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ARTICLE IN PRESS

Journal of Sound and Vibration 264 (2003) 1139–1153

JOURNAL OF SOUND AND VIBRATION

www.elsevier.com/locate/jsvi

The dynamic analysis of a cracked Timoshenko beam by the spectral element method

M. Krawczuka,b, M. Palaczb,*, W. Ostachowiczb

a

Department of Technical Sciences, University of Warmia and Mazury, Oczapowskiego 22, Olsztyn 10-736, Poland b ! Institute of Fluid Flow Machinery, Polish Academy of Sciences, Fiszera 14, Gdansk 80-231, Poland Received 12 March 2002; accepted 13 June 2002

Abstract The aim of this paper is to introduce a new ?nite spectral element of a cracked Timoshenko beam for modal and elastic wave propagation analysis. The proposed approach deals with the spectral element method. This method is suitable for analyzing wave propagation problems as well as for calculating modal parameters of the structure. In the paper, the results of the change in modal parameters due to crack appearance are presented. The in?uence of the crack parameters, especially of the changing location of the crack, on the wave propagation is examined. Responses obtained at different points of the beam are presented. Proper analysis of these responses allows one to indicate the crack location in a very precise way. This fact is very promising for the future work in the damage detection ?eld. r 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction In order to improve the safety, reliability and operational life, it is urgent to monitor the integrity of structural systems. Techniques of non-destructive damage detection in mechanical engineering structures are essential [1–3]. Previous approaches to non-destructive evaluation of structures to assess their integrity typically involved some form of human interaction. Recent advances in smart materials and structures technology has resulted in a renewed interest in developing advanced self-diagnostic capability for assessing the state of a structure without any human interaction. The goal is to reduce human interaction while at the same time monitor the

*Corresponding author. Tel.: +48-58-341-1271x109; fax: +48-58-341-6144. E-mail addresses: mk@imp.gda.pl (M. Krawczuk), mpal@imp.gda.pl (M. Palacz), wieslaw@imp.gda.pl (W. Ostachowicz). 0022-460X/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-460X(02)01387-1

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integrity of the structure. With this in mind, many researchers have made signi?cant strides in developing damage detection methods for structures based on traditional modal analysis techniques. These techniques are often well suited for structures, which can be modelled by discrete lumped-parameter elements where the presence of damage leads to some low-frequency change in the global behaviour of the system [4–7]. On the other hand, small defects such as cracks are obscured by modal approaches since such phenomena are high-frequency effects not easily discovered by examining changes in modal mass, stiffness or damping parameters. This is because at high frequencies, modal structural models are subject to uncertainty. This uncertainty can be reduced by increasing the order of the discrete model, however, this increases the computational effort of modal-based damage detection schemes. There is also a group of methods, which utilize thermodynamic damping for assessment of the structural integrity of vibrating structures [8–10]. Spectral analysis is a method for representing the dynamic solution in the form of a series of solutions at different frequencies [11]. The spectral element method reformulates these solutions as dynamic stiffness relations, making it suitable for assembly in a manner analogous to the ?nite element method. A wide variety of elements have been developed for structural members over ?nite and semi-in?nite regions. The spectral element method has several useful attributes. Since inertial properties are modelled exactly, few elements are needed to model large regions. The generalized nodes need to be placed only at structural discontinuities. Several spectral elements together can model structures having sections with different thicknesses or material properties [12]. The solution is obtained in terms of generalized displacements, subsequent calculations for velocity, acceleration, strain and stress for any applied load can then be found with relatively inexpensive post-processing calculations. The spectral element method directly computes a structure’s frequency response function and in this manner gives additional information that bridges the gap between modal methods based on free vibrations and time reconstructions based on direct integration [13]. Spectral elements may also be used to solve inverse problems such as force identi?cation. Finally, the core problem to be solved is incredibly small and is repeated many times in data-independent manner. This makes the spectral element method ideally suited for solution on computers with many processors. At the present many spectral models of structures are available in literature. One can ?nd spectral models of rods, beams and plates without any damage [13]. The spectral model of a cracked rod element is presented in Ref. [14]. The introduced model allows the use of propagating wave for precise localization of the crack. Spectral Bernoulli–Euler’s beam with a crack is presented by Ref. [15]. The model described gives proper results for identi?cation of crack parameters by analysis of the propagating wave. There is a model of a cracked Timoshenko beam available, but it is ?nite element method model [16]. This paper presents a new ?nite spectral Timoshenko beam element with a transverse open and non-propagating crack. This element can be applied for modal analysis and for examining the wave propagation process. Few numerical examples are presented in order to show the in?uence of the crack appearance on changes in modal parameters and wave propagation analysis. It is shown that the proposed element properly provides analysis of a damaged structure in order to localize and assess the size of the crack.

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2. Cracked Timoshenko beam spectral element A spectral Timoshenko beam ?nite element with a transverse open and non-propagating crack (crack velocity equal to zero) is presented in Fig. 1. The length of the element is L; and its area of cross-section is A: The crack is substituted by a dimensionless and massless spring, whose bending yb and shear ys ?exibilities are calculated using Castigliano’s theorem and laws of the fracture mechanics. This is brie?y presented in the next chapter. # # Nodal spectral displacements w and rotations f are assumed in the forms, for the left and right part of the Timoshenko beam as # w1 ?x? ? R1 A1 e?ik1 x ? R2 B1 e?ik2 x ? R1 C1 e?ik1 ?L1 ?x? ? R2 D1 e?ik2 ?L1 ?x? for xA?0; L1 ?; # f1 ?x? ? A1 e?ik1 x ? B1 e?ik2 x ? C1 e?ik1 ?L1 ?x? ? D1 e?ik2 ?L1 ?x? for xA?0; L1 ?; # w2 ?x? ? R1 A2 e?ik1 ?x?L1 ? ? R2 B2 e?ik2 ?x?L1 ? ? R1 C2 E ?ik1 ?L??L1 ?x?? for xA?0; L ? L1 ?; # 2 ?x? ? A2 e?ik1 ?x?L1 ? ? B2 e?ik2 ?x?L1 ? ? C2 e?ik1 ?L??L1 ?x?? f ? D2 e?ik1 ?L??L1 ?x?? for xA?0; L ? L1 ?; ?1? ? R2 D2 e?ik1 ?L??L1 ?x??

where L1 denotes the location of the crack, L is the total length of the beam, Rn is the amplitude ratios given by [13] Rn ? ikn GAS1 ; 2 ?GAS1 kn ? rAo2 ? n ? 1; 2; ?2?

where S1 ? ??0:87 ? 1:12n?=?1 ? n??2 is shear coef?cient for displacement [13], n is the Poisson ratio, G is shear modulus, r denotes density of the material, o is a frequency and i is imaginary p??????? unit given as i ? ?1: The wave numbers k1 and k2 are the roots of the characteristic equation in the general form ?GAS1 EJ?k4 ? ?GAS1 rJK2 o2 ? EJrAo2 ?k2 ? ?rJS2 o2 ? GAS1 ?rAo2 ? 0; ?3?

^ q1

^ q3

q2

^

1

2

q4

^

x

L1 L

y

Fig. 1. The model of the Timoshenko beam with a transverse open and not propagating crack simulated by elastic hinge.

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where S2 ? 12K1 =p2 is shear coef?cient for rotation [13], E denotes Young’s modulus and J is second moment of area. The coef?cients A1 ; B1 ; C1 ; D1 ; A2 ; B2 ; C2 and D2 can be calculated as a function of the nodal spectral displacements using the boundary conditions: At the left end of the element # # w1 ?x ? 0? ? q1 ; # # f1 ?x ? 0? ? q2 : ?4?

At the crack location (total change of displacements and rotation angle, compatibility of bending moments and shear forces) # @w1 # 1 ?x ? L1 ? ; # # ?x ? L1 ? ? f w2 ?x ? 0? ? w1 ?x ? L1 ? ? ys @x # @f1 ?x ? L1 ? # # ; f2 ?x ? 0? ? f1 ?x ? L1 ? ? yb @x # # # # @f1 ?x ? L1 ? @f2 ?x ? 0? @w1 ?x ? L1 ? # @w2 ?x ? 0? # ?5? ? ; ? f1 ?x ? L1 ? ? ? f2 ?x ? 0?: @x @x @x @x At the right end of the element # # w2 ?x ? L ? L1 ? ? q3 ; # # f2 ?x ? L ? L1 ? ? q4 : ?6?

By considering the formulae describing the nodal spectral displacements for the left and right part of the Timoshenko beam, the boundary conditions can be written in a matrix form as

2 R1 R2 ?R1 p1 6 6 1 1 p1 6 6 p ?R ? y ??iR k ? 1?? p ?R ? y ??iR k ? 1? y ??iR k ? 1? ? R s 1 1 2 2 s 2 2 s 1 1 1 6 1 1 6 6 p1 ?1 ? ik1 yb ? p2 ?1 ? ik2 yb ? 1 ? ik1 yb 6 6 ?p1 ik1 ?p2 ik2 ik1 6 6 6 p1 ??1 ? iR1 k1 ? p2 ??1 ? iR2 k2 ? ?1 ? iR1 k1 6 6 0 0 0 4 0 # q1 A1 6 7 6 7 6 B 1 7 6 q2 7 # 6 7 6 7 6C 7 6 0 7 6 17 6 7 6 7 6 7 6 D1 7 6 0 7 ?6 7 ? 6 7; 6A 7 6 0 7 6 27 6 7 6 7 6 7 6 B2 7 6 0 7 6 7 6 7 6C 7 6q 7 4 2 5 4 #3 5 # D2 q4 2 3 2 3 0 0 ?R2 p2 p2 ys ??iR2 k2 ? 1? ? R2 1 ? ik2 yb ik2 ?1 ? iR2 k2 0 0 0 0 ?R1 p1 ?p1 p1 ik1 p1 ?1 ? iR1 k1 ? R1 p5 p5 7 7 7 7 7 7 7 ?p2 ?p3 ?p4 7 7 p2 ik2 ?p3 ik1 ?p4 ik2 7 7 p2 ?1 ? iR2 k2 ? p3 ?1 ? iR1 k1 ? p4 ?1 ? iR2 k2 ? 7 7 7 R 2 p6 ?R1 ?R2 5 p6 1 1 0 ?R2 p2 0 R 1 p3 0 R2 p4 0 0 0 3

?7?

where p1 ? exp??ik1 L1 ?; p2 ? exp??ik2 L1 ?; p5 ? exp??ik1 L?; p3 ? exp??ik1 ?L ? L1 ??; p6 ? exp??ik2 L?; ?8?

p4 ? exp??ik2 ?L ? L1 ??;

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The formulae below relate the coef?cients A1 ; B1 ; C1 ; D1 ; A2 ; B2 ; C2 and D2 to the nodal spectral displacements by A1 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 11 # 12 # 17 # 18 # C1 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 31 # 32 # 37 # 38 # A2 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 51 # 52 # 57 # 58 # C2 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 71 # 72 # 77 # 78 # B1 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 21 # 22 # 27 # 28 # D1 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 41 # 42 # 47 # 48 # B2 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 61 # 62 # 67 # 68 # D2 ? D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ; 81 # 82 # 87 # 88 # ?9?

where D?1 denotes the elements of the inverse matrix of the matrix from Eq. (7). ij # # The nodal spectral forces (T the shear force, M the bending moment) can be determined by differentiating the spectral displacements with respect to x; and then can be expressed in the matrix form as 2 # 3 2 3 T1 r1 r2 p1 r 1 p2 r2 0 0 0 0 6 # 7 6 6 M1 7 6 iEJk1 iEJk2 ?p1 iEJk1 ?p2 iEJk2 0 0 0 0 7 7 7 ?6 6 7 6T 7 4 0 0 0 0 ?p5 r1 ?p6 r2 ?r1 ?r2 5 4 #2 5 0 0 0 0 ?p5 iEJk1 ?p6 EJk2 iEJk1 iEJk2 # M2 2 3 A1 6 7 6 B1 7 6 7 6C 7 6 17 6 7 6 D1 7 ?10? ? 6 7; 6A 7 6 27 6 7 6 B2 7 6 7 6C 7 4 25 D2

2 2 where r1 ? ?EJk1 ? rJo2 ; r2 ? ?EJk2 ? rJo2 : From relations (9) and (10), the square matrix (4 ? 4), which denotes the frequency-dependent dynamic stiffness for the Timoshenko beam spectral element with transverse open and nonpropagating crack, can be calculated. As the length of the spectral Timoshenko beam ?nite element can be very large, the computation of responses between nodes is necessary. Displacements at any point of the Timoshenko beam element can be calculated from the relationships for 0pxpL1

# w?x? ? R1 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik1 x 11 # 12 # 17 # 18 # ? R2 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik2 x 21 # 22 # 27 # 28 # ? R1 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik1 ?L1 ?x? 31 # 32 # 37 # 38 # ? R2 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik1 ?L1 ?x? ; 41 # 42 # 47 # 48 #

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for L1 oxpL2 # w?x? ? R1 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik1 ?L1 ?x? 51 # 52 # 57 # 58 # ? R2 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik2 ?L1 ?x? 61 # 62 # 67 # 68 # ? R1 ?D?1 q1 ? D?1 q2 ? D?1 q3 ? D?1 q4 ?e?ik1 ?L??x?L1 ?? 71 # 72 # 77 # 78 # ? R2 ?D?1 q1 ? D?1 q2 ? D: 81 # 82 # ?11?

3. Flexibilities at the crack location Coef?cients of the beam ?exibility matrix at the crack location (in general form) can be calculated using the Castigliano theorem [17] cij ? @2 U @Si @Sj for i ? 1; y; 6; j ? 1; y; 6; ?12?

where U denotes the elastic strain energy of the element caused by the presence of the crack and are the independent nodal forces acting on the element. For the analyzed beam, the elastic strain energy due to the crack appearance [18] can be expressed by Z 1 2 ?K 2 ? KII ? dA; ?13? U? E A I where A denotes the area of the crack, KI and KII are a stress intensity factors corresponding to the ?rst and second mode of the crack growth [19]. The stress intensity factors can be calculated as 6M p?????? a bT p?????? a paFII paFI ; KII ? ; ?14? KI ? BH 2 H BH H where M is a bending moment, b denotes shear factor [20], T is a shear force, B; H; a are dimensions—see Fig. 2, FI and FII are a correction function in the form [19] s???????????????????????? a tan?pa=2H? 0:752 ? 2:02?a=H? ? 0:37?1 ? sin?pa=2H??3 ; FI ? H pa=2H cos?pa=2H? a 1:30 ? 0:65?a=H? ? 0:37?a=H?2 ? 0:28?a=H?3 p????????????????????? FII ? : ?15? H 1 ? ?a=H? After simple transformations, the modelling of the ?exibilities of the elastic elements cracked cross-section of the Timoshenko beam spectral ?nite element, can be rewritten as Z % Z % 72p a 2 2bp a 2 cb ? cs ? ?16? aFI ?a? da; aFII ?a? da; % % % % % % BH 2 0 B 0 where a ? a=H; % a ? a=H (see Fig. 2). %

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M. Krawczuk et al. / Journal of Sound and Vibration 264 (2003) 1139–1153 1145

(a)

I

II

(b)

α dα z

a H

B

y

Fig. 2. (a) I and II crack propagation modes; (b) cross-section of the beam element at the crack location.

In the non-dimensional form the ?exibilities can be expressed as EJcb GAcs yb ? ; ys ? : L L

?17?

4. Numerical examples In the aim to demonstrate the validity of the proposed model, several numerical tests have been carried out for a cantilever beam with dimensions: length 2.0 m, height 0.02 m, width 0.02 m, Young’s modulus 210 GPa, and mass density 7860 kg/m3. In the numerical tests the model was ?xed at one end and impacted at the second. The model tested consisted of two elements, one with a crack and the second the so-called throw-off element [13]. This approach allowed forward and backward moving waves to be obtained. First and second numerical tests were done in order to show whether the proposed model is useful for modal analysis. According to the literature study and previous numerical investigations by authors using one spectral element, it is possible to analyze any range of frequencies. It is extremely important when there is a need to analyze high natural frequencies (very small defects cause detectable changes in high natural frequencies only). In such cases classical ?nite element models would require very dense grid, which is time consuming for computational calculations. The two examples presented show that the model allows for proper modal analysis as well as for the classical ?nite element one.

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Table 1 Changes in ?rst four natural frequencies (rad/s) 0.05 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.25 0.5 0.5 0.75

Relative crack depth

Relative crack location (from the ?xed end) 26.12 163.72 457.52 892.65 26.12 163.72 457.52 892.31 26.12 163.72 457.45 892.13 25.83 162.15 456.02 872.35 25.13 161.94 455.62 874.23 25.13 161.55 455.96 878.54

No crack

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Finite element method model [21] Frequency I 26.12 Frequency II 163.72 Frequency III 457.52 Frequency IV 892.75 Frequency XVIII Not available Frequency XIX Not available

25.196 162.09 454.82 861.43

25.76 161.71 455.12 860.54

25.83 161.67 449.36 859.8

M. Krawczuk et al. / Journal of Sound and Vibration 264 (2003) 1139–1153

Spectral model Frequency I Frequency II Frequency III Frequency IV Frequency XVIII Frequency XIX 26.138 164.24 459.62 899.69 21458 23851 26.138 164.24 459.62 899.63 21456 23854 26.138 164.24 459.55 899.63 21458 23851 26.075 164.18 458.04 896.30 21441 23759

26.138 164.24 459.62 899.75 21459 23854

26.138 163.36 456.62 895.23 21363 23852

26.138 163.93 457.16 895.54 21444 23763

25.196 163.99 446.8 873.43 21341 23295

25.887 157.21 459.62 864.5 21258 23845

26.138 161.67 439.26 867.9 21364 23308

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M. Krawczuk et al. / Journal of Sound and Vibration 264 (2003) 1139–1153

1.0

no crack ak/H=0.05

1147

1.0

no crack ak/H=0.05 ak/H=0.5

0.8

ak/H=0.5

0.5

I mode shape

0.6

II mode shape

0.4 0.8 1.2 1.6 2.0

0.0

0.4

-0.5

0.2

0.0 0.0

-1.0

0.0 0.4 0.8 1.2 1.6 2.0

(a)

1.0

Length [m]

(b)

1.0

no crack ak/H=0.05 ak/H=0.5

Length [m]

0.5

0.5

II mode shape

IV mode shape

no crack ak/H=0.05 ak/H=0.5

0.0

0.0

-0.5

-0.5

-1.0 0.0

-1.0

0.4 0.8 1.2 1.6 2.0 0.0 0.4 0.8 1.2 1.6 2.0

(c)

1.0

Length [m]

(d)

no crack ak/H=0.05 ak/H=0.5

Length [m]

1.0

no crack ak/H=0.05 ak/H=0.5

0.5

0.5

XVIII mode shape

0.0

XIX mode shape

0.0 0.4 0.8 1.2 1.6 2.0

0.0

-0.5

-0.5

-1.0

-1.0 0.0

0.4

0.8

1.2

1.6

2.0

(e)

Length [m]

(f)

Length [m]

Fig. 3. Mode shapes obtained for an uncracked beam (—), for a beam with 5% of the beam height crack (—E—) and for a beam with 50% of the beam height crack —K—: (a) I mode shape; (b) II mode shape; (c) III mode shape; (d) IV mode shape; (e) XVIII mode shape; (f) XIX mode shape.

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0.0008

0.0010

0.0012

0.0014

0.0016

0

400

800

1200

1600

Time [s]

Frequency [kHz]

1 20

Fig. 4. Excitation triangle signal lasting for 0.0925 ms, multiplied by sinusoidal signal lasting triangle signal and its FFT.

of the duration of the

(a)

L

(b)

0,7L

(c)

0,3L

Fig. 5. Location of measured points.

In order to examine whether the localization of the crack in?uences the natural frequencies, a ?rst numerical example was done. Table 1 shows the ?rst four and two higher computed natural frequencies for a beam without a crack and for a beam with nine different crack depths and locations. For numerical calculations it was assumed that frequencies are determined with precision 70.005 Hz. It is noticeable that changes in natural frequencies become more intensive with the growth of the crack depth. For higher natural frequencies, even for a very small crack, changes due to crack appearance are higher than the ?rst few ones. Results obtained for an elaborated spectral model of a Timoshenko beam were compared with results obtained for Timoshenko beam calculated from the ?nite element method [21]. Similar character of changes in natural frequencies for a cracked beam is reported in Ref. [22]. The next numerical example was to investigate whether the proposed model is useful for examining the changes in mode shapes due to the appearance of a crack. Fig. 3 illustrates changes in mode shapes for uncracked beams and for cracked beams with crack depths equal to 5% and 50% of the beam height precisely marked on the pictures. In both cases the crack was located in the middle of the beam. It is noticeable that the appearance of the crack leads to deformation of the calculated mode shapes. In the numerical tests described in Ref. [14], it appeared that the kind of excitation force in?uences the response of a structure. For the tests described in this paper, the excitation signal is presented in Fig. 4. It is a product of a triangle signal lasting for 0.0925 ms and a sinusoid 1 signal with the period equal to 20 of the duration of the triangle signal. This signal produces excitation of a wide range of frequencies, thus obtaining very clear response in the time domain.

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2.00

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Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

(a)

0.003

0.004

0.005

0.006

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

0.003

0.004

0.005

0.006

(b)

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

(c)

0.003

0.004

0.005

0.006

Time [s]

Fig. 6. Wave propagation, for a crack located at 20% of a beam height and at distance 25% of the length from the ?xed end. Measurements are taken in points shown in Fig. 5.

Fig. 5 presents the location of measurement points for the next numerical examples. The next three ?gures (Figs. 6–8) show the responses of the system obtained at different points of the beam for a crack of depth equal to 20% of the beam height, located at three different distances from the ?xed end of the beam. Fig. 6 shows the results obtained for a crack located at a distance equal to 25% of the beam length from the ?xed end. The curve in Fig. 6(a) illustrates the response measured at the impacted free node (Fig. 5(a)). It is noticeable that there are four responses on some pictures. The ?rst response represents the excitation signal, the second the crack, the third is the re?ection of the ?xed end and the fourth is the second re?ection from the crack location. The Fig. 6(b) presents the response of the systems shown in Fig. 5. In this case one sees again four signals, as for curve 6(a). However, for this example the excitation signal is recorded occurring later than in curve 6(a), because the wave needed to propagate from the

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2.00

Displacement [nm]

Reflections from the cracked place

0.00

-2.00 0.002

(a)

0.003

0.004

0.005

0.006

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

0.003

0.004

0.005

0.006

(b)

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

(c)

0.003

0.004

0.005

0.006

Time [s]

Fig. 7. Wave propagation, for a crack located at 20% of a beam height and at distance 50% of the length from the ?xed end. Measurements are taken in points shown in Fig. 5.

impacted node to the point of measurement. The curve shown in Fig. 6(c) is recorded at the point shown in Fig. 5(c). One can see three signals only. First stands for the excitation signal, the second re?ects the ?xed end, the third is the second re?ection from the crack. In this case the ?rst re?ection from the crack interfered with the second re?ection from the ?xed end. Fig. 7 introduces responses of the system recorded for a crack located at the middle of the beam and is 20% of its height. Curve of Fig. 7(a) corresponds to Fig. 5(a). In this case one sees four signals. First is the excitation signal, second one is a re?ection from the crack, third one is a re?ection from the ?xed end and fourth one is the second re?ection from the crack. Fig. 7(b) shows four signals as well. The ?rst one corresponding to Fig. 5(b) is recorded later than in Fig. 7(a) due to wave propagation from the impact point to the measurement point. Fig. 7(c) presents also four signals. Localization of recorded re?ections results from with the measurement

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2.00

1151

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

(a)

0.003

0.004

0.005

0.006

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

0.003

0.004

0.005

0.006

(b)

Time [s]

2.00

Displacement [nm]

Reflections from the crack place

0.00

-2.00 0.002

(c)

0.003

0.004

0.005

0.006

Time [s]

Fig. 8. Wave propagation, for a crack located at 20% of a beam height and at distance 75% of the length from the ?xed end. Measurements are taken in points shown in Fig. 5.

point location (Fig. 5(c)) and the distance, which needs to be reached by the propagating wave from the impact to the point of measurement. Fig. 8 presents results obtained for the 20% deep crack located at a distance of 75% of the beam length from the ?xed end. Fig. 8(a) corresponds to the result obtained from Fig. 5(a). The ?rst signal represents for the impact signal, the second one comes from the crack. The re?ected wave propagates for a short distance from the point of measurement and the crack, which is why the ?rst two re?ections are so close to each other. The third re?ection comes from the ?xed end and the last one is another re?ection from the crack place. The second picture (Fig. 8(b)), corresponding to Fig. 5(b) shows three re?ections only. This happens from interference between the ?rst re?ection from the crack place and the re?ection from the ?xed end. Fig. 6(c) illustrates the response at the measurement point shown in Fig. 5(c). The ?rst signal is the excitation signal,

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adjacent is the signal re?ected from the ?xed end, then the ?rst re?ection from the crack place and the last is the second re?ection from the crack place. From these results one concludes that knowing the wave propagation velocity and the time of appearance of additional re?ections, the location of the crack is easily calculated.

5. Conclusions This paper has presented a new model of a Timoshenko spectral ?nite beam element with a transverse open and non-propagating crack. The basic difference between the classical approach and the spectral method is clearly shown. It is easily seen that the spectral approach gives more information and is more suf?cient for damage detection. The elaborated element can be used not only for modal analysis, but also for wave analysis. The results obtained indicate that the current approach is capable of calculating high natural frequencies without any additional and timeconsuming numerical computation. Responses, measured at different points, from a cracked Timoshenko beam are presented. They can give information about the location of the crack and the in?uence of different locations of the crack on the wave propagation in the damaged structure was shown. Differences in the responses may be used for proper assessment of the crack location. Future research work will focus on extending the spectral element method to damaged structures with more complex geometry, like for example plates or constructions built up from beam-like structures.

References

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