Influence of Guideway Flexibility on Stability and Bifurcation of Maglev Vehicle Systems

Transcription

1 Columbia International Publishing Journal of Vibration Analysis, Measurement, and Control (5) Vol. 3 No doi:.776/jvamc.5.6 Research Article Influence of Guideway Flexibility on Stability and Bifurcation of Maglev Vehicle Systems Xinye Li *, Wei Hu, and Yanhui Qie Received: 6 January 5; Published online 5 July 5 The author(s) 5. Published with oen access at Abstract The dynamical analysis of the couled vehicle-guideway system in vertical direction is exlored with the consideration that the vehicle is modeled as a single-magnet susension system and the guideway is simlified as a simly suorted Euler-Bernoulli beam. By the Hurwitz criterion, the conditions for Hof bifurcation and Saddle-node bifurcation under roortional-derivative (PD) control are deduced to give the boundaries between the stable and unstable domains of the equilibrium. On one hand, the comrehensive comarison between the cases of rigid and flexible guideways is carried out through analytical and simulation results. It is concluded that, comared with the case of rigid guideway, the stable domain in the control arameters lane for the equilibrium is much smaller when the first mode of flexible guideway is considered, and the flexibility of guideway is essential for dynamical analysis and design of maglev vehicle systems. On the other hand, the comarison between the cases of the first mode and the first two modes of flexible guideway is also resented. It is illustrated that the influence of the high-order modes on the dynamic behavior of the Maglev system can be negligible. Furthermore, it is found that bifurcation of the equilibrium may result in a new equilibrium, eriodic or divergence motion after the system is erturbed when the control arameters are not in the stable domain. Finally, numerical simulations are in good agreement with the analytical redictions no matter the equilibrium is stable or not. Keywords: Maglev systems; Guideway flexibility; PD control; Stability; Bifurcation. Introduction As well known, maglev train at high seed is a novel guideway transortation system. Comared with the traditional wheel-guideway vehicles, a maglev train can offer the advantages of low energy consumtion, less environmental imact, lower noise and emissions, as well as safety and easier maintenance. The maglev train has no friction with the guideway. It is susended beneath the guideway by the attracting electromagnetic force and driven by linear electromotor. It uses *Corresonding xylihebut@63.com School of Mechanical Engineering, Hebei University of Technology, Tianjin 33, P.R. China State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University, Shanghai 4, P.R. China 3

2 (5) Vol. 3 No electromagnetic attractive force to balance the gravity of the vehicle, and active control to stabilize the susension system around a desired susension ga because the system is inherently unstable. As a result, the couled vibration of magnetically levitated vehicles and guideways, namely, the influence of flexibility of guideways on the dynamics of high-seed maglev vehicles, is one of the key considerations in design and oeration, and more comlex than that occurs in conventional railway systems. In the ast decades, numerous researchers have contributed their efforts to the study of modeling for an actual maglev vehicle. For examle, Chiu et al. (97) roosed a couled vehicle-bridge model, which was simlified as a -degree-of-freedom (-DOF) mass-sring-damer system, and the interactions between maglev vehicles and guideways were shown by the electromagnetic force. Biggers and Wilson (973) modeled the vehicle-guideway system as an arbitrary number of lumed, doubly-srung vehicle mass systems traveling in tandem along simly suorted beams. Wang (995) develoed a detailed maglev system model, which consists of a 5-DOF vehicle model, a suerconducting magnet model, a vertically flexible guideway model, and a feedback/feedforward control system. Zhao and Zhai () modeled a Transraid (TR) 6 carriage as a -DOF vehicle model with a rigid car body suorted by four sets of magnetic bogies to investigate the vertical random resonse and ride quality of a maglev traveling on elevated guideways. A numerical model to investigate the dynamic characteristics of maglev system was develoed by Ren et al. (), in which the modelings of maglev vehicle and guideway with the surface irregularity were described. Susension instabilities in maglev system with 3- and 5-DOF vehicles traveling on a double L- shaed set of guideway conductors were investigated by Cai and Chen (995), with various exerimentally measured magnetic force data incororated into theoretical models. An iterative interacting method for analyzing the dynamic resonse of a maglev train traveling on an elevated guideway suorted by iers embedded in soil was resented by Yang and Yau (8). The maglev train was idealized as a row of dimensional (D) rigid beams, each susended by levitation forces and controlled by onboard roortional-integral-derivative (PID) controllers. However, relatively a little attention has been aid in the literature to the electromagnets in lateral directions and secondary susensions. Hence, there have been some studies of modelings concerning these factors. Shen et al. (8) discussed modeling of a high-seed maglev train, including the whole mechanical system of one vehicle with otimized susension arameters and all controlled electromagnet airs in vertical and lateral directions. Shi et al. (7) resented a new dynamical model of high-seed maglev vehicle/guideway interaction, which considered vehicle and guideway as a whole system and couled the vertical interaction with the lateral interaction. Hong et al. (4) ointed out that the secondary susension arameters did not destroy the stability condition deduced from single-magnet susension control system, and the advanced virtual rototying technology was alied to set u a 3 dimensional (3D) model to study the dynamic resonse of the main comonents when the maglev vehicles went through different orbit curve. Obviously, the resonse analysis of a maglev train moving on a flexible guideway is related to not only the dynamics of vehicle/guideway interaction but also the control of the maglev system. Recently, there are many literatures on the study of couled vibration roblems in maglev systems. Dynamic interactions between the vehicle and guideway of a high seed ground transortation system based on magnetically levitated vehicles were extensively studied by Cai et al. (994), with 4

3 (5) Vol. 3 No an emhasis on the effects of vehicle and guideway arameters. In the literature review works conducted by Cai and Chen (997), various asects of dynamic characteristics, magnetic susension systems, vehicle stability, and susension control laws for maglev/guideway couling systems were discussed. A numerical model for dynamic interaction analysis of an actively maglev vehicle and flexible guideway structure was develoed by Lee et al. (9). Dynamic governing equations were derived by combining the 5-DOF maglev vehicle modal, the modal roerties of guideway structure and a linear quadratic gaussian(lqg) controller for electromagnetic susension. The effects of vibrational characteristics of guideway on the dynamics of Maglev vehicle UTM- were analyzed numerically and exerimentally by Han et al. (9). Their results indicated that natural frequency of the guideway was decreased by increasing mass density rather than by decreasing rigidity, and that its daming ratio was increased in the Maglev vehicle, emloying a five-state feedback control law to the levitation. The dynamic interaction between moving vehicles and two-san continuous guideway was discussed by Teng et al. (8). The results showed that vehicle seed, san length and rimary frequency of the guideway had an imortant influence on the dynamic resonses of the guideway. Modeling of vehicle/guideway interactions and resonse characteristics of maglev systems for a multicar, multiload vehicle traversing on a single-or double-san flexible guideway were considered by Cai et al. (996), with an emhasis on vehicle/guideway couling effects, comarison of concentrated and distributed loads, and ride comfort. Yau () modeled the guideway girder as a single-san susended beam and the maglev train traveling over it as a series of maglev masses. Numerical investigations demonstrated that when a controlled maglev train traveled over a susended guideway shaken by horizontal earthquakes, the roosed hybrid controller had the ability to adjust the levitation gas in a rescribed stable region for safety reasons and to reduce the vehicle s acceleration resonse for ride quality. Nonlinear systems may exhibit considerably comlex dynamic behaviour such as change in stability of resonse, quasieriodic motion and chaotic motion. Both the redictability and stability of maglev systems are rather imortant, which exlains the reason that research in the area of dynamics and its control of nonlinear systems has received a great deal of attention. For instance, numerical analyses for maglev control system with 5-DOF, which reresented five kinds of vehicle motions such as heave, sway, itch, roll and yaw, were resented by Zheng et al. (, 5). Analytical results showed the regularities of dynamic resonses of maglev vehicle to disturbance in each degree of freedom at various vehicle velocities. They also analyzed the dynamical behaviors and stability of the system theoretically and numerically, and gave the stable region of the system. In addition, they resented two kinds of vehicle/guideway couling models with controllable magnetic susension systems to investigate the vibration behavior of a maglev vehicle running on a flexible guideway, and observed the henomenon of divergence, flutter, and collision on the dynamic stability of a maglev vehicle traveling on a flexible guideway. Zhou et al. () develoed a couled model of steel track and magnetic levitation system, and emloyed the harmonic balance method to investigate the stability and amlitude of the self-excited vibration, which rovided an exlanation of the henomenon that track-induced self-excited vibration generally occurred at a secified amlitude and frequency. A factor that greatly affects the stability of a maglev vehicle system is the time delay caused by the dynamics of the electromagnets. Therefore, the time delay between the inut voltage and the outut attraction force should not be neglected. It is well known that a time delay may decrease the stability margin of a closed loo control system or even cause the system to become unstable. Zhang and Zhang (3) considered the flexible guideway system 5

4 (5) Vol. 3 No for maglev train with time-delayed velocity feedback control, and analyzed the stability and direction of Hof bifurcation by alying the normal form theory and the center manifold theorem. There is no doubt about the imortance to study the couled vibration that occurs between a stationary Electromagnetic Susension (EMS) maglev vehicle and a girder, which is a ractical roblem that greatly degrades the erformance of a maglev system. Yet as of today, to the authors knowledge, systematically comarative study on the cases of rigid and flexible guideways for maglev systems has been seldom reorted in available literatures. In this study, two kinds of couled vehicle guideway model of a maglev system under PD control is analyzed to investigate the influence of guideway flexibility on the maglev vehicle. In addition to the first mode, the third mode of the guideway is also considered. The reminder of this article is arranged as follows: In Section, the influence of guideway flexibility is discussed by establishing the couled dynamic equations of maglev vehicle and flexible guideway, stability analysis and numerical simulations, in which only the first mode of guideway is considered. In Section 3, the influence of high-order modes of flexible guideway on the dynamic behavior of the Maglev system is resented by similar rocedure, in which both the first and the third mode of guideway are considered. In the last section, a brief discussion and conclusion are rovided.. Influence of Guideway Flexibility.. Couled dynamic equations Considering the deformation of the elastic track, the schematic diagram of a single-degree-offreedom susension system with a controlled direct current electromagnet is shown in Fig., where F and mg denote the electromagnetic force and the weight of the electromagnet Fig.. The schematic of maglev vehicles with flexible guideway. 6

5 (5) Vol. 3 No resectively; z and z G denote the vertical dislacements of the electromagnet and the guideway at the osition x resectively, and s is the susension ga. According to the theory of Euler Bernoulli beams, the dynamic equation for the deflection of guideways in z direction is given by 4 zg zg zg EI A C F( x, t) () 4 x t t where E is Young s modulus, I is the moment of inertia of the beam cross section, C is the viscous daming coefficient, is the mass density of the beam, F( x, t) is the density of distributed load on the beam, and A is the cross-sectional area of the beam. The solution of the beam equation, based on the modal suerosition, is formulated as an infinite sum of the roduct of the mode shaes and time-varying modal coordinates. z ( x, t) ( x) q ( t) () G i i i where the eigenmodes are normalized to satisfy the orthonormal condition l A ( ) ( ) m x n x dx mn (3) where is the Kronecker delta. For a simly suorted beam, the orthonormal modes ( x) is in the form mn n i x i ( x) sin( ) x l (i =,..., ) (4) m l G where mg Al is the san mass, l is the san length. The modal coordinates under the concentrated load, a levitation force of the single electromagnet, can be derived by solving the second-order differential equation * q t q ( t) q ( t) F t ( i,..., n) (5) i i i i i i i where i and i are the circular frequency and modal daming ratio of the ith mode resectively, and given by i EI C i, i (6) l A A and the generalized load is determined by l F t ( x) F( x, t) dx F( t) ( ) (7) l * i i i when x l is assumed in Fig.. If and only if the first mode function is considered in Eq.(), one has l l zg (, t) ( ) q ( t) (8) Substituting Eq. (8) into Eq. (5) for i, we obtain the dynamic equation for the first mode l zg z G zg ( ) F( t) (9) i i 7

6 (5) Vol. 3 No Considering the geometrical relationshi among the dislacement of the magnet z, the midoint dislacement of the guideway z and the magnetic ga s, one concludes that G z z s () G where s is the desired susension ga and zg is the midoint dislacement of the guideway corresonding to s. And the relationshi can be rewritten as the following when the system is erturbed z zg s () where z, zg and s are the erturbations of z, zg and s resectively. Based on the first-order vibration mode of the guideway, the model of maglev system with flexible guideway can be exressed as F t zg - z G - zg - G - g m F t s z G zg (G -) - g m s s I I I k s kd s - R I s k s s () where is the daming ratio, is the natural frequency of the first-order mode, G = m mg, m is the carriage quality, I is the current through the coil of the electromagnet when air ga is the desired value s, I is the erturbation of I, g is the acceleration due to gravity, k and k d are the roortional and derivative feedback gains resectively, R is the resistance of the electromagnet, and the electromagnetic force is given by F t k I I ( s s ) N A where k m, is the sace ermeability, A m is the area of the electromagnetic ole, 4 and N is the number of turns of an electromagnet... Stability analysis Introducing the state variables T the overall system in state sace can be written as G G (3) x z z s s I, the couled equations of motion of 8

7 (5) Vol. 3 No x x k I x 5 x - x - x - G - g m s x3 x 3 x4 k I x x x x ( -) - g 5 4 G m s x3 s x I x x k x k x - Rx x d k s x3 From system (4), we can get the following equilibrium oint x the system at the equilibrium oint x, we can get the Jacobian matrix (4) T. Linearizing 4G g 4G g s I A (5) g g -G - -G s I s I s Rs k kd - k s k k The roerties of the modal and other arameters shown in Table are used in the following stability analysis and simulation. Table Parameters for the Maglev vehicle system based on the simly suorted guideway under first mode. μ m(kg) R(Ω) N(T) I(A) L(H) s(m) Am(m ) μg ω η 4π π. The characteristic equation of the linear system is A- I a a a a a (6) where

8 (5) Vol. 3 No a a.497kd a3.497k.557kd a4.57k kd a k If characteristic equation Eq.(6) has a air of ure imaginary roots, and all the other roots have negative real art, then Hof bifurcation will aear. On the other hand, if one eigenvalue is zero, saddle-node bifurcation will occur. According to the Routh-Hurwitz criterion, we could get the stable conditions of the equilibrium as follows.57k kd (8) k at the same time it is also needed to satisfy the following conditions a Δ (9) a a and a 3 a a a (7) 3 Δ 4 () a5 a4 a3 a a a 5 4 The stable domain of the equilibrium in control arameter lane can be obtained by combining Eqs.(8) through () as the triangle in Fig.. Fig.. Boundaries between the stable and unstable domains of the equilibrium based on the first mode of the guideway The comarison of the boundaries between the stable and unstable domains of the equilibrium in control arameter lane based on flexible and rigid guideways is shown in Fig. 3, where the stable

9 (5) Vol. 3 No domain for the case of rigid guideway including Domains and is formed by the horizontal line and the uer line. As shown in Fig., the stable domain for the case of flexible guideway includes only Domain. For convenience, results for the case of rigid guideway are resented in the Aendix..3 Simulation results Fig. 3. Boundary comarison between the cases of rigid and flexible guideways To verify the validation of the above analysis, some numerical simulations are resented in Fig. 4 through Fig.8. From Fig. 4, it can be seen that the time history of air ga is eriodic, which means that Hof bifurcation occurs no matter the guideway is rigid or flexible. Fig. 5 shows that the equilibrium of the system is stable for the cases of rigid and flexible guideways when the control arameters are in Domain. Fig.6 indicates that the equilibrium is stable for the case of rigid guideway and unstable(divergence motion occurs) for the case of flexible guideway. According to Fig.7, it is shown that saddle-node bifurcation of the equilibrium occurs when the control arameters are in the short lower boundary of Domain in Fig. 3, since it is a common boundary of the stable domain for rigid and flexible guideways. It is obvious that the original equilibrium moves to a new osition. Fig.8 shows that the equilibrium is stable if the guideway is rigid and unstable(eriodic motion occurs) if the guideway is flexible when the control arameters are in the boundary between the Domains and.

10 (5) Vol. 3 No x -3 rigid guideway flexible guideway Air ga(m) Time(s) Fig. 4. Time histories of air ga corresonding to the control arameters in uer boundary of Domain, k, k 837 d 6 x -3 4 rigid guideway flexible guideway Air ga(m) Time(s) Fig. 5. Time histories of air ga corresonding to the control arameters in Domain, k, k 7 d

11 (5) Vol. 3 No x -3 rigid guideway flexible guideway Air ga(m) Time(s) 8 x -3 (a) 6 rigid guideway flexible guideway 4 Air ga(m) Time(s) (b) Fig. 6. Time histories of air ga corresonding to the control arameters in Domain k, k 5, (b) k 35, k 38 d d (a) 3

12 (5) Vol. 3 No rigid guideway flexible guideway. Air ga(m) Time(s) Fig. 7. Time histories of air ga corresonding to the control arametersin the horizontal boundary of Domain, k 34, k 3 d.5 x -3 rigid guideway flexible guideway Air ga(m) Time(s) Fig. 8. Time histories of air ga corresonding to the control arameters in the boundary between Domains and, k 894, k 5 d 4

13 (5) Vol. 3 No Influence of High-order Mode of Flexible Guideway 3.. Couled dynamic equation In this section, the third mode of the flexible guideway is also considered since the middle oint of the guideway where the electronic force acts on is a node for the second mode of a simly suorted beam. From Eq.(5), one has the following equation corresonding to the third mode where the third mode of the beam is determined by () () 3 and 3 denote daming ratio and natural frequency of the third mode resectively, and given by (3 ) EI C 3, 3 l A A (3) 3 Thus the dislacement of the guideway can be written as z q q G 3 3 (4) 3. Stability analysis x = Δs Δs ΔI q q q q, one has the state vector Introducing the state variables as T equation 3 3 x x x x x x x k I x3 g -- 4G m s x s x I x3 x 3 k x kd x - Rx3 x k s x x 4 x5 I x x - x - x - k s x x 6 x7 I x x - x - x - k s x (5) 5

14 (5) Vol. 3 No From Eq. (5), we can get the following equilibrium oint mg 3mg x Linearizing the system (5) at x, we can get the Jacobian matrix T (6) g g - 4G - - 4G s I s s I Rs k kd - k k s k A (7) mg mg s I mg mg s I The characteristic equation of the linear system is a a a a a a a (8) where a a.83878kd a k kd a k 876.5kd (9) a kd 876.5k a k kd a k To avoid Hof bifurcation and saddle-node bifurcation of the equilibrium, which corresond to a air of ure imaginary eigenvalue and zero eigenvalue resectively, one can get the following conditions according to the Routh-Hurwitz criterion k kd (3) k and a Δ (3) a3 a 6

15 (5) Vol. 3 No a a a a 3 Δ 4 (3) a5 a4 a3 a a 3 a a 5 4 a a a a a a a a Δ 6 (33) a7 a6 a5 a4 a3 a a a a a a a 7 6 Stable domain of the equilibrium in control arameters lane can be obtained by combining Eqs. (3) through (33),which is resented in Fig.9. It can be seen that the Fig.9 is almost the same as Fig., which shows that the influence of the third mode of the guideway is negligible. Fig. 9. Stable domain of the equilibrium in the control arameter lane when both the first and the third mode of the guideway considered Simulation results are shown in Figs. through, from which it is can be seen that the influence is insignificant. 7

16 (5) Vol. 3 No x -3.5 first modal third modal Air ga(m) Time(s) x -4 (a) 8 6 first modal third modal 4 Air ga(m) Time(s) (b) Fig.. Time histories of air ga corresonding to the control arameters in the uer boundary, (a) k 3, k 489, (b) k 35, k 459. d d 8

17 (5) Vol. 3 No first modal third modal.8.6 Air ga(m) Time(s) Fig.. Time histories of air ga corresonding to the control arameters in the stable domain, k 3, k 4 d 4. Discussion and Conclusion In this article, the dynamic characteristics of the maglev system, which was simlified as a single magnet susention system with flexible guideway, was carried out analytically and numerically, with an emhasis on the influence of guideway flexibility. Based on the resent study, some observations on the dynamics of the system can be drawn as follows. Comared with the case of rigid guideway, the stable domain of the equilibrium of the couled vehicle-guideway system is much smaller when the first mode of the guideway is considered. Similar to the case of rigid guideway, Hof or saddle-node bifurcation occurs on the boundaries between the stable and unstable domains, which result in eriodic motion or a new equilibrium osition resectively. With the control arameters on the same boundary for both cases where Hof bifurcation occurs, the vibration amlitudes of the maglev vehicles with flexible guideway are larger than the case of rigid guideway. When the control arameters are located in the stable domain for both cases, the vibration resonse of the maglev vehicle with flexible guideway diminishes to zero with time and needs longer time than the case of rigid guideway. It is also found that the resultant motion can be divergence after the equilibrium is erturbed if the control arameters are not in the stable domain. For both cases when the first and first two modes are considered resectively, it is shown that the stable domain, the vibration amlitudes of the air ga with the control arameters located in the same boundary corresonding to Hof bifurcation, and the time resonse with the control arameters located in the stable domain are almost identical. As a consequence, the high-order modes have little influence on the dynamic behavior of the maglev vehicle system. 9

18 (5) Vol. 3 No Acknowledgments This research is suorted by the National Natural Science Foundation of China under Grant No The authors exress sincere thanks to the reviewers for their valuable suggestions. References Biggers, S. B., Wilson, J. F. (973). Dynamic interactions of high seed tracked air cushion vehicles with their guideways: art I of a arametric study, Journal of Dynamic System, Measurement, and Control, Transaction of the ASME, 95(), htt://dx.doi.org/.5/ Cai, Y., Chen, S. S., Rote, D. M., Coffey, H. T. (994). Vehicle/Guideway interaction for high seed vehicle on a flexible guideway. Journal of Sound and Vibration, 75(5), htt://dx.doi.org/.6/jsvi Cai, Y., Chen,S.S. (995). Numerical analysis for dynamic instability of eletrodynamic maglev system. Shock and Vibration, (4), htt://dx.doi.org/.55/995/649 Cai, Y., Chen, S. S., Rote, D. M. (996). Vehicle/guideway dynamic interaction in maglev system. Journal of Dynamic System, Measurement, and Control, Transaction of ASME, 8(3), htt://dx.doi.org/.5/.876 Cai, Y., Chen,S. S. (997). Dynamic characteristics of magnetically-levitated vehicle systems. Alied Mechanics Reviews, 5(), htt://dx.doi.org/.5/.3676 Chiu, W. S., Smith, R. G., Wormley, D. N. (97). Influence of vehicle and distributed guideway arameters on high seed vehicle-guideway dynamic interactions. Journal of Dynamic System, Measurement, and Control, Transaction of the ASME, 93(), htt://dx.doi.org/.5/ Han, H. S., Yim, B. H., Lee, N. J., Hur, Y. C., Kim, S. S. (9). Effects of the guideway's vibrational characteristics on the dynamics of a maglev vehicle. Vehicle System Dynamics, 47(3), htt://dx.doi.org/.8/ Hong, H. J., Li, J., Chang, W. S. (4). The levitation control simulation of maglev bogie based on virtual rototying latform and Matlab. In Proceedings of the 8th international conference on magnetically levitated systems and linear drives, Changsha, China, 4(. 6 ). Lee, J. S., Kwon, S. D., Kim, M. Y., Yeo, I. H. (9). A arametric study on the dynamics of urban transit maglev vehicle running on flexible guideway bridges. Journal of Sound and Vibration, 38(3), htt://dx.doi.org/.6/j.jsv.9.8. Ren, S., Romeijin, A., Kla, K. (). Dynamic simulation of the maglev vehicle/guideway system. Journal of Bridge Engineering, 5(3), htt://dx.doi.org/.6/(asce)be Shen, G., Meisinger, R., Shu, G.(8). Modeling of a high-seed maglev train with vertical and lateral control. Vehicle System Dynamics, 46 (Sul. ), htt://dx.doi.org/.8/ Shi, J., Wei, Q., Zhao, C. Y. (7). Analysis of dynamic resonse of the high-seed EMS maglev vehicle/guideway couling system with random irregularity. Vehicle System Dynamics, 45 (), htt://dx.doi.org/.8/

19 (5) Vol. 3 No Teng, Y. F., Teng, N. C., Kou, X. J. (8). Vibration analysis of continuous maglev guideway considering the magnetic levitation system. Journal of Shanghai Jiaotong University, 3(), 5. htt://dx.doi.org/.7/s Wang, S. K. (995). Levitation and guidance of a maglev vehicle using otional review control(ph.d Thesis). Pittsburgh: Carnegie Mellon University. Yang, Y. B., Yau, J. D. (8). An iterative interacting method for dynamic analysis of the maglev train guideway/foundation soil system. Engineering Structures, 33(3), 3 4. htt://dx.doi.org/.6/j.engstruct...4 Yau, J. D. (). Interaction resonse of maglev masses moving on a susended beam shaken by horizontal ground motion. Journal of Sound and Vibration, 39(), htt://dx.doi.org/.6/j.jsv Zhang, Z. Z., Zhang, L. L. (3). Hof bifurcation of time-delayed feedback control for maglev system with flexible guideway. Alied Mathematics and Comutation, 9(), htt://dx.doi.org/.6/j.amc...45 Zhao, C. F., Zhai, W. M. (). Maglev vehicle /guideway vertical random resonse and ride quality. Vehicle System Dynamics, 38(3), 85. htt://dx.doi.org/.76/vesd Zheng, X. J., Wu, J. J., Zhou, Y. H. (). Numerical analyses on dynamic control of five-degree-of -freedom maglev vehicle moving on flexible guideways. Journal of Sound and Vibration, 35(), htt://dx.doi.org/.6/jsvi Zheng, X. J., Wu, J. J., Zhou, Y. H. (5). Effect of sring non-linearity on dynamic stability of a controlled maglev vehicle and its guideway system. Journal of Sound and Vibration, 79(-), 5. htt://dx.doi.org/.6/j.jsv.3..5 Zhou, D. F., Li, J., Hansen, C. H. (). Alication of least mean square algorithm to suression of maglev track-induced self-excited vibration. Journal of Sound and Vibration, 33(4), htt://dx.doi.org/.6/j.jsv..7. 3

20 (5) Vol. 3 No Aendix Results Based on Rigid Guideways Fig. A. Structure of the maglev system with rigid guideway Introducing the following state variables for the electromagnetic susension system in Fig. A. the state equations of the maglev system are derived as follows T x = s s I (A.) x x k I x3 x g - (A.) m ( s x ) I x3 s x x 3 x k x kd x - Rx3 s x k In order to make the electromagnet susend at the certain air ga s, the necessary feedback is alied to the ort voltage of the electromagnet. A PD controller widely used in ractice is adoted. The static equilibrium of the susended bodies leads to I mg k (A.3) s where g is the acceleration of gravity. From system (A.), we can get the following equilibrium oint x (,, ). Linearizing (A.) at x, we can get the coefficient matrix g g A = - s I s I s s k kd - R k s k k The characteristic equation of the linear system is (A.4) 3

21 (5) Vol. 3 No Rs s g s g g A I k k R (A.5) k I k I k k 3 - = d - The stable domain of the equilibrium in control arameter lane based on rigid guideway is shown in Fig. A, where U and S reresents unstable and stable domain, resectively U K 3 S 5 U Kd Fig. A. Boundaries between the stable and unstable domains of the equilibrium in the control arameter lane for the case of rigid guideway 33