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Passive suspensions incorporatinginerters for railway vehiclesJason Zheng Jiang a , Alejandra Z. Matamoros-Sanchez b , Roger M.Goodall b & Malcolm C. Smith aa Department of Engineering , University of Cambridge ,Cambridge , CB2 1PZ , UKb Department of Electronic and Electrical Engineering ,Loughborough University , Leicestershire , LE11 3TU , UKPublished online: 16 Jul 2012.

To cite this article: Jason Zheng Jiang , Alejandra Z. Matamoros-Sanchez , Roger M. Goodall &Malcolm C. Smith (2012) Passive suspensions incorporating inerters for railway vehicles, VehicleSystem Dynamics: International Journal of Vehicle Mechanics and Mobility, 50:sup1, 263-276, DOI:10.1080/00423114.2012.665166

To link to this article: http://dx.doi.org/10.1080/00423114.2012.665166

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Vehicle System DynamicsVol. 50, Supplement, 2012, 263–276

Passive suspensions incorporating inerters for railway vehicles

Jason Zheng Jianga*, Alejandra Z. Matamoros-Sanchezb, Roger M. Goodallb andMalcolm C. Smitha

aDepartment of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK; bDepartment ofElectronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, UK

(Received 1 November 2011; final version received 5 February 2012 )

This paper investigates the possibility of improving the performance of railway vehicle suspensionsby incorporating a newly developed mechanical device known as the inerter. A comparative study ofseveral low-complexity passive suspension layouts is made. Improved performance for the lateral andvertical ride comfort, as well as lateral body movement when curving are demonstrated in comparisonwith the conventional suspension layout. The constraints imposed are to maintain the same level ofother performance metrics. The calculations and optimisations are based on linearised plan-view andside-view high-speed train mathematical models.

Keywords: inerter; train suspension; passive network; passenger comfort; gauging

1. Introduction

New possibilities for passive control of mechanical systems have been investigated recentlymaking use of the inerter [1] mechanical device. The inerter is a mechanical two-terminalelement with the property that the applied force at the terminals is proportional to the relativeacceleration across the terminals. The inerter, together with spring and damper, provides acomplete analogy between mechanical and electrical elements, which allows arbitrary passivemechanical impedances to be synthesised.

Applications of the method to vehicle suspension [2,3], control of motorcycle steeringinstabilities [4,5], vibration absorption [1] and building suspension control [6] have beenidentified. The inerter has been successfully deployed in Formula One racing since 2005 [7].

Some of the possible applications of the inerter device to the rail suspensions have beenexplored in [8–10]. In [8], the suspension in the vertical direction has been discussed toimprove the passenger comfort and wheel load. The lateral suspension has been studied in [9]to increase the critical speed; however, the ride comfort has not been considered as a constraint.Benefits of parallel and series inerter in lateral, longitudinal and vertical suspensions for thecritical speed and passenger comfort in the vertical direction for a full-train model with thepitching mode neglected are studied in [10].

*Corresponding author. Email: z.jiang@eng.cam.ac.uk

ISSN 0042-3114 print/ISSN 1744-5159 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/00423114.2012.665166http://www.tandfonline.com

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264 J.Z. Jiang et al.

The contribution of the present paper consists of three parts. First, based on a seven degree-of-freedom high-speed train plan-view model, the benefit of the inerter device for improvingthe ride comfort in the lateral direction is shown. Second, the potential benefit of reducing thesuspension deflection while maintaining the ride quality is also demonstrated. This possibilityis examined as a contribution to problems arising from the limited loading gauge (a particularissue on Britain’s railways in comparison with most other places around the world). Third, a sixdegree-of-freedom side-view train model with linearised airspring as the secondary suspensionis studied. It is shown that by adding a suspension structure with an inerter in parallel with theairspring, potential improvements in the ride comfort can be obtained.

The focus of this paper is an initial scientific assessment of the potential of inerters, and forthis reason a number of engineering practicalities are not considered at this stage. These are,however, identified at the relevant points of the paper and the concluding section summarisesfuture work that will be necessary in the context of the applications that appear to offer thegreatest potential.

2. High-speed vehicle plan-view model

As shown in Figure 1, the model used in Sections 2–4 consists of a half vehicle body, one bogieframe and two solid axle wheelsets. Longitudinal and lateral connections between wheelsetsand bogie have been included as well as lateral connections between bogie and half-vehiclebody. Vertical and roll modes are not modelled, and longitudinal connections are included onlyfor their contribution to the yaw modes. Hence, the model contains seven degrees-of-freedom:

Figure 1. Plan-view model of high-speed train.

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Vehicle System Dynamics 265

lateral and yaw modes for each wheelset (yw1, θw1, yw2, θw2) and for the bogie frame (yg, θg),and a lateral mode for the vehicle body (yv). The mathematical model is described by thefollowing equations:

mwyw1 = 2Kpy(yg − yw1) + 2Cpy(yg − yw1) − 2f22

Vyw1 + 2f22θw1 + 2Kpylwxθg

+ 2Cpylwx θg + mw

(V 2

R1− gθc1

), (1)

Iwθw1 = −2f11l2wy

Vθw1 − 2f11λlwy

r0yw1 + 2Kpxl2

x (θg − θw1) + 2Cpxl2x (θg − θw1)

+ 2f11l2wy

R1− 2f11λlwy

r0yt1, (2)

mwyw2 = 2Kpy(yg − yw2) + 2Cpy(yg − yw2) − 2f22

Vyw2 + 2f22θw2 − 2Kpylwxθg

− 2Cpylwx θg + mw

(V 2

R2− gθc2

), (3)

Iwθw2 = −2f11l2wy

Vθw2 − 2f11λlwy

r0yw2 + 2Kpxl2

x (θg − θw2) + 2Cpxl2x (θg − θw2)

+ 2f11l2wy

R2− 2f11λlwy

r0yt2, (4)

mgyg = 2Kpy(yw1 − yg) + 2Kpy(yw2 − yg) + 2Cpy(yw1 − yg) + 2Cpy(yw2 − yg)

+ 2Ksy(yv − yg) + 2Csy(yv − yg) + mgV 2

(1

2R1+ 1

2R2

)

− mgg

(θc1

2+ θc2

2

), (5)

Igθg = 2Kpylwx(yw1 − yg) + 2Kpylwx(yg − yw2) + 2Cpylwx(yw1 − yg)

+ 2Cpylwx(yg − yw2) + 2Kpxl2x (θw1 − θg) + 2Kpxl2

x (θw2 − θg)

+ 2Cpxl2x (θw1 − θg) + 2Cpxl2

x (θw2 − θg) − 4Kpyl2wxθg − 4Cpyl2

wx θg, (6)

mvyv = 2Ksy(yg − yv) + 2Csy(yg − yv) + mvV 2

(1

2R1+ 1

2R2

)

− mvg

(θc1

2+ θc2

2

), (7)

with parameters from Table 1. It is a slight modification of the model introduced in [11] withsome parameter values being modified and with suspension in the yaw direction added.

Although it is recognised that flexible modes of the vehicle are important when additionalcomponents are added across the suspension, at this stage the vehicle body flexible modesare excluded so that the fundamental potential of the inerter can be assessed. Normally thedamper will include a series spring, but this is not represented here because this can only beoptimised when body flexibility is included in the modelling.

A state-space form can be readily derived from Equations (1)–(7) as given in (8):

x = Ax + Bw, (8)

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266 J.Z. Jiang et al.

Table 1. Parameters and default settings of the 7-DOF train model described by Equations (1)–(7).

Symbol Parameter Unit Nominal value

V Vehicle speed ms−1 1 to 55mw Wheelset mass kg 1376Iw Wheelset yaw inertia kgm2 766mg Bogie frame mass kg 3477Ig Bogie frame yaw inertia kgm2 3200mv Half vehicle body mass kg 17230r0 Wheel radius m 0.445λ Wheel conicity − 0.3f11, f22 Longitudinal and lateral creepage coefficients N 1e7lwx Semi-longitudinal spacing of wheelsets m 1.225lwy Half gauge of wheelset m 0.75lx Semi-lateral spacing of steering linkages and primary longitudinal

suspensionm 1.2

Kpx Steering linkage stiffness plus primary longitudinal stiffness per axle box Nm−1 3.766 × 107

Cpx Steering linkage damping plus primary longitudinal damping per axle box Nsm−1 1.017 × 104

Kpy Primary lateral stiffness per axle box Nm−1 4.71 × 106

Cpy Primary lateral damping per axle box Nsm−1 1.2 × 104

Ksy Secondary lateral stiffness per axle box Nm−1 2.45 × 105

Csy Secondary lateral damping per axle box Nsm−1 2 × 104

R1, R2 Radius of the curved track at the front and rear wheelsets m 1000θc1, θc2 Cant angle of the curved track at the front and rear wheelsets rad 6 × π/180yt1, yt2 Straight track lateral stochastic displacement at the front and rear wheelsets m −g gravity ms−2 9.8Al Lateral track roughness factor m 1.886 × 10−9

where

x = [yw1, yw1, θw1, θw1, yw2, yw2, θw2, θw2, yg, yg, θg, θg, yv, yv]T,

w =[

1

R1, θc1, yt1,

1

R2, θc2, yt2

]T

.

The vector w is used to define the inputs from the railway track (curvature, cant and track lateralstochastic displacement). When entering a curve, the track cannot change from straight to thenominal value of the radius (R1, R2) and cant angle (θc1, θc2) immediately. It is assumed thatR1, R2 and θc1, θc2 are ramped with 3 s transition time. Some conventional railways have shortertransitions, whereas for high-speed trains a longer transition time is appropriate. Relevantinformation has appeared in [12]. The straight track lateral stochastic inputs (yt1, yt2) usedin this paper are of a broad frequency spectrum with a relatively high level of irregularities.Here, yt1(t) denotes the output of a filter H(s) = (21.69s2 + 105.6s + 14.42)/(s3 + 30.64s2 +24.07s) whose input is a process with a single-sided power spectral density

Sy(fs) = Al

f 2s

,

in which Al is the track roughness factor and fs is a spatial frequency in cycles/metre. Theidea is to provide an approximation to the 1/f 3 power spectrum. The body lateral accelerationis quantified in terms of the root mean square (r.m.s.) acceleration J1y, and evaluated usingthe covariance method, time domain simulation method and frequency calculation method.In practice frequency weighting to allow for human susceptibility is often used, but here theunweighted r.m.s. acceleration has been calculated so as not to obscure any undesirable effectsthat might emerge via the introduction of unconventional suspension devices. The results from

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Vehicle System Dynamics 267

the three methods are consistent. For the frequency calculation, J1 is expressed by

J21y =

∫ ∞

0|Gyt1→yv(jω)H(jω)(1 + e−jωTd )|2Sy dω,

≈ �ωSy

40π∑ω=0.01

|Gyt1→yv(jω)H(jω)(1 + e−jωTd )|2,

where

Sy = 2πAlV , (ms−1)2 (rad/s)−1 ,

Td equals 2lwx/V seconds, which is the time delay (Td) of the track input between the frontand rear wheelsets, and Gyt1→yv represents the transfer function from the track velocity inputyt1 to the lateral body acceleration yv.

The nominal speed is taken to be 55 ms−1. Using the default suspension layout and parametersettings, with velocities varying between 1 and 55 ms−1, it can be calculated that the leastdamping ratio (Ldmp) equals 6.45% (which is achieved at the nominal speed). Using thecovariance method [13], it can also be calculated that, with yt1 and yt2 as input, the maximumlateral body acceleration (Macc) equals 0.2204 ms−2 when the velocity equals 55 ms−1.

3. Potential benefits of inerters for improving the lateral ride comfort

In this section, a few simple networks are introduced as candidates for the suspension con-figurations. While this does not exploit the full possibility of passive suspensions, it providesa number of new possibilities to investigate which are relatively simple to realise in practice.Figure 2(a) shows the default suspension arrangement (S1). Figure 2(b) and (c) shows thedefault layout augmented by an inerter in parallel with the whole structure (S2) or in serieswith the damper (S3).

The inerter device is now combined in the secondary lateral suspension structure with theintention to minimise Macc while keeping the least damping ratio (Ldmp) above 6.45%. Allparameter values are constrained to be within physically reasonable ranges. Table 2 shows theoptimisation results for the layouts of Figure 2. In rows 1–3, the results of optimising only thesecondary lateral suspension are shown, while rows 4–6 represent the results of optimising

Layout S1 Layout S2

Layout S3

Figure 2. Candidate suspension layouts in the secondary lateral position.

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Table 2. Results for minimising Macc by incorporating the inerter in the secondary lateral suspension.

Layout Minimised Macc(m s−2) Impro. (%) Parameter values (Nm−1, Nsm−1, kg)

Conventional 0.2204 – Csy = 2 × 104

(opt. over Csy, same as default)Parallel bsy, Csy (layout S2) 0.2204 0 Csy = 2.01 × 104, bsy = 2

(opt. over sec. lateral)Series bsy, Csy (layout S3) 0.2172 1.45 Csy = 2.1025 × 104, bsy = 1.53 × 104

(opt. over sec. lateral)

Conventional 0.1997 9.4 Kpy = 6.33 × 106, Cpy = 5.24 × 103,

(opt. over pri. and sec. lateral) Csy = 1.61 × 104

Parallel bsy, Csy (layout S2) 0.1921 12.84 Kpy = 6.68 × 106, Cpy = 5.4 × 103,

(opt. over pri. and sec. lateral) Csy = 1.51 × 104, bsy = 256

Series bsy, Csy (layout S3) 0.1936 12.16 Kpy = 6.38 × 106, Cpy = 5.26 × 103,

(opt. over pri. and sec. lateral) Csy = 1.69 × 104, bsy = 9.74 × 103

Figure 3. Plot of (a) lateral body acceleration (b) least damping ratio against velocity for the schemes of Table 2.

both the primary and secondary suspension. It can be seen that when fixing the primary lateralspring stiffness and damper values, 1.45% improvement in ride comfort can be provided bylayout S3 (row 3), while layout S2 does not give any improvement. When optimising over boththe primary and secondary suspension parameters, up to 12.84% improvement can be obtainedby layout S2 (row 5). Figure 3(a) shows the secondary body accelerations across all velocitieswith five of the schemes shown in Table 2. From Figure 3(b), it can be seen that the constrainton Ldmp is satisfied. The optimisation results in rows 4–6 of Table 2 indicate that both theprimary lateral spring stiffness (Kpy) and damper value (Cpy) are increased from default.

4. Potential benefits of inerters for the gauging problem

In this section, the possibility of easing the gauging problem via incorporating the inerter devicein the secondary lateral suspension is discussed. The target is to minimise the maximum lateralmovement of the vehicle body Max(yv) when curving. Optimisations have been done withV = 55 ms−1. The restrictions are for Ldmp and Macc to be at least as good as their nominalvalues and for all parameter values to be within physically reasonable ranges. The optimisation

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Vehicle System Dynamics 269

Table 3. Results for minimising Max(yv) by incorporating the inerter in the secondary lateral suspension.

Layout Minimised Max(yv) (mm) Impro. (%) Parameter values (Nm−1, Nsm−1, kg)

Conventional 77.1 – Ksy = 2.45 × 105, Csy = 2 × 104

(opt. over Csy, same as default)Series bsy, Csy (layout S3) 74.3 3.63 Ksy = 2.61 × 105, Csy = 2.1 × 104,(opt. over sec. lateral) bsy = 1.37 × 104

Conventional 64.3 16.6 Kpy = 6.42 × 106, Cpy = 5.27 × 103,(opt. over pri. and sec. lateral) Ksy = 2.94 × 105, Csy = 1.84 × 104

Series bsy, Csy (layout S3) 61.6 20 Kpy = 6.50 × 106, Cpy = 5.33 × 103,(opt. over pri. and sec. lateral) Ksy = 3.13 × 105, Csy = 1.94 × 104,

bsy = 1.03 × 104

Figure 4. The time domain response of vehicle body lateral movement when running into a curve when V = 55 ms−1

with the schemes of Table 3.

results found no improvement with the parallel inerter-damper scheme (S2) and hence are notincluded.

It can be seen from Table 3 that when optimising over both the primary and secondarysuspension, up to 20% improvement of in the minimisation of Max(yv) can be obtained by theS3 layout (row 4). The time domain response when entering a curve associated with the fourschemes in Table 3 can be found in Figure 4. It has also been checked that when varying thevelocity around 55 ms−1, the percentage improvement of Max(yv) stays at the same level. Thisresult is a consequence of a higher lateral stiffness reducing the lateral deflection on curves,but where the suspension with the inerter is optimised to give the same ride comfort as withthe softer spring.

5. High-speed vehicle side-view model

In the following sections, potential benefits of the inerter in the vertical dynamics are exploredby focusing on bouncing and pitching motions of the railway vehicle travelling on a straight

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Figure 5. Side-view model of high-speed railway vehicle.

track. For this purpose, the side-view model is used with extra suspension layout in parallelwith the original airspring, as shown in Figure 5. This model allows only two-degrees-of-freedom for every single body, i.e. the two bogies and the vehicle body. There are two stagesof suspensions: the primary suspension, which is designed for vehicle stability and minimaltrack interaction; the secondary suspension, which is designed to provide a good ride qualityfor the passengers. The primary suspension links the wheelsets with the bogies frame, andconsists of a very stiff spring with a damper in parallel; the secondary suspension – which israther soft – is considered to be an airspring mounted on each bogie to support the vehicle body.

A simplified but equivalent model of the airspring, the Nishimura model [14], has beenused. The model consists of a mechanical network of linear springs and a damper modellingonly the vertical behaviour of the airspring. The vehicle has four wheelsets and so four contactpoints with the track; track irregularities are included as vertical displacements of the contactpoints and the vertical contact forces are neglected. Although ride quality of a vehicle runningon a straight track is not a major problem for high-speed railway vehicles in contrast with thestability, lateral ride quality and gauging problems presented in the previous sections, or thetilting problem [15], scientifically it is still interesting to investigate the possible benefits thatthe inerter can offer to this dynamic system.

For assessment of vertical ride quality, an incremental model of the system is formulatedaround the equilibrium condition. It is assumed that the amplitude of the pitching mode issmall enough to allow a linear approximation, and for simplicity it is written here in thefrequency domain using the Laplace transform (where ˆ over the variable is used to denotevariables in this domain). In this way, all the suspension systems can be represented in termsof their admittance functions directly.1

The equations modelling the dynamics are given in (9) with parameters from Table 4,and where G1 and G2 are defined in (10) and (11) and stand for the primary and secondarysuspension admittances, respectively. The variables z and θ represent the bouncing and pitchingmodes, and subindices v, b1, and b2 identify each body in this model (vehicle body, front bogieand rear bogie). Track irregularities enter into the system as variables δi, for i = 1, . . . , 4,being the index of every wheelset, where δi = δ1 e−τis, with τi = Li/V the time delay fordifferent wheelsets. The vehicle travels at the nominal speed V , and every wheelset i = 2, 3, 4is separated from the first wheelset by a distance Li, where L2 = 2lbx, L3 = 2lvx, and L4 =2(lvx + lbx)

m′vs2zv = G2s(zb1 + zb2 − 2zv),

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Vehicle System Dynamics 271

Table 4. Parameters and default settings of the 6-DOF railway vehicle model [15] described by Equation (9) andthe conventional suspension models in (10) and (11).

Symbol Parameter Unit Nominal value

m′v Vehicle body mass kg 38,000

mb Bogie mass kg 2500Ivy Vehicle pitch inertia kg m2 2.31 × 106

Iby Bogie pitch inertia kg m2 1500lvx Semi-longitudinal spacing of secondary suspension m 9.5lbx Semi-longitudinal spacing of wheelsets m 1.25Kaz Change of area stiffness Nm−1 0Ksz Airspring stiffness Nm−1 1.016 × 106

Krz Reservoir stiffness Nm−1 0.508 × 106

Crz Airspring damping Nsm−1 64.11 × 103

Kpz Primary suspension stiffness per axle box Nsm−1 4.935 × 106

Cpz Primary suspension damping per axle box Nsm−1 50.74 × 103

V Vehicle speed ms−1 55Av Vertical track roughness factor m 2.5 × 10−7

Ivzs2θv = lvxG2s(zb1 − zb2 − 2lvx θv(s)),

mbs2zb1 = G1s(δ1 + δ2 − 2zb1) − G2s(zb1 − (zv + lvx θv)),

Ibzs2θb1 = lbxG1s(δ1 − δ2 − 2lbx θb1),

mbs2zb2 = G1s(δ3 + δ4 − 2zb2) − G2s(zb2 − (zv − lvx θv)),

Ibzs2θb2 = lbxG1s(δ3 − δ4 − 2lbx θb2). (9)

According to Figure 5, the admittances for the primary and secondary suspension systemsare given by

G1 = 2Kpz

s+ 2Cpz, (10)

G2 = Gairspring + G0, Gairspring = γk/s + γc

s + α,

where

γk = Krz(Ksz + Kaz) + KszKaz

Crz, γc = Ksz + Kaz, α = Ksz + Krz

Crz(11)

and G0 is the admittance of the candidate suspension.For modelling the vertical track input, it is typical to consider both random irregularities and

specific deterministic profiles. Ride quality is assessed using random irregularities as trackinput, while critical deterministic profiles are used for assessment of suspension deflections.An ideal model with δ1 as pure white noise with a power spectral density

Sδ1= 2πAvV (ms−1)2(rad/s)−1

is used. This linear approximation of the random track unevenness is widely used in railwayresearch as an alternative to measure track irregularity data. For the deterministic input, atransitional section leading to a railway gradient of one second duration for the nominal speedof 55 ms−1, with a superimposed acceleration of 0.4 ms−2 (4%g approx.) is assumed.

The assessment of vertical ride quality is given by the index J1z which is quantified by theroot mean square (r.m.s.) acceleration at every position of interest, e.g. at the middle (M),

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272 J.Z. Jiang et al.

leading (L) and trailing (T) positions. An expression for J1z at the middle of the vehicle bodyis

J21z(M) = 1

2

∫ 40π

0|Gδ1→zv

(jω)(1 + e−τ2jω + e−τ3jω + e−τ4jω)|2Sδ1dω,

with Gδ1→zvbeing the transfer function from δ1 to the vehicle body vertical acceleration,

zv. Similarly, J1z(L) and J1z(T) can be obtained by including the measurement of the pitchingmotion, but is omitted here for brevity.

6. Candidate vertical suspension structures and optimisation

In order to investigate the potential benefit of the inerter device for vertical dynamic perfor-mance, four different suspension layouts are considered, as shown in Figure 6. In this case, theyare placed in parallel with the airspring. An end-stiffness in parallel with a damper has beenincluded in the layout S7 corresponding to the bushes attached to both ends of the structure.These parameters are fixed to typical values, ke = 5 × 106 Nm−1 and ce = 8 × 103 N s m−1.The reason for not including ke and ce in the other three layouts is that there is a spring inseries with all those structures which can be considered to incorporate the effect of the bushes.

The ride quality at the L, M and T positions with the default suspension layout is shown inTable 5. The vertical ride quality was minimised at the trailing position, J1z(T), which is theworst position in this Table 5, with a constraint on the stiffness of the spring in series in thelayouts S4, S5 and S6 given by ke.

The optimisation for layout S6 gave c1 = 0 N s m−1, making the ‘best’ settings for theconfiguration S6 equivalent to S5. It was also found from the optimisation procedure that a

Layout S4 Layout S5 Layout S6 Layout S7

Figure 6. Candidate suspension layouts in the secondary vertical position.

Table 5. Vertical ride quality of the vehicle withnormal suspension with default parameters.

Position Ride quality, J1z(%g)

Leading (L) 1.66Middle (M) 1.05Trailing (T) 2.22

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Vehicle System Dynamics 273

better performance would be obtained if the series spring in the candidate suspensions wasinfinitely stiff. Thus, a constraint on k was imposed given by the typical stiffness value of thebushes.

7. Potential benefits of inerters for improving the vertical ride comfort

According to the objective of design to improve the ride comfort in the trailing position, itcan be seen from Table 6 that the best improvement is provided by the layout S5. This layoutalso provides some improvement in the middle position and a small degradation in the front,the latter being normally explained by the geometric filtering phenomenon [16]. The optimalparameter settings are also presented in Table 6. It shows that an inerter in series with a damperwould have a much higher inertance value than one connected in parallel.

Figures 7 and 8 include the plots of the power spectral density (PSD) of the bouncing motionacceleration of the vehicle body for the proposed layouts in contrast with those for the normalsuspension. It can be seen that the main mode of vibration (the vehicle body bouncing mode)is attenuated by all the candidate suspensions, while the deterioration in the other modes isrelatively insignificant.

In addition, the ride comfort changes with the candidate suspensions are considered wherethe parameter values from Table 6, against variations in the damping parameter of the airspringCry. Results are shown in Figure 9. The plots show how the suspension layouts guarantee abetter performance in the trailing position still under variations and/or uncertainties of thisparameter, in comparison with airsprings only.

Table 6. Optimisation results for minimising J1T (trailing position) with the candidate suspensionstructures.

Improvement (%)

Layout min J1z(T)(%g) L M T Parameter values (kN s m−1, kg, kN m−1)

S4 2.03 −0.62 6.71 8.57 c = 11.049, b = 92, k = 5 × 106

S5 1.97 −0.37 4.62 11.46 c = 17.855, b = 4138, k = 5 × 106

S7 2.01 −1.16 7.86 9.69 c = 13.932, b = 165, k = 1.50 × 106

Figure 7. Power spectral density of the bouncing motion acceleration in the trailing position of the vehicle body.

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Figure 8. PSD of the bouncing motion acceleration in the leading position (left), and the middle position (right) ofthe vehicle body.

Figure 9. Ride quality variation in the leading (left), middle (centre) and trailing (right) position against the airspringdamping parameter, Crz(kN/ms−1) for the conventional suspension and the layouts S4, S5 and S7.

Figure 10. Secondary suspension deflection for the airspring only (default), and the layouts S4, S5 and S7.

Finally, from the assessment of the suspension deflection with the vehicle going onto a slopeinducing a 4%g vertical acceleration, it is found that the proposed suspensions are stiff enoughto keep the suspension deflection in a good range (as shown in Figure 10). Furthermore, theoverall maximum suspension deflection, i.e. considering both stochastic and deterministicinputs at the same time, is reduced 4–8% for the front suspension and 10–14% for the rearsuspension among the possibilities presented here. Although the normal trade-off betweenride comfort and suspension deflection still arises for the vehicle with the layouts S4–S7 (i.e.

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for a better comfort, higher suspension deflections), it is improved in comparison with thetrade-off for the vehicle with a conventional secondary suspension. This can be of interest forother suspension design objectives.

8. Concluding remarks

This paper has investigated the potential performance benefit of incorporating the inerterdevice in the secondary lateral and vertical suspension systems. Based on a 7-DOF plan-viewmodel, it is found that by using the series inerter–damper structure with proper parametervalues, the secondary body acceleration with stochastic track input and the lateral body move-ment when entering a curved track can be decreased. It is also worth pointing out that, withdifferent nominal parameter values (e.g. higher value of primary lateral spring stiffness) ofthe model studied here, using the inerter device in both the primary and secondary lateral sus-pension structure can provide larger performance benefits according to our calculations. Forthe side-view model, improvements in ride comfort can be obtained using suspension struc-tures incorporating the inerter device. This paper leaves open whether more substantial gainsare possible with inerters deployed in other locations or with different mechanical networks.As mentioned previously, various practicalities, in particular the need to avoid problems dueto flexibility in the vehicle body, need to be incorporated. Further study could make use ofmeasured track data and weighted r.m.s. accelerations.

Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council grant number EP/G066477/1.The research work also received support from the Universidad de Los Andes, Venezuela.

Note

1. Admittance is defined to be the ratio of force to relative velocity between the suspension terminals. It isthe reciprocal of impedance.. Here, the electrical terminology based on the force–current analogy betweenmechanical and electrical networks is used (see [1]).

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