Risk for fatigue failure - sensitivity analysis Igor Rychlik Chalmers Department of Mathematical Sciences I Motivation, introduction to sensitivity analysis II Example 1. moving vehicle on a rough road, linear responses to Gaussian - LMA loads III Example 2. Blade of a wind turbine, non-linear structure, square Gaussian load, if there is time.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Risk for fatigue failure - sensitivity analysis
Igor Rychlik
Chalmers
Department of Mathematical Sciences
I Motivation, introduction to sensitivity analysis
II Example 1. moving vehicle on a rough road, linear responses toGaussian - LMA loads
III Example 2. Blade of a wind turbine, non-linear structure,square Gaussian load, if there is time.
Presentation is based on:
Aberg S., Podgorski K. and Rychlik I. (2009) Fatigue damage assessmentfor a spectral model of non-Gaussian random loads, Prob. Eng. Mech,Vol 24, pp. 608-617.
Bogsjo, K., and Rychlik, I. (2009) Vehicle fatigue damage caused by roadirregularities. Fatigue and Fracture of Engineering Materials andStructures, Vol. 32, pp. 391-402.
Sarkar, S. Gupta, S. and Rychlik, I. (2010) Wiener Chaos Expansions for
Estimating Rain-flow Fatigue Damage in Randomly Vibrating Structures
with Uncertain Parameters. Mathematical Sciences rep. 2010:07.
If stress variability is stationary and its correlation has short memory thenV(DT ) ≈ σ2 T and E[DT ] = d T hence
R(DT )2 ≈ σ2
d2
1
T,
and is often neglected for new structures with long service time T , whileE[ln(DT )] ≈ ln(d) T . We call d - damage growth rate.
At the design stage d is estimated from mathematical models for externalloads and structures response (systems of linear, non linear diff.equation). Often the parameters (mass, stiffness and damping) may notbe exactly known and hence the damage growth rate d becomesuncertain.
Sensitivity analysis - Taylor expansion:
Corrections to account for the parameter uncertainties are oftenestimated by means of Gauss error propagation formula (accuratefor small parameter variations).
For simplicity assume that damage growth rate d depends only onone normally distributed error, Z ∈ N(0, 1), say.
Then E[d ] and V(d) can be approximated by means of
d(Z ) ≈ d(0) +∂d
∂z(0) Z +
1
2
∂2d
∂z2(0) Z 2.
Fourier expansion using Hermite polynomials1 :
For damage rate d(Z ) such that E[d(Z )2] <∞,
d(Z ) =∞∑j=0
cjHj(Z ) ≈n∑
j=0
cjHj(z) = dn(Z ),
cj = E[d(Z )Hj(Z )]. The truncated polynomial dn converges in L2 to d .The rate of convergence is quite fast for a smooth functions.
1In the one-dimensional case, the normalized Hermite polynomials are
H0(z) = 1, H1(z) = z , H2(z) = (z2 − 1)/√
2, H3(z) = (z3 − 3z)/√
6,
and higher order polynomials can be generated recursively
√n + 1Hn+1(z) = zHn(z)−
√nHn−1(x).
Example I: vehicle on a rough road.
Symbol Value Unitms 3400 kgks 270 000 N/mcs 6000 Ns/mmu 350 kgkt 950000 N/mct 300 Ns/m
The damage increase is of similar magnitude as conservatism of thenarrow band bound. Hence it can be safer to not correct the nb. boundby using some specific properties of the spectrum (and Gaussian model).
Future work:
I Sensitivity of damage rates and extreme responses for nonGaussian loading.
II Investigation how to choose (estimate) kernel f (x) -symmetrical, asymmetrical.
III Work on algorithms to make LMA load modelling accessiblefor engineers.
Example II: Wind load on a blade of a wind turbine - stallflutter.
U
r
V
pitch
2b
elastic axis
α′′ + α/U2 = 2Cm/(πµr 2α) + F sin(kτ),
α(τ), non dimensional time τ = tV /b
U(τ) =1
bωα
√V 2
rot + (Vg + V (t))2,
Cm(τ) is defined by a system of non linearequations.
Some dynamical properties of the blade
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
U
Response
The bifurcation plot U no gusts.
0 5 10 15 20 25 300
1
2
3
4
5
6 x 10 3
U
dam
age
rate
Damage rate no gusts.
Turbine rotating with 25.4 RPM and average head wind Vg below 25m/s, then 16.1 < U < 17.9.
For rotor at rest and the mean gusts 15 < Vg < 45 m/s, 4.8 < U < 14.5.
It seems that the blade parameters are well chosen in respect to minimize
risk for fatigue damages
Damage rate - gusts includedGusts are stationary zero mean Gaussian processes having Devenportspectrum, e.g. SV (ω) depends on Vg .
10 20 30 40 50 60 70 8010 8
10 7
10 6
10 5
10 4
10 3
10 2
Vg
Rotor at rest.
10 20 30 40 50 60 70 8010 4
10 3
10 2
Vg
Rotor working with 25.4 RPM.
Dots: damage rates obtained when gusts fluctuations are neglected;
Crosses: estimated damage rates with variable gusts; Solid line: Hermite
polynomial approximations of 7th, 5th order, respectively.
Acknowledgements:
This presentation uses results obtained by our group working onrandom loads from Chalmers TH, Lund TH and IIT Madras.
The group consists of:
Anastassia Baxevani; Anders Bengtsson (LTH); Klas Bogsjo(Scania); Thomas Galtier (CTH); Sayan Gupta (IITM); WengangMao (CTH); Krzysztof Podgorski (LTH); Igor Rychlik (CTH);Sunetra Sarkar (IITM); Joerg Wegener (LTH); Jonas Wallin (LTH)and Sofia Aberg.
THANK YOU FOR ATTENTION!
Example III: Wind load on antenna mast.
m1 m2m10
c1 c2 c10c3
k1 k2 k10k3
F1(t) F2(t) F10(t)
123
54
76
98
10
Loads: Fi (t) = ci (Vg + V (t))2
Vg - average wind speed,
V (t) - zero-mean wind gusts
V (t) =∫
f (t − x) dΛ(x)
Response:X (t) = c0 + c1
∫h(t − x)V (x) dx
+c2
∫h(t − x)V 2(x) dx .
X (t) = c +
∫q(t − x) dΛ(x) +
∫ ∫Q(t − x , t − y) dΛ(x) dΛ(y),
where q = c1 h ∗ f , Q(x , y) = c2
∫h(t)f (t − x)f (t − y) dt.
Measured wind gusts 200 km away from cyclone, left plot.
Mean speed Vg = 6.5 m/s; St. Dev. 2.52; Skew. 0.28; Exc. kurt. -0.04.
Possible model: asymmetric LMA load
V (t) =
∫f (t − x) dΛ(x), Λ(x) = ζ x + µΓ(x) + σ B(Γ(x)).
(we used κ = 0.01 > 0.) Simulations of 45 minutes of LMA-gusts, right
plot.
0 10 20 30 40−10
−8
−6
−4
−2
0
2
4
6
8
10
win
d gu
sts
spee
d m
/s
45 minutes of measured wind gusts
0 10 20 30 40−10
−8
−6
−4
−2
0
2
4
6
8
10
m/s
45 minutes of sim. wind gusts
Estimated spectrum compared with ”Devenport type” spectrum