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Analytical formulae for the design of a railway vehiclesuspension system

G R M Mastinu*, M Gobbiand G D Pace

Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

Abstract: A simple two-degree-of-freedom (2DOF) model is used to derive a number of analytical

formulae describing the dynamic response of railway vehicles to random excitations generated by

vertical track irregularities. The dynamic response is given in terms of standard deviations of a

number of relevant performance indices such as bodybogie suspension stroke and body acceleration.

The derived analytical formulae can be used either during preliminary design or for other special

purposes, especially when approximated results are acceptable. An estimation of the degree of

approximation oVered by analytical formulae is attempted and the results seem satisfactory. Byinspection of the analytical formulae a parameter sensitivity analysis may be readily performed.

In the second part of the paper, an optimization method for the improvement of the dynamic

behaviour of railway vehicles is introduced. The method, based on multiobjective programming

(MOP), is a general one and can be exploited for many engineering purposes. In the paper the method

has been applied with the aim of achieving the desired trade-oVamong conicting performances such

as standard deviation of the body acceleration versus standard deviation of the secondary suspension

stroke. As a result, new analytical formulae dening the settings of some relevant vehicle suspension

parameters have been derived.

Keywords: random excitation, symbolic calculus, railway vehicles, bogie, suspension stiVness,

suspension damping, multiobjective optimization

NOTATION

Ab track irregularity (1S-PSD)

0:9281010 (m1)Av track irregularity (2S-PSD)

0:035104 (m)B track irregularity factor (1S-PSD) as a

function of vehicle speed 2p2v3=2 Abpk1 primary suspension stiVness (referring to a

single axle box) (N/m)k2 secondary suspension stiVness (referring to a

single axle box) (N/m)

m1 one-quarter of the bogie mass (kg)

m2 one-eighth of the body mass (kg)

r1 primary suspension damping (referring to asingle axle box) (N s/m)

r2 secondary suspension damping (referring to a

single axle box) (N s/m)

s j!sc reference circular frequency (2S-PSD)

vc (rad/s)

Sx0 ! power spectral density of the vertical trackirregularity (x0) (m

2/rad s)

v vehicle speed (m/s)

V vertical force on the axle box (N)

WqS white noise power spectral densityq1 forthe 1S-PSD, q2 for the 2S-PSD)

x0 absolute vertical displacement of the axle box

(equal, by hypothesis, to the vertical track

irregularity) (m)

x1 absolute vertical displacement of massm1 (m)

x2 absolute vertical displacement of mass

m2 (m)

stiVness ratiok2=k1 mass ratio m2/m1V standard deviation of the vertical force on the

axle box (N)

x1x0 standard deviation of the stroke of theprimary suspension (m)

x2

x1 standard deviation of the stroke of the

secondary suspension (m)xx2 standard deviation of vertical body

acceleration (m/s2)

! circular frequency (rad/s)

The MS was received on 14 February 2000 and was accepted afterrevision for publication on 29 September 2000.* Corresponding author: Department of Mechanical Engineering,Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy.

683

C02000 IMechE 2001 Proc Instn Mech Engrs Vol 215 Part C


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C break wave number (2S-PSD)

0:99 (rad/m)

Subscripts/acronyms

r, opt reference, optimal

1S-PSD one-slope power spectral density[see equation (7)]

2S-PSD two-slope power spectral density

[see equation (8)]

1 INTRODUCTION

Designers often require simple reasoning for under-

standing relevant phenomena and making basic engi-

neering choices. This request is generally not satised by

complex vehicle system models which need, in order toprovide only a speciWc result, both a large amount of

processing eVort and many parameters, the latter often

being unknown at the very beginning of a design project.

During data acquisition in eld tests, technicians some-

times need simple formulae to set up their instruments or

check measurements. For academic purposes it may be

useful to introduce complex problems in a simplied

form. For all of the reasons listed above, in the present

paper the dynamic response of rail vehicles to random

excitation is dealt with by deriving very simple analytical

formulae. In addition to the study of the dynamic

response, an original method for the optimization ofvehicle parameters is presented which is based on multi-

objective programming (MOP), a very eVective method

for selecting parameters when a number of conicting

requirements have to be satised. The presented opti-

mization method is applied by using the simple model

introduced in the rst part of the paper, and gives basic

hints on how to select railway vehicle suspension para-

meters to obtain the best trade-oV between standard

deviation of vertical acceleration at the body and stan-

dard deviation of secondary suspension stroke. The

introduced optimization method, being a general one,

can be applied by using complex system models (insteadof the simple one used here) in order to obtain accurate

estimates of the parameters to be selected.

In the literature, the authors have not found papers

dealing with the problem of deriving simple analytical

formulae for the estimation of the dynamic response of

railway vehicles to random excitations generated by

vertical track irregularity. Two papers exist that deal

with the same problem for road vehicles [1, 2]. With

reference to vehiclebogie vibrations and to vehicle

track interaction, a number of authors have dealt with

the problem of deriving basic concepts useful for railvehicle design [36]. They usually have resorted to

numerical simulations even when dealing with simple

models.

Optimization of suspension parameters can be per-

formed by means of multiobjective programming, a

branch of operations research. In reference [7] the

adoption of MOP has been proposed to solve many

vehicle system engineering problems. Basically, optimi-

zation procedures based on MOP allow the best trade-oV

among user-dened performance indices to be found.

Given the model, the designer is often charged with thehard task of nding `one optimal solution by changing a

number of parameters. When many performance indices

have to be taken into account at the same time, and many

parameters may be changed, often the optimization

problem cannot be handled easily; i.e. a solution cannot

be found in a straightforward way. Moreover, in this

case, the concept of `optimal solution is not obvious and

requires a special denition (see, for example, reference

[8]). The concept of optimal solution to be considered

may be synthesized by stating that, if more than one

criterion (performance index) has to be satised by

changing one or more parameters, the possible optimal

solutions constitute a set. This implies that the designer

has to choose a preferred solution among those solutions

(and only those) belonging to the set. As the solutions of

the set are directly related to performance indices, the

task of the designer is to reason about performance

indices (or criteria) instead of reasoning about para-

meters. The methods (and related computer programs)

that allow such a way of operation (and of thinking) are

presented or reviewed in references [7] to [12]. Successful

applications of the method in the eld of ground vehicle

design are reported in references [7] and [13] to [16].In the rst part of the present paper, analytical for-

mulae are derived to describe the dynamic behaviour of

a railway vehicle. These formulae are then used to

derive, in a theoretically rigorous way, the best trade-oV

between the standard deviations of body acceleration

and secondary suspension stroke.

2 SYSTEM MODEL

2.1 Equations of motion and responses to stochastic

excitation

The adopted system model is shown in Fig. 1. The sys-

tem has two degrees of freedom. The massm1 represents

one-quarter of the mass of the bogie and the mass m2represents one-eighth of the mass of the body. The

excitation comes from the displacement, x0, which

represents the motion of the axle box, the track being

regarded as an uneven and innitely stiV structure. This

hypothesis had to be introduced to preserve both the

analytical formulation and the analytical solution to the

problem. In Section 3 the accuracy of the response of the

model will be discussed. The equations of motion maybe written in matrix form as

M xxR _xxKxf 1

684 G R M MASTINU, M GOBBI AND G D PACE

Proc Instn Mech Engrs Vol 215 Part C C02000 IMechE 2001


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where M, R and K are the mass, damping and stiVness

matrices respectively:

M m1 00 m2

" # 2

Rr1r2 r2r2 r2

" # 3

Kk1k2 k2

k2 k2

" # 4

f is the vector of external forces related to excitation

from the uneven track:

fr1 _xx0k1x0

0

( ) 5

and x is the vector of the independent variables:

xx1

x2

( ) 6

The displacementx0 (track irregularity) is assumed to be

a random variable dened by a stationary and ergodic

stochastic process. This assumption is consistent withthe results of studies, performed by many authors and

organizations (see, for example, references [17] and [18]),

on the stochastic properties of track (vertical) irregu-

larity. A number of analytical formulae have been

adopted (see references [3] and [4]) to interpolate the

measured data referring to the power spectral density

(PSD) of the stochastic process dening x0. In the pre-

sent paper, two of these analytical formulae have been

considered:

Sx0! Ab

2pv

3

!4 7

Sx0! Avs

2cv

!2!2 s2c 8

In a loglog scaled plot (abscissa !), the spectrum in

equation (7) (reported in reference [3]) takes the shape of

a line sloped at a rate of4. In the following, it will beindicated as a one-slope power spectral density (1S-

PSD). The PSD in equation (8) has been reported in

reference [4] and is widely used in railway vehicle

dynamics simulations. In the loglog scaled plot shown

in Fig. 2, equation (8) takes approximately the shape ofatwo-slopecurve, thus reference to it will be made by the

acronym 2S-PSD.

Fig. 1 System model

Fig. 2 Power spectral density (PSD) of the irregularity of the track in the vertical plane two-slope PSD

[2S-PSD, equation (8)] at 177km/h (adapted from reference [4])

ANALYTICAL FORMULAE FOR THE DESIGN OF A RAILWAY VEHICLE SUSPENSION SYSTEM 685



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System (1) can be rewritten in a more convenient

form:

_xxAxBuyCxDu

9

where x is the vector of state variables:

x

_xx1

_xx2

x1

x2

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

10

A and B are the state matrices:

A

r1r2m1

r2

m1 k1k2

m1

k2

m1

r2

m2 r2

m2

k2

m2 k2

m2

1 0 0 0

0 1 0 0

266666666664

377777777775

B

1=m1

0

0

0

26666664

37777775

11

u is the input variable, related to track irregularity, x0:

u fr1_xx0k1x0g 12

y is the vector of output variables:

yV

xx2

x2x1

8>>>>>:

9>>>=>>>;

13

the output variables representing, respectively, the force

on the axle box,V, the vertical acceleration of the body,

xx2, and the (vertical) relative bodybogie displacement

(stroke of the secondary suspension), x2x1, andmatrices C and D are as follows:

C

r1 0 k1 0r2

m2 r2

m2

k2

m2k2

m2

0 0 1 1

266664

377775

14

D1

0

0

26664

37775 15

The frequency response of the linear dynamic system (9)

is

Gy;us CsIA1BD 16

where I is the identity matrix. The input is representedby vector u, and the output is represented by vector y.

Given the PSD Su of the excitation input, the PSD Sjof the jth element of vector y can be computed as (see,

for example, reference [19])

Sjs GjsGjsSus 17

Gs G1sG2s

G3s

8>>>>>:

9>>>=>>>;

GV;usGxx2 ;us

Gx2x1;us

26664

9>>>=>>>;

18

Here Gjs, being a frequency response function of amechanical system, is obviously a ratio of two poly-

nomials (of variables), i.e.Gjs ns=ds, andSu andSx0 are related by the following expression:

Sus Hu;x0sHu;x0 sSx0s 19

where Hu;x0s k1sr1.In order to write Sjs [equation (17)] in a form

matching that of following equation (24) (the reason for

this need will be explained in the next section),Sx0 s hasto be written as

Sx0 s WqSHqSsHqSs; q1; 2 20

where for q1 reference is made to the 1S-PSD, and itfollows from equation (7), where sj!j

1p ,that H1Ss 1=s2 and W1SAb2pv3; for q2reference is made to the 2S-PSD [equation (8)] so that

H2Ss 1=ssscand W2SAvs2c v.The PSD Sj of the jth element of vector y can be

nally written as

Sjs WqSHqSsHu;x0sGjsGjsHu;x0sHqSs 21




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This form is convenient for computations that will be

presented in the next section [equations (23) and (24)].

The (vertical) relative bogieaxle box displacement

(stroke of the primary suspension), x1x0, and theforce on the axle box are related by the following

expression:

Hx1x0;Vs 1

k1sr1 1

Hu;x0 s

Therefore, the PSD ofx1x0 reads

Sx1x0 s WqSHqSsGV;usGV;usHqSs 22

2.2 Derivation of standard deviations in analytical form

By denition (see references [19] and [20]), the variance

of a random variable described by a stationary and

ergodic stochastic process is

2j 121

1Sjsds 23

In reference [21] it is shown that an analytical solution

exists for 2j ifSjcan be written as

SjNk1sNk1sDksDks 24

where Dk is a polynomial of degree k, and Nk1 is apolynomial of maximum degree k1k51. This isactually the case, and it can be veried that Sj may be

written as in equation (24) both by inspection of equa-

tions (21) and (22) and by analysing the expressions of

Gj and GV;u.

For example, considering the vertical acceleration of

the vehicle body, xx2, the following expression for

Nk

1

s

=Dk

s

can be obtained:

For the 1S-PSD, q1 [see equations (20) and (7)],k4

N3sD4s

W1Sp

H1SsHu;x0 sGxx2;us

where setting r10 and

W1SAb2pv3; H1Ss 1

s2; Hu;x0 s k1

Gxx2;us k2r2ss2

k1k2k1r2sk2m1s2 k1m2s2 k2m2s2 m1r2s3 m2r2s3 m1m2s4

gives

N3s Ab1=22pv3=2k1k2k1r2s

D4s k1k2k1r2sk2m1s2 k1m2s2

k2m2s

2

m1r2s

3

m2r2s

3

m1m2s

4

For the 2S-PSD, q2 [see equations (20) and (8)],k5

N4sD5s

W2Sp

H2SsHu;x0 sGxx2;us

where setting r10 and

W2SAvs2c v; H2Ss 1

ss

sc

gives

N4s Avs2cv1=2k1k2sk1r2s2

D5s k1k2k1r2sk2m1s2 k1m2s2 k2m2s2

m1r2s3 m2r2s3 m1m2s4ssc

The analytical formulae presented in the following

subsections have been derived by means of the analytical

solutions of the integral (23) reported in reference [21].

2.3 Complete formulae using the 1S-PSD [equation (7)]

The 1S-PSD [equation (7)] has been considered because

it allows a full and compact analytical solution of the

problem. For the standard deviations of interest, ana-

lytical solution of equation (23) gives

V

2p2v3=2 Abp f1

m1; m2; r1; k1; r2; k2

25

xx2 2p2v3=2

Ab

p f2m1; m2; r1; k1; r2; k2 26

x2x1 2p2v3=2

Abp

f3m1; m2; r1; k1; r2; k2 27

x1x0 2p2v3=2

Abp

f4m1; m2; r1; k1; r2; k2 28

where jdepend on the 3/2 power of vehicle speed v, on

the square root of track irregularity coeYcient Ab and

on analytical functions of the model parameters fj. The

analytical expressions of functions fjare rather complex.

For completeness, functions fjare reported in Appendix1. As the fj do not depend on v, according to this

formulation, the optimal settings of the suspension

parameters do not depend on vehicle speed.




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2.4 Formulae for vanishing primary damping using the

1S-PSD [equation (7)]

Primary dampers are commonly not tted into the

bogies of urban and suburban railway vehicles. This is

due to the fact that, with the primary stiVness, k1, being

relatively high (owing to the high ratio of payload to

tare mass), the pitching of the bogie is already limited bythe primary stiVness itself and dampers are not neces-

sarily needed to limit the dynamic pitching oscillations.

However, primary dampers can be adopted for reducing

track wear.

Setting the primary damping to zero, i.e. r10, thestandard deviations (25) to (28) assume relatively simple

expressions:

Force on the axle box:

V 2p2v3=2

Abp

k21m

21m

22k1m1m22r22m1m22k2m1m2k2m1m22k2m1m22 k1m22

r2k1m22

vuuuut29

Body acceleration:

xx2 2p2v3=2

Abp

r2

2k1m1m2 k22m1m22k1k2m22

r2k1m22s 30

Secondary stroke:

x2x1 2p2v3=2

Abp k2m1m22 k1m22

r2k1k2

s 31

The primary stroke is obviously x1x0 V=k1.

2.5 Simplied formulae using the 1S-PSD [equation (7)]

Equations (29) to (31) can be simplied by neglecting

those terms that vanish when parameter values refer toactual railway vehicles (see Appendix 2):

V B

k1m21

r212 22m1r23

s 32

xx2 B

r2

m2k2

r2

s 33

x2x1 B

m22

r2k2

s 34

A comparison has been made between the icomputed

by equations (29) to (31) and the i computed by

equations (32) to (34) (vehicle parameters in Table 1).

The absolute value of the error is 0.49 per cent for the

force on the axle box [equation (32)], 1.7 per cent for the

body vertical acceleration [equation (33)] and 0.37 per

cent for the secondary stroke [equation (34)].

2.6 Complete formulae using the 2S-PSD [equation (8)]

The analytical expressions of the standard deviations of

the force on the axle box V, the vertical acceleration of

the body, xx2, and the (vertical) relative bodybogie

displacement have been derived by analytical solution of

equation (23). They are not reported here because of

their extreme complexity. In this case it is more con-

venient to solve equation (23) numerically. It has to be

noted that, owing to the employed spectrum, contrary to

what happens in equations (25) to (28), the speed v is

mixed among the power spectral density and mechanical

parameters (Av, sc, mi, ri, ki). According to this for-mulation, the optimal suspension parameters set

depends on vehicle speed v:

Vf1DSv; Av; sc; m1; m2; r1; k1; r2; k2 35

xx2 f2DSv; Av; sc; m1; m2; r1; k1; r2; k2 36x2x1 f3DSv; Av; sc; m1; m2; r1; k1; r2; k2 37x1x0 f4DSv; Av; sc; m1; m2; r1; k1; r2; k2 38

2.7 Formulae for vanishing primary damping using the

2S-PSD [equation (8)]

If the primary damping vanishes (i.e. r1 0), the stan-dard deviations in equations (35) to (38) assume rela-

tively simple expressions:


VscpvAv

p

NV2m22r2k1k2k1r2sck2m1s2ck1m2s2c

k2m2s2cm1r2s3cm2r2s3cm1m2s4c

vuuut

Table 1 Data of the reference

railway vehicle taken

into consideration

(data from reference

[4])

m1r 773kgm2r 5217kg

k1r 28240000N/mk2r 162000N/mr1r 21890Ns/mr2r 14600Ns/m




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NV k21m21m

22k2r2scm2s2ck1m1m22k2m1m2m1r22m2r22k2m1k2m2m1r2scm2r2scm1m2s2c

k22m1m22k2m212k2m1m2k1m22k2m22m21r2sc2m1m2r2scm22r2scm21m2s2cm1m22s2c39

Body acceleration:

xx2

sc pvAvp

k1r

22k2m1k2m2m1r2scr2m2scm1m2s2c

k22k2m212k2m1m2k1m22k2m22m21r2sc2m1m2r2scm22r2scm21m2s2cm22m1s2c2m22r2k1k2k1r2sck2m1s2ck1m2s2ck2m2s2cm1r2s3cm2r2s3cm1m2s4c

vuuut 40Secondary stroke:

x2x1 scpvAv

p

k2m212k2m1m2k1m22k2m22m21r2sc2m1m2r2scm22r2scm21m2s2cm1m22s2c2r2k1k2k1r2sck2m1s2ck1m2s2ck2m2s2cm1r2s3cm2r2s3cm1m2s4c

s 41

The primary stroke is obviously x1x0 V=k1.2.8 Simplied formulae using the 2S-PSD [equation (8)]

Equations (39) to (41) can be simplied by neglecting those terms that vanish when parameter values refer to actual

railway vehicles (see Appendix 2):


VscpvAv

p

k21m21k1r2scm1s2c1=k1m12k1m1r22k1r2scm1s2c

2r2

k21

k1r2sc

m1k1s

2c

r2m1s

3c

m21s

4c

s 42

Body acceleration:

xx2 scpvAv

p

k1r2=m1k1r2scm1s2c 2k21=r2k1r2scm1s2c

2k21k1r2scm1k1s2c r2m1s3c m21s4c

s 43

Secondary stroke:

x2x1 scpvAv

p

m212k1r2scm1s2c

2r2k2

1k1r2sc

m1k1s

2

c

r2m1s3

c

m2

1

s4

c

s 44

The primary stroke is obviously x1x0 V=k1.A comparison has been made between the icomputed by equations (39) to (41) and the icomputed by equations

(42) to (44) (vehicle parameters in Table 1). In the vehicle speed range 20100m /s, the error varies from1:5 to 0.3 percent for the force on the axle box [equation (42)], from 2:2 to 1.2 per cent for the body vertical acceleration [equation(43)] and from0:2 to1:4 per cent for the secondary stroke [equation (44)].

3 VALIDATION

In order to validate the simple model described in Section 2, a comparison with the data presented in reference [ 4] is

performed. In reference [4] a model for the study of the vertical dynamics of railway vehicles has been proposed. The

model is rather complex as it accounts for the heave, pitch and roll of body and bogies. Vehicle body bending and

torsional modes of vibration have also been included. In reference [4] it is argued that the output model responses were

validated experimentally with satisfactory results. The data of the vehicle studied in reference [4] are reported in Table 1.




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In reference [4] the adopted PSD of the stochastic

process dening track irregularity in the vertical plane

[2S-PSD, see equation (8)] is

Sx0 ! Avs

2c v

!2!2 s2c Av

2cv

3

!2!2 v22c

where Av0:035cm2

rad/m is the track quality coe

Y-cient and c0:99 rad/m is the break wave number;

Sx0! is plotted in Fig. 2.In reference [4] an in-depth sensitivity analysis was

performed referring to the standard deviations of body

acceleration and of secondary stroke. Unfortunately, nodata are available on primary stroke and force on the

axle box.The following validation has been performed by

considering equations (36) and (37) which refer to the

model shown in Fig. 1 and described in Section 2. The

adopted PSD of the track irregularity is given by

equation (45). A steady speed v

equal to 177km/h hasbeen considered.

3.1 Primary stiVness

In reference [4] the primary suspension stiVness k1 was

rst increased by a factor of 4, and then decreased by thesame factor relative to the reference value (Table 1). In

Table 2 the values of standard deviations xx2 and x2x1computed by means of equations (36) and (37) are

compared with the corresponding values reported in

reference [4]. Body acceleration is substantially unaf-

fected by primary stiVness variation. The error given by

the model with respect to the corresponding gure in

reference [4] is always less than 10 per cent. For the

secondary stroke the error is always less than 2 per cent.

3.2 Natural frequency

The natural frequencies of the body rigid modes (heave,

pitch and roll) were varied in reference [4] without

changing their damping ratios. This was achieved by

suitable variations in the secondary suspension stiVness,

k2, and damping, r2. The eVect of halving and doubling

the reference natural frequencies was investigated. The

results are reported in Table 3. The eVect is relevant

both on body acceleration and on secondary suspension

stroke. Again, the simple model is able to give the

responses of the reference vehicle with a limited error.

3.3 Damping ratio

The eVect of the secondary damping on body accelera-

tion and secondary stroke was studied by varying the

coeYcient r2 so that the heave damping ratio

hr2=2

k2m2p of the body was increased from the

reference value (0.25) to 0.375, 0.50 and 0.707 (Table 4)

( is dened as if k1 were innity). Referring to the

Table 2 Comparison between computed results and data referring to an actual vehicle

(adapted from reference [4]). Variation in the primary stiVness k1 (k1r is

reported in Table 1)

xx2 (m/s2) xx2 (m/s

2) x2x1 (mm) x2x1 (mm)k1 from reference [4] equ ation (36) from reference [4] equation (37)

k1r=4 0.56 0.62 9.6 9.8k1r 0.72 0.68 9.5 9.64k1r 0.67 0.69 9.5 9.5

Table 3 EVect of the vehicle body natural frequency. Comparison between computed results and data referring to an actual vehicle

(adapted from reference [4])

fh k2 r2 xx2 (m/s2) xx2 (m/s

2) x2x1 (mm) x2x1 (mm)Natural frequency (Hz) (N/m) (N s/m) from reference [4] equation (36) from reference [4] e qua tion (37 )

Halved frequen cy 0.445 162 000/4 14 600/2 0.39 0.41 15.1 13.8Base frequency 0.885 162 000 14 600 0.72 0.68 9.5 9.6Doub le fre que nc y 1.76 9 1 62 0 00 4 14 600 2 1.17 1.27 5.3 6.6

Table 4 Comparison between computed results and data referring to an actual vehicle (adapted from reference

[4]). Variation in the secondary suspension damping r2 (r2r is reported in Table 1)

xx2 (m/s2 ) xx2 (m/s

2 ) x2x1 (mm) x2x1 (mm)Damping ratio h r2 (N s/m) from reference [4] equation (33) from reference [4] e qu atio n (34 )

r2 0.250 14 600 0.72 0.68 9.5 9.6r21:5 0.375 14 6001.5 0.85 0.81 7.5 7.7r22 0.50 14 6002 1.02 0.96 6.4 6.6r22

2

p 0.707 14 6002

2

p 1.20 1.19 5.4 5.5




9/16

vertical body acceleration, the error given by the model

is always less than 6 per cent. For the secondary stroke

the error is always less than 3 per cent.

4 PARAMETER SENSITIVITY ANALYSIS

The dynamic response of the railway vehicle systemmodel in Fig. 1 is analysed on the basis of equations (29)

to (31) and (39) to (41). The same analysis (not reported

here for the sake of space) has been performed by means

of equations (32) to (34) and (42) to (44). The dis-

crepancies were negligible.

A typical railway passenger vehicle for intercity ser-

vice is taken into consideration (Table 1). The results of

the parameter sensitivity analysis are shown in Figs 3 to

5. The parameters are varied within wide ranges. The

data are presented in non-dimensional form, i.e. the

standard deviation of interest j is divided by the cor-

responding deviation, jr, computed by considering the

parameters at their reference values (see Table 1):

Vr Vm1r; m2r; r1r; r2r; k1r; k2r 46xx2r xx2 m1r; m2r; r1r; r2r; k1r; k2r 47

x2x1r x2x1 m1r; m2r; r1r; r2r; k1r; k2r 48

The non-dimensional standard deviations derived from

equations (29) to (31) do not depend on vehicle speed.

The opposite occurs for the non-dimensional standard

deviations derived from equations (39) to (41) (referring

to 2S-PSD [equation (8)]). For this reason, the last non-

dimensional standard deviations are analysed at two

diVerent vehicle speeds: low speed (10 m/s) and high

speed (100m/s).

4.1 Standard deviation of the force on the axle box

Figure 3 shows that:

1. V depends almost linearly on the primary suspen-

sion stiVness k1.

2. Vdoes not depend signicantly on the secondary

suspension stiVness k2.

3. V

depends almost linearly on m1

(non-linearly at

the start).

4. V does not depend signicantly on m2 if the

excitation is given by equation (8) (2S-PSD).

5. The secondary suspension damping r2 has an

important inuence on the standard deviation V.

Some of the above considerations can be derived by a

simple inspection of equations (32) and (42).

Fig. 3 V=Vr , non-dimensional standard deviation of the force on the axle box as a function of the model

parameters. Data of the reference vehicle in Table 1. Each diagram has been obtained by varying a

single parameter, the other parameters being constant and equal to those of the reference vehicle




10/16

Fig. 4 xx2 =xx2 r, non-dimensional standard deviation of the body acceleration as a function of the model

parameters. Data of the reference vehicle in Table 1. Each diagram has been obtained by varying a


Fig. 5 x2x1 =x2x1r, non-dimensional standard deviation of the secondary stroke as a function of the modelparameters. Data of the reference vehicle in Table 1. Each diagram has been obtained by varying a





11/16

4.2 Standard deviation of body acceleration

Inspection of Fig. 4 shows that:

1. k1 does not signicantly inuence xx2 .

2. m1 does not signicantly inuence xx2 if the

excitation is given by equation (7) (1S-PSD).3. xx2 may decrease strongly as m2 increases.

4. The parameters of secondary suspension (k2, r2)have a remarkable inuence on xx2 . For a 2S-PSD

excitation, the eVect of an increase in the secondary

suspension damping r2 is positive at low speed but

negative at high speed. For a 1S-PSD excitation, the

eVect of an increase in the secondary suspension

damping r2 is positive at any speed.



4.3 Standard deviation of secondary stroke

Inspection of Fig. 5 shows that:

1. x2x1 is not inuenced by primary suspensionstiVness k1.

2. The stiVness of the secondary suspension, k2, has a

remarkable inuence on x2x1 . For a 2S-PSDexcitation, the eVect of a variation in k2 is less

relevant, increasing the vehicle speed.

3. x2x1 is not inuenced signicantly by m1.4. x2x1 depends almost linearly on m2 considering a

1S-PSD excitation. The relationship is non-linear

considering a 2S-PSD excitation.

5. x2x1 is inuenced remarkably by secondary sus-pension damping r2.



5 MULTIOBJECTIVE OPTIMIZATION

5.1 Problem formulation

Consider the optimization of the performances (i.e.

responses to a given input) of a system model by varying

the model parameters. For computational purposes, the

performances (or performance indices) should be at least

continuous functions of the model parameters. A gen-

eric multiobjective problem takes the form

min gz m in

g1z1; z2; . . . ; zng2z1; z2; . . . ; zn

.

.

.

gk

z1; z2; . . . ; zn

z jz1; z2; . . . ; znj Rn

49

where z1, z2; . . . ; zn are the n model parameters and g1,

g2; . . . ;gkare performance indices. The aim is to nd the

so-called eYcient or optimal solutions, i.e. those solu-

tions that minimize the vector gz as indicated above.EYcient solutions z* are those, and only those, that

should be taken into consideration for optimization

purposes.

A solution z* belonging to the feasible domain Z isPareto optimal if, and only if, for each j2 f1; . . . ; kg

gi4gj8g2Cj 50

where Cj fz2Z: gi4gi; i1; . . . ; k; i6 jg.EYcient solutions (often called Pareto optimal solu-

tions or simply Pareto solutions) are in general not

unique and constitute a set. Methods to nd the whole

set of eYcient solutions are reported in references [7] and

[9] to [12].

5.2 Constraints method to nd optimal solutions

One useful method to nd optimal solutions is the

`constraints method [812]. It may be introduced with

an example. Consider a problem in which two perfor-

mance indices gi and n parameters zi appear:

min

g1z1; z2; . . . ; zng2z1; z2; . . . ; zn

By constraining the performance index g2, the problem

can be transformed into

min g1z1; z2; . . . ; zng24

(

The values of the parameters that minimize g1 are eY-

cient (i.e. optimal) solutions. By varying properly the

value ofand searching for the new minimum ofg1, it is

possible to nd the whole set of optimal solutions. The

designer is aware of all possible choices when the whole

set of optimal solutions is known.

If there are two performance indicesgithat are related

to two system model parameters ziby means of analy-tical expressions, it is possible to use the constraints

method to nd the analytical expressions of both

the optimal performance indices and the optimal

parameters. In other words, it is possible to nd:

(a) g1g1(g2), i.e. the analytical expression that givesthe value of the performance index g1 when g2 is at

its best [or vice versa, which is conceptually the

same, g2g2(g1), i.e. the analytical expression thatgives the value of the performance index g2 wheng1is at its best];

(b) z1z1(z2) [or vice versa, z2z2(z1)], i.e. theanalytical expression that gives the values ofthe parameters z1 and z2 for a given optimal

performance index g1 or g2.




12/16

Remembering that g1(z1; z2) and g2(z1; z2) are the per-

formance indices and z1 and z2 are the system model

parameters, the scheme of the procedure to nd the

analytical expressions is as follows:

1. From the mathematical expression g1g1(z1; z2)the expression z2 z2(g1,z1) is derived by xing thevalue ofg1.2. By substituting the expression derived in step 1 into

the expression g2 g2(z1; z2), the expression g2g2(g1; z1) is obtained.

3. The minimum ofg2 is searched for by setting to zero

the derivative

dg2g1; z1dz1

0

and checking that

d2g2g1; z1dz21

>0

which corresponds to the search for the minimum of

the performance index g2, while the performance

index g1 is kept constant; from the expression of the

derivative, the expression z1 z1g1) can beobtained.

4. The expression z1z1g1) is substituted intog2g2g1; z1 and in this way it is possible toobtain the expression g2g2g1) which denes therelationship between the two optimal performance

indices.

5. The equation g2g2g1) is the image in the plane(g1g2) of the equationz1z1z2) in the plane (z1,z2); z1 z1(z2) may be obtained by substitution.

5.3 Optimal secondary suspension parameters

The mathematical procedure described above has been

used to optimize the parameters of the secondary sus-

pension of a railway vehicle described by the simple

system model in Fig. 1. The parameters to be optimized

were the stiVness, k2, and the damping, r2, of the sec-

ondary suspension, and the performance indices werexx2 and x2x1 .

5.3.1 Derivation of optimalxx2 , x2x1 and optimal k2,r2 using 1S-PSD

The performance indices are dened by equations (33)

and (34). The optimization procedure described in

Section 5.2 is applied as follows:

1. From equation (34) the expression of r2 as a

function ofx2x1 and k2 is derived:

r2 B2

m2

2

k22x2x151

2. The expression ofxx2 as a function ofr2 and x2x1 is

derived by substituting the expression of r2 [equa-

tion (51) into equation (33):

xx2 B

B2m2

k22x2x1 k

22

2x2x1

B2m22

s B

p 52

3. The following derivative, equal to zero, gives thestationary solution:

dxx2dk2

Bdp

dk2B 1

p d

dk2 0 53

The term 1=2p is always greater than zero, thus

solving with respect to k2:

B2m2

2x2x1 k22

2 k2B2m2

2

2x2x1 0 54

and therefore

k2

B4m32

24x2x1

3s

55

4. Finally, by substituting equation (55) into equations

(51) and (33), the expression ofxx2 as a function of

x2x1 is obtained:

xx2

27

4

B8

2x2

x1

6s

56

This equation denes the relationship between the

standard deviation of the acceleration of the body

and the standard deviation of the secondary

suspension stroke when the two standard deviations

are both minimized.

5. The equation that denes the optimal parameter set

is

r2r2opt

2k2m2p

57

For the system composed of the mass m2

, the

damper r2 and the spring k2 (mass m1 xed), the

critical damping may be dened as

r2crit

4k2m2p

58

and it follows that

r2opt 12

p r2crit 59

By setting the stiVness and the damping of the

secondary suspension as indicated above, the bestcompromise between the standard deviation of the

body acceleration and the standard deviation of the

secondary stroke is obtained.




13/16

5.3.2 Derivation of optimalxx2 , x2x1 and optimal k2,r2 using 2S-PSD

On the basis of equations (40) and (41), a numerical

search has been undertaken to nd both the optimal set

xx2 , x2x1 and the optimal set k2,r2. The correspondingplots are reported in Figs 6 and 7 respectively.

5.3.3 Optimalxx2 , x2x1 and optimal k2, r2

Both for the 1S-PSD [equations (56) and (57)] and for

the 2S-PSD (Section 5.3.2) excitations, optimal xx2 ,

x2x1 and optimal k2,r2 are plotted in non-dimensionalform in Figs 6 and 7 respectively. To obtain non-

dimensional values, reference is made to a railway pas-

senger vehicle, the relevant parameters of which are

reported in Table 1. Note that xx2 increases when x2x1decreases; i.e. these two performance indices are con-

icting. The designer should choose, on the basis of

given technical specications, the desired compromisebetween xx2 and x2x1 by selecting one point lying onthe curves plotted in Fig. 6, e.g. one of those marked

with special symbols (triangle, square, etc.). Having

chosen the preferred compromise between xx2 and

x2x1 , the corresponding values of the parameters k2and r2 are uniquely dened. This correspondence

between the points of the curves plotted in the xx2 ,x2x1plane (Fig. 6) and the points of the curves in the k2, r2plane (Fig. 7) are highlighted by special symbols in Figs

6 and 7. Inspection of Fig. 6 shows that for the 1S-PSD

the non-dimensional xx2 and x2x1 do not depend onvehicle speed. This is due to the fact that, for this exci-

tation spectrum, the speed parameter vis not mixed with

the parameters of the vehicle system k2, r2; . . . (see

Sections 2.3, 2.4 and 2.5). On the contrary, for the 2S-

PSD, which is very frequently found in actual applica-

tions, the non-dimensional xx2 and x2x1 do depend onvehicle speed (see Sections 2.6, 2.7 and 2.8). This sug-

gests that vehicle suspension parameters should vary

with vehicle speed in order to keep the optimality con-

ditions. This is technically easily achievable and hope-

fully, in the future, adaptive suspensions could be

adopted for railway vehicles.

In Fig. 7 the relationship between optimal stiVness k2

and optimal damping r2 is highlighted. To keep the bestcompromise between xx2 andx2x1 , the damping r2 hasto increase with the stiVnessk2, both for the 1S-PSD and

for the 2S-PSD excitation. The rate of change in r2 with

respect to k2 does not depend on vehicle speed for the

1S-PSD and varies considerably with vehicle speed for

the 2S-PSD excitation.

Fig. 6 Optimalxx2 and optimal x2x1 plotted in non-dimensional form. The curves are obtained by varyingk2 and r2, and the points highlighted by special symbols (triangle, square, etc.) refer to the points in

Fig. 7. Vehicle parameters in Table 1




14/16

6 CONCLUSION

Analytical formulae have been derived in order to

estimate the response of railway vehicles to random

excitations generated by the vertical track irregularity.

The accuracy of the derived formulae has been assessed

by comparison with data presented in the literature.

The analytical formulae should estimate with reason-

able accuracy the dynamic behaviour of an actual

railway vehicle running on rigid track. Referring to the

performed validation, the sensitivity of body accelera-

tion and secondary stroke to vehicle suspension para-

meters (primary and secondary stiVness, secondary

damping) is captured satisfactorily by the analytical

formulae. It has been found that analytical formulae (in

complete form) predicted the standard deviations of

both body acceleration and secondary stroke with an

error always less than 10 per cent, and often less than 2

per cent. On the basis of the validated analytical for-

mulae, a theoretical parameter sensitivity analysis has

been performed with reference to the standard devia-

tions of force on the axle box, body acceleration and

secondary suspension stroke. All these performanceindices are inuenced by secondary suspension para-

meters. In particular, secondary damping aVects the

body acceleration signicantly. A general result (con-

rmed by common experience) is the strong inuence of

the type of track irregularity on all the performance

indices considered. The bogie mass and the primary

stiVness do not seem to inuence the secondary stroke

signicantly.

By using the derived analytical formulae in the second

part of the paper, a method based on multiobjective

programming has been applied to nd the best trade-oV

between conicting requirements on performance indi-

ces such as xx2 (standard deviation of the body accel-

eration) and x2

x1 (standard deviation of the secondary

stroke). The parameters of the secondary suspension(stiVness k2 and damping r2) of a railway vehicle have

been optimized with the aim of minimizing both xx2 and

x2x1 . Simple analytical formulae have been derived forthe optimal xx2 , x2x1 and correspondingly optimal k2,r2. Optimal xx2 increases when both optimal k2 and

optimal r2 increase, and the opposite occurs for optimal

x2x1 . If the excitation is dened by the 1S-PSD, theoptimal secondary suspension settings do not depend on

vehicle speed. The opposite occurs for the more realistic

2S-PSD excitation, and thus it seems reasonable to

recommend, for future research, comprehensive studies

on the application of adaptive stiVness and damping

elements to railway vehicle secondary suspension

systems.

Fig. 7 Optimalk2 and optimalr2 plotted in non-dimensionalform for minimizingxx2 and x2 x1 . The pointshighlighted by special symbols (triangle, square, etc.) refer to the points in Fig. 6. Vehicle parameters

in Table 1




15/16

REFERENCES

1 Chalasani, R. Ride performance potentials of active

suspension systemsParts I and II. In ASME Symposium

on Simulation and Control of Ground Vehicles and

Transportation Systems (Eds L. Segel et al.), 1986, AMD

Vol. 80, DSC Vol. 2.

2 Thompson, A. G. Suspension design for optimum road-holding. SAE technical paper 830663, 1983.

3 Panagin, R. La Dinamica del Veicolo Ferroviario, 2nd

edition, 1997 (Levrotto & Bella, Turin).

4 Dukkipati, R. V. and Amyot, J. R. Computer-Aided

Simulation in Railway Dynamics, 1988 (Marcel Dekker,

New YorkBasel).

5 Grassie, S. L.,et al. The dynamic response of railway track

to high frequency vertical/lateral/longitudinal excitation.

J. Mech. Engng Sci., 1982,24, 77102.

6 Esveld, C. Modern Railway Track, 1989 (MRT Produc-

tions, Duisburg).

7 Mastinu, G.andGobbi, M.Advances in the optimal design

of mechanical systems. Course Notes, Hyderabad, India,

1999.

8 Miettinen, K. Nonlinear Multiobjective Optimization, 1999

(Kluwer Academic Publishers, Boston).

9 Grierson, D. and Hajela, P. (Eds) Emergent Computing

Methods in Engineering Design, 1996 (Springer-Verlag,

Berlin).

10 Ng, W. Y. Interactive Multi-Objective Programming as a

Framework for Computer-Aided Control System Design,

1989 (Springer-Verlag).

11 Stadler, W. Multicriteria Optimization in Engineering and

in the Sciences, 1988 (Plenum Press, New York).

12 Eschenauer, H., et al . Multicriteria Design Optimization,

1990 (Springer-Verlag).

13 Mastinu, G. A method for the optimal design of railway

vehicles. In World Congress on Railway Research,

Florence, 1997.

14 Gobbi, M. and Mastinu, G. Global approximation:

performance comparison of diVerent methods, with

application to road vehicle system engineering. In Innova-

tions in Vehicle Design and Development, ASME DE-Vol.

10, 1999.

15 Mastinu, G. Automotive suspension design by multi-

objective programming. In Proceedings of International

Symposium on Advanced Vehicle Control (AVEC94),

Tsukuba, 1994.

16 Gobbi, M., et al . Optimal and robust design of a road

vehicle suspension system. In Proceedings of 16th

IAVSD Symposium, Supplement ofVeh. Syst. Dynamics,

1999.

17 Hamid, A. andYang, T. L. Analytical description of trackgeometry variation. ENSCO, Transportation Technology

Engineering Division, Springeld, Virginia, 1981.

18 Power spectral density of track irregularities. ORE C116

RP1, Utrecht, 1971.

19 Mueller, P. C. and Schiehlen, W. O. Linear Vibrations,

1995 (Martinus NijoVPublishers, Dordrecht).

20 Crandall, S. (Ed.) Random Vibration, 1963 (MIT Press,

Cambridge, Massachusetts).

21 Newton, G. C., et al.Analytical Design of Linear Feedback

Controls, 1957, Appendix E (John Wiley, New York).

APPENDIX 1

Complete form of equation (25):

V2p2v3=2

Abp r21m1m2r1r22k1r21r2k2r1k22m1r2k21m2r1k22m2

m1m2r2m1r1m2r2m2m1m2r22k21r21k22 2k21k2m1m2 k1k2m1m22m1m2r2k1r1k2 r2m1r1m2r2m2r1r2k2m1k1m2k2m2

r2k1r1k22r1r2k1r1k2m1m2m1m2 r1r2m1r1r2m2k1m1m22m1m2r2k1r1k22 k1k2r2m1r1m2r2m22

r2k1r1k2r2m1r1m2r2m2r1r2k2m1k1m2k2m2

vuuuuuuuuut


xx2 2p2v3=2

Abp r21r32r2k1r1k2 r22k21r21k22r2m1r1m2r2m2

k1k2m1m2r2k1r1k2 r2m1r1m2r2m2r1r2k2m1k1m2k2m2m1m2r2k1r1k22 k1k2r2m1r1m2r2m22


vuuuuutComplete form of equation (27):

x2x1

2p2v3=2 Ab

p

m22

r21

k2r2m1r1m2r2m2 k1m1m2r2k1r1k2 r2m1r1m2r2m2r1r2k2m1k1m2k2m2

k2m1m2r2k1r1k22 k1k2r2m1r1m2r2m22r2k1r1k2r2m1r1m2r2m2r1r2k2m1k1m2k2m2

vuuuuut




16/16


x1x0 2p2v3=2

Abp k1r2k1r1k2m21m22k1m2m1r2m1r1m2r2m2r22m1r22m22k2m1m2

k2m1m22r2k1r1k2m1m2 r2m1r1m2r2m2r1r2k2m1k1m2k2m2k1m1m2r2k1r1k22 k1k2r2m1r1m2r2m22


vuuuuut

APPENDIX 2

For actual railway vehicles it is usually the case that

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