8 Appendix(Track)

8/12/2019 8 Appendix(Track)

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3 Power Spectral Density of Track Irregularity (PSD)

$ sample of real track irregularity is taken as an random process ，which is composed by a

%arity of harmonic wa%es， take %ertical profile irregularity as an e&ample，a random process is

described as：

In which： iΩ means spatial fre'uency of ith harmonic wa%e，described as：

The denominator represents wa%e length.

Fre'uencies in time domain and in spatial domain correspond a coefficient of speed v：

The %ariance of each harmonic wa%e is：

$ccording to the

random %ibration

theory ， The %ariance of

harmonic wa%e means

also its power at the

related fre'uency, which

describes the feature of a

random process in

fre'uency domain, and is

ust like auto correlation

function in time domain.

The ma&imum %alue of

auto correlation function

is %ariance. #ut %ariances

at each fre'uency in one plot, a power spectral is

thus obtained. To remo%e the effects of fre'uency inter%al，di%ided each power spectral by related

fre'uency inter%al, a #)* could be obtained. This process is shown in fig.!

#)* function (ω S is an e%en and real function of ω ，take a general form as below：

∑=

+Ω=k

i

iii xb xh1

cos(( ε

ii L/!π =Ω

v Lv ⋅Ω=→= ω π ω /!

!

1

! (1∑=

−=n

j

jhi hhn

σ

h1

Ω1

h1(X)

∆Ω1

Φh1

Φh(Ω) 高低不

平顺

h

顺道 x

x 0

h(X)

空顺角顺

率

高低不平顺

功率密度 Φ

h

fig.! #)* of track irregularity (principle description

+

,

-

!

!

1"

+

,

-

!

!

1"(ω ω ω

ω ω ω ω

aaaa

bbbbS

+++

+++=



4 Time domain irregularity sample from PSD

（1）mathematic priciple

$ shape filter ( ω j F is constructed from ( !ω S and satisfied flowing form：

)hape filter ( ω j F con%erts a white noise (t f to a colored noise （ con%olution

integral）：

λ λ λ d t h f t ut

∫ −="

(((

In which (t h means impulse response of ( ω j F ，or in other words ( ω j F is a Fourier

transform of (t h ： ∫ ∞

∞−

−= σ ω ωσ d eh j F j((

The #)* of obtained time domain sample is（note：the #)* of white noise e'ual 1，di%ided

by π ! means in unit rad ，and multiply by ! means one side spectral is used）：

((1

!!

11(((( "

!ω ω

π π ω ω ω ω j F j F j F j F S j F S u −⋅⋅=

××⋅−⋅==

)o, this pro%es that (t u is one of sample obtained by fre'uency domain track irregularity

#)*.

For speed v ， Ω⋅= vω ：

(1

( Ω= S v

S ω ，so：

( )( ) ( ) ( ) ,

,

!

!1"

1"(

1(

v

j

v

j

v

j

v

j

v

j

bbbb

aa j F v

j F ω ω ω

ω

ω ω

++⋅+

+=Ω=

=Ω

eneral coefficients in v!"m#s are gi%en.

（!）0oefficients of different shape filters of different track irregularities

)hape filter is represented in a general form as below：

,

,

!

!1"

1"

(((

((

ω ω ω

ω ω

jb jb jbb

jaa j F

+++

+=

Instruction of“ad!usted track ”and“need ad!usting track ”

(this track condition is used by Institute of Railway %ehicles, )2T3



$dusted track ：

4eed adusting track ：

For different track conditions, all F（ w）magnitudes are put in one figure to compare, asshown in fig.

"enerally speaking， t#e ad!usted track e$ui%alent to medium &#inese main line，and

need ad!usting track means 'ad track

$ccording to e&perience, $$R %ertical profile is too much big.

5imited speeds for $$R track of different classes are listed in following table：

Track class

+ 6 - ! 1

Freight car (km/h 17+ 1!8 9+ +- -" 16

#assenger car (km/h 17+ 1-- 1!8 9+ -8 !-

For locomoti%es, because of much hea%ier a&le loads and no empty/full different, the limited

speed should be higher than the table listed, but no %alues are proposed.

!(+9"-.1",-."

""!86."(

ω ω ω

j j j F

++=

!8-6!.""1+98+7+."

"""9!8"+6."(

ω ω ω

++=

j j F

,!((!8,!.1"77-."""7--"!"1."

""1!."(

ω ω ω

ω ω

j j j

j j F

+++=

Alignmen

t：

Vertical

Profile：

Cross lev

el：

!(+9"-.1",-."

""67."(

ω ω ω

j j j F

++

=

!(8-6!.""1+98+7+."

""11-,86+."(

ω ω ω

j j j F

++=

,!((!8,!.1"77-."""7--"!"1."

""16."(

ω ω ω

ω ω

j j j

j j F

+++=

Aligmnent

Vertical

Prifile

Cross Le

vel



本顺告采用顺，依次顺“ 顺整好” 和“ 需要顺整”

美 AAR顺，依次顺6，，!顺国

" 顺，依次顺“ 低#$” 和“ 高#$”国



" 顺，依次顺“ 低#$” 和“ 高#$”国



" 顺，依次顺“ 低#$” 和“ 高#$”国

%&不平顺，'

(&不平顺，)

高低不平顺，A*

本顺告(&不平顺顺

+" 顺,-./国

本顺告%&不平顺顺01

“ 顺整好” 2 3AAR顺当

“ 需要顺整” 4 AR!顺51

本顺告高低不平顺顺顺顺1

73AAR+AAR!8顺，9顺好

Fig.(green: adusted , need adusting track/ read: $$R class +,-,!/ blue: germany high, low



PSD coefficients of **+ track class

$$R track irregularities are gi%en in inch unit system（

*ynamics of Railway ;ehicle )ystem.;iay <. arg, Rao ;. *ukkipati, $cademic #ress, 198-）as shown in following table.

Table: #)* coefficients of $$R track class under inch unit system

=odel

c$f in!

#arameter ;alues of parameters by track class

)ymbol 3nit + 6 - ! 1

$lignmen

t (

((

!

!

!-

!

1

!!

!

φ φ φ

φ φ φ φ

+

+= A

S $

1φ

!φ

1">-in!/cpf

1">cpf

1">!cpf

". ".6 ".9 1.+ !.8 6."

1"." 1"." 1"." 1"." 1"." 1"."

6.+ 6.+ 6.+ 6.+ 6.+ 6.+

;ertical

#rofile (

((

!!!-

!

1

!!

!

φ φ φ

φ φ φ φ

+

+= A

S $

1

φ

!φ

1">-in!/cpf

1">cpf

1">!cpf

".6 ".8 1.- !.6 -.6 7.9

7.1 7.1 7.1 7.1 7.1 7.1

-." -." -." -." -." -."

0ross

le%el (((

!

!

!!

1

!

!

!

φ φ φ φ

φ φ

++= A

S $

1φ

!φ

1">-in!/cpf

1">cpf

1">!cpf

". ".6 ".7 1.1 1.+ !.

7.1 7.1 7.1 7.1 7.1 7.1

-." -." -." -." -." -."

For compare among different track conditions and for con%enient application in dynamic

software, I)? metric unit system is recommended. For I)? metric unit system, a general form of

#)* e%en and real function representation is used：

circlemm /! ⋅$$R track irregularities gi%en in metric unit system is gi%en in following table.

Take alignment irregularity as an e&ample，1m@.!8"8 ft ( k @.!8"8：

( )((

((

1--

1

1--

11

1--

1(

!!

!-

!1

!!!

!!

!-

!1

!

!!

,! φ

φ φ

φ

φ φ

k

k k A

k

k k

k A

k k S

k k S

+ΦΦ+Φ

⋅=

+

Φ

Φ

+

Φ

⋅=

Φ⋅=Φ m%m#circle

In principle coefficient k reduce %ertical coordinate （ 1 ft#circle@1/k m#circle）， magnify

horiAontal coordinate. Further more unit in! in #)* should be changed to m!：For cross le%el the con%ert formula is:

( )( )!

!

!!

1

!

!

!

!

!

!

!

1

!

!

!

,! ((

(

1--

1

1--

11

1--

1(

φ φ

φ

φ φ

φ

k k

k A

k

k k

A

k k S

k k S

+Φ+Φ⋅=

+

Φ

+

Φ⋅=

Φ⋅⋅=Φ

m!m/circle

Table: #)* coefficients of $$R track class under metric unit system

model

circlemm /! ⋅

#arameter ;alues of parameters by track class

symbol 3nit + 6 - ! 1



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8 Appendix(Track)