Analysis of the Load-Stress Response Characteristics of the Bogie Frame in Intercity Electric Multiple Unit

Quasi-static load-stress calibration is performed in a laboratory to study the load-stress transitive relation of the intercity EMU. The calibration site is shown in Figure 1. The assembly of the bogie frame follows the technical requirements of its actual operating condition, and the calibration load system is shown in Figure 2.

The bogie frame has many load systems, such as sink-float, side roll, torsion, lateral, motor bracket, gearbox bracket, anti-yaw, wheel brake, and traction loads. A global load and key stress points are adopted to analyze the transitive relation and completely verify the idea proposed in this paper. The measuring points utilized include D13, D15, D26, D34, D46, D48, D55, and D57, which represent key response points in various regions. The specific locations of these points are shown in Figure 3.

The calibration data of the laboratory bench are shown in Table 1.

As shown in Table 1, all eight selected measuring points have different response factors for all loads. Therefore, a closed computational model composed of eight stress points and nine loads is formed. The calculation methods for sink-float, side roll, and torsion loads are presented in a previous study [1]; the other loads are obtained with component sensors, which are calibrated by grouping the bridge circuits consisting of various sensitive points in each component part [20]. Repetitious data are not presented in the current paper.

The calibrated and well-assembled bogie frame is mounted on the intercity EMU; the operating area is Dazhou–Chengdu. Given that the online test involves other test projects, the braking system signals are removed, and the wheel braking coefficient of the transitive relation matrix is ignored.

The test bench indicates that the transitive relation load-stress response is

A group of test signals of the test run with a speed of 160 km/h are selected. The frequency spectral maps of some of the original signals are shown in Figures 4, 5, 6, 7.

As shown in the frequency spectral maps, the frequency spectra are mainly concentrated in the low- and medium-frequency responses below 100 Hz. This result is consistent with that in other studies [21‐23]. The frequency spectrum of the motor bracket load is concentrated at 20 and 60 HZ. The frequency spectrum of the gearbox bracket load is concentrated below 10 Hz. The torsion load spectrum is concentrated below 10 Hz. The frequency spectrum of the stress response is concentrated at 20 and 60 Hz. Accordingly, a 100 Hz low-pass filter and a 50 Hz band-pass filter are adopted to remove some random disturbances and power-frequency interferences.

If the load-stress response characteristics satisfy a linear relationship, then the stress value can be calculated using the load value, that is, the stress value is the product of the filtered load signal and the corresponding transitive relation coefficient. Subsequently, the calculated stress value and the actual measured stress value can be compared. In Figure 8, the calculated value of the assumed linear response at measuring point D46 is smaller than the measured value (blue line covers the red line; difference will be shown in Table 2). In Figure 9, the calculated value of the assumed linear response at measuring point D48 is greater than the measured value (red line covers the blue line; difference will be shown in Table 2).

Given that the adopted frequency is 1000 Hz and the maximum speed is maintained within 200 km/h, a 400 km run takes approximately 3 h, and the amount of sample data is approximately 10⁷. For simplification, the stress distribution evaluation method can be adopted for the fitting evaluation of such a large amount of data. However, the stress distribution function has no chronological order; consequently, it has poor relativity with the time-domain signal. Thus, this paper adopts percentage mean square error (PMSE) for fitting evaluation.where d(n) is the calculated stress and x(n) is the actual measured signal. Values that are less than 2 are disregarded to eliminate the interference error of the 0 value signal.

PMSE, expressed as follows, is adopted as an evaluation index:where N is the fitting sample size.

The fitting evaluation method can be utilized to identify the difference between the stress value calculated using the calibration coefficient and the actual stress value in the operation process.

The fitting error of each stress calculated according to the load time history signal and the load-stress calibration coefficient is shown in Table 2.

According to different transfer functions, the neural network can be divided into single-layer perceptron neural network, linear neural network, multi-node BP neural network, RBF/GRNN network, Hopfield neural network, random neural network, and so on. Single-layer perceptron neural networks are mainly utilized to solve binary pattern recognition problems. Linear neural networks can be used for simple linear separable stress values and linear fitting. BP networks can be employed for linear non-separable pattern recognition, function fitting, and optimization. RBF/GRNN networks can be used for the function approximation of small samples. Hopfield networks can be employed to solve complex pattern recognition problems and achieve prediction and optimization in time domain [22].

The RBF/GRNN network trains a number of network nodes according to sample size [24, 25]. If the sample size is larger than 105, the network will suffer from heavy operational burden, and a space overflow error will occur. Given that the amount of sample data obtained through the test is 107, an optimized BP neural network is adopted in this study for calculation.

The BP network algorithm is the most widely used algorithm for perceptron networks [26]; approximately 80% of neural network models adopt a BP network or a modification of a BP network. A multilayer perceptron network usually consists of three parts: input, hidden, and output layers. The input and output layers have one layer, whereas the hidden layer may have multiple layers. Currently, the two-layer hidden network is the most widely used network [26]; its structure is shown in Figure 10. For a multi-layer feedforward network, the number of nodes in a hidden layer is the key to success. If the number of nodes is too small, the network cannot obtain sufficient information to solve the problem; if the number of nodes is too large, the training time will be increased. More importantly, too many nodes in a hidden layer may lead to the so-called “overfitting” problem, that is, the test error will increase and lead to reduced fitting capability. Therefore, the number of nodes in a hidden layer should be rationally selected. Deciding on the number of hidden layers and nodes in a hidden layer is a complex task, but the general principle is to select a few nodes in a hidden layer to correctly reflect the input–output relationship and simplify the network structure. In this study, a network structure incremental pattern is adopted. That is, a few nodes are set to train the network and test the learning error. The number of nodes is then gradually increased until the learning error has no significant reduction.

The transfer function and number of network nodes are adopted as parameters to compare nonlinear and linear optimization.

The transfer function can be divided into three categories, as shown in Figure 11. Nonlinear logsig function is expressed as Eq. (4), and nonlinear tansig is expressed as Eq. (5). The linear transfer function is purelin, i.e., y = x, whose linear coefficient is the coefficient of the network node.

The calculation method is shown in the flowchart in Figure 12.

With D46 as the calculation object, the nonlinear network has eight inputs (lateral, sink-float, side roll, torsion, gearbox bracket, motor bracket, longitudinal, and anti-yaw loads) and one stress point as the output. Part of the data calculated in the optimization process is listed in Table 3. As shown in Table 3, the optimal result is that all hidden layers form a linear transitive relation network.

The optimal solution is the network structure with a linear transitive relation. The coefficient of the linear input layer with one node and the first layer of the hidden layer is

The coefficient of the first linear hidden layer and the second linear hidden layer is

Therefore, the transitive relation between the input layer, which includes eight input loads, and the output layer at the D46 measuring point is

Therefore, the neural network that contains two hidden layers can be converted into a single-layer linear network, the topology structure of which is shown in Figure 13.

The transitive relation of the two-node linear network structure can be simplified into a one-node single-layer linear network through the same method.

As shown in the process in Figure 13, the transitive relation coefficient between the stress at D46 and the eight loads is expressed as Eq. (9), which is optimized by the linear neural network. Similarly, a single-layer network computational method can be utilized to obtain the transitive relation coefficients between D13, D15, D26, D34, D48, D55, and D57 and the eight input loads.

A set of calculated stress values can be obtained by using the modified transitive relation in Eq. (10). The optimized stress errors are shown in Table 4. The local area model is zoomed in in Figure 14. The differences among the actual measured, unmodified calculated, and optimized calculated signals are clear. Compared with the unmodified calculated signal, the optimized calculated signal fits the actual measured signal more.

Mathematical Model Description [27]: If k factors, X₁, X₂, …, X _k , exist, and dependent variable y has the following linear relationship, then

Equation (11) is a multiple linear regression function, where β _i (i = 1, 2, …, k) is called the regression coefficient. If all X₁, X₂, …, X _k have no significant influence on y, the coefficient in the model (Eq. (11)) is β _i  =  0 (i = 1, 2, …, k). The significance test aims to assess whether

The three error statistics are total deviation square sum \(S_{T}^{2} = \sum\limits_{i = 1}^{n} {(y_{i} - \bar{y})^{2} } ,\) residual sum of squares \(S_{E}^{2} = \sum\limits_{i = 1}^{n} {(y_{i} - \hat{y}_{i} )^{2} } ,\) and the regression sum of squares \(S_{R}^{2} = \sum\limits_{i = 1}^{n} {(\hat{y}_{i} - \bar{y})^{2} } .\) If β₁ = β₂ = …=β _k  =  0, then \(\frac{{S_{T}^{2} }}{{\sigma^{2} }} \sim \chi^{2} (n - 1)\) and \(\frac{{S_{R}^{2} }}{{\sigma^{2} }} \sim \chi^{2} (k)\) is true.

The above property shows that the identified \(S_{T}^{2}\) should be small if H₀ is true and that large \(\frac{{S_{R}^{2} }}{{S_{E}^{2} }}\) is a low-probability event. The rejection region form is \(\left\{ {\frac{{S_{R}^{2} }}{{S_{E}^{2} }} > c} \right\}\). Therefore, if H₀ is correct, the critical value of the given significant level α is

When y and X₁, X₂, …, X _k have a significant linear relationship, the significance of each variable X _i (i = 1, 2, …, k) should be verified. If X _i exerts no significant effect on Y, then β _i should be zero, that is, H_0i: β _i  = 0 (i = 1, 2, …, k) should be tested. Its rejection region form is {|β _i | > c _i }, where

After substituting the stress time history data, relevant load time history data, and optimized coefficient of D46 into the above equation, the following can be obtained.

If \({\frac{{S_{R}^{2} }}{{S_{E}^{2} }} > c}\) then it falls in the H₀ rejection region; this condition indicates that the linear relationship coefficient of all X and Y cannot be zero. Thus, a linear relationship is established.

Each linear coefficient influences the significance level, which is presented in Table 5.

β _i > c _i , (i = 1,…, 8), indicating that H_0i (i = 1,…, 8) falls in the rejection region, and all variables (x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈) have a significant linear impact on y.

After obtaining the optimized transitive relation in the test run at an operating speed of 160 km/h, the load data obtained in a formal test at a speed of 200 km/h can be utilized to predict stress data. The coefficient calibrated in the laboratory and the actual measured signal can be used for comparison. The local area model is zoomed in in Figure 15. The stress data predicted by the modified transitive relation and measured signal error are shown in Table 6.

After modifying the transitive relation between load and stress, the load data of the intercity EMU at different working conditions, such as entering or departing the depot, upgoing line, downgoing line, and heavy vehicle load, are substituted into the aforementioned transitive relation and the transitive relation calibrated in the laboratory. Two calculated stress errors at the measuring points are shown in Tables 7 and 8.

As shown in Tables 7 and 8, all the stress value errors calculated with the optimized transitive relation are smaller than the stress value errors obtained with the laboratory-calibrated transitive relation. The optimized transitive relation can truly reflect the load-stress transitive relation of the bogie frame under the constraints of line operation. The data in Tables 7 and 8 also indicate that the modified load-stress transitive relation does not significantly vary in different working conditions.

The laboratory-calibrated load-stress transitive relation differs from the real load-stress transitive relation. This difference is caused by the difference between the constraints on the road and those on the laboratory bench, even if the laboratory bench installs the bogie, including spring, axle box, rod, anti-yaw bracket, and other related accessories, according to actual operating conditions. The most significant influencing factors are rigidity matching and gap issues. Different constraint stiffness values and contact gaps in the same direction may offset or increase the stress caused by different loads. Another cause for such a difference is the effect of the changing wheel-rail contact point on the arm of force. Therefore, from a statistical perspective, the optimized load-stress transitive relation is more reasonable than that calibrated by laboratory bench. The stress calculated by the optimized transfer coefficient still exhibits a certain degree of deviation from the measured signal. Such deviation is mainly caused by partial loads. For example, primary vertical damper loads and secondary horizontal damper loads are not measured in this test. Another reason is that elastic vibration phenomenon exists in the local structure (e.g., motor bracket). Overall, load and stress exhibit a linear relationship under the operating conditions.

The frequency-domain analysis indicates that when elastic vibration does not occur, the transitive relation can be described as a linear transitive relation, namely, x = ΣK _i F _i , where F _i denotes the load, K _i denotes the transitive relation coefficient, and x denotes the response strain. The linear relationship reflects the transitive relation between the load and stress of the vehicle under such assembly process conditions. When the line conditions deteriorate and the load changes, the strain response will change accordingly. When the vehicle weight increases and the load increases, the strain response will also increase. The linear relationship significance test and the comparison of data under different working conditions also prove that a linear transitive relation exists and remains unchanged.