Stationary response and first-passage failure of hysteretic systems under random excitations of...

Procedia Engineering 31 (2012) 1200 – 1205

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2012.01.1163

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Procedia

Engineering Procedia Engineering 00 (2011) 000–000

www.elsevier.com/locate/procedia

International Conference on Advances in Computational Modeling and Simulation

Stationary response and first-passage failure of hysteretic systems under random excitations of Poisson white noise and

its filtered processes Yan Zeng, Gang Li*

Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023,China

Abstract

Stationary response and first-passage failure of hysteretic systems, such as bilinear and Bouc-Wen hysteretic models, are investigated under random excitations of Poisson white noise and its filtered processes by Monte Carlo simulation. Results are compared with those of hysteretic systems under random excitations of Gaussian white noise and its filtered processes in the same intensity condition. It is found that stationary probability densities of responses of hysteretic systems subject to Poisson white noise and its filtered processes are close to those subject to Gaussian ones when mean arrival rate of random impulses increases and first-passage failures are more likely to happen when excitations are Poisson white noise and its filtered processes for different values of intensity of random excitation. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: Stationary response; First passage failure; Hysteretic system; Poisson white noise; Filtered Poisson white noise

1. Introduction

In the research of stochastic dynamics of structure engineering, random excitations are often modeled as Gaussian random processes[1, 2] or non-Gaussian ones obtained from nonlinear filtering or transformation of Gaussian random processes[3-5]. Although they are working well in modeling the random excitations with continuous sample functions, there are still many other random excitations with discontinue sample

* Corresponding author. Tel.: +86-411-84707267; fax: +86-411-84707267. E-mail address: [email protected].

1201Yan Zeng and Gang Li / Procedia Engineering 31 (2012) 1200 – 1205Yan Zeng et al./ Procedia Engineering 00 (2011) 000–000

functions[6] which affect the response and reliability of structures. Many approaches have been developed to study the response of linear/nonlinear systems subject to Poisson white noise[7-14], a typical model of random excitation with discontinue sample functions. But to the authors’ knowledge, theoretical result is still hard to obtain for nonlinear system subject to filtered Poisson white noise. Monte Carlo simulation has been successfully used to study the reliability of dynamical system subject to bounded noise, a typical non-Gaussian random excitation with continuous sample functions[15]. Therefore, stationary response and first-passage failure time of hysteretic system subject to Poisson white noise and its filtered processes are studied by Monte Carlo simulation in the present paper. Two classical hysteretic systems, such as bilinear hysteretic model and Bouc-Wen hysteretic model, are investigated in details. Base-band and Narrow-band filtering are used to obtain the filtered Poisson white noise. All these results are compared with those of systems subject to Gaussian white noise and its same filtered processes in the same intensity of random excitation.

2. Poisson white noise and its filtered processes

One versatile model for random pulse train is Poisson white noise ξ(t) which can be considered as the formal derivative of a homogeneous compound Poisson process C(t)[6]:

( ) ( )

1 1

( )( ) ( ) , ( ) ( )N t N t

k k k kk k

dC tt A t t C t A U t tdt

(1)

where δ(▪) is the Dirac delta function, U(t) is the step function and N(t) denotes a Poisson counting process given the number of pulses that arrive in the time interval [0, t] with the mean arrival rate λ > 0; {Ak, k≥1} is a collection of a real-valued identically distributed independent random variables; Ak represents the random magnitude of the impulse, which is independent of pulse arrival time tk. Without loss of generality, the random pulse amplitude Ak is assumed to be Gaussian distributed with zero mean and variance σA

2 in the present paper. Then the intensity of Poisson white noise ξ(t) is D = λσA

2. Filtered Poisson white noise X(t) can be generated by using 1-dimensional (1D) linear system (base-band filtering through a half oscillator) and 2-dimensional (2D) linear system (narrow-band filtering through a single-degree-of-freedom linear oscillator) separately as follows:

0 ( )X X t (2)

20 02 ( )X X X t (3)

where an apostrophe denotes one differentiation with respect to t. A sample function of Poisson white noise defined in Eq. (1) is illustrated with its filtered processes by using Eqs. (2) and (3) in Fig.1 in the parameters: λ = 1.0, D = 4.0, α0 = 1.0, ς0 = 0.01 and ω0

2 = 1.0.

0 5 10 15 20 25

t

-1,000

-500

0

500

1,000

1,500

2,000

t)

0 5 10 15 20 25

t

-5

-4

-3

-2

-1

0

1

2

3

4

5

X(t

)

0 5 10 15 20 25

t

-4

-3

-2

-1

0

1

2

3

4

X(t

)

1202 Yan Zeng and Gang Li / Procedia Engineering 31 (2012) 1200 – 1205Yan Zeng et al./ Procedia Engineering 00 (2011) 000–000

Fig. 1 Sample function of (a) Poisson white noise; (b) 1D filtered Poisson white noise (base-band filtering in Eq. (2)); (c) 2D filtered Poisson white noise (narrow-band filtering in Eq. (3))

3. Model of hysteretic systems

Consider a class of hysteretic systems governed by the equation of motion:

2 ( , ) ( )Y Y g Y Y t (4)

where g(Y,Y’)=αY+(1-α)Z, ζ(t) is a stochastic process which will be taken as Poisson white noise, Gaussian white noise or their filtered processes in what follows and the hysteretic component Z(t) can often be modeled by a first-order differential equation[16], for bilinear hysteretic model, we have

0 sgn 11

Z YZ

Y Z

(5)

and for Bouc-Wen hysteretic model, we have

1nZ aY Y Z Y Z Z (6)

If a sample function of random excitation ζ(t) in Eq. (4) is selected as Poisson white noise or its filtered processes illustrated in Fig. 1, hysteresis loops can be shown in Figs. 2 and 3 for bilinear and Bouc-Wen models separately, comparing with the case of Gaussian white noise and its filtered processes in the same intensity. Obviously, Poisson ones lead to more significant response.

-10 -5 0 5 10 15 20

Y

-1.5

-1

-0.5

0

0.5

1

1.5

2

g(Y

,Y')

Poisson white noise excited

Gaussian white noise excited

-6 -4 -2 0 2 4 6 8 10 12 14

Y

-1.5

-1

-0.5

0

0.5

1

1.5

2

g(Y

,Y')

1D filtered Poisson white noise excited

1D filtered Gaussian white noise excited

-2 -1 0 1 2 3 4 5

Y

-1.5

-1

-0.5

0

0.5

1

1.5

g(Y

,Y')



Fig. 2 Hysteresis loops of bilinear system excited by (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise (base-band filtering in Eq. (2)); (c) 2D filtered Poisson/Gaussian white noise (narrow-band filtering in Eq. (3))

-10 -5 0 5 10 15

Y

-1.5

-1

-0.5

0

0.5

1

1.5

2

g(Y

,Y')



-6 -4 -2 0 2 4 6 8 10 12 14

Y

-1.5

-1

-0.5

0

0.5

1

1.5

2

g(Y

,Y')



-2 -1 0 1 2 3 4 5

Y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

g(Y

,Y')



Fig. 3 Hysteresis loops of Bouc-Wen system excited by (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise (base-band filtering in Eq. (2)); (c) 2D filtered Poisson/Gaussian white noise (narrow-band filtering in Eq. (3))


4. Stationary response and first-passage failure

Using the same parameters of Fig. 1, probability density of stationary response Y and first passage failure time T of bilinear and Bouc-Wen hysteretic systems subject to Poisson/Gaussian white noise and their filtered processes for critical response |Yc| = 1 are investigated by Monte Carlo simulation. It is shown in Figs. 4-7 that stationary responses of hysteretic systems excited by Gaussian ones are more concentrated near zero than Poisson ones and Poisson ones are more dangerous than Gaussian ones especially when the mean arrival rate of random impulses decreases.

-100 -80 -60 -40 -20 0 20 40 60 80

y

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

p(y

)


Poisson whitenoise excited

-100 -80 -60 -40 -20 0 20 40 60 80

y

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045p

(y)

1D filtered Gaussianwhite noise excited

1D filtered Poissonwhite noise excited

-15 -10 -5 0 5 10 15

y

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

p(y

)



Fig. 4 Stationary probability densities of response of bilinear hysteretic system under (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise; (c) 2D filtered Poisson/Gaussian

-100 -80 -60 -40 -20 0 20 40 60 80

y

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

p(y

)

Gaussian whitenoise excited


-100 -80 -60 -40 -20 0 20 40 60 80

y

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

p(y

)



-15 -10 -5 0 5 10 15

y

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2p

(y)



Fig. 5 Stationary probability densities of response of Bouc-Wen hysteretic system under (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise; (c) 2D filtered Poisson/Gaussian

0 1 2 3 4 5 6 7 8 9 10

T

0

0.2

0.4

0.6

0.8

1

1.2

p(T

)

Bilinear modelGaussian white noisePoisson white noisePoisson white noise

0 1 2 3 4 5 6 7 8 9 10

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p(T

)

Bilinear model1D filtered Gaussian white noise1D filtered Poisson white noise1D filtered Poisson white noise

0 2 4 6 8 10 12 14 16 18 20

T

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

p(T

)

Bilinear model2D filtered Gaussian white noise2D filtered Poisson white noise2D filtered Poisson white noise

Fig. 6 Probability densities of first-passage failure time of bilinear hysteretic system under (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise; (c) 2D filtered Poisson/Gaussian

1204 Yan Zeng and Gang Li / Procedia Engineering 31 (2012) 1200 – 1205Yan Zeng et al./ Procedia Engineering 00 (2011) 000–000

0 1 2 3 4 5 6 7 8 9 10

T

0

0.2

0.4

0.6

0.8

1

1.2

p(T

)

Bouc-Wen modelGaussian white noisePoisson white noisePoisson white noise

0 1 2 3 4 5 6 7 8 9 10

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p(T

)

Bouc-Wen model1D filtered Gaussian white noise1D filtered Poisson white noise1D filtered Poisson white noise

0 2 4 6 8 10 12 14 16 18 20

T

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

p(T

)

Bouc-Wen model2D filtered Gaussian white noise2D filtered Poisson white noise2D filtered Poisson white noise

Fig. 7 Probability densities of first-passage failure time of Bouc-Wen hysteretic system under (a) Poisson/Gaussian white noise; (b) 1D filtered Poisson/Gaussian white noise; (c) 2D filtered Poisson/Gaussian

5. Conclusion remarks

The investigation reported in this paper focuses on the different effects of Poisson/Gaussian white noise and their filtered processes to statistics of random response of hysteretic systems. In the same intensity condition, random excitations with discontinuous sample functions, such as Poisson white noise and its filtered processes, induces more significant response than those with continuous sample functions, such as Gaussian ones. It can be concluded that it is necessary to pay more attention to random excitations with discontinue sample functions and more studies should be started around it, especially for the earthquake-resistance.

Acknowledgements

The work reported in this paper is supported by the National Natural Science Foundation of China (Grant Nos. 90815023 and 51021140006) and China Postdoctoral Science Foundation (Grant No. 20110491534).

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