Fatigue Life Prediction of Electromagnetic Brake Connection Device in High-Speed Maglev Train

As shown in Figures 1 and 2, the Device consists of a triangular connector, a triangle connector seat, a pin and a rubber elastic joint. Specifically, the rubber elastic joint is composed of a bearing outer ring, a gasket, a rubber bushing and a bearing inner ring (Figures 3 and 4).

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1-triangular connector; 2- triangular connector seat; 3- pin; 4- rubber elastic joint

Figure 1. Sketch map of Device structure

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Figure 2. Exploded view of the Device

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1-bearing outer ring; 2- gasket; 3- rubber bushing; 4- bearing inner ring

Figure 3. Sketch map of rubber elastic joint

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Figure 4. Exploded view of rubber elastic joint

Linking up the brake magnet and suspension frame, the Device is responsible for the rigid positioning of the electromagnetic brake, transmitting the braking force uniformly to the adjacent suspension frame, damping the vibration of the transmission system, reducing the attraction of the electromagnetic brake to the guide rail, and compensating the dynamic relative motion between adjacent suspension frame and brake magnet [10, 11].

2.1 Prediction method

As shown in Figure 5, the fatigue life of the Device was predicted in four steps [12].

Step 1: According to the Device structure and functions, calculate the static load and fatigue load and plot the curve of stress (S) against the number of cycles to failure (N) (hereinafter referred to as the S-N curve) of the components.

Step 2: Import the 3D solid model of the component to the finite-element analysis software and establish a finite-element model of the Device.

Step 3: Considering the loads of the components and constraints of finite-element analysis, perform static analysis to see if the stiffness and strength of the components meet the requirements of train operation.

Step 4: Through  finite element analysis, select the key points in the Device and determine their equivalent mean stress and stress amplitude. Based on the stress correction theory, analyse the fatigue strength of the Device and check if it satisfies the use requirements. Then, predict the fatigue life of each component.

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In addition, the rubber bushing is made of natural rubber, which is an incompressible, isotropic, super-elastic polymer with nonlinear stiffness features. For an isotropic material, the unit volumetric strain energy W is a scalar function of the deformation tensor [14]:

$W=W\left(I_{1}, I_{2}, I_{3}\right)$      (1)

where, I₁, I₂, and I₃ are three independent variables of the Cauchy–Green deformation tensor. These variables are orthogonal second-order tensor invariants:

$I_{1}=\operatorname{tr} C=C_{i i}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}$      (2)

$I_{2}=\frac{1}{2}\left[(\operatorname{tr} C)^{2}-\operatorname{tr} C^{2}\right]=\frac{1}{2}\left(I_{1}^{2}-C_{i j} C_{i j}\right)$

$=\left(\lambda_{1} \lambda_{2}\right)^{2}+\left(\lambda_{2} \lambda_{3}\right)^{2}+\left(\lambda_{3} \lambda_{1}\right)^{2} i, j=1,2,3$      (3)

$I_{3}=\operatorname{det} C=\left(\lambda_{1} \lambda_{2} \lambda_{3}\right)^{2}$       (4)

where, λ₁, λ₂ and λ₃ are the elongation rates along the X, Y and Z axes, respectively. Since rubber is incompressible, the following relationship is valid:

$I_{3}=\left(\lambda_{1} \lambda_{2} \lambda_{3}\right)^{2}=1$       (5)

Rubber materials are usually described by thermodynamic statistical model and continuous medium mechanical model based on phenomenological theory. Here, the latter model is adopted for subsequent analysis. According to this model, the rubber bushing of the Device is isotropic if not deformed, that is, long molecular chains in the materials point to random directions. The unit volumetric strain energy W of the rubber bushing can be expressed as a function of the three strain invariants of the main elongation ratio λ_i or the deformation tensor I_i:

$W=\left(I_{1}, I_{2}, I_{3}\right)$         (6)

$W=\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)$      (7)

The N-polynomial form model and Ogden form model are two popular phenomenological models. The unit volumetric strain energy W of the rubber bushing can be decomposed into strain partial energy and volumetric strain energy in the N-polynomial form model:

$W=f\left(\overline{I_{1}}-3, \overline{I_{2}}-3\right)+g(J-1)$       (8)

where, J is the ratio of pre-deformation volume to post-deformation volume. Assuming that $g=\sum_{i=1}^{N} \frac{1}{D_{i}}(J-1)^{2 i}$, the N-polynomial of the strain energy can be obtained through the expansion of Taylor formula:

$W=\sum_{i+j=1}^{N} C_{i j}\left(\bar{I}_{1}-3\right)^{i}\left(\overline{I_{2}}-3\right)^{j}+\sum_{i=1}^{N} \frac{1}{D}(J-1)^{2 i}$       (9)

The above N-polynomial is a complete polynomial representing the constitutive model of the rubber bushing, with N being the order of the polynomial. For the N-polynomial constitutive model, the initial shear modulus µ₀and the initial bulk modulus k₀ depend on the first-order coefficient of the polynomial:

$\mu_{0}=2\left(C_{10}+C_{01}\right), k_{0}=\frac{1}{D_{1}}$       (10)

where, N is 1, Mooney-Rivlin model of the rubber bushing can be obtained:

$W=C_{10}\left(\overline{I_{1}}-3\right)+C_{01}\left(\overline{I_{2}}-3\right)+\frac{1}{D_{1}}(J-1)^{2}$       (11)

The Mooney-Rivlin model is suitable for simulating the mechanical properties of rubber materials at small to medium strain. Here, the rubber bushing of the Device is analysed by the Mooney-Rivlin constitutive model, using the mechanical parameters in Table 2 [15].

Table 2. Mechanical parameters of rubber bushing

2.3 Finite-element model and loading conditions

The fatigue load of the Device was determined according to the actual working conditions of high-speed maglev train (Table 3).

Table 3. Fatigue load of the Device

The 3D model of the Device was imported to ANSYS workbench, and applied with fixed constraints and loading conditions (Figure 6).

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Figure 6. Loading conditions of the finite-element model on the Device

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Figure 7. Comparison between the results of each finite-element model and the experimental data

Next, the 3D model was discretized by three different methods, yielding three finite-element models. The finite-element models each consists of 64,289, 78,306 and 106,320 grids. Figure 7 compares the results of each model with the experimental data. It can be seen that the three finite-element analyses shared similar results with a relative error lower than 10%. In particular, the 78,306-grid model and 106,320-grid model had very close results. The comparison shows that the results of the 78,306-grid model were basically consistent with the experimental results. The relative errors was lower than 6%, which is within the allowable range for engineering application. Therefore, this model was selected for the subsequent analysis.

2.4 Static analysis

Under the fatigue load in Table 3, the stress results of the finite-element analysis on the Device are as follows: the maximum von Mises stress of the metallic components was 25.746MPa, occurring in the outer bearing ring (Figure 8); the maximum von Mises stress (1.004MPa) of the rubber bushing was observed on the inside (Figure 9); the maximum deformation of the Device took place in the rubber bushing, amounting to 1.384mm (Figure 10); the relative maximum displacement of the inner bearing ring along the Z axis was about 1.288mm (Figure 11).

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Figure 8. Stress distribution of the Device

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Figure 9. Stress distribution of the rubber bushing

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Figure 10. Deformation distribution of the rubber bushing

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Figure 11. Z-axis displacement of the inner bearing ring

2.5 Fatigue failure criterion for rubber bushing

The mechanical fatigue of rubber materials is essentially the reduction of the mechanical properties of rubber with the gradual propagation of cracks under dynamic stress. The previous studies have shown that rubber materials often suffer fatigue damages in areas of high stress and strain; these areas are prone to crack initiation under cyclic loading [16]. The fatigue failures of rubber materials mainly include abnormal cracking, glue failure and surface wrinkling. However, the occurrence of fatigue cracks does not necessarily mean that the rubber is in the state of fatigue failure [17]. With the increase in the number of loading cycles, the local damages of rubber will lead to continued decrease of the elastic modulus and strength of the material, until the strength fails to withstand the rated fatigue load [18, 19]. Since it is difficult to measure the elastic modulus of rubber bushing, the failure degree of the Device was determined by the static stiffness loss rate of the rubber element in the fatigue test [9]:

$\Delta K=\frac{K_{1}-K_{2}}{K_{1}} \times 100 \%$        (12)

where, ΔK is the static stiffness loss rate; K₁ is the initial static stiffness; K₂ is the post-test static stiffness.

At present, there is no uniform standard for static stiffness failure of rubber materials. According to load-bearing requirements of the Device, the fatigue failure criterion was set as 20% reduction in ΔK [20, 21].

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Figure 12. Predicted fatigue life of the Device

2.6 Fatigue life prediction

The stress analysis results of the S-N curves on the metallic and rubber materials were imported to the finite-element analysis software. Coupled with the loading conditions in Table 3, the fatigue life of the Device was predicted and recorded in Figure 12.

Targeting the Device of high-speed maglev train, this paper calculates the static strength of the Device through finite-element analysis, predicted the fatigue life of the Device, and verified the predicted results through experiment. The research findings lay a solid theoretical basis for the minimization of product development cycle, optimization of product structure and safe operation of high-speed maglev train.