þ rE ðx2 Þ ¼ 0; x2 2 BD;  II sn ðx2 Þ sE ðx2 Þ

ð20Þ

and the stress/displacement continuity conditions on AB become 1 X

C IIn U IIn ðx2 Þ 

1 X

n¼1

C In U In ðx2 Þ  U E ðx2 Þ ¼ 0;

x2 2 AB:

ð21Þ

energy and the sum of calculated reflected and transmitted energies never exceeds 1%, whereas on average, it is less than 0.5%. 4. Mode amplitudes in the grouted overlapping plates It is worth reiterating that the length of waveguides I, II and III (Fig. 2) is larger than their thickness. Since the evanescent modes that appear on a side of the grouted section decay exponentially they can be neglected on the other side and for this reason, we discuss only the amplitudes of propagating modes. Let M*(x) and N*(x) be the numbers of such modes in the waveguide I and II, respectively. Let the transducer placed in the boundary CD create the loading

(

n¼1

ð0; 0; 1; 0ÞT ; x2 2 CD;

One way to solve the infinite system of Eqs. (20) and (21) is to employ the orthogonality conditions

U T ðx2 Þ ¼

qAB ðU In ; U Im Þ ¼ F I ðnÞdnm ;

Solving the system of algebraic Eqs. (28) and (29) for UE = UT, in the steel plate, the amplitudes of the generated modes form an M* dimensional vector C IT , and in the three layer composite, they form an N* dimensional vector C IIT . If rather than (31) the loading is U E ¼ U ]i with ] = I, n > 0 or ] = II, n < 0, that is, if the loading is due to the guided waves, which are incident from within the structure, each of the incident modes interacts with the boundary AD and creates many more reflected and transmitted modes. Solving the system (28) and (29) gives us

qAD ðU IIn ; U IIm Þ ¼ F II ðnÞdnm ;

ð22Þ

where we have

~ ðU ] ; U ] Þ þ q ~S ðU ]m ; U ]n Þ; qS ðU ]n ; U ]m Þ ¼ q ZS n m   ~S ðU ]n ; U ]m Þ ¼ r]n u]m  s]n v ]m dx2 ; ] ¼ I; II q

ð23Þ ð24Þ

S

and FI(n), FII(n) – 0 for any n [14]. Applying the scalar product (23) to (20) and (21), with S = AD, and truncating the infinite set of Lamb modes to faIn gNn¼1 and faIIn gM n¼1 leads us to an approximate finite system of linear algebraic equations for coefficients fC In ; C IIn g. In particular, applying qAD ð; U IIm Þ to the first term in (21) we can write ! N X 0 ¼ qAD C IIn U IIn ; U IIm n¼1

¼ qAB

N X

! C IIn U IIn ; U IIm

þ qBD

n¼1

N X

; m ¼ 1; . . . ; N:

ð25Þ

n¼1

Using (21) again, the first term in the right-hand side of (25) can be expressed as

qAB

N X

! C IIn U IIn ; U IIm

¼

n¼1

N X

C In qAB ðU In ; U IIm Þ þ qAB ðU E ; U IIm Þ;

ð26Þ

n¼1

and using (20), the second term in right-hand side of (25) can be rewritten as

qBD

N X

!

C IIn U IIn ; U IIm

~BD ðU E ; U IIm Þ þ ¼q

n¼1

N X

~BD ðU IIm ; U IIn Þ: C IIn q

ð27Þ

n¼1

Substituting (26) and (27) into (25) yields N algebraic equations for N + M unknowns C In and C IIn , N X

~BD ðU IIm ; U IIn Þ C IIn q

n¼1



N X

C In qAB ðU In ; U IIm Þ

Applying equations

qAB ð; U Im Þ;

T I;II ; M  N  ;

T II;I ; N  M  ;

ð33Þ

where the elements of reflection matrix R ¼ are ! U Im conversion coefficients and the elements of reflection matrix II I RII;II ¼ RII;II nm are U n ! U m conversion coefficients. The transmission I,II II,I matrices T and T are defined in a similar manner. Whatever the loading, the resulting total elastodynamic field in the structure can be described in terms of vectors

¼

m ¼ 1; . . . ; N:

m > 0 to (21) leads to M additional algebraic

¼ qAB ðU

E

; U Im Þ;

m ¼ 1; . . . ; M:

ð29Þ

The numbers N and M have to be chosen carefully. The best approximations to the original infinite system are obtained when the highest attenuation coefficients in both waveguides match [14], that is, when we have II N Þ:

 Im ða

ð30Þ

In our calculations we specify N = 100 and the matching occurs at M = 30. For this choice of N, the difference between the incident

U In

ð34Þ

where C ]i is the amplitude of the U ]i mode. Let the grouted section illustrated in Fig. 2 have the length L and let the bounding steel plates be semi-infinite. Let us assume that e II . Its first reflection (from the initial excitation is described by C 0 either top or bottom of the composite) is described by the vector II II e II , where the matrix B has structure e II ¼ B C C 1 0

B ¼ ð28Þ

RI;I nm

  C I ¼ C I1 ; . . . ; C IN ; C I1 ; . . . ; C IN ;   C II ¼ C II1 ; . . . ; C IIM ; C II1 ; . . . ; C IIM ;

0

RII;II P II

RII;II P II

0

II

n¼1

Im ða

ð32Þ

!

;

II

II

P II ¼ diagðeia1 L ; . . . ; eiaN L Þ:

ð35Þ

The second reflection is described by the vector e II ¼ ðBII Þ2 C e II , etc., so that the total field in this structure is e II ¼ BII C C 2 1 0

e II ¼ DII C e II ¼ ½I þ BII þ ðBII Þ2 þ . . . C e II ; C 0 0

C IIn qAB ðU IIn ; U Im Þ

I MÞ

RII;II ; N  N

and

n¼1

~BD ðU X ; U IIm Þ; ¼ qAB ðU X ; U IIm Þ  q

1 X

RI;I ; M  M ;

ð31Þ

x2 2 AC:

I;I

! C IIn U IIn ; U IIm

ð0; 0; 0; 0ÞT ;

DII ¼ ðI  BII Þ1 :

ð36Þ

Now, let us modify the structure in Fig. 2, by assuming that the semi-infinite composite extends upwards and the end of the lower plate side extends the distance l below the composite bottom. In this new structure, the reflected modes are described by the matrix I

B ¼

RI;I P I

0 I;I

r P

I

0

! ;

I

I

P I ¼ diagðeia1 l ; . . . ; eiaM l Þ:

ð37Þ

Therefore, the total field in the plate is

e I ¼ DI C eI ; C 0

DI ¼ ðI  BI Þ1 :

ð38Þ

V. Zernov et al. / Ultrasonics 51 (2011) 57–64

eI . where the initial excitation is given by C 0 Let us combine the above two canonical structures into one, with the composite of length L and lower steel plate of length l. Let us assume the upper steel plate to be semi-infinite. This assumption is reasonable, since in Fig. 2, practically no energy is transmitted into this plate (see below). The transducer generates initial modes C T ¼ ð0; C IIT Þ, which reverberate in the composite producing the field C II1 ¼ DII C T . The modes transmit into the lower plate with the coefficients T II C II1 , where we have

T II ¼



0

0

0 T

II;I

 :

ð39Þ

Further reverberations in the steel plate lead to C I1 ¼ DI T II DII C T . The modes transmit back into the composite with the coefficients T I C I1 , where we have

TI ¼

T I;II

0

0

0

! ð40Þ

:

New reverberations lead to C II2 ¼ DII T I DI T II DII C T . By taking into account all possible reflections and transmissions we arrive at the final state

C II ¼ C II1 þ C II2 þ    ¼ ðI  DII T I DI T II Þ1 DII C T :

(a)

ð41Þ

61

5. Numerical results for a symmetric structure Let us consider two overlapping grouted steel plates of the same thickness H1 = H3. Let us study the propagation of energy in this structure, with the view to establishing the best operational regime for its condition monitoring. Clearly, small defects are best detected with the shortest wavelength, and thus the highest possible frequency. However, as is well known, at high frequencies, the waveguides support many highly dispersive modes. This makes the high frequency regimes hard to utilize. We seek a trade-off between the frequency and amount of energy delivered to a specific part of the structure. We begin with analysis of dispersion curves. We use the lower case letters s and a and capitals S and A to refer to symmetric and antisymmetric modes in the plates and three-layer composite, respectively. Using realistic dimensions H1 = H3 = 40 mm and H2 = 70 mm, the group velocity dispersion curves in the steel plates and steel/grout/steel composite are presented in Fig. 4. Clearly, in the frequency range above 30 kHz, in the steel/grout/steel composite, there are six propagating modes, which renders this frequency range impracticable. Fig. 5 shows the mode attenuation curves describing the energy loss in the waveguides submerged in water. In the steel plates, in the region below 30 kHz the attenuation of the s0 mode is reasonably low. In the steel/grout/steel composite,

(b)

Fig. 4. The group velocities in: (a) 40 mm thick steel plate and (b) 40/70/40 mm thick steel/grout/steel composite. Solid lines – symmetric modes. Dashed lines – antisymmetric modes.

(a)

(b)

Fig. 5. The attenuation curves when: (a) 40 mm thick steel plate is immersed in water and (b) 40/70/40 mm thick steel/grout/steel composite is immersed in water. Key as in Fig. 4.

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V. Zernov et al. / Ultrasonics 51 (2011) 57–64

(a)

(b)

Fig. 6. (a) The partition of energy radiated by the transducer. Solid black line – the energy that propagates into the composite. Solid gray line and dashed black line – the amounts of energy carried by S1 and A1 modes, respectively. (b) The energy composition of s0 mode in the lower plate. Solid gray line and dashed black line – the amounts of energy contributed to so by the S1 and A1 modes, respectively.

Fig. 7. The energy ‘‘beating” at the bottom of the composite section. Dashed line – the incident energy transmitted into the lower steel plate. Solid line – the energy reflected back into the composite.

Fig. 9. The group velocity dispersion curves for the asymmetric steel/grout/steel composite.

in the 0–7 kHz and 15–30 kHz frequency bands the least attenuative modes are S0 and S1, respectively. Thus, when the structure is immersed in water a reasonable choice of the inspection frequency band is 15–30 kHz. In these regime, the symmetric modes propagating in the steel plates and composite are s0 and S1, respectively. The partition of energy radiated by the transducer is illustrated in Fig. 6. When the frequencies are low some energy is transmitted into the upper steel plate. At frequencies f > 15 kHz, nearly all the energy propagates into the lower part of the structure. Fig. 6 also shows that in the [15, 40] kHz frequency band all transmitted

energy is equipartitioned between A1 and S1. This simplifies interpretation of the received signals. Fig. 7 reveals that when the transducer generated waves hit the bottom end of the composite the transmission/reflection energy exhibits a ‘‘beating” behavior: at some frequencies, the energy is totally reflected and at others, it is totally transmitted into the lower steel plate. This effect is caused by the interference of the A1 and S1 modes. Their energy equipartition gives rise a Rayleigh-like wave. When the energy of this wave is localized on the lower plate joint, it is fully transmitted into the lower steel plate. Localization near the other corner leads to total reflection.

(a)

(b)

Fig. 8. The pulse trains in the symmetric composite structure fully immersed in water. The input signal is 15 sine cycles. The carrying frequency is: (a) 24 kHz and (b) 26.5 kHz.

V. Zernov et al. / Ultrasonics 51 (2011) 57–64

(a)

63

(b)

Fig. 10. The attenuation dispersion curves for: (a) symmetric and (b) asymmetric composite immersed in water.

incident energy, respectively (see Fig. 7). We can see that for a narrow band inputs, the choice of the carrier frequency has a dramatic effect on arrivals. It enables us to suppress the reflection from the end of the grouted section, thus maximizing the reflection from the end of the bottom plate (Fig. 8a) and vice versa (Fig. 8b). 6. Numerical results for an asymmetric structure

Fig. 11. Energy ‘‘beating” at the bottom end of the asymmetric grouted section. Dashed line – the amount of incident energy transmitted into the lower steel plate. Solid line – the amount of energy reflected back into the composite.

The propagation of pulse trains is shown in Fig. 8. These are simulated using the harmonic synthesis (the inverse Fourier transform in time). The amplitudes are normalized by the amplitude of the wave radiated by the transducer. The length of the bottom plate is 3 m. The whole structure is submerged in water without support, which corresponds to a through crack 3 m below the end of the grouted section. This is an extreme situation, but the model provides qualitative information on whether the energy could be delivered to the bottom plate. In time domain, the beating effect can be utilized only for a narrow band input, say, 15 sine cycles. Fig. 8 compares the pulse trains generated by such input in the structure fully immersed in water for the carrying frequencies of 24 kHz and 26.5 kHz. These frequencies correspond to the maximal and minimal transmission of the

(a)

In this section we present numerical results for a realistic asymmetric structure H1 – H3. For definiteness, we assume that H1 = 80 mm, H2 = 70 mm, H3 = 48 mm, the length of the overlapping grouted section is 7.5 m and the end of the bottom plate is 3 m below the end of the grouted section. The group velocity dispersion curves are presented in Fig. 9. They look similar to the group velocity curves for the symmetric case (Fig. 4) but are scaled along the frequency axis, because the overall thickness of the three layer composite is greater. Clearly, there are five propagating modes in the frequency band below 25 kHz, as opposed to four propagating modes in the symmetric case. Unlike the symmetric case, there is no separation into the symmetric and antisymmetric modes. Fig. 10 compares the attenuation dispersion curves of the symmetric and asymmetric composites immersed in water. It shows that once the structural symmetry is violated, there is a significant increase in attenuation. When the immersion is full practically no energy can propagate through an asymmetric composite. The energy beating curves also sustain significant changes (see Fig. 11). There are no highly pronounced stop/pass frequency bands. However, an appropriate choice of the operating frequency can still reduce the amount of the reflected energy practically to zero. Fig. 11 also shows that the transmission coefficient is in the [0.5, 1] range. Since in the steel plate the attenuation of the s0 mode

(b)

Fig. 12. The received signal for the 15 sine cycle input pulse. The 1 m long asymmetric composite is immersed in water. The carrying frequency is: (a) 20 kHz and (b) 22 kHz.

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is almost negligible, this renders the underwater inspection of the bottom plate realistic. The propagation of the continuous wave consisting of 15 sine cycles is shown in Fig. 12. As predicted by the beating curves, at 22 kHz, there is little reflected signal from the bottom end of the grouted section (Fig. 12b). At 20 kHz the transmitted signal is somewhat minimized, but its total suppression is impossible (Fig. 12a). 7. Conclusions We have modeled propagation of guided waves in overlapping grouted steel plates. The model can be applied to investigating operating regimes for guided wave technology in large offshore structures, such as symmetric or asymmetric wind turbine monopiles. The reflection/transmission coefficients for the total field display an interesting beating effect that can be utilized to choose a carrier frequency to focus the ultrasonic energy on those portions of the structure that need to be monitored. The effect is present in both configurations, but it is less pronounced when the structure is asymmetric. The inspection/monitoring of asymmetric monopiles is complicated further by high attenuation of all modes when immersed in water, whereas in symmetric monopiles advantageous regimes can be found in which energy loss to water can be avoided almost entirely. The conclusions should be of interest to design teams working on future wind turbines. One final point: in today’s practice, favorable operating regimes involve just one advantageous waveguide mode. In this paper we have described a situation when an engineering advantage can be gained by exciting two specific modes. Acknowledgements This work has been carried out as part of an EU project OPCOM (Development of Ultrasonic Guided Wave Inspection Technology

for the Condition Monitoring of Offshore Structures), Contract No. NMP2-CT-2005-516993, coordinated by TWI. We are grateful to our industrial partners, particularly DONG Energy, Denmark (formerly Elsam Kraft A/S), for useful discussions.

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