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Abstract

It is difficult for the traditional methods to identify uncertain damage problems caused by noise. Therefore, a gray cloud rule generator algorithm based on cloud model and modal strain energy is presented to solve the problems. Cloud model can simulate both randomness and fuzziness with fixed parameters. Therefore, it is applicable for the uncertain damage problems. First, modal strain energy and modal strain energy dissipation ratio index are introduced. Then, numerical characteristics of a cloud model are described and some cloud generators are analyzed. Finally, a gray cloud rule is proposed and the gray cloud rule generator algorithm based on the gray cloud rule generator and modal strain energy is developed. The interference of uncertain noise is reduced through a large number of cloud droplets. A two-dimensional truss structure model has been used to verify the effectiveness of the algorithm. The results indicate that the proposed gray cloud rule generator algorithm is applicable to identify the uncertain damage caused by noise, and the identification results of the proposed method are relatively better than those of modal strain energy dissipation ratio index.

Keywords

Damage identification gray cloud rule modal strain energy generator cloud reasoning

Introduction

In the past two decades, the security of building structures has received increasing concern, especially in the damage detection for the structure itself. As a result of the structural damage, the local rigidity of the structure will descend while the modal parameters change. Most of the researchers make the damage detection analysis based on modal parameters and structural dynamic responses.¹ Therefore, structural damage identification is based on changes in dynamic characteristics of a structure to localize the damage zones and estimate the extent of damage. Generally, the structural damage identification methods can be divided into deterministic and uncertain methods. The deterministic method is relatively a traditional and conventional method, for example, the method based on structural vibration measurement. Most deterministic damage identification studies directly focus on dynamic characteristics obtained from dynamic tests. However, the uncertain damage identification method based on probability and statistics focus on the nature of uncertainty of the damage problems² such as structural damage detection following the Bayesian approach and the statistics theory.³ It is still necessary to utilize dynamic characteristics in the uncertain damage identification method.

Dynamic characteristics include natural frequencies, mode shapes, modal strain energy (MSE), and so on. They have been widely used in structural damage detection.⁴ Salawu⁵ identified structural damage through frequency changes. Chaudhari and Maiti⁶ proposed a numerical method for determining the location of a crack in a beam with varying depth by considering the three lowest natural frequencies of the cracked beam. Zhang et al.⁷ identified the delamination damage of composite beams using natural frequencies. Koh and Dyke⁸ used modal correlation to detect the damage of a suspension bridge. Shi et al.⁹ presented a modal strain energy change ratio (MSECR) for structural damage localization analysis by the modal displacement change before and after damage. Fan and Qiao¹⁰ detected the damage of the plate structures using the wavelet transform of structural vibration mode and modal shape, and obtained good results. Sazonov and Klinkhachorn¹¹ studied vibration-based methods for nondestructive damage detection by utilizing curvature and strain energy mode shapes. Liu et al.¹² reported a modal strain energy dissipation ratio index (MSEDRI), which can approximately detect the locations of damage and the severity of damage. Seyedpoor¹³ proposed a two-stage damage detection method based on MSE and particle swarm optimization. However, it is difficult for these traditional methods to solve the uncertain damage problem caused by measurement noise. Some robust methods were also developed to identify structural damage. Nobahari et al.¹⁴ presented an echolocation search algorithm to detect the site and extent of multiple damage cases in structural systems. We will also develop a new method to deal with uncertain damage problems.

Cloud model is a new cognitive model that can synthetically describe the fuzziness and randomness of concepts and implement the uncertain transformation between the qualitative concepts and the quantitative description.^15–17 It is applicable to uncertain damage identification problems. In statistics, probability distributions can be used to model the uncertainty of variables representing random phenomenon known as randomness. In fuzzy mathematics, the membership function is used to model the uncertainty of membership to fuzzy concepts known as fuzziness. Both randomness and fuzziness can be modeled by a cloud model with fixed parameters. Cloud models have been widely used in information sciences, including eutrophication assessment of water, intelligent control, and image segmentation.^18–20 The cloud model and the cloud generator can realize the transformation between the qualitative concept and the quantitative value. Therefore, a damage detection method based on the cloud model and the cloud generator was proposed to solve uncertain damage problems caused by noise.

Herein, we propose a gray cloud rule generator algorithm (GCRGA) based on a gray cloud model and condition cloud generators to identify structural damage. This article has six sections. Section “MSE” briefly describes the MSE and MSEDRI. Section “Cloud model and cloud generators” introduces the numerical characteristics of a cloud model, the normal and backward cloud generators. Section “Damage identification based on cloud rule generator” presents the GCRGA based on the gray cloud model and the condition cloud generator. Section “Numerical experiment” provides numerical examples. Finally, section “Conclusion” reports the conclusions of this study.

MSE

Structural damage will change the mass, stiffness, and damping properties of a structure, thus leading to the change of structural frequency and vibration mode. Therefore, various methods have been introduced to identify the damage in structures based on comparing the response data of the structure before and after damage. One of the most popular methods is the one based on using the strain energy of a structure. Assuming that a structure is divided into N elements, the MSE of the jth element and the ith mode before and after the occurrence of damage is expressed as follows

{MSE}_{ij}^{u} = \frac{1}{2} Φ_{i}^{T} K_{j} Φ_{i}

(1a)

{MSE}_{ij}^{d} = \frac{1}{2} Φ_{i}^{dT} K_{j} Φ_{i}^{d}

(1b)

where ${MSE}_{ij}^{u}$ is the undamaged MSE, $Φ_{i}$ is the ith undamaged mode shape, ${MSE}_{ij}^{d}$ is the damaged MSE, $Φ_{i}^{d}$ is the ith damaged mode shape, and K_j is the stiffness matrix of the jth element. When the first m mode shapes are considered, the undamaged and damaged MSE of the jth element can be expressed as follows

{MSE}_{j}^{u} = \frac{1}{2} \sum_{i = 1}^{m} Φ_{i}^{T} K_{j} Φ_{i}

(2a)

{MSE}_{j}^{d} = \frac{1}{2} \sum_{i = 1}^{m} Φ_{i}^{dT} K_{j} Φ_{i}^{d}

(2b)

Liu et al.¹² further developed an MSE method to identify the structural damage, which is named as MSEDRI. The damage coefficient of the jth element can be expressed as

MSEDR I_{j} = \frac{| {MSE}_{j}^{d} - {MSE}_{j}^{u} |}{| {MSE}_{j}^{d} - {MSE}_{j}^{u} | + {MSE}_{j}^{u}}

(3)

Cloud model and cloud generators

Cloud model

Generally, probability theory can be used to deal with randomness. The normal distribution can be used to approximate a large number of random phenomena. The normal membership function is widely used to describe the fuzzy concept. Thus, Li et al.²¹ proposed a normal cloud model to integrate the uncertainty processing advantages of the normal distribution and the membership function. In the cloud model, three numerical characteristics are used to describe the information uncertainty, which are Expectation (Ex), Entropy (En), and Hyper Entropy (He). The detailed descriptions are as follows:

Ex: a point that can represent the qualitative concept to the maximal degree in the universe. It is also the center position of gravity of the cloud.

En: the discrete degree of the cloud droplet that represents the qualitative concept reflected by this numerical characteristic, and it shows the connection between randomness and fuzziness.

He: this represents the measurement of the entropy. In other words, it is the entropy of the entropy. This numerical characteristic reflects the extent of uncertain aggregation of all the points that contribute to the qualitative concept in quantitative universe and the magnitude of the randomness in the degree of certainty.

In summary, the fuzziness and randomness are combined with three numerical characteristics, and the mutual mapping between the qualitative concept and quantitative value is shown. In this study, the three numerical characteristics are used to construct a normal cloud generator to achieve the transformation between the qualitative concept and quantitative value. The numerical characteristics of cloud model are shown in Figure 1.

Figure 1.

Numerical characteristics of cloud model.

Cloud generators

Cloud generators can be divided into normal and backward cloud generators. The position coordinate of a cloud droplet in the universe can be obtained in the quantitative and qualitative aspect by inputting the three numerical characteristics (Ex, En, and He), and the quantity of cloud droplets should be generated in the normal cloud generator. Similarly, the backward cloud generator is the inverse of the normal cloud generator. The normal cloud generator algorithm can be formulated as follows:¹⁹

Generate a normal random number Xr according to the three numerical characteristics (Ex, En, and He). The expectation of Xr is Ex and the standard deviation is En.

Generate a normal random number En′, where the expectation is En and the standard deviation is He.

Generate a cloud droplet (x_r,µ_r) by $μ_{r} = \exp [- (x_{0} - Ex)^{2} / (2 E n'^{2})]$ .

Repeat the above procedure until N cloud droplets are generated.

If the fluctuation range of the testing data is unknown, the cloud model can be established by a backward cloud generator at the sampling discrete testing level. The statistical parameters can be expressed as follows

\bar{X} = \frac{1}{N} \sum_{r = 1}^{N} x_{r}

(4)

E_{1} = \frac{1}{N} \sum_{r = 1}^{N} | x_{r} - \bar{X} |

(5)

S^{2} = \frac{1}{N - 1} \sum_{r = 1}^{N} {(x_{r} - \bar{X})}^{2}

(6)

where $\bar{X}$ is the sample mean, $E_{1}$ is the first-order absolute central moment, and $S^{2}$ is the sample variance. Thus, the backward cloud generator algorithm can be expressed as follows²²

\hat{E} x = \bar{X}

(7)

\hat{E} n = \sqrt{\frac{π}{2}} \times \frac{1}{N} \sum_{r = 1}^{N} | x_{r} - \hat{E} x |

(8)

\hat{H} e = \sqrt{| S^{2} - \hat{E} n^{2} |}

(9)

where N is the total number of testing samples. The normal and backward cloud generators are shown in Figure 2. Cloud generators are very useful for solving uncertain reasoning problems.

Figure 2.

Normal and backward cloud generators.

Damage identification based on cloud rule generator

Cloud rule generator

The normal cloud generators can be further divided into the X- and Y-condition cloud generators. Given three digital characteristics Ex, En, and He and a particular number x₀, the generator can produce the cloud droplet (x₀,μ_r). This kind of generator is called the X-condition cloud generator. The certainty can be calculated by

μ_{r} = \exp [- {(x_{0} - Ex)}^{2} / (2 E {n'}^{2})]

(10)

By contrast, given the three digital characteristics and a certainty μ₀, the generator can produce cloud droplet (x_r,μ₀). This kind of generator is called the Y-condition cloud generator. The value $x_{r}$ can be obtained by

x_{r} = Ex \pm En' \sqrt{- 2 \ln μ_{0}}

(11)

A qualitative rule can be described as if (a) then (b), only the single condition and the single rule generator are discussed here. The cloud rule generator algorithm is listed as follows. The detailed procedure is given below:

Generate a normal random number $E n'_{a}$ , where expectation is En_a, and the standard deviation is He_a.

Calculate the certainty by $μ = \exp [- (x_{a} - E x_{a})^{2} / (2 En'_{a}^{2})]$ .

Generate a normal random number $E n'_{b}$ , where expectation is En_b and the standard deviation is He_b.

If $x_{a} > E x_{a}$ , $x_{b} = E x_{b} + E n'_{b} \sqrt{- 2 \ln μ}$ .

If $x_{a} \leq E x_{a}$ , $x_{b} = E x_{b} - E n'_{b} \sqrt{- 2 \ln μ}$ .

From the above texts, we can know that the degree of uncertainty can be passed on by cloud generators. First, a quantitative value x_a is input into the X-condition cloud generator, then we can get the certainty µ by the cloud generator. Finally, the certainty µ is input into the Y-condition cloud generator then a quantitative value x_b is obtained.

Damage identification based on gray cloud rule generator

The cloud rule generator algorithm and gray cloud model will be used in this method. MSE with a random error is considered as dynamic parameters. Multicondition rules will be used in the cloud rule generator. The damage information can be obtained from the cloud reasoning. The steps are as follows:

1. Suppose damage patterns or levels. Here, s damage patterns are established: [c₁,c₂], [c₂,c₃], ……, [c_s, c_s_+ 1], in which c_i (i = 1, 2, …, s + 1) is the boundary. s damage patterns can be processed by s condition rule generators. Based on the finite element model and damage patterns, we can obtain some gray cloud numerical characteristics of damage levels and their MSE. Then, a gray cloud model is proposed to calculate the numerical characteristics of each damage pattern. The numerical characteristics of the gray cloud include peak (Cs), left boundary (Ls), right boundary (Rs), Entropy (En), and Hyper Entropy (He). The peak (Cs), Entropy (En), and Hyper Entropy (He) are given by

Cs = \frac{L_{s} + R_{s}}{2}

(12)

En = \frac{R_{s} - L_{s}}{6}

(13)

He = k

(14)

where k is dispersion of gray cloud entropy.

2. To avoid the unreliable detection results, lots of measurement data with random noise are used to identify the structural damage. The X-condition cloud generator is used to calculate the certainty value of cloud droplets under the s damage patterns.

3. Then, the Y-condition cloud generator is further used to calculate the damage value of cloud droplets under the s damage patterns. n cloud droplets can be produced based on the calculated value of certainty for each damage pattern.

4. In the cloud generator, many cloud droplets are produced with their certainty as ((x_r)_f, (μ_r)_f), where (x_r)_f is the quantitative output value of droplet in the fth rule activated and (μ_r)_f is the certainty of droplet in the fth rule. According to the weighting average theory, a multipattern weighting method is proposed. Thus, the value of the cloud droplet of the jth element can be written as

w_{j} = \frac{\sum_{f = 1}^{s} {(μ_{r})}_{fj} {(x_{r})}_{fj}}{\sum_{i = 1}^{s} {(μ_{r})}_{fj}}

(15)

5. Finally, calculate the damage information through the average method. If measurement data of m tests are considered and n weighted cloud droplets for every time are produced, the mean index of the jth element is given by

{\bar{x}}_{j} = \frac{\sum_{k = 1}^{m} \sum_{l = 1}^{n} {(w_{j})}_{kl}}{mn}

(16)

The proposed method is named as GCRGA in this article.

Numerical experiment

To investigate the effectiveness of the proposed method, a 31-bar truss structure^13,14 is used. The size of the similar truss structure is shown in Figure 3. The material properties of the structure are as follows: ρ = 2800 kg/m³, E = 72 GPa, and A = 0.001 m². The truss model consists of 31 bar elements, 14 nodes, and 25 degrees of freedom. The damage is simulated by a reduction in the stiffness of individual bars in the structure. The damage cases are shown in Table 1. The parameters of the gray cloud rule generator are as follows: damage patterns s = 7; boundary values are 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7; measurement times m = 20; dispersion of gray cloud entropy k = 0.1; and the number of cloud droplets generated for each pattern n = 100. The machine used for this research is a personal computer with an Intel i3-2120 3.30 GHz processor and a 4 GB memory. The operating system is Microsoft Windows XP. Mode shapes with random noise are used to form MSE. The first three mode shapes with complete mode data are applied to identify structural damage.

Figure 3.

Two-dimensional truss structure.

Table 1.

Damage cases for two-dimensional truss.

Case 1		Case 2
Element no. (1)	Damage (2)	Element no. (3)	Damage (4)
2	30%	5	25%
15	30%	8	20%
21	20%	24	25%

Generally, random measurement noise will cause an uncertain problem. Here, random measurement noise is considered and artificial noise is added into the mode data according to Udwadia’s method²³

{Φ_{ki}}^{(\exp)} = {Φ_{ki}}^{(cal)} \cdot (1 + NR \cdot ξ)

(17)

where $Φ_{ki}^{(\exp)}$ is the kth component of the ith experimental mode shape vector, $Φ_{ki}^{(cal)}$ is the kth component of the ith calculated mode shape vector, NR (noise ratio) is the percentage of noise added to the calculated data, and ξ is a random number of standard normal distribution. If there are only a small number of sensors to measure the modal data, the optimal sensor locations should be considered. Minimal root mean square method²⁴ and hybrid optimization algorithm²⁵ can be used to optimize sensor locations and acquire the adequate modal information.

To avoid the unreliable detection results, measurement data of 20 times with random noise is utilized to identify the structural damage. A cloud rule generator based on gray model and MSE is applied to solve the uncertain problem caused by noise. To compare different damage detection methods, the MSECR⁹ and MSEDRI¹² are also considered. The first three mode shapes with random noise are used to calculate the MSE.

The rule base of cloud rule generator is formulated. Suppose that the generator is composed of seven qualitative concepts: Damage pattern 1 to Damage pattern 7 Multiple damage patterns can deal with the uncertainties caused by random noise. Each digital characteristic of qualitative concept calculated by the gray cloud model is listed in Table 2.

Table 2.

Digital characteristics of qualitative concept in variable patterns.

	Pattern 1	Pattern 2	Pattern 3	Pattern 4	Pattern 5	Pattern 6	Pattern 7
Boundaries	(0,0.1)	(0.1,0.2)	(0.2,0.3)	(0.3,0.4)	(0.4,0.5)	(0.5,0.6)	(0.6,0.7)
Ex	0.05	0.15	0.25	0.35	0.45	0.55	0.65
En	0.0167	0.0167	0.0167	0.0167	0.0167	0.0167	0.0167
He	0.0017	0.0017	0.0017	0.0017	0.0017	0.0017	0.0017

Then, the gray cloud digital characteristic of MSE can be calculated by MATLAB software. Table 3 is the canonical matrix formulated by expectation of damaged MSE in variable degrees of damage. Entropy of MSE is also necessary to formulate the cloud model, which is shown in Table 4. And, the Hyper Entropy of damaged MSE is one-tenth of entropy according to the gray cloud model. Thus, the qualitative rule base between MSE and damage value based on damage patterns of two-dimensional truss is formulated. The X- and Y-condition cloud generators will use the qualitative rule base between MSE and damage: if A_ts, then B_ts, t = 1, 2, …, 31, s = 1, 2, 3, 4, 5, 6, 7.

Table 3.

Gray cloud expectation of MSE in variable patterns.

Element	Pattern 1	Pattern 2	Pattern 3	Pattern 4	Pattern 5	Pattern 6	Pattern 7
1	20,446.810	25,245.815	31,947.369	41,700.965	56,667.316	81,292.568	125,891.77
2	20,165.354	24,705.457	30,965.644	39,934.674	53,424.531	75,046.286	112,839.29
3	34,759.024	40,365.947	47,436.684	56,523.981	68,465.283	84,576.000	107,012.02
4	115,705.56	131,773.46	151,173.31	174,801.60	203,834.77	239,815.61	284,753.04
5	96,393.732	108,084.15	121,923.83	138,440.34	158,320.64	182,466.20	212,067.75
6	8750.644	10,451.843	12,701.729	15,764.192	20,083.029	26,451.828	36,405.020
7	135,454.27	158,541.15	187,870.60	225,818.83	275,949.11	343,748.99	437,903.52
8	1502.327	1746.737	2056.0224	2455.349	2983.383	3702.008	4715.306
9	19,458.538	22,439.684	26,159.684	30,883.393	37,005.283	45,133.218	56,242.451
10	72,575.533	82,748.317	95,197.706	110,649.01	130,136.91	155,175.51	188,045.80
11	4904.411	5869.210	7149.442	8899.233	11,379.779	15,062.775	20,872.402
12	84,841.778	98,793.626	116,421.73	139,108.56	168,932.66	209,118.42	264,845.55
13	51,027.576	59,365.372	69,895.129	83,443.813	101,259.40	125,289.10	158,689.31
14	33,339.083	38,599.670	45,202.152	53,640.968	64,658.682	79,409.816	99,763.954
15	22,743.591	26,525.799	31,331.479	37,563.927	45,845.103	57,173.707	73,233.064
16	3859.351	4628.791	5653.612	7060.7680	9067.173	12,068.703	16,851.855
17	32,987.623	38,614.558	45,801.157	55,175.552	67,712.747	84,988.591	109,675.30
18	55,633.371	64,569.343	75,812.186	90,215.503	109,058.55	134,321.36	169,183.56
19	33,655.564	39,291.790	46,457.106	55,751.917	68,098.243	84,966.553	108,808.34
20	56,521.308	66,975.208	80,562.991	98,642.409	123,369.53	158,309.24	209,629.71
21	2237.326	2689.205	3293.282	4126.499	5321.405	7122.457	10,022.196
22	63,761.771	76,135.326	92,415.496	114,392.11	144,972.45	189,091.27	255,547.76
23	3120.556	3643.4542	4309.9264	5177.896	6337.821	7937.444	10,231.768
24	1219.589	1420.552	1675.7527	2006.692	2446.741	3050.0130	3909.151
25	113,577.94	134,237.36	160,895.77	196,021.80	243,426.48	309,176.28	403,235.29
26	395.631	473.762	577.549	719.590	921.285	1221.401	1696.192
27	12,251.516	15,013.037	18,824.088	24,291.573	32,532.812	45,790.647	69,119.332
28	55,287.925	63,887.135	74,622.513	88,248.801	105,876.59	129,186.99	160,806.46
29	37,223.539	43,094.178	50,461.723	59,877.611	72,170.067	88,627.018	111,334.85
30	33,541.971	40,902.635	50,966.902	65,230.773	86,377.493	119,600.11	175,984.13
31	21,669.377	26,701.745	33,705.694	43,856.557	59,347.767	84,650.684	130,008.31

MSE: modal strain energy.

Table 4.

Gray cloud entropy of damaged MSE in variable patterns.

Element	Pattern 1	Pattern 2	Pattern 3	Pattern 4	Pattern 5	Pattern 6	Pattern 7
1	1152.420	1426.527	1984.176	2775.833	3955.860	5114.138	11,540.080
2	1118.030	1425.212	1650.142	2577.928	3746.653	4656.481	10,343.602
3	1765.537	2025.923	2430.271	2966.914	3738.748	4986.930	9809.435
4	5692.742	6260.557	7235.077	8422.478	9875.741	11,658.674	26,102.362
5	4382.482	4936.388	5656.216	6373.633	7938.885	8436.404	19,439.544
6	459.324	564.742	710.903	921.697	1241.292	1758.685	3337.127
7	6798.123	8106.409	9812.239	12,084.181	15,183.28	19,525.279	40,141.157
8	74.279	87.98500	105.856	129.7680	162.785	210.211	432.236
9	944.357	1106.028	1312.707	1582.673	2006.965	2445.067	5155.558
10	3431.316	3956.976	4968.735	5437.235	6799.190	7902.738	17,237.532
11	258.523	318.741	402.635	524.367	710.4110	1014.890	1913.304
12	4209.139	4989.293	5937.354	7343.296	9169.562	11,731.350	24,277.510
13	2526.425	2992.039	3596.073	4397.769	5490.796	7230.081	14,546.520
14	1634.120	1922.046	2294.715	2785.600	3940.610	5064.163	9145.029
15	1149.298	1405.611	1853.899	2004.641	2528.628	3285.291	6713.031
16	204.418	252.831	320.638	419.692	572.406	825.298	1544.753
17	1654.572	1978.540	2403.893	2980.108	3754.301	6494.880	10,053.569
18	2739.629	3234.194	3872.547	4714.776	5855.020	8090.477	15,508.494
19	1782.678	1999.116	2417.873	2980.555	3759.898	5261.388	9974.098
20	2915.485	3539.339	4380.435	5548.359	6930.024	9745.102	19,216.058
21	119.064	147.720	188.069	247.406	339.689	494.335	918.701
22	3344.733	4103.362	4785.279	6615.874	8154.763	12,127.880	23,425.212
23	155.751	185.570	224.853	278.081	352.752	462.229	937.912
24	60.545	71.906	86.801	106.866	134.8260	175.488	358.339
25	5588.494	7038.685	8654.532	10,860.45	13,958.217	18,448.401	36,963.236
26	20.884	25.772	32.592	42.507	57.699	82.645	155.484
27	679.566	866.731	1142.388	1571.502	2227.733	3621.456	6335.939
28	2695.595	3162.762	3759.089	4535.294	5568.728	7774.253	14,740.592
29	1822.644	2193.919	2561.257	3108.876	3849.808	6652.494	10,205.695
30	1686.552	2282.392	3041.465	4126.209	5892.734	7392.298	16,131.879
31	1199.275	1563.182	2080.037	2895.602	3964.988	5139.182	11,917.429

MSE: modal strain energy.

Case 1

In this case, elements 2, 15, and 21 have 30%, 30%, and 20% stiffness damage, respectively. The calculation procedures of the GCRGA are briefly shown as follows: first, the MSE are calculated using the measured modal shape. Then, the X- and Y-condition cloud generators are used to produce cloud droplets with different damage patterns. Finally, the weighting average method based on certainty is used to obtain damage information.

The MSECR, MSEDRI, and GCRGA are used to identify structural damage. The identification results of three kinds of methods are shown in Figure 4. From Figure 4, it can be observed that the MSECR, MSEDRI, and GCRGA can identify the damage locations. The values of damaged elements are further listed in Table 5. From Table 5, it can be seen that the identification precision of the proposed GCRGA is relatively better than those of the MSECR and MSEDRI when measurement noise is not considered. The MSECR hardly identifies the damage extent.

Figure 4.

Identification results of three kinds of methods when damage occurs in elements 2, 15, and 21.

Table 5.

Calculation results of damaged elements for case 1.

Methods	Element 2	Element 15	Element 21	Average error
MSECR	0.8751	0.5887	0.4145	0.3594
MSEDRI	0.4667	0.3706	0.2930	0.1101
GCRGA	0.2480	0.2802	0.1830	0.0296

MSECR: modal strain energy change ratio; MSEDRI: modal strain energy dissipation ratio index; GCRGA: gray cloud rule generator algorithm.

However, the measurement noise is inevitable. Here, measurement data of 20 tests with random noise are used to identify structural damage. The identification results of three kinds of methods are shown in Figure 5. An element with a relatively higher value indicates that it is more probable to be the true damage element. The run times of the MSECR, MSEDRI, and GCRGA are 2.1, 2.3, and 7.7 s, respectively. Figure 5 shows that neither the MSECR nor the MSEDRI can find the true damage locations, and the value of element 26 is obviously higher than those of the other elements. In addition, Figure 5 shows that GCRGA can identify the true damage locations even if the noise disturbance is considered. The mean values of damaged elements are higher than those of undamaged elements. Therefore, the GCRGA can approximately identify damage locations when measurement noise is considered.

Figure 5.

Identification results with mean value of three kinds of methods when damage occurs in elements 2, 15, and 21 (considering 3% noise).

The involved damage patterns and the membership cloud of GCRGA are shown in Figures 6 –8. Here, the patterns and cloud droplets of damaged elements 2, 15, and 21 are mainly considered. Figures 6 –8 show that multiple damage patterns are involved in damage detection. Due to the existence of measurement noise, some cloud drops often deviate from the true damage patterns. Therefore, multiple damage patterns are involved. Here, a large number of cloud droplets are used to reduce the noise interference.

Figure 6.

Membership cloud and damage patterns of element 2 for case 1 (3% noise).

Figure 7.

Membership cloud and damage patterns of element 15 for case 1 (3% noise).

Figure 8.

Membership cloud and damage patterns of element 21 for case 1 (3% noise).

Probability theory can be used to deal with randomness. The normal distribution can be used to approximate a large number of random phenomena. In fuzzy mathematics, the membership function is used to model the fuzziness. Cloud model can model both randomness and fuzziness with fixed parameters (Entropy (En) and Hyper Entropy (He)). The comparison results of En and He between MSEDRI and GCRGA are listed in Table 6. From Table 6, we can see that the randomness and fuzziness of the GCRGA are less than those of the MSEDRI. Therefore, the GCRGA is relatively better than the MSEDRI.

Table 6.

Comparison results of damaged elements between MSEDRI and GCRGA for case 1 (3% noise).

GCRGA/MSEDRI	Element 2	Element 15	Element 21	Average value
En _(GCRGA)/En_(MSEDRI)	0.8767	0.9175	0.7387	0.8443
He_(GCRGA)/He_(MSEDRI)	1.0216	0.9678	0.7234	0.9043

GCRGA: gray cloud rule generator algorithm; MSEDRI: modal strain energy dissipation ratio index.

Case 2

In this case, elements 5, 8, and 24 have 25%, 20%, and 25% stiffness damage, respectively. The MSECR, MSEDRI, and GCRGA are used to identify the damage. The identification results of three kinds of methods are shown in Figure 9. From Figure 9, we can observe that the MSECR, MSEDRI, and GCRGA can identify the damage locations. The calculation values of damaged elements are listed in Table 7. From Table 7, we can see that the identification precision of the proposed GCRGA is still better than those of the MSECR and MSEDRI when measurement noise is not considered. The MSECR hardly identifies the damage extent.

Figure 9.

Identification results of three kinds of methods when damage occurs in elements 5, 8, and 24.

Table 7.

Calculation results of damaged elements for case 2.

Methods	Element 5	Element 8	Element 24	Average error
MSECR	0.3374	0.4372	0.4380	0.1709
MSEDRI	0.2523	0.3042	0.3046	0.0537
GCRGA	0.2504	0.2428	0.2415	0.0172

MSECR: modal strain energy change ratio; MSEDRI: modal strain energy dissipation ratio index; GCRGA: gray cloud rule generator algorithm.

Measurement noise must be considered. Here, measurement data of 20 tests with random noise are used to identify structural damage. The identification results of three kinds of methods are shown in Figure 10. The run times of the MSECR, MSEDRI, and GCRGA are 2.2, 2.5, and 8.6 s. Figure 10 shows that neither the MSECR nor the MSEDRI can find the true damage locations. Elements 21 and 26 have relatively higher values, but they are not the true damaged elements. Figure 10 also shows that the GCRGA can obviously identify the true damage locations even if the noise disturbance is considered. The mean values of damaged elements are higher than those of undamaged elements. Therefore, the GCRGA is the best one of the three methods.

Figure 10.

Identification results with mean value of three kinds of methods when damage occurs in elements 5, 8, and 24 (considering 3% noise).

And, 5% noise is also considered. The identification results with 5% noise are shown in Figure 11. The run times of the MSECR, MSEDRI, and GCRGA are 2.5, 2.6, and 8.7 s. Figure 11 shows that neither the MSECR nor the MSEDRI can find the true damage locations. Elements 16, 21, and 26 have relatively higher values, but they are not the true damaged elements. Figure 11 also shows that the GCRGA can approximately find the true damage locations even if the noise disturbance is considered. The mean values of damaged elements are higher than those of undamaged elements. Therefore, the GCRGA can approximately identify damage locations when measurement noise is considered.

Figure 11.

Identification results with mean value of three kinds of methods when damage occurs in elements 5, 8, and 24 (considering 5% noise).

The membership cloud and damage patterns of GCRGA are shown in Figures 12 –14. Here, the involved patterns and cloud droplets of damaged elements 5, 8, and 24 are mainly considered. Figures 12 –14 show that multiple damage patterns are integrated to identify the damage. Due to the existence of measurement noise, some cloud drops often deviate from the true damage patterns. Therefore, multiple damage patterns are involved. Here, a large number of cloud droplets are produced to avoid a large deviation from the actual damage. The gray cloud rule generator and the multipattern weighting method are also used to deal with the uncertain problems caused by noise. Therefore, the GCRGA can solve the uncertain damage problems. Cloud model can model both randomness and fuzziness with fixed parameters (Entropy (En) and Hyper Entropy (He)). The comparison results of En and He between MSEDRI and GCRGA are listed in Table 8. From Table 8, we can see that the randomness and fuzziness of the GCRGA are less than those of the MSEDRI. Therefore, the GCRGA is relatively superior to the MSEDRI.

Figure 12.

Membership cloud and damage patterns of element 5 for case 2 (3% noise).

Figure 13.

Membership cloud and damage patterns of element 8 for case 2 (3% noise).

Figure 14.

Membership cloud and damage patterns of element 24 for case 2 (3% noise).

Table 8.

Comparison results of damaged elements between MSEDRI and GCRGA for case 2.

GCRGA/MSEDRI	Element 5	Element 8	Element 24	Average value
En _(GCRGA) / En _(MSEDRI)	0.9785	0.9090	0.9136	0.9337
He_(GCRGA)/He_(MSEDRI)	0.9834	0.5361	0.9501	0.8232

GCRGA: gray cloud rule generator algorithm; MSEDRI: modal strain energy dissipation ratio index.

In the mentioned earlier, we can see that it is difficult for the MSECR and MSEDRI to deal with the uncertain damage problems caused by random measurement noise. Therefore, the GCRGA is proposed to determine these uncertain factors. Both the cases show that the GCRGA not only identifies the true damage locations but also reduces the measurement noise disturbance. The identification results of the GCRGA are relatively better than those of the MSECR and MSEDRI. Therefore, the GCRGA improves the reliability and effectiveness of identification results.

Conclusion

A damage identification method based on the gray cloud rule generator and MSE is developed to identify structural damage. The GCRGA utilized the X- and Y-condition cloud generators to improve the identification reliability and produce cloud droplets. Due to the existence of measurement noise, many cloud drops often deviate from the true damage patterns. Therefore, multiple damage patterns are involved in the calculation procedure. Here, a large number of cloud droplets are used to reduce the interference of noise. Finally, a weighting method is used to obtain uncertain damage information. Considering the identification results of the numerical examples, the following conclusion can be drawn: (1) the quantification results of the proposed GCRGA are better than those of MSECR and MSEDRI if measurement noise is not considered. (2) The localization results of the GCRGA are still relatively superior to those of MSECR and MSEDRI if measurement noise is considered. Therefore, the proposed GCRGA can identify the uncertain damage problems caused by measurement noise. The identification principle of the proposed GCRGA is applicable for other types of structures. Therefore, the method should be able to identify other types of structures. In general, the changes of dynamic characteristics caused by small damage are easily submerged in the noise. Therefore, for small damage, the miss probability will be higher. In addition, the better the measuring accuracy of sensor is, the less the miss probability is.

Footnotes

Handling Editor: Daxu Zhang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant nos 51578094 and 51468058) and the Fundamental Research Funds for the Central Universities (grant no. 106112014CDJZR200011).

ORCID iD

Hui-Yong Guo

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Structural damage identification based on gray cloud rule generator algorithm

Abstract

Keywords

Introduction

MSE

Cloud model and cloud generators

Cloud model

Cloud generators

Damage identification based on cloud rule generator

Cloud rule generator

Damage identification based on gray cloud rule generator

Numerical experiment

Case 1

Case 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References