Time-frequency analysis of wheel-rail shock in the presence of wheel flat

Journal of Traffic and Transportation Engineering ( English Edition) 2014,1(6) :457-466

Time-frequency analysis of wheel-rail shock in the presence of wheel flat

Jianming Ding1 ' • , Jianhui Un 1 , Guangming Wang', Jie Zhao' 1

State Key Uzboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan, China 2

CNR Tangshan Railway Vehicle Co., Ltd., Tangshan, Hebei, China

Abstract: Against the deficiencies of traditional time domain and frequency domain analysis in detecting wheel-rail (W-R) system hidden risks which wheel flats generate, the time-frequency characteristics of W-R shock caused by wheel flat are analyzed and the vehicle-rail dynamic model with wheel flat is investigated. The 10 degrees of freedom (DOF) vehicle model is built up. 90-DOF rail model is constructed. The wheel flat excitation model is built up. The vehicle-track coupling dynamic model including wheel flat excitation is set up through nonlinear Hertzian contact theory. The vertical accelerations of axle box are calculated at different speeds and flat sizes based on the vehicle-track coupling dynamic model with wheel flat. Frequency slice wavelet transform ( FSWT) is employed to analyze timefrequency characteristics of axle box accelerations to detect the W -R noncontact risks, which the traditional time domain or frequency domain method does not analyze. The results show that the small flat size and high running speed lead to high frequency W -R impact. Large flat size and high running speed result in momentary loss of W-R contact, and there exist security risks between wheel and rail. The conclusion that the phase of axle box accelerations is same to W-R forces lays a theoretical foundation of mouitoring W-R contact safety from axle box acceleration instead of traditional W-R force detection. Key words: railway vehicle; wheel flat; wheel-rail shock; frequency slice wavelet transform; timefrequency characteristic

1

Introduction

speed. Statistically, the dynamic wheel-rail ( W -R)

It is unavoidable that wheel flats appear due to wheel

impact forces caused by wheel flat are several times larger than static W -R forces (Bray et al. 1973; Duk-

lock and sliding along the rail when railway vehicle brakes or runs in low friction conditions. The wheel

kipati and Dong 1999 ; Jergens et al. 1999; Belotti et al. 2006; Baeza et al. 2006a, 2006b; Uzzal et al.

flats circularly shock W -R system and induce large periodic W-R shock forces when wheel rotates at high

2009; Brizuela et al. 2010, 2011; Zhao et al. 2012;

• Corresponding author: Jianming Ding, PhD, Assistant Researcher. E-maD, fltingjianming@126. com.

Ding et al. 2013; Ding et al. 2014). As a result,

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Jianming Ding et al.

high cycle fatigue , crack appearance , crack fast propagation of the key components , such as rails , sleepers , wheels , bearings , will seriously affect and reduce vehicle and track service lives and endanger vehicle operation safety. At the same time , a periodic impact noise is produced in addition to the usual rolling noise, which is more random and non-stationary in character due to wheel flat shock ( Newton and Clark 1979 ; Fermer and Nielsen 1994 ; Johansson and Nielsen 2003 ; Stratman et al. 2007) . Therefore, it is necessary and urgent to analyze W -R system shock characteristics caused by wheel flats and study its corresponding detection methods which have great significance on vehicle maintenance and safe operation. Domestic and foreign scholars have made a lot of research work and achieved fruitful results in the vibration characteristics analysis of the wheel flats. Researchers established vehicle-track coupling dynamic model to analyze dynamic responses at different operation speeds and different flat sizes and drew three very important conclusions through the time domain or frequency domain analysis ( Bracciali and Cascini 1997 ; Zhai et al. 2001 ; Wu and Thompson 2002; Madej ski 2006; Ding et al. 2013) . The first is that W-R force increases with the increase of flat size. The second is that the W-R force firstly increases then decreases with the increase of operation speed. Finally, peak frequency distribution of W -R force is analyzed. These conclusions provide a scientific basis for wheel removal, reasonable running speed control and optimization design of vehicle track system. But in time or frequency domain, the classical analysis meth-

10-DOF vehicle model

ods do not analyze the W -R contact characteristics under the course of W -R dynamic interaction. In this paper , the time-frequency analysis using the frequency slice wavelet transform ( FSWT ) is employed to extract the character of losing contact between wheel and rail. The vehicle-track coupling dynamic model is set up in the second section. The dynamic response is solved under different running speeds and flat sizes in the third section. In the fourth section, the FSWT is employed to analyze the timefrequency characteristics of the axle box vibration acceleration caused by wheel flats and to detect the hidden risks of W -R contact loss. Finally, the conclusions are drawn.

2

Vehicle-track coupling dynamic model simulating wheel flat

Dynamics simulation mechanism of wheel flat is shown in Fig. 1. It mainly includes 10 degrees of freedom ( DOF) vehicle dynamic model, 90-DOF track dynamic model and wheel flat excitation model. The vehicle-track coupling dynamic model including wheel flat excitation is built up through nonlinear Hertzian contact theory. The W-R normal compression amount is calculated through the displacement of wheel , rail and track irregularity including the wheel flat effect. The W-R forces are obtained by applying the W-R normal compression amount and nonlinear Hertzian contact theory and are regarded as feedback of vehicle model and track model to execute the dynamic simulation.

Wheel flat excitation model

Vertical displacement of axle box

90-DOF track model

Vertical displacement of track

W-R normal compression amount

Nonlinear Hertzian contact theory

W-Rforces Fig. 1

Dynamics simulation mechanism of wheel flat

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Journal of Traffic and Transportation Engineering( English Edition)

Based on the dynamic simulation mechanism of wheel flat, the vehicle-track coupling dynamic model including the wheel flat is set up. The 10-DOF vehicle model and 90-DOF track model ( Zhai 2007) are adopted. The fast numerical calculation method ( Zhai 1996 ) is adopted. 2.1

10-DOF vehicle dynamic model

The 10-DOF vehicle dynamic model includes the vertical motion of car body , the front and rear frames and four wheel sets , the roll motion of car body and the front and rear frames. The vibration differential equation is described as Eq. ( 1 ) Mx+Cx+Kx=F (1) where M, C, K, F represent the quality matrix , damping matrix, stiffness matrix , and force vector, respectively ; i, x describe the acceleration, velocity and displacement of corresponding 10-DOF, respectively.

x,

2.2

90-DOF track dynamic model

The Euler beam model with discrete support is adopted to build up track model. Its vibration differential equation is represented in Eq. ( 2) N

!~(~k) 4 q,(t)

+ 6K,,Y,(x,) x

2.4

Non Ii near Hertzian contact theory

The W-R normal compression amount in the jth wheel set is described as Eq. ( 6) L1 (x, t) = Z.;(t)- Z.;(t) -z0 (x, t) (6) The W -R force is obtained through noulinear Hertzian contact theory, when L1 ( x, t) is larger than 0 1 (x, t) ]'·' p1 (t) = [ GL 1

3. 86 x

(2)

j=l

NM

Z,(x, t) = I,Y,(x)q,(t)

=

10-sR-o.m.

4

wherep,(t), p,(t), p,(t) andp 4 (t) represent the corresponding W -R force from the first wheel set to fourth wheel set, respectively; C" is the fastener damping; K" is the fastener stiffness; N is the discrete support amount; NM is the track mode number; q, ( t) is the regular modal coordinate of rail; Y, ( x1 ) represents rail modal ; E, I and l describe the elasticity modulus , section inertia and the length of rail , respectively ; m, represents the line density of rail. The vertical vibration displacement is described as Eq. ( 3) (3)

k=l

2.3

where ( ( x, t) is the equivalent track irregularity caused by wheel flat; x is the displacement along the scratch; L is the scratch length of wheel flat; D1 is the scratch depth of wheel flat, D1 =L2 /16R. The wheel flat simulation model not ouly describes the geometry but also reflects the high frequency characteristics of W-R shock. The new track irregularity z, ( x, t) is reduced through the superposition of the test track irregularity z., ( x) and the corresponding equivalent track irregularity ( ( x, t) caused by wheel flat. (5) z0 (x, t) = z..,(x, t) + ((x, t)

p1 (t) = 0 where G is the W -R contact parameter, G

N

I,Y,(x,)q,(t) = LP;(t)Y,(x.,)

(4)

or else

h=l

NM

h=l

2 ((x, t) = ; DAl- cos( ;"')]

NM

ij,(t) + L c,,Y,(x,) I,Y,(x,)q,(t) + i:=l

model is represented as Eq. ( 4)

Equivalent track irregularity input model containing wheel flat

The equivalent track irregularity displacement caused by wheel flat is used to simulate the wheel flat dynamic response. The concrete displacement input

3

Dynamic response at different running speeds and flat sizes

In order to analyze the time-frequency characteristics

of W -R shock caused by wheel flat, the dynamic response is simulated at different runuiog speeds and flat sizes. The concrete simulation conditions are as follows. The flat sizes ( L) are 10, 20, 30, 40, 50, 60, and 70 mm, the runuiog speeds ( V) are 50, 70, 90, 110, 130, and 150 km/h, respectively. Because there are a lot of simulation conditions , several simulation conditions are selected to analyze the dynamic behaviors. When the runuiog speed is 50 km/h and wheel flat size is 10 rnm, the time histories of W-R force and axle box acceleration are shown in Fig. 2. When the runuiog speed is 90 km/h and wheel flat size is 20 rnm, the time histories of W-R force and axle box acceleration are shown in Fig. 3. When the runuiog speed is 150 km/h and wheel flat

460

Jianming Ding et al.

size is 70 mm, the time histories of W-R force and

axle box acceleration are shown in Fig. 4. 25

100

B

20

D

90

~

:§,"'

15

01

10

.9 ~ ....


....Q


C'

~


80

""

E'

5

Q Q

~

><

~

0

E

,J:1

70

-5

~ -10

<

c

60 0.320

-15

B'

D'

-20L------L------L-----~------~----~

0.324

0.328

0.332

0.336

0.340

0.320

0.324

0.328

Time (s) (a) W-R force

140 D

C'

120

~

!

F

100

.9 ~ ....

A

80

~

G'


....

~

E'

40

01

Q


0.340

80 B


0.336

Time histories of W-R force and axle box acceleration ( V =50 km/h, L =10 mm)

Fig. 2

~

0.332

Time(s) (b) Axle box acceleration

"" Q Q

G

60

0

A'

><

F'

0

40

,J:1

E

..2 -40 ><

20

<

c

0 0.10

0.20

0.30

0.40

-80 0.10

0.50

0.50

Time histories of W-R force and axle box acceleration ( V =90 km/h, L =20 mm)

100

400

~

300

0

:§,

0

01

.E -1oo

D'

..2
200

Q

E

~

~

~

0.40

(b) Axle box acceleration


....

0.30

Longitudinal displacement (m)

B


0.20

(a) W-R force

500

Q

D'

Longitudinal displacement (m)

Fig. 3

~

B'

100

-200

0

A

,J:1
~

0

-300

B'

-100

-400L-----~------~------L-----~------~

0

0.2

0.4

0.6

Longitudinal displacement (m) (a) W-R force

Fig. 4

0.8

1.0

0

0.2

0.4

0.6

Longitudinal displacement (m) (b) Axle box acceleration

Time histories of W-R force and axle box acceleration ( V =150 km/h, L =70 mm)

0.8

1.0

461

Journal of Traffic and Transportation Engineering( English Edition)

From Figs. 2-4 , the peak value time and the zero value time of the axle box acceleration are same to those of W-R forces, which indicates that the phase of axle box acceleration is consistent with that of W-R force. So the W-R contact situation is characterized by the axle box acceleration instead of traditional W-R force analysis. A lot of wheel situation judgment methods based on axle box acceleration are developed ( Bracciali and Cascini 1997 ; Barke and Chiu 2005 ; Madej ski 2006). But those literatures did not tell the basic principle of applying axle box acceleration to detect the wheel situation instead of W-R forces. In addition, the W-R forces between point C and point D, between point D and point F, between point F and point G are equal to 0 (Fig. 4), which proves W-R contact is lost during those times. If there is large lateral W-R force, the derailment will easily occur. The

hidden risks which endanger vehicle operation safety exist in the W-R system. So it is necessary to detect the wheel flat and W-R contact characteristics. It is noteworthy that the axle box accelerations between point C and point D, between point D and point F, between point F and point G are constant and change gradually. There are suspension forces and not W-R forces. Additionally, W-R forces are larger and change much fiercer than suspension forces. So the constant acceleration at a period is the feature of W-R noncontact characteristics. The W-R contact characteristics are added according to all the simulation conditions. The statistical results are shown in Fig. 5. The domain surrounded by points A-B-C-D-E-F-F' -E' -D'c' -B' -A' is loss domain of W-R contact, and the domain surrounded by points A-B-C-D-E-F-F" -E" -D" -c"B" -A" is the domain keeping W-R contact.

80

B'

A'

C'

D'

E'

D

E

F

D"

E"

F"

F'

60

Keeping domain ofW-R contact

20

B"

A"

C"

oL---------~--------~----------~--------~--------~L---------~

40

60

100

80

120

140

160

Running speed (km/h) Fig. 5

Statistical result of W-R contact situation

When the running speed of vehicle is larger than a certain value and the flat size is bigger than a value , the loss of W-R contact will appear. The hidden risks exist in the W-R system. The study results may be used to determine reasonable running speed according to exact flat size without loss of W-R contact. At the same time, the high running speed of railway vehicle and big flat size of wheel would lead to the W-R noncontact. Therefore, the FSWT is introduced to analyze the axle box acceleration to exact the time-frequency

feature of shock caused by wheel flat.

4

4.1

Time-frequency analysis of W-R shock caused by wheel flat Basic theory of FSWf

FSWT is a new time-frequency analysis method ( Yan et al. 2009; Duan et al. 2013) . FSWT combines the characteristics of fast Fourier transform, short time Fourier transform and wavelet transform. FSWT can

462

Jianming Ding et al.

complete the time-frequency analysis through the fre-

can not be executed by applying Eq. ( 8 ) . So it is

quency slice function to reform the traditional Fourier transform. FSWT has overcome the shortage of con-

necessary to define the frequency domain expression of frequency selection function. The frequency selec-

stant frequency resolution in the short time Fourier transform, the difficulty of selecting the wavelet func-

tions (DC) (Yan et al. 2009).

tion, and defect of the cross-term interference in the Wigner-Vill distribution of the multi-component sig-

at the position of 0 is not eqnal to 0.

nals. FSWT is suitable for the analysis of the non-stationary signal, such as the instantaneous impact signal , transient signal and time-varying frequency signal. So in this paper, FSWT is used to analyze the time-frequency characteristics of W -R shock caused by wheel flat.

(1) FSWT For any f(t) e L' (R), the Fourier transform p(w) of p ( t) exists , and the FSWT is stated as 1 J+O W(t,w,,\ ,u) = .,/ -• j(u) 2

tion function must satisfy the following design condiDC1 : the amplitude of frequency selection function

j;(O) = 1, j;(O) ;o' 0 DC2: frequency selection function bas a finite energy.

r

DC3 : the amplitude of frequency selection function

p( ± .. )

DC4 : frequency selection function is a symmetric

I j;( ±

are constant or they are the functions of w and t;

p( u)

oo)

I .;; j;(O)

According the four design conditions of frequency

(7)

u where the scale u( ;o'O) and energy coefficient ,\ ( ;o'O)

tion of p( u) ;

= 0

even function about the zero frequency.

X

p' (u- "')e""du

p' (u)

oo

at the position of ± oo is equal to 0.

A

is Fourier transform off( t) ;

I j;(w) l'dw <

selection function, the frequency selection function is described in Eq. ( 9 )

j ( u)

is conjugate func-

e-+-'

j;(w) =

(9)

( 3 ) Scale determination of FSWT Without loss of generality , assume that energy co-

is frequency selection function.

The general wavelet is a microscope in time do-

efficient,\ =1, scale coefficient u =!"_,I< >0. Eqs. I(

main , but FSWT is a microscope in frequency domain. So this transform is called the FSWT. w is used to complete the shift of frequency selection function along the frequency axle , similarly , the Fourier transform in the time domain is executed. u is applied

(7) and (8) are translated into Eqs. (10) and (11)

W(t,w,I<) = _1_,\J+• j(u) 2"'T _..,

p' (I< u- "')e""du (10) (lJ

W(t ,w,I<) = i"_e""J+•f( 7' )e""p' [ !"_( 7' K

-

t)] dr ( 11)

K

-oo

to realize the dilation of frequency selection function

According to the Heisenberg uncertain principle , it

at the frequency scale , similarly, wavelet transform is

is not possible that the high time resolution ratio and

executed in the time domain of signal. So, the fre-

the high frequency resolution ratio exist at the same

quency selection function plays an important role in

time. The time resolution ratio and frequency resolu-

analyzing time-frequency characteristic of signal ap-

tion ratio are set to satisfy the analysis need.

plying traditional Fourier transform.

The frequency resolution ratio is recorded as 11

( 2) Characteristic and selection of frequency selection function The frequency domain expression of Eq. ( 7 ) is


(12) w The amplitude ratio of expectation response is re1/ = -

corded as {

stated as

W(t,w,,\,u)

W(t,w +
=,\ue""f:J(r)e"" X

p'[u(r-t)]dr

I (8)

When the time domain expression of frequency selection function is ouly known , the wavelet transform

I.;; {

(13 )

Whenf( t) = e"'' , the Eq. ( 13 ) is translated into Eq. (14)

I p(K'T/) I< {I p(O) I

(14)

463

Journal of Traffic and Transportation Engineering( English Edition)

When f( t) =8 ( t - t0 ) into Eq. ( 15)

lp(~)

,

the Eq. ( 13) is translated

I<

Cl p(O) I

4.2

(15) The FSWT is used to analyze the time-frequency characteristics of the axle box vertical acceleration signals under different conditions. When the running speed of vehicle is 50 km/h and the size of wheel flat is 10 mm, the time history of axle box vertical acceleration and its time-frequency analysis are shown in Fig. 6. When the running speed of vehicle is 110 km/h, and the size of wheel flat is 10 mm, the time history of axle box vertical acceleration and its time-frequency analysis are shown in Fig. 7. When the running speed of vehicle is 130 km/h, and the size of wheel flat is 40 mm, the time history of axle box vertical acceleration and its time-frequency analysis are shown in Fig. 8. When the running speed of vehicle is 150 km/h and the size of wheel flat is 70 mm , the time history of axle box vertical acceleration and its time-frequency analysis are shown in Fig. 9.

= !:J.w!:J.t

Jl-

Because frequency selection function is Gaussian function in this paper, Jl- =0 . 5. The Eqs. ( 14) and ( 15 ) are used to construct Eq. ( 16 ) K~

/2ln( 1/C)

I

~

(16)

J.L

K,:::

--= ~/2ln( 1/C) The solving of Eq. ( 16) gets the following result

l

K

= /i -

2~

Time-frequency analyss of axle box acceleration

(17)

C = e-t The relationship of coefficient u with time resolution ratio and frequency resolution ratio is constructed through Eq. ( 17) to determine the scale coefficient of FSWT. 20-

3000 0 456.3

10-

912.5

~

~

!"'

.s" '@

N

2000

23.

1369

"" "' " ~

1825

» (.)

0

0" (.)

0'

(.)

<

2281 1000

2738

- ]03194 3650 - 20L---~---~---~--~

0

0.01

0.02

0.03

0.04

Time (s) (a) Acceleration Fig. 6

0

0.01

0.02

0.03

0.04

Time (s) (b) Analysis ofFSWT

Time history of of acceleration and its time-frequency analysis ( V =50 km/h, L =10 mm)

From Figs. 6 and 7 , the W-R shock caused by wheel flat does not lead to the loss of W-R contact because the time history of axle box acceleration has no constant part. There is a single energy peak in the time-frequency image. But in Figs. 8 and 9, the time history of acceleration has a constant part, loss of W-R contact occurs, and multiple energy peaks appear. Loss of W-R contact will lead to many shocks

between the wheel and rail. The result of time-frequency analysis is constant with the real W-R shock situation. It is proved that the time-frequency analysis can detect W-R system noncontact hidden risks which the traditional time and frequency analysis methods can not detect. In Fig. 10, the frequency of the maximum peak value is added. The conclusion that higher running speed and smaller flat size will lead to higher

464

Jianming Ding et al.

frequency shock is drawn. The small flat size can be seized during the high speed operation, the sample frequency would be raised. 25 -

The following section will discuss the wheel flat detection method through the combination of the time-frequency characteristic, running speed and wheel radius. 3000

15 ~

i

5-

Cl

.g "...

"...:" u u

-5 -

- 15 -

- 25 0

0.02

0.01

0.03

0

0.04

0.01

Time (s) (a) Acce leration Fig. 7

0.02

0.03

0.04

Time (s) (b) Analysis ofFSWT

Time history acceleration and its time-frequency analysis ( V = 110 km/h, L = 10 mm)

100 0

50 0

"'

~ Cl .s

3.550X 10' 7.100Xl0'

0

1.065 X 10' - 50

1.420 X 10'

"...: - 100

1.775 X 10'

'§

"u u

2.130 X 10' - 150

2.485 X 10' 2.840 X 10'

- 200 0

0.0 1

0.02

0.03

0.04

Time (s) (a) Acceleration Fig. 8

5

0

0.01

0.02

0.03

0.04

Time (s) (b) Analysis ofFSWT

Time history of acceleration and its time-frequency analysis ( V =130 km/h, L =40 mm)

Conclusions

The combination of FSWT and vehicle-rail dynamic model is used to study the time-frequency of W-R shock caused by wheel flat. In this paper, the method mainly has the following characteristics. The phase of the W-R force is consistent with axle

box acceleration. So the contact characteristic can be detected through axle box acceleration, instead of the traditional W-R detection. Because the acceleration measurement is easier than W-R forces , the market outlook of acceleration measurement is better than WR forces detection of wheel flat. Loss of W -R contact easily occurs when the run-

465

Journal of Traffic and Transportation Engineering( English Edition) 100 0

3.950X 10 3

0

7.900 X 10 3

~

s';;' - 100-

1.185X 10'

.s

>, (.)

'§

]

1.580X 10'

"g. 0)

0)

- 200-

0)

~

(.)

..::

1.975 X 10' 1000 2.370X 104

- 300-

2.765X 10' 3.160X 10'

- 400L-______ 0.01 0

L __ _ _ _ _ _L __ _ _ _ _ _~----~

O.D3

0.02

0.02

0.01

0

0.04

0.03

Time (s)

Time (s)

(a) Acceleration

(b) Analysis ofFSWT

Fig. 9

0.04

Time history of acceleration and its time-frequency analysis ( V =150 km/h, L =70 mm)

5500 5000 4500 4000

g 3500 g 3000

~L=lOmm

---e-- L=20 mm ~ L=30

mm

-----T-- L=40 mm

______._____ L=50 mm _____.,____ L=60 mm

---+--- L=70 mm

0)

" if

2500

<.!::

~ 2000 p.,

1500 1000

500 OL----------L----------~--------~-----------L----------~--------~

40

60

100

80

120

140

160

Running speed (km/h) Fig. 10

Frequency distribution under different conditions

ning speed of vehicle is high and the wheel flat size is big. The frequency of W-R shock caused by small flat size at high speed is higher. FSWT is used to analyze the time-frequency characteristics and realize the location of time-frequency characteristics along the circumference through the combination of the time-frequency characteristic, running speed and wheel radius. The multiple energy peaks of axle box accelerations would appear when loss of W-R contact occurs, which provide a powerful tool to detect W-R contact situation.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 51305358, 61134002). References Baeza, L. , Roda, A. , Carballeira, J. , et al. , 2006a. Railway traintrack dynamics for wheel flats wilh improved contact models. Nonlinear Dynamics, 45(3/4) : 385-397. Baeza, L. , Roda, A. , Nielsen, J. C. 0. , 2006b. Railway vehicle/ track interaction analysis using a modal substructuring approach. Journal of Sound and Vibration, 293(112): 112-124.

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Belotti, V. , Crenna, F. , Michelini, R. C. , et al. , 2006. Wheel-flat diagnostic tool via wavelet transform. Mechanical Systems and

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116.

Bray, D. E., Dalvi, N. G., Fmch, R. D., 1973. Ultrasonic flaw detection in model railway wheels. IDtrasonics, 11 (2): 66-72.

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Duan, C. D., Gao, Q., Xu, X. F., 2013. Generatorunitfaultdiagnosis using the frequency slice wavelet transform time- frequency analysis method. Proceedings of !be CSEE, 33(32), 96-103. Dukkipati, R. V. , Dong, R. G. , 1999. Impact loads due to wheel

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