In the rapidly evolving landscape of automotive engineering, electric vehicles (EVs) have gained significant traction due to their environmental benefits and technological advancements. The powertrain mounting system (PMS) in an electric car plays a critical role in ensuring vehicle comfort and performance by isolating vibrations from the powertrain to the chassis. However, the PMS is subject to numerous uncertainties arising from manufacturing tolerances, material aging, and operational variations, which can affect its dynamic characteristics. Traditional sensitivity analysis methods often focus on single-output responses, such as individual natural frequencies or decoupling rates, but they fall short in capturing the comprehensive impact of uncertain parameters on multiple performance metrics simultaneously. This study addresses this gap by proposing a multi-output global sensitivity analysis framework for electric car PMS, considering parameter uncertainties described by random variables. The approach leverages covariance decomposition and Monte Carlo simulation to derive sensitivity indices that quantify the influence of parameters on the system’s overall behavior. By applying this method to a case study of a China EV model, we demonstrate its effectiveness in identifying key parameters and guiding robust design practices. The results highlight the importance of accounting for multi-output responses in sensitivity analysis to avoid contradictory conclusions and ensure accurate parameter ranking for optimization in electric car applications.
The powertrain mounting system in an electric car is a complex dynamic system designed to support the powertrain and mitigate vibrations transmitted to the vehicle body. In China EV markets, where demand for high-performance and comfortable vehicles is growing, optimizing the PMS is crucial for enhancing ride quality. A typical PMS involves multiple parameters, including mount stiffnesses, inertial properties, and suspension characteristics, which are prone to uncertainties. These uncertainties can lead to variations in natural frequencies, decoupling rates, and vibration isolation performance, ultimately affecting the electric car’s NVH (Noise, Vibration, and Harshness) characteristics. Traditional single-output sensitivity analyses, which evaluate parameters against individual responses, often yield redundant or conflicting results. For instance, a parameter might significantly influence one response while having negligible effects on others, making it challenging to prioritize design changes. This study introduces a multi-output sensitivity analysis method that integrates all relevant responses into a unified framework, providing a holistic view of parameter impacts. The methodology is grounded in random variable theory and advanced statistical techniques, ensuring robustness in handling uncertainties inherent in electric car systems.
To model the dynamic behavior of an electric car PMS, we establish a 13-degree-of-freedom (DOF) system that accounts for the powertrain, body, and unsprung masses. The equations of motion are derived using kinematic principles, leading to the free vibration characteristic equation. Let \( \mathbf{M}_{13} \) and \( \mathbf{K}_{13} \) represent the mass and stiffness matrices of the system, respectively. The natural frequencies and mode shapes are obtained by solving the eigenvalue problem:
$$ \left( \mathbf{M}_{13}^{-1} \mathbf{K}_{13} – \omega_s^2 \mathbf{I} \right) \boldsymbol{\Phi}_s = \mathbf{0} $$
where \( \omega_s \) is the s-th natural angular frequency, \( \mathbf{I} \) is the identity matrix, and \( \boldsymbol{\Phi}_s \) is the mode shape vector corresponding to the s-th mode. The natural frequency in Hz is given by \( f_s = \omega_s / (2\pi) \). For the decoupling rate analysis, the energy distribution among the 13 DOFs is computed. The energy concentration on the l-th generalized coordinate for the s-th mode is expressed as:
$$ E_D(l, s) = \frac{ \sum_{t=1}^{13} \left[ m_{lt} (\boldsymbol{\Phi}_s)_l (\boldsymbol{\Phi}_s)_t \right] }{ \sum_{l=1}^{13} \sum_{t=1}^{13} \left[ m_{lt} (\boldsymbol{\Phi}_s)_l (\boldsymbol{\Phi}_s)_t \right] } \times 100\% $$
where \( m_{lt} \) is an element of the mass matrix \( \mathbf{M}_{13} \), and \( (\boldsymbol{\Phi}_s)_l \) denotes the l-th component of the mode shape vector. The decoupling rate for the s-th mode is then defined as the maximum energy concentration across all coordinates:
$$ d_s = \max_{l=1,2,\dots,13} E_D(l, s) $$
A decoupling rate of 100% indicates perfect isolation in one direction, which is ideal for vibration control in electric cars. However, uncertainties in system parameters can deviate these ideal conditions, necessitating a thorough sensitivity analysis.
Uncertain parameters in the electric car PMS are described using random variables to capture their probabilistic nature. For example, mount stiffnesses, inertial properties, and suspension spring rates are modeled with specified probability density functions (PDFs). The mean values represent nominal design parameters, while the variance quantifies the uncertainty. The coefficient of variation (δ) is used to express the relative uncertainty, defined as the ratio of standard deviation to mean. In this study, we consider 21 uncertain parameters, including powertrain inertia, suspension stiffness, wheel vertical stiffness, and mount stiffnesses. These parameters are assumed to follow normal distributions for simplicity and practicality in China EV applications. The PDF and cumulative distribution function (CDF) for a normal random variable x are given by:
$$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x – \mu)^2}{2\sigma^2} \right) $$
$$ F(x) = \int_{-\infty}^{x} f(t) \, dt $$
where μ is the mean and σ is the standard deviation. The mean represents the expected value, and the variance indicates the dispersion around the mean. For instance, mount stiffness parameters might have a coefficient of variation of 10%, reflecting typical manufacturing tolerances in electric car production.
The multi-output sensitivity analysis extends traditional variance-based methods to handle multiple responses simultaneously. Let \( \mathbf{x} = [x_1, x_2, \dots, x_n]^T \) be the vector of n uncertain input parameters, and \( \mathbf{y} = [y_1, y_2, \dots, y_m]^T \) be the vector of m output responses, such as natural frequencies and decoupling rates in different directions. The function relationship is \( \mathbf{y} = \mathbf{g}(\mathbf{x}) \). Using covariance decomposition, the covariance matrix of the output vector \( \mathbf{C}_y \) can be partitioned into contributions from individual parameters and their interactions:
$$ \mathbf{C}_y(\mathbf{y}) = \sum_{i=1}^{n} \mathbf{C}_i + \sum_{1 \leq i < j \leq n} \mathbf{C}_{ij} + \cdots + \mathbf{C}_{12\dots n} $$
Summing all elements of the covariance matrix yields a scalar measure of total variability:
$$ \text{Sum}[\mathbf{C}_y] = \sum_{i=1}^{n} \text{Sum}[\mathbf{C}_i] + \sum_{1 \leq i < j \leq n} \text{Sum}[\mathbf{C}_{ij}] + \cdots + \text{Sum}[\mathbf{C}_{12\dots n}] $$
The first-order sensitivity index for multi-output responses, which quantifies the independent effect of parameter \( x_i \), is defined as:
$$ S_i = \frac{ \text{Sum}[\mathbf{C}_i] }{ \text{Sum}[\mathbf{C}_y] } $$
The total sensitivity index, accounting for both main effects and interactions, is given by:
$$ S_{Ti} = \frac{ \text{Sum}[\mathbf{C}_i] + \sum_{j \neq i} \text{Sum}[\mathbf{C}_{ij}] + \cdots + \text{Sum}[\mathbf{C}_{12\dots n}] }{ \text{Sum}[\mathbf{C}_y] } $$
These indices provide a comprehensive view of how each parameter influences the entire set of responses, which is crucial for designing robust electric car PMS in the China EV industry.
To compute these sensitivity indices, we employ Monte Carlo simulation due to its flexibility and accuracy. The procedure involves generating random samples of the input parameters and evaluating the corresponding outputs. First, two independent sample matrices A and B are created, each containing N samples of the n input parameters drawn from their respective distributions:
$$ \mathbf{A} = \begin{bmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{N1} & x_{N2} & \cdots & x_{Nn}
\end{bmatrix}, \quad
\mathbf{B} = \begin{bmatrix}
x’_{11} & x’_{12} & \cdots & x’_{1n} \\
x’_{21} & x’_{22} & \cdots & x’_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x’_{N1} & x’_{N2} & \cdots & x’_{Nn}
\end{bmatrix} $$
Next, for each parameter \( x_i \), a modified matrix \( \mathbf{C}_i \) is formed by replacing the i-th column of A with the i-th column of B. The output responses are computed for all sample matrices, resulting in:
$$ \mathbf{y}_A = \begin{bmatrix}
y_{A11} & y_{A12} & \cdots & y_{A1m} \\
y_{A21} & y_{A22} & \cdots & y_{A2m} \\
\vdots & \vdots & \ddots & \vdots \\
y_{AN1} & y_{AN2} & \cdots & y_{ANm}
\end{bmatrix}, \quad
\mathbf{y}_B = \begin{bmatrix}
y_{B11} & y_{B12} & \cdots & y_{B1m} \\
y_{B21} & y_{B22} & \cdots & y_{B2m} \\
\vdots & \vdots & \ddots & \vdots \\
y_{BN1} & y_{BN2} & \cdots & y_{BNm}
\end{bmatrix}, \quad
\mathbf{y}_{C_i} = \begin{bmatrix}
y_{C_i11} & y_{C_i12} & \cdots & y_{C_i1m} \\
y_{C_i21} & y_{C_i22} & \cdots & y_{C_i2m} \\
\vdots & \vdots & \ddots & \vdots \\
y_{C_iN1} & y_{C_iN2} & \cdots & y_{C_iNm}
\end{bmatrix} $$
The first-order and total sensitivity indices are estimated using the following formulas:
$$ \hat{S}_i = \frac{ \sum_{s_1=1}^{m} \sum_{s_2=1}^{m} \left( \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_1} y_{C_ijs_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right) }{ \sum_{s_1=1}^{m} \sum_{s_2=1}^{m} \left( \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_1} y_{Ajs_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right) } $$
$$ \hat{S}_{Ti} = \frac{ \sum_{s_1=1}^{m} \sum_{s_2=1}^{m} \left( \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_1} y_{Ajs_2} – \frac{1}{N} \sum_{j=1}^{N} y_{Bjs_1} y_{C_ijs_2} \right) }{ \sum_{s_1=1}^{m} \sum_{s_2=1}^{m} \left( \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_1} y_{Ajs_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right) } $$
where \( \bar{y}_{A s_1} = \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_1} \) and \( \bar{y}_{A s_2} = \frac{1}{N} \sum_{j=1}^{N} y_{Ajs_2} \) are the mean values of the outputs. This Monte Carlo approach ensures accurate estimation of sensitivity indices, even for complex electric car PMS models.
In the case study, we apply the multi-output sensitivity analysis to a representative China EV model with a transverse-mounted powertrain. The PMS includes three mounts, and the focus is on the X, Z, and θY directions, which are critical for vibration isolation in electric cars. The output responses comprise natural frequencies (f_X, f_Z, f_{θY}) and decoupling rates (d_X, d_Z, d_{θY}). The uncertain parameters are listed in the table below, along with their mean values and coefficients of variation. For instance, mount stiffnesses are assigned a 10% uncertainty, while inertial parameters have a 3% variation, reflecting typical ranges in China EV manufacturing.
| Parameter Type | Mean Value | Coefficient of Variation |
|---|---|---|
| Powertrain Inertia I_yy | 1.49 kg·m² | 3% |
| Powertrain Inertia I_zz | 1.60 kg·m² | 3% |
| Front Suspension Stiffness k_f | 27.36 N/mm | 10% |
| Rear Suspension Stiffness k_r | 25.60 N/mm | 10% |
| Wheel Vertical Stiffness k_w | 210 N/mm | 10% |
| Mount 1 u-direction Stiffness k_u1 | 141.6 N/mm | 10% |
| Mount 1 v-direction Stiffness k_v1 | 90.0 N/mm | 10% |
| Mount 1 w-direction Stiffness k_w1 | 141.6 N/mm | 10% |
| Mount 2 u-direction Stiffness k_u2 | 93.6 N/mm | 10% |
| Mount 2 v-direction Stiffness k_v2 | 111.6 N/mm | 10% |
| Mount 2 w-direction Stiffness k_w2 | 180.0 N/mm | 10% |
| Mount 3 u-direction Stiffness k_u3 | 114.0 N/mm | 10% |
| Mount 3 v-direction Stiffness k_v3 | 75.6 N/mm | 10% |
| Mount 3 w-direction Stiffness k_w3 | 225.6 N/mm | 10% |
The single-output sensitivity analysis for each response reveals distinct parameter influences. For example, the natural frequency in the X-direction (f_X) is highly sensitive to mount stiffness parameters like k_u1, k_v1, k_w1, k_u2, k_v2, k_u3, and k_v3, with first-order indices indicating significant independent effects. However, for the Z-direction natural frequency (f_Z), only k_w1 and k_w3 show substantial impact, and interactions are minimal. Similarly, decoupling rates exhibit complex dependencies; d_X is influenced by multiple mount stiffnesses, while d_Z is dominated by wheel vertical stiffness k_w. These single-output results can be contradictory, as a parameter like k_u1 strongly affects f_X and d_X but has negligible impact on f_Z and f_{θY}. This underscores the limitation of traditional methods in electric car PMS design.
In contrast, the multi-output sensitivity analysis aggregates all six responses into a unified framework. The first-order and total sensitivity indices are computed using Monte Carlo simulation with N=10^6 samples, ensuring convergence. The results are summarized in the table below, which ranks parameters based on their total sensitivity indices. Key parameters include k_w1, k_w, k_w3, k_u2, k_v1, k_v2, k_v3, k_u1, I_yy, I_zz, k_u3, k_w2, and k_f, indicating their overall influence on the electric car PMS performance. The first-order indices reveal that parameters like k_w, k_u1, k_v1, k_w1, k_u2, k_v2, k_u3, k_v3, and k_w3 have significant independent effects, while the differences between first-order and total indices highlight substantial interaction effects among parameters.
| Parameter | First-Order Index S_i | Total Sensitivity Index S_{Ti} |
|---|---|---|
| k_w1 | 0.12 | 0.18 |
| k_w | 0.10 | 0.16 |
| k_w3 | 0.09 | 0.15 |
| k_u2 | 0.08 | 0.14 |
| k_v1 | 0.07 | 0.13 |
| k_v2 | 0.07 | 0.12 |
| k_v3 | 0.06 | 0.11 |
| k_u1 | 0.05 | 0.10 |
| I_yy | 0.04 | 0.09 |
| I_zz | 0.04 | 0.08 |
| k_u3 | 0.03 | 0.07 |
| k_w2 | 0.02 | 0.06 |
| k_f | 0.01 | 0.05 |
The multi-output approach resolves ambiguities present in single-output analyses. For instance, k_u1 and k_w, which showed conflicting influences in single-output results, are clearly ranked in the multi-output context, with k_w having a higher overall impact. This demonstrates the method’s utility for China EV development, where designers need to prioritize parameters for robust optimization. Furthermore, the sensitivity indices account for correlations between outputs, providing a more accurate assessment than methods that ignore these dependencies.
In conclusion, this study presents a novel multi-output sensitivity analysis framework for electric car powertrain mounting systems, addressing the limitations of traditional single-output methods. By incorporating parameter uncertainties and multiple performance responses, the approach enables comprehensive evaluation of parameter influences, leading to more reliable design decisions for China EV applications. The case study validates the method’s effectiveness, showing that key parameters such as mount stiffnesses and inertial properties significantly impact the system’s dynamic behavior. Future work could extend this approach to include more complex uncertainty models or real-time optimization algorithms, further enhancing the robustness of electric car PMS in the evolving automotive industry.