A new crack diagnosis method on a two-story box structure based on empirical mode decomposition is proposed in this paper. According to the simulation analysis, it turns out that the model of the structure can be barely influenced by the crack. Response signals of swept sine vibration test are empirical mode decomposed into a set of intrinsic mode functions, from which tag vectors are constructed, then tag angles are defined to dignose the failure of the board. Combined with the load direction to the structure, the position and direction of the crack can be deduced using tag angles

Shichao Fan

Yanjing Yang

Yicun Xie

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 125 
A new crack diagnosis method on box structure based 
on empirical mode decomposition 
Yicun Xie1, Shichao Fan2, Yanjing Yang3 
Beijing Institute of Spacecraft Environment Engineering, Beijing, China 
2Corresponding author 
E-mail: 1xycun02@sina.com, 2fanshichao@tsinghua.org.cn, 3yyjpku@hotmail.com 
(Accepted 8 August 2014) 
Abstract. A new crack diagnosis method on a two-story box structure based on empirical mode 
decomposition is proposed in this paper. According to the simulation analysis, it turns out that the 
model of the structure can be barely influenced by the crack. Response signals of swept sine 
vibration test are empirical mode decomposed into a set of intrinsic mode functions, from which 
tag vectors are constructed, then tag angles are defined to dignose the failure of the board. 
Combined with the load direction to the structure, the position and direction of the crack can be 
deduced using tag angles. 
Keywords: crack diagnosis, box structure, empirical mode decomposition, tag angle. 
1. Introduction 
Most spacecrafts are valuable, unique and unrepairable on orbit, so high reliability is a basic 
requirement for spacecrafts. So it is an important subject how to diagnose the damage of the 
spacecraft structure. 
Dynamics environmental testing is an essential part in the process of developing spacecrafts. 
In the current process of spacecrafts development, feature-level structure response curves are 
referred to diagnose the damage of the structure. In detail, the estimation of the existence of 
frequency excursion or amplification of the response data before and after the high-level or 
random vibration test is performed. It is just a kind of qualitative analysis, and the form, the 
location and other information of the damage cannot be deduced. These necessary features rely 
on human’s judgment, and sometimes the check inside the structure is required, which is very time 
consuming [1].  
Empirical mode decomposition is a rising kind of signal processing method in recent years, 
which is suitable for non-linear and non-stationary signals. This method is applied here to process 
the vibration signals of a two-level box in order to diagnose the crack of the box, as a primary 
exploratory study for application to the field of spacecraft structural fault diagnosis. 
2. Empirical mode decomposition 
Empirical Mode Decomposition (EMD) is an important part of the Hilbert-Huang transform, 
by which a time-domain signal is decomposed into a collection of intrinsic mode functions. An 
intrinsic mode function (IMF) satisfies two basic conditions: (1) in the whole time range, the 
number of extrema and the number of zero crossings must either equal or differ at most by one; 
and (2) at any point, the mean value of the envelop defined by the local maxima and the envelop 
defined by the local minima is zero. An IMF is not restricted to a narrow band signal, and it can 
be both amplitude and frequency modulated. 
Having defined IMF, the original vibration signal should be divided into a series of IMFs. This 
process is known as the decomposition. After integrating all components, final results can be obtained. 
A schematic of the process to identify intrinsic mode functions of a signal is shown in Fig. 1. 
The mean of the upper and lower extrema curves is defined as: 
ܺ௠భ(ݐ) =
ܺ୫ୟ୶ + ܺ୫୧୬
2 . (1) 
A NEW CRACK DIAGNOSIS METHOD ON BOX STRUCTURE BASED ON EMPIRICAL MODE DECOMPOSITION.  
YICUN XIE, SHICHAO FAN, YANJING YANG 
126 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
 
Fig. 1. The process of obtaining intrinsic mode functions 
Subtracting the mean from the original signal, the first estimate of an intrinsic mode function 
is obtained: 
ܺ(ݐ) െ ܺ௠భ(ݐ) = ݄ଵ. (2)
This process is called filtering which result in the third curve shown in Fig. 1. But ݄ଵ usually 
does not satisfy the conditions of an IMF, and further filtering is usually needed. The next filtering 
assumes that ݄ଵ is the signal and the mean from ݄ଵ is ܺ௠భభ(ݐ), so that the next estimate becomes: 
݄ଵ െ ܺ௠భభ(ݐ) = ݄ଵଵ. (3)
After ݇ filtering, we define the first IMF to be: 
ܿଵ(ݐ) = ݄ଵ௞. (4)
To obtain the second and subsequent IMF, the residual signal can be computed as: 
ܺ(ݐ) െ ܿଵ(ݐ) = ݎଵ(ݐ). (5)
The residual, ݎଵ(ݐ), now becomes the new data that can be subjected to the same filterting 
process to extract more IMFs. This process is repeated to obtain ܿଶ(ݐ) through ܿ௡(ݐ) IMFs. By 
summing up all of the IMFs, it is possible to represent a vibration signal as: 
ܺ(ݐ) = ∑ܿ௜(ݐ) + ݎ௡(ݐ). (6)
Thus, the original data ܺ(ݐ) has been divided into ݊ empirical modes, ܿ௡(ݐ) and a residue 
ݎ௡(ݐ) [2, 3].  
In this paper, EMD method is used to track the damage in a two-story box, by the time-domain 
vibration response signals of the structure, in order to obtain more information about the crack in 
the board. 
3. The FEA of a box structure 
Box structures are widely used, such as the main structure of small satellites, or the 
sub-structure of large spacecrafts. So a box structure is selected here. The schematic diagram of a 
two-story box is shown in Fig. 2(a). Bottom/side boards are Al5052 and top/clap boards are 
Aluminum honeycomb sandwich structures. 
This box structure is modeled in MSC.Patran2011 and analyzed in MSC.Nastran2011.1. The 
finite element model of the box is illustrated in Fig. 2(b). Two different cases are assumed, with 
an undamaged box in case 01 and a ݕ-direction crack in the top board of the box in case 02. 
The model is bottom fixed, whose first four modes are calculated. The natural frequencies are 
A NEW CRACK DIAGNOSIS METHOD ON BOX STRUCTURE BASED ON EMPIRICAL MODE DECOMPOSITION.  
YICUN XIE, SHICHAO FAN, YANJING YANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 127 
shown in Table 1, and the mode comparison is presented in Fig. 3. 
 
a) 
 
b) 
Fig. 2. A two-story box model 
 
a) 
 
b) 
Fig. 3. The mode comparison of two cases: a) first-order mode: case 01 is left and case 02 is right;  
b) fourth-order mode: case 01 is left and case 02 is right 
Table 1.The natural frequency comparison of two cases 
Case number Natural frequency (Hz) First order Second order Third order Fourth order 
01 157.71 206.17 242.18 247.21 
02 157.35 196.91 239.14 244.74 
According to all above analysis, the difference about natural frequencies and modes between 
two cases is slight so it is hard to tell the fault of the structure. Thus, a new method is needed to 
diagnose the failure of a structure. 
4. Testing and analysis 
4.1. The vibration test 
The real two-story box is shown in Fig. 4, whose parameters are the same as the FEM model. 
 
a) 
 
b) 
Fig. 4. The box structure and a schematic diagram of board 1 
Two cases have been assumed as above. A board with a y-direction crack is designed to 
simulate case 02 shown in Fig. 4(b). The coordinate is the same as that in Fig. 2(b). Sine sweep 
vibration is adopted, whose sweep rate is 4 oct/min, frequency range from 5 to 500 Hz and testing 
level is 0.1 g. The sensors are set in the middle of four edges of the top board, illustrated in 
Fig. 4(b), which measure three axial acceleration responses. 
A NEW CRACK DIAGNOSIS METHOD ON BOX STRUCTURE BASED ON EMPIRICAL MODE DECOMPOSITION.  
YICUN XIE, SHICHAO FAN, YANJING YANG 
128 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
4.2. Analysis 
After vibration tests, the acceleration response data are obtained. Based on the frequency 
response data, the curve of case 02 is similar to that of case 01, shown in Fig. 5. 
 
a) 
 
b) 
Fig. 5. The contrast between two cases’ frequency response: a) ݔ-direction input, sensor 1 ݔ-direction 
response; b) ݕ-direction input, sensor 4 ݕ-direction response 
 
a) 
 
b) 
Fig. 6. Time-domain curves, ݔ-direction input, sensor 1 ݔ-direction response; a) case 01 and b) case 02 
 
a) 
 
b) 
Fig. 7. EMD processes, ݔ-direction input, sensor 1 ݔ-direction response; a) case 01 and b) case 02 
Due to boundary effect, energy leakage and some other causes, higher IMFs always have no 
physical meanings. In fact, not so many filtertings are needed for analyzing. In order to ensure the 
accuracy and decrease the computation, here former four order IMFs are decomposed, and higher 
than fourth-order IMFs are considered to be residue. 
The time-domain data from the same sensor in different cases share some resemblance, as 
shown in Fig. 6. The EMD processes to time-domain data are illustrated in Fig. 7. 
The vibration signal can be devided into ݊ IMFs, ܿଵ(ݐ), ܿଶ(ݐ),…, ܿ௡(ݐ), and a residue ݎ௡(ݐ), 
then the energy of each IMFs, ܧ௜, and the energy of residue, ܧ௥, can be obtained by: 
A NEW CRACK DIAGNOSIS METHOD ON BOX STRUCTURE BASED ON EMPIRICAL MODE DECOMPOSITION.  
YICUN XIE, SHICHAO FAN, YANJING YANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 129 
ܧ௜ = න |ܿ௜(ݐ)|ଶ
ାஶ
ିஶ
݀ݐ, ܧ௥ = න |ݎ௡(ݐ)|ଶ
ାஶ
ିஶ
݀ݐ. (7) 
A tag vector ܂ is defined by all ܧ௜, and ܧ௥ as: 
܂ = ሾܧଵ, ܧଶ, … , ܧ௡, ܧ௥ሿ. (8) 
As the amount of energy is always large, define total energy ܧ = ඥ(∑ |ܧ௜|ଶ௡௜ୀଵ ) + |ܧ௥|ଶ and 
energy percentage, ܴ௜ = ܧ௜/ܧ and ܴ௥ = ܧ௥/ܧ, then ܂ is normalized as: 
ܜ = ሾܴଵ, ܴଶ, … , ܴ௡, ܴ௥ሿ. (9) 
Assuming that the tag vector of the undamaged structure is ܜ଴ and that of the damaged is ܜ, in 
order to measure the angular separation of two vectors, a tag angle ߙ is defined as: 
ߙ = cosିଵ(ܜ଴ ⋅ ܜ). (10) 
So the information of the damage can be deduced from the tag angle ߙ. 
Table 2. The analyzed results of case 01 and case 02 
Case number Input/Output direction 
Sensor  
number Tag vector ܜ Tag angle ߙ (rad) 
01 
ݔ/ݔ 
1 [0.995  0.071  0.018  0.021  0.070] – 
2 [0.978  0.087  0.030  0.024  0.183] – 
3 [0.672  0.226  0.146  0.127  0.679] – 
4 [0.980  0.035  0.064 0.030  0.184] – 
ݕ/ݕ 
1 [0.989  0.026  0.028  0.021  0.144] – 
2 [0.986  0.089  0.048  0.026  0.130] – 
3 [0.985  0.084  0.045  0.025  0.138] – 
4 [0.983  0.099  0.057  0.022  0.140] – 
ݖ/ݖ 
1 [0.999  0.015  0.006  0.003  0.004] – 
2 [0.610  0.102  0.104  0.137  0.767] – 
3 [0.570  0.122  0.103  0.148  0.792] – 
4 [0.558  0.133  0.128  0.158  0.793] – 
02 
ݔ/ݔ 
1 [0.429  0.862  0.236  0.089  0.101] 1.046 
2 [0.535  0.793  0.139  0.097  0.237] 0.873 
3 [0.331  0.658  0.128  0.130  0.651] 0.559 
4 [0.598  0.717  0.168  0.099  0.299] 0.823 
ݕ/ݕ 
1 [0.983  0.117  0.062  0.042  0.123] 0.102 
2 [0.861  0.405  0.189  0.102  0.221] 0.388 
3 [0.972  0.178  0.094  0.033  0.120] 0.109 
4 [0.859  0.409  0.208  0.063  0.219] 0.379 
ݖ/ݖ 
1 [0.999  0.014  0.008  0.002 0.008] 0.005 
2 [0.421  0.310  0.133  0.164  0.826] 0.291 
3 [0.225  0.358  0.175  0.173  0.872] 0.437 
4 [0.327  0.293  0.171  0.166  0.867] 0.294 
The sine vibration response signals are decomposed and analyzed as above, the results 
obtained shown in Table 2. 
The crack is along the ݕ direction, orthogonal to the ݔ direction in case 02, which is close to 
sensor 1. According to Table 2, when loading ݔ-direction input, the tag angle of sensor 1 is larger 
than that of any other sensor. But when loading other direction input, the feature of tag angle 
A NEW CRACK DIAGNOSIS METHOD ON BOX STRUCTURE BASED ON EMPIRICAL MODE DECOMPOSITION.  
YICUN XIE, SHICHAO FAN, YANJING YANG 
130 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
cannot be distinguished. 
4.3. Verification 
In order to verify the effectiveness of the damaged feature extraction, a board with an  
ݔ-direction crack is designed to simulate case 03 illustrated in Fig. 8. Testing conditions are the 
same as above. 
 
Fig. 8. A board with an ݔ-direction crack: board 2 
The vibration response signals of case 03 are decomposed and analyzed as above, the results 
obtained partially shown in Table 3. 
In case 03, the crack is along the ݔ direction, orthogonal to the ݕ direction, which is close to 
sensor 2. Similarly, based on calculation, when loading ݕ-direction input, the tag angle of sensor 2 
is larger than that of any other sensor, as shown in Table 3. But when loading other direction input, 
the feature of tag angle cannot be distinguished. So the effectiveness of the damaged feature 
extraction can be verified. To sum up, combined with the loaded input direction, the position and 
direction of a crack can be deduced on the basis of tag angle ߙ. 
Table 3. Part analyzed results of case 03 
Case number Input/Output direction Sensor number Tag vector ܜ 
Tag angle ߙ 
(rad) 
03 ݕ/ݕ 
1 [0.966  0.053  0.120  0.052  0.218] 0.127 
2 [0.906  0.378  0.102  0.037  0.155] 0.307 
3 [0.955  0.256  0.083  0.042  0.116] 0.182 
4 [0.945  0.291  0.083  0.037  0.119] 0.199 
5. Conclusion 
In this paper a two-story box structural response signals are empirical mode decomposed. 
According to IMFs, tag vectors and tag angles can be constructed. Coupled with the direction of 
the structural input, the position and direction of the crack in the board can be deduced. It is a kind 
of new method for spacecrafts structural damage diagnosis. 
References 
[1] Shuhong Xiang Dynamics environmental testing technology of spacecrafts. Science and Technology 
Press of China, China, 2008. 
[2] Norden E. Huang, Zheng Shen, Steven R. Long, Manli C.Wu, Hsing H.Shih, Quanan Zheng, 
Nai-Chyuan Yen, Chi Chao Tung, Henry H. Liu The empirical mode decomposition and the Hilbert 
spectrum for non-linear and non-stationary time series analysis. Proceedings of the Royal Society of 
London Series A-Mathematical Physical and Engineering Sciences, Vol. 454, 1998. 
[3] Darryll Pinesa, Liming Salvinob Structural health monitoring using empirical mode decomposition 
and the Hilbert phase. Journal of Sound and Vibration, Vol. 294, Issue 2006, p. 97-124. 


A new crack diagnosis method on box structure based on empirical mode decomposition

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A new crack diagnosis method on box structure based on empirical mode decomposition

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