Although various types of anti-roll torsion bars have been developed to inhibit excessive roll angle of the electric multiple unit (EMU) car body, it is critical to ensure the reliability of structural design due to the complexity of the problems involving time and uncertainties. To address this issue, a multi-objective fuzzy design optimization model is constructed considering time-variant stiffness and strength reliability constraints for the anti-roll torsion bar. A hybrid optimization strategy combining the design of experiment (DoE) sampling and non-linear programming by quadratic lagrangian (NLPQL) is presented to deal with the design optimization model. To characterize the effect of time on the structural performance of the torsion bar, the continuous-time model combined with Ito lemma is proposed to establish the time-variant stiffness and strength reliability constraints. Fuzzy mathematics is employed to conduct uncertainty quantification for the design parameters of the torsion bar. A physical programming approach is used to improve the designer's preference and to make the optimization results more consistent with engineering practices. Moreover, the effectiveness of the proposed method has been validated by comparing with current methods in a practical engineering case.
Reliability-based design optimization (RBDO) is an important way to improve the reliability of products under uncertainty in the design stage. Due to the time-variant property of design parameters, working conditions and uncertainties affecting the products’ performance, time-variant reliability-based design optimization (TRBDO) has become urgent to ensure the operating reliability and safety during the products’ lifecycle. Currently, TRBDO has attracted more attentions and is being applied in engineering practices [
For a complicated engineering problem, one main challenging task is to build the TRBDO model sufficiently considering the complexity of the working conditions and uncertainty quantification as well as the nested relationship between objectives and constraints. In addition, several objectives should be balanced and therefore the multi-objective optimization should be performed during the procedure of the TRBDO. So far, there are several preliminary developments by integrating the TRBDO and the multi-objective optimization, which will be one of the important trend of the design technique under uncertainty. Multidisciplinary design optimization (MDO) mainly focus on the coupled relationship between different disciplines or subsystems [
For the structure of EMU, the secondary suspension system usually uses the air spring to obtain better vertical performance and improve the riding comfort. However, the low stiffness of the air spring will lead to the reduction of the roll stiffness of the vehicle and then the roll angle increases. Especially when the vehicle is operating on the curve track with superelevation, or when the vehicle encounters a large lateral wind, the smaller roll stiffness will raise the operating risk greatly. In order to address the issue well, the anti-rolling device is used to increase the counter-torque against the lateral rolling of the vehicle body. The overturning safety is then improved by reducing the inclination angle of the vehicle without increasing the vertical stiffness of the spring. As the most part of the anti-rolling device, the time-variant reliability of the anti-roll torsion bar will directly affect the operating safety of the vehicle.
However, many studies are about the stiffness and strength analysis under the time-invariant conditions as well as the fatigue experiments for the anti-roll torsion bar. Duan et al. [
In this paper, a time-variant reliability-based multi-objective fuzzy design optimization (TRBMFDO) method for the anti-roll torsion bar of the EMU is proposed. The TRBDO constraints related to the stiffness and strength are first established by accounting for the time-variant stochastic working conditions and the working principle of the anti-rolling torsion bar. The optimal rage of design variables is then determined by employing the membership function combined with the fuzzy allowable interval. The physical programming method is then presented to transform the multi-objective fuzzy design optimization problem to a single-objective design optimization problem under the satisfaction of the reliability constraints. The results of the TRBMFDO are finally achieved by using the combinatorial optimization strategy.
The remainder of the paper is organized as follows. Mechanical performance analysis of the torsion bar are presented in
The mechanical performance analysis of the torsion bar is the foundation of the design optimization. Basically, the anti-roll torsion bar can be divided into the built-in bar and external bar according to the different position of the support seat. The support seat of the built-in anti-roll torsion bar is generally composed of the upper part and lower part, where the rubber joint is used. The support seat of the external anti-roll torsion bar is an entire structure, where the metal joint or integral polymer wear-resistant bushing is used. In the paper, we will study the external anti-roll torsion bar, whose installation position in the bogie frame is shown in
As the most important performance index of the anti-roll torsion bar device, anti-roll stiffness can ensure the roll angle of the vehicle body and the flexibility coefficient of the vehicle. Proper anti-roll stiffness can effectively improve the safety, stability, and comfort of operation. The anti-roll stiffness can be derived from the force analysis shown in
For the given parameters in
Considering the working principle of the anti-roll torsion bar, the restoring moment can be expressed as
Therefore, the anti-roll stiffness of the torsion bar can be derived by
The torsion bar includes three areas, namely the working area, transition area, and connection area. Therefore, the stiffness of the bar is integrated by the stiffness of the working area, transition area, and connection area. Provided that the cross-section of the torsion bar is circular and its diameter is
When the transition area is an arc, the length of the transition section can be provided by [
Considering the relationship of the stiffness between the torsion bar and the three areas, the torsional stiffness of the torsion bar is
When the anti-roll torsion bar is operating, it will suffer from the bending stress and torsion shear stress. Generally, the torsion shear stress is greater than the bending stress. The strength of the bar can be derived with the help of the third strength theory by combining the bending stress and torsion shear stress, where the bending moment and torque are shown in
From
With the combination of the bending and shear conditions, the stress of the torsion bar is therefore expressed by
The stiffness, strength, and allowable stress of the anti-roll torsion bar are usually time-variant due to several uncertain factors from the working conditions, loadings and also the degradation of the material. The time-variant factors will result in the time-variant change of the reliability of the bar. If the time-variant property is ignored, the reliability only ensure the safety of the bar at the initial time, namely
Since the diameter
Mean and variance of
Based on
Provided
Take the logarithm of the function according to
Mean and variance of
The material strength of the anti-roll torsion bar is random and time-variant, and therefore non-stationary random process can be employed to describe the material strength. The material strength of the torsion bar can be quantified as the product of the initial material strength and attenuation function.
Then the material strength can be expressed with the given expression of the attenuation function
Mean and variance of
According to the time-variant stress-strength interference (SSI) model, the reliability of the anti-roll torsion bar can be expressed by [
We define
Since
The time-variant reliability of the torsion bar is the provided by
Physical programming is an effective approach to dealing with the multi-objective optimization problem, since the designer can express his preferences according to the experience. Actually, there are four kinds of preference function, namely Class 1S, Class 2S, Class 3S, and Class 4S, to quantify the designer's requirements [
In
The quantitative preference function can be obtained with the piecewise function curve fitting method based on
Due to the existence of incomplete information, fuzzy variable is used to describe the uncertainty of the design variables. It is important to select the proper membership function for the fuzzy variables, since the shape of the membership function will affect the design optimization results. According to the boundary constraint of the design variables, the linear membership function is usually used and provided by
The upper and lower limits of the upper and lower bounds in
The framework of the proposed TRBMFDO method of the anti-roll torsion bar is shown in
The detailed structure of the anti-roll torsion bar is shown in
The initial stiffness of the torsion bar in the working area, transition area, and connection area can be calculated, respectively
In order to obtain the distribution parameters of d(
With the collected data in
When the drift and volatility rate are substituted into
Then the time-variant stiffness of the torsion bar is
According to the CRH3-350-PS-021 technical specification, the loadings of the torsion bar mainly include the static loading
Load type | |||
---|---|---|---|
Value | 2 |
The value of the loading is usually determined by the vehicle's center of gravity deviation and the outside lateral force. The expressions of the loading normal condition and special condition are provided by
When designing the torsion bar, three conditions should be satisfied: (1) the maximum stress should be less than the fatigue limit of the material; (2) the maximum stress is not greater than the yield strength of the material; (3) the maximum shear stress is less than the allowed shear stress of the material under the special load case. The stresses of the torsion bar under the normal and special conditions are calculated based on
The stresses of
With the collected data in
When the drift and volatility rate are substituted into
Due to the existence of the failure modes of fatigue, wear and corrosion, the strength of the bar will degrade with time. The collected historical data of the allowable stress is provided in
When the data in
The mean and standard deviation of the yield limit can be expressed by
The time-variant reliability constraints of the stresses are provided
The diameters of the working area, transition area, and connection area for the anti-roll torsion bar of the EMU are usually uncertain, because the uncertainty is from the design, manufacturing and the working conditions. According to the design rule, the upper and lower bounds of expansion coefficients for diameter are
The optimal level cut set obtained by the fuzzy comprehensive evaluation is
Upper and lower limit | ||||
---|---|---|---|---|
Upper bound | Upper limit | 77.70 | 67.20 | 64.05 |
Lower limit | 74.00 | 64.00 | 61.00 | |
Lower bound | Upper limit | 65.00 | 55.00 | 52.00 |
Lower limit | 58.50 | 49.50 | 46.80 |
In this example, the objective is to maximize the strength and stiffness, and meanwhile to minimize the mass of the anti-roll torsion bar under the satisfaction of the reliability and other performance constraints.
The mass can be calculated by
The stresses are given by
The preference function interval of optimization objectives can be obtained based on relevant standards and engineering experience, provided in
Optimization objectives | Sign | Unit | |||||
---|---|---|---|---|---|---|---|
Mass | Kg | 36.42 | 41.47 | 46.52 | 51.58 | 56.61 | |
Stesss 1 | MPa | 378.26 | 450.20 | 522.13 | 594.07 | 666 | |
Stesss 2 | MPa | 633.85 | 755.29 | 876.32 | 997.56 | 1118.79 |
By using
The expression of preference function for the mass of the anti-roll torsion bar is
The expression of preference function for the stress
The expression of preference function for the stress
The fitting curve of preference functions for the mass, stress
With the obtained preference functions of the objective functions and the related constraints, the TRBMFDO model is
Since the complicated and coupled relationship exists in the TRBMFDO model, it is difficult to obtain the highly efficient and accurate results of the design by using general optimization strategies. In order to obtain the global optimal results effectively, a hybrid optimization strategy is implemented with integrating DOE sampling and numerical optimization. This hybrid optimization strategy can reduce the probability of trapping in the local optimal solution. The main task of the proposed strategy is that DOE method is employed to draw samples in the design region evenly and the parameter optimization module is to conduct design optimization for the optimal design results. The flowchart of the hybrid optimization strategy is provided in
Three combinations based on the DOE including DOE+NLPQL, DOE+MIGA and DOE+ASA are used for the problem. The related results are provided in
Optimization strategy | Optimization method | Computation time/times | ||||||
---|---|---|---|---|---|---|---|---|
DOE+NLPQL | Non-fuzzy optimization | 0.065 | 0.055 | 0.0574 | 56.689 | 0.900 | 0.9837 | 17 |
Fuzzy optimization | 0.0624 | 0.0528 | 0.0574 | 56.265 | 0.900 | 0.9837 | 17 | |
DOE+MIGA | Non-fuzzy optimization | 0.065 | 0.0552 | 0.0574 | 56.698 | 0.906 | 0.9850 | 1001 |
Fuzzy optimization | 0.0625 | 0.0531 | 0.0574 | 56.290 | 0.900 | 0.9838 | 1001 | |
DOE+ASA | Non-fuzzy optimization | 0.065 | 0.055 | 0.0574 | 56.689 | 0.900 | 0.9837 | 2527 |
Fuzzy optimization | 0.0624 | 0.0528 | 0.0574 | 56.265 | 0.900 | 0.9837 | 3370 |
It can be seen from
To further verify the differences between the three optimization strategies in the optimization process, the design space and the optimal solution set are shown in
In this paper, a TRBMFDO method for the anti-roll torsion bar of the EMU is proposed, which will be an effective tool to design the anti-roll torsion bar optimally under the satisfaction of the lifecycle reliability and safety. The time-variant reliability models of the torsion bar are first built related to the stiffness and strength considering the uncertainty and time-variant property. The physical programming method is then presented to handle the multi-objective design optimization to avoid the selection of the weight factors. The fuzzy allowable interval of design variables is then estimated based on the fuzzy theory. The comprehensive preference function is solved by the DOE+NLPQL hybrid optimization strategy. With the practical example, it is testified that the reliability of the anti-roll torsion bar increases by 14% during the lifecycle, and the weight of the torsion bar decrease by 0.6%. For the proposed DOE+NLPQL optimization strategy, only 17 function calls are needed for the global optimal solution, and therefore the computational efficiency increased by over 58.8 times compared with the DOE+ASA optimization strategy and the DOE+MIGA optimization strategy. In our future work, we will consider more time-variant failure modes including wear, corrosion and so on.