In the rapidly evolving electric vehicle industry, the powertrain mounting system (PMS) plays a critical role in ensuring vehicle comfort and performance by isolating vibrations from the electric drivetrain. As electric vehicles, particularly in the China EV market, become more prevalent, addressing uncertainties in PMS design is paramount. These uncertainties arise from manufacturing tolerances, material variations, and operational conditions, often leading to correlated parameters that complicate analysis. Traditional methods frequently assume parameter independence, which overlooks real-world correlations and can result in suboptimal designs. This study focuses on developing a robust optimization framework for electric vehicle PMS that accounts for both parametric uncertainty and correlation, enhancing system reliability and performance. We propose novel analytical techniques to efficiently handle these complexities, ensuring that electric vehicle designs meet stringent vibration and noise standards. By integrating probabilistic models and correlation-aware methods, we aim to improve the inherent characteristics of PMS, such as natural frequencies and decoupling rates, under uncertain conditions. The approach is validated through a case study, demonstrating significant improvements in robustness for electric vehicle applications.
The powertrain mounting system in an electric vehicle supports the drivetrain, limits its movement, and dampens vibrations. Unlike internal combustion engines, electric vehicles lack masking effects from engine noise, making PMS-induced vibrations more noticeable. Thus, optimizing PMS for inherent characteristics like natural frequencies and decoupling rates is crucial. However, parameters such as mount stiffnesses are subject to uncertainties due to factors like material inhomogeneity and assembly errors. Moreover, these parameters often exhibit correlations, e.g., stiffnesses in different directions may vary together due to shared manufacturing processes. Ignoring these aspects can lead to inaccurate predictions and reduced system robustness. In this work, we address these challenges by modeling uncertainties probabilistically and incorporating correlations into the analysis and optimization processes. Our goal is to provide a comprehensive methodology that enhances the electric vehicle PMS design, ensuring consistent performance across varying conditions.
To analyze the PMS under uncertainty, we first establish a six-degree-of-freedom model for the electric vehicle powertrain. The system’s mass and stiffness matrices are derived, and the natural frequencies and mode shapes are obtained by solving the eigenvalue problem. The energy distribution in each mode is used to compute decoupling rates, which indicate how well vibrations are isolated in specific directions. For instance, a decoupling rate of 100% in the vertical direction means all vibrational energy is confined to that axis, minimizing coupling to other directions. This is particularly important for electric vehicles, where uncoupled vibrations can lead to audible noise and reduced comfort. The fundamental equations are given by:
$$(M^{-1}K – \omega_i^2 I) \phi_i = 0$$
$$f_i = \frac{\omega_i}{2\pi}$$
$$E(k,i) = \frac{\phi_{k,i} \sum_{j=1}^6 M_{k,j} \phi_{j,i}}{\phi_i^T M \phi_i}$$
$$d_i = \max_{k=1,2,\ldots,6} E(k,i)$$
Here, \(M\) is the mass matrix, \(K\) is the stiffness matrix, \(\omega_i\) is the angular frequency, \(\phi_i\) is the mode shape, \(f_i\) is the natural frequency, \(E(k,i)\) is the energy in the k-th generalized coordinate for the i-th mode, and \(d_i\) is the decoupling rate. These equations form the basis for evaluating PMS performance, but when parameters are uncertain, deterministic analysis falls short.
We model uncertain parameters, such as mount stiffnesses, as random variables with specified distributions. For example, stiffnesses may follow log-normal distributions to ensure positivity. The mean and standard deviation of each parameter are estimated from data, and correlations between parameters are quantified using Pearson correlation coefficients. The correlation between two variables \(x_\alpha\) 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, \(e_{x_\alpha}\) and \(\sigma_{x_\alpha}\) are the mean and standard deviation of \(x_\alpha\), respectively. To handle correlated variables, we employ the Nataf transformation, which maps correlated random variables to independent standard normal variables. This transformation is essential for efficient uncertainty propagation. The Nataf model relates the correlated vector \(x\) to an independent standard normal vector \(u\) through:
$$x = T(u)$$
The transformation involves computing the covariance matrix \(C\) and its Cholesky decomposition \(L\), such that \(C = LL^T\). Then, the correlated standard normal vector \(h\) is obtained as \(h = L u\), and \(x\) is derived using the inverse cumulative distribution functions. This allows us to generate samples that preserve the correlation structure, enabling accurate uncertainty analysis.
For uncertainty analysis, we propose two methods: the Nataf-Monte Carlo (NMC) method and the more efficient Nataf-Arbitrary Polynomial Chaos Expansion (NAPCE) method. The NMC method involves sampling from the transformed space and evaluating the PMS model for each sample to compute response statistics. While accurate, it requires numerous simulations, making it computationally expensive for complex systems. In contrast, the NAPCE method uses polynomial chaos expansion to approximate the response as a series of orthogonal polynomials in the independent variables, significantly reducing computational cost. The response \(Y(x)\) is expanded as:
$$Y(x) \approx \sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} c_{i_1,\ldots,i_n} \phi_{i_1,\ldots,i_n}(u)$$
where \(c_{i_1,\ldots,i_n}\) are expansion coefficients, and \(\phi_{i_1,\ldots,i_n}(u)\) are multivariate polynomial bases. The coefficients are determined using Gaussian quadrature, which involves evaluating the model at specific quadrature points. The mean and standard deviation of the response are then derived from these coefficients:
$$e_Y = c_{0,\ldots,0}$$
$$\sigma_Y = \sqrt{\sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} c_{i_1,\ldots,i_n}^2 – c_{0,\ldots,0}^2}$$
Additionally, the correlation between responses is computed based on the quadrature points, providing a comprehensive uncertainty characterization. The NAPCE method efficiently handles correlated inputs and outputs, making it suitable for robust design optimization.
In robust optimization, we consider both the mean and variability of responses, such as natural frequencies and decoupling rates. For electric vehicle PMS, key objectives include maximizing decoupling rates in critical directions (e.g., bounce and pitch) while ensuring natural frequencies avoid excitation ranges. We formulate a multi-objective optimization problem where weights are assigned to objectives based on their correlations, using a correlation-based weighting scheme. This approach ensures that highly correlated responses do not dominate the optimization, leading to balanced improvements. The weight for response \(Y_\alpha\) is computed as:
$$\delta’_\alpha = \frac{\sum_{i=1, i \neq \alpha}^N |\rho_{Y_\alpha,Y_i}|}{N-1}$$
$$\bar{\delta}’_\alpha = \frac{1}{\delta’_\alpha}$$
$$v’_\alpha = \frac{\bar{\delta}’_\alpha}{\sum_{i=1}^N \bar{\delta}’_i}$$
Subjective weights from engineering judgment are combined with these objective weights to form composite weights. The robust optimization model then aims to maximize the weighted sum of response means minus six times their standard deviations (following the Six Sigma principle), subject to constraints on natural frequencies and decoupling rates. For example, natural frequencies should avoid ranges that coincide with common excitations in electric vehicles, such as road-induced vibrations or motor harmonics. The optimization variables are the nominal stiffness values, with bounds set to feasible ranges. The model ensures that even under uncertainties, the system meets performance targets with high probability.
To demonstrate the methodology, we apply it to a case study of an electric vehicle PMS with three mounts. The drivetrain mass is 91 kg, and the moments of inertia are given in the table below. The mount stiffnesses are modeled as log-normal random variables with means and standard deviations derived from manufacturing data. Correlations between stiffnesses are assumed based on empirical observations. We analyze the system using both NMC and NAPCE methods to validate the efficiency and accuracy of NAPCE. The results show that NAPCE provides comparable results to NMC with significantly fewer function evaluations, making it ideal for optimization loops.
| Parameter | Value (kg·m²) |
|---|---|
| I_XX | 0.59 |
| I_YY | 1.49 |
| I_ZZ | 1.60 |
| I_XY | -2.45e-16 |
| I_YZ | -0.11e-16 |
| I_ZX | 1.77e-16 |
The initial analysis reveals that under uncertainty, some responses, like decoupling rates, do not meet robustness criteria. For instance, the bounce direction decoupling rate has a mean below the target and a high standard deviation, indicating sensitivity to parameter variations. Similarly, the pitch direction natural frequency may fall into critical ranges. After optimization, the stiffness values are adjusted, leading to improved mean decoupling rates and reduced variability. The table below summarizes the optimization results, showing significant enhancements in system robustness for the electric vehicle application.
| Response | Before Optimization (Mean ± Std Dev) | After Optimization (Mean ± Std Dev) |
|---|---|---|
| Bounce Frequency (Hz) | 9.63 ± 0.13 | 10.80 ± 0.16 |
| Pitch Frequency (Hz) | 16.42 ± 0.25 | 16.75 ± 0.26 |
| Bounce Decoupling Rate (%) | 84.55 ± 3.07 | 96.59 ± 1.02 |
| Pitch Decoupling Rate (%) | 85.38 ± 1.23 | 91.61 ± 1.01 |
The optimization process involves iteratively evaluating the robust objective function using the NAPCE method, which efficiently computes response statistics. The constraints ensure that natural frequencies remain within safe limits, and decoupling rates are maximized. The improved design demonstrates the effectiveness of our approach in enhancing electric vehicle PMS robustness, contributing to quieter and more comfortable electric vehicles, especially in the competitive China EV market.
In conclusion, this study presents a comprehensive framework for robustness optimization of electric vehicle powertrain mounting systems under parametric uncertainty and correlation. The NAPCE method offers a computationally efficient alternative to Monte Carlo simulation, accurately capturing response statistics and correlations. The robust optimization model, incorporating correlation-based weighting, ensures balanced improvement across multiple objectives. The case study validates the methodology, showing significant gains in performance and robustness. Future work could extend this approach to dynamic analysis or include other uncertain parameters, further advancing electric vehicle design. As the electric vehicle industry grows, particularly in China, such methods will be crucial for developing high-quality, reliable vehicles that meet consumer expectations for comfort and performance.
The integration of uncertainty analysis and robust design is essential for modern electric vehicle development. By addressing correlations and variability early in the design process, manufacturers can reduce the need for costly iterations and ensure consistent quality. This approach not only benefits electric vehicle PMS but can also be applied to other automotive systems where uncertainties play a critical role. As electric vehicle technology evolves, continued research in this area will drive innovations in vibration control and noise reduction, enhancing the overall driving experience for electric vehicle users worldwide.