International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

An approximate approach for assessing the reliability of a stochastically excited softening Duffing oscillator Zhang, Yuanjin; Kougioumtzoglou, Ioannis A. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 1 An Approximate Approach for Assessing the Reliability of a Stochastically Excited Softening Duffing Oscillator Yuanjin Zhang  Institute for Risk & Uncertainty, University of Liverpool, Liverpool, UK Ioannis A. Kougioumtzoglou Assistant Professor, Dept. of Civil Engineering and Engineering Mechanics, Columbia University, New York, USA ABSTRACT: An approximate analytical technique for assessing the reliability of a softening Duffing oscillator subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging treatment of the problem the oscillator time-varying survival probability is determined in a computationally efficient manner. In comparison with previous techniques that neglect the potential unbounded response behavior of the oscillator when the restoring force acquires negative values, the herein developed technique readily takes this aspect into account by introducing a special form for the oscillator non-stationary response amplitude probability density function (PDF). A significant advantage of the technique relates to the fact that it can readily handle cases of stochastic excitations that exhibit strong variability in both the intensity and the frequency content. An illustrative numerical example of a softening Duffing oscillator subject to earthquake excitation is included. Comparisons with pertinent Monte Carlo simulation data demonstrate the efficiency of the technique.  1.  INTRODUCTION It is often desirable for risk assessment applications to estimate the probability (also known as survival probability) that the system response stays within a prescribed domain over a given time interval. Indicatively, advanced Monte Carlo simulation (MCS) methodologies such as importance sampling, subset simulation and line sampling have been developed for reliability assessment applications; see Bucher (2011), and Au and Beck (2001) for some indicative references. Note, however, that there are cases of complex systems where MCS can be a computationally demanding, or even a prohibitive task; thus, there is a need for developing efficient approximate analytical and/or numerical techniques for addressing the problem such as Poisson distribution based approximations (e.g. Vanmarcke (1975)), probability density evolution schemes (e.g. Li and Chen (2009)), stochastic averaging/linearization approaches (e.g. Spanos and Kougioumtzoglou (2014a, 2014b)), as well as Wiener path integral techniques (e.g. Kougioumtzoglou and Spanos (2014a), Zhang and Kougioumtzoglou (2014)) and numerical path integration schemes (e.g. Naess and Johnsen (1993), Di Paola and Santoro (2008)).  The softening Duffing oscillator is a nonlinear oscillator possessing a linear-plus-cubic restoring force so that the spring has a softening characteristic. This oscillator has received considerable attention in the literature primarily due to its importance in describing the roll motion of a ship model in beam seas (e.g. Spyrou and Thomson (2000), Belenky and Sevastianov (2007)). Note, however, that the softening Duffing oscillator has found applications in diverse other fields of engineering dynamics such as structural system vibration isolation (e.g. Fu et al. (2014)), energy harvesting (e.g. Vandewater and Moss (2013)) and dynamics of timber structures (e.g. Reynolds et al. (2014)).   12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 Further, although several research efforts have focused on studying the oscillator response under deterministic excitation (e.g. Nayfeh and Sanchez (1989), Brennan et al. (2008)), limited results exist regarding the response analysis of the oscillator when it is subjected to stochastic excitation (e.g. Roberts and Vasta (2000), Cottone et al. (2010)). Specifically, most of the results are based on rather heuristic approaches which inherently assume stationarity and that the probability the response leaves the stable region is extremely small; thus, neglecting important aspects of the analysis such as the possible unbounded response behavior when the restoring force acquires negative values. Recently, a numerical path integral approach was developed in Kougioumtzoglou and Spanos (2014b) for determining the survival probability of a softening Duffing oscillator subject to stochastic excitation. The unbounded character of the response was rigorously taken into account by introducing a special form for the conditional response PDF, while the solution was propagated by utilizing a discrete version of the C-K equation.  In this paper, an efficient approximate analytical technique for determining the survival probability of a softening Duffing oscillator subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging treatment of the problem and introducing a special form for the oscillator response PDF, the technique developed in Spanos and Kougioumtzoglou (2014b) is adapted and generalized herein to account for the special case of the softening Duffing oscillator. A significant advantage of the technique is that it can readily handle cases of evolutionary stochastic excitation with arbitrary evolutionary power spectrum (EPS) forms, even of the non-separable kind.  2.  MATHEMATICAL FORMULATION 2.1.  Softening Duffing oscillator response analysis Consider the softening Duffing oscillator whose motion is governed by the equation ?̈? + 2𝜁0πœ”0?Μ‡? + πœ”02π‘₯ + πœ€πœ”02π‘₯3 = 𝑀(𝑑),   (1) where a dot over a variable denotes differentiation with respect to time 𝑑 ; πœ€ < 0 denotes a negative constant representing the magnitude of the nonlinearity degree; 𝜁0 is the ratio of critical damping; πœ”0  is the natural frequency corresponding to the linear oscillator (i.e. πœ€ = 0) and 𝑀(𝑑)  represents a Gaussian, zero-mean non-stationary stochastic process possessing an evolutionary broad-band power spectrum 𝑆𝑀(πœ”, 𝑑) . Examining Eq.(1), it can be readily seen that there exist values of the response displacement π‘₯(𝑑)  for which the oscillator restoring force 𝐹(π‘₯) = πœ”02π‘₯ + πœ€πœ”02π‘₯3 =πœ”02π‘₯(1 + πœ€π‘₯2)  reaches zero, and even negative values. Clearly, this may lead to unbounded system response, and a special treatment is necessary to account for this behavior. Next, bearing this qualitative behavior in mind, and focusing on lightly damped systems (i.e. 𝜁0 β‰ͺ 1 ), it can be argued (e.g. Spanos and Lutes (1980)) that for 𝐹(π‘₯) = πœ”02π‘₯(1 + πœ€π‘₯2) β‰₯ 0 , or equivalently π‘₯2 β‰₯ βˆ’1/πœ€, the oscillator response exhibits a pseudo-harmonic behavior described by the equations  π‘₯(𝑑) = π‘Žcos[πœ”(π‘Ž)𝑑 + πœ™(𝑑)],                 (2) and  ?Μ‡?(𝑑) = βˆ’πœ”(π‘Ž) π‘Žsin[πœ”(π‘Ž)𝑑 + πœ™(𝑑)].         (3) In Eqs.(2-3), πœ™  and π‘Ž  represent a slowly varying with time phase and a slowly varying with time response amplitude, respectively. Manipulating Eqs.(2-3) yields an expression for the oscillator response amplitude; that is, π‘Ž(𝑑) = √π‘₯2(𝑑) +?Μ‡?2(𝑑)πœ”(π‘Ž).                     (4) It is primarily the assumption of light damping that allows a combination of deterministic and stochastic averaging to be performed next and to approximate the second-order stochastic differential equation 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 3 (SDE) (Eq.(1)) by a first-order SDE governing the response amplitude process π‘Ž. A more detailed presentation/discussion of the assumptions involved and the corresponding assumed pseudo-harmonic behavior of the response process π‘₯(𝑑)  can be found in references such as Spanos and Lutes (1980) , Roberts and Spanos (1986), and Kougioumtzoglou and Spanos (2009). Next, following a stochastic averaging/linearization approach (e.g. Kougioumtzoglou and Spanos (2009)) a linearized version of Eq.(1) becomes ?̈? + 2𝜁0πœ”0?Μ‡? + πœ”2(π‘Ž)π‘₯ = 𝑀(𝑑),               (5) where the equivalent natural frequency πœ”(π‘Ž) is given by the expression πœ”2(π‘Ž) =πœ”02πœ‹π‘Žβˆ« π‘π‘œπ‘ πœ“((π‘Ž cosπœ“2πœ‹0+ πœ€(π‘Ž cosπœ“)3)π‘‘πœ“= πœ”02 (1 +34πœ€π‘Ž2).                          (6) Examining Eq.(6) it can be readily seen that the stiffness element of the equivalent linear oscillator becomes zero at the critical response amplitude value π‘Žπ‘π‘Ÿ = βˆšβˆ’43πœ€. In this regard, the requirement π‘₯2 β‰₯ βˆ’1/πœ€  for the oscillator of Eq.(1) to have a bounded response is equivalently expressed in the following by the requirement π‘Ž < π‘Žπ‘π‘Ÿ . Bearing this qualitative aspect in mind, a special form for the non-stationary response amplitude PDF 𝑝(π‘Ž, 𝑑) is introduced next; that is, 𝑝(π‘Ž, 𝑑) =π‘Žπ‘(𝑑)exp (βˆ’π‘Ž22𝑐(𝑑)) π‘Ÿπ‘’π‘π‘‘(π‘Ž)+ 𝑆(𝑑)𝛿(π‘Ž βˆ’ π‘Žβˆž),                            (7) where π‘Ÿπ‘’π‘π‘‘(π‘Ž) = 𝑒(π‘Ž) βˆ’ 𝑒(π‘Ž βˆ’ π‘Žπ‘π‘Ÿ) , 𝑒(. ) denotes the unit step function, 𝑐(𝑑) is a time-dependent coefficient to be determined, 𝛿(. ) denotes the Dirac delta function, and π‘Žβˆž represents an arbitrary response amplitude value with the property π‘Žβˆž ≫ π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ] . Further, the time-dependent factor 𝑆(𝑑)  can be determined by applying the normalization condition ∫ 𝑝(π‘Ž, 𝑑)π‘‘π‘Žβˆž0= 1; this yields  𝑆(𝑑) = 1 βˆ’ βˆ«π‘Žπ‘(𝑑)exp (βˆ’π‘Ž22𝑐(𝑑))π‘‘π‘Žπ‘Žπ‘π‘Ÿ0= exp (βˆ’π‘Žπ‘π‘Ÿ22𝑐(𝑑)).                           (8) Examining the form of the non-stationary response amplitude PDF of Eq.(7), it can be readily seen that it comprises two conceptually different terms. The first one represents a truncated Rayleigh PDF for amplitude values in the range [0, π‘Žπ‘π‘Ÿ] , whereas the factor 𝑆(𝑑)  in the second term represents the probability at a specific time instant that the response grows unbounded, namely the system response asymptotically approaches infinity. The rationale behind the choice of the truncated time-dependent Rayleigh PDF of Eq.(7) relates to the fact that the linear oscillator stationary response amplitude PDF is a Rayleigh one. In fact, as it was shown in Spanos and Lutes (1980), the non-stationary response amplitude PDF of a linear oscillator subject to Gaussian white noise excitation is a time-dependent Rayleigh PDF of the form 𝑝(π‘Ž, 𝑑) =π‘Žπ‘(𝑑)exp (βˆ’π‘Ž22𝑐(𝑑)) with the property limπ‘‘β†’βˆž 𝑝(π‘Ž, 𝑑) =π‘ŽπœŽ2𝑒π‘₯𝑝 (βˆ’π‘Ž22𝜎2) ; where 𝜎2  represents the linear oscillator stationary response variance. In Kougioumtzoglou and Spanos (2009), it was further shown that the Rayleigh representation is suitable for nonlinear oscillators also and under evolutionary stochastic excitation as well. It is pointed out that a significant difference between adopting a PDF of the form 𝑝(π‘Ž, 𝑑) =π‘Žπ‘(𝑑)exp (βˆ’π‘Ž22𝑐(𝑑)) in Kougioumtzoglou and Spanos (2009) and introducing a PDF form of Eq.(7) in the herein developed technique, is that in the former case 𝑐(𝑑) accounts for the variance of the non-stationary response process π‘₯ , whereas in the latter case 𝑐(𝑑)  is simply a time-varying coefficient to be determined.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 4 Next, relying on Eq.(7), it can be argued that an alternative to Eq.(5) equivalent linear system is given in the form ?̈? + 2𝜁0πœ”0?Μ‡? + πœ”π‘’π‘ž2 (𝑑)π‘₯ = 𝑀(𝑑),              (9) where the time-dependent stiffness element πœ”π‘’π‘ž2 (𝑑)  is defined as (see also Spanos and Kougioumtzoglou (2014b), Kougioumtzoglou and Spanos (2009)) πœ”π‘’π‘ž2 (𝑑) = 𝐸[πœ”2(π‘Ž)] = ∫ πœ”2(π‘Ž)𝑝(π‘Ž, 𝑑)π‘‘π‘Žβˆž0.     (10) Note that taking into account the form of the amplitude PDF of Eq.(7), the time-varying equivalent stiffness element of Eq.(10) also has two parts. Specifically, for π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ] , πœ”π‘’π‘ž2 (𝑑)  has a bounded part, i.e. πœ”π‘’π‘ž,𝐡2 (𝑑) , whereas for  π‘Ž > π‘Žπ‘π‘Ÿ  the stiffness element πœ”π‘’π‘ž2 (𝑑)  exhibits negative values; thus, yielding negative restoring force values resulting potentially in an unbounded system response behavior. In this regard, utilizing Eq.(7) the bounded part πœ”π‘’π‘ž,𝐡2 (𝑑)  is determined as πœ”π‘’π‘ž,𝐡2 (𝑑) = ∫ πœ”2(π‘Ž)𝑝(π‘Ž, 𝑑)π‘‘π‘Žπ‘Žπ‘π‘Ÿ0.           (11) Analytical determination of the integral in Eq.(11) yields  πœ”π‘’π‘ž,𝐡2 (𝑑) = πœ”02 (1 +32πœ€π‘(𝑑)(1 βˆ’ 𝑆(𝑑))).       (12) Examining Eq.(12) it can be readily seen that the stiffness element πœ”π‘’π‘ž,𝐡2 (𝑑)  is bounded between the values 0  and πœ”02 . Specifically, assuming that the oscillator is initially at rest yields lim𝑑→0+ 𝑝(π‘Ž, 𝑑) = 𝛿(π‘Ž0) , or in other words, lim𝑑→0+ 𝑐(𝑑) = 0 , which yields lim𝑐(𝑑)β†’0+ πœ”π‘’π‘ž,𝐡2 (𝑑) = πœ”02 . This means that for the very early part of the oscillation duration the oscillator features an approximately linear restoring force. Further, as time increases and the transient phase progresses, the truncated Rayleigh PDF of Eq.(7) broadens as the oscillator exhibits higher amplitude values π‘Ž(𝑑) . Equivalently, the time-varying coefficient  𝑐(𝑑)  increases with time, whereas the equivalent stiffness part πœ”π‘’π‘ž,𝐡2 (𝑑)  decreases with time. Taking into account Eqs.(7) and (12) it can be readily shown that in the extreme case lim𝑐(𝑑)β†’βˆžπœ”π‘’π‘ž,𝐡2 (𝑑) = 0. Thus, the equivalent stiffness part πœ”π‘’π‘ž,𝐡2 (𝑑) is a non-negative and bounded quantity varying with time between the values 0  and πœ”02 . This is in agreement with the fact that πœ”π‘’π‘ž,𝐡2 (𝑑)  corresponds to amplitude values π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ]  where the oscillator response is assumed to behave in a bounded manner.  Further, focusing on the case where π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ]  and based on a stochastic averaging approach Eq.(9) can be cast in a first-order SDE governing the evolution in time of the amplitude π‘Ž(𝑑); see Spanos and Lutes (1980) , Roberts and Spanos (1986), and Kougioumtzoglou and Spanos (2009) for a more detailed presentation. Related to this SDE is the Fokker-Planck (F-P) partial differential equation  πœ•π‘(π‘Ž, 𝑑|π‘Ž1, 𝑑1)πœ•π‘‘= βˆ’πœ•πœ•π‘Ž[𝐾1(π‘Ž, 𝑑)𝑝]+12πœ•2πœ•π‘Ž2[𝐾2(π‘Ž, 𝑑)𝑝],                      (13) where 𝐾1(π‘Ž, 𝑑) = βˆ’πœ0πœ”0π‘Ž +πœ‹π‘†(πœ”π‘’π‘ž,𝐡(𝑑), 𝑑)2π‘Žπœ”π‘’π‘ž,𝐡2 (𝑑) ,     (14) and 𝐾2(π‘Ž, 𝑑) =πœ‹π‘†(πœ”π‘’π‘ž,𝐡(𝑑), 𝑑)πœ”π‘’π‘ž,𝐡2 (𝑑).                 (15) The F-P Eq.(13) governs the evolution in time of the transition PDF 𝑝(π‘Ž, 𝑑|π‘Ž1, 𝑑1)  for π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ]  and π‘Ž1 ∈ [0, π‘Žπ‘π‘Ÿ] . Next, a solution of the associated F-P equation 𝑝(π‘Ž, 𝑑|π‘Ž1 = 0, 𝑑1 = 0) =  𝑝(π‘Ž, 𝑑) is attempted in the form of the truncated Rayleigh PDF of Eq.(7). Specifically, substituting the truncated Rayleigh PDF into the associated F-P equation, assuming that the oscillator is initially at rest (i.e. 𝑝(π‘Ž, 𝑑 = 0) = 𝛿(π‘Ž)), and manipulating yields the first-order nonlinear differential equation ?Μ‡?(𝑑) = βˆ’2𝜁0πœ”0𝑐(𝑑) +πœ‹π‘†(πœ”π‘’π‘ž,𝐡(𝑑), 𝑑)πœ”π‘’π‘ž,𝐡2 (𝑑),      (16)  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 5 to be solved numerically for the time-varying coefficient 𝑐(𝑑) . Obviously, once the time-varying coefficient 𝑐(𝑑)  is determined, the time-dependent coefficient 𝑆(𝑑)  can be evaluated via Eq.(8). Further, equations similar to Eq.(16) can be derived for the case of the response amplitude transition PDF in a straightforward manner. Specifically, following a similar analysis as in Spanos and Solomos (1983), the transition amplitude PDF 𝑝(π‘Ž, 𝑑|π‘Ž1, 𝑑1) is sought in the form 𝑝(π‘Ž, 𝑑|π‘Ž1, 𝑑1)= {π‘π‘‘π‘Ÿ(π‘Ž, 𝑑|π‘Ž1, 𝑑1) + 𝑅(𝑑, 𝑑1)𝛿(π‘Ž βˆ’ π‘Žβˆž) , π‘Ž1 ∈ (0, π‘Žπ‘π‘Ÿ)𝛿(π‘Ž βˆ’ π‘Žβˆž),     π‘Ž1 > π‘Žπ‘π‘Ÿ,                (17) where  π‘π‘‘π‘Ÿ(π‘Ž, 𝑑|π‘Ž1, 𝑑1) =π‘Žπ‘(𝑑, 𝑑1)exp (βˆ’π‘Ž2 + β„Ž2(𝑑, 𝑑1)2𝑐(𝑑, 𝑑1)) 𝐼0 (π‘Žβ„Ž(𝑑, 𝑑1)𝑐(𝑑, 𝑑1)) π‘Ÿπ‘’π‘π‘‘(π‘Ž), (18) and 𝑐(𝑑, 𝑑1)  and β„Ž(𝑑, 𝑑1)  are time-varying coefficients to be determined. Further, applying the normalization condition ∫ 𝑝(π‘Ž, 𝑑|π‘Ž1, 𝑑1)∞0π‘‘π‘Ž = 1  yields  the time-varying coefficient 𝑅(𝑑, 𝑑1) = 1 βˆ’βˆ« π‘π‘‘π‘Ÿ(π‘Ž, 𝑑|π‘Ž1, 𝑑1)π‘‘π‘Žπ‘Žπ‘π‘Ÿ0,      (19) where 𝐼0(. )  denotes the modified Bessel function of the first kind and of zero order. In a similar manner as before, under the condition that π‘Ž ∈ [0, π‘Žπ‘π‘Ÿ]  and π‘Ž1 ∈ [0, π‘Žπ‘π‘Ÿ] substituting the bounded part of Eq.(17) into Eq. (13) and manipulating yields the first-order differential equations (see Spanos and Solomos (1983) for a more detailed derivation) 𝑑𝑐(𝑑, 𝑑1)𝑑𝑑+ 2𝜁0πœ”0𝑐(𝑑, 𝑑1) βˆ’πœ‹π‘†(πœ”π‘’π‘ž,𝐡(𝑑), 𝑑)πœ”π‘’π‘ž,𝐡2 (𝑑)= 0, (20) and π‘‘β„Ž(𝑑, 𝑑1)𝑑𝑑+ 𝜁0πœ”0β„Ž(𝑑, 𝑑1) = 0.              (21) Eqs.(20-21) are subject to the initial condition 𝑝(π‘Ž2, 𝑑1|π‘Ž1, 𝑑1) = 𝛿(π‘Ž2 βˆ’ π‘Ž1)  which states that no change of state can occur if the transition time is zero.  2.2 Softening Duffing oscillator reliability assessment In this section the approach developed in Spanos and Kougioumtzoglou (2014b) is adapted and generalized herein to account for the special case of the softening Duffing oscillator and to determine the oscillator time-dependent survival probability. This is defined as the probability 𝑃𝐡(𝑑)  that the amplitude π‘Ž  stays below the threshold π‘Žπ‘π‘Ÿ over a given time interval [𝑑0, 𝑇] ; that is, π‘ƒπ‘Ÿπ‘œπ‘[π‘Ž(𝑑) ≀ π‘Žπ‘π‘Ÿ , π‘œπ‘£π‘’π‘Ÿ [𝑑0, 𝑇]|π‘Ž(𝑑0)  < π‘Žπ‘π‘Ÿ]. In the following, adopting the dicretization scheme applied in Spanos and Kougioumtzoglou (2014b), the time domain is divided into intervals of the form  [π‘‘π‘–βˆ’1, 𝑑𝑖  ], 𝑖 = 1,2… ,𝑀, 𝑑0 = 0, 𝑑𝑀 = 𝑇  π‘Žπ‘›π‘‘ 𝑑𝑖 = π‘‘π‘–βˆ’1 + 𝑑𝑇 π‘‡π‘’π‘ž(π‘‘π‘–βˆ’1),            (22) where π‘‡π‘’π‘ž  denotes the equivalent natural period of the oscillator given by π‘‡π‘’π‘ž(𝑑) =2πœ‹πœ”π‘’π‘ž,𝐡(𝑑) ,                       (23) and 𝑑𝑇   is a constant to be selected with the property 𝑑𝑇  ∈ (0,1]. In the ensuing analysis, the survival probability is determined assuming that it is approximately constant over the time interval [π‘‘π‘–βˆ’1, 𝑑𝑖 ]. Clearly, for 𝑑𝑇 = 1 the time interval [π‘‘π‘–βˆ’1, 𝑑𝑖] corresponds to the equivalent time-dependent natural period of the oscillator. The choice is justified by the fact that the response amplitude π‘Ž  is assumed to be approximately constant over the interval [π‘‘π‘–βˆ’1, 𝑑𝑖] , owing to its slowly varying character with respect to time (see section 2.1). Thus, the survival probability 𝑃𝐡(𝑇)  is assumed to be constant over [π‘‘π‘–βˆ’1, 𝑑𝑖]  as well. Of course, if higher accuracy is required a smaller value for 𝑑𝑇 can be chosen.  Further, taking into account the discretization of Eq.(22), the survival probability 𝑃𝐡(𝑇) is given by the equation  𝑃𝐡(𝑇 = 𝑑𝑀) =∏(1 βˆ’ 𝐹𝑖)𝑀𝑖=1,                 (24) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 6 where 𝐹𝑖  is defined as the probability that  π‘Ž will cross the barrier π‘Žπ‘π‘Ÿ in the time interval [π‘‘π‘–βˆ’1, 𝑑𝑖] , given that no crossings have occurred prior to time π‘‘π‘–βˆ’1 . Next, invoking the Markovian property for the process π‘Ž and utilizing the standard definition of conditional probability yields 𝐹𝑖 =π‘ƒπ‘Ÿπ‘œπ‘[π‘Ž(𝑑𝑖) β‰₯ π‘Žπ‘π‘Ÿ β‹‚π‘Ž(π‘‘π‘–βˆ’1) ≀ π‘Žπ‘π‘Ÿ]π‘ƒπ‘Ÿπ‘œπ‘[π‘Ž(π‘‘π‘–βˆ’1) ≀ π‘Žπ‘π‘Ÿ]=π‘„π‘–βˆ’1,π‘–π»π‘–βˆ’1(25) where π›¨π‘–βˆ’1 = ∫ 𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–βˆ’1π‘Žπ‘π‘Ÿ0,           (26) and, by utilizing the relationship 𝑝(π‘Ž1, 𝑑1; π‘Ž2, 𝑑2) = 𝑝(π‘Ž1, 𝑑1)𝑝(π‘Ž2, 𝑑2|π‘Ž1, 𝑑1), π‘„π‘–βˆ’1,𝑖 = ∫ (∫ 𝑝(π‘Žπ‘– , 𝑑𝑖|π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–+βˆžπ‘Žπ‘π‘Ÿ)π‘Žπ‘π‘Ÿ0 𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–βˆ’1.     (27) Next, taking into account Eqs.(7) and (17), Eqs.(26-27) become  π»π‘–βˆ’1 = 1 βˆ’ exp (π‘Žπ‘π‘Ÿ22𝑐(π‘‘π‘–βˆ’1)),               (28) and π‘„π‘–βˆ’1,𝑖= ∫ (∫ (π‘π‘‘π‘Ÿ(π‘Žπ‘– , 𝑑𝑖|π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)+βˆžπ‘Žπ‘π‘Ÿπ‘Žπ‘π‘Ÿ0+ 𝑅(𝑑𝑖, π‘‘π‘–βˆ’1)𝛿(π‘Ž βˆ’ π‘Žβˆž))π‘‘π‘Žπ‘–) 𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–βˆ’1. (29) respectively. Taking into account the properties of the Dirac delta function, Eq.(29) becomes π‘„π‘–βˆ’1,𝑖 = ∫ 𝑅(𝑑𝑖, π‘‘π‘–βˆ’1)π‘Žπ‘π‘Ÿ0𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–βˆ’1,   (30) and utilizing Eq.(19) yields  π‘„π‘–βˆ’1,𝑖 = ∫ 𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘Žπ‘π‘Ÿ0π‘‘π‘Žπ‘–βˆ’1 βˆ’βˆ« (∫ π‘π‘‘π‘Ÿ(π‘Žπ‘– , 𝑑𝑖|π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–π‘Žπ‘π‘Ÿ0)π‘Žπ‘π‘Ÿ0     𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1) π‘‘π‘Žπ‘–βˆ’1.                                   (31) Next, considering Eqs.(26), Eq.(31) takes the form π‘„π‘–βˆ’1,𝑖 = π»π‘–βˆ’1 βˆ’βˆ« ∫ π‘π‘‘π‘Ÿ(π‘Žπ‘– , 𝑑𝑖|π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘Žπ‘π‘Ÿ0π‘Žπ‘π‘Ÿ0 𝑝(π‘Žπ‘–βˆ’1, π‘‘π‘–βˆ’1)π‘‘π‘Žπ‘–π‘‘π‘Žπ‘–βˆ’1.       (32) Relying further on the assumption that πœ”π‘’π‘ž,𝐡(𝑑) follows a slowly varying with time behavior, the following approximation over a small time interval [π‘‘π‘–βˆ’1, 𝑑𝑖] is introduced; i.e., πœ”π‘’π‘ž,𝐡(𝑑) = πœ”π‘’π‘ž,𝐡(π‘‘π‘–βˆ’1)  for 𝑑 ∈ [π‘‘π‘–βˆ’1, 𝑑𝑖] . Next, based on the slowly varying with time behavior of the EPS, 𝑆𝑀(πœ”, 𝑑) is also treated as a constant over the interval [π‘‘π‘–βˆ’1, 𝑑𝑖] . Further, based on the above assumptions, introducing the variables πœπ‘– = 𝑑𝑖 βˆ’ π‘‘π‘–βˆ’1 , π‘Ÿπ‘–2 =𝑐(π‘‘π‘–βˆ’1)𝑐(𝑑𝑖)(1 βˆ’ 2𝜁0πœ”0πœπ‘–), and applying a first-order Taylor expansion around point πœπ‘– = 0  for various quantities, analytical treatment of the involved double integral of Eq.(32) is possible yielding  π‘„π‘–βˆ’1,𝑖 = π»π‘–βˆ’1 βˆ’ (𝐴0 +βˆ‘π΄π‘›π‘π‘›=1),             (33) where 𝐴0 = (1 βˆ’ exp (βˆ’π‘Žπ‘π‘Ÿ22𝑐(𝑑𝑖)(1 βˆ’ π‘Ÿπ‘–2))) (1 βˆ’exp (βˆ’π‘Žπ‘π‘Ÿ22𝑐(𝑑𝑖)(1 βˆ’ π‘Ÿπ‘–2)))(1 βˆ’ π‘Ÿπ‘–2),    (34)  𝐴𝑛 =π‘Ÿπ‘–2𝑛(1 βˆ’ π‘Ÿπ‘–2)(𝑛!)2𝐿𝑛 ,                                    (35) and 𝐿𝑛 = (Ξ“[1 + 𝑛, 0] βˆ’ Ξ“ [1 + 𝑛,π‘Žπ‘π‘Ÿ22𝑐(π‘‘π‘–βˆ’1)(1 βˆ’ π‘Ÿπ‘–2)])   (Ξ“[1 + 𝑛, 0] βˆ’ Ξ“ [1 + 𝑛,π‘Žπ‘π‘Ÿ22𝑐(𝑑𝑖)(1 βˆ’ π‘Ÿπ‘–2)]) . (36) In Eq.(36) Ξ“[𝛾, 𝑧]  represents the incomplete Gamma function defined as Ξ“[𝛾, 𝑧] =∫ π‘‘π›Ύβˆ’1π‘’βˆ’π‘‘π‘‘π‘‘+βˆžπ‘§. A more detailed presentation of the derivations in this section can be found in Spanos and Kougioumtzoglou (2014b).  3.  NUMERICAL EXAMPLES As noted in the introductory section, although the softening Duffing oscillator has been widely utilized to model the nonlinear ship rolling motion in beam seas (e.g. Spyrou and Thomson (2000), Belenky and Sevastianov (2007)), it has also been used in conjunction with structural dynamics/earthquake engineering applications such as structural system vibration isolation (e.g. Fu et al. (2014)), energy harvesting (e.g. Vandewater 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 7 and Moss (2013)) and dynamics of timber structures (e.g. Reynolds et al. (2014)).  In this regard, the non-separable excitation EPS of the form  𝑆𝑀(πœ”, 𝑑)= 𝑆 (πœ”5πœ‹)2𝑒π‘₯𝑝(βˆ’0.2𝑑)𝑑2𝑒π‘₯𝑝 (βˆ’ (πœ”10πœ‹)2𝑑),          (37) is considered in this example. This spectrum comprises some of the main characteristics of seismic shaking, such as decreasing of the dominant frequency with time (e.g. Sabetta and Pugliese (1996)). Further, survival probabilities determined via the herein developed approximate technique are compared with pertinent Monte Carlo simulation data (10,000 realizations). To this aim, realizations compatible with the EPS of Eq.(37) are generated based on a spectral representation approach (e.g. Liang et al. (2007)), while a standard fourth-order Runge-Kutta scheme is employed for solving the nonlinear equation of motion (Eq.(1)). The initial distribution chosen for the response amplitude PDF is the Dirac delta function, i.e., 𝑝(π‘Ž0, 𝑑0 = 0) = 𝛿(π‘Ž0), assuming the system is initially at rest. In the ensuing analysis the value 𝑁 = 60  is chosen for the terms to be included in the expansion.  Fig.(1). Bounded equivalent time-varying natural frequency πœ”π‘’π‘ž,𝐡(𝑑) for a softening Duffing oscillator (𝑆 = 1, πœ”02 = πœ‹2, 𝛽0 = 0.0628) In Fig.(1), the bounded equivalent natural frequencies (Eq.(12)) of the oscillators with parameter values (𝑆 = 1,πœ”02 = πœ‹2, 𝛽0 =0.0628, πœ€ = βˆ’1),  (𝑆 = 1,πœ”02 = πœ‹2, 𝛽0 =0.0628, πœ€ = βˆ’2), and (𝑆 = 1, πœ”02 = πœ‹2, 𝛽0 =0.0628, πœ€ = βˆ’3)  are plotted. In Fig.(2) the survival probabilities determined by Eqs.(24) are plotted for various barrier levels  π‘Žπ‘π‘Ÿ =βˆšβˆ’43πœ€; comparisons with MCS (10000 realizations) demonstrate a quite satisfactory agreement.   Fig.(2). Survival probability for a softening Duffing oscillator (𝑆 = 1, πœ”02 = πœ‹2, 𝛽0 = 0.0628, 𝑑𝑇 = 0.125) ; comparisons with MCS (10,000 realizations) 4.  CONCLUSION In this paper, an approximate analytical technique based on stochastic averaging has been developed for determining the survival probability of a softening Duffing oscillator subject to evolutionary stochastic excitation. In this regard, by introducing a special form for the non-stationary response amplitude PDF that takes into account the potential unbounded behavior of the oscillator, the time-varying survival probability has been determined in a computationally efficient manner. Approximate technique based results regarding a softening Duffing oscillator under earthquake excitation have been compared with pertinent MCS data demonstrating a satisfactory level of accuracy.   5.  REFERENCES Au S.-K., Beck J.L., 2001, Estimation of small failure probabilities in high dimensions by subset simulation, Prob Eng Mech, 16 (4):  263–277. Belenky V. L., Sevastianov N. B., 2007. Stability and safety of ships: Risk of capsizing, The Society of Naval Architects and Marine Engineers, 2nd Edition. Brennan M. J., Kovacic I., Carrella A., Waters T. P., 2008. On the jump-up and jump-down frequencies of the Duffing oscillator, Journal of Sound and Vibration, vol. 318: 1250-1261. Bucher C., 2011. Simulation methods in structural reliability, Marine Technology and Engineering, vol. 2: 1071-1086. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 8 Cottone G., Di Paola M., Ibrahim R., Pirrotta A., Santoro R., 2010. Stochastic ship roll motion via path integral method, International Journal of Naval Architecture and Ocean Engineering, vol. 2: 119-126. Di Paola, M., and Santoro, R. 2008. Path integral solution for nonlinear system enforced by Poisson white noise. Probab. Eng. Mech., 23(2–3), 164–169. Fu Niu, et al, 2014, Design and analysis of a quasi-zero stiffness isolator using a slotted conical disk spring as negative stiffness structure, Journal of Vibroengineering, 16(4): 1392-8716. Kougioumtzoglou I. A., Spanos P. D., 2009, An approximate approach for nonlinear system response determination under evolutionary stochastic excitation, Current Science, 97:1203-1211. Kougioumtzoglou I. A., Spanos P. D., 2014a, Nonstationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism, J. Eng. Mech., 04014064-1~ 04014064-14.  Kougioumtzoglou I. A., Spanos P. D., 2014b. Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a path integral approach, Probabilistic Engineering Mechanics, vol. 35: 67-74. Li J., Chen J., 2009. Stochastic dynamics of structures, New York: John Wiley & Sons. Liang J., Chaudhuri S. R., Shinozuka M., 2007. Simulation of non-stationary stochastic processes by spectral representation, Journal of Engineering Mechanics, vol. 133: 616-627. Naess, A., and Johnsen, J. M. 1993, Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probab. Eng. Mech., 8(2), 91–106. Nayfeh A. H., Sanchez N. E., 1989. Bifurcations in a forced softening Duffing oscillator, Int. J. Non-Linear Mech., 24: 483-497. Reynolds Thomas, Harris Richard, Chang Wen-Shao, 2014, Nonlinear pre-yield modal properties of timber structures with large-diameter steel dowel connections, Engineering Structures 76 :235–244. Roberts, J. B., and Spanos, P. D. 1986. Stochastic averaging: An approximate method of solving random vibration problems.  Int. J. Nonlinear Mech., 21(2), 111–134. Roberts J. B., Vasta M., 2000. Markov modeling and stochastic identification for nonlinear ship rolling in random waves, Phil. Trans. R. Soc. Lond. A, vol. 358: 1917-1941. Sabetta, F., and Pugliese, A., 1996, β€œEstimation of Response Spectra and Simulation of Non-Stationary Earthquake Ground Motions,” Bull. Seismol. Soc. Am., 86, pp. 337–352. Spanos, P. D., and Lutes, L. D. 1980. β€œProbability of response to evolutionary process.” J. Engrg. Mech. Div., 106(2), 213–224. Spanos P. D., Solomos G. P., 1983. Markov approximation to transient vibration, Journal of Engineering Mechanics, vol. 109: 1134-1150. Spanos P. D., Kougioumtzoglou I. A., 2014a. Galerkin scheme based determination of first-passage probability of nonlinear system response, Structure and Infrastructure Engineering, vol. 10: 1285-1294. Spanos P. D., Kougioumtzoglou I. A., 2014b. Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation, ASME Journal of Applied Mechanics, vol. 81, 051016: 1-9. Spyrou K. J., Thomson J. M. T., 2000. The nonlinear dynamics of ship motions: a field overview and some recent developments, Phil. Trans. R. Soc. Lond. A, vol. 358: 1735-1760. Vandewater L. A. and Moss S. D., 2013, Probability-of-existence of vibro-impact regimes in a nonlinear vibration energy harvester, Smart Mater. Struct. 22: 094025.  Vanmarcke E. H., 1975. On the distribution of the first-passage time for normal stationary random processes, ASME Journal of Applied Mechanics, vol. 42: 215-220. Zhang Y., Kougioumtzoglou I. A., Nonlinear oscillator stochastic response and survival probability determination via the Wiener path integral, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B. Mechanical Engineering (Accepted).  

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