# Coupling vibration analysis of full-vehicle vehicle-guideway for maglev train.

1. IntroductionUnlike rail trains, maglev train is a new type of traffic mode without wheel-rail contact, and it has advantages of smooth operation, better ride comfort and low noise, good market prospects (Lee, H W, K C, and Lee 2006). Suspension control is based on electromagnetic force, which couples the electromagnetic force, the suspension gap and the electromagnet coil current together (Zhou, Hansen, and D F, C H, and Li 2010). If the suspension control performance is slightly worse, it easily leads to vehicle-guideway coupling vibration. In the early studies, a single electromagnet was usually used as the object of study. The PSO algorithm was used to optimise the PID control parameters (Wu, Cole, and Mcsweeney 2016; Wu et al. 2016), so that the single iron suspension system can obtain a better static suspension (R R and Chen 2015). Wang, X B, and Shen (2015) adopting a small proportion of single iron suspension system as the controlled object, realised the stable static suspension on a elastic track beam. A vertical coupling vibration model of a single iron vehicle-guideway was established, and the dynamic characteristics of the system were studied when the system ran on a simply supported beam or a two-span continuous beam (Liang, S H, and Ma 2012). The reason of the self-excited vibration of the vehicle-guideway coupling vibration system was explored by establishing a single iron suspension model (Li, Li, and Zhou 2014).

Although the static stability of single-iron system is the basis for the stable operation of the maglev vehicle system, it cannot completely reflect the dynamic characteristics of the maglev vehicle system. As a result, a single suspension frame model was established to study the relevant dynamic characteristics of the suspension system. The dynamic model of the single frame of CMS04 type maglev train was established, and the effects of geometrical decoupling and elastic decoupling on the anti-roll and decoupling functions were compared for the suspension frame (Liu Y Z, W X, and Li 2014). The kinetic equation of a single suspension frame was deduced, and the condition of mechanical decoupling of maglev train through small radius curve was studied with D-H transform (Zhang, Li, and Li 2012). Further, the full-vehicle dynamics model was established respectively, and the dynamic characteristics of vehicle-guideway coupling vibration were studied, respectively (Yau 2009; Ren, Romeijn, and Klap 2009; Shi and Wang 2011; K J, B H, and Han 2015).

Although the correlation analysis of the dynamic characteristics was carried out for different maglev models, single iron suspension models were the most commonly used. Because single iron maglev system usually treats only one suspension point as a studied object, ignoring the coupling among different suspension points in a single suspension frame and the coupling among different suspension points of different suspension frame, static levitation is unstable or vehicle-guideway coupling vibration is more serious in the vehicle test debugging process. As there is an inductance, the suspension current of different suspension points of suspension electromagnets have different perturbation and time lag to a certain extent. And suspension control actuator has a certain degree of delay. As a result, all of these factors will have varying degrees of impact on the stability of the full-vehicle suspension system. However, according to the open literature, there are few studies on this aspect, so it is necessary to carry out the relevant research in this area.

For the problem, a vertical dynamic coupling vibration model of HSST type low-speed maglev train is established with five suspension frames. In order to make the research results more general, the double-loop control algorithm is adopted as the suspension control algorithm, which is the practical application algorithm in engineering. The effects of the magnitude of the feedback gain coefficient P of the internal loop current, the existence of a certain degree of delay in the suspension control actuator and the existence of different degrees of perturbation of the suspension control current on the suspension stability are studied for the maglev vehicle. The research can provide some essential guidances for the study of maglev vehicle static suspension.

2. Vehicle-track coupling model

2.1. Track beam model

Because there is a tight coupling between Maglev train and track beam via vertical levitation force for the track beam, it is inevitable to mainly produce vertical vibration rather than lateral vibration. So this paper only studies the vertical vibration characteristics of track beam. In the vehicle-track coupling vibration analysis, the track beam is generally simplified as BE beam, and the kinetic characteristics can be described by the superposition of the product of vibration mode function and modal coordinate. The left end of the track beam is the origin of the coordinates, and the vertical vibration dynamic equation of the track beam can be established (Wang et al. 2013) as shown in the following equation (1):

[mathematical expression not reproducible] (1)

Here, EI is the bending stiffness of the track beam, [delta]is the track beam damping coefficient, [rho]is the track density. [sigma]is the position function, which is used to transmit the position information of the force, FE is the external force of the bridge, x is the relative position coordinate of the electromagnet and the track beam. Using the modal superposition method, the vertical deflection of the elastic bridge can be expressed as:

[mathematical expression not reproducible] (2)

Here, [q.sub.i] is the i-order modal coordinates of the beam, [[DELTA].sub.i]is the i-order mode function of the beam as shown in the following Eq. (2).

[mathematical expression not reproducible] (3)

Here, l is the length of the beam. Substituting Eq. (1) into (2) and multiplying[[DELTA].sub.i] at both ends of the equation, then integrating for m 0 to l for the equation, there is Eq. (4) according to modal orthogonality condition.

[mathematical expression not reproducible] (4)

Here, [w.sub.i] is the i-order modal frequency of the beam as shown in the following Eq. (5).

[mathematical expression not reproducible] (5)

Here,[[lambda].sub.i]is the i-order modal wavelength of the beam.

2.2. Vertical dynamic model of HSST-type maglev vehicle

The maglev vehicle is taken as a rigid body and structural symmetry, and a single maglev vehicle is taken as the research object. As shown in Figure 1, a single vehicle body is supported by five suspension frames through 20 air springs. The air springs are numbered 1-20. The single suspension frame consists of two sets of anti-lateral girders, and four independent suspension control points provide levitation force. Here, [f.sub.n] is the levitation force, which is numbered n. [m.sub.1] and [m.sub.2] are the mass of the vehicle body and suspension frame, respectively. [[beta].sub.2]and[[alpha].sub.2]are the pitching and side roll angle acceleration of suspension frame, respectively. [J.sub.3] and [J.sub.4] are the pitching and side roll angle moment inertial of suspension frame, respectively. [[beta].sub.1]and[[alpha].sub.1]are the pitching and side roll angle acceleration of vehicle body, respectively. [J.sub.1] and [J.sub.2] are the pitching and side roll angle moment inertial of vehicle body, respectively. [k.sub.1], [k.sub.2], [c.sub.1] and [c.sub.2] are the spring stiffness, anti-lateral roll stiffness, air spring damping and anti-later roll beam damping. [x.sub.n] and [y.sub.n] are the length of the arm at the corresponding position. [z.sub.1n] and [z.sub.2n] are the absolute displacement of the vehicle body and the suspension frame for the serial number n. Assuming that the vehicle body and the suspension frame have ups and downs, pitching, side roll 3 degrees of freedom, these can be described as shown in the following Eqs. (6-12) (S Q, L., and K. L. Zhang. 2015).

Vehicle body ups and downs:

[mathematical expression not reproducible] (6)

Vehicle body pitching:

[mathematical expression not reproducible] (7)

Vehicle body side roll:

[mathematical expression not reproducible] (8)

Suspension frame ups and downs:

[mathematical expression not reproducible] (9)

Suspension frame pitching:

[mathematical expression not reproducible] (10)

Suspension frame side roll:

[mathematical expression not reproducible] (11)

Levitation force is shown as the following Eq. (12):

[mathematical expression not reproducible] (n = 1 - 20)(12)

In the formula (12), [[mu].sub.0]is the vacuum permeability, N is the number of turns, [A.sub.m] is the pole area of the suspension electromagnet, c is the suspension gap, [I.sub.e] is the measured current for the suspension electromagnet.

The side view of the suspension frame is shown in Figure 2.

3. Single iron suspension control algorithm

Maglev trains are generally supported by multiple suspension points, and the suspension system can be decomposed into a single suspension electromagnet control by decoupling. Because a single iron stable suspension is the basis of maglev vehicle running smoothly, as a result, single iron levitation control algorithm is often used to design a suspension control system (Y Z, W X, and Gong 2014). Engineering applications typically set up four independent suspension control points in each suspension module. Generally, there are four separate suspension control points in each suspension module. Therefore, there are 20 independent suspension control points for the maglev vehicle with five suspension frames. Finally, the stability levitation of the maglev vehicle will be achieved through independently controlling the 20 floating points. As described above, a more mature double-loop control algorithm is used as the suspension control algorithm, which is more engineering significance in this research.

3.1. Double loop control principle

In the double-loop control, the levitation gap is generally used as the outer loop control target, and the levitation current is generally used as the inner loop control target. The schematic diagram of double loop control is shown in Figure 3.

In this paper, PD control algorithm is selected as the outer loop control algorithm, and to improve the dynamic response characteristics of the levitation system, the acceleration of suspension gap is also introduced into the feedback control system. As a result, the target control current is as shown in the following Eq. (13).

[I.sub.m] = [I.sub.0] + [K.sub.p][DELTA]c + [K.sub.d][DELTA][??] + [K.sub.c][DELTA][??] (13)

Here, [I.sub.m], [I.sub.0], [K.sub.p], [K.sub.d], [K.sub.c], [DELTA]c, [DELTA][??] and [DELTA][??] are represented for target levitation current, nominal levitation current, levitation gap feedback coefficient, speed feedback coefficient of levitation gap, acceleration feedback coefficient of levitation gap, the change value of levitation gap relative to the nominal position, the change value of levitation gap speed relative to the nominal speed, the change value of levitation gap acceleration relative to the nominal acceleration, respectively.

The electrical equation of the suspension electromagnet is shown as the following Eq. (14) (Zhou, Hansen, and D F, C H, and Li 2010).

[mathematical expression not reproducible] (14)

Here, U is the voltage across the suspension electromagnet, R is the equivalent resistance of the suspension electromagnet, L is the effective inductance of the suspension electromagnet, [??]is the effective flux flowing through the suspension electromagnet. From Eq. (14), we can see that due to the presence of large inductors in the current loop for suspension electromagnets, the actual current [I.sub.e] in the suspension electromagnet can not track the target current [I.sub.m] faster. Hence, for the purpose of achieve rapid tracking of the target current, P control mode (P is the ratio coefficient of the PID control and the value of P is defined as the inner loop gain factor) is generally adopted in the inner loop control, and the inner loop control algorithm is shown as Eq. (15).

[DELTA]I = P([I.sub.m] - [I.sub.e]) (15)

Here, [DELTA]Iis the difference between the target levitation current [I.sub.m] and the actual levitation current [I.sub.e].

4. Dynamic simulation analysis

Adopting the formula form (1)-(15), the vehicle model with five suspension frames and the track beam model are established by software Matlab/Simulink. Kp, Kd and Kc are selected empirical values 6600, 50 and 0.2 as outer loop control feedback coefficient, respectively. In this section, the influence of the inner loop gain coefficient P, actuator delay and levitation current with a certain degree of perturbation on the static suspension stability for the vehicle are studied. There are some symmetries in the dynamic model as shown in Figure 1, so the levitation gaps corresponding to the centre of the suspension frame 1, suspension frame 3 and suspension frame 5 are researched. Since the mid-span deflection of the track beam is the largest, it is more likely to cause vehicle-guideway coupling vibration, as a result, the dynamic characteristics of the track beam at the mid-span position is taken as the research object. Part parameters of the system are shown in Table 1.

4.1. Influence of inner loop gain coefficient p on suspension stability

The selection of the inner loop gain coefficient will directly determine that the actual levitation current of the suspension electromagnet tracks the speed of the target current in the inner loop control. As a result, three kinds of cases are selected to study the effect of different inner loop gain coefficients on the static suspension stability for the maglev vehicle, and in Section 3.1 the following three cases are defined.

Case 1: The inner loop gain coefficient of the suspension controller is 300 for all the levitation points of the suspension frames.

Case 2: The inner loop gain coefficients of suspension controllers are 80 and 150 for suspension frame 1 and suspension 3, respectively, and the inner loop gain coefficients of the suspension controllers are still 300 for the levitation points of the remaining suspension frames.

Case 3: The inner loop gain coefficients of suspension controllers are 30 and 150 for suspension frame 1 and suspension frame 3, respectively, and the inner loop gain coefficients of the suspension controllers are still 300 for the levitation points of the remaining suspension frames.

The dynamic response characteristics of levitation gap of the suspension frame 1, 3 and 5 are shown in Figures 4-6, respectively, in cares 1, 2 and 3, respectively.

It can be seen from Figure 4 that the levitation gap of suspension frame 1, 3 and 5 will be able to get stable levitation about one second adopting the inner loop gain coefficient corresponding to case 1. It can be known that the maglev vehicle can achieve a better static suspension when the inner loop gain coefficient is chosen to be a larger value of 300.

It can be seen from Figure 5 that although the vehicle system can obtain a stable static suspension adopting the inner loop gain coefficient corresponding to case 2, maglev system need to go through about 3 s to get a stable suspension. Further analysis can be seen that the dynamic response time of the levitation gap of the suspension frame 1, 3 and 5 are increased, relative to the case 1, and the suspension gap stability of the suspension frame 1 is the worst with the minimum inner loop gain coefficient.

It can be seen from Figure 6 that the entire maglev system will lose stability adopting the inner loop gain coefficient corresponding to case 3. In fact, the difference between the case 2 and the case 3 is only that the inner loop gain coefficient of the suspension frame 1 for case 3 is reduced from the value 80 to 30 compared to the case 2.

Since the vehicle system will lose stability adopting the inner loop gain coefficient corresponding to case 3, the influence of inner loop gain coefficients corresponding to case 1 and 2 on the vibration characteristics of track beam are studied, respectively. The dynamic response characteristics of the mid-span displacement and the mid-span vibration acceleration of the track beam are shown in Figures 7 and 8, respectively, when the inner loop gain coefficients are adopted the case 1 and case 2.

It can be seen from Figure 7 that the displacement of the track beam has a greater vibration adopting the inner loop gain coefficient corresponding to case 2 and the track beam needs about 3s to get stable. However, only 1 s is needed to obtain stability for track beam adopting the inner loop gain coefficient corresponding to case 1.

It can be seen from Figure 8 that the track beam has a greater vibration acceleration adopting the inner loop gain coefficient corresponding to case 2, compared with case 1.

To sum up, the inner loop gain coefficient is best to apply a larger value into the levitation controller for better suspension stability, such as the value of 300. While the inner loop gain coefficient of one suspension point is taken a smaller value, such as the value of 30, it is a real possibility, the whole system will be unstable. Meanwhile, in order to suppress the vehicle-guideway coupling vibration, the inner loop gain coefficient can not too small, otherwise it will increase the vehicle-guideway coupling vibration.

4.2. Influence of actuator delay on suspension stability

Suspension controller actuator delay is unavoidable for the reason of aging hardware, overheated hardware and so on. As a result, setting the value of the inner loop gain coefficients of suspension controllers to 300 (as can be seen from Section 3.1, this value can make the system stable), the suspension stability of the maglev vehicle and the dynamic response characteristics of the track beam are studied when the actuators of suspension frame 1, 3 and 5 have the following two kinds of time delay cases.

Case 4: The actuator of suspension frame 1, 3 and 5 have a time lag of 0.001, 0.002 and 0.003 s, respectively, and the actuators of the remaining suspension frames do not have time lag.

Case 5: The actuator of suspension frame 1, 3 and 5 have a time lag of 0.001, 0.002 and 0.004 s, respectively, and the actuators of the remaining suspension frames do not have time lag.

The dynamic response characteristics of levitation gap of the suspension frame 1, 3 and 5 are shown in Figures 9 and 10), respectively, in cases 4 and 5, respectively.

It can been seen from Figure 9 that although the actuators of suspension frame 1, 3 and 5 have different time lag to a certain extent, respectively, the suspension gaps of the suspension frame 1, 3 and 5 have almost the same dynamic response characteristics.

It can been seen from Figure 10 that the entire maglev vehicle will be unstable, when the actuator delay of the suspension frame 5 is increased from 3 to 4 ms.

To sum up in conclusion, as long as the maglev system does not lose stability, the actuator time lag has little effect on the dynamic response characteristics for the maglev system, however, one actuator time lag is larger, which will lead to the entire suspension system unstably.

4.3. Influence of levitation current perturbation on suspension stability

Due to the actuator ageing and other factors, it is inevitable that levitation current has a certain degree of perturbation. For the problem, the relevant dynamic characteristics of the maglev vehicle will be studied in this section when the levitation control current has the following three perturbation cases.

Case 6: Assuming that the frequency of the suspension current is relatively faster, that is 500 Hz, the levitation current perturbation value of the suspension frame 1, 3 and 5 are 10%, 30% and 50%, respectively, and the levitation current of the remaining suspension frame are ideal conditions.

Case 7: Assuming that the frequency of the suspension current is relatively gentle, that is 100 Hz, the levitation current perturbation value of the suspension frame 1, 3 and 5 are still 10%, 30% and 50%, respectively, and the levitation current of the remaining suspension frame are still ideal conditions.

Case 8: Assuming that the frequency of the suspension current is relatively slower, that is 20 Hz, the levitation current perturbation value of the suspension frame 1, 3 and 5 are still 10%, 30% and 50%, respectively, and the levitation current of the remaining suspension frame are still ideal conditions.

When the range of suspension current perturbation are adopted, the case 6, case 7 and case 8, respectively, the dynamic response characteristics of the suspension gap of suspension frame 1, 3 and 5 are shown in Figures 11-13, respectively. For better explaining the problem, Figures 11 and 1 have a small plot to describe the original picture.

It can been seen from Figure 11 that although the suspension control current of suspension frame 1, 3 and 5 have different degrees of perturbation, it has a little effect on the stability of the suspension gap while the perturbation frequency of the levitation current is high. The dynamic response characteristics of levitation gap of suspension frame 1, 3 and 5 are almost the same.

It can been seen from Figure 1 that the dynamic response characteristics of the levitation gap of the suspension frame 1, 3 and 5 are still better. It only takes 1 s to get stable suspension, when suspension current perturbation frequency is 100 Hz.

It can been seen from Figure 13 that the maximum fluctuation range of the suspension gap of suspension frame 5 is about 9 mm relative to the nominal suspension gap. It can be drawn that maglev system has lost stability at this moment.

Further analysis from the Figures 11-13, it can be seen that the greater the degree of perturbation of the levitation current, the greater the fluctuation range of the suspension gap.

The mid-span displacement and mid-span vibration acceleration of the track beam are shown in Figures 14 and 15 with the cases 6, 7 and 8.

It can been seen from Figures 14 and 15 that the higher the current perturbation frequency, the smaller the vibration displacement and the vibration acceleration of the track beam, when the levitation current perturbation value of the suspension frame 1, 3 and 5 are 10%, 30% and 50%, respectively.

To sum up, because the frequency of the current perturbation is related to the frequency of the suspension controller, as a result, when the suspension current perturbation is too large, such as about 50%, appropriately increasing the frequency of the suspension controller is conducive to suppress the coupling vibration of the track and improve the suspension stability of the maglev system.

5. Conclusion

In this paper, the vertical dynamic model of the maglev vehicle with five suspension frames is established. The effects of inner loop current gain coefficient P, actuator delay and suspension current perturbation on static stability are studied, and it can be concluded as follow.

In order to suppress the vehicle-guideway coupling vibration of the maglev system, the inner loop gain coefficient should take a larger value, such as a value of 300, so that the actual levitation current value of the suspension electromagnet can quickly track the target current value. Otherwise, it may lead to more violent vehicle-guideway coupling vibration, even levitation instability.

One suspension controller has a larger time lag, such as a value of 4 ms, which can make the whole maglev system occur more intense vehicle-guideway coupling vibration, even levitation instability.

While the suspension controller has a time lag to a certain extent, the value of which is under 3 ms. As long as the maglev system does not lose stability, the time lag of the actuator has a little effect on the dynamic response characteristics for the maglev system.

Increasing the suspension controller frequency to a certain extent, such as a value of 500 Hz, even if the levitation current has a certain degree of perturbation, such as a value of 50%, maglev system can be better stable suspension. However, when the suspension current has a larger range perturbation and the controller frequency is lower, such as a value of 20 Hz, maglev system will lose stability.

Acknowledgments

The authors acknowledge the support of State Key Laboratory of Traction Power, Southwest Jiaotong University.

Disclosure statement

No potential conflict of was reported by the authors.

Funding

This work was supported by the National key R & D program funded projects [Project number 2016YFB1200601-A03 and 2016YFB1200602-13] and State Key Laboratory of independent research topics [Project number 2016TPL_T03].

Notes on contributors

Keren Wang is a PhD student at State Key Laboratory of Traction Power, Southwest Jiaotong University. His main research fields include dynamic analysis and control of maglev train.

Weihua Ma is a researcher at State Key Laboratory of Traction Power, Southwest Jiaotong University. His main research fields include dynamic analysis and control of maglev train and longitudinal train dynamics.

Shihui Luo is a professor at State Key Laboratory of Traction Power, Southwest Jiaotong University. His main research fields include the design and control of maglev train and longitudinal train dynamics.

Ruiming Zou is a PhD student at State Key Laboratory of Traction Power, Southwest Jiaotong University. His main research fields include dynamic analysis of maglev train and longitudinal train dynamics.

Xin Liang is an engineer at CRRC QINGDAO SIFANG Co., Ltd. His main research fields include dynamic analysis and design of maglev train.

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Keren Wang (a), Weihua Ma (a), Shihui Luo (a), Ruiming Zou (a) and Xin Liang (b)

(a) State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, China; (b) CRRC QINGDAO SIFANG CO., Ltd, National Engineering Research Center for High-speed EMU, Maglev R&D Department, Qingdao, China

CONTACT Weihua Ma [??] mwh@swjtu.cn

https://doi.org/10.1080/14484846.2018.1486794

Table 1. System parameters. Parameter Value Parameter Value [m.sub.1] (kg) 11,800 [k.sub.1] (kN [m.sup.-1]) 300 [m.sub.2] (kg) 1800 l (m) 24 Am ([m.sup.2]) 0.021 [rho](Kg/m) 2200 N 365 [u.sub.0] 3.14*4e-7

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Title Annotation: | ARTICLE |
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Author: | Wang, Keren; Ma, Weihua; Luo, Shihui; Zou, Ruiming; Liang, Xin |

Publication: | Australian Journal of Mechanical Engineering |

Date: | Jun 1, 2018 |

Words: | 4874 |

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