Introduction

Supercapacitor is an efficient power supply device that stores electrical energy by utilizing the polarization of the electrolyte1. Compared to traditional energy storage power sources, it offers advantages such as high energy density, rapid charge and discharge rates, long service life, small size, and light weight2. It is widely applied in braking energy recovery systems for urban rail vehicles3.

As an indispensable component, the safety of the energy storage supercapacitor box is of crucial importance. During the operation of urban rail vehicle, it is subjected to complex and highly random external vibration loads transmitted from the track surface4. Long term exposure to such random vibrations could lead to fatigue damage in the bracket, module, and other components within the supercapacitor box. To ensure sufficient fatigue resistance, fatigue assessments are generally conducted according to relevant standards during the structural design stage. This allows for identifying the fatigue-prone locations within the structure and assessing whether they meet the operational requirements. Therefore, fatigue performance analysis under random vibration is of great significance for the design and life prediction of the supercapacitor box.

The methods for structural random vibration fatigue researches primarily consist of two approaches: the time-domain method and the frequency-domain method. The time-domain method demands high-quality signals, necessitating a long duration of the signal, extensive computational resources, and longer calculation times. Additionally, it is challenging to identify risk points prior to analysis. Performing vibration fatigue analysis using the frequency-domain method could effectively avoid these shortcomings. The calculation of the frequency response function provides more valuable insights compared to the time consuming computation of dynamic responses in the time-domain5.

Several studies have been carried out on the effect of battery random vibration. Yoon C et al.6 analyzed the electrochemical performance of lithium-ion battery pack through random vibration test. Joshy N et al.7 investigated the impact of vibration frequencies on battery thermal control systems. And they found that battery surface temperature increased with the increase of vibration frequencies. Akbulut M et al.8 performed random vibration tests based on European Cooperation for Space Standardization on a selected battery pack. It was noted that the battery pack maintained its structural soundness without any mechanical failures. Berg P et al.9 conducted random vibration assessments on lithium-ion batteries according to the SAE J2380 standard. The findings indicate that, for potential current or future uses in harsher vibration conditions, the internal structure of the cells and their orientations should be taken into account. However, a reliable and safe design of the battery pack’s structure under random vibration environment is still required.

About the fatigue life of the battery pack under random vibration environment, researches are primarily focused on simulation of fatigue performance with the help of structural finite element analysis software. These efforts aim to investigate the fatigue life of power battery packs. Wang W et al.10 analyzed the structural response in a random vibration environment using the frequency-domain method by Hypermesh and Ansys. Based on Palmgren-Miner damage theory, they conducted simulation analysis on the fatigue life of the power battery under random vibration excitation. They proposed a method for analyzing the random vibration fatigue of power battery box structures. Huang P et al.11 investigated the safety of battery pack under random vibration and shock conditions. By constructing a finite element analysis model of the battery pack, they conducted random vibration analyses along X, Y, and Z axis. Subsequently, a 25 g half-sine wave was applied at the connection between the battery pack and the vehicle body along Z-direction. Stress and acceleration data were extracted from the analysis results. Based on these findings, hazardous contact locations were identified, and optimizations were proposed. Dai J et al.12 employed frequency-domain analysis method and utilized the finite element software nCode to calculate the fatigue life of battery pack. The validity of the method was verified through comparisons between the analysis results and experimental findings. Kim H et al.13 conducted a comprehensive finite element analysis of a battery pack. The cumulative fatigue damage of the battery pack was estimated. Based on the findings from the simulated fatigue assessments, the battery pack design is mostly secure, apart from specific regions of the support frames and mounting plate. Zhang X et al.14 introduced a dual-goal optimization technique to assess the crushing stress and the vibration fatigue duration of the battery pack system. This technique could provide good combination of the thicknesses of the battery pack system components, which could also helps engineers to achieve robustness and efficient designs.

At present, most studies only focus on analyzing the mode, crushing and strength performance of battery packs, with a few researchers examining the fatigue characteristics of battery packs. Only a few researches15,16 are concentrated on the cooling analysis of the energy storage supercapacitor box. And there are almost no studies on the fatigue performance of energy storage supercapacitor box structures under random vibration loads. This paper takes the energy storage supercapacitor box applied to urban rail vehicle as the research object, and establishes a finite element model of the supercapacitor box. It analyzes the first 10 modes of the supercapacitor box, studies its frequency response under unit gravitational acceleration excitation, and investigates its fatigue characteristics within the frequency domain using acceleration spectral density (ASD) as the excitation load. Finally, the energy storage supercapacitor box is manufactured using lightweight aluminum alloy material, and the fatigue damage of the aluminum alloy supercapacitor box is analyzed to provide a basis for the structural design optimization of the energy storage supercapacitor boxes.

Theory

Frequency response theory

Frequency response analysis, also known as frequency sweep analysis, is a method used to calculate the response of a structure under excitation. Its purpose is to study the dynamic characteristics of the system from a frequency domain perspective and to determine whether the structure will suffer damage due to phenomena such as resonant fatigue17.

There are generally two methods for solving frequency response: mode superposition and direct integration. The direct integration method solves the equations through small time steps in physical space to obtain highly accurate calculation results. However, the solving process is complex and time-consuming, making it suitable for small scale calculations. The mode method, on the other hand, transforms the physical space into mode space for solving, allowing for larger time steps. It could reduce calculation scale without compromising accuracy. Based on the above principles, the mode superposition method is more suitable for large scale finite element analysis18. Therefore, this paper adopts mode superposition method for solving frequency response analysis.

The differential equation of vibration for a general damping system with multiple degrees of freedom is

$$M\mathop {x\left( t \right)}\limits^{..} + C\mathop {x\left( t \right)}\limits^. + Kx\left( t \right) = f\left( t \right)$$
(1)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, x(t) is displacement response vector, and f(t) is the external load vector.

The mode superposition method assumes that x(t) could be represented by a linear superposition of the modes of the structure, then

$$x\left( t \right) = \left[ \Phi \right]\left\{ y \right\}$$
(2)

where [Φ] is the matrix of the mode, and {y} is the matrix transforming the physical space into mode space.

Substitute Eq. (2) into Eq. (1), and decouple by multiplying [Φ]T in front, we could obtain Eq. 

$${\left[ \Phi \right]^T}M\left[ \Phi \right]\left\{ {\mathop y\limits^{..} } \right\} + {\left[ \Phi \right]^T}C\left[ \Phi \right]\left\{ {\mathop y\limits^. } \right\} + {\left[ \Phi \right]^T}K\left[ \Phi \right]\left\{ y \right\} = {\left[ \Phi \right]^T}f\left( t \right)$$
(3)

According to the orthogonality of natural modes, [Φ]TM [Φ] = 1, [Φ]TK [Φ] = ω2, [Φ]TC [Φ] = 2ξω, Eq. (3) could be simplified as

$$\mathop y\limits^{..} + 2\xi \mathop y\limits^. + {\varpi ^2}y = {\left[ \Phi \right]^T}f\left( t \right)$$
(4)

where ω is the angular velocity of vibration, and ξ is the damping coefficient of the system.

When the system is subjected to a periodic excitation load, it is necessary to perform a Fourier transform on the periodic load. After obtaining the system’s response at various orders, they are superposed to derive the total response of the entire system. When solving frequency response using finite element simulation, a unit forced excitation frequency is usually applied to obtain the transfer function in the frequency domain.

Random vibration fatigue theory

Supercapacitor box is often subjected to variable amplitude stress cycles. When each stress cycle subjects the component to the fatigue limit, supercapacitor box would suffer damage. A single instance of damage is insufficient to reach the material’s failure limit. However, as the number of operating cycles increases, the damage would accumulate. When the cumulative damage reaches the material’s fatigue failure limit, structural failure occurs. This is the cumulative damage criterion often used to predict lifetime.

The theory of fatigue cumulative damage mainly encompasses three major categories: cumulative linear damage theory, modified linear fatigue damage theory, and nonlinear fatigue cumulative damage theory. Among these three, the cumulative linear damage theory is the most widely used and typical damage theory19. The Palmgren-Miner linear cumulative damage criterion is widely applied in lifetime prediction due to its ease of use, high prediction accuracy, and good agreement with experimental results.

The fundamental assumptions of Palmgren-Miner damage theory include:

  1. (1)

    The material undergoes a certain amount of damage during each stress cycle, and absorbs energy. When the accumulated damage energy reaches the material’s fatigue failure limit, structural failure occurs.

  2. (2)

    The energy absorbed when the material fails is independent of the amplitude of the fatigue load, but is related to the accumulated energy.

  3. (3)

    The energy absorbed by the material during each stress cycle is independent of the sequence of stress levels and exists independently.

Based on the Palmgren-Miner damage theory, the damage caused to a structure by stress cycles could be linearly accumulated. For a structure composed of a single material, its cumulative damage is

$$D = \sum\limits_{i = 1}^n {\frac{{{n_i}}}{{{N_i}}}}$$
(5)

where Ni is the number of fatigue failure cycles for a structure under a specific stress σi, and ni​ is the actual number of cycles the structure undergoes under the stress σi.

To ensure that a structure does not suffer structural failure within a specified period, its cumulative damage value D should be smaller than 1.

This paper conducts random vibration fatigue analysis using the three-band method proposed by Steinberg, which combines the Gaussian distribution with Palmgren-Miner damage theory. The three-interval method assumes that the stress response of the structure follows a Gaussian distribution, as shown in Fig. 1. The probabilities of stress distribution within each interval are presented in Table 1.

Fig. 1
figure 1

Schematic diagram of 3σ intervals.

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Table 1 3σ probability distribution.
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As can be seen from Table 1, there is a probability of only 0.27% that the structure will experience a stress greater than 3σ during random vibration. And it is assumed that the stress occurring at this probability will not cause any damage to the structure.

f+µ in Table 1 represents the average frequency at which the random stress passes through the mean value µ with a positive slope. It is the frequency where the energy is most concentrated in the spectrum, which is related to the 0th and 2nd moments of inertia of the one-sided acceleration spectral density G(f) of the stress response. Then

$$f_\mu ^ + = \sqrt {\frac{{{m_2}}}{{{m_0}}}} = \frac{{\sqrt {\int_0^\infty {{f^2}G\left( f \right)df} } }}{{\sqrt {\int_0^\infty {{f^0}G\left( f \right)df} } }}$$
(6)

where m2 is the 2nd moment of inertia of G(f), and m0 is the 0th moment of inertia of G(f).

According to Palmgren-Miner damage theory, combined with the Gaussian three-band method, the overall damage could be simplified based on the principle of cumulative damage.

$$D = \frac{{{n_{1\sigma }}}}{{{N_{1\sigma }}}} + \frac{{{n_{2\sigma }}}}{{{N_{2\sigma }}}} + \frac{{{n_{3\sigma }}}}{{{N_{3\sigma }}}}$$
(7)

where n1σ, n2σ and n3σ represent the number of cycle within the stress level intervals equal to or less than 1σ, 2σ and 3σ respectively, and N1σ, N2σ and N3σ represent the allowable number of cycles corresponding to the stress levels of 1σ, 2σ and 3σ respectively, obtained from the SN curve of the material.

Substituting the data in Table 1 into Eq. (7), then

$$D = \frac{{{\rm{0}}{\rm{.683f}}_{\rm{\mu }}^ + }}{{{N_{1\sigma }}}} + \frac{{{\rm{0}}{\rm{.271f}}_{\rm{\mu }}^ + }}{{{N_{2\sigma }}}} + \frac{{{\rm{0}}{\rm{.043f}}_{\rm{\mu }}^ + }}{{{N_{3\sigma }}}}$$
(8)

The fatigue life before structural failure T is

$${\rm{T}} = \frac{1}{D}$$
(9)

Method

Frequency domain analysis based on finite element software is used for fatigue analysis of the supercapacitor box. Along with the Palmgren-Miner damage theory and the SN curve of the material, a fatigue analysis of the supercapacitor box is conducted according to the requirements of random vibration testing. Firstly, a mode response analysis is conducted on the structure to obtain the transfer function between structural stress and input frequency. Subsequently, fatigue life prediction is performed by combining ASD and the material’s SN curve. This analysis calculates its lifetime and predicts the location of potential fatigue failure. The analysis process is illustrated in Fig. 2.

Fig. 2
figure 2

Flow chart of fatigue life under random vibration.

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Finite element model of the supercapacitor box

The supercapacitor box concerned in this paper has a total weight of 70 kg, including a supercapacitor module weighing 22.4 kg. The supercapacitor box has a length of 630 mm, a width of 471 mm, and a height of 450 mm. The overall parameters of the supercapacitor box are shown in Table 2.

Table 2 The overall parameters of the supercapacitor box.
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The entire box is fixed to the bottom of the urban rail vehicle body with four bolts. The main frame and the supercapacitor box body are connected via bolted joints, while the cover plate is connected to the supercapacitor box body through riveting. The detailed structure is illustrated in Fig. 3.

Fig. 3
figure 3

Structure of the supercapacitor box.

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The features in the supercapacitor box structure that have minor impacts on the overall structure’s weight and natural frequency are simplified, with the main measures including:

  1. (1)

    Smaller structural components (such as rivets) are omitted to avoid excessively small mesh sizes and excessively large aspect ratios, which would increase calculation time and reduce calculation accuracy.

  2. (2)

    The structure of the supercapacitor module is complex. If meshing is performed based on its actual structure, it will lead to a significant increase in the number of mesh elements. The definition of the constraints become extremely complicated, resulting in a sharp rise in the number of mesh elements. This would cause excessive computational load, increase calculation time, and affect the calculation accuracy. Therefore, the supercapacitor module is treated as a rigid body. Only the outer surface of the capacitor module is taken, and solid filling is performed. Its density is calculated based on its weight and the volume after solid filling.

  3. (3)

    To facilitate mesh generation, ensure the quality of meshing, and reduce calculation time, the chamfers with relatively small geometric dimensions on the supercapacitor box are removed.

The materials of the supercapacitor box are all Q235, with specific material properties detailed in Table 3.

Table 3 Material parameters of the supercapacitor box parts.
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Based on the structural characteristics of the supercapacitor box, the simulation primarily employs tetrahedral elements, with an element size of approximately 20 mm (with mesh refinement in areas such as bolt holes, resulting in a minimum element size of 2 mm). The element type used here is the C3D10 ( ten-node second-order tetrahedral element ). The entire model consists of a total of 305,067 elements. The meshing result is shown in Fig. 4.

Fig. 4
figure 4

Meshing results of the supercapacitor box.

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Random vibration fatigue analysis

GB/T 21,563 − 2018 “ Railway applications - Rolling stock equipment - Shock and vibration tests ” stipulates various requirements for random vibration and random shock test items for rail vehicle equipment. It could be used to verify the equipment’s ability to withstand vibrations, and therefore it could also be the basis to conduct vibration fatigue analysis on supercapacitor box.

Fig. 5
figure 5

ASD spectrum of Class 1, Category B and body mounted vehicles.

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The prerequisite for conducting frequency response analysis is to obtain the ASD based on GB/T 21,563 − 2018, which serves as the vibration load excitation for the simulated structure. The ASD of vibrations for the supercapacitor box refers to the relevant instructions in GB/T 21,563 − 2018 regarding the installation of equipment on Class 1, Category B vehicle bodies, which is shown in Fig. 5.

The relevant provisions regarding the frequencies of rail vehicle equipment in the GB/T 21,563 − 2018 standard are shown in Table 4.

Table 4 The specifications regarding the frequencies of rail vehicle equipment.
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The mass of the supercapacitor box is 70 kg, which is less than 500 kg. According to the specifications in Table 4, then f1 = 5 Hz and f2 = 150 Hz. With an acceleration scale factor of 7.83, the ASD magnitude and Root Mean Square (RMS) value for the simulated long life testing could be calculated, as shown in Table 5.

Table 5 ASD magnitudes and RMS values for the simulated long life testing.
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Based on the relationship that the area enclosed by the ASD curve and the horizontal axis equals the RMS value, the ASD for the long life testing of the supercapacitor box could be calculated, as shown in Table 6.

Table 6 ASD for the long life test along X, Y and Z axis.
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Results and discussion

Mode analysis

Mode analysis is an important reference for the evaluation of fatigue strength of structures under random vibration. If the mode frequency is too small, actions should be taken to strengthen the structure, so as to avoid resonance caused by the overlapping of structural mode frequency and external excitation frequency.

Since the energy storage supercapacitor box is bolted to the lower body of the subway structure, the constrained mode simulation is carried out this time. And all the six degrees of freedom of the mounting bolt holes are constrained. Lanczos method is adopted to extract the first 10th order modal values of the supercapacitor box structure. The displacement result is output to assess the vibration mode. Table 7 shows the first 10 mode values and relevant data.

Table 7 The first 10 vibration modes of the supercapacitor box structure.
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It can be seen from the data that when the external excitation frequency reaches around 82 Hz, the supercapacitor box reaches the resonance critical value, and resonance deformation is prone to happen to the supercapacitor box.

The first 4 typical vibration modes of the supercapacitor box are selected for analysis, as shown in Fig. 6. The vibration mode of the 1st order mode ( Fig. 6(a) ) is the longitudinal vibration of the front cover plate of the box along the Z axis. And the vibration modes of the 2nd ( Fig. 6(b) ), 3rd ( Fig. 6(c) ), and 4th ( Fig. 6(d) ) order modes are all vertical vibrations of the upper plate of the supercapacitor module along the Y axis.

Fig. 6
figure 6

The first 4 typical vibration modes of the supercapacitor box. (a)1st order; (b) 2nd order; (c) 3rd order; (d) 4th order.

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The low frequency resonance of the urban rail vehicles refers to the resonant when the rail vehicles are running in the tunnel, which is caused by the air vibration between the train and the tunnel. It occurs at low frequency range between 10 and 30 Hz20. Based on the frequency values from the 1st to the 10th order in Table 6, it can be spotted that the frequencies in its constrained state are all greater than 30 Hz. Therefore, this energy storage supercapacitor box will not experience resonance during the normal operation.

Frequency response analysis

Frequency response analysis is performed based on the established finite element model of the supercapacitor box. All the degrees of freedom of the bolt installation holes between the supercapacitor box and the train are constrained. Unit gravitational acceleration g ( 9810 mm/s² ) loads along X, Y and Z axis are applied respectively. The structural damping parameter is defined to be 0.05.

Figure 7 shows the Mises stress distribution of the supercapacitor box along X axis under a unit gravitational acceleration load. The maximum vibration stress is 181.6 MPa, occurring at node number of 23,050. This node is located at the contact point between the mounting bracket of the supercapacitor module and the bottom bracket of the supercapacitor box. The primary reason for this stress concentration is that the supercapacitor box is subjected to lateral excitation, experiences an inertial forward tilt, which subsequently leads to stress concentration at the mounting bracket of the supercapacitor module.

Fig. 7
figure 7

Stress distribution along X axis under a unit gravitational acceleration load.

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Fig. 8
figure 8

Stress distribution along Y axis under a unit gravitational acceleration load.

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Figure 8 illustrates the stress distribution of the supercapacitor box along Y axis under a unit gravitational acceleration load. The maximum vibration stress is 18.58 MPa, occurring at node number of 38, which is located at the lower edge of the bottom bracket of the supercapacitor box. This high stress is primarily due to the vertical excitation applied to the supercapacitor box, causing it to shake up and down, thus tending to generate high stress.

Fig. 9
figure 9

Stress distribution along Z axis under a unit gravitational acceleration load.

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Figure 9 illustrates the stress distribution of the supercapacitor box along Z axis under gravitational acceleration load. The maximum vibration stress is 32.84 MPa, occurring at node number of 632, which is located at the upper latch of the supercapacitor box cover. This high stress at the latch position is primarily due to the longitudinal excitation applied to the supercapacitor box, causing it to undergo torsional deformation.

Fig. 10
figure 10

Mode stress curves from the frequency response analysis.

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The mode stress curves from the frequency response analysis at the nodes with maximum stresses along X, Y, and Z axis are shown in Fig. 10. It can be observed that the stress generated by the unit load along X axis is bigger with a maximum stress reaching 181.6 MPa, indicating harsher working conditions along X axis. Node 23,050 at approximately 200 Hz, node 38 at around 175 Hz, and node 704 at around 80 Hz, exhibit significant amplitude points, which could correspond to the numerical results of the 10th, 5th, and 1st modes in Table 7, respectively. This also confirms the accuracy of the mode calculation results to a certain extent.

Fatigue analysis results under random vibration

The simulated long life testing at increased random vibration levels in GB/T 21,563 − 2018 is used as the input acceleration excitation for the fatigue analysis. Then the overall fatigue damage of the supercapacitor box under random vibrations along X, Y, and Z axis is obtained.

During random vibration fatigue analysis, the SN curve of the material is required as input. The SN curve is a polygonal line showing the relationship between failure life and stress amplitude under controlled stress conditions. It is crucial for estimating the fatigue life of supercapacitor box. Based on the welding method of the structure, referring to the IIW 2016 Recommendations for Fatigue Design of Welded Joints and Components ( Fig. 11 ), and the FAT St.36 SN curve is selected for fatigue analysis.

Fig. 11
figure 11

SN curves for different FATs.

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According to Fig. 11, the SN curve’s data of the material Q235 is shown in Table 8.

Table 8 SN data of material Q235 for random vibration fatigue analysis.
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When the material undergoes random vibration along the X, Y, and Z axes, the maximum 1σ stresses are 0.13 MPa, 2.63 MPa, and 3.80 MPa, respectively, all of which are far less than 20 MPa. Referring to Fig. 5, we could get.

$$f_\mu ^ + = \frac{{{f_1} + {f_2}}}{2} = \frac{{5 + 150}}{2} = 77.5(Hz)$$

And N1σ, N2σ, and N3σ are all taken as 107. Then the overall fatigue damage is.

$$D = \frac{{0.683 \times 77.5}}{{{{10}^7}}} + \frac{{0.271 \times 77.5}}{{{{10}^7}}} + \frac{{0.043 \times 77.5}}{{{{10}^7}}} = 7.73 \times {10^{ - 6}} < 1$$

According to Eq. (9) in paragraph 1.2 of this paper, the fatigue life of the supercapacitor box under random vibration could be calculated by.

$$T = \frac{1}{D} = \frac{1}{{7.73 \times {{10}^{ - 6}}}} = 1.29 \times {10^5}\quad\rm(cycles)$$

As a component of urban rail vehicle, the fatigue life of the supercapacitor box is generally assessed by low stress high cycle fatigue characteristics, with a cyclic life requirement of 1 × 104 cycles or more. According to the calculation results, the fatigue damage of the supercapacitor box is minimal with an order of 10− 6. And the number of fatigue cycles is on an order of 105. The fatigue life of the supercapacitor box structure could meet the requirements for low stress high cycle life of urban rail vehicle components.

Fig. 12
figure 12

Simulation results of fatigue damage of the supercapacitor box structure under random vibration.

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The fatigue simulation results of the supercapacitor box under random vibration mode are shown in Fig. 12. The maximum fatigue damage is 6.24 × 10− 6. Comparing with the fatigue life calculated based on equivalent stress, it is found that the simulation result is slightly bigger. The final difference in fatigue damage is about 2.39%.The reason of the difference is mainly due to the fact that the values taken from the material’s SN curve are not exactly the same.

Lightweight design

The analysis results of fatigue damage indicate that the fatigue failure of the supercapacitor box structure made from steel Q235 is minimal, at the order of magnitude of 10− 6, demonstrating a significant design margin. Lightweighting is of great significance in improving the energy efficiency of urban rail vehicles, extending their driving range, and enhancing their overall performance.

Aluminum alloy has a much lower density than steel, allowing the supercapacitor box made from aluminum alloy to significantly reduce weight while maintaining sufficient strength. Additionally, aluminum alloy maintains good corrosion resistance at high temperatures, which is crucial for the operation of supercapacitor box in harsh environments. Therefore, considering aluminum alloy 6063-T5 to replace Q235 for manufacturing the supercapacitor box could reduce its mass, save energy and reduce emission. The material parameters of the aluminum alloy profile 6063-T5 used are shown in Table 9.

Table 9 Material parameters of the aluminum alloy 6063-T5.
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After replacing with lightweight aluminum alloy 6063-T5, the first 10 order modes of the supercapacitor box structure are shown in Table 10.

Table 10 The comparison of first 10 vibration modes of 6063-T5 and Q235.
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As seen from the mode analysis results, after the material replacement of 6063-T5, the first 10 order modes’ frequencies do not change significantly, varying from 63 to 198 Hz, which are still much greater than 30 Hz. Therefore, resonance will not occur in the supercapacitor box.

Fig. 13
figure 13

Mode stress curves from the frequency response analysis of 6063-T5.

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The mode stress curves from the frequency response analysis at the nodes with maximum stresses along X, Y, and Z axis are shown in Fig. 13. It can be observed that the unit load generates relatively bigger stress along X axis, with a maximum stress reaching 114.6 MPa. Compared to the other two directions, the working condition along X axis is more severe. The stress level is smaller than the Q235 energy storage supercapacitor box. It is because the mass of the aluminum energy storage supercapacitor box is smaller. Under the same acceleration, the reacted external load is smaller. Then the stress level is smaller. There are significant amplitude points at nodes 23,050 at 125 Hz and 150 Hz, node 38 at 125 Hz, and node 704 at around 135 Hz, which correspond to the numerical results of the 3rd, 6th, 3rd, and 4th modes in Table 9, respectively.

Fig. 14
figure 14

Fatigue simulation of 6063-T5 under random vibration.

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The fatigue simulation result of 6063-T5 under random vibration is shown in Fig. 14. The maximum fatigue damage is 1.47 × 10− 4, with a fatigue cycle life of about 104 times. The fatigue life of the supercapacitor box of 6063-T5 could meet the requirements for low stress high cycle life of the urban rail vehicle components.

Conclusion

The fatigue life of an energy storage supercapacitor box applied to urban rail vehicle is studied in this paper. A finite element model of the supercapacitor box is established. The first 10 modes of the supercapacitor box structure is calculated to assess resonance phenomenon. The fatigue characteristics within the frequency domain under random vibrations defined by ASDs is also analyzed. Finally, the energy storage supercapacitor box is manufactured using lightweight aluminum alloy. And the fatigue damage of the aluminum alloy supercapacitor box is analyzed. The main conclusions of this paper are as follows:

(1) The frequencies in constrained state of the energy storage supercapacitor box are all greater than 30 Hz. It will not experience resonance during the normal operation of the urban rail vehicle.

(2) The stress generated by the unit gravitational acceleration load along X axis is bigger than Y and Z axes with a maximum stress reaching 181.6 MPa, occurring at the contact point between the mounting bracket of the supercapacitor module and the bottom bracket of the supercapacitor box.

(3) The maximum fatigue damage of the energy storage supercapacitor box is 6.24 × 10− 6 under random vibration loading along X, Y and Z axis. And the number of fatigue cycles is on an order of 105. The fatigue life of the supercapacitor box structure could meet the requirements for low stress high cycle life of urban rail vehicle components.

(4) The maximum fatigue damage of the aluminum energy storage supercapacitor box is 1.47 × 10− 4, with a fatigue cycle life of about 104 times. The fatigue life of the supercapacitor box of 6063-T5 could meet the requirements for low stress high cycle life of the urban rail vehicle components.