Keywords

1 Introduction

Rubber bushes in suspension systems are used to absorb vibrations from the road surface and improve the ride comfort of the vehicle. In addition, the deformation of rubber bushes changes the wheel alignment, which also affects the handling characteristics. Thus, it is important to consider the characteristics of rubber bushes in the analysis of a suspension system. Since rubber bushes have high stiffness, a numerical analysis of a suspension system which considers the effect of rubber bushes requires a small step size [1]. This increases the computational costs, making it difficult to apply the suspension model to real-time analysis such as driving simulators. Therefore, the aim of this study is to generate a surrogate model of a suspension system with high stiffness elements for real-time analysis using machine learning. The Long Short-Term Memory Network (LSTM) was used as the machine learning method because it is capable of long-term time series prediction [2]. The response and computational time of the LSTM suspension model were evaluated by comparison with those of the quarter car model with a bushing element.

2 Analysis Using Suspension Models with High Stiffness Elements

In this study, a three-degree-of-freedom quarter car model with a bush in the lower part of damper shown in Fig. 1 was created to evaluate the effect of the bush on ride comfort. The model parameters are shown in Table 1. The parameters were determined based on actual automobiles. Equations (1), (2) and (3) define the equations of motion for the quarter car model.

$$ m_1 \ddot{x}_1 = k_b (x_b - x_1 ) + c_b (\dot{x}_b - \dot{x}_1 ) - k_1 (x_1 - x_0 ) $$
(1)
$$ m_b \ddot{x}_b = k_2 (x_2 - x_b ) + c_2 (\dot{x}_2 - \dot{x}_b ) - k_b (x_b - x_1 ) - c_b (\dot{x}_b - \dot{x}_1 ) $$
(2)
$$ m_2 \ddot{x}_2 = - k_2 (x_2 - x_b ) - c_2 (\dot{x}_2 - \dot{x}_b ) $$
(3)
Table 1. Model Parameters
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Fig. 1.
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3 DOF quarter car model

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The quarter car model was subjected to the road surface displacement shown in Fig. 2a to examine the response of body acceleration when the stiffness values of the bush was changed. Figure 2b shows the body acceleration during the simulation, and Fig. 2c shows the deformation of the bush. It was confirmed that the peak value of the body acceleration increases as the bushing stiffness increases, while the deformation of the bush became smaller.

Fig. 2.
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Simulation results of the bump run using the 3DOF model

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3 Procedure for Generating Surrogate Model

3.1 Construction of Surrogate Models

The objective of this study is to develop a surrogate model that can predict body acceleration, when the stiffness values of the bush was changed. A similar approach was taken in reference [3], but this study examined three different surrogate modeling methods as shown in Table 2. The machine learning model 1 predicts the body acceleration for the road surface inputs. On the other hand, the machine learning model 2 and model 3 predict the response of the bush for the displacement of the unsprung mass. The response of the body acceleration was calculated based on the machine learning prediction results and Eq. (3). Equation (2) that defines the equations of motion for the bush has a high natural frequency, which increases the calculation time of the system. Machine learning models 2 and 3 were examined to replace equations with a large computational load. The model 2 predicts the displacement and velocity of the bush, the model 3 predicts the bush deformation and bush deformation rate. The response of the body acceleration is influenced by various forces such as those from the spring, damper, and bush. The changes caused by the effect of the bush on body acceleration are small as shown in Fig. 2b, so it is difficult to predict the body acceleration directly with machine learning. It is expected to predict the changes in body acceleration caused by the bush with good accuracy by predicting bushing deformation and using the equations of motion to calculate body acceleration.

Table 2. Surrogate Model inputs and outputs
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3.2 Machine Learning Method

The long short-term memory network (LSTM) was used as the machine learning method, which can learn long time series data and ensure long-term dependability. Figure 3 shows the relationship between hyperparameter settings and performance in the machine learning model 3. The root mean squared error (RMSE) was used as the metric for prediction accuracy. Figure 3 shows that the machine learning model with a large number of units and layers makes more accurate predictions, but it also takes more computation time. The number of hidden layers was set to 2 and the number of units to 40 in each layer from the perspectives of the computation time and the accuracy. The other hyperparameters are shown in Table 3.

Table 3. Hyperparameters setting
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Fig. 3.
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Calculation time and RMSE with hyperparameters

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3.3 Preparation of Road Surface Input Data

Two types of random road profiles were generated according to the ISO 8608 standard as shown in Fig. 4. Figure 4a shows the road surface for machine learning training, and Fig. 4b shows the road surface used to obtain predict data. Simulations were conducted on a quarter-car model, assuming a speed of 60 km/h on these road surfaces. The simulation results were used as training and predict data for machine learning.

Fig. 4.
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Road surface profile used for simulations

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4 Surrogate Modeling and Accuracy Validation

Figure 5a shows the calculated body acceleration for machine learning models, and Fig. 5b shows its power spectral density (PSD). The hyperparameters of each machine learning model were set to the values shown in Table 3 to align the conditions. Figure 5b shows that the machine learning model 1 has a lower PSD than the quarter car model above 100 Hz, while the machine learning model 3 maintains good accuracy in predicting body acceleration up to 200 Hz. Figure 6 shows the ratio of the maximum body acceleration predicted by each machine learning model to that of the quarter car model for different bushing parameters. It was confirmed that calculating body acceleration based on the model 3, which predicts bushing deformation, is more accurate than predicting vehicle body acceleration based on the model 1.

Fig. 5.
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Output data used for machine learning test data

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Fig. 6.
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Error ratio of the body acceleration for different bush parameter

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In addition, the computational time for the quarter car model and machine learning model 3 were evaluated on the MATLAB environment on Windows 11. Both analyses were performed with the fixed-time step solver ode4 (Runge-Kutta). Figure 7 shows the RMSE calculated for each time step solution against the solution with a time step of 10e−8 s. RMSE of the body acceleration increases with a large step time setting, and eventually the solution will diverge. If the stiffness of the bushing is constant at kb = 2.0e6 N/m, the time step at which the solution diverges depends on the damping coefficient. On the other hand, the machine learning model was stable, meaning that the time step of this model can be adjusted according to the purpose. In this study, the machine learning model was set up with three different time steps of 0.1 ms, 1 ms, and 10 ms. Table 4 shows the computation time for a 5-s simulation for each model. These results show that even with large time steps, the predictions of the machine learning model were accurate and computationally fast. In particular, when the time step was 1 ms or 10 ms, the machine learning model was able to compute faster compared to the quarter-car model calculated at the maximum time step that can be analyzed with the settings in Table 1. Thus, for real-time analysis, the machine learning model with prediction accuracy has an advantage over quarter car model in terms of computational time.

Fig. 7.
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RMSE of the body acceleration when the time step is changed

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Table 4. Calculation time and accuracy for each model
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5 Conclusion

In this study, the surrogate modeling methods of a suspension system with high stiffness elements using machine learning was investigated. It was confirmed that the response of the body acceleration changed depending on the stiffness and damping characteristics of the bush with a three-degree-of-freedom quarter car model with a bush element. Three surrogate modeling methods were examined to predict body acceleration. As a result, it was confirmed that the response of the body acceleration was predicted with good accuracy by predicting the bush deformation and calculating the body acceleration based on the prediction results, instead of predicting the body acceleration. It was also found that the machine learning model was stable and had an advantage over the quarter car model in terms of computation time.