Damage Identification of Plate Structure Based on the Method of Modal Flexibility Curvature Difference

According to the higher order modes of the structure are difficult to extract and the lower order are easy to obtain in practice, it is put forward that only uses the parameters of lower or first-order modal to constitute the modal flexibility curvature difference (MFCD) as the index of damage identification, which based on the flexibility matrix of structure being sensitive to the structure modal characteristics. Numerical analysis is made on different damage conditions of a plate structure, and the result shows that the MFCD can not only accurately identify the single-damage and multi-damage position of plate structure, but also reflect the size of damage degree. It has a great significance to apply the index into the range of damage identification of actual structures.


Introduction
In recent years, the technology of damage identification based on structural vibration is becoming a hot topic. It is found that the damage of structure will generally cause the change of its stiffness, which will change the modal parameters (natural frequency, modal vibration mode and damping) of the structure. Based on the above understanding, some scholars used the frequency and the mode of structural vibration signal to identify the damage early [1][2][3][4]. But the frequency and vibration mode are not sensitive to the structural damage, the identification index which is more sensitive to damage is proposed later, such as the modal strain energy method [5], the residual force vector method [6], the flexibility matrix method [7] and so on. Among these indexes, the flexibility matrix method can get accurate damage identification results only need lower order modal parameters, so it has more practical application value (Because the higher order modes of the structure are difficult to extract and the lower order are easy to obtain in practice). According to the study, the flexure value of the structure changes greatly in the damaged position or the adjacent area, and the flexure value changes differently with different degree of damage [8]. By comparing several methods of modal flexibility matrix which are used to identify the structure damage, it is found that [9][10][11][12][13][14] whatever damage, only the MFCD can clearly indicate the location of the single-damage and multi-damage, and can reflect the size of damage degree, the effect is better than others. Most of the above researches are based on the damage identification of beam structures, but few studies have been done on the application of the MFCD to plate structures or shell structures.
In this paper, the MFCD is proposed to identify the plate damage, and numerical analysis is made on different damage conditions of a plate structure. Firstly, according to the theory of Kirchhof thin plate, the plate structure is divided and the flexibility matrix of structure response is constructed. Secondly, the change of flexibility value is amplified by the second order differential, and the difference of flexibility curvature matrix which are before and after structural damage is further obtained. Finally, through numerical calculation, it is shown that the damage can be accurately located only with lower order or first-order modal parameters, and the size of damage degree can be effectively determined. It has a great significance to apply the index into the range of damage identification of actual structures.

Fundamental
According to the modal parameters of plate structure, the flexibility matrix of structure is: Where: ij a --the displacement or rotation value of the node in column j of row i , n a b   ; r is the total number of modes, and n is the number of nodes.
We can get the transition matrix ta F from formula (1): Let k C be the maximum value of each column element in the above matrix, so we get the transition matrix tb F : The maximum flexibility matrix is constructed from For each element in p F of the plate structure, it corresponds to the z direction displacement and , x y direction rotation angle of each node, that is, each row or column in the matrix is the flexibility of the coordinate function. Near the point of structural damage, the flexibility will change, and this change will become more pronounced with the action of the second derivative. The transverse flexibility curvature matrix ch F of the structure is: Where: a a E  --Order a unit matrix; S --The unit area.
From the above, the flexural curvature difference between before and after structural damage is obtained, that is, the transverse damage index Among them: chw F is the transverse flexibility curvature of intact structure, chd F is the transverse flexibility curvature of damage structure. It can be seen from the above that the damage index after the second order differential is amplified and more sensitive to modal parameters. The longitudinal flexibility curvature matrix cv F of the structure is: Where:     The flexural curvature difference before and after structural damage is obtained, that is, the longitudinal  In the finite processing [15], the following matrix representation is applied relation to the node number in the figure: 111 121 As opposed to the unit number in the diagram, the following matrix representation is applied: 90 100 In the processing of dealing with the matrix of flexibility maximum value and the flexibility curvature matrix difference that is damage index matrix, the following matrix is arranged (I'm just going to give you the horizontal direction, so you get the vertical direction):

Damage analysis
In this paper, we take the reduction of element stiffness to simulate the damage of structure, and divide the damage into single-position damage and multi-position damage. The identification of single damage is analyzed by unit 45 damaging in the middle of the plate or unit 95 damaging at the edge of plate respectively. Multi-position damage identification is divided into unit 45 and unit 95 in the same row or unit 23 and unit 77 in different columns in different rows. We analyze the unit 45 damage 20%, 50%, 70% respectively to judge the degree of damage and compare the damage degree of multi-position damage. The details is in table 1.

Numerical analysis
When it comes to numerical calculation, you need to get the modal information of the structure to extract i  which is the modal frequency and i  which is the corresponding displacement or corner vector, and then to deal with i  to obtain i  that is the displacement or rotation vector of the mass normalization. We take the transverse flexibility curvature difference as damage index in this paper, and then extract the frequency and displacement vector of foregoing three orders modal, numerical analysis is as follows:

Single-position damage identification
We have calculated the damage index of the first order, that was shown in Fig 2, and the damage index of foregoing three orders, that was shown in Fig 3, that the unit 45 in the middle of the plate is damaged by 50%. The Fig 2 is the figure of the first order damage index when the unit 45 in the middle of the square aluminum alloy plate is damaged by 50%. Among them, Fig. 2(a) is a three-dimensional view of the damage index, from which it can be clearly seen that unit 45 has damage; In order to further determine the damage location, the section curves of the 5 th row and 6 th row of Fig. 2(a) are drawn, as shown in Fig. 2(b). We can see obviously from Fig. 2(b), the damage indexes all have a step at the position of node 5 and node 6 which are belong to 5 th , 0 (2019) MATEC Web of Conferences https://doi.org/10.1051/matecconf/201925602015 256 ICMME 2018 2015 curve and 6 th curve, exactly corresponding to the node 49, node 50, node 60 and node 61 in the finite element structure, that are the four nodes of the unit 45. So we can sure that the unit 45 has damage. The Fig. 3 is the figure of the foregoing three orders damage index when the unit 45 in the middle of the square aluminum alloy plate is damaged by 50%. Similar to the above analysis, the damage of unit 45 was determined. Comparing Fig.2 and Fig.3, it can be seen that there are more peak areas in the 3D view of the foregoing three orders damage index. If the damage occurs in the area where the peak belongs to, it is easy to cause identification error. Therefore, the damage position can be determined by using the modal frequency of the first order and the corresponding displacement vector based on the damage identification index of flexibility curvature difference. In the analysis below, we only use the index obtained from the first order modal parameters to identify the damage.
The first order damage index of unit 95 at the edge of the plate is calculated when the damage is 50%, as shown in Fig. 4.
The Fig. 4 is the figure of the first order damage index when the unit 95 at the edge of the square aluminum alloy plate is damaged by 50%. Among them, Fig 4(a) is a three-dimensional view of the damage index, from which it can be clearly seen that unit 95 has damage; the damage location was further determined in Fig. 4(b). Because the four corners of square plate are fixed, the damage of four corner elements are not identified and analyzed in this paper. According to the above analysis, whether the damage is in the middle or edge of the plate, this index can identify the damage correctly. MFCD ×10 -4 Fig. 2(a). Damage index of three-dimensional view. Fig.2(b). The 5th and 6th section curves.  Fig. 3(a). Damage index of three-dimensional view. Fig. 3(b). The 5th and 6th section curves.   Fig. 4(a). Damage index of three-dimensional view. Fig. 4(b). The 10th and 11th section curves.

Multi-positions damage identification
When multiple damage occurs in the structure, can the index accurately identify multiple damage locations? It is assumed that unit 45 and unit 95 both are damaged by 50% at the same time, the figure of damage index as shown in Fig. 5. Furthermore, it is supposed that unit 23 and unit 77 both are damaged by 50% at the same time, that is, the damage is located in different rows, and in different columns, the figure of damage index as shown in Fig. 6.     We can see from Fig.5 and Fig.6, for multiple injuries, wherever the damage is, the index can accurately identify them. However, when multiple injuries are of the same degree of damage, there are some differences in the recognition of damage degree.

Damage degree assessment
The judgment of damage degree is mainly based on comparing the size of damage indexes that are at the same position of the structure and under different damage degree. On the premise of confirmed the damage location, in order to distinguish the damage size more clearly, we select one of the two section curves related to the damage element in the damage index diagram. It is assumed unit 45 is damaged by 20%, 50% and 70% respectively, the index size of the damage degree are shown in Fig. 7.   Fig. 7 is the 6 th row section curves in damage index graph that related to the damage element when unit 45 is damaged by 20%, 50%, and 70% respectively. We can see from the Fig. 7 clearly, with the increase of damage degree, the value of damage index curve at the node of damage unit (that is node 5 and node 6) also increased significantly, this indicates that the index can reflect the size of a single damage degree. Fig. 8 is the 3 rd , 4 th , 8 th , and 9 th rows section curves in damage index graph when unit 23 is damaged by 20%, and unit 77 is damaged by 70% at the same time. A comparison between Fig. 8 and Fig. 6(b) shows that, when both unit 23 and unit77 are damaged by 20%, the index value of unit 23 is about 4 1.5 10   , and the index value of unit 77 is about 4 2.5 10   , there are not much difference; but when unit 23 is damaged by 20%, and unit 77 is damaged by 70% at the same time, the index value of unit 23 is still about 4 1. 5 13 10   correspond to 70%, this indicates that the index can fully reflect the size of damage degree. But we can know from Fig. 8, when the two degree of structure damage differ greatly, the small one is easy to be ignored, or we may confuse the single damage and multiple damage, because of the magnitude difference between the two damage indexes being large. For the damage identification index of longitudinal flexure curvature difference, the recognition result is the same as above, which is not repeated here.

Conclusion
It has been present that use the damage identification index of MFCD to identify the location of damage and determine the extent of damage. The plate had been divided by finite element method according to the theory of Kirchhof thin plate. In this article, the single-position damage and multi-position had been simulated and calculated with reducing the unit stiffness. We had found that the MFCD can not only accurately identify the single-damage and multi-damage position of plate structure, but also reflect the size of damage degree by analyzing. It has a great significance to apply the index into the range of damage identification of actual structures.