CT5870 Dynamica.e

8/17/2019 CT5870 Dynamica.e

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Track Dynamics

Lectures 1995/96

DYNAMICSin

Railway Engineering

Prof.dr.ir

. C. Esveld


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Track Dynamics

Lectures 1995/96

Dynamic aspects

Loads variation in time

Structure

• periodic

• impact

• stochastic•

mass• damping

• stiffness

natural frequencies

interaction

vehicle/track


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Track Dynamics

Lectures 1995/96

K d mvdt

( )

Force = change of momentum

Principle of dynamics


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Track Dynamics

Lectures 1995/96

K maForce = mass * acceleration

NewtonWith constant mass:


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Track Dynamics

Lectures 1995/96

1 mass

-

spring system


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Track Dynamics

Lectures 1995/96

Response of 1 mass

-

spring system

for harmonic loads the followwing

applies:


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Track Dynamics

Lectures 1995/96

Unit impulse response function

h t em

t

t

( ) sin( ( ))

( )

11

22


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Track Dynamics

Lectures 1995/96

Duhamel-integral

Steady state solution via convolution:

x t F( h t d t

( ) ) ( ) 0


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Track Dynamics

Lectures 1995/96

Frequency domain

Response via multiplication:

X f H f F( f ( ) ( ) )

Much more simple than via,

however requires Fourier transform


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Track Dynamics

Lectures 1995/96

Relationship between h and H

h FT H ( )

H FT h 1( )


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Track Dynamics

Lectures 1995/96

Track loads • Wavelength l • Frequency f

l v f

l[ m ]


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Track Dynamics

Lectures 1995/96

EI Q x

w(x)k

-3L -L 3L

x

-0.2

0.5

1

-0.208

-0.043

L4

p 2 L

p

L34

pL

54

pL

p

h , m

L

deflection

w(x) = h(x)Q2kL

moment

M(x) =m

(x)QL4

h

m(x) = e

(x) = e -|x/L| sincos xL

+ }{ |x|L

-|x/L|sincos

x

L- }{

|x|

L

-

Beam on an elastic foundation


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Track Dynamics

Lectures 1995/96

EIy x t my x t cy x t ky x t iv ( , ) ( , ) ( , ) ( , ) 0

EIy t Qe y t y t i ft ( , ) . ; ( , ) ; ( , )0 0 5 0 0 02p

Boundary conditions:


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Track Dynamics

Lectures 1995/96

As static!


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Track Dynamics

Lectures 1995/96


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Track Dynamics

Lectures 1995/96

L=characteristic length

static


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Track Dynamics

Lectures 1995/96


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Track Dynamics

Lectures 1995/96


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Track Dynamics

Lectures 1995/96

H f kL

f

f

f

f er

n n

i( ) [ ]

1

21 4

2

2

2 2

2

2

3

8

3

4

2

12

2a

f f

f f

n

n

tan


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Track Dynamics

Lectures 1995/96

Analogy with 1 mass spring system


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Track Dynamics

Lectures 1995/96

EI

EI2

Q -

1

a

K 1

K 2

2 Layer system


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Track Dynamics

Lectures 1995/96

Hertzian spring

stiffness of Hertzian spring

Liniarized

For a wheel diameter of 1 m and a wheel loadof 75 kN


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Track Dynamics

Lectures 1995/96

r

Hertzian spring k H


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Track Dynamics

Lectures 1995/96

For high frequencies it follows from a

equilibrium consideration of the wheel:

The transfer function of the wheel is

found

With


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Track Dynamics

Lectures 1995/96

Relationship FH and differential displacementHertzian spring:

In the frequency domain this becomes:


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Track Dynamics

Lectures 1995/96

With:

we find for the geometry;

k


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Track Dynamics

Lectures 1995/96

Relationship between axle box acceleration and

track geometery:

T k D


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Track Dynamics

Lectures 1995/96

1

2

3

4

5

6

02 4 6 8 10 12 14 16 18 20 22

time [ms]

DQ

Q= Dynamic amplification

P 1

P 2

Impact loads

T k D i


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Track Dynamics

Lectures 1995/96

Impact loads due to welds

P Q v k mm m st H e

e u1 2 1

P Q vm

m m

c

k m mk m st

u

u t

t

t u t

t u2 2 14

p

T k D i


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Track Dynamics

Lectures 1995/96

P Q v m k st u l 2

Impact loads due to welds

In practice:

T k D i


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Track Dynamics

Lectures 1995/96

10.00

1.00

0.10

0.010.1 1.0 10.0

Excitation frequency, f

Undamped natural frequency, f 0

t r a n s m i s s i o n

z 1.0

2

z 0.5

z 0.2

z 0.1z 0.05

z 0

K / F

natural frequency:

2pf

0=

1mk

transmission factor:

damping ratio:

z =2 km

c

wk

m

F

K

c

Vibration transmission

T k D i


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Track Dynamics

Lectures 1995/96

Frequency domain

X f x t e dt i ft ( ) ( )

2p

x t X f e df i ft ( ) ( )

2p

Via Fast Fourier Transform FFT

T k D i


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Track Dynamics

Lectures 1995/96

S f T

X f X f ij i j( ) ( ) ( ) 1

S f autospectrumii( ) S f

ii

D D 2

(complex)

(real)

T k D i


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Track Dynamics

Lectures 1995/96

Estimation of transfer functions

[ ] Sxy Sxx H

[ ] H Sxx Sxy 1

2

y x

T

f

H Sxy

Syy. ( )

For reliable results it is required that: 0 85 102. .

T k D i


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Track Dynamics

Lectures 1995/96

1 2 ... ... n 1 2 ... ... n

1

...

p

1

q

m

L L

hb

hg

ondergrond

betonplaat

kurkrubber

plaatjes

rail

TILLY model tram structure

Track Dynamics


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Track Dynamics

Lectures 1995/96

Impulse load

0 0.5 1 1.5-5

0

5verplaatsing (Tilly)

tijd [s]

verpl[mm]

0 50 100 150 200 250 300 350 4000

10

20

30

40 Autospectrum verplaatsing (zoom tot 400 Hz)

freq [Hz]

verpl[mm

^2s]

Track Dynamics


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Track Dynamics

Lectures 1995/96

Moving

load0 0.5 1 1.5-0.6-0.4

-0.2

0

0.2verplaatsing (Tilly)

tijd [s]

verpl[mm]

0 0.5 1 1.5-0.05

0

0.05verplaatsing (getaperd)

tijd [s]

verpl[mm]

0 20 40 60 80 1000

0.05

0.1 Autospectrum verplaatsing (zoom)

freq [Hz]

verpl[mm^2s]

Track Dynamics


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Track Dynamics

Lectures 1995/96

mx cx kx cy ky

mz cz kz my

k c

m

x

y

z = y - x

Geometrical excitation

Track Dynamics


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Track Dynamics

Lectures 1995/96

k c

m

x

F

Track Dynamics


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Track Dynamics

Lectures 1995/96

1 2 3 4

+

+ x

z

q

onderlegplaatje

dwarsligger

ballast

bak

draaistel

wielstel

Hertz'se veer

primaire vering

secundaire vering

+

Dynamic vehicle/track model

Track Dynamics


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Track Dynamics

Lectures 1995/96

G _

r c =T

wave propagation:

soft soils: wave propagation speed can approach train speed

liquefaction type of phenomena

Problems to be considered:

Track Dynamics


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Track Dynamics

Lectures 1995/96

High Speeds

Critical speed: v

m

kEI cr 2

2

y

y v

v

dyn

stat

cr

1

1

2

Dynamic amplification:

Track Dynamics


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Track Dynamics

Lectures 1995/96

Measurements in England on soft soils

-14

-13-12

-11

-10

-9

-8

-7

-6

-5

120 150 180 210 240

running speed [km/h] v e r t i c

a l d i s p l a c e m e n t [ m m ]

High speed train

IC train

critical train speed

225

Track Dynamics


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Track Dynamics

Lectures 1995/96

DIANA Model

VAR1 1.2 m dikker zandbed

VAR0 model Sliedrecht0.1750.35

1.70

1.30

1.20

3.50

0.70

1.80

10.55

19.0

100 kN 100 kN

lengte 29.4 m

M9

M8

M7

M6 M5

M4

M3

M1

M2

M12

M11

VAR2 1.2 m dunner zandbed

VAR3 betonplaat 0.3 mM10

Track Dynamics


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Track Dynamics

Lectures 1995/96

Calculational results

Concrete plate

+1.2 m sand

-1.2 m sand

Track Dynamics


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Track Dynamics

08.55-09.40 Introduction10.00-10.45 Track maintenance

10.55-11.40 Track maintenance11.50-12.35 Measuring systemsBreak 13.40-14.25 VRA system14.35-15.20 Track loads15.30-16.15 Decision support systems

08.55-09.40 Dynamics10.00-10.45 Dynamics10.55-11.40 High speed operation

11.50-12.35 Rail defectsBreak 13.40-14.25 ERRI D20214.35-15.20 Experimental research15.30-16.15 MINIPROF system

CT5870 Dynamica.e

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