Optimal Active Suspension Applied to Repulsive Magnetically Levitated Vehicle Systems

Co pyright © 1FAC 11 th Trie nni al Wo rld Congress, Talli nn , Esto ni a, USS R, 1990

OPTIMAL ACTIVE SUSPENSION APPLIED TO REPULSIVE MAGNETICALLY LEVITATED VEHICLE SYSTEMS M. Nagai, A. Moran and S. Tanaka Department of M echanical Engineering, Tokyo University of Agriculture and Technology, Koganei-shi, Tokyo 184, Japan Abstract. For high speed ground trans porta t ion systems significant attention must be given to the secondary suspension as it determines t he r unning stabil ity and ride quality . This paper analyzes the performance of an active secondary s uspension applied to a high speed ' Maglev ' vehicle running on a rough guideway . The dynamics of the s uspension and the question of the optimal control policy are analyzed and r eferred to an adequate performance index .

INTRODUCTION

passe nge r ca r s , a nd on l y few st udies foc u s on Mag l ev vehi cles ap pl ications [8 ] . The Mag l ev vehic l e systems are characterized by high veloc it ies , lightweight cabins , low car body to truck mass ratio , low undamped primary natural f r eq uency and , sometimes , negative primary suspen sion dam ping [ 9 ] . This parameters , having an impo r tant ro l e on the s uspension performance , are different to those of conventional ground vehicles and can af f ect the dynamic behaviour and design considerations of the applied active suspensions . Moreove r, while in wheel-supported vehicles is necessary t o maintain the wheel - road contact fo r c~ in Mag l ev ve hicles is important to minimize t he varia t io ns of t he magnetic levitation gap to avoid da nge r ous contact between the magnets and guideway given t he smal l magnetic clearances required to red uce the energy req uirements .

The need for a secondary suspensio n mode in conventional ground vehicles is based on t he fact that the suspension should f ulfill two conf l icting requirements : good roadlguideway tracking and acceptable ride comfort. Reliable tracking demands " stiff " support of the vehicle to mantain constant the wheel - road contact force, while good rirle comfort requires weak co u pling between the passenger cabin and the road . Also the s uspension s ho u ld sa t isfy vario u s design constraints to p rev~nt contact between the vehicle body and t he s us pension com ponents. Different s u spensions satisfy the above r eq uirime nts to differing degrees : passive , semi active and f u lly active suspensions , Passive suspe nsions , characterized by the a bsence of energy sources , have fixed parameters a nd limited performance. Active suspensions , requiring energy sources to generate forces in response to feedback signals , offer increased design flexibility and can greatly improve the perfo r mance of the system over a wide variety of operating conditions . These suspensions have been subject to much research in the past several years with special emphasis being placed in optima l suspension con trol [ 1 , 2 ], s us pension response sensiti vity .[ 3 , 4] an d capa bili ties and limitations of the active and semi- active systems [5 , 6] .

This paper analyzes the effectiveness of the active control air suspensions applied to repulsive - ty pe Maglev vehicle systems . Both a small sized and a f u ll sized vehicle models subject t o dist u rbing forces sim u lating the guideway r oughness are examined . The small model is a na l yzed theoretically and experimentally for wh ich a l aboratory - scale suspension eq uipped with a microcompu t er controlled pneumatic actuator was buil t , The large model sized according to JNR MLU001 parameters is examined by computer simulation . The step response , frequency response and power spectral density of the vehicle variables are exa mined t o investigate the advantages and limitations of the active suspensions on the ride q uality and r unning stability . The different config ur ation of both models illustrates the i n flue nce of the suspension parameters on the dy namic behaviour of the overall system . Fi na ll y , the sensitivity of t he suspension perfor ma nce t o var iations in t he mass ratio is di sc ussed in th e las t pa rt of t he pape r .

The concept of using a ' magnetic cushion ' to support vehicles on purpose-built guideways has been intenSively researched during the past two decades . This technology which has acquired the gene r ic name ' Maglev ' (short fo r magnetic levitation) , now offers a new transpo r tation system which can compete with the rai l ways and fi t , with considerable environmental appea l a nd f ue l efficiency into the 250- 500 Kmlh s l ot where railways are too slow and short-h a ul aircrafts too ex pe nsive [ 7]. This tec hnology is r eac hing the phE se of practical application and research oriented t o improve the ride quality and runni ng stab i li t y becomes necessary . Here is where the Maglev a nd Active Control technologies can meet to gain a vehicle suspension with improved gl oba l pe r for mance . The most of research on active s uspensions deal specially with conventional railways and

TIIEORETICAL MODEL Suspension Dynamics The general configuration of a full - scale repulsive type Maglev vehicle using supercon -

41 9

Table 1

Specifica ti ons of th e ex perimental model.

Upper mass Lower mass

Superconduct i ve magnet

Secondary stiffness Damping coefficient Equiv. primary s tiffn ess Piston cross section

Cylinde r volume

Ai r Guidance and propulsion coi I

Equivalent capacity Valve r esista nce Initial pr ess ur e

18 . 2 Kg 7.23 Kg 1.57xl0 3 N/m 43 . 0 N. s/m 4.85xl0 3 N/m 2 . 64xl0- 4 m2 3.96xl0- 5 m3 1. 2xl0 - 1O m3 /Pa 1.80xl0 9 ra . s / m3 1.01xl0 5 Pa

m2 ml kl c2 kl Ao Vs Cs Rp Po

floss ratio

2.52 1.48 4. 12 0 . 74

undamp . natur . freq . Pri ma. undamp . natur. [req. Actuator cut - off frequency

Seeon .

Fig. I

Sectional view of a full scale repulsive Maglev vehicle .

Upper magnet

50x50x20 100x70xl0 1.22 24.0

Lower magnet Flux density Levitation gap

Hz Ilz Hz

mm mm Tesla mm

[equation of truck)

The equivale nt stiffness of the primary sus pension kl is obtained by linearizing the magnetic force law aro und the nominal l evitation gap . Neglecting some nonlinear and high frequency effects , the equation that governs the pneumatic actuator can be formulated as : (3)

where u is the control signal actuating on the electro magnetic valves and together with the pressure P determine the air flow rate . Combining Eqs . (1), (2) and (3) and equating the corresponding terms, the state space equation of the system dynamics can be written as :

1 :Pneumatlc cylinder, 2 : Spring, 3:0sci Ilator, q:Accelerometer, 5:Upper mass, S:Servo valve, 7:Microcomputer, 8:Air compressor, 9:Potentlometer, IO: A/D converter, 11:1/0 board, 12:Lower mass , 13:Pe rmanent magnets, lq : Gap sensor Fig . 2

x=

A X + B u + W ~o

(4)

where the state variable matrix is

Two degrees of freedom suspension model . Structure of the experimental apparatus .

ducting magnets and a passive secondary suspension is shown in Fig . l. The equivale nt two degrees of freedom suspension model equipped with an active device analyzed in this paper is shown in Fig . 2. The model in itself is not a realistic desc ript ion of a complete vehicle but its simplicity permit to und erstand the most basic problems of t h e suspension design . In the model, the magnetic levitation is r ealized by two repulsive and permanent magnets and the secondary suspension is constituted by a passive linear spring and a cont r olled pneumatic actuator which provides the damping and active forces. Assuming there is not dynamic coupling between the vehic l e suspension and guideway motion,and considering linear suspension components , the equations describing the suspension dynamics when the system is excited by guid eway irregularities are expressed as : [equation of car body)

There a r e many control strategies that can be used to achieve an optimal performance of the composite system : various state variables can be determined and related in several ways to obtain diffe r ent suspension characteristics . This paper considers a linear control law defined as : u =

-F X

(5)

where the optimal feedback gain

is calcula t ed by minimizing the following performance index selected from a dynamical ene rg y viewpoint and reflecting suspension performance:

(6)

(1)

420

w2 wl

...

AX2

car body natural frequency, w22 2 truck natural frequency, w1 mass ratio m2/ ml : Xl - Xo A Xl x2 - Xl ,

The pressure in the cylinder chamber and the air flow rate are controlled by two electromagnetic valves (No . 6) d riven in a way of pulse width modulatio n ( PWM) . The magne t ic l evitatio n is rea l ized by two r ep u lsive and permanent rare - earth magnets attached to the lower mass and the test stand (No . 3) which simulates the r ough guideway . The use of two simple magnets for levitation gives constructiona l simplicity since the primary objective of th is design i s to analyze the system dynamics .

k2/m2 kl/ml

, is a weighting coefficient of the dynamical energy of the secondary suspension against that of the primary suspension . H is an appropriate val ue for evaluating vibration isolation performance and indicates the relative importance of the dynamical energy of the overall system against the value of the respective input signal that controls the pneumatic actuator . By analyzing the closed loop transfer function (no included in this paper), it can be deduced that the car absolute vertical velocity feedback gain (f 2 ) has the same effect as providing ' skyhook ' damping to the sprung mass attenuating the resonant frequency vibrations. The acceleration feedback gain (f 3 ) has the same effect as increasing the 'apparent' car body mass attenuating the acceleration transmissibility; however, since the static weight of the vehicle is unchanged, the 'locomotive power requirements remain unchanged. The Laplace transform block diagram representing the relations between the suspension parameters including feedback control variables is shown in Fig . 3 .

Control Strategy

Car body

The feedback control system is designed as follows . The vertical acceleration and relative displacement of the upper and lower masses are measured by two accelerometers (No . 4), a linear potentiometer (No.9) and a gap sensor (No . 14) as are shown in Fig. 2. These four Signals are inputs to the microcomputer (No.7) through an A/D converter (No . lO) . Calculating the velocity from the measured acceleration and according to optimal regulator principles the microcomputer produces an output signal which actuates on the elec tromagnetic valves through an I/O board (No . ll). Considering that an important frequency range of the suspension systems is generally less than 10 Hz , the sampling freq uency of the measured signal is determined to be 50 Hz so that the sampling time 20 msec is long enough to realize the advanced control in the microcomputer. Vi bra t ion Excit a t i on

Secondarr suspension

TrLlck ~

....:Pri • .1tr , k Cl x _: suspension , ................ 1. .................... '

x, Tuct

Fig. 3

Laplac e block diagram of the system dynamics and control variables.

The perfo r mance of the suspension is examined by three methods of vibration excitation step input, sinusoidal wave and random wave to investigate the suspension stability, frequency response of transmissibility and ride comfort. For vehicles running at constant velocity, the power spectral density of the guideway roughness can be considered be proportional to the minus second power of the temporal wave number of the road surface profile. In the experiment, this relation is obtained approximately by designing a shaping filter subject to a zero-mean Gaussian white-noise [10] .

EXPERIMENT RESULTS OF TIlE SMALL SIZED MODEL Experimental model structure The analyses of this suspension with a mass ratio equal to 2 . 52 indicates that , in the analyzed range, the effectiveness of the active suspension improves with increasing coefficient H (Eq.(6)). The overshoots are reduced significantly and the speed of response of the system is greatly improved. In the following, the car body and truck step responses and the car PSD vertical acceleration obtained with the greatest analyzed coefficient H = 3 . 2xl0 12 are discussed and compared with those obtained by using a passive secondary suspension. The influence of the active suspension on the magnet-guideway clearance is analyzed through the truck vertical displacement.

The fundamental structure of the experimental set up and its speCifications are shown in Fig . 2 ana Table I, respectively . A controlled pneumatic actuator (No.l) and a passive spring (No . 2) are located in parallel between the upper mass (No . 5) and lower mass (No.12) which simulate the car body and truck respectively. This configuration has the following advantages : (1) The passive spring supports the static vehicle weight while the air cylinder compensates for dynamical energy only so that the energy supply is not required as much . (2) Even if the actuator or the control circuit happens to break down , the passive spring can support the vehicle weight . Therefore, this suspension has a fail-safe performance .

Ste p Response. According to the experimental results shown in Fig . 4 , it is apparent that the step response of the system is improved by the

421

Table 2

Car body mass Truck mass Secondary stiffness Damping coefficent Equiv . primary stiffness

. 0.0' \

.... 0 .005 N

><

0.0 _, 0.0' \

Piston cross section

Equivalent capacity Valve resistance

~O·::L___-L~_ \_·,·".r·· ~:',O'-____~jr.~o ~,r.~O----~Sr.~O--__· __

____

\( . , 01

Fig. 4

Experimental step response of the car body (x2) and truck (xl)'

Table 3

N

::c

-... -~ 0.1

Cl)

A

(/)

O.O l

0.-

'"

. 0.001

.

..,

.

OJ

/

Q,OOQl

lIz

300 1.5xl0- 6

Km/h m

1.0

0.5x10 14

1.0x10 14

5.0x10 14

10.0xl0 14

f1

0.277xI08 0 . 600x10 7

o .I.60x108

0.130xl0 9 0.206x10 8

0.195x10 9 0.429x10 8

f3

0.215x106

0.990x10 7 O. 315x 106

f4

-0.179xI0 9 -0.405x10 7

-0.263xl0 9 -0.515x10 7

f5

0.66lx10 6

0.867xl0 6

-0.608xl0 9 -0.717x10 7

-0 . 861xl0 9 -0.730x l 0 7

superconducting Maglev test vehicle developed by the Japan Railway Technical Research Institute capable of carrying SO passengers and with 420 Km/h as maximum attainable speed . For this model, the optimal feedback gains (Eq.(5» and the corresponding evaluating coefficients (Eq.(6» are shown in Table 3. In this system the mass ratio 0 .67 is lower than the ratio of the preceding small model 2 . 52 to illustrate the influence of the mass distribution on the dynamic behaviour of the system. Reduced mass ratio results in a relatively increased dynamical energy of the primary suspension which has a greater influence on the composite system .

\\ \\

\\111\ \\1,

>

Hz Hz

11

f2

I~


Kg Kg N/m N.s/m N/m 1.13xl0- 2 m2 1.25xl0- lO m3 /Pa 5.01xl0 8 Pa . s/m 3

Optimal feedback gains and respective weighting coefficients .

{J

Simulation -·- ·- Active Experiment ······ ·· ·Passive ----- Active . ...... ,''\.-. 12 h/'---:.:.c'~:"~,=3 . 2x 10

//,/ /1

rl

OJ
_

4000 6000 3.55xl0 5 4 . 70xl0 4 1.00xl0 6

0.66 1. 50 2.05 2.54 v A

Running speed Guideway roughness

Figure 5 shows the Power Spectral Density (PSD). experimental PSD of the car body vertical acceleration when the system is excited by random wave. This result expresses the effectiveness of the active suspension reducing significantly the peak value of the acceleration and , therefore, improving the ride comfort . The deviations of the experimental response at high frequencies may be caused by nonlinearities in the pneumatic actuator

~

m2 ml k2 c2 kl Ao Cs Rp

Mass ratio Seeon. undamp. natur. freq. Prima. undamp. natur. freq. Actuator cut-off frequency

active suspension. The percentage overshoot of the car body response is reduced from 85% to less than 25%, the rise time is slightly increased and the settling time is reduced in 25%. Although the percentage overshoot of the truck response is almost the same for both suspensions, the settling time is reduced significantly and the nominal magnetic levitation gap is fast recovered .

" -... E

Specifications of the full sized model.

10

frequency [IIzJ

Fig. 5

Differing from the small sized model, the step response of the car body displacement, Fig. 6(a), is characterized by the presence of undershoot which increases whereas the overshoot decreases with increasing coefficient H. There are not undershoots in the truck response, Fig. 6(b), and therefore, the elements of the secondary suspension are overloaded when the car body is in 'undershoot movement'. Both the car body and truck resp onses have a fast recovering which is desirable to maintain the nominal levitation gap reducing the risk of contact between the magnets and the guideway .

Experimental power spectral density of the car body vertical acceleration .

The above results illustrate the effect of the active suspension improving the transient resp onse characteristics of the system and reducing the car body acceleration improving the ride comfort. However, at high frequencies the active and passive suspensions have similar responses .

RESULTS OF THE FULL SIZED MODEL

The power spectral density of the car body vertical acceleration and the U. T .A. C. V. comfort specification are illustrated in Fig. 7. The PSD of the guideway roughness is determined by the analytical expression :

Table 2 shows the main parameters of the vehicle suspension model sized according to JNR MLU-OOI specifications. The MLU-OOI is a

422

2 .0

Passive ....... Active -----H

.\ i '


UJ

--H

iA

c: o p.

;''\

UJ

...
'vi

P.


The influence of the active suspension reducing the PSD of the car acce l eration aro und th e r esonant frequency is clearly noted; this effect increases with increasing coeffici ent H. However, at high frequencies the PSD increases by the active suspension and there is a negative effect on the capa bilities of the system to isolate high frequency vibrations. Figure 7 shows also that all the curves pass through a common point . This fixed point cons titut es a limitation of the active suspension since the cha r acteristics of the response before and after this point are different . However. fixed points can also be found in passive and active s uspensions with high mass ratio [ 6) .

O.5x10 14 5.0x10 14

'-"

'-'

(/)

0 .0

1.0

3 .0

2.0

5 .0

'.0

V

time [sec] INFLUENCE OF MASS RATIO ON SUSPENSION PERFORMANCE

(a) car body

Passive ......... Active

- - -- - fi

'c:" UJ

-- H

~

0

0-

UJ

...'" 1.0 Cl)

O.5x10 14 5 .0x10 14

~,

f/"'"~

M

p.

..,'"

The sensitivity of the performance of the full sized model to variations in the car to truck mass ratio ( 0<. ) is analyzed by setting the primary and secondary undamped natural frequencies . the secondary damping coefficient, the actuator cutoff frequency and the following coefficient b which relates actuator parameters with the car body mass :

\ ../

'"

~

\,/

This coefficient acts directly on the control signal u (Eq . (4)) and plays an important role in deter mining the optimal feedbak gains . Change the mass ratio fixing this coefficient has the same effect as change the mass ratio maintaining constant the car body mass (m2) ' Given the particular characteristics of the analyzed active suspension and differing from the passive and other types of active suspension models [13). variations in the mass ratio by changing the sprung or unsprung masses no always conduce to the same results [14]. The step response of the variations in the magnetic levitation gap depicted in Fig. 8 shows the special influence of the mass ratio on the transient response characteristics . The peak value and the settling time of the responses decrease with increasing mass ratio for 0« 2 . 5 but increase for higher values . The speed of response always decrease with increa sing rati o .

0 . 0 L---~--'~.~ 0 -------2~.0--------3. 0-------'~.0 -------5-.0--~

time (sec] (b) truck Fig. 6

Step response of the full sized model.

~(w)

=

A.v/w 2

(7)

where A is a statistical roughness parameter. v is the longitudinal velocity of the vehicle and w is the temporal angular frequency . From Eq . (7) is deduced that high velocities en l arge the PSD of the guideway roughness and could make the vehicle difficult and uncomfortable to ride [11,12].

_._T_."_.C ]t-3t-____U _._ y

::::::

(8)

b

_. - - - - - - . .

, Passive ......... \ Active ---- H -.-.-. H

"Cl>

O. 5xlO 14 l.Ox1 0 14 5 .0x 10 14

H

5 . 0x10 14

Cl) 10

UJ

c: o

lE-4

0-

Cl

UJ

Cl)

"'"

0..

0-

..,
lE- S

Cl)

10

Frequency [Hz)

Fig. 7

Power spectral density of the car body vertica l accele r ation .

Fig. 8

423

Step response of the variations in the magnetic levitation gap .

dynamical configuration of the overall system. Considering that the effectiveness of the active suspension and the global performance of the system improve with increasing mass ratio , Maglev systems with as high as possible ratios are r ecommend . Although the active suspensions has no important effect on the capability of the system to isolate high frequency vibrations, the range of effectiveness can be widen by using adequate active elements with high cut- off frequencies and little response delay to achieve a good performance in the desired operating range .

The frequency response of the magnetic gap variations in Fig. 9 shows that the capabilities of the system to attenuate low frequency vibrations dec r ease with increasing mass ratio. High ratios displace the peak value to the left and reduce the frequency at which the response reach the unitary value.

H= 5 . 0xl0 14

0 .1

REFERENCFS [1) Thompson A. G. and Davis B.R.: Optimal Linear Active Suspensions with Derivative Constraints and Output Feedback Control . Vehicle System Dynamics, 17, pp . 179-192. (1988) . (2) Hrovat D., Margolis D.L . and Hubbard M.: An Approach Toward the Optimal Semi-Active Suspension Trans. of ASME, Journal of Dynamic Systems , Measurement, and Control, vol . ll0 , pp.288-296. (1988) . ( 3) Redfield R.C. and Karnopp D. : Optimal Performance of Variable Component Suspension . Vehicle System Dynamics, 17, pp.231-253. (1988). (4) Thompson A. G.: Optimal and Suboptimal Linear Suspensions for Road Vehicles. Vehicle System Dynamics, 13, pp. 61-72 . (1984) . (5) Sharp R.S . and Hassan S . A.: On the performance capabilities of Active Automobile suspensions . Vehi cle System Dynamics, 16, pp. 213-225. (1987) . (6) Karnopp D. : Theoretical Limitations in Active Vehicle Suspensions . Vehicle System Dynamics, IS, pp . 41-54. (1986). (7) Sinha P. K.: Electromagnetic Suspension , Dynamics and Control . lEE Control Engineering Series 30 . (1987). (8) Nagai M. and Sawada Y.: Active Suspension for Flexible Structure Control of High Speed Ground Vehicles . 10th World Congress on Automatic Control, Munich, 1987. IFAC, vol . 3, pp. 197-202. (9) Yamamura S . : the State- of-the-Art of Maglev Systems Development. Journal of IEEEJ, 96-10, p.887 . (1976). (10) Nagai M. , Shioneri T. and Tanaka S.: Active Suspension with Microcomputer Controlled Pneumatic Actuator. Proc. 28th SICE Annual Conference , 1988, pp. 823-826. [ll) Wilkie D. and Borcherts R. : Dynamic Characracteristics and Control Requirements of Maglev Levitation Systems, ASt1E Publications 73-ICT-17 . (1973). (12) Guide for the Evaluation of Human Exposure to Whole-Body Vibration . International Standard ISO. Ref . No . 2631-1978 (E) . (13) Thompson A. G.: Design of Active Suspensions. Proc . Inst. of Mech . Engrs . 1970-1971, vol.185 , 36/71 , pp . 553- 563 . (14) Cho D. and Hedrick J .K.: Pneumatic Actuators for Vehicle Active Suspension Applications. Trans . of ASME, Journal of Dynamic Systems, Measurement , and Control, vol . 107, pp . 67-72 . (1985) . [15) Nagai M. , Moran A. and Tanaka S . : Optimal Active Suspension to Improve the Dynamic Stability of Repulsive Maglev Systems . 11th Intern. Confer . in Maglev Systems and Linear Drives . Japan, 1989 .

10

Frequency [HzJ

Fig. 9

Frequency response of the variations in the magnetic levitation gap .

H = 5.0xl0 14

1E-4

1£-5

Frequency [Hz) Fig. 10

Power spectral density of the car body vertical acceleration.

The notable influence of the mass ratio on the ride quality can be clearly appraised in Fig . 10. Low imcrements in the mass ratio produce great improvements in the ride comfort . This effect is specially emphasized with low mass ratios and high frequency waves.(15)

CONCLUSIONS The results obtained by analyses of the small and full sized suspension models show the influence of the activelly controlled air sus pensions on the performance and dynamic behaviour of the repulsive-type magnetically levitated vehicle systems . In general, the active suspensions improve the running stability and ride quality of the vehicle at not very high frequencies and specially aro und the resonance points ; however, this improvement is affected by the mass distribution and

424