The development of the automotive industry is intrinsically linked to two paramount global challenges: environmental protection and energy security. In this context, the battery electric vehicle (EV) has emerged as a pivotal solution, addressing the dual issues of petroleum resource dependency associated with internal combustion engine vehicles and their contribution to environmental pollution. Consequently, there has been a significant global push, particularly in rapidly developing nations like China, to accelerate the adoption and innovation of battery EV cars. The dynamic characteristics of a battery EV car, however, differ markedly from its traditional counterparts due to distinct operational excitation sources. The vehicle body structure is a critical component, directly influencing safety, ride comfort, and energy economy. Therefore, a primary objective in contemporary automotive design is to achieve body lightweighting while rigorously ensuring structural integrity, safety, and superior NVH (Noise, Vibration, and Harshness) performance. This paper presents a comprehensive methodology integrating experimental modal analysis with finite element (FE) simulation to evaluate and subsequently optimize the body-in-white (BIW) structure of a battery EV car.
The optimization of battery EV car body structures has been the focus of considerable research. Predominantly, studies rely on finite element analysis (FEA) to assess modal parameters, stiffness, and subsequently guide design changes. For instance, topology and size optimization techniques are frequently employed with mass minimization as the objective, constrained by performance metrics like natural frequency and stiffness. While these simulation-based approaches are powerful, validation through physical testing is essential for ensuring model fidelity. Some studies have performed experimental modal testing on battery EV car prototypes or similar structures to correlate with FE models, but a fully integrated process from test-based model validation to subsequent structural optimization is less commonly detailed. This work aims to bridge that gap by establishing a reliable FE model through experimental correlation and then using that validated model as the basis for a systematic structural optimization of the battery EV car body.
1. Modal Parameter Identification via Experimental Testing
Accurate identification of the inherent dynamic characteristics is the first step in evaluating the battery EV car body. Experimental modal analysis was conducted to extract the natural frequencies, damping ratios, and mode shapes of the BIW in a free-free boundary condition.
1.1 Experimental Setup and Procedure
A dedicated vibration testing platform was established for the battery EV car body. The core of the data acquisition and analysis system was LMS Test.Lab 14A software coupled with a 24-channel data acquisition front-end. To approximate a free-state condition, the battery EV car body was suspended using four soft springs attached to its underside longitudinal members, ensuring the suspension frequency was significantly lower than the body’s fundamental frequencies.
Excitation was provided by two electrodynamic shakers. A burst random signal generated by the LMS software was fed through charge amplifiers to the shakers. The shakers were positioned to apply excitation at the front and rear longitudinal rails of the battery EV car body, as this effectively excites global bending and torsional modes. The response was measured using six tri-axial PCB 356A16 accelerometers. A total of 203 measurement points were defined on the battery EV car body shell, forming a comprehensive wireframe model. Due to hardware limitations, the accelerometers were moved in 34 setup positions, with 10 averages taken per setup to enhance measurement accuracy. The analysis frequency was set to 128 Hz with a resolution of 0.0625 Hz, adequately capturing the lower-order global modes of the battery EV car body structure which typically reside below 100 Hz.
1.2 Parameter Identification Using the PolyMax Method
The modal parameters were identified from the measured Frequency Response Functions (FRFs) using the PolyMax (Least-Squares Complex Frequency-Domain) method. This method is favored for its clear stabilization diagrams and robustness against generating spurious modes. The fundamental dynamics of the battery EV car body structure, excluding external forces, are governed by the equation of motion:
$$ \mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} + \mathbf{Kx} = \mathbf{0} $$
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices, respectively, and \(\mathbf{x}\) is the displacement vector. The corresponding eigenvalue problem is:
$$ \left( \mathbf{K} + \lambda\mathbf{C} – \lambda^2\mathbf{M} \right) \mathbf{v} = \mathbf{0} $$
where \(\lambda\) are the complex eigenvalues and \(\mathbf{v}\) the eigenvectors. The displacement FRF matrix \(\mathbf{H}(\omega)\) can be expressed in pole-residue form:
$$ \mathbf{H}(\omega) = \sum_{r=1}^{N_m} \left( \frac{\mathbf{v}_r \mathbf{l}_r^T}{j\omega – \lambda_r} + \frac{\mathbf{v}_r^* \mathbf{l}_r^H}{j\omega – \lambda_r^*} \right) + \frac{\mathbf{U}}{\omega^2} + \mathbf{L} $$
In this equation, \(N_m\) is the number of modes, \(\lambda_r\) is the pole for mode \(r\), \(\mathbf{v}_r\) is the mode shape vector, \(\mathbf{l}_r\) is the modal participation vector, and \(\mathbf{U}\) and \(\mathbf{L}\) are the upper and lower residual terms, respectively. The superscripts \(*\) and \(H\) denote complex conjugate and Hermitian transpose. The PolyMax algorithm fits this model to the measured FRF data using a least-squares approach, yielding estimates for the poles \(\lambda_r = -\zeta_r \omega_r + j\omega_r\sqrt{1-\zeta_r^2}\), the undamped natural frequencies \(\omega_r\), the damping ratios \(\zeta_r\), and the mode shapes \(\mathbf{v}_r\).
Analysis of the summed FRFs and stabilization diagrams allowed for clear identification of the first three global modes of the battery EV car body. The experimental results are summarized in the table below.
| Mode Order | Natural Frequency (Hz) | Damping Ratio (%) |
|---|---|---|
| 1 | 37.6 | 0.59 |
| 2 | 44.4 | 0.68 |
| 3 | 55.9 | 0.85 |
2. Finite Element Modeling and Simulation
2.1 Development of the Finite Element Model
A high-fidelity finite element model of the battery EV car body-in-white was developed to enable detailed analysis and optimization. The process began with a detailed 3D geometric model comprising 302 sheet metal parts, including the roof, floor pan, front and rear compartments, and left/right side frames. This CAD geometry was imported into a pre-processing software (e.g., HyperMesh) for mid-surface extraction and meshing. A balance between accuracy and computational efficiency was struck by using predominantly shell elements with a mesh size ranging from 4 mm to 8 mm, resulting in a model with 823,212 elements.
The connections between parts, critical for representing the structural integrity of the battery EV car body, were modeled using different techniques appropriate for joint types: Rigid Body Elements (RBE) for spot-weld clusters, ACM (Area Contact Method) spring elements for individual spot welds, and adhesive elements for bonded areas. A total of 10,230 connection elements were defined. The material properties assigned were typical for automotive steel: Young’s Modulus \(E = 210\) GPa, Poisson’s ratio \(\nu = 0.3\), and density \(\rho = 7850\) kg/m³.
2.2 Modal and Stiffness Analysis
The established FE model was first used to perform a normal modes analysis. The Lanczos method was employed to extract the first three natural frequencies and corresponding mode shapes. The results from this computational modal analysis are presented in the following section for comparison.
Furthermore, the torsional stiffness of the battery EV car body, a key indicator of its structural rigidity and handling performance, was calculated. A standard torsion test was simulated by constraining the rear suspension mounting points and applying two equal and opposite vertical forces (\(F = 2000\) N) at the front longitudinal rails. The relative twist angle \(\theta\) was calculated from the vertical displacements \(Z_L\) and \(Z_R\) at the application points, separated by distance \(L\):
$$ \theta = \arctan\left( \frac{Z_L – Z_R}{L} \right) $$
The torsional stiffness \(K_{torsion}\) is then given by:
$$ K_{torsion} = \frac{T}{\theta} = \frac{F \cdot D}{\theta} $$
where \(T\) is the applied torque and \(D\) is the wheelbase. The FE simulation yielded a torsional stiffness value of 23,275 N·m/deg for the baseline battery EV car body design.
3. Model Correlation and Validation
The accuracy of the finite element model is paramount for conducting reliable optimization. A direct comparison between the experimental modal analysis (EMA) and the computational modal analysis (CMA) was performed. The results for the first three natural frequencies and damping ratios are compared in the table below.
| Parameter | EMA | CMA | Error |
|---|---|---|---|
| 1st Freq. (Hz) | 37.6 | 39.2 | +4.3% |
| 1st Damp. (%) | 0.59 | 0.69 | +16.9% |
| 2nd Freq. (Hz) | 44.4 | 49.9 | +12.4% |
| 2nd Damp. (%) | 0.68 | 0.76 | +11.8% |
| 3rd Freq. (Hz) | 55.9 | 59.0 | +5.5% |
| 3rd Damp. (%) | 0.85 | 0.72 | -15.3% |
The frequency errors are within an acceptable range for complex structures like a battery EV car body, with the largest discrepancy being 12.4% for the second mode. Damping, which is challenging to model accurately in FE, showed higher relative errors but is often considered less critical for stiffness-dominated structural optimization. More importantly, the visual comparison of the experimental and computational mode shapes showed excellent agreement in deformation patterns:
- Mode 1 (~38 Hz): Predominantly involved torsion of the rear section of the battery EV car body.
- Mode 2 (~45-50 Hz): Characterized by front-end vertical bending and rear upper section stretching/curling.
- Mode 3 (~56-59 Hz): Featured lateral swaying motion of the front section of the battery EV car body.
This strong correlation in both frequency and mode shape validated the FE model as a sufficiently accurate representation of the physical battery EV car body’s dynamic behavior, making it suitable for subsequent optimization studies.
4. Structural Optimization of the Battery EV Car Body
With a validated model in hand, a structural optimization was undertaken with the primary goal of lightweighting the battery EV car body while maintaining its dynamic performance.
4.1 Optimization Problem Formulation
The optimization was formulated as a size optimization problem, where the thicknesses of key sheet metal panels were chosen as the design variables. The objective was to minimize the total mass of the battery EV car body. Constraints were applied to ensure the vehicle’s NVH and structural performance did not degrade beyond acceptable limits.
Design Variables: A set of ‘n’ thickness variables, \(d_1, d_2, …, d_n\), representing major panels (floor sections, side rails, roof components, etc.). Each variable had a lower bound \(d_{min}\) (e.g., 0.5 mm for manufacturability) and an upper bound \(d_{max}\) (often the original thickness).
Objective Function: Minimize the total mass \(M\) of the battery EV car body:
$$ \text{Minimize: } M(\mathbf{d}) = \sum_{i=1}^{N_{parts}} \rho A_i(d_i) t_i(d_i) $$
where \(\mathbf{d}\) is the vector of thickness variables, \(\rho\) is density, \(A_i\) is the area, and \(t_i\) is the thickness of part \(i\).
Constraints:
- Modal Frequency Constraints: The first three natural frequencies (\(f_1, f_2, f_3\)) of the optimized battery EV car body must remain within a specified range to avoid resonance with major excitation sources (road inputs ~20 Hz, e-motor inputs >150 Hz):
$$ 30 \text{ Hz} \leq f_j(\mathbf{d}) \leq 100 \text{ Hz}, \quad j=1,2,3 $$ - Torsional Stiffness Constraint: The torsional stiffness \(K_{torsion}(\mathbf{d})\) must not fall below a minimum acceptable threshold to ensure structural integrity and handling:
$$ K_{torsion}(\mathbf{d}) \geq 20,000 \text{ N·m/deg} $$
In summary, the mathematical optimization problem is:
$$ \begin{aligned}
& \underset{\mathbf{d}}{\text{minimize}}
& & M(\mathbf{d}) \\
& \text{subject to}
& & g_1(\mathbf{d}): 30 – f_j(\mathbf{d}) \leq 0,\\
& & & g_2(\mathbf{d}): f_j(\mathbf{d}) – 100 \leq 0, \quad j=1,2,3\\
& & & g_3(\mathbf{d}): 20,000 – K_{torsion}(\mathbf{d}) \leq 0\\
& & & d_{min} \leq d_i \leq d_{max}, \quad i=1,…,n
\end{aligned} $$
4.2 Optimization Process and Results
The optimization was performed using the validated FE model within an optimization solver. The process involved an iterative loop: modifying thickness variables, running FE analyses (modal and static torsion), evaluating constraints and objective, and updating the design until convergence. A gradient-based optimization algorithm was typically used for such problems.
The optimization yielded a new set of thickness values for the battery EV car body panels. A subset of the design variable changes is illustrated in the table below.
| Variable (Panel Location) | Initial Thickness (mm) | Optimized Thickness (mm) | Change |
|---|---|---|---|
| d₁ (Underbody Frame) | 1.0 | 1.00 | No change |
| d₂ (Underbody Cross-member) | 1.0 | 0.70 | -30.0% |
| d₃ (Underbody Rail) | 1.0 | 0.79 | -21.0% |
| d₄ (Right Side Frame) | 1.0 | 0.75 | -25.4% |
| d₅ (Right Side Inner) | 1.0 | 0.68 | -32.3% |
| d₂₂ (Left Side Pillar) | 0.6 | 0.60 | No change |
| d₄₆ (Roof Panel) | 0.6 | 0.59 | -1.7% |
The most significant outcome was the reduction in the total mass of the battery EV car body. The mass decreased from 456.6 kg for the baseline design to 431.3 kg for the optimized design. This represents a mass saving of 25.3 kg, corresponding to a 5.5% reduction in the body-in-white mass, a significant achievement in lightweighting for a battery EV car.
The performance of the optimized battery EV car body was re-evaluated to ensure all constraints were satisfied. The results are summarized and compared with the baseline below.
| Performance Metric | Baseline Design | Optimized Design | Change | Constraint Status |
|---|---|---|---|---|
| Total Mass (kg) | 456.6 | 431.3 | -25.3 kg (-5.5%) | — |
| 1st Nat. Freq. (Hz) | 39.2 | 38.7 | -0.5 Hz | Within 30-100 Hz |
| 2nd Nat. Freq. (Hz) | 49.9 | 46.9 | -3.0 Hz | Within 30-100 Hz |
| 3rd Nat. Freq. (Hz) | 59.0 | 54.5 | -4.5 Hz | Within 30-100 Hz |
| Torsional Stiffness (N·m/deg) | 23,275 | 21,725 | -1,550 | > 20,000 |
The optimization successfully achieved its primary goal. The mass of the battery EV car body was reduced by 5.5%. As expected, the reduction in material led to a decrease in both natural frequencies and torsional stiffness. However, the optimization algorithm strategically reduced thickness in less critical areas, ensuring that the first three natural frequencies of the battery EV car body remained well within the allowable range of 30-100 Hz, thus avoiding potential resonance with dominant excitations. The torsional stiffness saw a reduction of 1,550 N·m/deg but remained above the stringent constraint limit of 20,000 N·m/deg, preserving the structural rigidity required for the battery EV car.
5. Conclusions
This study presented an integrated experimental-simulation methodology for the analysis and optimization of a battery EV car body structure. The key findings and contributions are as follows:
- Model Validation: The dynamic characteristics of the battery EV car body were accurately captured through experimental modal testing. The developed finite element model showed good correlation with experimental results in terms of natural frequencies (errors <12.5%) and mode shapes, establishing its credibility for subsequent engineering analysis.
- Effective Lightweighting: A size optimization framework was successfully implemented, utilizing panel thicknesses as design variables. The primary objective of mass minimization was achieved, resulting in a 5.5% weight reduction (25.3 kg) for the battery EV car body-in-white. This directly contributes to improved energy efficiency and extended range for the battery EV car.
- Performance Trade-off Management: The optimization process effectively managed the inherent trade-off between mass reduction and dynamic performance. While the natural frequencies and torsional stiffness decreased slightly, the final optimized design of the battery EV car body met all specified performance constraints: natural frequencies stayed within the safe range away from major excitations, and torsional stiffness remained above the critical threshold.
- Practical Methodology: The workflow demonstrated—from physical testing and model correlation to FE-based optimization—provides a robust and practical blueprint for the design and development of battery EV car bodies. It ensures that weight reduction efforts are grounded in physical reality and do not compromise essential NVH and structural attributes.
A limitation of this work is the lack of physical validation of the optimized design via testing a prototype. Future work should focus on manufacturing a prototype based on the optimized thickness parameters and conducting a final experimental verification of its modal and stiffness properties. Furthermore, the optimization could be extended to include other performance metrics like bending stiffness, crashworthiness, and multi-disciplinary constraints to yield an even more holistic optimal design for the next generation of battery EV car bodies.