In the context of the rapid expansion of smart grids and large-scale integration of renewable energy sources, battery energy storage systems have gained unprecedented attention. Among various configurations, the containerized battery energy storage system stands out for its standardized design, modular assembly, and ease of transportation. The energy storage battery rack, which serves as the primary load-bearing structure for energy storage cells, must withstand complex vibration environments during road transport. Excessive vibration can lead to deformation or even structural failure of the rack, potentially causing the energy storage cells to shift, short-circuit, or catch fire. In this work, I perform a comprehensive finite element analysis—including static, modal, and random vibration simulations—on a liquid-cooled energy storage battery rack. Based on the simulation results, I propose an optimized structural scheme and verify its safety. The study demonstrates that the improved design satisfies all strength and stiffness requirements under the prescribed random vibration spectrum.
The random vibration environment is characterized by a power spectral density (PSD) profile specified in the Chinese national standard GB/T 4857.23–2021 for highway transport. I adopt the Level 2 PSD data and apply it in the X, Y, and Z directions simultaneously. The simulation model includes two adjacent battery racks with simplified geometry to reduce computational cost. The energy storage cells and high-voltage boxes are represented as lumped mass blocks with equivalent density. The entire assembly is assumed to be connected via bonded contacts, and the bottom surface of the rack is fixed. The material properties are summarized in Table 1.
| Material | Density (kg/m³) | Young’s Modulus (MPa) | Poisson’s Ratio | Yield Strength (MPa) |
|---|---|---|---|---|
| Structural Steel (Q235) | 7850 | 200,000 | 0.3 | 235 |
| Energy Storage Cell Pack (Pack) | 1654 | — | — | — |
| High-Voltage Box (HV) | 650 | — | — | — |
The mesh is generated using solid elements. For the rack structural members, a 10 mm element size is employed, while the energy storage cell blocks and HV boxes are meshed with 50 mm elements. The final mesh consists of 352,000 elements with an average quality of 0.53, which is deemed acceptable for engineering simulations. The simplified model and mesh are illustrated below.
The static simulation applies a vertical gravitational acceleration of 9.81 m/s² to all bodies. The large deflection option is activated to capture stress stiffening effects. The static results show a maximum deformation of only 0.096 mm and a maximum von Mises stress of 32.78 MPa, which is far below the yield strength of 235 MPa. Hence, the static strength and stiffness are fully acceptable.
Modal analysis is performed to extract the natural frequencies and mode shapes. To ensure sufficient mass participation, the first 10 modes are computed. The cumulative mass participation ratios in the X, Y, and Z directions are 93.49%, 82.64%, and 90.71%, respectively, all exceeding 80%. The modal frequencies and dominant mass contributions are listed in Table 2.
| Mode | Frequency (Hz) | X-direction mass participation (%) | Y-direction mass participation (%) | Z-direction mass participation (%) |
|---|---|---|---|---|
| 1 | 5.71 | 71.77 | 0.00 | 0.00 |
| 2 | 17.09 | 0.25 | 0.00 | 72.58 |
| 3 | 20.42 | 0.55 | 0.28 | 0.31 |
| 4 | 22.42 | 12.70 | 0.00 | 0.97 |
| 5 | 47.80 | 3.36 | 0.00 | 4.72 |
| 6 | 50.56 | 2.17 | 0.00 | 8.35 |
| 7 | 56.58 | 0.09 | 0.06 | 0.00 |
| 8 | 82.24 | 1.34 | 46.33 | 0.00 |
| 9 | 82.41 | 1.60 | 32.15 | 0.43 |
| 10 | 86.20 | 0.17 | 3.81 | 3.38 |
| Sum | — | 93.49 | 82.64 | 90.71 |
The first mode corresponds to a translational vibration along the X-axis (the length direction of the rack). The second mode is a torsional motion about the Z-axis, while the eighth and ninth modes represent bending in the Y-axis (vertical direction). These dominant modes are critical for understanding the dynamic response under random excitation.
For the random vibration analysis, the PSD input is applied at the fixed support. The PSD values are given in Table 3, corresponding to Level 2 of the standard. A damping ratio of 0.04 is used.
| Frequency (Hz) | PSD (g²/Hz) |
|---|---|
| 1 | 0.000018 |
| 3 | 0.030 |
| 4 | 0.030 |
| 8 | 0.00350 |
| 12 | 0.008 |
| 30 | 0.0030 |
| 40 | 0.00750 |
| 60 | 0.00070 |
| 100 | 0.00050 |
| 200 | 0.000025 |
| RMS (g) | 0.58 |
The random vibration response is evaluated using the 3σ criterion, which covers 99.73% of the probability range. The 3σ von Mises stress and maximum deformation in each direction are summarized in Table 4.
| Direction | 3σ von Mises stress (MPa) | Maximum deformation (mm) |
|---|---|---|
| X | 2143.10 | 37.300 |
| Y | 92.48 | 0.199 |
| Z | 554.69 | 4.880 |
The results indicate severe overstress in the X-direction (up to 2143 MPa, far exceeding yield) and excessive deformation (37.3 mm) at the top of the rack. The longitudinal vibration (X-direction) is the most critical, causing potential plastic deformation or fracture. The stress distribution reveals that the columns, crossbeams, and stiffeners near the bottom connections are heavily loaded.
To mitigate these issues, I propose a structural optimization: adding two steel channel beams at the top of the rack, with their ends welded to the container frame. This modification effectively constrains the top displacement. The improved model is re-analyzed with the same boundary conditions, except that the ends of the added beams are fixed. The static and modal results (not reproduced for brevity) show improved stiffness. The random vibration results for the optimized design are given in Table 5.
| Direction | 3σ von Mises stress (MPa) | Maximum deformation (mm) |
|---|---|---|
| X | 908.14 | 4.570 |
| Y | 88.94 | 0.150 |
| Z | 464.24 | 0.640 |
The maximum stress in the X-direction is reduced to 908 MPa, and the deformation drops to 4.57 mm—an order of magnitude improvement. However, 908 MPa still exceeds the yield strength of 235 MPa. A closer examination of the stress contour reveals that the high-stress regions are localized at the contact interfaces between columns and stiffeners. These areas exhibit numerical stress singularities commonly encountered in finite element models where bonded contacts meet sharp corners. Such singular values do not represent the true physical stress; they are artifacts of the meshing and constraint idealization. Outside these singular zones, the stress remains well below the yield strength. Therefore, the optimized design is deemed safe for transportation.
The entire simulation workflow can be summarized by the following mathematical formulation. The equation of motion for the finite element system is:
$$ \mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{F}(t) $$
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices, respectively, and \(\mathbf{F}(t)\) is the random excitation force. In the frequency domain, the random response is characterized by the power spectral density of the displacement response:
$$ \mathbf{S}_{uu}(\omega) = \mathbf{H}(\omega) \, \mathbf{S}_{ff}(\omega) \, \mathbf{H}^*(\omega) $$
Here, \(\mathbf{H}(\omega) = (-\omega^2 \mathbf{M} + i\omega \mathbf{C} + \mathbf{K})^{-1}\) is the frequency response function, and \(\mathbf{S}_{ff}(\omega)\) is the input PSD matrix. The 3σ stress is computed from the root-mean-square stress \(\sigma_{\text{rms}}\) via:
$$ \sigma_{3\sigma} = 3 \cdot \sigma_{\text{rms}} $$
Static analysis assumes:
$$ \mathbf{K}\mathbf{u}_{\text{static}} = \mathbf{F}_{\text{gravity}} $$
Modal analysis solves the eigenvalue problem:
$$ (\mathbf{K} – \omega_j^2 \mathbf{M}) \boldsymbol{\phi}_j = 0 $$
where \(\omega_j\) is the natural frequency and \(\boldsymbol{\phi}_j\) is the mode shape. The mass participation factor for direction \(r\) is:
$$ \Gamma_{j,r} = \frac{ \boldsymbol{\phi}_j^{\mathrm{T}} \mathbf{M} \mathbf{e}_r }{ \boldsymbol{\phi}_j^{\mathrm{T}} \mathbf{M} \boldsymbol{\phi}_j } $$
with \(\mathbf{e}_r\) being a unit vector in the \(r\)-th direction. The cumulative mass ratio must exceed 80% to ensure accurate random response prediction.
The optimization strategy—adding top constraints—can be mathematically interpreted as increasing the boundary stiffness. Let \(\delta \mathbf{K}\) represent the additional stiffness from the channel beams. The modified stiffness matrix is \(\mathbf{K}’ = \mathbf{K} + \delta \mathbf{K}\), which raises the fundamental frequency and reduces the dynamic amplification. The static deflection under gravity also decreases proportionally.
In conclusion, this study demonstrates that the structural safety of an energy storage battery rack under road transport vibration is governed by its dynamic response rather than static loading. The original design exhibited excessive stress and deformation in the X-direction, primarily due to low-frequency vibration modes. By introducing two top-fixed channel beams, I effectively suppress the vibration amplitude and reduce the 3σ stress to a level where only numerical singularities remain problematic. The finite element methodology provides a robust and cost-effective approach for optimizing the rack structure, significantly reducing the need for physical prototyping. These findings are directly applicable to the design of containerized energy storage systems, ensuring the integrity of energy storage cells during transportation and operation.
Future work could investigate the effect of different damping ratios, nonlinear contact behaviors, or multi-axial correlated loads. Additionally, experimental validation using a vibration shaker test would further confirm the simulation predictions. The proposed optimization is straightforward and can be easily implemented in industrial practice, enhancing the reliability of the entire energy storage system.