An approximation of the first passage probability of systems under

Appl. Math. Mech. -Engl. Ed. 30(2), 255–262 (2009) DOI: 10.1007/s10483-009-0213-y c Shanghai University and Springer-Verlag 2009

Applied Mathematics and Mechanics (English Edition)

An approximation of the first passage probability of systems under nonstationary random excitation ∗

)

Jun HE (

(Department of Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, P. R. China) (Communicated by Li-qun CHEN)

Abstract An approximate method is presented for obtaining analytical solutions for the conditional first passage probability of systems under modulated white noise excitation. As the method is based on VanMarcke’s approximation, with normalization of the response introduced, the expected decay rates can be evaluated from the second-moment statistics instead of the correlation functions or spectrum density functions of the response of considered structures. Explicit solutions to the second-moment statistics of the response are given. Accuracy, efficiency and usage of the proposed method are demonstrated by the first passage analysis of single-degree-of-freedom (SDOF) linear systems under two special types of modulated white noise excitations. Key words first passage probability, D-type barrier, decay rate, nonstationary excitation, envelope process Chinese Library Classification O29 2000 Mathematics Subject Classification

60K10

Introduction In many engineering applications, it is important to determine the first passage probability of systems under random excitations: P (t) = P (t, b)fB (b) db, (1) B

where fB is the joint probability density function of barrier vector B, and P (t, b) is the conditional first passage probability given B = b, in which b is the given value of B. To evaluate (1), P (t, b) needs to be calculated repeatedly. It contributes significantly to P (t)[1-3] . Therefore, it is important that the calculation of P (t, b) is simple and accurate enough. Various approximate techniques have been proposed, which are divided into three categories, i.e., out-crossing approaches, the Monte Carlo simulation methods, and diffusion methods. It has been stated that either plain simulation or importance sampling would make any computational effort unfeasible[4-5] . Diffusion methods were proposed 30 years ago[6], in which Fokker-Planck-Kolmogorov equation (FPKE) governing the reliability function ∗ Received May 6, 2008/Revised Dec. 1, 2008 Project supported by the National Natural Science Foundation of China (No. 50478017) Corresponding author Jun HE, Associate Professor, Ph. D., E-mail: [email protected]

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Jun HE

should be solved. However, most numerical solutions of the FPKE belong to low-dimensional problems[7-8] . The simplest approximation belongs to out-crossing approaches, in which out-crossings of the considered processes are assumed to be independent and constitute a Poisson process[9]. The Poisson-type assumption has been proved to be inappropriate for narrow band processes. VanMarcke proposed a commonly used improved formula, accounting for the dependence between the crossing events and the time that the process spends above the barrier[10]. Besides, Lutes’ approximation[11], the integral equation method[12] , Langley’s approximation[13], and the bounds method[14] have also been proposed. Out-crossings approaches have been also extended to non-linear systems[15] , non-Gaussian systems[16] and multi-component systems[17] . In general, current out-crossing approaches focus on systems under stationary excitation. However, in practical engineering, excitations are often nonstationary, such as an earthquake ground motion and a wind force. The objective of the present study is to develop an outcrossing approach for linear systems under nonstationary random excitation. Numerical examples demonstrate the accuracy and effectiveness of the approach.

1

Formulation of the problem Consider a linear oscillator with the motion equation, ˙ ¨ + 2ζωn X(t) + ωn2 X(t) = F (t). X(t)

(2)

Here, ωn and ζ represent, respectively, the un-damped natural frequency and the damping ratio. F (t) = A(t)W (t) is the nonstationary random excitation, in which A(t) is a modulating function and W (t) is a white noise with the spectrum density S0 . The second-moment statistics of the system response are t 2 σX (t) = 2πS0 h2 (t − τ )A2 (τ ) dτ, (3) 0 t 2 (t) = 2πS (4) h˙ 2 (t − τ )A2 (τ ) dτ, σX 0 ˙ 0 t ˙ − τ )A2 (τ ) dτ, h(t − τ )h(t (5) cX X˙ (t) = 2πS0 0

t cX Xˆ˙ (t) = 2S0

0

0

t

˙ − τ2 )A(τ1 )A(τ2 ) h(t − τ1 )h(t dτ1 dτ2 , τ2 − τ1

(6)

˙ respectively. cX X˙ (t) is where σX (t) and σX˙ (t) are the standard deviations of X(t) and X(t), ˙ the covariance of X(t) and X(t). cX Xˆ˙ (t) is the covariance of X(t) and its Hilbert transform: ∞ 1 t W (s) ˆ X(t) = dsdτ, (7) h(t − τ )A(τ ) π 0 −∞ τ − s where h(t) = e−ζωn t sin(ωd t)/ωd is the impulse response function, and the damped frequency ωd = 1 − ζ 2 ωn . ˆ˙ The derivative X(t) then becomes ∞ 1 t˙ W (s) ˙ ˆ X(t) = dsdτ. (8) h(t − τ )A(τ ) π 0 −∞ τ − s Substituting the impulse response function into Eq. (6) and simplifying gives[18-19] 2S0 t −ζωn s sin(ωd s) t −2ζωn (t−τ ) cX Xˆ˙ (t) = e e A(τ − s)A(τ ) dτ ds. ωd 0 s s

(9)

Approximation of the first passage probability of systems

257

For a scalar quantity b, empirical evidence suggests that P (t, b) can be written as P (t, b) = 1 − e−

Rt 0

α(τ ) dτ

,

(10)

where α(t) is called the expected decay rate. If one replaces α(t) over the time increment Δt by its value at the midpoint of that interval, then, for tn , one obtains 1

P (tn , b) = 1 − [1 − P (tn−1 , b)] e−α[(n− 2 )Δt]Δt with the initial condition P (0, b) = 0. From VanMarcke’s formula, the expected decay rate can be written as + νR (t,b) 1 − exp − νX (t,b) α(t) = νX (t, b) , + νX (t,b) 1 − ν + (t,0)

(11)

(12)

X

+ − (t, b)+νX (t, b) denotes the expected rate of the response beyond the domain. where νX (t, b) = νX The expected rates of the up- and down-crossings are[20] ∞ + ˙ νX (t, b) = (x˙ − b)f ˙ dx, ˙ (13) ˙ (t, b; t, x) XX b˙

and − νX (t, −b)

b˙

= −∞

(b˙ − x)f ˙ X X˙ (t, −b; t, x) ˙ dx, ˙

(14)

˙ where fX X˙ (t, x; t, x) ˙ is the joint probability density function (JPDF) of X(t) and X(t). + In Eq. (12), νR (t, b) denotes the expected up-crossing rate of the envelope process R(t), which can be defined as[21] ˆ 2 (t). R(t) = X(t) + X (15) + ˙ is available, one can calculate νR (t, b) from Eq. (13) after replacing If the JPDF fRR˙ (t, r; t, r) x˙ and fX X˙ (t, x; t, x) ˙ with r˙ and fRR˙ (t, r; t, r), ˙ respectively.

2

Evaluation of crossing rates It is convenient to compute the above crossing rates in terms of the normalized variables, X(t) , σX (t) R(t) Q(t) = Y 2 (t) + Yˆ 2 (t) = , σX (t)

Y (t) =

and η(t) =

b , σX (t)

where Yˆ (t) is the Hilbert transform of Y (t). Because Y (t) and Y˙ (t) are uncorrelated, Eqs. (13) and (14) can be rewritten as η˙ η˙ η˙ + νX (t, b) = νY+ (t, η) = ω0 φ(η) φ( ) − Φ(− ) , ω0 ω0 ω0 η ˙ η ˙ η ˙ − (t, b) = νY− (t, η) = ω0 φ(η) φ( ) + Φ( ) , νX ω0 ω0 ω0

(16) (17)

(18)

(19) (20)

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where φ(·) and Φ(·) are, respectively, the standard normal density function and the standard normal distribution function, and ω0 is the standard deviation of Y˙ (t): ω02 = E Y˙ 2 (t) ⎡ 2 ⎤ ˙ σX (t)X(t) − X(t)σ˙ X (t) ⎦ =E⎣ 2 (t) σX =

2 σX ˙ (t)

−2

2 (t) σX

2 σ˙ X σ˙ X (t) ˙ (t) c . (t) + ˙ XX 3 2 σX (t) σX (t)

(21)

˙ For a normal process, Q(t) and Q(t) are mutually independent with the Rayleigh and normal [22] distribution, respectively , q2 ), 2 1

fQ (q) = q exp(−

(22)

1 q˙2 fQ˙ (q) ˙ = ), exp(− 2 ω02 − λ2 2π(ω02 − λ2 )

(23)

˙ ˆ ˆ where λ(t) is the covariance of Y (t) and Yˆ (t). Because X(t) and X(t) are uncorrelated, X(t) ˙ˆ and X(t) are also uncorrelated. One can obtain easily ˙ λ(t) = E Y (t)Yˆ (t) = E

ˆ˙ ˆ σ˙ X (t) c ˆ˙ (t) X(t) σX (t)X(t) − X(t) = X2X . 2 σX (t) σX (t) σX (t)

(24)

Then the up-crossing rate for the envelope is + + νR (t, b) = νQ (t, η) = fQ (η) η˙

∞

(s − η)f ˙ Q˙ (s) ds

2 = ω02 − λ2 ηe−η /2 φ

η˙

ω02 − λ2

η˙

η˙

Φ − 2 − 2 ω0 − λ2 ω0 − λ2

.

(25)

2 2 (t), σX If σX ˙ (t), cX X˙ (t), and cX X ˆ˙ (t) are available, then the crossing rates of X(t) and R(t) can be evaluated directly.

3

Evaluation of second-moment statistics The second-moment statistics can be written as[18] πS0 (I3 − I1 ) , ωd2 πS0 2 2 σX ωd − ζ 2 ωn2 I1 − 2ζωn ωd I2 + ωn2 I3 , ˙ (t) = 2 ωd πS0 cX X˙ (t) = 2 (ζωn I1 + ωd I2 − ζωn I3 ) , ωd

2 σX (t) =

(26) (27) (28)


where

t

I1 =

259

e−2ζωn (t−τ ) cos [2ωn (t − τ )] A2 (τ ) dτ,

(29)

e−2ζωn (t−τ ) sin [2ωn (t − τ )] A2 (τ ) dτ,

(30)

e−2ζωn (t−τ ) A2 (τ ) dτ.

(31)

0

t

I2 = 0

t

I3 = 0

For the special case of the results are I1 = C

2

e

−2Bt

I2 = C

−e

2

I3 = C

e

n+1

[Dk cos(kθ)] −

k=1 n+1 −2Bt

2

A(t) = Ctm e−Bt ,

−2Bt

(−1)n n! e−2ζωn t n+1 χ

[Dk sin(kθ)] −

k=1 n+1

[Ek (t)] −

k=1

(32) cos [2ωd t − (n + 1)θ] ,

(−1)n n! e−2ζωn t n+1 χ

(−1)n n! e−2ζωn t n+1 γ

(33)

sin [2ωd t − (n + 1)θ] ,

(34)

,

(35)

where B and C are the real-valued and positive “model” constants, 2m is a positive integer, n = 2m, and γ = 2ζωn − 2B, χ =| γ − 2iωd |,

(36) (37)

θ = arg(γ − 2iωd ),

(38)

k−1

(−1) n! tn−k+1 , k = 1, 2, · · · , n + 1, χk (n − k + 1)! k χ Ek (t) = Dk (t), k = 1, 2, · · · , n + 1. γ

Dk (t) =

(39) (40)

The approximate solutions for cX Xˆ˙ (t) will be developed in the following examples.

4

Examples

4.1 Example 1 It is assumed that A(t) = 1 for t ≥ 0, then Eqs. (33)–(40) yield ωd ζ cos(2ωd t) + sin(2ω t) + I1 = e−2ζωn t − d 2ωn 2ωn2 ωd ζ I2 = e−2ζωn t − sin(2ωd t) − cos(2ω t) + d 2ωn 2ωn2 1 1 − e−2ζωn t . I3 = 2ζωn

ζ , 2ωn ζ , 2ωn2

(41) (42) (43)

2 2 Substituting Eqs. (41)–(43) into Eqs. (26)–(28) leads to σX (t), σX ˙ (t). A(t) = 1 ˙ (t), and cX X [19] is substituted into Eq. (9), then cX Xˆ˙ (t) can be expressed as 2ζ 1 + e−2ζωn t , (44) cX Xˆ˙ (t) ≈ σS2 ωn 1 − e−2ζωn t − π

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Jun HE

where σS2 = S0 π/2ζωn3 denotes the variance of the limiting stationary response. Figures 1 and 2 show the analytical solutions of α(t) compared with Lutes’ results. Figures 3 and 4 show the analytical solutions of P (t, b) compared with Lutes’ results and the simulation results. The present method has good agreements with Lutes’ approximation and simulation. 1.2

1.0

b=σs

P(t, b)

0.8

20

N

30

0.0 0

50

Fig. 2

1.0

b=σs

Simulation with 10 000 simples Present method Lutes’ approximation b=3σs 10

20

N

30

40

b=3σs 10

20

N

50

The conditional first passage probability of a linear oscillator with ζ = 0.05

30

40

50

Expected decay rate as functions of numbers of N = ωn t/π with ζ = 0.1

b=σs

b=2σs

0.8

b=2σs

0.2

Fig. 3

40

Expected decay rate as functions of numbers of N = ωn t/π with ζ = 0.05

0.4

b=2σs

0.2

b=3σs 10

0.6

0.0 0

Present method Lutes’ approximation

0.6 0.4

b=2σs

0.2

1.0

α(t)

Present method Lutes’ approximation

0.4

Fig. 1

b=σs

0.8

0.6

0.0 0

1.0

P(t, b)

α(t)

0.8

0.4

Simulation with 10 000 simples Present method Lutes’ approximation

0.2

b=3σs

0.6

0.0 0

Fig. 4

10

20

N

30

40

50

The conditional first passage probability of a linear oscillator with ζ = 0.1

4.2 Example 2 In this example, the modulating function is considered to be A(t) = Cte−Bt , which was used in models of nonstationary earthquake ground motion. The three basic integrals can be evaluated as follows: 2 2t cos(2θ) 2 cos(3θ) 2e−2ζωn t cos(3θ − 2ωd t) 2 −2Bt t cos(θ) I1 = C e − + − , χ χ2 χ3 χ3 2 t sin(θ) 2t sin(2θ) 2 sin(3θ) 2e−2ζωn t sin(3θ − 2ωd t) − + + , I2 = C 2 −e−2Bt χ χ2 χ3 χ3 2 t 2t 2 2e−2ζωn t − 2+ 3 − , I3 = C 2 e−2Bt γ γ γ γ3

(45)

(46) (47) (48)


261

2 2 from which σX (t), σX ˙ (t) can be evaluated, respectively. ˙ (t) and cX X The approximations for cX Xˆ˙ (t) was obtained as[19] 2 cX Xˆ˙ (t) ≈ ωn σX −

2ζωn 2 2 2 −2Bt 2e−2ζωn t σS C t e + . π 3ωn2

(49)

It is assumed that B = 0.15π, then the constant C is chosen to be C = Be = 1.281 so that the maximum of A(t) is unity at the time t = 1/B = 2.122 s. The analytical and simulation results are shown in Figs. 5 and 6. The present method has good agreements with simulations. Both results asymptotically converge to the constant values. At large N , the absolute errors between the present method and the simulation are less than 5 × 10−2 , which are allowed in engineering applications. 1.0

0.8

b=σs

Simulation with 10 000 simples Present method

0.6 0.4

b=2σs

0.2 0.0 0

Fig. 5

5

P(t, b)

P(t, b)

0.8

1.0

8

N

12

Simulation with 10 000 simples Present method

0.6

b=2σs

0.4 0.2

b=3σs 4

b=σs

16

20

Conditional first passage probability of a linear oscillator with ζ = 0.05

0.0 0

Fig. 6

b=3σs 4

8

N

12

16

20

Conditional first passage probability of a linear oscillator with ζ = 0.1

Conclusions

An approximate method for the first passage probability of systems under nonstationary random excitation has been developed. The expected decay rates in the approximation are evaluated by VanMarcke’s formula. However, instead of the spectrum density functions, variance of the response and its derivative, covariance of the response and its time derivative, and covariance of the response and its Hilbert transformation are used to evaluate the expected decay rates. Explicit solutions for these second-moment statistics can be obtained by the foregoing efforts. Therefore, the method can be applied on systems under nonstationary random excitation, in which the spectrum density function of the response is unavailable. Two different modulating functions are considered, respectively, in two examples. Numerical results are compared with those of Lutes’ approximation or simulation. Numerical examples show a high degree of the accuracy and effectiveness of the method. Acknowledgements

The writer wishes to thank the referees of this paper for their critical com-

ments and suggestions.

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