Method to estimate the crossing statistics of a linear structure subjected to quadratic transformations of non-Gaussian loadings.

1. Estimating Crossing Statistics of Second Order
Response of Structures Subjected to LMA
Loadings
Jithin Jith
NA07B031
Under the guidance of
Dr. Sayan Gupta
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

2. Introduction
Motivation:
Accurate modelling of environmental forces like waves, wind,
etc.
Reliability assessment of weakly non-linear systems
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

3. Moving Average
Gaussian MA
X(t) = f (t − s)dB(s). (1)
dB(s) - increments of Brownian motion - Gaussian.
Laplace driven MA
Λ(s) = ζs + µΓ(s) + σB(Γ(s)) (Laplace motion) (2)
X(t) = f (t − s)dΛ(s). (3)
dΓ(s) = Γ(s + ds) − Γ(s) ∼ Γ(ds/ν, 1) (Gamma distribution)
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

4. LMA advantages
S(ω) =
1
2π
σ2
+ µ2
ν
|F[f (t)](ω)|.
All parameters fitted by method of moments.
Advantages
Captures mean, variance, skewness and kurtosis
Conditioned on the Gamma process, Γ(·) = γ(·), becomes
Gaussian
λ(s) = ζs + µγ(s) + σB(γ(s)) (4)
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

5. Gaussian MA vs LMA (skewness=0.9,kurtosis=7.5)
(a) Time history (GMA) (b) Histogram (GMA)
(c) Time history (LMA) (d) Histogram(LMA)
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

6. Second Order Response
Volterra series:
Z(t) = h1(s)X(t − s)ds + h2(s1, s2)X(t − s1)X(t − s2)ds1ds2
= k1(t − s)dΛ(s) + k2(t − s1, t − s2)dΛ(s1)dΛ(s2).
(5)
Eigenvalue equation:
k2(t, s)φ(s)ds = λφ(t). (6)
Kac-Siegert technique:
Z(t) =
∞
i=1
ci Wi (t) + λi Wi (t)2
; Wi (t) = φi (t − s)dΛ(s).
(7)
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

7. Crossing statistics
Mean upcrossing intensity
ν+
(u) =
∞
0
˙zfZ ˙Z (u, ˙z)d˙z. (8)
Intensity of local maxima
µ(u) = −
0
−∞
∞
u
¨zfZ ˙Z ¨Z (z, 0, ˙z)dzd¨z. (9)
Joint pdfs fZ ˙Z (u, ˙z), fZ ˙Z ¨Z (z, 0, ˙z) not easy to determine for
quadratic transformations.
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

8. Determination of joint pdfs
Z(t) =
∞
i=1
ci Wi (t) + λi Wi (t)2
˙Z(t) =
∞
i=1
(ci + 2λi Wi (t)) ˙Wi (t)
¨Z(t) =
∞
i=1
(ci + 2λi Wi (t)) ¨Wi (t) + 2λi
˙Wi (t)2
.
1 Condition Wi (t) on Γ(·) = γ(·).
2 Condition ˙Z on Wi , and ¨Z on Wi & ˙Wi .
3 Transformation of variables.
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

9. Example 1: SDOF System
SDOF system: natural frequency(ω0) = 0.6 rad/s, damping
ratio = 0.05, mass = 1 kg
Loading:
F(t) = α(X(t) + βX(t)2
) (α = ω2
0, β = 1). (10)
X(t): skewness = 0.5, kurtosis = 4.5
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

10. Example 1: SDOF System
(e) Spectrum of X(t) (f) LMA kernel f (t)
(g) Upcrossing intensity (h) Intensity of maxima
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

11. Example 2: Offshore Jacket Platform
120 m water depth
Subjected to small amplitude waves that follow P-M
spectrum, with Hs = 10 m, ωp = 0.5 rad/s
Sηη(ω) =
5
16
H2
s
ω4
p
ω5
exp −
5ω4
p
4ω4
. (11)
Loading:
F(t) = kM
˙X(t) + kD|X(t)|X(t). (12)
X(t): skewness = -0.2, kurtosis = 4.5
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

12. Example 2: Offshore Jacket Platform
Figure: Lumped mass model of the jacket platform
Jithin Jith Crossing Statistics, Second Order Response, LMA loading

13. Example 2: Offshore Jacket Platform
(a) Spectrum of X(t) (b) LMA kernel f (t)
(c) Upcrossing intensity (d) Intensity of maxima
Jithin Jith Crossing Statistics, Second Order Response, LMA loading