In the rapidly evolving landscape of electric vehicles, the powertrain mounting system (PMS) plays a critical role in ensuring vehicle comfort and performance. As electric cars become more prevalent, particularly in markets like China EV, the need for robust design strategies that account for real-world uncertainties has never been more pressing. In this study, I explore the challenges posed by parametric uncertainties and correlations in electric car PMS and propose a novel methodology to enhance system robustness. The inherent characteristics of PMS, such as natural frequencies and decoupling rates, are highly sensitive to variations in mount stiffness parameters, which often exhibit probabilistic distributions and interdependencies. Traditional optimization approaches frequently overlook these correlations, leading to suboptimal designs that may fail under uncertain operating conditions. Through this research, I aim to bridge this gap by integrating advanced uncertainty analysis techniques with multi-objective optimization, ultimately contributing to the development of more reliable and efficient electric vehicles.
The foundation of this work lies in the six-degree-of-freedom model of an electric car powertrain mounting system. The system’s dynamic behavior is governed by the eigenvalue equation derived from the equations of motion. Specifically, the free vibration characteristics are captured by the following equation:
$$(M^{-1}K – \omega_i^2 I) \phi_i = 0$$
where \(M\) represents the mass matrix, \(K\) is the stiffness matrix, \(\omega_i\) denotes the circular frequency for the \(i\)-th mode, and \(\phi_i\) is the corresponding mode shape. The natural frequency \(f_i\) is then calculated as \(f_i = \omega_i / (2\pi)\). To assess the energy distribution across different generalized coordinates, the energy for the \(k\)-th coordinate in the \(i\)-th mode is given by:
$$E(k,i) = \frac{\phi_{k,i} \sum_{j=1}^{6} M_{k,j} \phi_{j,i}}{\phi_i^T M \phi_i}$$
The decoupling rate for the \(i\)-th mode, which indicates the concentration of vibrational energy, is defined as the maximum value of \(E(k,i)\) across all coordinates. A decoupling rate of 100% signifies complete isolation of vibration in a specific direction, which is a key objective in PMS design for electric cars to minimize noise and vibration.
In practical applications, the parameters of the PMS, such as mount stiffnesses, are subject to uncertainties due to manufacturing tolerances, material variations, and environmental factors. These uncertainties are often modeled probabilistically, and in many cases, the parameters are correlated. For instance, in a typical China EV setup, the stiffness parameters of rubber mounts may follow log-normal distributions with specific means and standard deviations. Consider a system with \(n\) uncertain parameters represented by a vector \(\mathbf{x} = [x_1, x_2, \ldots, x_n]^T\). The mean and standard deviation of each parameter are denoted as \(e_{x_\alpha}\) and \(\sigma_{x_\alpha}\), respectively. The correlation coefficient between any two parameters \(x_\alpha\) and \(x_\beta\) is computed as:
$$\rho_{x_\alpha, x_\beta} = \frac{1}{l-1} \sum_{i=1}^{l} \left( \frac{x_{\alpha,i} – e_{x_\alpha}}{\sigma_{x_\alpha}} \right) \left( \frac{x_{\beta,i} – e_{x_\beta}}{\sigma_{x_\beta}} \right)$$
where \(l\) is the number of observations. However, when parameters are correlated, the Nataf transformation is employed to handle the dependency structure. This transformation maps correlated random variables to independent standard normal variables, facilitating easier analysis. The correlation in the standard normal space is related to the original correlation through empirical coefficients, and the covariance matrix \(\mathbf{C}\) is decomposed using Cholesky factorization to derive a lower triangular matrix \(\mathbf{L}\) such that \(\mathbf{C} = \mathbf{L} \mathbf{L}^T\). This allows the transformation of independent standard normal vector \(\mathbf{u}\) to correlated vector \(\mathbf{h}\) via \(\mathbf{h} = \mathbf{L} \mathbf{u}\), and ultimately to the original correlated parameters \(\mathbf{x} = T(\mathbf{u})\).
To analyze the uncertainty and correlation in PMS responses, I first developed the Nataf-Monte Carlo (NMC) method. This approach involves generating samples from the independent standard normal distribution, transforming them to correlated samples using the Nataf transformation, and then evaluating the system responses through Monte Carlo simulation. While accurate, this method is computationally intensive, especially for complex systems like those in electric cars, where numerous simulations are required. For example, in a typical analysis, up to \(10^6\) samples may be needed to achieve convergence, leading to significant computational costs. The steps of the NMC method are outlined below:
| Step | Description |
|---|---|
| 1 | Compute the lower triangular matrix \(\mathbf{L}\) from the covariance matrix of correlated parameters. |
| 2 | Generate independent samples \(\mathbf{u}_s\) from the standard normal distribution. |
| 3 | Transform \(\mathbf{u}_s\) to correlated samples \(\mathbf{h}_s\) using \(\mathbf{h}_s = \mathbf{L} \mathbf{u}_s\). |
| 4 | Map \(\mathbf{h}_s\) to the original parameter space \(\mathbf{x}_s\) via inverse cumulative distribution functions. |
| 5 | Evaluate PMS responses for each sample and compute statistics (mean, standard deviation, correlations). |
Despite its accuracy, the computational burden of NMC motivated the development of a more efficient technique: the Nataf-Arbitrary Polynomial Chaos Expansion (NAPCE) method. This approach combines the Nataf transformation with arbitrary polynomial chaos (APC) expansion to approximate the system responses with minimal function evaluations. The APC expansion represents the response \(Y(\mathbf{x})\) as a series of orthogonal polynomials in the independent standard normal variables \(\mathbf{u}\):
$$Y(\mathbf{x}) \approx \sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} c_{i_1,\ldots,i_n} \phi_{i_1,\ldots,i_n}(\mathbf{u})$$
where \(c_{i_1,\ldots,i_n}\) are the expansion coefficients, and \(\phi_{i_1,\ldots,i_n}(\mathbf{u})\) are the multivariate polynomial bases formed as products of univariate polynomials. The univariate polynomials \(\phi_i(u_\alpha)\) for each variable \(u_\alpha\) are constructed using recurrence relations based on statistical moments. The coefficients are determined through Gaussian quadrature, which involves evaluating the system at specific quadrature nodes. The mean and standard deviation of the response are then derived from the coefficients:
$$e_Y = c_{0,\ldots,0}, \quad \sigma_Y = \sqrt{\sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} c_{i_1,\ldots,i_n}^2 – c_{0,\ldots,0}^2}$$
Additionally, the correlation between different responses, such as natural frequencies and decoupling rates, is computed using the Gaussian quadrature nodes and weights. The NAPCE method significantly reduces the number of system evaluations—for instance, in our case, only 512 evaluations were needed compared to \(10^6\) for NMC—while maintaining high accuracy. The following table compares the computational efficiency of NMC and NAPCE for a typical electric car PMS analysis:
| Method | Number of Function Evaluations | Computational Time (seconds) | Accuracy (Relative Error) |
|---|---|---|---|
| NMC | 1,000,000 | 118.20 | Reference |
| NAPCE | 512 | 0.66 | < 3.31% |
To validate the NAPCE method, I conducted extensive uncertainty analyses under varying levels of parameter uncertainty and correlation. For example, the stiffness parameters of the mounts were assumed to follow log-normal distributions with means and standard deviations derived from empirical data. The table below summarizes the distribution properties for a three-mount system in a representative China EV model:
| Mount | Stiffness Direction | Distribution Type | Mean (N/mm) | Standard Deviation (N/mm) |
|---|---|---|---|---|
| Mount 1 | Ku | Log-normal | 87.98 | 0.93 |
| Kv | Log-normal | 60.12 | 0.50 | |
| Kw | Log-normal | 132.94 | 0.95 | |
| Mount 2 | Ku | Log-normal | 144.67 | 1.52 |
| Kv | Log-normal | 98.73 | 0.82 | |
| Kw | Log-normal | 84.56 | 0.61 | |
| Mount 3 | Ku | Log-normal | 79.18 | 0.83 |
| Kv | Log-normal | 78.69 | 0.65 | |
| Kw | Log-normal | 149.56 | 1.07 |
I evaluated the system’s natural frequencies and decoupling rates for the bounce and pitch directions, which are critical for electric car comfort. The results showed that as the standard deviation of the stiffness parameters increased, the response boundaries expanded linearly. For instance, the bounce direction natural frequency \(f_B\) had a mean of approximately 9.63 Hz with a standard deviation ranging from 0.02 to 0.13 Hz across different uncertainty levels. The decoupling rates, however, often fell below the desired robustness thresholds, highlighting the need for optimization. The correlation analysis revealed that response correlations are influenced by both system inherent properties and parameter correlations. For example, the correlation between pitch direction frequency \(f_P\) and decoupling rate \(d_P\) was negative, while other response pairs showed positive correlations. As parameter correlation increased from 0 to 0.9, the response boundaries initially narrowed and then expanded, indicating a complex dependency that must be accounted for in design.
Building on the uncertainty analysis, I formulated a robustness optimization model for the electric car PMS. The objective is to maximize the weighted sum of the decoupling rates for bounce and pitch directions, considering their uncertainties and correlations. The 6Sigma criterion is applied to ensure that the responses remain within specified limits under uncertain conditions. The optimization problem is stated as:
$$\text{maximize} \quad v_B (e_{d_B} – 6\sigma_{d_B}) + v_P (e_{d_P} – 6\sigma_{d_P})$$
subject to:
$$d_{i,\min} + 6\sigma_{d_i} \leq e_{d_i}, \quad i=1,2,\ldots,6$$
$$f_{i,\min} + 6\sigma_{f_i} \leq e_{f_i} \leq f_{i,\max} – 6\sigma_{f_i}$$
$$t_j^L \leq t_j \leq t_j^U, \quad j=1,2,\ldots,9$$
Here, \(v_B\) and \(v_P\) are the combined weights for the bounce and pitch decoupling rates, determined using a correlation-based weighting method. The weights account for both subjective engineering priorities and objective correlations between responses. The correlation between responses \(Y_\alpha\) and \(Y_\beta\) is computed as:
$$\rho_{Y_\alpha, Y_\beta} = \frac{1}{Q-1} \sum_{i=1}^{Q} \left( \frac{Y_{\alpha,i} – e_{Y_\alpha}}{\sigma_{Y_\alpha}} \right) \left( \frac{Y_{\beta,i} – e_{Y_\beta}}{\sigma_{Y_\beta}} \right)$$
where \(Q\) is the number of quadrature nodes. The objective weight for each response is derived from the average correlation with other responses, ensuring that highly correlated responses have reduced weights to avoid over-emphasis. The combined weight \(v_\alpha\) is then calculated as the average of subjective and objective weights.
In the optimization process, the nominal values of the mount stiffnesses are treated as design variables, with bounds set to ±40% of their initial values. The initial and optimized stiffness values are summarized below:
| Mount | Stiffness Direction | Initial Nominal Value (N/mm) | Optimized Nominal Value (N/mm) |
|---|---|---|---|
| Mount 1 | Ku | 87.98 | 60.34 |
| Kv | 60.12 | 75.98 | |
| Kw | 132.94 | 189.67 | |
| Mount 2 | Ku | 144.67 | 141.25 |
| Kv | 98.73 | 125.77 | |
| Kw | 84.56 | 106.64 | |
| Mount 3 | Ku | 79.18 | 79.89 |
| Kv | 78.69 | 80.07 | |
| Kw | 149.56 | 150.13 |
The optimization results demonstrated significant improvements in system robustness. The mean decoupling rates for bounce and pitch directions increased from 84.55% and 85.38% to 96.59% and 91.61%, respectively, while their standard deviations decreased from 3.07% and 1.23% to 1.02% and 1.01%. This indicates that the optimized design not only enhances performance but also reduces sensitivity to parameter variations. The natural frequencies remained within acceptable limits, satisfying the robustness constraints. The following table compares the response statistics before and after optimization:
| Response | Before Optimization (Mean ± Std Dev) | After Optimization (Mean ± Std Dev) |
|---|---|---|
| Bounce Frequency \(f_B\) (Hz) | 9.63 ± 0.13 | 10.80 ± 0.16 |
| Pitch Frequency \(f_P\) (Hz) | 16.42 ± 0.25 | 16.75 ± 0.26 |
| Bounce Decoupling Rate \(d_B\) (%) | 84.55 ± 3.07 | 96.59 ± 1.02 |
| Pitch Decoupling Rate \(d_P\) (%) | 85.38 ± 1.23 | 91.61 ± 1.01 |
The response boundaries, computed using the 6Sigma criterion, showed that all optimized responses now lie within the desired robustness limits, whereas pre-optimization, some boundaries exceeded acceptable ranges. This underscores the effectiveness of the proposed methodology in enhancing the reliability of electric car powertrain mounting systems, particularly in the context of China EV applications where stringent performance standards are enforced.
In conclusion, the integration of Nataf transformation and arbitrary polynomial chaos expansion into the uncertainty analysis and optimization framework provides a powerful tool for addressing parametric uncertainties and correlations in electric car PMS. The NAPCE method offers high computational efficiency and accuracy, making it suitable for iterative design processes. The robustness optimization model, incorporating response correlations through objective weighting, ensures balanced and reliable design outcomes. This approach not only improves the inherent characteristics of the PMS but also contributes to the overall comfort and durability of electric vehicles. Future work could explore the application of these techniques to other vehicle subsystems or extend them to account for dynamic loading conditions, further advancing the state of the art in electric car design.