Proceedings of the 45th IEEE Conference on Decision & ControlManchester Grand Hyatt HotelSan Diego, CA, USA, December 13-15, 2006

The Performance Improvements

of Train Suspension Systems with InertersFu-Cheng Wang, Chung-Huang Yu, Mong-Lon Chang, and Mowson Hsu

Abstract-This paper investigates the performance benefitsof train suspension systems employing a new mechanicalnetwork element, called Inerter. Combined with traditionalpassive suspension elements - dampers and springs, Inerteris shown to be capable of improving the performance, interms of the passenger comfort, system dynamics andstability (safety), of the train suspension systems.Furthermore, a motor-driven platform is constructed totest the properties of suspension struts with inerters.

I. INTRODUCTIONT HE analogy between mechanical and electrical1 network systems is well known. By comparing thedynamic equations there are two analogies, namely the"force-voltage" and the "force-current" analogies,between the mechanical and electrical systems. Theformer analogy was frequently used in the old times whenvoltage was normally considered as electromotive force.The later analogy is more popular now because thecurrent in electrical systems, as the force in mechanicalsystems, is regarded as a through variable in networks.The standard force-current analogy is illustrated in Fig. 1.

Fig. 1: The old force-current mechanical/electrical network analogy [10].

Manuscript received on February 22, 2006, and revised on August 28,2006. This work was supported in part by the National Science Council ofTaiwan under Grant 93-2218-E-002-115 and 94-2218-E-002-063.

Fu-Cheng Wang is with the Mechanical Engineering Department ofNational Taiwan Univeristy, No.1, Sec. 4, Roosevelt Road, Taipei 10617,Taiwan. (phone: +886-2-33662680; fax: +886-2-23631755; e-mail:fcw@ntu.edu.tw).

Chung-Huang Yu is with the Institute of Rehabilitation Science andTechnology at National Yang-Ming University, 155 Li-Nang Street, Section2, Taipei 112, Taiwan. (e-mail: chyu@ym.edu.tw).

Mong-Lon Chang was with the Mechanical Engineering Department ofNational Taiwan Univeristy, and is now a research engineer with the MitacInternational corp., Taipei, Taiwan. (e-mail: r92522824@ntu.edu.tw).

Mowson Hsu was with the Mechanical Engineering Department ofNational Taiwan Univeristy, and is now a research specialist with Min AikTechnology Co.,Ltd., Taoyun, Taiwan. (e-mail: r92522818@ntu.edu.tw).

From Fig. 1 it is noted that one terminal of "mass" isalways grounded, so that electrical networks withungrounded capacitors do not have a directspring-mass-damper analogy. As a result, it potentiallynarrows the class of passive mechanical impedanceswhich can be physically realized [11]. It was from theappreciation of the gap in the old mechanical/electricalanalogy that a new mechanical element, called Inerter,was proposed. A new network analogy is shown in Fig. 2,with the Inerter symbol and the defining equation asfollows:

F bd(v2 v-)dt

in which F, v and b represent the force, velocity andinertance of the system respectively [10].

F.:henchanical /e Electrical

?S2L Lt r~~~(S) s 2 l Y(S) =LdtF k(V2 V1) sprinLg dt=!L

Fdt v l dt= -LI(V2 V1) capacitor

0

V2 L, LV" Y(8) = bs v2rs)= R8F = bd(V2-VI) dal r i =Cd(V2-Vl) capaistor

digd:Tenwmcaia/lcrclntwranlg[1]

With the introduction of Inerter, mechanical and electricalnetwork systems become really analogous to each other,and all passive network impedance/admittance can bephysically realized via three mechanical elements -springs, dampers and inerters. Consequently it allows abroader use of passive network impedance/admittance topotentially increase the performance of passivemechanical systems. The first successful application ofInerter is to vehicle suspension design [11,12], whereseveral combing layouts of inerters, dampers and springswere optimized for various performance criteria. It wasconcluded that some layouts are more suitable than othersfor particular performance criterion. In [9] theoptimization was further carried out by the LMI (Linear

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

Matrix Inequalities) method, where all passive transferfunctions with fixed order were optimized for variousperformance measurements. The resulting passivenetworks were then synthesized by Bott-Duffinrealization method. It was shown that the systemperformance can be further improved by allowing higherorder passive impedance, with the drawback of verycomplicated network synthesis. The second application ofInerter is the mechanical steering compensator ofhigh-performance motorcycles [2], where Inerter wasused to replace a conventional steering damper in order tostabilize the system in both of the "wobble" and "weave"modes.In this paper we discuss the performance benefits to thetrain systems by applying inerter to the suspension design.The train suspension system is more complicated than thevehicle suspension system in that there is one morecomponent- bogie - between the body and the wheel.Two suspension struts are placed to connect the body andbogie, and the bogie and wheel. This paper is arranged asfollows: in section II a one-wheel train suspension systemis introduced. The dynamics of the system with threesuspension layouts, namely the conventional, the paralleland the serial layouts, is described and optimized for twoperformance measurements. It is shown that thoseperformance criteria can be improved by applying inertersto the suspension design. In Section III the lateral stabilityof train suspension systems is discussed. It is illustratedthat the critical speed is significantly increased byapplying inerter to the lateral suspension design. Insection IV, Inerter is realized by the ball-screw structure,and a motor-driven testing platform is constructed toverify the properties of this model. Finally, someconclusions are drawn in the last section.

II. One-Wheel Train Suspension SystemA. One-Wheel Train Suspension Model

(a) free-body diagram. (b) suspension arrangement.Fig. 3: A one-wheel train suspension system.

A one-wheel train suspension system is shown in Fig. 3,where mS, mb and m, represent the masses of the trainbody, bogie and wheel respectively. Q, represents thesuspensions between the body and the bogie withstiffness k, and damping rate c,; Q2 represents thesuspensions between the bogie and the wheel with

stiffness kb and damping rate Cb. The vertical reaction ofthe wheel and rail track is modeled as a parallelcombination of spring kw and damper c,. The systeminputs are F, and z, which represent the vertical force andtrack irregularities respectively, while the system outputsare Zs, Zb and z, which represent the vertical displacementsof the body, bogie and wheel respectively. Threesuspension layouts, namely the conventional, paralleland serial arrangements as illustrated in Fig. 4, areconsidered in both of the Q, and Q2 cases.

Fig. 4: Three suspension layouts: conventional, parallel, serial arrangements.

B. Dynamic Equations(a). The Conventional SuspensionThe dynamic equations of a one-wheel train system

with traditional suspension, as shown in Fig. 5(a), are asfollows:

MsZs= Fs - Us 1MbZb Us- Ub,

mWzW Ub- Uw;in which

Us(=s(Z-b) + ks (Zs Zb )Ub = Cb (Zb )+kb (Zb Zw),uw =C(Zw - Zr) +kw (Zw - Zr).

Taking Laplace transformation of (1-6) resultstransfer matrix:

LY(s) = RU(s),where Y(s) [Zs Zb ]w U(s) Ps[Fs Zr]

m s2 +CcSs + k -C- k 0L=

-css - ks Mbs2+ (Cs + COs + ks + k - k,_

-Cbs - kb mws + (cb + CW)

(1)(2)(3)

(4)(5)(6)

in the

(7)

b

)s+ kb + kw

R=O O

Cws+kw]and " " represents the Laplace transform of thecorresponding variables. Since L is nonsingular, (7) canbe simplified as:

Y(s) =L 1RU(s) = G(s)U(s) . (8)Note that G(s) is a three-by-two system transformationmatrix.(b). Parallel Inerter between the Body and the Bogie

There are several ways of applying inerter to thesuspension design. First of all, considering the parallelinerter arrangement between the body and the bogieillustrated in Fig. 5(b), the dynamic equations of thesystem are as (1-6) except (4) is modified as follows:

us =b(s-Zb)+Cs(Zs -Zb)+ks(Zs- Zb)

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with the Laplace transform u, = (bs + css + ks )(zs Zb) .

(a) (b) (c)

(d) (e)Fig. 5: Five suspension arrangements of the one-wheel train system.

(c). Serial Inerter Between the Body and the BogieNow considering the serial arrangement of Fig. 5(c),

where the spring in parallel with a serial damper-inerterset, the system dynamic equations are as in (1-6) withone more equation:

b(-Zbb) = Cs(4bb Zb)and the suspension force us in (4) is modified as:

(cbs2US =(s + ks)(S- Zb)Cs + bs

(d). Parallel Inerter Between the Bogie and the WheelConsidering the parallel arrangement of inerter

between the bogie and the wheel, as shown in Fig. 5(d),the dynamics equations of the system are similar to (1-6)except the suspension force ub is modified as follows:

Ub = b(4 -w) + Cb(b -w)+k(Zb( Zw)with the Laplace transform Mb = (bs2 + CbS + kb )(Zb -w)(e). Serial Inerter between the Bogie and the WheelNow considering Fig. 5(e) where a serial arrangement

of inerter is placed between the bogie and the wheel, thedynamic equations of the system can be expressed as(1-6) with one more equation:

b(Zb -Z ~w) =cs (Z'ww -Zw )7and the suspension force us in (4) is modified as:

ebbs2Cb= b ks)Ub- w)cb +bs

The above transfer functions are derived by Maple.It is noted that the system dynamics can also bedescribed by the state-space form with state variablesx =[Z z Zb Zb z T . And an extra state, Zbb orZWW illustrated in Fig.5(c), (e) needs be used with theserial arrangements. Actually, for complicated models,such as the full train model, state-space representation ismore convenient than than transfer function models.Another way to obtain the dynamic models ofcomplicated systems is to use multi-body system

TM ~~~TMpackages, such as AutoSimT and SimMechanicsC. Performance IndexThere are several indexes to evaluate performance oftrain suspension systems. Among them, the passengercomfort and system damping ratio are chosen to illustratethe performance benefits by Inerter. The passengercomfort index is defined as the r.m.s. of body verticalacceleration, which was derived in [11] as:

J1-=Tz.ZS 11 -Another performance index is the system damping ratio.By adjusting the suspension settings the minimaldamping ratio of the system can be maximized as:

4min = sup min{Mj.D. Performance Benefits k,c,bIn this section, the performance benefits by employinginerter to the suspension design are investigated. Thefollowing parameters from [6] are used for numericalsimulations: ms=3500 kg, mb=250 kg, mw=350 kg,k5=141 N/mm, cs=8.87 Ns/mm, kb=1260 N/mm, Cb=7.1Ns/mm, kw = 8X106 N/mm, cw=670 Ns/mm. Thetesting models are illustrated in Table 6, where in groupA the suspension elements between the body and thebogie are optimized while in group B the suspensionelements between the bogie and the wheel are optimized.The optimization is carried out numerically in Matlab'TMby adjusting the damping and inerter settings for variousspring stiffness values.Models Descriptions

A1 Fig. 5(a): Conventional suspension in Q(Optimization over cs

A2 Fig. 5(b): Parallel suspension in Q(Optimization over cs and b

A3 Fig. 5(c): Serial suspension in Q(Optimization over cs and b

B1 Fig. 5(a): Conventional suspension in Q2Optimization over Cb

B2 Fig. 5(d): Parallel suspension in Q2Optimization over Cb and b

B3 Fig. 5(e): Serial suspension in Q2Optimization over Cb and bTable 6: The testing models.

(a). Optimization of J1Using group A suspensions, the optimization results

are shown in Fig. 7, where the horizontal axis is the

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static stiffness of k, and the four separated plots illustratethe optimal J1, the percentage performance improvement,the corresponding optimal damping rate c, and inertanceb. It is noted that up to 12% performance improvement isachieved by serial arrangement (A3).

120

100

80

0 60

40

20

X,- ~~A1__A3

0.5 1 1.5 2 2.5ksst x 10o

14

10

-E 8-a)E) 6>

E) 4-' 2/

ol

5

'b 4-.4 _'E3.c 2-In

-1 _Q-

x 10o2

1.5 X

51

0.5

0.5 1 1.5 2 2.5ksst x 10o

Fig. 7: Optimization of J1 using grout

Similarly, the optimization resultsuspensions are shown in Fig. 8,improvement of J1 is achieved by seriCompared the values of J1 in Fig. 7improve the passenger comfort (J11inerter between the bogie and the wh(

501 6

45

40

35

30

25

200

B

B32 3

kbSt X l o'

LA

0.5 1 1.5 2 2.5ksst x 10o

responses of the body vertical displacement zs to thesystem inputs Fs=IN and Zr =1mm are shown in Fig. 9.

N1.5

InLL

1a)In

In

o- 0.5U)

x 10-,2 1.8

E 1.6

1.4

1.2

081

0604

026

time (sec)

A1A2A,

time (sec)

Fig. 9: Step responses of TF )Z , TZ )Z with optimal group A suspensions.

Similarly, with group B suspensions the improvementA2 of 4min is achieved by up to 49 %. For example, at

k,=1800 N/mm, the system minimal damping ratios are---------------- =0.21168 for B1 with c,=487.68Ns/mm,

Smin =0.21173 for B2 with c,=487.01Ns/mm and051 kSt 15 2 265 b=207.44kg, and 4Min =0.31595 for B3 withksst x0xlO c,=399.33Ns/mm and b=61873kg. That is, 4min is notpA suspensions. noticeably improved by the parallel arrangement (B2),

while the serial arrangement (B3) is more useful in thiss using group B case. Compared with the optimization of group A, thewhere up to 5% best way to improve system damping ratio is to

ial arrangement (B3). implement the serial suspension between the bogie and8, the best way to the wheel (B3). The step responses of the body verticalis to adopt serial displacement z, to the system inputs F,= IN and zr =1mm

zel (B3). are shown in Fig. 10.

BB3

=0.20231

)kg. The step

x 1 01.2

-4

E 3a)E 2-

kbSt x lo'

E 11nNT 0.8InLL

0.6a)

04a0a)U) 0.21-

x 108 F-

BB2B3

2 3time (sec)

1.6E

1.41InNT 1.2

0806 1

In 0.4U) 0.6020.2

B1B2B,

1 2 3

time (sec)4 50

L0

-n4zB

2 3kbSt x lo'

ob2.5'3 2a)

.c 1.5-uzs

1

20.5-zo

O _0

Fig. 8: Optimization of J1 using group B susp(

(b). Optimization of 4minTo optimize the minimal damping ratio

170% improvement is achieved by group AFor example, at k,=3150 N/mm, the sysdamping ratios are 4min =0.1263 fcc,=237.35Ns/mm, 4min =0.20231 forc,=193.27Ns/mm and b=6501.3kg, and 4for A3 with c,=193.56Ns/mm and b=19289

MAYi = - Yt + 2filYri - WaYi -2Vft a)j + Isyj - Fti )1475

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B Fig. 10: Step responses ofT T with optimal group B suspensions.

III. Lateral StabilityApart from the passenger comfort, lateral stability is also

kbSt 10 an important issue for train suspension design. Using asix D.O.F. (degree of freedom) full-train model shown inFig. 11, the increase of critical speed (the maximumallowable speed with lateral stability) by applying inerterto the lateral suspension design are investigated. The

)Sm,Up to dynamics of the model with conventional suspensionsuspensions. was derived in [7] as follows:tem minimalr Al with mtyX F (9)

A2 with I~< M,(10)

O0

oI

O _0

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

jki =- W3 Yi + vf12 Y>+(-2f12+a2W)y.,V V ,(12)(

+W +(r+ V -0 Yi + M'Zi

in which i= 1,2 and V represents the forward speed, and

F 2Kpy + 2Cy Yj + 2Ky Y2 + 2Cy Y2 + (

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

resolution). Furthermore, the force and displacementsignals are collected by a NIO pci-6071E card and

TMrecorded in LabVieW for analyses.

Fig. 14: A motor-drive testing platform.

suspension layout with ball-screw inertermodel was constructed and experimentallyverified to match with the theoretical model atlow frequency. Although only a one-wheeltrain suspension was presented in this paper,it is discussed in [1] that for complex models,from the two-wheel to the full-train systems,inerter is also potentially capable ofimproving the system performance.

x 105 k2 b3 c

-10-o

18

= 4

2

C. Experimental ResultsFor the parallel arrangement of Fig. 13 with inertance b,damping rate c and stiffness k, the theoretical transferfunction from the displacement to suspension force canbe expressed as:

G(s) (s) = (bs2 + cs + k)(4, - z2),(12 z2)where zl, Z2 are the displacements of two terminals.Using the collected force and displacement signals, asillustrated in Fig. 15, system identification methodsdescribed in [8,11] are utilized to compare the transferfunctions from theorem and experiments. For example,when inertance b=1 15.31kg, damping rate c=2Ns/mmand spring stiffness k=30.94N/mm, the theoretical andpractical transfer functions are compared in Fig. 16. It isnoted that at low frequency the suspension layout isclose to the theoretical model, while at the higherfrequency range it is drifting away. The difference mightbe resulted from some mechanical nonlinearity, such asbacklash and friction, and will be discussed in details byother articles. Therefore, it is important to check theinerter property in the concerned frequency range of thesystems.

Fig. 15: The collected force and displacement signals in LabViewTM.

V. ConclusionIn this paper inerter has been applied to trainsuspension designs. It was shown that both ofthe system performance and stability areimproved by combining inerter with thetraditional suspension elements. It was notedthat the resulting suspension layouts ispassive, i.e. no energy input is required toachieve those performance benefits. A parallel

,

(D 150

100uz

Frequency Hz10 15

(- 50

0 5 1 0 15Fig. 16: Comparison of the theoretical (dashed) and practical (solid) Inerters.

References[1].M.L. Chang, "The Application of Inerter to Train Suspension

Systems", Master Thesis, National Taiwan University, 2005.[2].S. Evangelou, D.J.N. Limebeer, R.S. Sharp and M.C. Smith,

"Steering Compensation for High-Performance Motorcycles", 43rdIEEE Conference on Decision and Control, Atlantis, ParadiseIsland, Bahamas, December 14-17, 2004

[3].V.K. Garg, and R.V. Dukkipati. Dynamic of Railway VehicleSystem. New York: Academic Press, 1984

[4].Y. He, and J. Mcphee, "Optimization of the Lateral Stability ofRailVehicles", Vehicle System Dynamics, Vol.38, No.5, 361 390,2002.

[5].M. Hsu, "The Realisations of Inerter Concepts and the Applicationto Building Suspension", Master Thesis, National TaiwanUniversity, 2005.

[6].C.G. Koh, J.S.Y. Ong, D.K.H. Chua and J. Feng. "Moving ElementMethod for Train-Track Dynamics", International Journal fornumerical methods in engineering, 56, 1549-1567, 2003.

[7].W.- S. Y. Lee and Yung-Chung Cheng, "Hunting Stability AnalysisofHigh-Speed Railway Vehicle Trucks on Tangent Tracks", Journalof Sound and Vibration, 282 (3-5): 881-898, 2005.

[8].L. LJUNG, System Identification Theory for the User (SecondEdition), Prentice-Hall, 1999.

[9].C. Papageorgiou and M.C. Smith "Positive Real Synthesis UsingMatrix Inequalities for Mechanical Networks: Application toVehicle Suspension ", 43rd IEEE Conference on Decision andControl, Atlantis, Paradise Island, Bahamas, December 14-17,2004.

[10].M.C. Smith. "Synthesis of Mechanical Networks: the Inerter."IEEE Transactions on Automatic Control, 47, 1648-1662, 2002.

[11].M.C. Smith and F.-C. Wang. "Performance Bene_ts in PassiveVehicle Suspensions Employing Inerters", Vehicle SystemDynamics 42 (4): 235-257 OCT 2004.

[12].M.C. Smith and F.-C. Wang. "Performance Bene_ts in PassiveVehicle Suspensions Employing Inerters. ", 42nd IEEE Conferenceon Decision and Control, Hawaii, USA, December 9-12, 2003.

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