Keywords

1 Introduction

In order to ensure the safety and reliability of high-speed Electrical Multiple Units (EMU), the structure strength analysis and fatigue life prediction are the important contents in structural design and health monitoring. Strength design and fatigue life prediction are related to structure stress status. The structure stress cannot be obtained directly through measurement but indirectly through displacement or strain measurement because there are the relationship between stress, strain and displacement. The displacement mode analysis method can be used to establish the displacement response model of EMU structure and analyze the structural vibration response. And then, the strain mode of vehicle response is obtained by displacement-strain calculation. However, it is a differential process from displacement to strain, and the error of displacement will be amplified [1].

Therefore, many scholars have studied to apply the displacement mode method to the strain mode, and obtain the structural strain mode. Hillary et al. measured the force-strain transfer function with resistance strain gauge and proposed the concept of strain mode [2]. Li [3], Bernasconi [4] and Yam et al. [5] used the differential operation method of displacement mode to deduce and discuss the strain mode theory. Tsang [6], Li et al. [7] used the finite element method to verify the strain mode theory, and compared it with the calculation simulation and experimental test. Li et al. [7,8,9] also demonstrated the orthogonality of the strain mode. These studies are focused on the construction of strain mode theory, but it is rarely applied to the damage identification and fault diagnosis of complex structures.

In this paper, the equivalent vertical model of a type of EMU is used to simulate and compare the displacement mode and strain mode. So the local damage or fatigue position of the structure is determined, and the fatigue crack size is determined according to the damage degree. With the strain mode of the car body structure, the strain of all structure, such as the strain near the window, can be studied, which provides the basis for the study of structural stress concentration, the influence of local structural improvement on the vicinity of the changing region, and the health detection of car body structure.

2 Strain Mode Theory

In the three-dimensional coordinate system, a displacement vector is set as,

$${\mathbf{x = (u}},{\mathbf{v}},{\mathbf{w)}}^{T}$$
(1)

where x is the displacement vector, u,v,w is the displace vector in u,v,w directions. The ith displacement mode is

$${\varvec{\upvarphi }}_{i} {\mathbf{ = }}(\varphi_{i}^{u} ,\varphi_{i}^{v} ,\varphi_{i}^{w} )^{T}$$
(2)

the displacement response can be expressed through the superposition of vibration modes and mode coordinates of each order, namely,

$${\mathbf{x}} = ({\mathbf{u}},{\mathbf{v}},{\mathbf{w}})^{T} = \sum\limits_{i = 1}^{n} {q_{i} (\varphi_{i}^{u} ,\varphi_{i}^{v} ,\varphi_{i}^{w} )^{T} = {\varvec{\Phi \mathbf{q}}}}$$
(3)

where \({{\varvec{\Phi}}}\) is the displacement mode shape, q is the mode coordinates. Assuming the strain vector is

$${\mathbf{E = ({{\varvec{\upvarepsilon}}}_{x} {{\varvec{,\upvarepsilon}}}_{y} {{\varvec{,\upvarepsilon}}}_{z} )}}^{T}$$
(4)

The ith strain mode is

$${{\varvec{\uppsi}}}_{{\text{i}}}^{\varepsilon } {\mathbf{ = }}(\psi_{i}^{u} ,\psi_{i}^{v} ,\psi_{i}^{w} )^{T}$$
(5)

Among them,

$${{\varvec{\upvarepsilon}}}_{x} = \frac{\partial u}{{\partial x}} = \frac{\partial }{\partial x}\sum\limits_{i = 1}^{n} {{\varvec{\varphi }}_{i} (x)q_{i} }$$
(6)

Because \(q_{i}\) is a function related to time and its partial derivative to position x is 0, the above equation can be written as

$${{\varvec{\upvarepsilon}}}_{x} = \sum\limits_{i = 1}^{n} {q_{i} } \frac{{\partial {\varvec{\varphi }}_{i} (x)}}{\partial x} = \sum\limits_{i = 1}^{n} {q_{i} } {{\varvec{\uppsi}}}_{i}^{\varepsilon } (x)$$
(7)

The formula \({{\varvec{\uppsi}}}_{i}^{\varepsilon } (x) = \frac{{\partial {\varvec{\varphi }}_{i} (x)}}{\partial x}\) is called strain mode. Writing the above formula as a discrete form and getting the following result,

$${{\varvec{\upvarepsilon}}}_{x} = \sum\limits_{i = 1}^{n} {q_{i} } {{\varvec{\uppsi}}}_{i}^{\varepsilon } (x) = \sum\limits_{i = 1}^{n} {\frac{{{{\varvec{\uppsi}}}_{i}^{\varepsilon } {\varvec{\varphi }}_{i}^{T} }}{{k_{i} - \omega^{2} m_{i} + j\omega c_{i} }}} {\mathbf{F}}e^{j\omega t}$$
(8)

Similarly, the strain mode expression in other directions can be obtained, which is the same as formula (8). According to the superposition principle, the strain response can be written as

$${\mathbf{E = ({{\varvec{\upvarepsilon}}}_{x} {{\varvec{,\upvarepsilon}}}_{y} {{\varvec{,\upvarepsilon}}}_{z} )}}^{T} { = }\sum\limits_{i = 1}^{n} {q_{i} } (\psi_{i}^{u} ,\psi_{i}^{v} ,\psi_{i}^{w} )^{T} { = }\sum\limits_{i = 1}^{n} {{{\varvec{\uppsi}}}_{{\text{i}}}^{\varepsilon } } q_{i}$$
(9)

From foumula (8), the strain mode response matrix is

$${\text{H}}^{\varepsilon } { = }\sum\limits_{i = 1}^{n} {\frac{{{{\varvec{\uppsi}}}_{i}^{\varepsilon } {\varvec{\varphi }}_{i}^{T} }}{{k_{i} - \omega^{2} m_{i} + j\omega c_{i} }}}$$
(10)

It can be seen from the above formula that the characteristics of the strain frequency response function matrix are as follows.

  1. (1)

    because of \({\text{H}}_{ij} \ne {\text{H}}_{ji}\), the strain frequency response function matrix is an asymmetric matrix.

  2. (2)

    Each element in the matrix contains \(c_{i} ,m_{i} ,k_{i}\) information.

  3. (3)

    Each row of the matrix contains all information of the displacement mode, and each column of the matrix contains all information of the strain mode. Therefore, for the strain mode analysis, if you want to get all the mode parameters of the displacement mode and the strain mode, you must measure one line and one column of \({\text{H}}^{\varepsilon }\).

The ultimate goal of obtaining the structural strain mode is to carry out stress analysis. After obtaining the strain mode, the mode stress can be obtained according to Hooke’s law, and the stress response can be calculated by mode superposition method. According to the stress-strain relationship of elastic mechanics

$$\begin{gathered} \sigma_{x} = \lambda \theta + 2G\varepsilon_{x} , \, \tau_{xy} = G\gamma_{xy} \hfill \\ \sigma_{y} = \lambda \theta + 2G\varepsilon_{y} , \, \tau_{xy} = G\gamma_{yz} \hfill \\ \sigma_{z} = \lambda \theta + 2G\varepsilon_{z} , \, \tau_{zx} = G\gamma_{zx} \hfill \\ \end{gathered}$$
(11)

where, \(\theta\)—volume strain, \(\lambda\)—lame constant, G—Shear modulus.

Using the strain mode superposition method,

$${{\varvec{\uptheta}}} = {{\varvec{\upvarepsilon}}}_{x} + {{\varvec{\upvarepsilon}}}_{y} + {{\varvec{\upvarepsilon}}}_{z} = \sum\limits_{i = 1}^{n} {q_{i} } ({{\varvec{\uppsi}}}_{i}^{{\varepsilon_{x} }} + {{\varvec{\uppsi}}}_{i}^{{\varepsilon_{y} }} + {{\varvec{\uppsi}}}_{i}^{{\varepsilon_{z} }} ) = \sum\limits_{i = 1}^{n} {q_{i} } {{\varvec{\Theta}}}_{i}$$
(12)

The above formula is brought into formula (11), obtained

$$\sigma_{x} = \lambda \sum\limits_{i = 1}^{n} {q_{i} } \Theta_{i} + G\sum\limits_{i = 1}^{n} {q_{i} {{\varvec{\uppsi}}}_{i}^{{\varepsilon_{x} }} } = \sum\limits_{i = 1}^{n} {q_{i} } (\lambda {{\varvec{\Theta}}}_{i} + G{{\varvec{\uppsi}}}_{i}^{{\varepsilon_{x} }} ) = \sum\limits_{i = 1}^{n} {q_{i} } {{\varvec{\Xi}}}_{i}^{x}$$
(13)

where, \({{\varvec{\Xi}}}_{i}^{x}\)—x direction stress mode of the ith, \(q_{i}\)—the ith mode coordinates, Similarly, y, z direction stress modes can be obtained.

3 Strain Mode Analysis of EMU Car Body

The car body structure is welded by large and light hollow aluminum alloy. In order to meet the needs of the structural design, the windows, doors and other structures be installed. Therefore, these locations are prone to stress concentration and structural fatigue. Because the strain mode is more sensitive to the stress change and local damage caused by local structural changes than the displacement mode, the numerical simulation method is used to analyze this.

According to the structural parameters of an EMU car body, just considering the vertical equivalent section stiffness and mode, according to the principle of equivalent of vertical stiffness and mode frequency, the car body is equivalent to an equal section free homogeneous Euler-Bernoulli beam with rectangular section. After the equivalent treatment of the car body, the section size is as follows, the width of the beam is 0.32 m, the height is 1.25 m, and the length is 24.5 m. The basic properties of car body material are as follows, Elastic modulus, Poisson’s ratio and density are 69 GPa, 0.3, 2700 kg/m3, respectively.

The numerical simulation results of displacement mode and strain mode are obtained when the structure has no damage and middle positions of the equivalent beam has damage respectively. The damage treatment process of the model beam is as follows, assuming that defect groove on the upper surface of the model beam is equaled to the width of the model, the length of the defect groove is 5 mm, and the depth of the defect groove is 5%, 10%, 15%, 20% and 0% of the height of the model beam respectively, as shown in Fig. 1.

Fig. 1
figure 1

Diagram of the damage of car body structure

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The 8-node solid, 185 element model is used to establish finite element by ANSYS software. The width mesh number of the beam is 4, the vertical mesh number is 13, and the longitudinal is divided into 245 elements. The finite element model of the beam is shown in Fig. 2, which is divided into 17,220 nodes and 12,740 elements. Through the finite element post-process, the mode displacement and mode strain of each node of the model beam can be extracted, and the displacement mode and strain mode of the model beam can be obtained.

Fig. 2
figure 2

Finite element analysis model

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The frequency is the inherent characteristic of the structure. The first three mode frequency of the equivalent model of the car body is as shown in Table 1. From Table 1, the mode frequency changes little before and after the damage, so the frequency can’t be used as the index of damage diagnosis, and the damage location can’t be determined by the frequency.

Table 1 The mode frequency comparison of car body model
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Therefore, in the case of the first three displacement modes and strain modes under the damage amount of 0, 5, 10, 15, 20 and 30% are extracted, as shown in Figs. 3 and 4. It can be seen from the figure that the mode shapes of displacement modes are basically unchanged in the case of intact and damaged, that is, the sensitivity of displacement modes to damage is low. Therefore, the damage of location and degree cannot be determined by the displacement vibration mode. In the case of intact and damaged strain mode, the change of vibration mode shape is obvious, especially the first-order and third-order mode shape is significant. The second-order strain mode change is not obvious because the second-order mode node is in the middle position. So, the strain mode is sensitive to damage, but if the damage position is the mode node, the sensitivity will decrease.

Fig. 3
figure 3

Mode shape of displacement mode

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Fig. 4
figure 4

Strain mode shape

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In order to obtain the damage location, the variation curve of strain mode change rate can be obtained by applying the derivative, and the damage location can be determined from the strain mode change rate curve. Because the strain mode curve is not smooth, the difference method in numerical calculation is a method to calculate the rate of change of discrete value.

Using numerical difference calculation method of the strain mode curve, the strain mode difference curve is shown in Fig. 5. It can be seen from the figure that the strain mode curves of no damage structure change continuously, and the difference curve is also the same. But in damaged structure, the strain mode curve will have a sudden change at the damage position, and the difference curve will change drastically at the damage position. For the damage at the mode nodes, the strain mode difference curves will also change sharply. The difference curves for each damage amount almost intersect at the same point, which is the damage point. So the strain mode difference value can be used as a judgment index for damage location.

Fig. 5
figure 5

Differential curve of the strain mode

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The simulation analysis results using the equivalent model of the car body show that the strain mode and its differential curve can be used as an effective method for car body structure strength calculation and damage location diagnosis. However, the car body is actually a complex welded structure, and its strength calculation and damage location diagnosis are more complicated than the equivalent model. Therefore, the finite element model of the car body structure is used to extract the displacement mode and strain mode, and the strain mode method is used to analyze the strength of the car body structure.

The car body of high-speed train is mainly welded by large hollow and wide aluminum alloy. It consists of the under frame, roof, end walls and side walls. Figure 6 is a finite element model of a certain type of EMU car body. The mesh of the model is divided into 49,503 nodes and 82,527 elements.

Fig. 6
figure 6

Car body finite element model

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Extracting the first-order vertical bending displacement mode and strain mode is extracted, which is shown in Fig. 7. It can be seen from the figure that the displacement mode curve continuously changes without obvious change, indicating that the displacement mode of the car body structure is not sensitive to structural changes such as windows, and the structural strength calculated by the displacement mode is not reflect the influence of the window structure of the car body. However, the strain mode curve at the position of the window has changed, indicating that the window structure has a great influence on the strain of the car body structure. The strain changes alternately in the length direction of the car body. It can be seen from formula (13) that the car body will have stress changes at this position, and structural defects such as stress concentration and insufficient strength will easily occur, the attention should be paid to the structural design.

Fig. 7
figure 7

The first-order vertical bending displacement mode and strain mode of car body

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4 Conclusion

Using the equivalent model and the finite element model of the car body, the displacement mode and strain mode of the car body are analyzed, and the application of the strain mode in strain and stress acquisition is studied. The main conclusions of body strength design and damage diagnosis are as follows,

  1. (1)

    Strain mode and displacement mode are two manifestations of the same energy balance state. Strain mode can also be solved by superposition of strain mode shape and mode coordinate.

  2. (2)

    Compared with the displacement mode, the strain mode is more sensitive to structural damage. This conclusion is verified by the car body equivalent model and the finite element model. The strain mode difference curve can assist in determining the location of structural damage.

  3. (3)

    For complex structures such as car bodies, the strength design is an important part of the structural design. The strain and stress calculation methods of the car body based on the strain mode have certain applicability and feasibility, which can be used for car body fatigue strength analysis and load spectrum research.