A study on the vibration analysis of a maglev vehicle A theoretical investigation of the effect of magnetic damping on a vibration control system

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1 International Journal of Applied Electromagnetics and Mechanics 13 (2001/2002) IOS Press A study on the vibration analysis of a maglev vehicle A theoretical investigation of the effect of magnetic damping on a vibration control system Ken Watanabe a,, Yasuhiro Ohta b, Masao Nagai b and Takayoshi Kamada b a Railway Technical Research Institute, Hikari-cho, Kokubunji-shi, Tokyo, Japan b Tokyo University of Agriculture and Technology, Naka-cho, Koganei-shi, Tokyo, Japan Abstract. The superconducting Maglev system is conceptualized as the next generation high-speed public transportation system. For practical use, it is important to improve the ride comfort particularly in high-speed running. The Maglev vehicle is composed of lightweight car bodies and bogies which are mounted with SCMs (superconducting magnets), on-board refrigerating system, etc. The vibration isolation performance is lower than in conventional railways, since the mass ratio of sprung mass to un-sprung mass in the Maglev vehicle is lower than in the typical railway car. In this magnetically levitated system, the passive electromagnetic damping between the SCMs and ground coils is very small. Therefore, it is important to add active electromagnetic damping to the primary suspension between the SCMs and ground coils, and to adjust the secondary suspension between the car body and bogie. This paper examines lateral and rolling vibration control of the Maglev vehicle using LQ (linear quadratic) control theory. Moreover, the estimated electromagnetic damping, which interacts between the SCMs and the guideway, is also considered in the model to improve the ride comfort. 1. Introduction The superconducting Maglev vehicle is composed of lightweight car bodies and bogies mounted with SCMs (superconducting magnets) and on-board refrigerating systems. The vibration isolation performance is lower than in conventional railways, since the mass ratio of sprung mass to un-sprung mass in the Maglev vehicle is higher than in the typical railway car. In this system, the electromagnetic damping between the SCMs and ground coils is very small. This paper focuses on the reduction of the lateral and rolling vibrations of the Maglev vehicle using a vibration control system designed based on LQ (linear quadratic) control theory. Moreover, the case considering the effect of the electromagnetic damping is investigated. Corresponding author: Ken Watanabe, Tel.: ; Fax: ; ken w@rtri.or.jp /01/02/$ /2002 IOS Press. All rights reserved

2 80 K. Watanabe et al. / A study on the vibration analysis of a maglev vehicle Active F y or c y Car Body θ c Electromagnetic Spring Irregularity of Guideway Air Spring y c y 0 Bogie θ b y b SCM Electromagnetic Damping Active F z or Guideway z c m c z y Fig. 1. Simulation model. 2. Simulation model Figure 1 shows the 4-degree of freedom simulation model, which consists of the car body and the bogie. The car body and bogie are connected in secondary suspension, which consists of air springs and vertical and lateral dampers. Vehicle vibrations are excited by the irregularities of the guideway through the primary suspension of the electromagnetic springs. The primary suspension is generated by the interaction between the SCMs and levitation coils of the guideway. In this paper, we examine the following two methods to improve the ride comfort. (1) LQ control theory applied to the secondary suspension. (2) LQ control theory applied to the secondary suspension, with electromagnetic damping added to the primary suspension. We adopt lateral actuators instead of lateral dampers and regard the output force of the actuators as control input. We also adopt vertical actuators instead of vertical dampers and regard the output forces of the actuators as control input. We assume these actuators to be ideal ones that have no time lag in response. 3. Control system design The objective of this control system is to reduce the lateral acceleration and rolling angle of the car body and relative displacement between the bogie and guideway. In addition, the lateral relative displacement of the minimum gap between the bogie and guideway must be within 40 mm [1].

3 K. Watanabe et al. / A study on the vibration analysis of a maglev vehicle c 0 Gain y / y [m/ s 2 / m] passive Type B Type A Gain θ c / y 0 [rad / m] passive Type B Type A 10-6 Fig. 2. Comparison of vibration characteristics with different patterns of weighting factors. The state variables of the system are as follows: x =[ẏ c y c ẏ b y b θc θ c θb θ b ] T (1) The state equation of the system is as follows: ẋ = Ax + Bu + Wy 0 For the system given in Eq. (2), according to LQ control theory, the cost function is expressed as: [ ) 2 ( ) (ÿc yb y 2 ( ) 2 ( ) ] 2 0 θc θb ( u ) 2 J = dt (3) r 0 q 1 q 2 q 3 The objective is to minimize the above cost function J, where, u is control input that is determined by state feedback as: q 4 (2) u = Kx (4) 4. Simulation results 4.1. Without electromagnetic damping (a) Lateral actuators First, we control the lateral acceleration of the car body. For the cost function given in Eq. (3), we set the weighting factors of the rolling angles q 3 and q 4 to infinity and design the controller. We call this case Type A. Figure 2 shows a comparison between Type A and the passive system without control. From this figure, we confirm that the gain of the lateral acceleration of the car body near 1.3 Hz is lower in Type A than in the passive system. However, the gain of the rolling angular acceleration of the car body increases near 1 Hz, since we do not evaluate the rolling angles in the controller. Second, we consider the effects of the rolling angles on the lateral acceleration. We call this case Type B. The simulation results of Type B are shown in Figs 2 and 3. We confirm that the gain of the rolling angles is improved, except for frequencies lower than 1.2 Hz.

4 82 K. Watanabe et al. / A study on the vibration analysis of a maglev vehicle (Type C) 10 3 actuator Lateral actuator (Type B) Gain y c / y 0 [m/ s 2 / m] Gain θ c / y 0 [rad / m] (Type C) actuator Lateral actuator (Type B) 10-6 Fig. 3. Comparison of vibration characteristics with different patterns of actuators Gain y c / y 0 [m/ s 2 / m] actuator + ξ : 10 % Lateral actuator + ξ : 10 % Gain θ c / y 0 [rad / m] actuator + ξ : 10 % Lateral actuator + ξ : 10 % 10-6 Fig. 4. Comparison of vibration characteristics considering electromagnetic damping. (b) Lateral and rolling actuators As we mentioned above, the improvement of the gain of the rolling angles is insufficient with the lateral actuators alone. Therefore, we add rolling actuators to control the rolling angle of the car body. We call this case Type C. The cost function is expressed as: [ ) 2 ( ) (ÿc yb y 2 ( ) 2 ( ) 2 ( ) 2 ( ) ] 2 0 θc θb ula uve J = dt (5) 0 q 1 q 2 q 3 where, u la and u ve are control inputs of the lateral and rolling actuators, respectively. Figure 3 compares the vibration characteristics of different patterns of actuators. From this figure, it is clarified that the peak of the lateral acceleration of the car body in the passive system near 1.3 Hz is reduced in Type C. It is also important to note that the gain of the rolling angular acceleration of the car body for frequencies less than 1.2 Hz is reduced in Type C. q 4 r l r v

5 4.2. With electromagnetic damping K. Watanabe et al. / A study on the vibration analysis of a maglev vehicle 83 In the superconducting Maglev system, there is small electromagnetic damping between the SCMs and guideway. There are several methods of generating electromagnetic damping in the primary suspension. One of them is the electromagnetic damping control system with the application of the distributed-type linear generator [2]. This system generates lateral force and rolling moment by changing the current phase of the on-board power collecting coils. There is a merit in which this system needs no exclusive damping coils. In this study, the lateral force and rolling moment are assumed as parameters of each damping coefficient. We add 10% damping coefficient to the primary suspension, and control the lateral and rolling actuators of the secondary suspension. Figure 4 shows a comparison of the vibration characteristics considering electromagnetic damping. From this figure, it is effective for the improvement of the ride comfort to combine the electromagnetic damping and the actuators. Especially, it is more effective for the car body acceleration in the area of 6 Hz, which is near the natural frequency of lateral electromagnetic spring. 5. Conclusions It is shown that the lateral acceleration and rolling angle of the car body on the superconducting Maglev vehicle can be effectively controlled using LQ theory applied to the secondary suspension. It is also found that adding electromagnetic damping to the primary suspension further enhances the control system performance. References [1] M. Azakami, The Development of Maglev Bogie System on the First Train Set for Yamanashi Test Line, RTRI Report 10(1) (1996), (Japanese). [2] T. Murai, H. Hasegawa, T. Yamamoto and S. Fujiwara, Active Magnetic Damper Using Linear Generator, T. IEE Japan 119-D(11) (1999), (Japanese).