Under the global push for carbon neutrality, the adoption of electric vehicles has surged, placing unprecedented emphasis on the safety and durability of EV battery packs. As the core energy storage component, the structural integrity of the EV battery pack directly impacts vehicle safety and performance over its entire lifecycle. The prevalent assembly method—from individual cells to modules, to the casing, and finally the complete pack—demands that the battery pack structure withstands complex dynamic loads encountered during operation. This article, from my engineering perspective, delves into the application of random vibration analysis using the modal superposition method to assess and enhance the structural strength of an EV battery pack, adhering to the vibration test specifications outlined in standard GB38031-2020.
Random vibration refers to oscillatory motion where the amplitude and frequency are uncertain and vary randomly over time. For an EV battery pack mounted on a vehicle, the excitation from road irregularities throughout its service life is precisely such a non-deterministic, chaotic load. Unlike deterministic vibrations, random vibrations cannot be described by precise mathematical functions and must be analyzed using statistical methods. In engineering practice, the frequency-domain spectral analysis method is widely employed for structural random vibration analysis. This approach derives the probability density function of stress amplitudes from the stress Power Spectral Density (PSD), forming the basis for random vibration fatigue assessment.
The Power Spectral Density is a statistical measure of the mean-square value of a random variable. In the context of random vibration, it represents the distribution of power (or energy) per unit frequency. For a stationary ergodic random process \( x(t) \), the one-sided PSD, \( S_{xx}(f) \), is defined as the Fourier transform of its autocorrelation function \( R_{xx}(\tau) \):
$$ S_{xx}(f) = 2 \int_{0}^{\infty} R_{xx}(\tau) \cos(2\pi f \tau) d\tau \quad \text{for} \quad f \geq 0 $$
where \( R_{xx}(\tau) = E[x(t) \cdot x(t+\tau)] \) and \( E[\cdot] \) denotes the expectation operator. For acceleration input, the PSD has units of \( g^2/Hz \). The mean square value \( \sigma_x^2 \) of the response is obtained by integrating the PSD over all frequencies:
$$ \sigma_x^2 = \int_{0}^{\infty} S_{xx}(f) df $$
The standard deviation \( \sigma_x \) is the root-mean-square (RMS) value. In structural analysis, the \( n\sigma \) (e.g., \( 3\sigma, 9\sigma \)) rule is often used for peak stress estimation, where the probability of exceeding \( n\sigma \) is very low for Gaussian processes.
The equation of motion for a multi-degree-of-freedom model of an EV battery pack under base excitation is:
$$ [M]\{\ddot{u}\} + [C]\{\dot{u}\} + [K]\{u\} = -[M]\{r\}\ddot{u}_g(t) $$
Here, \( [M] \), \( [C] \), and \( [K] \) are the global mass, damping, and stiffness matrices, respectively. \( \{u\} \) is the vector of nodal displacements relative to the base, \( \{r\} \) is the influence vector mapping the base acceleration \( \ddot{u}_g(t) \) to nodal forces, and \( t \) is time. Using the modal superposition method, the physical response is expressed as a linear combination of modal coordinates \( \{q\} \):
$$ \{u\} = [\Phi]\{q\} $$
where \( [\Phi] \) is the matrix of mass-normalized mode shapes. For proportionally damped systems, the equations decouple into a set of single-degree-of-freedom equations in modal space:
$$ \ddot{q}_i + 2\zeta_i \omega_i \dot{q}_i + \omega_i^2 q_i = -\gamma_i \ddot{u}_g(t), \quad i=1,2,…,N_m $$
In this equation, \( \omega_i \) is the natural frequency of mode \( i \), \( \zeta_i \) is the modal damping ratio, and \( \gamma_i = \{\phi_i\}^T[M]\{r\} \) is the modal participation factor for mode \( i \). The PSD of the modal response \( S_{q_i q_i}(f) \) is related to the input acceleration PSD \( S_{aa}(f) \) by:
$$ S_{q_i q_i}(f) = |H_i(f)|^2 \gamma_i^2 S_{aa}(f) $$
where \( H_i(f) = 1 / (\omega_i^2 – (2\pi f)^2 + j \cdot 2 \zeta_i \omega_i (2\pi f)) \) is the frequency response function. The PSD of physical stress \( \sigma \) at a location can then be computed by summing contributions from all significant modes:
$$ S_{\sigma\sigma}(f) = \sum_{i=1}^{N_m} \sum_{j=1}^{N_m} \{\psi_\sigma\}_i \{\psi_\sigma\}_j S_{q_i q_j}(f) $$
Here, \( \{\psi_\sigma\}_i \) is the stress transformation vector for mode \( i \), and \( S_{q_i q_j}(f) \) is the cross-PSD between modal coordinates, which for uncorrelated inputs simplifies to the sum of individual modal PSDs. The root-mean-square (RMS) stress is finally:
$$ \sigma_{RMS} = \sqrt{\int_{0}^{\infty} S_{\sigma\sigma}(f) df} $$
The finite element model of the EV battery pack was developed to perform this analysis efficiently. Key components included the lower casing (often made of aluminum or steel), the upper cover, simplified equivalent modules representing the mass and stiffness of the cell assemblies, module metal sheets (brackets), and module tie rods. To conserve computational resources while maintaining accuracy, non-critical details were simplified. The mesh primarily consisted of CTRIA3 and CQUAD4 shell elements for thin-walled structures and HEX8 solid elements for thicker parts, with a global element size of 2 mm. The simplified module models were assigned appropriate mass and stiffness properties to represent the dynamic behavior of the actual battery cells and their interconnections. Weld lines were modeled using shell elements sharing nodes with parent components, with material properties reduced to 80% of the base material to simulate the heat-affected zone. Bolt connections were modeled using rigid beam elements (RBE2) or multipoint constraints to simulate fastened joints accurately. Material properties used in the analysis are summarized in the table below.
| Component | Material | Young’s Modulus, E (GPa) | Poisson’s Ratio, ν | Density, ρ (kg/m³) | Yield Strength, σ_y (MPa) |
|---|---|---|---|---|---|
| Lower Casing | Aluminum Alloy | 70 | 0.33 | 2700 | 250 |
| Upper Cover | Aluminum Alloy | 70 | 0.33 | 2700 | 250 |
| Module Metal Sheets | Steel | 210 | 0.30 | 7850 | 350 |
| Module Tie Rods | Steel | 210 | 0.30 | 7850 | 500 |
| Equivalent Module Core | Epoxy Composite | 10 | 0.35 | 1800 | 80 |
The analysis sequence comprised two steps: a modal analysis step followed by a random vibration response step. The modal analysis was performed using the Lanczos method with the EV battery pack’s installation points fully constrained to simulate a fixed boundary condition on the vehicle chassis. The first 50 modes were extracted to ensure adequate modal participation in the frequency range of interest. The subsequent random vibration analysis step applied the acceleration Power Spectral Density (PSD) specified in GB38031-2020 at the same installation points. The vehicle coordinate system was defined with X as the longitudinal direction (front-rear), Y as the lateral direction (left-right), and Z as the vertical direction (perpendicular to the road). The PSD profiles for the three orthogonal directions are detailed below.
| Frequency, f (Hz) | Z-axis PSD (g²/Hz) | Y-axis PSD (g²/Hz) | X-axis PSD (g²/Hz) |
|---|---|---|---|
| 5 | 0.0150 | 0.00200 | 0.00600 |
| 10 | – | 0.00500 | – |
| 15 | 0.0150 | – | – |
| 20 | – | 0.00500 | – |
| 30 | – | – | 0.00600 |
| 65 | 0.0010 | – | – |
| 100 | 0.0010 | – | – |
| 200 | 0.0001 | 0.00015 | 0.00003 |
The modal analysis results for the initial EV battery pack design revealed the first 10 natural frequencies, as shown in the following table. The mode shapes indicated that the first six modes were local deformations of the casing and covers, while the seventh mode, at 29.6 Hz, was the first global bending mode of the battery module assembly itself. This frequency is critical because road-induced excitations often have significant energy content in the 20-30 Hz range, posing a risk of resonance and high dynamic stress.
| Mode Number, i | Natural Frequency, f_i (Hz) | Description (Based on Mode Shape) | Modal Participation Factor (Z-direction), γ_i |
|---|---|---|---|
| 1 | 19.8 | Local torsion of lower casing | 0.12 |
| 2 | 20.7 | Local bending of upper cover | 0.08 |
| 3 | 21.5 | Combined local plate bending | 0.15 |
| 4 | 24.1 | Lateral deformation of side rails | 0.05 |
| 5 | 26.9 | Complex casing distortion | 0.10 |
| 6 | 28.8 | High-order local mode | 0.03 |
| 7 | 29.6 | First global bending of modules | 1.45 |
| 8 | 32.1 | Second global module bending | 0.92 |
| 9 | 32.3 | Casing rocking mode | 0.21 |
| 10 | 33.1 | Combined module/casing deformation | 0.67 |
Random vibration analysis was first conducted for the Z-direction, as vertical road inputs typically induce the most severe loading on an EV battery pack. A modal damping ratio of ζ=0.03 (3%) was assumed for all modes, based on typical values for welded metal structures. The stress response PSD was computed, and the 1σ (RMS) and 3σ stress values were obtained. However, for high-reliability applications like an EV battery pack, a 9σ peak stress estimate is often used for design assessment, corresponding to an extremely low probability of exceedance. The von Mises stress was the primary metric. The initial analysis showed that under the Z-direction PSD, the maximum 9σ von Mises stress in the lower casing was 130 MPa, and in the module metal sheets (brackets) it was 118 MPa. Given the yield strength of the materials (250 MPa for aluminum casing, 350 MPa for steel sheets), these stresses, while below yield, represented high utilization factors and posed a potential risk for fatigue failure over the target lifetime, which often corresponds to millions of stress cycles. The stress concentration was primarily at the junctions between the module tie rods and the casing, and at the edges of the metal sheet brackets.
The root cause was identified through the modal analysis: the first global mode of the modules at 29.6 Hz was too low, risking resonance with common road excitation frequencies. The module assembly’s stiffness was insufficient because the primary connection between individual modules and the lower casing was through several tie rods positioned near the top of the modules. This configuration left the bottom section of the modules with relatively low constraint, allowing bending deformation. The optimization goal was to shift this fundamental frequency significantly higher, preferably above 35 Hz for passenger vehicles, to avoid the high-energy low-frequency band.
The proposed optimization involved enhancing the stiffness of the module assembly by adding a continuous layer of epoxy resin board (0.5 mm thick) bonded to the bottom surface of all modules within the EV battery pack. Epoxy resin offers strong adhesion and low shrinkage, creating a unified, stiff base that effectively couples the modules together and to the lower casing. This design change increases the system’s overall bending stiffness. The finite element model was updated by adding shell elements with the material properties of epoxy composite (E=10 GPa, ν=0.35, ρ=1800 kg/m³) connected to the module nodes via rigid links simulating perfect adhesion. The modified EV battery pack model was then subjected to the same modal and random vibration analysis sequence.
The results confirmed the effectiveness of the optimization. The first global modal frequency of the module assembly increased from 29.6 Hz to 55 Hz, a significant improvement that places it well outside the critical excitation range. The table below compares key parameters before and after the optimization for the EV battery pack.
| Parameter | Initial Design | Optimized Design | Improvement / Change |
|---|---|---|---|
| Module 1st Bending Frequency | 29.6 Hz | 55.0 Hz | +85.8% |
| Lower Casing Max 9σ von Mises Stress (Z-dir) | 130 MPa | 49 MPa | -62.3% |
| Module Metal Sheets Max 9σ von Mises Stress (Z-dir) | 118 MPa | 23 MPa | -80.5% |
| RMS Acceleration Response at a Key Point (Z-dir) | 4.2 g | 2.1 g | -50.0% |
| Stress Utilization Factor (Casing, σ_9σ / σ_yield) | 0.52 | 0.20 | Reduced by 60% |
The drastic reduction in dynamic stress can be explained analytically. The response of a single-degree-of-freedom system to random vibration is proportional to the integral of the product of the PSD input and the square of the modulus of the Frequency Response Function (FRF). For a system with natural frequency \( f_n \) much higher than the dominant frequencies in the input PSD, the FRF modulus \( |H(f)| \) is small. Quantitatively, the mean square stress can be approximated for a dominant mode as:
$$ \sigma_{RMS}^2 \approx \psi_\sigma^2 \gamma^2 \int_{0}^{\infty} |H(f)|^2 S_{aa}(f) df $$
where \( |H(f)| = 1/\sqrt{(1-(f/f_n)^2)^2 + (2\zeta f/f_n)^2} \). By increasing \( f_n \) from 29.6 Hz to 55 Hz, the value of \( |H(f)|^2 \) within the frequency band where \( S_{aa}(f) \) is significant (primarily below 30 Hz) is greatly reduced, leading to a lower \( \sigma_{RMS} \). Furthermore, the optimized design redistributes the load path, reducing stress concentrations. The optimized EV battery pack structure now comfortably meets the 9σ stress criterion, with all peak stresses well below the material yield strengths, ensuring a high safety margin for both static overload and long-term fatigue.
To generalize the approach, the process for assessing and optimizing an EV battery pack against random vibration can be summarized in a step-by-step formulaic procedure:
1. Model Development: Create a finite element model of the EV battery pack with appropriate simplifications. Define materials, connections, and constraints.
$$ \text{Model} = f(\text{Geometry}, \text{Materials}, \text{Connections}, \text{Mesh}) $$
2. Modal Extraction: Perform eigenvalue analysis to obtain natural frequencies \( \omega_i \) and mode shapes \( \{\phi_i\} \).
$$ ([K] – \omega_i^2 [M]) \{\phi_i\} = 0 $$
3. PSD Input Definition: Define the acceleration PSD profile \( S_{aa}(f) \) based on relevant standards (e.g., GB38031-2020) for each directional axis.
4. Random Vibration Analysis: Compute the response PSD for stress \( S_{\sigma\sigma}(f) \) and integrate to find RMS and peak (\( n\sigma \)) values.
$$ \sigma_{n\sigma} = n \cdot \sqrt{\int_{0}^{f_{max}} S_{\sigma\sigma}(f) df} $$
5. Assessment: Compare peak stresses with allowable limits (yield strength divided by a safety factor, or fatigue strength).
$$ \text{Pass if: } \sigma_{n\sigma} \leq \frac{\sigma_{allowable}}{SF} $$
6. Optimization: If assessment fails, identify weak modes (low frequency, high participation) and modify stiffness/mass distribution. Re-evaluate.
$$ \text{Optimize: } \max(f_{critical}), \min(\sigma_{peak}) \text{ subject to mass/space constraints} $$
In conclusion, the structural integrity of an EV battery pack under dynamic road loads is paramount. Through detailed finite element modeling and random vibration analysis based on the modal superposition method, potential weaknesses in the initial design of an EV battery pack were identified. The low first bending frequency of the module assembly was the primary contributor to high dynamic stresses. The implementation of a relatively simple yet effective design change—adding an epoxy resin board to stiffen the module base—resulted in a dramatic increase in modal frequency and a corresponding significant reduction in stress responses. This optimization ensures that the EV battery pack meets stringent safety standards and possesses enhanced durability for its service life. This analytical approach provides a robust and efficient framework for the design validation and iterative improvement of EV battery pack structures in the virtual domain, reducing reliance on physical prototyping and testing while accelerating development cycles for safer electric vehicles.