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Dynamic response of structures to random exitation
F. Poirion, I. Zentner
To cite this version:
F. Poirion, I. Zentner. Dynamic response of structures to random exitation. 9th International
Con-ference on Structural Dynamics (EURODYN 2014), Jun 2014, PORTO, Portugal. �hal-01078518�
Dynamic response of
structures to random
exitation.
F. Poirion, I. Zentner *
9th International Conference on Structural
Dynamics (EURODYN 2014)
PORTO, PORTUGAL
30 juin-2 juillet 2014
Dynamic response of structures to random exitation.
Analyse de la réponse des structures soumises à des excitations aléatoires.
par
F. Poirion, I. Zentner *
* LaMSID, UMR EDF-CNRS, Clamart
Résumé traduit :
On se propose de caractériser les sorties d'un système dynamique excité par un processus aléatoire quelconque à l'aide de la représentation de Karhunen Loève de la sortie construite à partir d'une base de donnée de l'excitation. Deux exemples académiques permettent d'illustrer la pertinence d'une telle approche.
Dynamic response of structures due to random exitation.
F. Poirion1, I. Zentner2,
1ONERA, 1, av. du G´en´eral Leclerc, Chatillon
2LaMSID, UMR EDF-CNRS, 1 av. du G´en´eral de Gaulle, 92141 Clamart, France
email: irmela.zentner@edf.fr, poirion@onera.fr
ABSTRACT: In the context of random environment, time-domain simulations of structural responses are often necessary when the structure is nonlinear, but not only : when the structure is linear and the excitation modelled as a non-stationnary non-Gaussian process, Monte Carlo approaches are necessary to identify the probabilistic distribution of the output. It is for example the case in seismic analysis. In this paper we use a stochastic model developped earlier by the authors for non-stationnary and non-Gaussian random processes. It is based on an empirical Karhunen-Lo`eve expansion of the process constructed from an available data basis. In a first part we study the particular case of linear structures. The Monte Carlo becomes particularely simple: one needs only to consider the construction of a low number of responses, corresponding to the deterministic eigen functions of the Karhunen-Lo`eve expansion. The KL expansion of the response is then obtained directly. In a second part we study the case of nonlinear structures. We propose here to construct the Karhunen Lo`eve model of the response, using the experimental input data basis for constructing a data basis for the response.
KEY WORDS: Non-Gaussian; Non-linear; Non-stationary; Stochastic process; Dynamical system. 1 INTRODUCTION
Natural hazards are random phenomena that can have an important impact on civil engineering structures such as bridges, dams, power plants, wind turbines etc. Since they are essentially unpredictable in a deterministic sense, it is difficult to account for them in the definition of safety regulations such as for instance the aircraft certification rule for turbulence and gusts. However, these phenomena have to be modelled accurately for evaluating risks related to natural phenomena and their impacts on structural design. Crude modelling of random phenomena and the approximate evaluation of its impact on structures can introduce over conservative security margins.The manufacturer has to bear a real economical extra cost, this is important essentially in the aeronautic industry where the weight is a critical factor for the exploitation cost and for the environmental impact.
High fidelity stochastic models are therefore needed in order to take into account the effects of such unpredictable phenomena on structures and engineering products and structure during their conception stage or to assess their reliability for existing ones. Another important goal is to propose user-friendly models which are easy to build up and not CPU-intensive since they have to be used in general together with Monte Carlo procedures in order to deal with complex nonlinear structures and systems. The construction of such models is not an easy task since the description of a non Gaussian non stationary process requires the knowledge of its uncountable, time varying family of marginal distributions.
In the last years, several papers focusing on the construction of non-Gaussian models from experimental data have been published [2], [5], [4], [9], [8]. The construction of probabilistic models is achieved either by fitting the variability of the
phenomenon to a given analytical probabilistic model (based on physics, Bayesian considerations or information theory) [5], [8] or by constructing empirical models reproducing the observed statistical characteristics, without making any assumptions on the probability distribution [4], [9].
What we propose in this paper is to show that when using a Karhunen-Lo`eve expansion together with a Gaussian kernel estimator construction, it is relatively very simple to construct a stochastic model for any time dependant random phenoma as soon at it can be observed and measured. Using the particular simple form of the Karhunen Lo`eve expansion, we show that it is particularly adapted to the response analysis of high dimension linear codes to random excitation. Finally, in the context where the structure is nonlinear, we show that the model construction can be done for the system response, instead of modeling the excitation, avoiding to solve a high number of nonlinear problems in order to construct statistical estimators of the response.
2 MODEL CONSTRUCTION
Karhunen-Lo`eve expansion provides a suitable framework for modelling a non-stationary scalar random process X(t,ω):
∀t ∈ D, X(t) = limN →+∞ N
∑
α=1 λα ξαφα(t), (1)in which ξ1,ξ2,...,ξα,... are uncorrelated random variables
given by
ξα=√1λ
α
D< X(t),φα(t) > dt. (2) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4
2
where theλαandφα(t) are the solutions of the integral equation
DRX(t,t
)φ(t)dt=λφ(t) (3)
and where the limit in (1) is taken in the space L2(Ω,R).
Let be given N measures of time histories of the studied phenomenon, for instanceN accelerograms of an earthquake: {X(l)(ti) ; i = 1,...,N} ; l = 1,N . The first step to
the K-L expansion construction is to estimate the empirical autocorrelation of X: R(ti,tj) =N1 N
∑
l=1 X(l)(ti)X(l)(tj) (4)Solving the discretized eigenvalue problem (3) yields theλαand
φα(ti), forα,i = 1,N .
The second step is to construct the samplesξα(l)using relation
(2) which gives an explicit relation :
ξα(l)=√1λ α N
∑
i=1 X(l)(ti)φα(ti)Δt ; l = 1,N ;α= 1,N (5)whereΔt is the sampling time step.
In order to be able to use the KL expansion for simulating the random process X, the last step is to identify the distribution of the random vector ξ = (ξ1,...,ξN), which in the general
case, is not Gaussian. This multivariate distribution will be approximated by a kernel density estimator of theξdistribution. Definition 2.1: Let K :Rd→ R+an integrable function such
thatRdK(u)du = 1. K is called a kernel. For each n ∈ N∗, and
each h> 0; the kernel estimator ˆfnof a function f is defined by:
ˆfn,h(x) = ˆfn,h(x1,...,xd) =nh1d n
∑
i=1 K Xi− x h . h is called the window bandwidth of the kernel estimator.Example 1: The Gaussian kernel:
K(x) = 1
(2π)−d/2exp(−
1
2xTx) ; x ∈ Rd
Letξ(), = 1,N , the observed samples of random variableξ.
In the following we shall consider the Gaussian kernel. The density estimator is therefore given by:
ˆfN ,h(x) = 1 N hN(2π)N/2 N
∑
=1 exp (x−ξ())T(x −ξ()) 2h2 (6)The choice of the type of kernel is not crucial in density estimation. The main difficulty being the determination of the window bandwith σ which has to be chosen in order to balance smoothness and accuracy. We will show below that the bandwidth can be chosen following the scalar Silverman’s rule of thumb even if the Gaussian kernels are a priori introduced to construct an estimator for the ξ multi-dimensional density. Details on kernel density estimators construction can be found in the following references [3], [7], [6].
Simulation of a multivariate random variableξ described by the kernel density (6) is straightforward:
• generate a random integer J uniformly distributed on
{1,2,...,N }
• generate a N -dimensional normalized Gaussian random
variableG
• construct the sampleξ=ξ(J)+ h × G .
Remark 2.2: The use of a Gaussian kernel implies implicitely that the distribution ofξ has a C∞density.
In practice, the approximation due to truncation of the KL expansion is evaluated considering the total energy
[0,T]E(X(t)2)dt of the process :
[0,T]RX(t,t)dt = ∑αλα, the
relative errorεbeing then defined by:
ε=
∑
NT α=1λα/ ∞∑
α=1λα= NT∑
α=1λα/ [0,T]RX(t,t)dt (7)Numerically, the total energy can be determined from the empirical autocorrelation function of the N trajectories:
[0,T]RX(t,t)dt = N
∑
i=1 RX(ti,ti)Δti2-1 Probability distribution of the Gaussian kernel model Using a Gaussian kernel approximation for the distribution of random variable ξ = (ξ1,...,ξN) allows to derive a closed
analytical expression for the probability distribution of X(t). Indeed, let pξ(x1,...,xN) the probability density ofξ, then the
characteristic function of X(t) is: ΨX(t)(u) = E[eiu(∑ N α=1 √ λαξαφα(t))] therefore, ΨX(t)(u) = RNe iu(∑N α=1 √ λαxαφα(t))pξ(x1,...,xN)dx1...dxN (8)
Using the kernel approximation: ΨX(t)(u) =N1 N
∑
=1 RNe iu(∑N α=1 √ λαxαφα(t))× ×∏
N α=1g(ξ () α ,h;x)dx1...dxN (9)where g(ξα(),h;x) is the density of the N(ξα(),h) Gaussian
variable. Moreover, RNe iu(∑N k=1 √ λkxkφk(t)) N
∏
k=1 g(ξk(),h;xk)dx1...dxN= E(eiuN(m(t),σh(t))) (10) with m(t) =∑
N α=1 λαxαφα(t) σh(t) = h N∑
α=1λαφ 2 α(t)1/2= h ×σX(t)Hence the distribution of X(t) is: pX(t)(x) =N1
∑
N=1
g(m(t),σh(t);x) (11)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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It appears then that the distribution of X(t) is given as a Gaussian kernel estimator.
Remark 2.3: Since our goal is to model X(t), the value for the bandwidth h is chosen relatively to the scalar problem (11) but not relatively to problem of constructing a kernel estimator for the N-dimensional density of vectorξ(ω). Here we use the value given by the Silverman rule of thumb, which yields:
σh(t) = h ×σX(t)≈ 1.06 × n−1/5×σX(t)
hence
h= 1.06 × n−1/5 2-2 Spline approximation of sampled data
In general experimental databases such as the ones existing in geophysics contain only sampled trajectories {X(ti) ; i =
1,...,N} ; = 1,N . The autocorrelation function estimator is known only for the N2points(t
i,tj) ∈ I × I. We shall denote R the matrix: R= (Ri j) ; Ri j=N − 11 N
∑
k=1 Xk(ti)Xk(tj) (12)It is then natural to interpolate a function through the N points where the trajectory X(t,ω) is known and to construct then the KL expansion of the interpolated trajectory. So let be given N interpolation functions ej(t) such that ej(tj) = 1 and
ej(tk) = 0 if j = k. The interpolation functions can be for
instance polynomial or step functions. The interpolation ˜Xof a trajectory Xis then defined as:
˜X(t) =
N
∑
i=1
X(ti)ei(t) (13)
The above function belongs to a function vector space of dimension N. Let{ fi(t)} a basis of this space, then:
ei(t) = N
∑
j=1φi j
fj(t) (14)
Due to the assumption on the interpolating functions ei(t) it
can be shown that matrix Φ = (φi j) is the inverse of matrix
F= ( fi(tk)). Expressed in this basis, the interpolated process
is written: ˜
X(t,ω) =
∑
Ni, j=1φi j
X(ti,ω) fj(t) (15)
The autocorrelation function of the interpolated process is given by:
˜R(t,s) =
∑
Ni, j=1
γi jfi(t) fj(s) (16)
where(γi j) = ΦTRΦ. It can then be easily checked that the
eigen function u(t) of the operator defined by the autocorrelation function ˜R(t,s) can be written
u(t) =
∑
Ni=1
uifi(t) (17)
where vector U = (u1,...,uN)T is an eigen vector of matrix
ΦTRΦΨ and where Ψ is the scalar product matrix of the basis
functions fi(t):
Ψi j=
Ifi(t) fj(t)dt (18)
Remark 1: Because of the matrixΨ appearing in the above product, one can see that it is not correct to conctruct first the KL expansion on the discretized trajectories and then use interpolated simulated trajectories. There exist some cases where this matrix is the identity matrix: when the basis ( fi)
is orthonormal, or when the discretization step is constant and one use a constant value between two discretization times to interpolate the function.
Besse, in his paper Ref. [1] gives a very general presentation of spline approximation of Hilbertian random variables PCA. He proves furthermore a convergence theorem.I
We shall denote ˜λα, ˜φα(t) and ˜ξα(ω) the principal values,
principal factors and principal factors of the interpolated process ˜
X(t,ω). The empirical model XN will be modeled by the
Karhunen-Lo`eve expansion of the interpolating process ˜X(t,ω): XN(t,ω) =
∑
α≥1
˜
λα ξ˜α(ω) ˜φα(t) (19)
3 RESPONSE OF A DYNAMICAL SYSTEM TO KL MODELS
We show here how the particular expression of the Karhunen Lo`eve expansion of a stochastic excitation can significantly reduce the computation time of the response analysis.
3-1 The linear context
There exist many real-life systems which are described using industrial linear codes: in structures, aeronautics, etc. The use of such linear approximations of real structures and systems are still today mandatory in the conception phase or for optimization because high fidelity models are too CPU intensive. This is particularly true for probabilistic analysis based on Monte Carlo approaches. Even in the case of linear systems, there exist no analytical methods to characterize the system output when the excitation is modelled through a nonGaussion non stationary process. One has to go through numerical simulations in order to construct various statistical estimators.
Consider a linear system described through a differential equation
Y(t) = A(t)Y(t) + X(t) ; Y (0) = Y0; t∈ [0,T]. (20)
It is well known that its solution can be written Y(t) = H(t,0)Y0+
t
0 H(t,s)X(s)ds (21)
where H is the resolvent matrix. Function H is the impulse response function and represents the linear system. It is generally obtained using an industrial code (Nastran, Marc, ...) and its derivation can be computationally expensive for high dimensionnal models such as the ones encountered in aeronautics (106 degrees of freedom). When X is a general
second order stochastic process defined on a given probability
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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space (Ω,T ,P), Monte Carlo approaches have to be used in order to construct statistical estimators of Y , and, even in a linear context, Monte Carlo based probabilistic analysis can become non practical for some applications. When X is approximated through a truncated KL expansion denoting mX(t)
its expectation, relation 21 can be written, Y(t,ω) = H(t,0)Y0+ t 0 H(t,s)mX(s)ds+ N
∑
α=1 λαξα t 0 H(t,s)φα(s)ds, ; t ∈ [0,T ] (22)The output characterization necessitates only N+ 1 calls of the linear codes, whith N generally less than 100 in order to construct the N deterministic response functions ψα(t) =
t
0H(t,s)φα(s)ds and the second term of equation (22). The
probabilistic analysis is then acheived generating random linear combinations of these N functions.
3-2 The nonlinear context
We want to characterize the output of a nonlinear system when its input is modelled as a random process:
Y(t,ω) = f (t,X(t,ω)). (23) In this situation Monte Carlo approaches are generally mandatory and their applicability depends on the computation time needed for solving the nonlinear system (23). As in the linear situation it is numerically much more efficient to construct a KL model of the output in order to construct statistical estimates directly simulating Y(t,ω) than to use an input KL model in order to feed input simulated trajectories in equation 23. Moreover, any error on the input model (due for instance to a low number of samples) could be amplified by the system nonlinearity. We shall give one illustration based on an hysteretic oscillator.
3-2.1 Hysteretic oscillator
We consider the following nonlinear system
¨Y + 2ζω˙Y + (1 −γ)ω2Z= X(t,ω) (24)
˙Z = A ˙Y −α| ˙Y|Z|Z|n−1−β˙Y|Z|n, (25)
with ζ = .02,ω = 1,γ = .5,A = 1,α = .35 and β = −.65. Denoting B(t,ω) the Brownian motion, the input is modelled as :
X(t,ω) = aB2(t,ω) + bB(t,ω) ; a,b ∈ R, (26)
with a = .2,b = 1. Figure (1) shows two trajectories of the input processX. Figure (2) illustrates a typical hysteresis loop. A database of 50 trajectories of X(t,ωj), j = 1 : 50 is
built from which the corresponding 50 responses Y(t,ωj) are
constructed solving equations (24) and (25). Then KL models are constructed for modeling process X and process Y . From the input X KL model we generate 2000 trajectories of the excitation and construct various estimates for the response. In a second stage, the response KL model is used to generate 2000 trajectories of Y(t,ω) from which the same statistics are built. Finally a reference solution is constructed using 10000
10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 Data time (s) X(t, ω )
Fig. 1. Input trajectories
−6 −4 −2 0 2 4 6 8 10 12 −30 −20 −10 0 10 20 30 Y(t) Z(t)
Fig. 2. Hysteretic loop
trajectories of X. In order to illustrate the numerical effects of the number of samples on convergence, the same numerical experiment is performed using 150 trajectories instead of 50. Figure (3) shows the mean trajectory t → (E(Y (t)),E( ˙Y (t))). We now look at the maximum response maxt∈[0,T](Y(t,ω)) and
construct an estimator of its distribution. We plotted on figure (4) this distribution for the reference solution (in red), for the case corresponding to the 50 samples model (green) and for the 150 samples model (blue). Not surprisingly, the model constructed with 150 samples gives a better result than the one constructed with 50. For this last case, the result obtained from the model constructed from the output observations is more precise than the ones obtained from the input model. Now when the models are constructed using 150 samples, the results are similar.
The same numerical experience is repeated when considering a time modulated excitation sin(2πt/T) × X(t,ω), using a database of 150 trajectories. Looking again at the maximum response distribution we see now that both models give solution very similar to the reference one as it can be checked on figure (5). Finally a real world excitation is applied to the nonlinear
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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−0.5
0
0.5
1
1.5
2
−0.1
0
0.1
0.2
0.3
E(Y(t))
E(Y’(t))
Mean trajectory
Fig. 3. Mean response trajectory
0 5 10 15 20 25 0 0.1 MAXt(Y(t,ω)) 0 2 4 6 8 10 0 0.2 0.4 MAXt(Y’(t,ω)) output 150 input 150 reference output 50 input 50
Fig. 4. Maximum distribution
0 5 10 15 20 0 0.1 0.2 Maximum distribution maxt(X(t,ω)) 0 2 4 6 8 10 0 0.2 0.4 maxt(X’(t,ω))
Fig. 5. Maximum distribution
0 5 10 15 20 −0.5 0 0.5 time (s) acceleration (m/s/s)
Fig. 6. seismic input
system: using one hundred accelerograms from a seismic dabase, a Karhunen Lo`eve model is constructed for simulating a seismic excitation, which typical simulated trajectories are illustrated on figure (6). Figure (7) shows once again the comparison of the maximum distributions obtained by each KL model.
4 CONCLUSION
Karhunen Lo`eve models are very efficient for constructing probabilistic models of random phenomena such as the one encountered in the natural hazards domain. But we have shown in this work that it is also possible to use such approaches to build a KL stochastic model for outputs of linear and nonlinear systems when a stochastic excitation is applied to them. In the case of linear systems, the probabilistic solution necessitates the construction of a low number of responses related to the deterministic eigen function appearing in the KL expansion. When the system is nonlinear, the approach is particularly efficient when the cost of solving the nonlinear system renders Monte Carlo approaches unworkable.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
0 2 4 6 8 10 12 14 0 0.2 0.4 Maximum distribution maxt Y(t,ω) 0 10 20 30 40 50 0 0.05 0.1 maxt Y’(t,ω) output KL model input KL model
Fig. 7. Maximum distribution REFERENCES
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