EPC procedure for PDF solution of nonlinear. oscillators excited by Poisson white noise

Transcription

1 * Manuscript Click here to view linked References EPC procedure for PDF solution of nonlinear oscillators excited by Poisson white noise H.T. Zhu,G.K.Er,V.P.Iu,K.P.Kou Department of Civil and Environmental Engineering, University of Macau, Macao SAR, P.R. China Abstract The stationary probability density function (PDF) solution of the responses of nonlinear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, nonlinear oscillators are analyzed with different impulse arrival rates, degree of oscillator nonlinearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being. Key words: Stochastic nonlinear oscillator; Poisson white noise; Probability density function; Exponential-polynomial function Corresponding author. Tel.: ; Fax: address: ya5741@umac.mo (H.T. Zhu). Preprint submitted to Elsevier 4 November 8

2 1 Introduction Poisson white noise is widely applied in many engineering fields as a typical discrete non-gaussian process. The process represents a train of random impulses arriving at random time instants. It can be employed to model earthquake ground motion, traffic load, shock wave and wind, etc. Under such excitation, the evaluation on statistical moments or probability density function (PDF) of oscillator responses is a critical issue because it provides necessary information about system behavior in reliability analysis. Although extensive efforts have been devoted to this subject about Poisson white noise [1], it is difficult to obtain the PDF solutions of nonlinear stochastic oscillators exactly or approximately. Some researches have been reported on this issue [ 6]. If the statistical moments of responses are concerned, the most widely employed approximation method is equivalent linearization (EQL) procedure [7,8]. It was initially proposed for analyzing nonlinear stochastic dynamic systems in the case of Gaussian white noise excitation [9,1]. Subsequently, it was extended and examined for Poisson white noise [11 15]. In the case of low impulse arrival rate or high system nonlinearity, the error in the PDF obtained with the extended EQL procedure can be big. Alternatively, cumulant-neglect closure procedure was also developed [16 18] to seek the response moments under Poisson white noise. In order to evaluate non-gaussian behavior, either fourth order or sixth order cumulant-neglect closure needs to be adopted to obtain the approximate moments. This method is inappropriate when an oscillator is highly nonlinear or the impulse arrival rate of the Poisson white noise is low. If the PDF of system responses is concerned, the generalized Fokker-Planck- Kolmogorov (FPK) equation or Kolmogorov-Feller (KF) equation governing

3 the PDF solution of a stochastic oscillator needs to be solved [,3,19,]. Based on the KF equation, some exact stationary solutions were obtained for a highly restricted class of systems [4 6]. Numerical or approximation methods were therefore developed for the PDF solutions of more general oscillators. Perturbation method was outlined to solve the truncated KF equation [] and further improved [3]. With the perturbation method, a suitable initial solution and a small perturbation parameter need to be preliminarily determined. Petrov-Galerkin method which behaves well in the case of high mean arrival rate of Poisson white noise was also proposed for solving the KF equation [1,4]. Cell-to-cell mapping technique which is effective in the case of low mean arrival rate of Poisson white noise is another numerical procedure for obtaining the PDF of system responses [ 4]. Finite difference method was also utilized for analyzing the nonlinear oscillators with the external excitation of Poisson white noise [5,6]. With the finite difference method, negative PDF value in the tail regions can be possibly obtained. As described above, the accuracy of the obtained PDFs from various techniques relies heavily on the degree of system nonlinearity or the magnitude of the impulse arrival rate of Poisson excitation. In this paper, the exponentialpolynomial closure (EPC) method is employed and extended to analyze the PDF solutions of the nonlinear stochastic oscillators with the external excitation being Poisson white noise. The EPC method was initially proposed for obtaining the PDF solutions of the nonlinear stochastic oscillators with Gaussian excitation [7 3]. In order to assess the effectiveness of the EPC method in the case of Poisson excitation, nonlinear oscillators are numerically analyzed. The influence of the impulse arrival rate of Poisson excitation, the degree of system nonlinearity and excitation intensity on the obtained PDF 3

4 solutions is investigated, respectively. Compared with the results from Monte Carlo simulation, good agreement is observed from the results obtained with the EPC method. Problem formulation Consider the following nonlinear stochastic oscillator: Ẍ + h (X, Ẋ) =W (t), (1) where X Rand Ẋ Rare stochastic processes; R denotes real space; h is a function of X and Ẋ; W (t) is the Poisson white noise formulated as W (t) = N(T ) k=1 Y k δ(t τ k ), () where N(T ) is the total number of pulses that arrive in the time interval (,T], Y k is the impulse amplitude of the kth pulse arriving at time τ k,and δ(t) is Dirac delta function. In this paper, N(T ) is assumed to be a Poisson process with constant impulse arrival rate λ. The impulse amplitudes Y k are independent and identically distributed (i.i.d.) random variables with zero mean, and also independent of the pulse arrival time τ k. When the arrival rate λ is constant and the impulse amplitudes Y k are i.i.d., the process becomes stationary [31]. Setting X = x 1 and Ẋ = x, Eq. (1) can be expressed as: ẋ 1 = x, (3) ẋ = h (x 1,x )+W (t). (4) 4

5 The response vector {x 1,x } T is Markovian and the PDFs of the oscillator responses are governed by the following KF equation in series form or generalized FPK equation [3]: p t = x p + (h p)+ 1 x 1 x! λe[y ] p x 1 3! λe[y 3 ] 3 p + 1 x 3 4! λe[y 4 ] 4 p +..., (5) x 4 where E[ ] denotes the expectation of ( ). Furthermore, if only the stationary PDF solution is considered, the term on the left side of Eq. (5) vanishes and the KF equation is reduced to be p x + (h p)+ 1 x 1 x! λe[y ] p x 1 3! λe[y 3 ] 3 p + 1 x 3 4! λe[y 4 ] 4 p +...=. (6) x 4 It is assumed that the PDF p(x 1,x ) of the stationary responses of the random oscillator is subjected to the following conditions: p(x 1,x ), x 1,x R, lim x i ± p(x 1,x )=, i =1,, + + p(x 1,x )dx 1 dx =1. (7) These requirements should be fulfilled by the PDF solution of the oscillator expressed by Eq. (1). Generally the exact solution of Eq. (6) is not obtainable. Therefore, an approximate PDF subjected to the conditions (7) needs to be formulated. Here, the approximate PDF is expressed as an exponentialpolynomial function of state variables. The approximate PDF solution p (x 1,x ; a) 5

6 to Eq. (6) is assumed to be p (x 1,x ; a) =ce Qn(x 1,x ;a), (8) where c is a normalization constant and a is an unknown parameter vector containing N p entries. The polynomial Q n (x 1,x ; a) is expressed as Q n (x 1,x ; a) = n i i=1 j= a ij x i j 1 x j, (9) which is an n degree polynomial in x 1 and x. To fulfill the conditions (7), it is required that Q n (x 1,x ; a) =, x 1,x Ω, (1) where Ω = [m 1 c 1 σ 1,m 1 + d 1 σ 1 ] [m c σ,m + d σ ] R in which m i, σ i denote mean values and standard deviations of x i (i =1, ), respectively. c i > andd i > are constants; The values of c i and d i can be selected such that m i c i σ i and m i + d i σ i locate in the tail regions of the PDF of x i.this means that the approximate PDF is assumed to be valid only in Ω and vanish beyond Ω. Generally, the KF equation (6) cannot be satisfied exactly with p (x 1,x ; a) because p (x 1,x ; a) is only an approximation of p(x 1,x ) and the number of the unknown parameters N p is always limited in practice. Substituting p (x 1,x ; a) forp(x 1,x ) leads to the following residual error: p Δ(x 1,x ; a)= x + (h p )+ 1 x 1 x! λe[y ] p 1 3! λe[y 3 ] 3 p x ! λe[y 4 ] 4 p x 4 x +... (11) 6

7 Because there are an infinite number of higher order derivative terms in Eq. (11), it is difficult to obtain the solution if it is possible. Here only the terms up to fourth order derivative are retained for analysis in view that the contribution of high order terms is small to the whole equation. Hence, p Δ(x 1,x ; a)= x + (h p )+ 1 x 1 x! λe[y ] p 1 3! λe[y 3 ] 3 p x ! λe[y 4 ] 4 p x 4 x. (1) Substituting Eq. (8) into Eq. (1) and noting E[Y 3 ]=fory k being Gaussian with zero mean, it gives Δ(x 1,x ; a) =F (x 1,x ; a) p (x 1,x ; a), (13) where Q n Q n F (x 1,x ; a)= x + h + 1 ( ) x 1 x! λe[y ] Q n Qn + x x + 1 4! λe[y 4 ] 4 Q n +4 Q n 3 ( Q n ) Q n +3 x 4 x x 3 x ( ) Qn ( ) 4 Q n Qn h. (14) x x x x Because p (x 1,x ; a), hence the only possibility for p (x 1,x ; a) tofulfill Eq. (6) is F (x 1,x ; a) =. However, usually F (x 1,x ; a) because p (x 1,x ; a) is only an approximation of p(x 1,x ). In this case, another set of mutually independent functions H s (x 1,x ) that span space R Np can be introduced to make the projection of F (x 1,x ; a) onr Np vanish, which leads to + + F (x 1,x ; a)h s (x 1,x )dx 1 dx =, (15) 7

8 or + + x Q n Q n + h + 1 ( ) x 1 x! λe[y ] Q n Qn + x x + 1 4! λe[y 4 ] 4 Q n +4 Q n 3 ( Q n ) ( ) Q n Qn Q n x 4 x x 3 x x x ( ) 4 Qn + + h x x H s(x 1,x )dx 1 dx =. (16) This means that the reduced KF equation is fulfilled with p (x 1,x ; a) in the weak sense of integration if F (x 1,x ; a)h s (x 1,x )isintegrableinr Np. Substituting Eq. (9) into Eq. (16), the following equations are obtained. + + n i n i (i j)a ij x i j 1 1 x j + ja ij h x i j 1 x j 1 i=1 j= i=1 j= + 1 n i λe[y ] j(j 1)a ij x i j 1 x j i=1 j= + 1 n i n q λe[y ] jra ij a qr x i j+q r 1 x j+r i=1 j= q=1 r= + 1 n i 4! λe[y 4 ] j(j 1)(j )(j 3)a ij x i j 1 x j 4 i=1 j= + 4 n i n q 4! λe[y 4 ] jr(r 1)(r )a ij a qr x i j+q r 1 x j+r 4 i=1 j= q=1 r= + 3 n i n q 4! λe[y 4 ] jr(j 1)(r 1)a ij a qr x i j+q r 1 x j+r 4 i=1 j= q=1 r= + 6 n i n q n ī 4! λe[y 4 ] jr j( j 1)a ij a qr a ī jx i j+q r+ī j 1 x j+r+ j 4 i=1 j= q=1 r= ī=1 j= + 1 n i n q n ī n q 4! λe[y 4 ] jr j ra ij a qr a ī ja q r x i j+q r+ī j+ q r 1 x j+r+ j+ r 4 i=1 j= q=1 r= ī=1 j= q=1 r= + h } H s (x 1,x )dx 1 dx =. (17) x 8

9 Selecting H s (x 1,x )as: H s (x 1,x )=x k l 1 x l f 1(x 1 )f (x ), (18) where k = 1,,...,n; l =, 1,,...,k and s = 1 (k +)(k 1) + l +1; N p nonlinear algebraic equations in terms of N p unknown parameters can be formulated. The algebraic equations can be solved with any available method to determine the parameters. Numerical experience shows that a convenient and effective choice for f 1 (x 1 )andf (x ) is the PDF obtained with the EQL procedure under Gaussian excitation with the intensity λe[y ] as follows f 1 (x 1 )= f (x )= { 1 exp x 1 πσ1 σ 1 }, (19) { } 1 exp x. () πσ σ Because of the particular choice for f 1 (x 1 ) and f (x ), the integration in Eq. (17) can be easily calculated by taking into account the relationships between higher and lower order moments of Gaussian random variables. As a result, the following nonlinear algebraic equations are obtained. n i n q n ī n q i=1 j= q=1 r= ī=1 j= q=1 r= n i n q n ī + + in which i=1 j= q=1 r= ī=1 j= n i n q i=1 j= q=1 r= γ kl ijqra ij a qr + α kl ijqrī j q r a ija qr a ī ja q r β kl ijqrī j a ija qr a ī j n i i=1 j= η kl ij a ij + ψ kl =, (1) α kl ijqrī j q r = 1 4 λe[y 4 ]jr j ri i j+q r+ī j+ q r+k l 1 I j+r+ j+ r+l 4, () β kl ijqrī j = 1 4 λe[y 4 ]jr j( j 1)I i j+q r+ī j+k l 1 I j+r+ j+l 4, (3) 9

10 γijqr kl = 1 8 λe[y 4 ]jr(j 1)(r 1)I i j+q r+k l 1 I j+r+l λe[y 4 ]jr(r 1)(r )I i j+q r+k l 1 I j+r+l λe[y ]jri i j+q r+k l 1 I j+r+l, (4) ηij kl = (i j)i i j+k l 1 1 I j+l 1 ψ kl = λe[y 4 ]j(j 1)(j )(j 3)I i j+k l 1 I j+l λe[y ]j(j 1)I i j+k l 1 I j+l + ji kl ij, (5) + I kl ij = h x x k l 1 x l f 1(x 1 )f (x )dx 1 dx, (6) h (x 1,x )x i j+k l 1 x j+l 1 f 1 (x 1 )f (x )dx 1 dx, (7) + I1 m = x m 1 f 1(x 1 )dx 1, (8) I m = + where m =, 1,,... x m f (x )dx, (9) 3 Numerical analysis To assess the effectiveness of the above solution procedure, the PDF solutions of a Duffing oscillator with different degree of system nonlinearity and excitation intensity are analyzed and compared with those from Monte Carlo simulation (). The simulation procedure in [3,11] is followed. The sample size in the simulation is 1 7. The Duffing oscillator is expressed as Ẍ +ζω Ẋ + ω (X + εx3 )=W (t), (3) 1

11 in which W (t) is Poisson white noise as expressed in Eq. () and Y k in Eq. () is assumed to be Gaussian with zero mean. In this case, Eqs. (6) and (7) can be expressed as ψ kl =ζω I1 k l I, l (31) Iij kl I i j+k l 1 I j+l + ω Ii j+k l+1 1 I j+l 1 + ω εii j+k l+3 1 I j+l 1. (3) The parameters in the oscillator and excitation are listed in Table 1 with four different cases. 3.1 Case 1: High impulse arrival rate and slight nonlinearity In this case, the oscillator excited by the Poisson white noise with high impulse arrival rate (λ = 1) is analyzed. The PDFs obtained with the EPC method from n =, 4, 6 are compared in Fig. 1 for displacement and velocity, respectively. Numerical results show that the results obtained with EPC (n = ) is same as those obtained with the EQL procedure under Gaussian white noise with the intensity being λe[y ]. Therefore, the EPC method with n =is same as the EQL procedure. Figs. 1(a) and 1(b) show the PDFs and logarithmic PDFs of displacement obtained with the EPC method and. In the case of n =,thepdf obtained with the EPC method or the EQL procedure deviates a lot from that obtained with. It is seen that even in the case of weak system nonlinearity, the PDF of displacement obtained with the EQL procedure deviates much from that obtained with. If n =4orn = 6, the PDF of displacement obtained with the EPC method is much improved, particularly in the tail regions of the PDFs. 11

12 The PDFs of velocity are shown in Figs. 1(c) and 1(d). The PDFs obtained with EPC (n=, 4 and 6) are very close to those obtained with. It means that the PDF of velocity is close to being Gaussian in this case. This behavior is caused by the fact that the Poisson white noise with high impulse arrival rate in this case is close to Gaussian white noise. It is known that the PDF of the velocity of Duffing oscillator is Gaussian in the case of excitation being Gaussian white noise. 3. Case : Low impulse arrival rate and slight nonlinearity In this case, the impulse arrival rate is 1 and hence the Poisson white noise deviates much from Gaussian white noise. The PDFs of displacement obtained with various methods are presented in Figs. (a) and (b). Compared to Case 1, the PDF of displacement obtained with EPC (n = ) or the EQL procedure deviates more from that obtained with in this case as shown in Fig. (a). As the value of n increases to either 4 or 6, the PDFs of displacement become closer to those obtained with as shown in Figs. (a) and (b). As n increases to 6 with the EPC method, improvement can also be observed in Figs. (c) and (d) about the PDFs of velocity. The PDF of velocity is not close to being Gaussian any more because the Poisson white noise is not close to Gaussian white noise in this case. Hence the PDF of velocity is obviously different from that of the Duffing oscillator excited by Gaussian white noise. 1

13 3.3 Case 3: Low impulse arrival rate and high nonlinearity The analysis on the PDF solution of highly nonlinear systems is a challenging problem. In order to further evaluate the effectiveness of the presented solution procedure, a highly nonlinear system is considered with lower impulse arrival rate of the Poisson excitation. The PDFs and logarithmic PDFs of displacement obtained with the EPC method and are presented in Figs. 3(a) and 3(b). The PDF of displacement obtained with EPC (n = ) or the EQL procedure in this case deviates more from that obtained with than that in the above two cases. Such inaccuracy is more pronounced in the tail regions as shown in Fig. 3(b). Improvement can be observed with the EPC method as n =4andn =6.The behavior of the PDFs of velocity obtained with the EPC method and is shown in Figs. 3(c) and 3(d). The PDF of velocity can still be improved as n increases to 6 with the presented solution procedure. 3.4 Case 4: Low impulse arrival rate, high nonlinearity and high excitation intensity This case is about a highly nonlinear system with lower impulse arrival rate and strong excitation. Figs. 4(a) and 4(b) shows that the PDF of displacement differs much from being Gaussian. Hence, the PDF obtained with EPC (n =) or the EQL procedure is much different from that obtained with. Much improvement can still be observed as the polynomial degree n increases to 4 or 6. 13

14 Figs. 4(c) and 4(d) show the PDFs of velocity obtained with the EPC method and. The PDF of velocity obtained with EPC (n =6)isalsomuch improved compared to the PDF of velocity obtained with the EQL procedure. 4 Conclusions From the above numerical analysis and discussion, it can be concluded that the presented solution procedure can be used for the PDF analysis of nonlinear oscillators under external Poisson white noise excitation. With n being 6, the obtained PDFs of system responses are much close to those obtained with even in the tail regions of the PDFs. It is known that a good estimation of the tail behavior of the PDFs is important in reliability analysis. In the case of lower impulse arrival rates, the PDFs obtained with EPC (n =6) agree well with the simulated results no matter whether system nonlinearity is slight or high. When the external excitation is strong, the PDFs obtained with EPC (n = 6) are still close to those obtained with. Hence the presented solution procedure is effective for nonlinear oscillators under external excitation regardless of system nonlinearity and excitation intensity in the case of excitation being Poisson white noise. It is also observed that even in the case of slight system nonlinearity and high impulse arrival rate, the PDF of displacement obtained with EPC (n = ) or the EQL procedure deviates much from that obtained with. 14

15 5 Acknowledgments The results presented in this paper are obtained in the course of research supported by the funding of the Research Committee of University of Macau (Grant Nos. RG6/5-6S/KKP/FST, RG6/5-6S/7R/EGK/FST and RG6/5-6S/8R/EGK/FST). 15

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20 Table 1 Parameters for case study Items λ ε λe[y ] Case Case Case Case ζ=.5 ω =1. Y k : Gaussian process

21 .5. 1 PDF.15.1 Log(PDF) 3.5 EPC n= EPC n= x 1 /σ 1 x 1 /σ 1 (a) (b) PDF.1 Log(PDF) EPC n= EPC n= x /σ x /σ (c) (d) Fig. 1. Comparison of PDFs in Case 1, λ =1, ε =.1, λe[y ]=1.: (a) PDFs of displacement; (b) Logarithmic PDFs of displacement; (c) PDFs of velocity; (d) Logarithmic PDFs of velocity. 1

22 .5. 1 PDF.15.1 Log(PDF) 3.5 EPC n= EPC n= x 1 /σ 1 x 1 /σ 1 (a) (b) PDF.1 Log(PDF) EPC n= EPC n= x /σ x /σ (c) (d) Fig.. Comparison of PDFs in Case, λ =1., ε =.1, λe[y ]=1.: (a) PDFs of displacement; (b) Logarithmic PDFs of displacement; (c) PDFs of velocity; (d) Logarithmic PDFs of velocity.

23 PDF Log(PDF) EPC n= EPC n= x 1 /σ 1 x 1 /σ 1 (a) (b) PDF.1 Log(PDF) EPC n= EPC n= x /σ x /σ (c) (d) Fig. 3. Comparison of PDFs in Case 3, λ =1., ε =1., λe[y ]=1.: (a) PDFs of displacement; (b) Logarithmic PDFs of displacement; (c) PDFs of velocity; (d) Logarithmic PDFs of velocity. 3

24 PDF EPC n= Log(PDF) EPC n= x 1 /σ 1 x 1 /σ 1 (a) (b) PDF.1 Log(PDF) 3.5 EPC n= EPC n= x /σ x /σ (c) (d) Fig. 4. Comparison of PDFs in Case 4, λ =1., ε=1., λe[y ] = 1.: (a) PDFs of displacement; (b) Logarithmic PDFs of displacement; (c) PDFs of velocity; (d) Logarithmic PDFs of velocity. 4