Dynamic excitationsof transport structures
Prof. Ing. Ladislav Frýba, DrSc., Dr.h.c.
UNC 2008
Institute of Theoretical and Applied Mechanics, v.v.i.Academy of Sciences of the Czech Republic, Prague
• Introduction• Movement of loads• Rails• Random vibration of structures• Degradation of structural materials under
dynamic loads• Conclusions
Outline
Introduction
Movement of the load
Transport structures – moving load– earthquake– wind
Moving load
[ ] [ ]2
2
( ),( , ) ( )
d v u t tf x t x u t F m
dtδ
⎧ ⎫⎪ ⎪= − −⎨ ⎬⎪ ⎪⎩ ⎭
m
v(x,t)
x u t= ( )
[ ] 22 2 2 2 2
2 2 2 2
( ),2
d v u t t v du v du v d u vdt u dt u t dt u dt t
∂ ∂ ∂ ∂⎛ ⎞= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
Uniform movement at velocity c2
2( ) , 0du d ux u t ct cdt dt
= = = =
2 2 2 22
2 2 2
( , ) ( , ) ( , ) ( , )2x ct
d v ct t v x t v x t v x tc 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 2d w w du w dv w w du dvdt x dt y dt t x y dt dt
⎡∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + + + +⎢ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢⎣2 2 2 2
2 2( )( )
2 2x u ty v t
w du w dv w d u w d vx t dt y t dt x dt y dt =
=
⎤∂ ∂ ∂ ∂+ + + + ⎥∂ ∂ ∂ ∂ ∂ ∂ ⎦
Rails
Diferencial equation
0( , ) ( ),v x t v v s=
( , ) ( , ) 2 ( , ) ( , ) ( )IVbEIv x t v x t v x t kv x t x ct Pμ μω δ+ + + = −&& &
Steady state vibration
0 38 2P PvEI k
λλ
= =
1/ 4
4kEI
λ ⎛ ⎞= ⎜ ⎟⎝ ⎠
P
v(x,t)
x t= c
-x +xk
s
0
Moving coordinate:
Speed parameter:
Damping parameter:
Critical speed:
( )s x ctλ= −1/ 2
2cr
c cc EI
μαλ
⎛ ⎞= = ⎜ ⎟⎝ ⎠
1/ 2( / )b kβ ω μ=1/ 2
2crEIc λμ
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Ordinary differential equation:2( ) 4 ( ) 8 ( ) 4 ( ) 8 ( )IV II Iv s v s v s v s sα αβ δ+ − + =
Solution
1 1 2 12 21 1 2
2( ) ( cos sin )( )
bsv s e D a s D a sa D D
−= ++
3 2 4 22 22 3 4
2( ) ( cos sin )( )
bsv s e D a s D a sa D D
= ++
for s > 0
for s < 0
Random vibration of a beam on elestic foundation under moving force
Equation of the beam
[ ]( , ) ( , ) ( , ) 2 ( , ) ( ) ( , )IVbL v x t EIv x t v x t v x t k x v x tμ μω≡ + + + =&& &
( ) ( )x ct F tδ= −
Foundation ( ) ( )k x k k xε= +o
[ ]( )k E k x=
Force 1ε( ) ( )F t F F tε= +
o
Moving coordinate
Static deflection and bending moment underforce F
[ ]( )F E F t= 1/ 4
( ) ,4ks x ctEI
λ λ ⎛ ⎞= − = ⎜ ⎟⎝ ⎠
0 03 ,8 2 4
F F Fv MEI k
λλ λ
= = =
Coeficient of variation VV and VM as a function of speed α,β = 0.2, γi = 0, σ = 0.2, D = 10/λ
COVARIANCEPOWER SPECTRAL DENSITY
Random vibration of a beamBernoulli-Euler equation
Normal mode analysis
Generalized deflection
Generalized force
( , ) ( , ) 2 ( , ) ( , )IVbEI v x t v x t v x t p x tμ μω+ + =&& &
1( , ) ( ) ( )j j
jv x t v x q t
∞
=
= ∑
1( , ) ( ) ( )j j
jp x t v x Q tμ
∞
=
= ∑2( ) 2 ( ) ( ) ( )j b j j j jq t q t q t Q tω ω+ + =&& &
0
1 ( , ) ( ) 0( ) for
00
L
jjj
p x t v x dxMQ t t
⎧≥⎧⎪= ⎨ ⎨<⎩⎪
⎩
∫
Solution
( ) ( ) ( ) ( ) ( )j j j j jq t h t Q d h Q t dτ τ τ τ τ τ∞ ∞
−∞ −∞
= − = −∫ ∫
Impulse function (weighting function)1( ) ( )
2i
j jh H e dτωτ ω ωπ
∞
−∞
= ∫
Frequency response function
( ) ( ) ij jH h e dωτω τ τ
∞−
−∞
= ∫Load
[ ]( , ) ( , ) ( , )p x t E p x t p x t= +o
Stationary vibration of a beam undermoving continuous random load
Load ( , ) ( ) 1 ( )p x t p p s r tε ε⎡ ⎤ ⎡ ⎤= + ⋅ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
o o
[ ]( , )p E p x t=2( , ) ( ) ( ) ( ) ( )p x t p s p r t p s r tε ε ε= + +
o o o o o
Covariance of the moving load
/s t x c= −1ε
2 2 2 21 2 0 1( , , ) ( ) ( ) ( / )pp pp rr prC x x C p C pC x cτ ε τ τ ε τ ε τ= + + + + +
2 3 3 3 3 42( / )rp ppr prr ppr prr pprrpC x c C pC C pC Cε τ ε ε ε ε ε+ − + + + + +
1 20
x xc
τ −=
Non-stationary vibration of a beamunder moving random force
Load( , ) ( ) ( )p x t x ct P tδ= − ( ) ( )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 tδ δ= − −
, ,
1 2 1 1 2 2 1 2 1 2 1 21( , ) ( ) ( ) ( ) ( ) ( , )
j kq q j k j k ppj k
C t t h t h t v c v c C d dM M
τ τ τ τ τ τ τ τ∞ ∞
−∞ −∞
= − −∫ ∫
Speed parameterDamping parameter
Simple beam
1/(2 )c f lα =
1/ /(2 )bβ ω ω ϑ π= =3
0( ) sin ,48j
j x Plv x vl EIπ
= =
Degradation of structural materialsunder dynamic loads
Dynamic loads
Fatigue tests
Wöhler curve
Concept of fatigue assessment
Static system
Material properties(Wöhler curve)
“Real” LoadsP (t)
Stress curveσ (t)
Spectrumni (σo, σn)
Calculation valuefor damage DSd
ProofDSd ≤ Dlim
Simulation oftrain passages
Counting procedure(Rainflow)
Damage hypothesis(Palmgren-Miner)Δσ = σmax - σmin
ρ = σmin / σmax
Strategy of maintenance, repairs and reconstructions,
yes – the conditionsare fulfilled,
not – the conditionsare not fulfilled
Effect of traffic loads
Stress-time recordsstatistical evaluation
estimation of residual life
yes
yes
yes
yes
yes
Modal analysis
criteria of dereriorationnot
not
not
not
Monitoring of cracks
criteria of cracks
Inspections
Technical economic decision
Maintenance
Repair
Reconstruction
Conclusions
The dynamic loads, i.e. in time varying loads, increase the stresses in transport structureswith increasing speed
The dynamic loads are varying in time eitherregularly or randomly
The response of structural materialsdeteriorates their properties in time