# A simplification of railway vehicle lateral vibration model based on LQG control strategy.

1. IntroductionIt is doubtless that the vibration induced by track irregularity has a significant influence on the dynamic performance of a railway vehicle (Garg and Dukkipati 1984; Cai et al. 2015), especially the lateral vibration of high-speed railways. Lateral running stability is very important for the highspeed running of a railway vehicle. So far, passive suspension systems have been applied widely, but there is still room for improvement in train ride comfort and running safety by using innovative suspension techniques. Therefore, active and semi-active suspension systems are receiving more and more attention worldwide.

The multi-indexes of semi-active suspension based on a linear quadratic Gaussian (LQG) control strategy can be controlled simultaneously, and the control effect is very obvious and steady. Moreover, the control effect of every control index can be adjusted subjectively. Therefore, the LQG control strategy is very suitable for railway vehicle multi-body dynamic models.

Zhang et al. (2010), Chen and Changfu (2015), and Luo and Yang (2013) developed an LQG controller based on models with low degrees of freedom (DOFs). Their focus was more on optimisation of the weighting coefficients of the control indices, but there was not enough detail to confirm the model accuracy.

In the process of designing an LQG controller for a railway vehicle, accuracy of the controlled model is quite important. However, if we want to get accurate calculation results, the vehicle must be modelled in detail as well as possible, which means that the model may have many DOFs. For an accurate railway vehicle model, Zhai (2007) established a complete full-size model of railway vehicle lateral vibration they analysed movement of the wheelset, the bogie, and the car body and defined accurate primary and secondary suspension forces. According to his complete model, there are 21 DOFs for lateral vibration. (Note that all the models mentioned in this paper are given in Appendix 1.) The calculation of the model can be quite difficult. If a full-size vehicle dynamic model is integrated with the LQG control strategy in the simulation, the solution scale will be so large that the number of vehicle model DOFs has to be reduced. As a result, some new problems occur.

Although the controlled model needs to be simplified, the wheelsets cannot be left out directly, because the wheel-rail non-linear relation is very important for model accuracy. The wheel-rail non-linear relation can be excited and changed by track irregularities. Steenbergen (2006) analysed the differences in the wheel-rail relations among wheel-rail models with a single-point contact, a transient double contact, and a multipoint contact. Wang et al. (2017) analysed the distribution rule of the random wheel-rail force under the condition of random track irregularity. Liu and Wang (2017) and Liu et al. (2016) developed a complicated train-track coupled dynamic model and investigated the influence of braking and line excitations on wheel-rail lateral dynamic forces. In the analysis of railway vehicle running performance, the wheel-rail relation and system excitation should be considered in detail as much as possible.

By taking into consideration the vibration characteristics of the railway vehicle and the LQG control strategy, a 7-DOF simplified model is proposed in this paper based on a transitive calculation method. This means that the full-size vehicle model can be divided into two sub-models: sub-1 (the wheelset part) and sub-2 (the bogie and car body part). Therefore, the calculation has been divided accordingly, and the calculation results from sub-1 can be used as the transitive excitation to calculate sub-2. The 7-DOF simplified model proposed in this paper consists of sub-2 and the transitive excitation. On this basis, the LQG controller is designed based on the simplified model. Then the number of DOFs of the controlled model can be reduced and the wheel-rail non-linear relation can also be considered.

In this way, the complex calculation can be simplified to a great extent and secondary goals can be removed to debug and control the main goals. This idea can provide a reference for research of other complex systems.

2. Modelling and calculation of the simplified railway vehicle model

2.1. Modelling procedure

In general, there are 17 DOFs in the lateral vibration of a full-size railway vehicle: the lateral, yaw and roll motions of the car body; the lateral, yaw and roll motions of two bogies; and the lateral and yaw motions of the four wheelsets. Wang and Liao (2009) and Zong etal. (2013) modelleda17-DOF controlled model of railway vehicle lateral vibration and then designed an LQG controller based on this model. However, if we were to design the LQG controller with a 17-DOF vehicle model, the coefficient matrices A, B, C, D and Q would at least be 34 dimensions, which can be very difficult and tedious to calculate. Moreover, if there are too many control indexes, the control effect of the main indexes can be quite difficult to perfect. Therefore, in this paper we propose a 7-DOF simplified model that accounts for the lateral, yaw, and roll motions of the car body and the lateral and yaw motions of the two bogies, without considering the DOFs of the wheelsets.

The 7-DOF simplified model of railway vehicle lateral vibration is shown in Figure 1.

The lateral vibration differential equations for the vehicle are established according to the analysis of the model:

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

where [M.sub.c] and [M.sub.2], respectively, represent the mass of the car body and the bogie;[J.sub.cz] and [J.sub.cx], respectively, represent the yaw and roll rotational inertia of the car body and the [J.sub.tz] represents the yaw rotational inertia of the bogie;[k.sub.2y], [c.sub.2y], [k.sub.2x], [c.sub.2x], and [k.sub.1y], respectively, represent double the lateral stiffness and damping of the secondary spring, the longitudinal stiffness and damping of the secondary spring, and the lateral stiffness of the primary spring; L represents the longitudinal distance between the bogie and the centre of the vehicle; [L.sub.1] represents the longitudinal distance between the wheel-set and the centre of the bogie; [h.sub.1] represents the vertical distance between the centre of the car body and the stiffness of the secondary spring; [b.sub.2] represents the lateral distance between the centre of the car body and the secondary spring; [b.sub.3] represents the lateral distance between the centre of the car body and the secondary damping; [u.sub.1] and [u.sub.2], respectively, represent the control forces of the front and rear bogies; [y.sub.c], [[??].sub.c], and [[theta].sub.c], respectively, represent the lateral, yaw, and roll displacements of the car body; [y.sub.t1] and [y.sub.t2] respectively represent the lateral displacements of the front and rear bogies; [y.sub.[omega]1], [y.sub.[omega]2], [y.sub.[omega]3] and [y.sub.[omega]4], respectively, represent the lateral displacements of wheelset-1, wheelset-2, wheelset-3, and wheelset-4; and [[??].sub.[omega]1], [[??].sub.[omega]1], [[??].sub.[omega]1], and [[??].sub.[omega]1], respectively, represent the four transitive excitations of the system.

2.2. Calculation of the transitive excitation

Based on Equations (1-7), the state variable X can be selected as follows:

[mathematical expression not reproducible]

The state equations for the semi-active system are given as follows:

[??] = A.X + B.U + F.W Y = C.X + D.U (8)

where [??] is the intermediate variable matrix, Y is the observed quantity of the system, U = [[[u.sub.1], [u.sub.2]].sup.T] is the vector quantity of control forces, W is the excitation matrix of the system, and the matrices A, B, C, D, and F are all coefficient matrices.

The passive system can be described by means of the following state equation:

[??] = A.X + F.W (9)

To get the excitation matrix of the system, W, the state variable X is put into Equation (9). The specific expression is as follows:

In Equation (10), we see that the excitation matrix of the system is W = [[[??].sub.[omega]1], [??].sub.[omega]2], [??].sub.[omega]3], [??].sub.[omega]4]].sup.T]. This matrix gives the lateral velocities of the four wheelsets. Therefore, to improve the computational accuracy of the 7-DOF simplified model without wheelsets, the lateral velocities of the wheelsets of the full-size three-dimensional vehicle model in the multi-body dynamic software ADAMS/Rail are exported as the excitation matrix W = [[[??].sub.[omega]1], [??].sub.[omega]2], [??].sub.[omega]3], [??].sub.[omega]4]].sup.T] of the 7-DOF simplified model.

First, a 62-DOF full-size three-dimensional railway vehicle model is established in ADAMS/Rail. The railway vehicle model is shown in Figure 2.Thisvehiclemodel contains car body, two bogies, two traction rods, four wheelsets, and eight axle boxes. The car body, bogies, wheelsets, and traction rods have six DOFs each, and the axle boxes have only one DOF each. Therefore, the vehicle model has 62 DOFs. The main parameters of the simulation are given in Tables 1-3.

The mass, rotational inertia, and other simulation parameters of this model are the same as those in the 7-DOF model that we analysed. The German railway spectrum of low irregularity (GRSLI) is set up as the excitation of track irregularities. The GRSLI can be expressed as follows (Cai et al. 2015):

[S.sub.v]([OMEGA]) = [[A.sub.v][[OMEGA].sub.c.sup.2]/([[OMEGA].sup.2] + [[OMEGA].sub.r.sup.2])([[OMEGA].sup.2] + [[OMEGA].sub.c.sup.2])] ([m.sup.2]/(rad/m)) (11)

[S.sub.a]([OMEGA]) = [[A.sub.a][[OMEGA].sub.c.sup.2]]/[([[OMEGA].sup.2] + [[OMEGA].sub.r.sup.2])([[OMEGA].sup.2] + [[OMEGA].sub.c.sup.2])] ([m.sup.2]/(rad/m)) (12)

[S.sub.L]([OMEGA]) = [[A.sub.v][b.sup.-2][[OMEGA].sub.c.sup.2][[OMEGA].sup.2]/([[OMEGA].sup.2] + [[OMEGA].sub.r.sup.2])([[OMEGA].sup.2] + [[OMEGA].sub.c.sup.2])([[OMEGA].sup.2] + [[OMEGA].sub.s.sup.2]) (1/(rad/m)) (13)

where [OMEGA] represents the spatial circular frequency; [[OMEGA].sub.r], [[OMEGA].sub.s], and [[OMEGA].sub.c] are the cut-off frequencies, which are, respectively, 0.0206, 0.4380, and 0.8246; [A.sub.v] and [A.sub.a] are the roughness coefficients, which are respectively 4. 032 x [10.sup.-7] and 2.119 x [10.sup.-7] [m.sup.2] * rad/m; and b is half the gauge length, which has a typical value of 0.75 m.

After the simulation, the lateral velocities [[??].sub.[omega]1], [[??].sub.[omega]2], [[??].sub.[omega]3] and [[??].sub.[omega]4] of wheelsets 1, 2, 3 and 4 are exported respectively from ADAMS/Rail as the transitive excitation of the 7-DOF simplified vehicle model. Then, the modelling of the simplified model is complete. If we take the condition with 200 km/h as an example, the lateral velocities of the four wheelsets are shown in Figure 3.

3. Verification of the simplified vehicle model

For active and semi-active suspension systems, the control effect is very important. The accuracy of the controlled model that we analysed is also quite important. Only by setting up an accurate model can the design of the controller be more meaningful.

The 7-DOF simplified model does not account for the DOFs of the wheelsets. However, the wheelsets are only important parts for the calculation results of the railway vehicle model. Therefore, to verify the importance of the wheelsets and the accuracy of the 7-DOF simplified vehicle model, three models, calculated under different velocity conditions, are established: Model-1, Model-2 and Model-3. The details are listed in Table 4.

The excitations of Model-2 and Model-3 are both obtained by converting the power spectral density (PSD) of the GRSLI into the time domain by means of the inverse fast Fourier transform (IFFT) method. The excitation of Model-1 is obtained in Section 2.2, it comprises just the lateral velocities of the four wheelsets in Model-3.

To be more convincing, the three models are simulated with different velocities: 200, 250, 300 and 350 km/h. The root mean square (RMS) value and the maxima (Max) of the car body lateral acceleration ([[??].sub.c]), the car body yaw angular acceleration ([[??].sub.c]), and the car body roll angular acceleration ([[??].sub.c]) are listed in Table 5.

Model-3 is the 62-DOF full-size three-dimensional vehicle model, which accounts for vertical, lateral and longitudinal coupled vibration. In the calculation process for the three-dimensional vehicle, vibration in any direction would be influenced by that in the other two directions. Therefore, the results of the lateral vibration for full-size three-dimensional vehicle model would be more accurate than the result for the 17-DOF vehicle model, which only accounts for the lateral vibration. It is meaningful to take the lateral vibration results of Model-3 as a criterion to analyse the accuracy of the simplified models, Model-1 and Model-2.

From Table 5, we can obtain the variation trend of the RMS value and maxima of [[??].sub.c],[[??].sub.c] and [[??].sub.c] of the three models. An analysis and a comparison of the results are shown in Figures 4-6.

From Figures 4(a), 4(b), 5(a), 5(b), 6(a) and 6(b) we see that the RMS value and maxima of the car body lateral acceleration, yaw angular acceleration and roll angular acceleration in Model-1 and Model-3 are a factor of 10 greater than those values in Model-2. Therefore, in Figures 4(c), 5(c) and 6(c) we compare only the RMS values and maxima between Model-1 and Model-3. From Figures 4(c), 5 (c) and 6(c), it can be seen that the deviations of the RMS values and maxima between Model-1 and Model-3are all <16%. Especially for the 300 km/h case, the deviations of [[??].sub.c] and [[??].sub.c] are <10%.

To make a more intuitive comparison, we take the 200 km/h velocity condition as an example. The time and frequency domain calculation results are shown in Figures 7-9.

From Figures 7(a), 8(a) and 9(a), it can be seen that the vibration amplitudes of the indexes in Model-1 and Model-3 have the same tendency. The amplitudes of the results in Model-1 and Model-3 are both a factor of 10 greater than those of Model-2, which is consistent with the conclusion obtained from Figures 4(a), 4(b), 5(a), 5(b), 6(a) and 6(b).

From Figures 7(b), 8(b) and 9(b), we can see that in Model-1, Model-2 and Model-3, the main frequencies of the car body lateral acceleration are respectively 3.71, 3.27 and 3.52 Hz. The main frequencies of the car body yaw angular acceleration are respectively 2.53, 2.83 and 1.96 Hz. The main frequencies of the car body yaw angular acceleration are respectively 1.07, 3.52 and 0.85 Hz. Then, the main frequency deviations of the three indexes between Model-1 and Model-3 are, respectively, 5.4%, 29.1% and 25.9%.

In conclusion, the reason why the calculation results from Model-1 and Model-3 are both a factor of 10 greater than the results from Model-2 is that Model-2 is asimplified model, which does not account for the wheelsets. Its excitation is the railway spectrum. Therefore Model-2 does not take the wheel-rail nonlinear relation into consideration, and the wheel-rail non-linear relation is quite important for accuracy of the model calculation results. For example, when a high-speed vehicle passes a pit excitation on the rail surface, the excitation will be magnified by the rigid impact between the wheel and rail. Therefore, the excitation applied to the primary suspension is quite a bit larger than that applied to the wheelsets. The larger the excitation on the rail is, the more obvious the phenomenon will be.

The excitations of Model-1 are transitive excitations, which are the lateral velocities of the wheelsets obtained from the results of Model-3. Although Model-1 does not contain the wheelsets, the wheel-rail non-linear relation is considered in the calculation process of transitive excitation of Model-1. Therefore, the results of Model-1 are quite accurate.

4. Design of the LQG controller based on the simplified vehicle model

4.1. Modelling of the target function

In the process of designing the LQG controller based on Model-1, the eight DOFs of the wheelsets that account for the lateral motion and yaw motion can be left out. This means that the dimensions of the coefficient matrices A, B, C, D, and Q will be reduced by at least 16. Consequently, the difficulty in calculation and debugging will be reduced significantly.

The control indexes can be selected from the state variable X established in Section 2.2.the lateral acceleration, the yaw angular acceleration, and the roll angular acceleration of car body,[[??].sub.c], [[??].sub.c], and [[??].sub.c]; the lateral accelerations of the front and rear bogies,[[??].sub.t1] and [[??].sub.t2]; the yaw angular accelerations of the front and rear bogies, [[??].sub.t1] and [[??].sub.t2];the carbody lateral displacements relative to the front and the rear bogies, ([y.sub.c] - L. [[??].sub.c] - [h.sub.1] [[theta].sub.c] - [y.sub.t1]) and ([y.sub.c] + [L[??].sub.c] - [h.sub.1][[theta].sub.c] - [y.sub.t2]); and the yaw angular displacements of the car body relative to the front and rear bogies ([[??].sub.c] - [[??].sub.t1]) and ([[??].sub.c] - [[??].sub.t2]).

The target function J can be established based on these indexes:

[mathematical expression not reproducible] (14)

4.2. Weighting coefficients based on an analytic hierarchy process

The weight of the indexes in target function J can be changed by adjusting the weighting coefficients, [f.sub.1], [f.sub.2],... and so on. The weighting coefficients are defined according to the value of these indexes. However, there are great differences in terms of units and magnitudes among the control index values, and the same standard cannot be applied to directly adjust the weight of these indexes. Therefore, the control index values should be converted to the same level. Then, the weighting coefficients can be obtained after designing the subjective weighting coefficients.

4.2.1. Quantification with the same scale

First, the passive system is simulated in the software MATLAB. The RMS values of the 11 indexes that we analysed in Equation (14) were then obtained. Next, the control indexes were named in the order of Equation (14): [[sigma].sub.1.sup.2], [[sigma].sub.2.sup.2], [[sigma].sub.3.sup.2], [[sigma].sub.4.sup.2], [[sigma].sub.5.sup.2], [[sigma].sub.6.sup.2, [[sigma].sub.7.sup.2], [[sigma].sub.8.sub.2], [[sigma].sub.9.sub.2], [[sigma].sub.10.sub.2], and [[sigma].sub.11.sub.2]. The proportionality coefficient with the same scale [[beta].sub.i] can be obtained according to the following equation (Chen et al. 2008):

[[sigma].sub.1.sup.2] * 1 = [[sigma].sub.i.sup.2] * [[beta].sub.i] [??]15[??]

4.2.2. Subjective weighting coefficients

The ratios of relative importance between two control indexes [h.sub.ij] are listed in Table 6. Comparison of the importance of the indexes we analysed can be quantified by selecting the ratio [h.sub.ij]. For the relative importance between two values, interpolation should be conducted between them to find the ratio. Table 6 (a,b,c,d,e) represents the following levels, respectively: equally important, slightly important, relatively important, important, and very important.

The subjective judgment matrix H can be obtained as follows:

H = [[h.sub.ij].sub.11x11] (16)

Then the vector quantity M of the product of the row elements, then-root vector quantity [??] of M, and regular vector V are obtained based on H:

M = [[M.sub.1], [M.sub.2], [M.sub.3]...., [M.sub.n]].sup.T] [[??].sub.i] = [nth root of Mi] (17)

V = [??]/[n.summation over (i=1)] (18)

The weighting proportionality coefficient of the car body lateral acceleration, [r.sub.1], has been defined as 1. Then the weighting proportionality coefficients of other indexes [r.sub.i] can be obtained according to (Chen et al. 2008)

[V.sub.1] = [V.sub.i]/[r.sub.i] (19)

Therefore, the weighting proportionality coefficient [f.sub.i] in the final functional is obtained as follows:

[f.sub.i] = [[beta].sub.i] [r.sub.i] (20)

The final results of the weighting coefficients are given in Table 7.

4.3. Calculation of the control force

The control force is one of the key components in the process of designing the LQG controller. The excitation of the controlled model can influence the control force significantly.

Guo et al. (2013), Chen et al. (2005) and Guan et al. (2004) proposed a semi-active suspension adaptive LQG control strategy, in which they mainly researched the control effects of the model under different excitation conditions. Therefore, the input excitation is very important for designing an active or semi-active controller. Therefore, two LQG controllers are designed based on Model-1 and Model-2, respectively, to compare and analyse the control forces.

According to the quadratic form principle, Equation (9) can be converted into the following form:

[mathematical expression not reproducible] (21)

The form of the Riccati equation is as follows:

[(SA).sup.T] + SA - (SB + N)[R.sup.-1][(SB + N).sup.T] + Q = 0, (22)

where S is the solution of the Riccati equation and the matrices A and B are, respectively, the coefficient matrix of the state variable X based on Equations (1-7) and the coefficient matrix of control forces U*.Matrices Q, R, and N can be obtained from Equation (21). The feedback matrix K of optimum control can be obtained based on the Riccati equation. Then, the optimum control forces U* = [[[u.sub.1], [u.sub.2]].sup.T] can be obtained as follows:

U* = -K * X (23)

Under the condition in which the velocity of the railway vehicle is 200 km/h, the control forces [u.sub.1] and [u.sub.2] of Model-l are shown in Figure10.Controlforcesu1 and u2 of Model-2 are shown in Figure11.

From Figures 10 and 11,wecanseethattheamplitude of control force [u.sub.1] is a little bigger than that of control force [u.sub.2] in both Model-1 and Model-2. However, the amplitudes of the control forces in Model-1 are much bigger than those of Model-2. The comparison is shown in Figure 12 and the specific data analyses are listed in Table 8.

As is given in Table 4, the DOFs, dynamic parameters and simulation parameters of Model-1 are identical to those of Model-2 except for the excitation. However, it is not hard to see by analysing the results in Table 8 that the RMS and maximum of control force [u.sub.1] in Model-1 are both a factor of 10 greater than those of Model-2. Therefore, in the process of designing the LQG controller, the accuracy of the excitation can seriously influence the calculation results for the control force. Because the non-linear wheel-rail relation has been taken into consideration in the calculation process of the transitive excitation in Model-1, the excitation of Model-1 is quite accurate. So, the control force in Model-1 is very accurate and meaningful.

4.4. Modelling of the semi-active system based on the LQG controller in MATLAB/simulink

A semi-active system can be established in MATLAB/Simulink according to the state equation, which is expressed in Equation (8).

The module state-space in MATLAB/Simulink is used to build the vibration differential equations of the simplified vehicle model. The excitation of Model-1 obtained from ADAMS/Rail is imported into MATLAB/Simulink through the module From Workspace. The control forces are then added in the model as the feedback element. The integrator used in simulation is ode4 (Runge--Kutta), and the fixed-step size is 0.005. Throughout the whole process, it is very important that the matrices calculated together must have the same dimensions. The model in MATLAB/Simulink is shown in Figure 13.

5. Analysis of simulation results

For the control effect of the semi-active suspension based on the LQG strategy, there is little difference among the vehicle models with different simulation velocities. Consequently, we use the condition with 250 km/h as an example. The results under other velocity conditions are given in Appendix 2.The calculation results for the control indexes are obtained by designing the LQG controller based on Model-1. The most important three indexes are analysed first.

The simulation results for lateral, yaw angular and roll angular accelerations of the car body are shown in Figures 14-16.

From Figures 14(a), 15(a) and 16(a), we see that the control effect of the results with large amplitude are much more obvious than those with low amplitude, which means that most of the energy of the control forces is used to control the peak value of the car body vibration. From Figures 14(b), 15(b), and 16 (b), we see that, when the frequencies are 1.37 and 3.71 Hz, the control effect of the car body lateral acceleration is most obvious. In addition, the frequencies with the most obvious control effect of car body yaw angular acceleration are 1.17 and 2.53 Hz. For the car body roll angular acceleration, the frequencies with the most obvious control effect are 1.07 and 3.51 Hz.

The specific data for the lateral acceleration, yaw angular acceleration, and roll angular acceleration of the car body before and after the control are given in Table 9.

From Table 9, we see that compared with a passive suspension system, the RMS value and the maximum of the car body lateral acceleration for the semi-active suspension system drop, respectively, 48.09% and 46.48%, those of the car body yaw angular acceleration drop, respectively, 39.34% and 49.88%, and those of the car body roll angular acceleration drop, respectively, 44.35% and 41.49%.

Therefore, we get the result that the LQG controller of the model can greatly reduce the value of the critical multi-indexes for a high-speed railway vehicle simultaneously. The control effects are quite obvious. The average value of the control effect is ~45%.

Although the LQG controller can effectively reduce the value of the critical control indexes such as lateral acceleration, yaw angular acceleration, and roll angular acceleration of the car body, the control indexes with less important weighting coefficients may increase after control. The calculation results of the car body lateral displacements relative to front and rear bogies, [[y.sub.c] - L[[??].sub.c] - [h.sub.1][[theta].sub.c] - [y.sub.t1]) and ([y.sub.c] + L[[??].sub.c] - [h.sub.1][[theta].sub.c] - [y.sub.t2]), are shown in Figures 17 and 18. The specific data analyses are listed in Table 10.

From Table 10, we see that, the RMS and the maximum of the car body lateral displacement relative to the front bogie in the semi-active suspension system increase, respectively, 9.10% and 2.86% compared to their values for the passive suspension system. The indexes of the car body lateral displacement relative to the rear bogie change by, respectively, 9.09% and -10.71%.

6. Conclusion

With the vibration characteristics of the railway vehicle and the control strategy of the LQG controller taken into consideration, a 7-DOF simplified model is proposed in this paper. The model is proved to be quite accurate by comparison with the 62-DOFfull-size three-dimensional vehicle model. The LQG controller is then designed based on this simplified model. By analysing the verification process and the control process, some conclusions are obtained:

(1) Although the simplified model does not account for the wheelsets, the non-linear wheel-rail relation has been taken into consideration in this model. By comparing the results of the simplified model with the full-size three-dimensional model under different velocity conditions, it is found that the maximum deviations of the RMS value and maxima of the car body lateral acceleration, yaw, and roll angular acceleration are all <16%. Especially, in the 300 km/h case, the deviations of the car body yaw and roll angular accelerations are <10%.

(2) To simplify the calculation and debugging of the LQG controller, it is designed based on a 7-DOF simplified model. The eight DOFs of the four wheelsets can be left out by comparing with the 17-DOF full-size vehicle, which means that the dimensions of coefficient matrices A, B, C, D, Q, and N will be at least reduced by 16.

For the control effect, there is little difference among the vehicle models with different simulation velocities. Therefore, the condition with 250 km/h is given as an example. By means of analysing the results, it can be seen that the RMS value and the maximum of the car body lateral acceleration drop, respectively, 48.09 and 46.48%, those of the car body yaw angular acceleration drop, respectively, 39.34% and 49.88%, and those of the car body roll angular acceleration drop, respectively, 44.35% and 41.49%.

Although the control effect of the control indexes with much more important weighting coefficients can be very significant, control indexes with less important weighting coefficients may increase after control.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by the National Natural Science Foundation of China [Grant Nos. 51605315, 11172183, and 11572206], the Open Project Program of the State Key Laboratory of Traction Power [Grant No. TPL1707], and the Science and Technology Research Project of Hebei Province [Grant No. BJ2016047].

Notes on contributors

Wang Yi-xuan, who was born in August 1992, Hebei province, China. And he is currently studying as an postgraduate student in Shijiazhuang Tiedao University. On the other hand, his research interests are multi-body dynamics analysis and semi-active control.

Chen En-li, he is currently working as a professor and doctoral supervisor in Shijiazhuang Tiedao University. And his research interests: vibration signals analysis and vehicle dynamics analysis.

Liu Peng-fei, who received his Ph.D. from Southwest Jiaotong University. And he is currently working as a teacher in Shijiazhuang Tiedao University.

Zhang Lin, she is currently studying as an postgraduate student in Shijiazhuang Tiedao University.

References

Cai, X., L. Zhao, A. L. L. Lau, S. Tan, and R. Cui. 2015. "Analysis of Vehicle Dynamic Behavior under Ballasted Track Irregularities in High-Speed Railway." Noise & Vibration Worldwide 46 (10): 10-17. doi:10.1260/0957-4565.46.10.10.

Chen, S.-A., F. Qiu, R. He, and S. Lu. 2008. "A Method for Choosing Weights in a Suspension LQG Control." Journal of Vibration and Shock 27 (02): 65-68+176.

Chen, S., and Z. Changfu. 2015. "Genetic Particle Swarm LQG Control of Vehicle Active Suspension." Automotive Engineering 37 (2): 189-193.

Chen, W.-W., Y. Wang, Q. Wang, L. Yang, J. Huang, and Z. Xu. 2005. "Adaptive LQG Control for the Electric Power Steering System of an Automobile." Chinese Journal of Mechanical Engineering 41 (12): 167-172. doi:10.3901/JME.2005.12.167.

Garg,V.K., and R. V. Dukkipati. 1984. Dynamics of Railway Vehicle Systems. New York, NY: Academic Press.

Guan, J.-F., L. Gu, and C. Hou. 2004. "The Adaptive LQG Control for the Semi-Active Suspension Vehicle." Journal of System Simulation 16 (10): 2340-2343.

Guo, K.-H., W.-H. Yu, X. Zhang, F. Ma, and F. Zhao. 2013. "Semi Active Suspension Adaptive Control Strategy." Journal of Hunan University (Natural Sciences) 40 (02): 39-44.

Liu, P., and K. Wang. 2017. "Effect of Braking Operation on Wheel-Rail Dynamic Interaction of Wagons in Sharp Curve." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-Body Dynamics 231 (1): 252-265.

Liu, P., W. Zhai, and K. Wang. 2016. "Establishment and Verification of Three-Dimensional Dynamic Model for Heavy-Haul Train-Track Coupled System." Vehicle System Dynamics 54 (11): 1511-1537. doi:10.1080/00423114.2016.1213862.

Luo, X.-Y., and S.-W. Yang. 2013. "Design of a LQG Controller for a Vehicle Active Suspension System Based on AHP." Journal of Vibration and Shock 32 (02): 102-106.

Steenbergen, M. J. M. M. 2006. "Modelling of Wheels and Rail Discontinuities in Dynamic Wheel-Rail Contact Analysis." Vehicle System Dynamics 44 (10): 763-787. doi:10.1080/00423110600648535.

Wang, D. H., and W. H. Liao. 2009. "Semi-Active Suspension Systems for Railway Vehicles Using Magnetorheological Dampers. Part I: System Integration and Modelling." Vehicle System Dynamics 47 (11): 1305-1325. doi:10.1080/00423110802538328.

Wang, J., G. Jirong, C. Long, and L. Xiangguo. 2017. "Distribution Characteristics of Wheel-Rail Contact under Random Parameters." Australian Journal of Mechanical Engineering (3): 1-7. doi:10.1080/14484846.2017.1299664.

Zhai, W. 2007. Vehicle-Track Coupling Dynamics (Third Edition). Beijing: Science Press.

Zhang, Y., H. Zhang, M. Xia, and J. Qin. 2010. "Integrated Optimization for Vehicle Active Suspension System Based on Vibration Control." Noise & Vibration Worldwide 41 (10): 49-58. doi:10.1260/0957-4565.41.10.49.

Zong, L.-H., X.-L.Gong,S.-H. Xuan, and C.-Y. Guo. 2013. "Semi-Active H[infinity] Control of High-Speed Railway Vehicle Suspension with Magnetorheological Dampers." Vehicle System Dynamics 51 (5): 600-626. doi:10.1080/00423114.2012.758858.

Appendix 1

Table A1. All the models mentioned in this paper. Railway vehicle model Description of the model 62-DOF model Full-size three-dimensional vehicle, which contains the vertical, the lateral, and the longitudinal coupled vibration 21-DOF model Complete full-size model of vehicle lateral vibration, which only contains the lateral vibration 17-DOF model Full-size model of vehicle lateral vibration, which only contains the lateral vibration 7-DOF model Simplified model of vehicle lateral vibration, which only contains the lateral vibration

Appendix 2

The results for the 250 km/h condition have been analysed in Section 5. The results for the other conditions listed in Table 5 are given as follows: The weighting coefficients are listed in Table A2, and comparisons of the car body critical indexes between passive and semi-active suspensions are given in Figures A1-A3 and Tables A3-A5.

Table A2. Weighting coefficients under different simulation velocity conditions. Simulation velocity (km/h) [f.sub.1] [f.sub.2] [f.sub.3] [f.sub.4] [f.sub.5] 200 1 40 21 0.021 0.006 300 1 50 35 0.012 0.009 350 1 65 55 0.006 0.009 Simulation velocity (km/h) [f.sub.6] [f.sub.7] [f.sub.8] [f.sub.9] [f.sub.10] 200 0.012 0.012 1800 1200 900 300 0.006 0.021 1200 900 1500 350 0.0045 0.021 1500 1200 1800 Simulation velocity (km/h) [f.sub.11] 200 821.7 300 900 350 1200 Table A3. Comparison of mcar body critical dynamic indexes between passive and semi-active suspensions. [[??].sub.c] (m*[s.sup.-2]) RMS Max Passive suspension 0.1580 0.6131 Semi-active suspension 0.0866 0.3543 Control effect -45.19% -42.21% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0300 0.1121 Semi-active suspension 0.0196 0.0593 Control effect -34.67% -47.10% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0348 0.1292 Semi-active suspension 0.0198 0.0804 Control effect -43.10% -37.77% Table A4. Comparison of car body critical dynamic indexes between passive and semi-active suspensions. [[??].sub.c] (m*[s.sup.-2]) RMS Max Passive suspension 0.2011 0.7185 Semi-active suspension 0.1012 0.3482 Control effect -49.68% -51.54% [[??].sub.c] (rad*[s.sub.-2]) RMS Max Passive suspension 0.0430 0.1960 Semi-active suspension 0.0280 0.1321 Control effect -34.88% -32.60% [??].sub.c (rad*[s.sup.-2]) RMS Max Passive suspension 0.0485 0.1850 Semi-active suspension 0.0283 0.1023 Control effect -41.65 -44.70% Table A5. Comparison of car body critical dynamic indexes between passive and semi-active suspensions. [[??].sub.c] (m*[s.sup.-2]) RMS Max Passive suspension 0.2461 0.7822 Semi-active suspension 0.1182 0.3837 Control effect -51.97% -50.95% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0502 0.2320 Semi-active suspension 0.0315 0.1488 Control effect -37.25% -35.86% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0581 0.2031 Semi-active suspension 0.0383 0.1274 Control effect -34.08% -37.27%

WANG Yi-Xuan, CHEN En-Li, LIU Peng-Fei, QI Zhuang and ZHANG Lin

School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China

CONTACT WANG Yi-Xuan [??] 15733179692@163.com

https://doi.org/10.1080/14484846.2018.1486793

Table 1. Structural parameters of the main parts. Item Size (mm) Size of the car body 24,500 x 3380 x 3700 Centre distance of bogie 17,500 Wheelbase 2500 Lateral span of primary suspension 2000 Lateral span of air spring 2460 Lateral span of anti-hunting damper 2700 Lateral span of lateral damper of secondary 630 suspension Table 2. Suspension parameters of the main parts. Item Value Lateral damping (secondary) 58.8 (kN*s/m) Vertical damping (secondary) 9.8 (kN*s/m) Longitudinal damping (secondary) 9.8 (kN*s/m) Lateral/longitudinal stiffness (secondary) 151.6 (kN/m) Vertical stiffness (secondary) 176.4 (kN/m) Lateral/longitudinal stiffness (primary) 980 (kN/m) Vertical stiffness (primary) 1176 (kN/m) Table 3. Main inertial parameters of the simulation. Items Mass(t) Rotational inertia (t*m2) [l.sub.xx] = 93.312 Carbody 28.8 [l.sub.yy] = 1411 2 [l.sub.zz] = 1331.712 [l.sub.xx] = 2.106 Bogie 2.6 [l.sub.yy] = 1 424 Wheelset 1.97 [l.sub.ZZ] = 2.600[l.sub.yy] = 0.623[l.sub.yy] = 0.078[l.sub.zz] = 0.623 Table 4. Vehicle models used in verification. Model Description Model-1 7-DOF simplified model excited by lateral velocities of the wheelsets Model-2 7-DOF simplified model excited by the railway spectrum directly Model-3 62-DOF full-size three-dimensional model obtained in ADAMS/Rail Table 5. Car body critical dynamic indexes under different conditions. [[??].sub.c] (m*[s.sup.-2]) Velocity (km*[h.sup.-1]) Model RMS Max Model-1 0.1572 0.6123 200 Model-2 0.0102 0.0401 Model-3 0.1391 0.5604 Model-1 0.1621 0.6059 250 Model-2 0.0122 0.0428 Model-3 0.1490 0.5469 Model-1 0.2013 0.7172 300 Model-2 0.0134 0.0447 Model-3 0.1747 0.6213 Model-1 0.2453 0.7812 350 Model-2 0.0146 0.0414 Model-3 0.2121 0.6738 [[??].sub.c] (rad*[s.sup.-2]) Velocity (km*[h.sup.-1]) RMS Max 0.0299 0.1112 200 0.0019 0.0068 0.0268 0.1195 0.0363 0.1511 250 0.0022 0.0071 0.0331 0.1616 0.0424 0.1957 300 0.0024 0.0086 0.0399 0.1961 0.0504 0.2311 350 0.0025 0.0085 0.0477 0.2065 [[??].sub.c] (rad*[s.sup.-2]) Velocity (km*[h.sup.-1]) RMS Max 0.0347 0.1286 200 0.0018 0.0085 0.0391 0.1321 0.0414 0.1598 250 0.0028 0.0090 0.0474 0.1455 0.0483 0.1820 300 0.0030 0.0096 0.0506 0.1768 0.0572 0.2028 350 0.0032 0.0098 0.0539 0.1821 Table 6. Ratios of importance between control indexes. Ratio i/j A B C D E [h.sub.ij] 1 3 5 7 9 Table 7. Calculation results of the weighting coefficients. [f.sub.1] [f.sub.2] [f.sub.3] [f.sub.4] [f.sub.5] [f.sub.6] 1 51 31 0.015 0.012 0.018 [f.sub.7] [f.sub.8] [f.sub.9] [f.sub.10] [f.sub.11] 0.015 2000 1600 1200 600 Table 8. Comparison of control forces of the two models. Model Root mean square (N) Maximum (N) Model-1 3507.6 11,598.5 Model-2 220.6 949.2 Table 9. Comparison of car body critical dynamic indexes between passive and semi-active suspensions. [[??].sub.c] (m*[s.sup.-2]) RMS Max Passive suspension 0.1645 0.6841 Semi-active suspension 0.0854 0.3661 Control effect -48.09% -46.48% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0305 0.1201 Semi-active suspension 0.0185 0.0602 Control effect -39.34% -49.88% [[??].sub.c] (rad*[s.sup.-2]) RMS Max Passive suspension 0.0363 0.1422 Semi-active suspension 0.0202 0.0832 Control effect -44.35% -41.49% Table 10. Comparison of car body lateral displacement relative to front and rear bogies between passive and semi-active suspensions. ([y.sub.c] - L[[??].sub.c] - [h.sub.1] [[theta].sub.c] - [y.sub.t1]) RMS Max Passive suspension 0.0011 0.0035 Semi-active suspension 0.0012 0.0036 Control effect 9.10% 2.86% ([y.sub.c] + L[[??].sub.c] - [h.sub.1] [[theta].sub.c] - [y.sub.t2]) RMS Max Passive suspension 0.0022 0.0093 Semi-active suspension 0.0024 0.0084 Control effect 9.09% -10.71%

Printer friendly Cite/link Email Feedback | |

Title Annotation: | ARTICLE |
---|---|

Author: | Yi-Xuan, Wang; En-Li, Chen; Peng-Fei, Liu; Zhuang, Qi; Lin, Zhang |

Publication: | Australian Journal of Mechanical Engineering |

Geographic Code: | 9CHIN |

Date: | Jun 1, 2018 |

Words: | 6797 |

Previous Article: | A discontinuous model simulation for train start-up dynamics. |

Next Article: | An efficient approach for the diagnosis of faults in turbo pump of liquid rocket engine by employing FFT and time-domain features. |

Topics: |