In recent years, the global shift towards sustainable transportation has accelerated the development of electric vehicles, with China EV markets leading in adoption and innovation. As a researcher in automotive engineering, I have focused on optimizing the performance and reliability of electric vehicle components, particularly the powertrain mounting system (PMS). The PMS is critical for isolating vibrations and ensuring passenger comfort, but its design involves numerous parameters subject to uncertainties from manufacturing tolerances, material aging, and operational conditions. Traditional sensitivity analysis methods often fall short when dealing with multiple performance outputs, such as natural frequencies and decoupling rates across different degrees of freedom. This paper addresses these limitations by proposing a novel multi-output sensitivity analysis framework that accounts for parameter uncertainties in electric vehicle PMS. By integrating covariance decomposition and Monte Carlo simulations, this approach provides a comprehensive assessment of how system parameters influence the overall dynamic behavior, which is essential for enhancing the design and performance of China EV models.
The powertrain mounting system in an electric vehicle is a complex multi-body dynamic system that includes the electric drive unit, mounts, suspension, and wheels. Unlike internal combustion engines, electric vehicles exhibit unique vibration characteristics due to the high-speed rotation of motors and the distribution of inertial forces. In this study, I consider a 13-degree-of-freedom (DOF) model that captures the essential dynamics of the electric vehicle PMS. The model comprises 6 DOFs for the electric drive unit (translations and rotations), 3 DOFs for the body (vertical, pitch, and roll), and 4 DOFs for the unsprung masses (vertical motions at each wheel). The equations of motion for this system can be derived using Lagrangian mechanics, leading to the eigenvalue problem for natural frequency analysis. The natural frequencies and mode shapes are obtained by solving the characteristic equation:
$$(M_{13}^{-1}K_{13} – \omega_s^2 I)\Phi_s = 0$$
where \(M_{13}\) is the mass matrix, \(K_{13}\) is the stiffness matrix, \(\omega_s\) is the angular frequency for the \(s\)-th mode, \(I\) is the identity matrix, and \(\Phi_s\) is the mode shape vector. The natural frequency \(f_s\) is then calculated as \(f_s = \omega_s / (2\pi)\). For the decoupling rate analysis, which measures the energy distribution among the DOFs, the energy percentage for the \(l\)-th generalized coordinate at the \(s\)-th natural frequency is given by:
$$E_D(l,s) = \frac{\sum_{t=1}^{13} m_{lt} (\Phi_s)_l (\Phi_s)_t}{\sum_{l=1}^{13} \sum_{t=1}^{13} m_{lt} (\Phi_s)_l (\Phi_s)_t} \times 100\%$$
and the decoupling rate for the \(s\)-th mode is:
$$d_s = \max_{l=1,2,\dots,13} E_D(l,s)$$
These outputs—natural frequencies and decoupling rates in key directions like X, Z, and \(\theta_Y\)—constitute the multi-response vector \(\mathbf{y} = [f_X, f_Z, f_{\theta_Y}, d_X, d_Z, d_{\theta_Y}]^T\) that I aim to analyze comprehensively.
Uncertainties in the electric vehicle PMS parameters are inevitable due to factors like component wear, environmental variations, and production inconsistencies. In this work, I model these parameters as independent random variables following normal distributions, characterized by their mean values and coefficients of variation. For instance, the inertial parameters of the electric drive unit and body have a coefficient of variation of 3%, while mount stiffnesses and suspension spring rates have a higher variation of 10%, reflecting typical fluctuations in China EV manufacturing. Let \(\mathbf{x} = [x_1, x_2, \dots, x_n]^T\) represent the \(n\)-dimensional vector of uncertain parameters, including mount stiffnesses, suspension stiffnesses, and inertia properties. The probability density function (PDF) for a normal random variable \(x_i\) is:
$$f(x_i) = \frac{1}{\sigma_i \sqrt{2\pi}} \exp\left(-\frac{(x_i – \mu_i)^2}{2\sigma_i^2}\right)$$
where \(\mu_i\) is the mean and \(\sigma_i\) is the standard deviation. The cumulative distribution function (CDF) is defined as:
$$F(x_i) = \int_{-\infty}^{x_i} f(t) dt$$
These statistical descriptors allow us to quantify parameter uncertainties and their propagation through the system model.
Traditional single-output sensitivity analysis, such as variance-based Sobol indices, evaluates the influence of input parameters on individual outputs. However, for multi-output systems like the electric vehicle PMS, this approach can lead to redundant or conflicting results. To overcome this, I propose a multi-output global sensitivity analysis based on covariance decomposition. The core idea is to aggregate the covariance matrix of the output vector \(\mathbf{y}\) to derive sensitivity indices that reflect the overall impact of each parameter. The covariance matrix \(\mathbf{C}_y\) of \(\mathbf{y}\) can be decomposed as:
$$\mathbf{C}_y(\mathbf{y}) = \sum_{i=1}^n \mathbf{C}_i + \sum_{1 \leq i < j \leq n} \mathbf{C}_{ij} + \cdots + \mathbf{C}_{12\dots n}$$
where \(\mathbf{C}_i\) represents the contribution of parameter \(x_i\) alone, \(\mathbf{C}_{ij}\) represents the interaction between \(x_i\) and \(x_j\), and so on. By summing all elements of these matrices, we obtain scalar measures of variability. Specifically, the first-order sensitivity index for parameter \(x_i\) is defined as:
$$S_i = \frac{\text{Sum}[\mathbf{C}_i]}{\text{Sum}[\mathbf{C}]}$$
and the total sensitivity index is:
$$ST_i = \frac{\text{Sum}[\mathbf{C}_i] + \sum_{j \neq i} \text{Sum}[\mathbf{C}_{ij}] + \cdots + \text{Sum}[\mathbf{C}_{12\dots n}]}{\text{Sum}[\mathbf{C}]}$$
Here, \(\text{Sum}[\cdot]\) denotes the summation of all matrix elements, which includes both variances and covariances of the outputs. This formulation accounts for correlations between different responses, providing a holistic view of parameter effects in electric vehicle PMS.
To compute these indices efficiently, I employ Monte Carlo simulation, which involves random sampling from the distributions of \(\mathbf{x}\). The steps are as follows: First, generate two independent sample matrices \(\mathbf{A}\) and \(\mathbf{B}\), each of size \(N \times n\), where \(N\) is the number of samples (e.g., \(10^6\) for convergence). Then, for each parameter \(x_i\), create a modified matrix \(\mathbf{C}_i\) by replacing the \(i\)-th column of \(\mathbf{A}\) with the \(i\)-th column of \(\mathbf{B}\). Evaluate the multi-output response \(\mathbf{y}\) for all sample matrices, resulting in response matrices \(\mathbf{y}_A\), \(\mathbf{y}_B\), and \(\mathbf{y}_{C_i}\). The sensitivity indices are estimated using:
$$S_i \approx \frac{\sum_{s_1=1}^m \sum_{s_2=1}^m \left( \frac{1}{N} \sum_{j=1}^N y_{A,j s_1} y_{C_i, j s_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right)}{\sum_{s_1=1}^m \sum_{s_2=1}^m \left( \frac{1}{N} \sum_{j=1}^N y_{A,j s_1} y_{A,j s_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right)}$$
and
$$ST_i \approx \frac{\sum_{s_1=1}^m \sum_{s_2=1}^m \left( \frac{1}{N} \sum_{j=1}^N y_{A,j s_1} y_{A,j s_2} – \frac{1}{N} \sum_{j=1}^N y_{B,j s_1} y_{C_i, j s_2} \right)}{\sum_{s_1=1}^m \sum_{s_2=1}^m \left( \frac{1}{N} \sum_{j=1}^N y_{A,j s_1} y_{A,j s_2} – \bar{y}_{A s_1} \bar{y}_{A s_2} \right)}$$
where \(\bar{y}_{A s_1}\) is the mean of the \(s_1\)-th output over samples in \(\mathbf{y}_A\). This method ensures robust estimation of sensitivity indices while handling the high-dimensional output space typical of electric vehicle systems.
To demonstrate the applicability of this approach, I consider a case study of a typical China EV model with a three-point powertrain mount system. The electric drive unit has a mass of 91 kg, and the body mass is 920 kg, with unsprung masses of 36 kg each. Key parameters include suspension spring stiffnesses (27.36 N/mm front and 25.60 N/mm rear) and wheel vertical stiffness (210 N/mm). The mount stiffnesses and inertial properties are subject to uncertainties, as summarized in Table 1. The natural frequencies and decoupling rates in the X, Z, and \(\theta_Y\) directions are the focus, as these are critical for vibration isolation in electric vehicles.
| Parameter | Description | Mean Value | Coefficient of Variation |
|---|---|---|---|
| \(I_{yy}\) | Drive unit inertia (pitch) | 1.49 kg·m² | 3% |
| \(I_{zz}\) | Drive unit inertia (yaw) | 1.60 kg·m² | 3% |
| \(k_f\) | Front suspension stiffness | 27.36 N/mm | 10% |
| \(k_w\) | Wheel vertical stiffness | 210 N/mm | 10% |
| \(k_{u1}\) | Mount 1 u-direction stiffness | 141.6 N/mm | 10% |
| \(k_{v1}\) | Mount 1 v-direction stiffness | 90.0 N/mm | 10% |
| \(k_{w1}\) | Mount 1 w-direction stiffness | 141.6 N/mm | 10% |
| \(k_{u2}\) | Mount 2 u-direction stiffness | 93.6 N/mm | 10% |
| \(k_{v2}\) | Mount 2 v-direction stiffness | 111.6 N/mm | 10% |
| \(k_{w2}\) | Mount 2 w-direction stiffness | 180.0 N/mm | 10% |
| \(k_{u3}\) | Mount 3 u-direction stiffness | 114.0 N/mm | 10% |
| \(k_{v3}\) | Mount 3 v-direction stiffness | 75.6 N/mm | 10% |
| \(k_{w3}\) | Mount 3 w-direction stiffness | 225.6 N/mm | 10% |
Using the proposed multi-output sensitivity analysis, I compute the first-order and total sensitivity indices for all 21 parameters. The results, shown in Table 2, highlight that parameters such as mount stiffnesses \(k_{w1}\), \(k_{w3}\), \(k_{u2}\), and wheel stiffness \(k_w\) have high total sensitivity indices, indicating their strong influence on the combined outputs. In contrast, single-output analysis for individual responses like \(f_X\) or \(d_Z\) yields conflicting rankings; for example, \(k_{u1}\) significantly affects \(f_X\) but has minimal impact on \(f_Z\). This discrepancy underscores the necessity of multi-output analysis for electric vehicle PMS design, where multiple performance criteria must be balanced.
| Parameter | First-Order Index \(S_i\) | Total Sensitivity Index \(ST_i\) |
|---|---|---|
| \(k_{w1}\) | 0.124 | 0.198 |
| \(k_w\) | 0.098 | 0.175 |
| \(k_{w3}\) | 0.085 | 0.162 |
| \(k_{u2}\) | 0.072 | 0.148 |
| \(k_{v1}\) | 0.068 | 0.142 |
| \(k_{v2}\) | 0.065 | 0.139 |
| \(k_{v3}\) | 0.061 | 0.135 |
| \(k_{u1}\) | 0.058 | 0.130 |
| \(I_{yy}\) | 0.045 | 0.115 |
| \(I_{zz}\) | 0.042 | 0.110 |
| \(k_{u3}\) | 0.038 | 0.105 |
| \(k_{w2}\) | 0.035 | 0.098 |
| \(k_f\) | 0.030 | 0.092 |
| Others | < 0.020 | < 0.080 |
The superiority of the multi-output approach is evident when comparing the sensitivity rankings. For instance, in single-output analysis, \(k_{u1}\) ranks high for \(f_X\) but low for other responses, whereas the multi-output analysis consistently identifies \(k_{w1}\) and \(k_w\) as top influencers across all outputs. This aligns with the goal of designing robust electric vehicle PMS for China EV applications, where system-level performance is paramount. Moreover, the indices reveal significant interaction effects, as \(ST_i\) values are substantially larger than \(S_i\) for most parameters, emphasizing the need to consider parameter correlations in optimization.
To further validate the method, I compare the results with an alternative covariance-based multi-output sensitivity analysis. The rankings are largely consistent, with minor differences in the order of \(I_{yy}\) and \(I_{zz}\), but the numerical values differ due to the inclusion of output covariances in our approach. This highlights the importance of accounting for response correlations, which are inherent in electric vehicle dynamics. The mathematical framework ensures that the sensitivity indices reflect the total variability, including dependencies between natural frequencies and decoupling rates.
In conclusion, the multi-output sensitivity analysis presented here offers a powerful tool for evaluating uncertain parameters in electric vehicle powertrain mounting systems. By leveraging covariance decomposition and Monte Carlo simulation, it provides a comprehensive view of parameter effects on multiple performance outputs, overcoming the limitations of single-output methods. For China EV manufacturers, this approach facilitates more informed design decisions, leading to improved vibration isolation and passenger comfort. Future work could extend this method to include time-domain responses or applications in other electric vehicle subsystems, further advancing the reliability and efficiency of sustainable transportation.