A novel method for single and multiple damage detection in beams using relative natural frequency changes

Mechanical Systems and Signal Processing 132 (2019) 335–352

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A novel method for single and multiple damage detection in beams using relative natural frequency changes Ganggang Sha a,b, Maciej Radzien´ski b, Maosen Cao c,a,⇑, Wiesław Ostachowicz b a

Department of Engineering Mechanics, Hohai University, Nanjing 210098, China Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdansk 80-231, Poland c Jiangxi Provincial Key Laboratory of Environmental Geotechnical Engineering and Disaster Control, Jiangxi University of Science and Technology, Ganzhou 341000, China b

a r t i c l e

i n f o

Article history: Received 31 October 2018 Received in revised form 7 June 2019 Accepted 20 June 2019

Keywords: Damage localization Damage growth monitoring Severity estimation Relative natural frequency change Bayesian data fusion

a b s t r a c t Concentrated damage such as cracks is one of the most common types of damage in beams. It is essential to detect such damage early to avoid structural failure. Vibration-based damage detection methods that employ changes in structural dynamic parameters such as natural frequencies have been extensively studied, and it is commonly acknowledged that natural frequencies depicting structural global dynamic properties are incompetent to portray local damage. Differing from extant work, this study presents a concept of relative natural frequency change (RNFC) curves for local damage characterization and especially ascertains the relationship between RNFC curves and mode shapes, leading to an explicit equation of RNFC. With RNFC curves and measured values of RNFCs, a two-step method for localizing and quantifying damage is created: a novel probabilistic damage indicator is developed using Bayesian data fusion for localizing single and multiple damage; moreover, a damage severity factor defined as the stiffness reduction ratio of the damaged element is formulated to quantify damage. The proposed method features localization, quantification, and evolution monitoring of damage, relying solely on natural frequencies. The efficacy of the method is verified numerically and then validated experimentally on cracked beams. The numerical and experimental results demonstrate the capability of the method to localize single and multiple damage and to estimate damage severity. This mechanism of characterizing damage relying solely on natural frequencies provides the foundation for developing practical local damage detection and monitoring technologies for beam-type structures. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Structural damage detection has a variety of applications in the civil, mechanical, and aerospace industries [1,2]. In particular, vibration-based methods have been widely studied, as they are nondestructive, inexpensive, and expedient [3,4]. These methods are generally based on damage-induced changes in structural modal parameters, such as natural frequencies, mode shapes, and damping ratios [5,6]. Mode shapes require measurements at numerous locations, entailing a timeconsuming and complex process. Damping ratios are difficult to measure and sensitive to environmental factors such as variations in humidity and temperature [7]. In contrast, natural frequencies are easily measured using a single sensor at any ⇑ Corresponding author. E-mail address: [email protected] (M. Cao). https://doi.org/10.1016/j.ymssp.2019.06.027 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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accessible point in the structure (except for a node of any particular mode of vibration) and can be monitored with superior accuracy and reproducibility [8,9]. Thus, natural frequency-based damage detection has been presented in many studies [10–34]. Among these, some attempts to detect single damage in beam-like structures are as follows. Adams et al. [10] found that the ratio of natural frequency changes in two modes is only a function of damage location. This property has been used for localization of single damage in beam-like structures [11,12]. Kam et al. [13] used natural frequencies and mode shapes to detect a single crack in a cantilever beam with damage modeled using a reduced stiffness element. Nikolakopoulos et al. [14] and Nahvi et al. [15] utilized contours extracted from three-dimensional plots of the relationships between natural frequency changes and crack parameters to detect a single crack in beams. Lee et al. [16] combined finite element (FE) analysis with rank ordering of the first four natural frequencies of a cantilever beam to locate a single crack. Sayyad et al. [17] used the first two natural frequencies to identify crack location and size. Lee [18] presented a method for detecting the location and size of a crack in a tapered cantilever pipe-type beam using natural frequency changes. Yang et al. [19] developed a damage detection approach, using a modal frequency surface that was generated by attaching a point mass at different locations. The detection of multiple damage in beams using natural frequencies has attracted much more attention from researchers owing to the greater challenge and significance compared to single damage detection [20]. Vestroni et al. [21] presented a damage detection procedure based on a priori knowledge of possible damage locations. Morassi et al. [22] presented a technique for detecting two cracks of equal severity in a simply supported beam relying on changes in the first three natural frequencies. They discovered that, even if symmetrical positions are kept away, cracks with different severity in two sets of different locations could produce identical natural frequency changes. Rubio et al. [23] proved that the appropriate use of natural and antiresonant frequencies could avoid the non-uniqueness problem of damage localization. Patil et al. [24] developed a method for identifying multiple cracks in a beam using a transfer matrix method to simulate the transverse vibration of the beam, with each crack modeled using a rotational spring. Lin et al. [25] proposed a natural frequency change index and used it to determine the depths of two cracks in beams. Singh et al. [26] derived a damage indicator from a probability density function to characterize multiple cracks in shafts. Mazanoglu et al. [27] used natural frequency contour lines to detect double cracks in beams. Chang et al. [28] and Xiang et al. [29] presented a two-step multiple damage detection method that included mode shape-based crack localization and natural frequency-based severity estimation. Khiem et al. [30] treated the inverse diagnostic problem as an optimization problem in which damage location and severity were estimated by minimizing the difference between theoretical and experimental natural frequencies. The optimization algorithms used in such methods include genetic algorithm [31], bee algorithm [32], particle swarm optimization [33], and hybrid Cuckoo-NelderMead optimization algorithm [34]. These optimization methods are effective but time-consuming, especially for multiple damage detection. Moreover, it is difficult to determine the error criterion used to control the convergence of the optimization procedure. Differing from extant methods, an ingenious procedure is provided in this study for damage identification in beams based on natural frequencies. The rest of the paper is organized as follows. Section 2 presents the damage-induced relative natural frequency change (RNFC) curves, on which basis a novel damage detection method is proposed that combines RNFC curves and measured values of RNFCs. Section 3 provides numerical proof of the method. Section 4 presents the experimental validation of the applicability of this method by detecting cracks in a beam. Conclusions are presented in Section 5. 2. Method formulation 2.1. Damage-induced RNFC curves The presence of concentrated damage, e.g., crack and notch in a beam, can cause changes in natural frequencies. Herein, damage-induced RNFC curves on a fixed-fixed beam are analyzed using a FE model (Fig. 1). In the model, damage is modeled by a constant stiffness reduction in an element, where the stiffness reduction is represented by a stiffness reduction ratio (SRR), defined as the ratio between the reduced and original modulus. The jth natural frequency of the intact beam is xj with j being the number of modes. When damage is at the ith element, the jth natural frequency of the damaged beam is denoted as xdi;j . The RNFC, denoted by Dxi;j , is calculated as

Dxi;j ¼

xj  xdi;j xj

ð1Þ

Fig. 1. Fixed-fixed beam model with damage at the dimensionless coordinate f.

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A sequence of damage cases is produced by changing the SRR from 0% to 50% in 2% increments successively. For these damage cases, RNFCs of the first four modes are evaluated and plotted in Fig. 2. In the figure, RNFCs take on a twodimensional function of damage severity, SRR, and location, fi , the dimensionless coordinate of the ith element. Moreover, the function of RNFCs versus damage location and severity can be represented in an approximate form [35,36]:

Dxi;j ¼ s  g j ðfi Þ

ð2Þ

where s is a damage severity factor that is independent of the mode number, and g is a function of the damage location for each mode. For a constant damage severity factor s, Dxi;j of the jth mode can be normalized into the [0, 1] range as

Dxi;j  minðDxi;j Þ



Dxi;j ¼

i

maxðDxi;j Þ  minðDxi;j Þ i

ð3Þ

i





In terms of Eq. (2), Dxi;j is only related to damage location. From Eqs. (2) and (3), the RNFC curve can be represented by

Dxi;j as 

g j ðfi Þ ¼ Dxi;j

ð4Þ

The first four RNFC curves are individually displayed as red dashed lines in Fig. 3. 2.2. Relationship between RNFC curves and mode shapes A relation of RNFCs with mode shapes was derived in [37,38]:

Fig. 2. The RNFCs versus damage location and severity for 1st–4th modes of the fixed-fixed beam.

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Fig. 3. First four RNFC curves,

r curves and mode shapes.


2  Dxj ¼ s  /00j

ð5Þ



where s corresponds to damage severity, /j is the jth mode shape, and /00j is the mode shape curvature. From Eqs. (2) and (5), 2

2

each RNFC curve can be estimated by normalizing the corresponding mode of ð/00 Þ . Normalization of ð/00 Þ , denoted by r, is calculated as


 2 2 ð/00 Þ  min ð/00 Þ

 r¼ 2 2 max ð/00 Þ  min ð/00 Þ

ð6Þ

In a fixed-fixed beam, the mode shapes can be expressed as [39]

  coskj  coshkj /j ðfÞ ¼ C j ðcoskj f  coshkj fÞ  ðsinkj f  sinhkj fÞ sinkj  sinhkj

ð7Þ

kj  ð2j þ 1Þp=2

ð8Þ

where C j is an arbitrary constant. The first four r curves are individually plotted in Fig. 3 as black lines. It is clear that the r curves almost coincide with the RNFC curves; this relationship indicates the potential for obtaining RNFC curves using mode shapes. In Fig. 3, mode shapes are also presented as blue lines for reference.

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2.3. Damage detection algorithm The natural frequencies of the intact and damaged beams are denoted as xj and xdj . The RNFC curve, g j ðfi Þ, can be obtained via numerical simulation or theoretical analysis as described in Section 2.2. xdj is measured from the damaged beam. The novel damage detection algorithm involves the following two steps. 2.3.1. Damage localization An indicator for damage localization is developed as follows. (1) Calculation of the normalized RNFCs:

Dxj ¼ 

Dxj ¼

xj  xdj ; xj

ð9Þ

Dxj  minðDxj Þ : maxðDxj Þ  minðDxj Þ

ð10Þ

(2) Definition of the damage position function (DPF) at the element i of the jth mode:

     DPFi;j ¼ 1  g j ðfi Þ  Dxj :

ð11Þ

(3) Data fusion of multiple DPFs: Bayesian theory is an effective method for data fusion and has been used in damage detection [40,41]. The following is a brief introduction to data fusion using Bayesian theory. Consider that there are m information sources, S1, S2,. . ., Sm, and n events, A1, A2,. . ., An, that will be identified. The prior probability of each event is denoted as P(Ai). The conditional probability of Ai is known as PðS1 ; S2 ; :::; Sm jAi Þ. In accordance with the Bayesian formula, the Bayesian fusion can be given as

PðS1 ; S2 ; :::; Sm jAi ÞPðAi Þ u¼1 ðPðS1 ; S2 ; :::; Sm jAu ÞPðAu ÞÞ

PðAi jS1 ; S2 ; :::; Sm Þ ¼ Pn

ð12Þ

When the decision from each information source can be considered independent, Eq. (12) can be written as

PðAi jS1 ; S2 ; :::; Sm Þ ¼ P

Qm PðSj jAi ÞPðAi Þ
j¼1  Qm n u¼1 j¼1 PðSj jAu ÞPðAu Þ

ð13Þ

To identify damage, mode j is considered as information source Sj and damage occurring at element i is denoted as event Ai. The prior probability of each event is set as P(Ai) = 1/n. The conditional probability of Ai is obtained as

 PðSj Ai Þ ¼ DPFi;j

ð14Þ

According to Eq. (13), the Bayesian fusion can be given as

PðAi jS1 ; S2 ; :::; Sm Þ ¼ P

Qm DPFi;j
j¼1  Qm n u¼1 j¼1 DPFu;j

ð15Þ

where PðAi jS1 ; S2 ; :::; Sm Þ represents Bayesian probability and is written as P i for simplicity. (4) Improvement of Bayesian probability:

Qi ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pi Pnþ1i

ð16Þ

(5) Z-score normalization:

Z-score ¼

Q  meanðQ Þ SDðQ Þ

where SDðQ Þ is the standard deviation of Q .

ð17Þ

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(6) Calculation of the probabilistic damage indicator (PDI):

 PDI ¼

Z-score; 0;

if Z-score P 0 : if Z-score < 0

ð18Þ

Damage locations are pinpointed by the peaks of PDI. 2.3.2. Severity estimation Suppose that k damage locations have been identified at f1 , f2 ,. . ., and fk . If the damaged beam remains a linear system, the following equation is obtained:

Dxj ¼

k X

sr  g j ðfr Þ

ð19Þ

r¼1

where sr is the damage severity factor of the rth damage. Severity factor s can be solved with k RNFCs. The SRR of the damaged element is adopted as a more general physical measure of damage severity than the geometrical one e.g., depth, because a crack usually has an irregular shape that cannot be exactly described by the depth. SRR can be obtained from the corresponding s via the relationship between s and SRR established using FE analysis. 2.4. Temperature effect Natural frequency is a quantity that is easy to measure while susceptible to temperature variation that occurs in usual circumstances during the course of structural damage monitoring [42,43]. For this reason, it is necessary to investigate the ability of the proposed method to accommodate temperature fluctuation. The effect of temperature on natural frequencies is widely reported. In structures with boundary conditions released or in which the internal force is insignificant, the effect of temperature on natural frequencies can be commonly described by the fact [44,45] that temperature fluctuation causes changes in the elastic modulus and geometrical parameters and such changes alter the natural frequencies of the structure. The underlying mechanism can be interpreted by the analytical expression of natural frequencies [39], taking a beam as an example:

xj ¼

k2j

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Ebh 3

12Ml

ð20Þ

where l, b, and h are the length, width, and height of the beam, respectively, E is the elastic modulus, and M is the mass of the beam. When variables l, b, h, and E are the functions of temperature, the variation of natural frequencies can be calculated as

dxj

xj

¼

dE db 3dh 3dl þ þ  2E 2b 2h 2l

ð21Þ

where d represents an increment in corresponding variables. Assuming that the coefficient of linear thermal expansion of the material is hT and the temperature coefficient of the elastic modulus of the material is hE , one can obtain

dE db dh dl ¼ hE DT; ¼ hT DT; ¼ hT DT; ¼ hT DT E b h l where DT is the difference between the actual and reference temperatures. Combining Eqs. (21) and (22) yields dxj

xj

¼

1 ðhE þ hT ÞDT 2

ð22Þ

ð23Þ

The RNFCs estimated from Eq. (23) are independent of mode order and can be removed by the normalization manipulation in Eq. (10). As a result, when only the effect of temperature on elastic modulus and geometrical parameters is considered, the proposed method is competent to locate damage in beam-type structures under temperature fluctuations. It is noteworthy that, when the effect of temperature on an internal force such as the axial force in a beam is considered, the proposed method is inappropriate for damage detection. In a fixed-fixed beam, when the temperature changes, an axial force occurs due to constrained thermal extension [46] and therefore the natural frequencies change. In this situation, the natural frequency change is nonlinear and complex. The RNFCs due to temperature-induced axial force are mode dependent and thus cannot be eliminated by normalization, as deduced in the Appendix. The proposed method is incapable of locating damage under fluctuating temperatures. 3. Numerical proof of method 3.1. Specimen description A FE model of a fixed-fixed beam is built using 300 elements. The geometrical parameters and material properties of the beam are taken as length 1000 mm, width 10 mm, height 20 mm, elastic modulus 210 GPa, and density 7860 kg/m3. Damage

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is introduced by reducing the stiffness of some elements. Damage cases with different locations and size of damage are elaborated as shown in Table 1. The first six natural frequencies of intact and damaged beams are obtained via modal analysis (Table 2).

Table 1 Damage cases with different locations and size of damage. Case no.

Damage location and severity f1

SRR1

f2

SRR2

f3

SRR3

1 2 3 4 5

0.30 0.50 0.15 0.15 0.15

5% 5% 5% 5% 5%

– – 0.30 0.30 0.45

– – 5% 10% 5%

– – – – 0.70

– – – – 5%

Table 2 Natural frequencies of intact and damaged beams. Case no.

Intact 1 2 3 4 5

Natural frequencies (Hz) 1

2

3

4

5

6

105.99 105.99 105.98 105.98 105.98 105.97

291.28 291.23 291.28 291.23 291.18 291.22

568.74 568.71 568.64 568.67 568.64 568.60

935.50 935.49 935.50 935.36 935.34 935.29

1389.26 1389.05 1389.03 1388.85 1388.61 1388.74

1927.22 1926.99 1927.20 1926.82 1926.57 1926.59

Fig. 4. PDIs of case 1 by fusing first (a) 5 and (b) 6 modes.

Fig. 5. PDIs of case 2 by fusing first (a) 5 and (b) 6 modes.

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3.2. Damage detection results 3.2.1. Single and multiple damage detection The PDIs for cases 1–5 are obtained by fusing the first 5 and 6 modes, shown in Figs. 4–8 with red dashed lines marking actual damage locations. In Figs. 4–8, the red dashed lines seem to symbolize peaks of PDIs, thereby the latter pinpointing damage locations for both single and multiple damage cases. Fusion using more modes leads to higher resolution of damage localization. This method invariably yields symmetrical fake damage locations because the damage at symmetrical locations causes identical amounts of natural frequency change. This is a common phenomenon that has been reported in the

Fig. 6. PDIs of case 3 by fusing first (a) 5 and (b) 6 modes.

Fig. 7. PDIs of case 4 by fusing first (a) 5 and (b) 6 modes.

Fig. 8. PDIs of case 5 by fusing first (a) 5 and (b) 6 modes.

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literature [10,22,47]. Although the solution of damage locations is not unique, this method has the ability to condense damage locations into an extremely small region consisting of real damage locations and their fake counterparts at symmetrical positions. When considering the prominent merits of natural frequency endowing the method with reliability and easy

Fig. 9. The relationship between s and SRR for case 5.

Fig. 10. PDIs of case 3–4 by fusing first (a) 5 and (b) 6 modes.

Fig. 11. PDIs of case 1–4 by fusing first (a) 5 and (b) 6 modes.

Fig. 12. A steel beam with three cracks.

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Table 3 Measured natural frequencies of a beam with three cracks at f1 , f2 , and f3 in dissimilar depth scenarios. Crack scenarios

Intact (0-10-0)% (single crack) (10-10-0)% (double crack) (10-10-10)% (1st triple crack) (20-20-10)% (2nd triple crack)

Natural frequencies (Hz) 1

2

3

4

5

6

43.17 43.14 43.14 43.14 43.07

119.0 118.99 118.96 118.87 118.74

233.3 233.16 233.0 232.93 232.14

385.6 385.52 385.29 385.2 384.26

576.1 575.91 575.82 575.38 574.55

804.6 804.17 804.15 803.7 802.29

Fig. 13. First 6 RNFC curves and DPFs of the single crack scenario.

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acquisition, the limitation of fake counterparts cannot impair the highlight of the proposed method. Moreover, the fake counterparts of actual damage locations can be identified by nondestructive methods since the possible damage locations are recognized. When damage occurs in the middle of the beam, there is no fake symmetrical peak (Fig. 5). Fig. 9 shows the relationship between s and SRR obtained by numerical simulations. Case 5 is used to confirm the feasibility of estimating damage severity. In accordance with Eq. (19), (s1 ; s2 ; s3 ) are solved by using the RNFCs from the 1st to 6th modes, with the resulting solution being (0.031, 0.036, 0.033)%. For this solution of (s1 ; s2 ; s3 ), the SRRs are (4, 5, 5)%, showing good agreement with the actual damage severity.

Fig. 14. PDIs of the single crack scenario by fusing first (a) 5 and (b) 6 modes.

Fig. 15. PDIs of the double crack scenario by fusing first (a) 5 and (b) 6 modes.

Fig. 16. PDIs of the 1st triple crack scenario by fusing first (a) 5 and (b) 6 modes.

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3.2.2. Damage growth monitoring Along with detecting existing damage, the proposed method offers the benefit of monitoring the growth of damage. First, case 3 is taken as the reference to monitor the growth of existing damage from cases 3 to 4; this is denoted as case 3–4. Following this, case 1 is taken as the reference to monitor both the growth of existing damage and initiation of new damage from cases 1 to 4; this is denoted as case 1–4. The PDIs of cases 3–4 and 1–4 produced by fusing the first 5 and 6 modes are presented in Figs. 10 and 11. The red dashed lines mark the actual locations of damage growth. These results show that both the growth of existing damage and initiation of new damage can be effectively monitored. This means that natural frequencies measured at a particular stage of the life of the beam can act as the reference for monitoring possible growth of damage.

Fig. 17. The relationship between s and SRR for the 1st triple crack scenario.

Fig. 18. PDIs of the double–triple crack scenario by fusing first (a) 5 and (b) 6 modes.

Fig. 19. PDIs of the 1st–2nd triple crack scenario by fusing first (a) 5 and (b) 6 modes.

G. Sha et al. / Mechanical Systems and Signal Processing 132 (2019) 335–352

(a)

(b)

Saw cut

Fig. 20. Experimental setup: (a) Electromechanical shaker and (b) SLDV.

Fig. 21. FRFs for the intact beam and the damaged beam at Stages I, II, and III.

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4. Experimental validation 4.1. General method validation 4.1.1. Experimental description The experiment on a fixed-fixed beam is taken from Khiem et al. [48]. The parameters of the test beam (Fig. 12) are: length 1040 mm, width 20 mm, height 9 mm, elastic modulus 200 GPa, and density 7855 kg/m3. Three cracks were fabricated by saw cuts at f1 = 0.2, f2 = 0.45, and f3 = 0.7, with dissimilar depths. Table 3 provides the first six measured natural frequencies for crack scenarios concerned.

4.1.2. Results (1) Single and multiple damage detection Fig. 13 presents the first 6 RNFC curves and DPFs of the single crack scenario. Figs. 14–16 show the PDIs of single, double, and 1st triple crack scenarios. These PDIs are produced by fusing the first 5 and 6 modes. The red dashed lines mark the actual crack locations. Clearly, peaks of PDIs in Figs. 14–16 coincide with the red dashed lines, indicating the crack locations

Table 4 Natural frequencies of the intact beam and the damaged beam at Stages I, II, and III. Stage

T (°C)

Intact I II III

17 17 33 41

Natural frequencies (Hz) 1

2

3

4

5

7.04 6.88 6.87 6.87

44.07 43.52 43.44 43.43

123.44 120.32 120.78 120.78

241.41 240.63 240.31 239.37

423.29 418.75 417.81 417.03

Fig. 22. PDIs by fusing first (a) 2, (b) 3, (c) 4, and (d) 5 modes for Stage I: 17 °C.

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for both single and multiple crack scenarios. Better results of crack detection are obtained as more modes are fused. As shown in Figs. 14–16, some small peaks of PDIs appear near the boundary areas for each scenario and the peaks are not well separated for the 1st triple crack scenario. Possible reasons are insufficient modes fused as well as measurement error in the measured natural frequencies. The relationship between s and SRR is established using a FE model of the experimental sample with 200 elements as shown in Fig. 17. The 1st triple crack scenario is used to confirm the applicability of damage severity estimation. In accordance with Eq. (19), (s1 ; s2 ; s3 ) are solved by using the RNFCs from the 1st to 6th modes, with the resulting solution being (0.160, 0.156, 0.165)%. For this solution of (s1 ; s2 ; s3 ), the SRRs are (14, 13, 14)%. This result means that the (10, 10, 10)% depth of cracks in the experiment are equivalent to the (14, 13, 14)% stiffness reduction of the damaged element. (2) Damage growth monitoring The double crack scenario is taken as the reference for monitoring the initiation of new damage from the double to 1st triple crack scenarios; this is denoted as the double–triple crack scenario. The 1st triple crack scenario is taken as the reference for monitoring the growth of existing damage from the 1st to 2nd triple crack scenarios; this is denoted as the 1st–2nd triple crack scenario. The PDIs of the double–triple and 1st–2nd triple crack scenarios produced by fusing the first 5 and 6 modes are presented in Figs. 18 and 19. The red dashed lines mark the actual locations of damage growth. The experimental results confirm the capability of the proposed method to monitor both the initiation of new damage and growth of existing damage. 4.2. Experimental validation: temperature effect 4.2.1. Experimental setup An experiment is elaborated to validate the ability of the proposed method to locate damage under varying temperatures. The specimen is an aluminum cantilever beam. The dimensions of the beam are 480 mm length, 15 mm width, and 2 mm height. A through-height crack, as enclosed by the red circle in Fig. 20(a), is created by saw cut at the position 161 mm (f  0:34) from the fixed end with the depth of 2 mm. The temperature variation of the beam is realized and controlled

Fig. 23. PDIs by fusing first (a) 2, (b) 3, (c) 4, and (d) 5 modes for Stage II: 33 °C.

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Fig. 24. PDIs by fusing first (a) 2, (b) 3, (c) 4, and (d) 5 modes for Stage III: 41 °C.

by using a heater to adjust the environmental temperature. The sequence of temperatures, 17 °C, 33 °C, and 41 °C, is considered for investigation of the method, with 17 °C being the reference temperature with regard to the intact state of the beam. Three temperature situations of the damaged beam are elaborated: Stage I at 17 °C, Stage II at 33 °C, and Stage III at 41 °C. An electromechanical shaker is used to excite the beam near the fixed end at the centerline. A scanning laser Doppler vibrometer (SLDV, Polytec PSV-400) is used to measure the velocity frequency response function (FRF) of the intact beam and the damaged beam at Stages I, II, and III. The experimental setup is shown in Fig. 20. 4.2.2. Results The FRFs for the intact beam and the damaged beam at Stages I, II, and III are illustrated in Fig. 21, from which the first five natural frequencies (Table 4) are determined by the peaks of the FRFs. Clearly, the natural frequencies decrease with the increase in temperature. The PDIs for Stages I, II, and III, obtained by fusing the first 2, 3, 4, and 5 modes individually, are presented in Figs. 22–24, respectively. It is evident from Figs. 22–24 that the localization accuracy almost maintains an increasing trend with the increase in fused modes and the localization resolution reaches a high level when the first 5 modes are fused. Clearly, the peak of PDIs near the red dashed line pinpoints the location of damage, regardless of temperature variation. Thus it is demonstrated that the proposed method can localize damage in such beam-type structures under varying temperatures. 5. Conclusions The RNFC curve representing concentrated damage in a beam is analyzed using FE analysis. The relationship between RNFC curves and mode shapes is clarified, providing an explicit expression of RNFC with damage location and severity factor. A two-step method for damage localization and severity estimation is proposed based on the RNFC curves and measured values of RNFCs. The numerical and experimental results show that the proposed method is capable of localizing and quantifying single and multiple damage in a beam. Furthermore, the proposed method offers the following main advantages: (i) It identifies damage, taking advantage of natural frequencies for facilitation and reliability in measurement; (ii) it can not only localize but also quantify multiple damage, relying solely on natural frequencies; (iii) it offers the benefit of monitoring damage growth in structures. Finally, a possible limitation of the method is that it always yields possible damage locations sym-

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metrically. Such limitation is common in natural frequency-based methods. To resolve the symmetrical localization problem, the method can be used in combination with conventional nondestructive testing methods, such as local inspection using ultrasonic or radiography equipment. In future work, we intend to fuse PDIs with mode shape-based damage indicators to address the symmetrical localization problem. Acknowledgements The authors are grateful for partial support from the National Key Research and Development Program of China (2018YFF0214705), the National Natural Science Foundation of China (11772115), the Fundamental Research Funds for the Central Universities (2016B45014), and a scholarship from the China Scholarship Council (201706710025). Appendix In a fixed-fixed beam, when the temperature changes, an axial force FT occurs due to constrained thermal extension [46]:

F T ¼ h T  E  A  DT

ðA1Þ

In the case of compression force, the natural frequencies of the beam are [49]

xDj T

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u FT l ¼ xj t1  2 nj EI

ðA2Þ

where xjDT denote the natural frequencies including the temperature-induced axial force, and nj are the buckling eigenvalues. The RNFCs due to the temperature-induced axial force are calculated as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u FT l Dxj ¼ 1  t1  2 nj EI

ðA3Þ

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