Dynamic Excitations

8/12/2019 Dynamic Excitations

1/23

Dynamic excitationsof transport structures

Prof. Ing. Ladislav Frba, DrSc., Dr.h.c.

UNC 2008

Institute of Theoretical and Applied Mechanics, v.v.i.

Academy of Sciences of the Czech Republic, Prague


2/23

Introduction Movement of loads

Rails Random vibration of structures

Degradation of structural materials under

dynamic loads

Conclusions

Outline


3/23

Introduction

Movement of the load

Transport structures moving load

earthquake

wind

Moving load

[ ] [ ]2

2

( ),( , ) ( )

d v u t t

f x t x u t F m dt

=

m

v(x,t)

x u t= ( )

[ ] 22 2 2 2 22 2 2 2

( ),2

d v u t t v du v du v d u v

dt u dt u t dt u dt t

= + + +


4/23

Uniform movement at velocity c2

2( ) , 0

du d ux u t ct c

dt dt = = = =

2 2 2 22

2 2 2( , ) ( , ) ( , ) ( , )2

x ct

d v ct t v x t v x t v x t c cdt t x t x

=

= + +

Motion in (x,y) plane x= u(t), y= v(t)load f(x,y,t) m(x,y,t) d2w/ dt2

2 22 2 2 2 2

2 2 2 2 2

d w w du w dv w w du dv

dt x dt y dt t x y dt dt

= + + + + 2 2 2 2

2 2( )

( )

2 2x u t

y v t

w du w dv w d u w d v

x t dt y t dt x dt y dt ==

+ + + +


5/23

Rails

Diferencial equation

0( , ) ( ),v x t v v s=

( , ) ( , ) 2 ( , ) ( , ) ( )IV bEIv x t v x t v x t kv x t x ct P + + + = && &

Steady state vibration

0 38 2P PvEI k

= =

1/ 4

4

k

EI

=

P

v(x,t)

x t= c

-x +xk

s

0


6/23

Moving coordinate:

Speed parameter:

Damping parameter:

Critical speed:

( )s x ct = 1/ 2

2cr

c c

c EI

= =

1/ 2( / )b k =

1/ 2

2crEI

c

=

Ordinary differential equation:

2( ) 4 ( ) 8 ( ) 4 ( ) 8 ( )IV II Iv s v s v s v s s + + =


7/23

Solution

1 1 2 12 2

1 1 2

2( ) ( cos sin )

( )

bsv s e D a s D a s

a D D

= +

+

3 2 4 22 2

2 3 4

2( ) ( cos sin )( )

bsv s e D a s D a s

a D D= +

+

for s > 0

for s < 0


8/23


9/23

Random vibration of a beam on

elestic foundation under moving force


10/23

Equation of the beam

[ ]( , ) ( , ) ( , ) 2 ( , ) ( ) ( , )IV bL v x t EIv x t v x t v x t k x v x t + + + =&& &

( ) ( )x ct F t=

Foundation ( ) ( )k x k k x= + o

[ ]( )k E k x=

Force 1

( ) ( )F t F F t = +

o

Moving coordinate

Static deflection and bending moment under

force F

[ ]( )F E F t = 1/ 4( ) ,

4

ks x ct

EI

= =

0 03 ,8 2 4

F F F

v MEI k

= = =


11/23

Coeficient of variation VV and VM as a function of speed,

= 0.2,i= 0, = 0.2, D = 10/


12/23

COVARIANCE

POWER

SPECTRALDENSITY


13/23

Random vibration of a beam

Bernoulli-Euler equation

Normal mode analysis

Generalized deflection

Generalized force

( , ) ( , ) 2 ( , ) ( , )IV bEI v x t v x t v x t p x t + + =&& &

1

( , ) ( ) ( )j jj

v x t v x q t

=

=

1

( , ) ( ) ( )j jj

p x t v x Q t

=

= 2( ) 2 ( ) ( ) ( )j b j j j jq t q t q t Q t + + =&& &

0

1( , ) ( ) 0

( ) for

00

L

j

jj

p x t v x dxMQ t t

=


14/23

Solution

( ) ( ) ( ) ( ) ( )j j j j jq t h t Q d h Q t d

= =

Impulse function (weighting function)1

( ) ( )2

i

j jh H e d

=

Frequency response function

( ) ( ) ij jH h e d

=

Load[ ]( , ) ( , ) ( , )p x t E p x t p x t= +

o


15/23

Stationary vibration of a beam under

moving continuous random load


16/23

Load ( , ) ( ) 1 ( )p x t p p s r t

= + +

o o

[ ]( , )p E p x t=2

( , ) ( ) ( ) ( ) ( )p x t p s p r t p s r t = + +

o o o o o

Covariance of the moving load

/s t x c= 1

2 2 2 2

1 2 0 1( , , ) ( ) ( ) ( / )pp pp rr prC x x C p C pC x c = + + + + +

2 3 3 3 3 4

2( / )rp ppr prr ppr prr pprr pC x c C pC C pC C + + + + + +

1 2

0

x x

c

=


17/23

Non-stationary vibration of a beam

under moving random force


18/23

Load

( , ) ( ) ( )p x t x ct P t= ( ) ( )P t P P t = + o

[ ]( )E P t P=

Covariance of the force

Covariance of moving load

Covariance of generalized deflection

1 2 1 2( , ) ( ) ( )ppC t t E P t P t =

o o

1 2 1 2 1 1 2 2 1 2( , , , ) ( ) ( ) ( , )pp ppC x x t t x ct x ct C t t =

, ,

1 2 1 1 2 2 1 2 1 2 1 2

1( , ) ( ) ( ) ( ) ( ) ( , )

j kq q j k j k pp

j k

C t t h t h t v c v c C d d M M

=

Speed parameter

Damping parameter

Simple beam

1/(2 )c f l=

1/ /(2 )b = =3

0( ) sin , 48j

j x Pl

v x vl EI

= =


19/23

Degradation of structural materials

under dynamic loads

Dynamic loads

Fatigue tests

Whler curve


20/23

Concept of fatigue assessment

Static system

Material properties

(Whler curve)

Real Loads

P (t)

Stress curve

(t)

Spectrumni(

o,

n)

Calculation value

for damageDSd

Proof

DSd

Dlim

Simulation of

train passages

Counting procedure(Rainflow)

Damage hypothesis

(Palmgren-Miner) = max -min = min / max


21/23

Strategy of maintenance,

repairs and reconstructions,

yes the conditions

are fulfilled,not the conditions

are not fulf il led

Effect of traffic loads

Stress-tim e recordsstatis tical evaluation

estimation of residual l ife

yes

yes

yes

yes

yes

Modal analysi s

crit eria of dereriorationno t

not

no t

not

Monitoring of crack s

criteria of c racks

Inspections

Technical ec onomic decis ion

Maintenance

Repair

Reconstruction


22/23

Conclusions

The dynamic loads, i.e. in time varying loads,increase the stresses in transport structures

with increasing speed

The dynamic loads are varying in time either

regularly or randomly

The response of structural materials

deteriorates their properties in time


23/23

Documents

Dynamic Excitations