Abstract
This study addresses the complex scenario where parameters of the powertrain mounting system (PMS) in electric vehicles exhibit both uncertainty and correlation. We propose a robust design optimization framework by integrating Nataf transformation, Monte Carlo sampling, and arbitrary polynomial chaos expansion (APC). The Nataf-Monte Carlo (NMC) method and Nataf-Arbitrary Polynomial Chaos Expansion (NAPCE) method are developed to analyze response uncertainties and correlations. A correlation coefficient weighting approach is adopted to formulate the optimization model, which is validated via a case study. Results show that the NAPCE method offers high computational accuracy and efficiency, and the proposed optimization method significantly improves system robustness.
1. Introduction
The powertrain mounting system (PMS) in electric vehicles (EVs) serves as an elastic connection between the electric drive unit and the vehicle frame, performing functions of support, limit, and vibration isolation . Unlike internal combustion engine vehicles, EVs lack engine masking effects, making PMS-induced vibration and noise more prominent. Parameters of PMS, particularly rubber mounts, exhibit significant uncertainty and correlation due to manufacturing processes, material properties, and operational conditions .
Existing studies on PMS uncertainty analysis primarily focus on independent parameters, ignoring correlations. For instance, LYU et al. [3] treated mount stiffness as interval-random variables, while WU et al. [5] used uniform and normal distributions for reliability optimization . Non-probabilistic models like ellipsoidal and multidimensional parallelepiped models have been applied to handle correlated parameters, but probabilistic approaches integrating both uncertainty and correlation remain limited .
This study aims to develop a probabilistic framework for robust PMS optimization in EVs, considering both parameter uncertainty and correlation. The objectives are to: (1) analyze PMS response uncertainties using NMC and NAPCE methods; (2) formulate a robust optimization model with correlation-aware weighting; and (3) validate the framework through a case study.
2. PMS Modeling and Parametric Uncertainties
2.1. PMS Dynamic Model
For centralized drive EVs, the PMS is modeled as a six-degree-of-freedom (6-DOF) system, treating the electric drive unit as a rigid body and mounts as three-directional elastic elements . The coordinate system \(G_0-XYZ\) is defined with the X-axis opposite to the vehicle’s forward direction, Z-axis vertically upward, and Y-axis following the right-hand rule .
The free vibration characteristic equation of the PMS is:\((M^{-1}K – \omega_i^2 I)\phi_i = 0 \quad (1)\) where M is the mass matrix, K is the stiffness matrix, \(\omega_i\) is the circular frequency of the i-th natural frequency, and \(\phi_i\) is the mode shape. The natural frequency \(f_i\) is:\(f_i = \frac{\omega_i}{2\pi} \quad (2)\)
The vibration energy at the k-th generalized coordinate for the i-th natural frequency is:\(E(k,i) = \frac{\phi_{k,i} \sum_{j=1}^{6}(M_{k,j}\phi_{j,i})}{\phi_i^T M \phi_i} \quad (3)\) The decoupling ratio \(d_i\) for the i-th mode is defined as the maximum energy concentration:\(d_i = \max_{k=1,2,\dots,6} E(k,i) \quad (4)\)
2.2. Parametric Uncertainty and Correlation
Consider n correlated uncertain parameters in the PMS, described by the vector \(x = [x_1, x_2, \dots, x_n]^T\). The correlation coefficient between parameters \(x_\alpha\) and \(x_\beta\) is:\(\rho_{x_\alpha x_\beta} = \frac{1}{l-1} \sum_{i=1}^{l} \left(\frac{x_{\alpha,i} – e_{x_\alpha}}{\sigma_{x_\alpha}}\right)\left(\frac{x_{\beta,i} – e_{x_\beta}}{\sigma_{x_\beta}}\right) \quad (5)\) where \(e_{x_\alpha}\) and \(\sigma_{x_\alpha}\) are the mean and standard deviation of \(x_\alpha\), and l is the number of observations .
Using Nataf theory, the correlation coefficient can be expressed in terms of standard normal variables \(h = [h_1, h_2, \dots, h_n]^T\):\(\rho_{x_\alpha x_\beta} = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{F_{x_\alpha}^{-1}(\Phi(h_\alpha)) – e_{x_\alpha}}{\sigma_{x_\alpha}} \cdot \frac{F_{x_\beta}^{-1}(\Phi(h_\beta)) – e_{x_\beta}}{\sigma_{x_\beta}} \cdot \varphi(h_\alpha, h_\beta, \rho_{h_\alpha h_\beta}) \, dh_\alpha dh_\beta \quad (6)\) where \(\Phi(\cdot)\) is the standard normal distribution function, \(F_{x}^{-1}(\cdot)\) is the inverse cumulative distribution function of x, and \(\varphi(\cdot)\) is the joint probability density function of \(h_\alpha\) and \(h_\beta\) .
The covariance matrix C is:\(C = \begin{bmatrix} \sigma_{x_1}^2 \rho_{h_1 h_1} & \sigma_{x_1}\sigma_{x_2} \rho_{h_1 h_2} & \dots & \sigma_{x_1}\sigma_{x_n} \rho_{h_1 h_n} \\ \sigma_{x_2}\sigma_{x_1} \rho_{h_2 h_1} & \sigma_{x_2}^2 \rho_{h_2 h_2} & \dots & \sigma_{x_2}\sigma_{x_n} \rho_{h_2 h_n} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{x_n}\sigma_{x_1} \rho_{h_n h_1} & \sigma_{x_n}\sigma_{x_1} \rho_{h_n h_2} & \dots & \sigma_{x_n}^2 \rho_{h_n h_n} \end{bmatrix} \quad (7)\) which can be decomposed as \(C = LL^T\) to obtain the lower triangular matrix L, transforming independent standard normal vectors u to correlated vectors h via \(h = Lu\) . Thus, x is expressed as \(x = T(u)\), where \(T(\cdot)\) denotes the Nataf transformation .
3. Response Uncertainty and Correlation Analysis
3.1. Nataf-Monte Carlo (NMC) Method
The NMC method combines Nataf transformation with Monte Carlo sampling to analyze PMS responses under correlated uncertainties. Key steps include:
- Calculate matrix L using equations (7)–(9) based on parameter standard deviations and correlations .
- Sample independent standard normal vectors \(u_s\) and transform them to correlated vectors \(h_s\) via \(h_s = Lu_s\) .
- Convert \(h_s\) to correlated parameter samples \(x_s\) using inverse cumulative distribution functions .
- Substitute \(x_s\) into the PMS model to compute natural frequencies and decoupling ratios .
- Calculate response statistics (mean, standard deviation, correlation coefficients) from l samples .
NMC serves as a reference method for validating other approaches but has low computational efficiency due to extensive sampling .
3.2. Nataf-Arbitrary Polynomial Chaos Expansion (NAPCE) Method
To improve efficiency, the NAPCE method integrates Nataf transformation with APC. The PMS response function \(Y(x)\) is expanded as:\(Y(x) \approx \sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} c_{i_1,\dots,i_n} \varphi_{i_1,\dots,i_n}(u) \quad (8)\) where \(c_{i_1,\dots,i_n}\) are expansion coefficients, and \(\varphi_{i_1,\dots,i_n}(u)\) are polynomial bases for independent standard normal vectors u . The polynomial bases satisfy the recurrence relation:\(b_i \varphi_i(u_\alpha) = (u_\alpha – a_i)\varphi_{i-1}(u_\alpha) + b_{i-1}\varphi_{i-2}(u_\alpha) \quad (9)\) with \(\varphi_{-1}(u_\alpha) = 0\) and \(\varphi_0(u_\alpha) = 1\) .
The statistical moments of \(u_\alpha\) are:\(\mu_{x_\alpha}^i = \int_{\Omega} u_\alpha^i w(u_\alpha) \, du_\alpha \quad (10)\) where \(w(u_\alpha)\) is the probability density function . Hankel matrices \(H_\alpha\) are constructed from moments and decomposed via Cholesky factorization to determine recurrence coefficients \(a_i\) and \(b_i\) .
Gaussian integration nodes \(\hat{u}_\alpha\) and weights \(w_\alpha\) are obtained from the eigenvalue decomposition of the Jacobi matrix \(J_\alpha\). Substituting \(\hat{u}_\alpha\) into \(x = T(u)\) gives correlated parameter nodes \(\hat{x}_\alpha\) . The expansion coefficients are:\(c_{i_1,\dots,i_n} = \sum_{j_1=1}^{q_1} \cdots \sum_{j_n=1}^{q_n} Y(\hat{x}_{1,j_1}, \dots, \hat{x}_{n,j_n}) \cdot \varphi_i(\hat{u}_{1,j_1}, \dots, \hat{u}_{n,j_n}) \cdot \hat{w}_{1,j_1} \cdots \hat{w}_{n,j_n} \quad (11)\) where \(q_\alpha\) is the number of Gaussian nodes for \(x_\alpha\) . The mean and standard deviation of \(Y(x)\) are:\(e_Y = c_{0,\dots,0}, \quad \sigma_Y = \sqrt{\sum_{i_1=0}^{s_1} \cdots \sum_{i_n=0}^{s_n} (c_{i_1,\dots,i_n})^2 – (c_{0,\dots,0})^2} \quad (12)\)
Response correlation coefficients are calculated as:\(\rho_{Y_\alpha Y_\beta} = \frac{1}{Q-1} \sum_{i=1}^{Q} \left(\frac{Y_{\alpha,i} – e_{Y_\alpha}}{\sigma_{Y_\alpha}}\right)\left(\frac{Y_{\beta,i} – e_{Y_\beta}}{\sigma_{Y_\beta}}\right) \quad (13)\) where \(Q = \prod_{i=1}^n q_i\) .
4. Robustness Optimization Model
4.1. Objective Function Weighting
In multi-objective PMS optimization, weights for sub-objectives are determined using a correlation coefficient weighting method [15]. The mean correlation of response \(Y_\alpha\) with others is:\(\delta_\alpha’ = \frac{\sum_{i=1, j \neq \alpha}^{N} |\rho_{Y_\alpha Y_i}|}{N-1} \quad (14)\) where N is the number of responses. The weight is inversely proportional to correlation:\(\overline{\delta}_\alpha’ = \frac{1}{\delta_\alpha’} \quad (15)\) Normalizing gives the objective weight:\(v_\alpha’ = \frac{\overline{\delta}_\alpha’}{\sum_{i=1}^{N} \overline{\delta}_i’} \quad (16)\) Combining subjective weights \(v_\alpha”\) (based on engineering experience), the comprehensive weight is:\(v_\alpha = 0.5(v_\alpha’ + v_\alpha”) \quad (17)\)
4.2. Optimization Model Formulation
Focusing on the Bounce and Pitch directions, the robust optimization model minimizes response uncertainties while satisfying constraints. The objective function is:\(\max \, v_B(e_{d_B} – 6\sigma_{d_B}) + v_P(e_{d_P} – 6\sigma_{d_P}) \quad (18)\) Subject to:\(\begin{cases} d_{i,\text{min}} + 6\sigma_{d_i} \leq e_{d_i}, & i=1,2,\dots,6 \\ f_{i,\text{min}} + 6\sigma_{f_i} \leq e_{f_i} \leq f_{i,\text{max}} – 6\sigma_{f_i} \\ t_j^L \leq t_j \leq t_j^U, & j=1,2,\dots,9 \end{cases}\) where \(v_B\) and \(v_P\) are weights for Bounce and Pitch decoupling ratios, \(e_{d_i}\) and \(\sigma_{d_i}\) are mean and standard deviation of decoupling ratios, \(f_{i,\text{min}}\) and \(f_{i,\text{max}}\) are natural frequency bounds, and \(t_j\) are optimization variables .
5. Case Study and Results
5.1. PMS Model Description
A three-mount PMS for a centralized drive EV is analyzed, with the electric drive unit mass of 91 kg. Table 1 lists the powertrain moments of inertia and products of inertia .
| Inertia | Values (kg·m²) | Inertia Product | Values (10⁻² kg·m²) |
|---|---|---|---|
| \(I_{XX}\) | 1.60 | \(I_{XY}\) | -2.45 |
| \(I_{YY}\) | 1.49 | \(I_{YZ}\) | -0.11 |
| \(I_{ZZ}\) | 0.59 | \(I_{ZX}\) | 1.77 |
Table 1. Powertrain Moments of Inertia and Products of Inertia
Mount stiffness parameters are treated as uncertain, with distributions, means, and standard deviations listed in Table 2 .
| Mount | Stiffness | Distribution | Mean (N·mm⁻¹) | Std Dev (N·mm⁻¹) |
|---|---|---|---|---|
| Mount 1 | \(K_{u1}\) | Lognormal | 87.98 | 0.93 |
| \(K_{v1}\) | Lognormal | 60.12 | 0.50 | |
| \(K_{w1}\) | Lognormal | 132.94 | 0.95 | |
| Mount 2 | \(K_{u2}\) | Lognormal | 144.67 | 1.52 |
| \(K_{v2}\) | Lognormal | 98.73 | 0.82 | |
| \(K_{w2}\) | Lognormal | 84.56 | 0.61 | |
| Mount 3 | \(K_{u3}\) | Lognormal | 79.18 | 0.83 |
| \(K_{v3}\) | Lognormal | 78.69 | 0.65 | |
| \(K_{w3}\) | Lognormal | 149.56 | 1.07 |
Table 2. Mount Stiffness Distributions and Statistics
5.2. Uncertainty Analysis Results
Using NMC and NAPCE, we analyze PMS responses under six uncertainty levels (\(\sigma_x, 2\sigma_x, \dots, 6\sigma_x\)) with a correlation coefficient of 0.3. Table 3 shows Bounce direction results, indicating that \(f_B\) meets robustness requirements, while \(d_B\) means fail to satisfy \(85\% + 6\sigma_{d_B}\) .
For Pitch direction (Table 4), \(f_P\) mean under \(6\sigma_x\) is below the lower bound, and \(d_P\) means fail to meet robustness criteria .
The relative errors of NAPCE compared to NMC (Table 5) show maximum errors of 1.68% for means and 3.31% for standard deviations, verifying high accuracy . Computationally, NAPCE takes 0.66 s with 512 system calls, versus 118.20 s and 10⁶ calls for NMC, demonstrating superior efficiency .
5.3. Correlation Analysis
Varying parameter correlation coefficients from 0 to 0.9999, we analyze response correlations. Figure 1 (textual summary) shows that NAPCE closely matches NMC results, with maximum relative error 1.73% for \(\rho_{f_P d_P}\) . Responses \(f_P\) and \(d_B\), \(f_P\) and \(d_P\) exhibit negative correlation, while others are positively correlated. Increasing parameter correlation enhances correlations between \(f_B\) and \(f_P\), \(f_P\) and \(d_B\), \(d_B\) and \(d_P\), but decreases other correlations .
5.4. Robustness Optimization Results
With a parameter correlation of 0.3 and \(6\sigma_x\) uncertainty, the optimization adjusts mount stiffnesses (Table 6) to improve system robustness .
| Mount | \(K_u\) (N·mm⁻¹) | \(K_v\) (N·mm⁻¹) | \(K_w\) (N·mm⁻¹) |
|---|---|---|---|
| Mount 1 | 75.98 | 60.34 | 189.67 |
| Mount 2 | 106.64 | 125.77 | 141.25 |
| Mount 3 | 150.13 | 80.07 | 79.89 |
Table 3. Optimized Mount Stiffnesses
Optimization results (Table 7) show significant improvements: \(d_B\) and \(d_P\) means increase to 96.59% and 91.61%, with standard deviations reduced to 1.02% and 1.01% . The response boundaries (Figure 2, textual summary) narrow, with \(f_P\), \(d_B\), and \(d_P\) now meeting robustness requirements .
| Response | \multicolumn{2}{c | }{Before Optimization} | \multicolumn{2}{c | }{After Optimization} |
|---|---|---|---|---|
| Mean | Std Dev | Mean | Std Dev | |
| \(f_B\) (Hz) | 9.63 | 0.13 | 10.80 | 0.16 |
| \(f_P\) (Hz) | 16.42 | 0.25 | 16.75 | 0.26 |
| \(d_B\) (%) | 84.55 | 3.07 | 96.59 | 1.02 |
| \(d_P\) (%) | 85.38 | 1.23 | 91.61 | 1.01 |
Table 4. Mean Values and Standard Deviations Before and After Optimization
6. Conclusions
- The NAPCE method efficiently captures PMS response uncertainties and correlations in electric vehicles, with maximum relative errors <3.31% compared to NMC .
- Parameter correlation significantly affects PMS responses: response boundaries first narrow (correlation 0–0.9) then expand (correlation 0.9–0.9999), highlighting the need for correlation-aware analysis .
- The proposed robust optimization method enhances decoupling ratio means by 12–6% and reduces standard deviations by 67–18%, effectively improving EV PMS robustness against parametric uncertainties .
This study provides a comprehensive framework for uncertainty-informed PMS design in electric vehicles, integrating probabilistic analysis and robust optimization to address real-world engineering challenges.