Damage detection by mode shape squares extracted from a passing vehicle

Damage detection by mode shape squares extracted from a passing vehicle

Journal of Sound and Vibration 331 (2012) 291–307 Contents lists available at SciVerse ScienceDirect Journal of Sound and Vibration journal homepage...

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Journal of Sound and Vibration 331 (2012) 291–307

Contents lists available at SciVerse ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Damage detection by mode shape squares extracted from a passing vehicle Yao Zhang, Longqi Wang, Zhihai Xiang n Applied Mechanics Laboratory, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China

a r t i c l e i n f o

abstract

Article history: Received 26 January 2011 Received in revised form 9 July 2011 Accepted 4 September 2011 Handling Editor: L.G. Tham Available online 23 September 2011

The success of damage detection heavily depends on the quality of damage features, such as mode shapes and power mode shapes, etc. However, these features are usually difficult and inconvenient to be accurately obtained in practice. To solve this problem, this paper develops a simple method to approximately extract structural mode shape squares from the acceleration of a passing tapping ‘‘vehicle’’, which serves as a ‘‘message carrier’’ of the dynamic properties of the structure. Based on the approximately obtained mode shape square, a new damage index is proposed to improve the sensitivity to damage. Numerical simulations and simple experiments demonstrate the validity of the proposed method. Compared with traditional methods, it is easier to be implemented and more accurate in noisy environment, because it requires neither preinstalling many sensors on the structure, nor solving eigenvector or singular value problems and uses only the information of point impedance. In addition, the density of measurement points can be flexibly adjusted since the tapping vehicle scans the structure continuously. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Vibration based non-destructive damage detection has been widely studied for many years [1–4]. For example, the mode shape curvature method and the flexibility matrix method proposed by Pandey et al. [5,6], the strain energy method proposed by Stubbs et al. [7], and the flexibility curvature index defined by Zhang and Aktan [8], etc. Recently, Fang and Perera [9] suggested using the power mode shape, acquired from distributed sensors on structures, for early damage detection. These methods were effective if the quality of mode shapes or power mode shapes could be guaranteed. People do spend much effort to improve the mode shape acquisition methods [10]. Either based on forced vibration or ambient vibration, these methods usually require many sensors fixed on the structure and have to solve a certain eigenvalue or singular value problem. This could be troublesome and costly in practice. Recently, Yang et al. [11–15] got an idea of extracting natural frequencies of bridge structures from the acceleration of a passing vehicle. This method needs only one sensor installed on the vehicle, which greatly facilitates its implementation. Inspired by Yang’s work, Bu et al. [16] proposed an innovative bridge condition assessment method based on a finite element model of the bridge and the dynamic response of the passing vehicle. Following Yang’s method, it further deduces in this paper that if mounting a tapping device on the passing vehicle, the power spectrum of the vehicle acceleration contains peaks at structural natural frequencies and its amplitude at the ith natural frequency is approximately proportional to the square of the ith mode shape. Therefore, it is easy to extract the MOde Shape Square (MOSS) n

Corresponding author. Tel.: þ86 10 62796873; fax: þ 86 10 62772902. E-mail address: [email protected] (Z. Xiang).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.09.004

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approximately from the power spectrum. Because the acceleration is continuously measured when the vehicle passes through the structure, the interval between two adjacent measurement points can be easily adjusted at will. In addition, since the acceleration signal is measured at the same location of the tapping force, the point impedance actually plays an important role in this method. Consequently, the extracted MOSS is not sensitive to environment noises. Similar to the mode shape, the extracted MOSS also contains damage information, and hence can be used to detect the damage. However, it is known that structural global properties, such as the natural frequency and the mode shape, are not very sensitive to local damage [1,17,18]. Therefore, a new damage index is introduced in this paper to dominate the difference between the damaged and intact MOSS curve. The following text is organized as: Section 2 presents the theoretical analysis of the interaction system of a passing vehicle and a plate or a beam. Based on this, the method of extracting the approximate MOSS is proposed. Section 3 gives some numerical examples to prove the accuracy and validity of this method, discusses several factors that influence the accuracy of the extracted MOSS, and shows the potential of using the MOSS to detect damage. Section 4 conducts two experimental validations for this method. Finally, conclusions are presented in Section 5. 2. The damage detection method based on the extracted MOSS 2.1. The interaction between a passing vehicle and a plate or a beam Fig. 1 is a simplified model for the proposed method to extract the MOSS. In this model, a vehicle of mass M is supported on a spring of stiffness kV, and excited by a tapping force F sin ot. This vehicle passes through a plate with ~ per arbitrary boundary condition along the line y¼y0 at constant speed v. This plate has bending stiffness D and mass m unit area. For simplicity, some practical factors, such as damping, are temporarily ignored in the theoretical analysis. The governing equations of the vehicle and the plate can be written as ~ u€ þ Dr4 u ¼ f ðtÞdðxvt,yy0 Þ m

(1)

M q€ d þkqd ¼ kuðvt,y0 ,tÞF sin ot

(2)

where qd(t) and u(x,y,t) are the vertical displacements of the vehicle and the plate measured from the static equilibrium position, respectively. f(t)d(x vt,y  y0) is the contact force between the vehicle and the plate: (3) f ðtÞdðxvt,yy0 Þ ¼ k½qd uðvt,y0 ,tÞMg where d is the Dirac delta function, which indicates the contact point at position x ¼vt, y¼ y0; g is the gravity acceleration; and k is an effective stiffness which represents the series connection between the vehicle spring stiffness kV and the plate stiffness kP: kV kP ðx,yÞ (4) kðx,yÞ ¼ kV þ kP ðx,yÞ The kP can be evaluated as the ratio of a load acting at the contact point (x,y) divided by the deflection generated at the same location. Using the modal superposition method, the displacement of the plate can be represented as XX uðx,y,tÞ ¼ jn,m ðx,yÞqn,m ðtÞ (5) n

m

where jn,m(x,y) is the (n,m)th mode shape and qn,m(t) is the corresponding modal coordinate. Substituting Eq. (5) into Eq. (1), multiplying jp,l on both sides and integrating over the whole area, obtains Z aZ b Z aZ b Z aZ b XX XX ~ jp,l ðjn,m q€ n,m Þ dx dy þ Djp,l r4 ðjn,m qn,m Þ dx dy ¼ f ðtÞdðxvt,yy0 Þjp,l dx dy m 0

0

n

m

0

0

n

m

Fig. 1. A passing vehicle on a plate.

0

0

(6)

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Using the orthogonality property of the mode shapes, Eq. (6) becomes f ðtÞjn,m ðvt,y0 Þ f ðtÞ q€ n,m þ o2n,m qn,m ¼ ¼ j ðvt,y0 Þ Ra Rb ~ 2 m On,m n,m ~ m 0 0 jn,m dx dy

(7)

where on,m is the (n,m)th natural frequency of the plate; and On,m is an integration constant Z aZ b On,m ¼ j2n,m dx dy 0

(8)

0

Considering Eqs. (2) and (3), Eq. (7) can be written as q€ n,m þ o2n,m qn,m ¼ 

ðM q€ d þMg þ F sin otÞ jn,m ðvt,y0 Þ ~ On,m m

(9)

Generally q€ d 5 g, otherwise one cannot ensure the firm contact between the vehicle and the plate. Therefore, Eq. (9) can be approximated as ðMg þ F sin otÞ jn,m ðvt,y0 Þ (10) q€ n,m þ o2n,m qn,m   ~ On,m m Generally, it is impossible to solve Eq. (10) if jn,m(vt,y0) is unknown. However, if the vehicle moves slowly enough, jn,m(vt,y0) can be approximately regarded as a constant of jn,m(vt0,y0) in a short time interval of [t0  Dt, t0 þ Dt]. Thus, Eq. (10) becomes q€ n,m þ o2n,m qn,m  

ðMg þ F sin otÞ jn,m ðvt0 ,y0 Þ ~ On,m m

(11)

In this case, the solution to Eq. (11) can be easily obtained as Fo F Mg qn,m ðtÞ ¼ j ðx ,y Þsin on,m t j ðx ,y Þsin ot j ðx ,y Þ ~ On,m ðo2n,m o2 Þ n,m 0 0 ~ On,m o2n,m n,m 0 0 ~ On,m on,m ðo2n,m o2 Þ n,m 0 0 m m m t 2 ½t0 Dt,t0 þ Dt

(12)

Substituting Eq. (12) into Eq. (5), yields the plate vertical displacement # " XX Fo F Mg uðx,y,tÞ ¼ sin on,m t sin ot j ðx ,y Þj ðx,yÞ ~ On,m ðo2n,m o2 Þ ~ On,m o2n,m n,m 0 0 n,m ~ On,m on,m ðo2n,m o2 Þ m m m n m t 2 ½t0 Dt,t0 þ Dt

(13)

Similarly, Eq. (2) can also be approximately written as M q€ d þkqd  kuðvt0 ,y0 ,tÞF sin ot ¼ kuðx0 ,y0 ,tÞF sin ot Substituting Eq. (13) into Eq. (14) yields q€ d þ o

2 d qd

¼

XX n

An,m ðt0 Þsin on,m t

m

XX n

m

! XX F Bn,m ðt0 Þ þ Cn,m ðt0 Þ sin ot M n m

where

od ¼ An,m ðt0 Þ ¼

(14)

t 2 ½t0 Dt,t0 þ Dt

rffiffiffiffiffi k M

(15)

(16)

F oo2d j2 ðx ,y Þ ~ On,m on,m ðo2n,m o2 Þ n,m 0 0 m

(17)

F o2d j2 ðx ,y Þ ~ On,m ðo2n,m o2 Þ n,m 0 0 m

(18)

Mg o2d j2 ðx ,y Þ ~ On,m o2n,m n,m 0 0 m

(19)

Bn,m ðt0 Þ ¼

Cn,m ðt0 Þ ¼

In a similar way, the solution of Eq. (15) is X X An,m sin on,m t ðP P Bn,m þ ðF=MÞÞsin ot P P Cn,m n m   n m2 qd ðtÞ ¼ o2d o2n,m o2d o2 od n m " P P # oð n m Bn,m þ ðF=MÞÞ X X on,m An,m  þ sin od t t 2 ½t0 Dt,t0 þ Dt 2 2 od ðo2d o2 Þ n m od ðod on,m Þ

(20)

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From Eq. (20), it is easy to obtain the vertical acceleration of the vehicle " P P # X X o2n,m An,m sin on,m t o2 ðP P Bn,m þ ðF=MÞÞsin ot oð n m Bn,m þ ðF=MÞÞ X X on,m An,m n m  þ  o q€ d ðtÞ ¼  sin od t d 2 2 ðo2d o2 Þ o2d o2n,m o2d o2 n m n m ðod on,m Þ t 2 ½t0 Dt,t0 þ Dt

(21)

The plate degenerates to an Euler beam when b 5a. Thus, Eq. (1) changes to mu€ þEIuðIVÞ ¼ f ðtÞdðxvtÞ

(22)

where m denotes the mass per unit length, E the elastic modulus and I the moment of inertia of the beam. Similarly, the acceleration of the vehicle passing through a beam can be obtained as " # P X X o2 An sin obn t o2 ðP Bn þ ðF=MÞÞsin ot obn An oð n Bn þ ðF=MÞÞ 2 n bn  þ þ o q€ d ¼  sin od t d 2 2 o2d o2bn o2d o2 od ðo2d o2 Þ n n od ðod obn Þ

An ðt0 Þ ¼

t 2 ½t0 Dt,t0 þ Dt

(23)

F oo2d j2 ðx0 Þ mO0n obn ðo2bn o2 Þ n

(24)

Fig. 2. Comparison of the extracted MOSS with the mode shape squares from analytical solution and modal testing method: (a) mode 1 and (b) mode 2.

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Bn ðt0 Þ ¼

F o2d j2 ðx0 Þ mO0n ðo2bn o2 Þ n

295

(25)

where obn and jn denotes the nth natural frequency and mode shape of the beam, respectively; and O0n is the corresponding integration constant. 2.2. Extracting the MOSS It observes from Eq. (21) that the vehicle acceleration q€ d contains three frequency components: the natural frequencies of the plate on,m, the vehicle frequency od, and the frequency of the exciting force o. Applying the Short Time Fourier Transformation (STFT) to Eq. (21), obtains the spectrum of q€ d at position x0 ¼vt0 Z 1 X X An,m ðt0 Þo2n,m Z 1 ~ Þ¼ q€ d ðtÞwðtt0 Þejo~ t dt ¼  Fy0 ðx0 , o ðsin on,m tÞwðtt0 Þejo~ t dt o2d o2n,m 1 1 n m P P Z ð n m Bn,m ðt0 Þ þ ðF=MÞÞo2 1 ðsin otÞwðtt0 Þejo~ t dt þ o2d o2 1 " P P #Z 1 oð n m Bn,m ðt0 Þ þ ðF=MÞÞ X X on,m An,m ðt0 Þ  ðsin od tÞwðtt0 Þejo~ t dt (26) od 2 2 2 2 ðod o Þ 1 n m ðod on,m Þ where w(t t0) is a window function.

Fig. 3. The extracted MOSS at different vehicle speed: (a) mode 1 and (b) mode 2.

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Then, the amplitude of the spectrum at frequency on,m and position x0 is 9Fy0 ðx0 , on,m Þ9 ¼

An,m ðt0 Þo2n,m

o2d o2n,m

(27)

Substituting Eq. (17) into Eq. (27), one can find that 9Fy0 ðx0 , on,m Þ9 ¼ Gn,m ðvt0 ,y0 Þj2n,m ðvt0 ,y0 Þ ¼ Gn,m ðx0 ,y0 Þj2n,m ðx0 ,y0 Þ

(28)

where Gn,m ðx0 ,y0 Þ ¼

F oon,m ~ On,m ðo2n,m o2 Þ½1ðo2n,m =ðo2d ðx0 ,y0 ÞÞÞ m

(29)

If the vehicle is very stiff, od b on,m , so that Gn,m could be regarded as a constant over the whole plate approximately. Thus, the amplitude of the spectrum is approximately proportional to the MOSS

j2n,m ðx1 ,y0 Þ : j2n,m ðx2 ,y0 Þ :    : j2n,m ðxN ,y0 Þ  9Fy0 ðx1 , on,m Þ9 : 9Fy0 ðx2 , on,m Þ9 :    : 9Fy0 ðxN , on,m Þ9

(30)

where x1 ,x2 ,. . .,xN are the test points along the line y ¼y0, which could be flexibly chosen since the tapping vehicle scans the structure continuously. Based on Eq. (30) , one can extract the approximate MOSS using certain normalization method (normalizing to the maximum value is adopted in this paper) if test points are dense enough.

Fig. 4. The extracted MOSS under different exciting frequency at the speed of 1 m/s: (a) mode 1 and (b) mode 2.

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If the condition of od b on,m cannot be satisfied, Gn,m(x0,y0) certainly varies over the whole structure. Hence, the extracted MOSS from Eq. (30) will have some biases to the true mode shape square. However, as discussed in the following, one can still detect the damage with this biased MOSS. Actually, no matter how close the extracted MOSS to the true mode shape square, it just gives a pattern of the intact structure. This pattern will change when the structure is damaged. Supposing a single damage occurred at position (x0,y0), kP(x0,y0) could be different from its intact state, which would result in the change of od at that location. This change can be exaggerated in the extracted MOSS if the exciting frequency o is close to on,m, according to Eq. (29). Thus, it is possible to detect the damage by comparing the damaged MOSS with the intact one.

2.3. The damage index based on the extracted MOSS As explained in the above section, the damage could be represented as a local change in the extracted MOSS curve. However, this change could not be very clear if directly comparing the normalized MOSS with and without damage, especially when measurement noises were presented. To dominate the local change, the ith MOSS of the damaged structure, MOSSdi , can be modified by multiplying a scale factor a, so that it has the minimum distance to MOSSui , the ith MOSS of the undamaged structure: argmin :MOSSui aMOSSdi :2 a

Fig. 5. The extracted MOSS under different exciting frequency at the speed of 2 m/s: (a) mode 1 and (b) mode 2.

(31)

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where a can be easily obtained as



MOSSdi UMOSSui MOSSdi UMOSSdi

(32)

Then a new damage index Di can be proposed by calculating the discrepancy between the ith modified and intact MOSS at each point:

Di ¼ 9MOSSui aMOSSdi 9

(33)

Thus, the peak value of the damage index could indicate the damage location. 3. Numerical examples This section uses numerical simulations to check the validity of the method proposed in Section 2. For this purpose, a simply-supported beam similar to that in Ref. [11] and a passing vehicle with mass M¼1200 kg and the spring stiffness kV ¼2  109 N/m are adopted. This beam has the cross-sectional area A ¼4.71 m2, the moment of inertia I¼1.57 m4, the length L¼25 m, the elastic modulus E¼27.5 GN/m2 and the density r ¼2400 kg/m3. The numerical simulations are conducted by ABAQUS implicit Finite Element Method (FEM) package with the damping ratio of 5% and the time step of 0.001 s. During the simulation, the beam is modeled by 2-node plane beam elements in length of 0.5 m. The spectrum of the vehicle acceleration is obtained by the STFT method with the Hanning window of width Lw ¼1 s.

Fig. 6. The extracted MOSS under different exciting frequency at the speed of 5 m/s: (a) mode 1 and (b) mode 2.

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3.1. Verification of the MOSS extracting method To evaluate the quality of the extracted MOSS from Eq. (30), here just simply set the vehicle speed v ¼1 m/s and add two exciting forces with the same amplitude of F ¼6000 N but at different frequencies of f¼3 Hz, 18 Hz, which are close to the first two natural frequencies (4.9 Hz and 18.9 Hz) of the beam. Fig. 2 compares the MOSS extracted by the proposed Passing Vehicle (PV) method with the mode shape squares obtained from the Analytical Solution (AS) and the traditional modal testing method based on Frequency Response Functions (FRFs) [10]. The modal testing is numerically conducted by applying an impact load of 6000 N at 10 m away from the left end of the beam, and the calculated acceleration at each node is used to construct the FRFs. In these comparisons, measurement noises are also considered. The two polluted accelerations of the passing vehicle are with Signal to Noise Ratio (SNR) of 40 dB and 10 dB, respectively. And in the modal testing, the acceleration at each node is added with SNR ¼10 dB noises. It observes that (1) The extracted MOSS is almost the same as the analytical one in mode 1. But some biases can be detected in mode 2. This is because Eq. (30) is just an approximate relation, and mode 1 varies much smoothly than mode 2. (2) The mode shape square from FRFs approximately matches the analytical solution, while because of measurement noises, some irregularities appears at the locations of large curvature. (3) The extracted MOSS is not sensitive to measurement noise, even at the very poor noise level of SNR ¼10 dB. This is benefited from using the information of point impedance.

Fig. 7. The extracted MOSS by different window functions of the width Lw ¼ 1 s: (a) mode 1 and (b) mode 2.

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3.2. Influence of vehicle speed on the extracted MOSS As mentioned in Section 2.1, Eq. (10) can be approximated as Eq. (11) with the assumption that the vehicle moves at low speed. Thus, it is possible to obtain the analytical solution of the vehicle acceleration. Therefore, it is necessary to check the influence of vehicle speed on the quality of the extracted MOSS. For this purpose, the extracted MOSSes in different speed scenarios of v¼1,2,5 m/s are compared in Fig. 3, while keeping the same exciting force as that in Section 3.1. It is a natural observation that the lower the speed, the more accurate the extracted MOSS. In more details, the extracted MOSS of mode 1 is accurate enough even the speed grows up to 5 m/s; while the accuracy of mode 2 is just acceptable at the speed of 2 m/s. This coincides with the fact that mode 1 varies much smoothly than mode 2.

3.3. Influence of exciting frequency on the extracted MOSS According to Eqs. (28) and (29), if the exciting frequency o is close to the natural frequency on,m, the amplitude of vehicle acceleration spectrum is more sensitive to the mode shape square, and hence, it is easier to detect the damage with the extracted MOSS. However, if o was too close to on,m, the quality of the MOSS could not be so good due to the resonance. Therefore, it is necessary to discuss the optimal distance between o and on,m. Since it is very difficult to give an analytical criterion, here just uses numerical simulations to compare the extracted MOSSes with the analytical ones under different exciting frequencies at the vehicle speed of v ¼1,2,5 m/s, respectively. From Figs. 4–6, it observes that (1) At all speeds, the extracted MOSSes of mode 1 under exciting frequencies f¼3,4,5 Hz are more accurate than those under f¼2,6 Hz; and the extracted MOSSes of mode 2 under exciting frequencies f¼16,18,20 Hz are more accurate

Fig. 8. The extracted MOSS by the Hanning window of different widths: (a) mode 1 and (b) mode 2.

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Fig. 9. The damage detection results by (a) mode 1, (b) mode 2, and (c) the sum of modes 1 and 2 at the vehicle speed of 1 m/s.

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than those under f ¼14,22 Hz. This coincides with the conclusion that the exciting frequency should be close to the natural frequency to get better results. (2) Sometimes, the exciting frequency most close to the natural frequency cannot generate the best results, e.g. the curve of f¼ 5 Hz in Figs. 4(a) and 5(a), and the curve of f¼18 Hz in Figs. 5(b) and 6(b). This is probably due to the resonance effect. 3.4. Influence of window functions on the extracted MOSS Figs. 7 and 8 show the influence of the type and width of window functions on the extracted MOSS at the speed of 1 m/s. As shown in Fig. 7 different types of windows almost give the same results. This is probably because the exciting force is sinusoidal and periodic. It is well known that the frequency resolution is poor with the narrow window. Therefore, it can be observed from Fig. 8(a) that the extracted MOSS is problematic when the window width is only 0.5 s. However, even the frequency resolution can be improved with wide windows; cares should be taken about the spatial resolution in damage detections. 3.5. Damage detection based on the extracted MOSS To evaluate the damage detection capability of the extracted MOSS, an artificial damage is introduced into the beam discussed in Section 3.1 by reducing 20% height at the location between 15 m and 15.5 m. Some measurement noises of SNR 40 dB and SNR 10 dB are added into the calculated vehicle acceleration. And the Hanning window with the width of Lw ¼1 s is adopted in the STFT. With the extracted MOSS, the damage indices defined in Eq. (33) can be calculated to detect the damage. Fig. 9 plots the distribution of normalized damage indices calculated from the extracted MOSSes with different measurement noises and the clean mode shape squares obtained by the FEM. It seems that the first mode performances better than the second mode. This is probably because it has higher amplitude at the damaged location and as Section 3.1 shows, it has fewer biases to the true mode shape square. To avoid the casual results due to the coincidence of the locations of damage and the peek value of the MOSS, it is better to use a few modes to detect damage. As Fig. 9(c) shows, the sum of the damage indices of the first two modes could show acceptable performance, especially when measurement noises are presented. Fig. 10 compares the damage detection results by the sum of modes 1 and 2 at different speeds without measurement noises. It is clear that the slower the vehicle moves, the better damage detection result can be obtained. And the damage acceptably detected at the speed of 2 m/s, which is reasonable in practice. 4. Experimental verification With the confidence obtained in numerical simulations, two experiments were carried out to give further verification for the proposed method. 4.1. The setup of experiments The photographs and profiles of two plywoods in dimension of 150 mm  80 mm  5 mm are shown in Figs. 11 and 12, respectively. These two plates are supported at two ends with the span of 100 mm (see Figs. 12 and 13). In Plate 1, single

Fig. 10. The damage detection results by the sum of modes 1 and 2 at different vehicle speeds.

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impact damage is located on Line 1 at 25 mm away from the left support. In Plate 2, two impact damages are located on Line 2 at 35 mm and 70 mm away from the left support. All impact damages are about 1 mm deep with the diameter of 8 mm. As Fig. 13 shows, a simple tapping device (‘‘vehicle’’) is designed for these experiments. The base and wheels are all made of steel, which can offer hard stiffness. A mini shaker (JZK-2) is mounted on the top of the base to provide harmonic exciting forces. An accelerometer (YD-36) is fixed on the base to acquire the vertical acceleration at the sampling frequency of 1000 Hz. During the experiment, this vehicle passes along Line 1 or Line 2 on the smooth side of the plywood without the impact damage (see Fig. 12) at a constant speed of 5mm/s under the support of a linear motion unit. Since the natural frequencies of the plate were not clear, the sweep-frequency method (sinusoidal sweep from 20 Hz to 200 Hz at the sweep frequency of 10 Hz) was applied to generate the exciting force, which can cover the natural frequencies of the

Fig. 11. The damaged surface of plywood: (a) Plate 1 and (b) Plate 2.

Fig. 12. The profiles of the smooth surface of plywood: (a) Plate 1 and (b) Plate 2.

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plate while avoids resonance. The spectrum of the ‘‘vehicle’’ acceleration was obtained by the STFT with the Hanning window of Lw ¼1 s in width. 4.2. Experimental results The first two MOSSes of Line 1 before and after damage are plotted in Fig. 14, from which one can find the slight discrepancy between these two curves. These differences can be clearly dominated by the modified MOSS of the damaged plate, with which one can obtain the damage indices shown in Fig. 15. The position of the peak values indicates the

Fig. 13. Photograph of experiment setup.

Fig. 14. The first two MOSSes of Line 1 before and after damage on Plate 1: (a) mode 1 and (b) mode 2.

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Fig. 15. The damage detection results of Plate 1.

Fig. 16. The first two MOSSes of Line 2 before and after damage on Plate 2: (a) mode 1 and (b) mode 2.

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Fig. 17. The damage detection results of Plate 2.

location of damage. It observes that satisfactory results can be obtained either using only mode 1 or mode 2 or their combination for this single damage scenario. The similar results can also be found in Plate 2. One can marginally detect the difference between the MOSSes along Line 2 in Fig. 16. These differences are clearer in terms of damage indices in Fig. 17.

5. Conclusion This paper proposes a method to approximately extract the MOSS of plate and beam structures from the acceleration of a passing ‘‘vehicle’’ with tapping devices. Based on the extracted MOSS, a new damage index is introduced to detect the damage. Since only one sensor mounted on a passing vehicle is needed, a large number of measurement points can be more easily acquired by this method than by traditional modal testing methods, which usually require pre-installing many sensors on the structure. In addition, this method is robust in noisy environment, because it is based on the idea of point impedance. Because the proposed method is based on the assumption of low moving speed, the extracted MOSS could have more biases and the consequent damage detection result could be worse when the vehicle moves faster. However, the numerical simulations show acceptable damage detection results when the speed comes up to 2 m/s, which is reasonable for its implementation to short bridges. And the experimental results also show the potential of this method to the damage detection of composites. Besides the vehicle speed, the damage detection is also affected by the exciting frequency, which should be close to the natural frequency of the structure while avoiding the resonance. In practice, the sweep-frequency method is recommended to generate the ideal exiting force. The mode shapes could also be extracted by this method with additional information. Since the MOSS is always positive, the absolute value of mode shapes can be easily obtained. However, a positive or negative sign should be determined in order to reconstruct the phase information of the mode shape. And this information could be obtained from experimental experience for simple structures as what was discussed in Ref. [9].

Acknowledgments This work is support by National Science Foundation of China with Grant no. 10802040). This support is gratefully acknowledged. References [1] [2] [3] [4]

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