Nonlinear Oscillator Stochastic Response and

Yuanjin Zhang Institute for Risk and Uncertainty, University of Liverpool, Liverpool L69 3GH, UK e-mail: [email protected]

Ioannis A. Kougioumtzoglou Department of Civil Engineering and Engineering Mechanics, The Fu Foundation School of Engineering and Applied Science, Columbia University, New York, NY 10027 e-mail: [email protected]

Nonlinear Oscillator Stochastic Response and Survival Probability Determination via the Wiener Path Integral A Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first, the nonlinear oscillator is cast into an equivalent linear one with timevarying stiffness and damping elements. Next, relying on the concept of the most probable path, a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman–Kolmogorov (C–K) equation is utilized for advancing the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered as numerical examples. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique. [DOI: 10.1115/1.4029754] Keywords: stochastic process, nonlinear system, path integral, survival probability, first-passage problem, variational formulation, Monte Carlo simulations

1

Introduction

MCS (e.g., [1]) has been, perhaps, the most versatile tool for determining response and reliability statistics of stochastic systems. Further, assessing the risk of failure, or performing a reliabilitybased analysis of dynamical systems, is closely related to the determination of the probability that the response of the system stays below a prescribed threshold over a given time interval. This time-dependent probability is also known as survival probability. The aforementioned problem, known as the first-passage problem in the literature, has been a persistent challenge in the field of stochastic dynamics. In this regard, several research efforts have been focused on developing versatile MCS-based techniques, such as importance sampling, subset simulation, and line sampling for reliability assessment applications; see [2–5] for some indicative references. Nevertheless, there are cases, especially of large-scale complex systems or when the quantity of interest has a relatively small probability of occurrence, where MCS techniques can be computationally prohibitive. Thus, there is a need for developing alternative efficient approximate analytical and/or numerical solution techniques [6–10]. One of the promising frameworks relates to the concept of the WPI. In this regard, note that although the WPI has strongly impacted the field of theoretical physics, the engineering community has ignored its potential as a powerful uncertainty quantification tool. The concept of path integral was introduced by Wiener [11] and was reinvented in a different form by Feynman [12] to reformulate quantum mechanics. A more detailed treatment of path integrals, especially of their applications in physics, can be found in a number of books such as the one by Chaichian and Demichev [13]. Recently, Kougioumtzoglou and Spanos [14] developed an approximate analytical WPI technique based on a variational formulation Manuscript received September 1, 2014; final manuscript received January 20, 2015; published online April 20, 2015. Assoc. Editor: Athanasios Pantelous.

and on the concepts of stochastic averaging/linearization for addressing certain stochastic engineering dynamics problems. In this regard, relying on the concept of the most probable trajectory, an approximate expression was derived for the nonstationary response PDF. Further, the aforementioned technique was extended by Kougioumtzoglou and Spanos [15] to treat multi degree-offreedom (MDOF) systems and hysteretic nonlinearities. In Di Matteo et al. [16], the technique was further enhanced and generalized to treat linear and nonlinear systems endowed with fractional derivatives terms (e.g., [17]). The aforementioned WPI technique should not be confused with alternative numerical schemes (commonly referred to as numerical path integral schemes) which constitute, in essence, a discrete version of the C–K equation (e.g., [18–23]). Note that these schemes can be computationally demanding potentially; this is due to the fact that the solution needs to be advanced in short-time steps, while multidimensional numerical integration needs to be performed at every time step as well. In this paper, a variational formulation-based WPI technique is developed together with a stochastic averaging/linearization treatment of the problem for determining response and reliability statistics of nonlinear oscillators subject to stochastic excitation. Specifically, first, the nonlinear oscillator is cast into an equivalent linear time-variant oscillator. Next, relying on the concept of the most probable path and considering a small time interval, an approximate closed-form expression is derived for the oscillator joint transition PDF. Further, the joint transition PDF is used in conjunction with a discrete version of the C–K equation to propagate the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the survival probability and first-passage PDF of the nonlinear oscillator are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus,

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the computational cost is kept at a minimum level. Numerical examples include the hardening Duffing and the bilinear hysteretic oscillators. Further, pertinent MCSs are included to demonstrate the reliability of the developed technique.

nonstationary oscillator response amplitude PDF, equivalent timevarying damping and stiffness elements can be defined by taking expectations on Eqs. (6) and (7); i.e., Z ∞ βðAÞpðA; tÞdA ð8Þ β eq ðtÞ ¼ E½βðAÞ ¼

2

and

0

Mathematical Formulation

2.1 Stochastic Averaging Treatment. The basic elements of an approximate analytical technique developed in [9] are reviewed in this section for completeness. Consider a nonlinear single degree-of-freedom (SDOF) oscillator whose motion is governed by the stochastic differential equation (SDE) x¨ ðtÞ þ β x˙ ðtÞ þ zðt; x; x˙ Þ ¼ wðtÞ

ð1Þ

where a dot over a variable denotes differentiation with respect to time t; x, x˙ , and x¨ denote the response displacement, velocity, and acceleration, respectively; zðt; x; x˙ Þ is the restoring force, which can be either hysteretic or depend only on the instantaneous values of x and x˙ ; β is a linear damping coefficient so that β ¼ 2ζ 0 ω0 ; ζ 0 is the ratio of critical damping; ω0 is the natural frequency corresponding to the linear oscillator (i.e., zðt; x; x˙ Þ ¼ ω20 x); and wðtÞ represents a Gaussian zero-mean white noise process with a power spectrum value equal to S0 . Focusing next on lightly damped systems (i.e., ζ 0 ≪ 1), it can be argued (e.g., [24]) that the response x of the oscillator of Eq. (1) exhibits a pseudoharmonic behavior described by the equations xðtÞ ¼ AðtÞ cos½ωðAÞt þ ϕðtÞ

ð2Þ

ω2eq ðtÞ ¼ E½ω2 ðAÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x˙ 2 ðtÞ AðtÞ ¼ x2 ðtÞ þ ωðAÞ

ð4Þ

Further, relying primarily on the assumption of light damping, a combination of deterministic and stochastic averaging is performed in this section for approximating the second-order SDE (Eq. (1)) by a first-order SDE governing the response amplitude process A. A more detailed presentation/discussion of the assumptions involved and the corresponding assumed pseudoharmonic behavior of the response process x can be found in references such as [24–26]. Applying, next, a stochastic averaging/linearization procedure [9,27], a linearized version of Eq. (1) becomes x¨ ðtÞ þ βðAÞ˙xðtÞ þ ω2 ðAÞxðtÞ ¼ wðtÞ

ð5Þ

0

ω2 ðAÞpðA; tÞdA

x¨ ðtÞ þ β eq ðtÞ˙xðtÞ þ ω2eq ðtÞxðtÞ ¼ wðtÞ

ð9Þ

ð10Þ

It can be readily seen that the linear time-variant oscillator of Eq. (10) is an alternative to the Eq. (5) linearized version of Eq. (1). Further, based on a stochastic averaging approach, Eq. (10) can be cast in a first-order SDE governing the evolution in time of the amplitude A; see [8,9,24,25] for a more detailed presentation. Related to this SDE is the Fokker–Planck (F–P) partial differential equation ∂ ∂ pðA2 ; t2 jA1 ; t1 Þ ¼ − ½K ðA; tÞpðA2 ; t2 jA1 ; t1 Þ ∂t ∂A 1 1 ∂2 ½K 2 ðA; tÞpðA2 ; t2 jA1 ; t1 Þ þ 2 ∂A2 2

ð11Þ

where

ð3Þ

In Eqs. (2) and (3), ϕ and A represent a slowly varying with time phase and a slowly varying with time response amplitude, respectively. Manipulating Eqs. (2) and (3) yields the following expression for the oscillator response amplitude:

∞

respectively. Note that due to the definition of the equivalent linear elements of Eqs. (8) and (9), it can be argued that they inherently are slowly varying functions with respect to time. Considering next Eqs. (8) and (9), the equivalent linear system of Eq. (5) can be cast in the form

and x˙ ðtÞ ¼ −ωðAÞAðtÞ sin½ωðAÞt þ ϕðtÞ

Z

and

1 πS0 K 1 ðA; tÞ ¼ − β eq ðtÞA þ 2 2Aω2eq ðtÞ

ð12Þ

sffiffiffiffiffiffiffiffiffiffiffiffi πS0 K 2 ðA; tÞ ¼ ω2eq ðtÞ

ð13Þ

Equation (11) governs the transition PDF pðA2 ; t2 jA1 ; t1 Þ of the response amplitude A. In Ref. [9] (see also [22]), it has been shown that Eq. (11) is satisfied by a solution of the form A A2 exp − ð14Þ pðA; tÞ ¼ cðtÞ 2cðtÞ for pðA2 ; t2 jA1 ¼ 0; t1 ¼ 0Þ ¼ pðA; tÞ. In Eq. (14), cðtÞ accounts for the variance of the nonstationary oscillator response process x. Specifically, substituting Eq. (14) into the associated F–P Eq. (11) and assuming that the oscillator is initially at rest (i.e., pðA; t ¼ 0Þ ¼ δðAÞ, where δð·Þ is the Dirac delta function) yields c˙ ðtÞ ¼ −β eq ðcðtÞÞcðtÞ þ

πS0 ω2eq ðcðtÞÞ

ð15Þ

ð7Þ

Note that the representation of Eq. (14) is suitable not only for the herein considered white noise excitation process but also for nonstationary stochastic excitations of arbitrary evolutionary power spectrum forms (e.g., [8,9]). Equation (15) constitutes a simple firstorder ordinary differential equation (ODE) that can be solved efficiently by standard numerical schemes, such as the Runge–Kutta. Once solved, the nonlinear oscillator (Eq. (1)) nonstationary response variance is obtained, and the time-varying equivalent linear damping and stiffness elements are determined via Eqs. (8) and (9), respectively.

To derive Eqs. (6) and (7), an error between Eqs. (1) and (5) has been defined, and a minimization procedure has been applied in the mean square sense. Further, assuming that pðA; tÞ denotes the

2.2 Wiener Path Integral Formulation. According to the WPI technique (e.g., [13]), the joint transition PDF pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ of the oscillator response going from a state

where βðAÞ ¼ β þ

− π1

R 2π 0

sin ψ · zðt; A cos ψ; −ωðAÞA sin ψÞdψ AωðAÞ

ð6Þ

and R

ω2 ðAÞ

¼

1 2π π 0

cos ψ · zðt; A cos ψ; −ωðAÞA sin ψÞdψ A

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ðxm−1 ; x˙ m−1 ; tm−1 Þ at t ¼ tm−1 to a new state ðxm ; x˙ m ; tm Þ at t ¼ tm , with tm > tm−1 can be expressed as a functional integral over the space of all possible paths Cfxm−1 ; x˙ m−1 ; tm−1 ; xm ; x˙ m ; tm g of the form Z fx ;˙x ;t g m m m W½xðtÞ½dxðtÞ pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ ¼ fxm−1 ;˙xm−1 ;tm−1 g

ð16Þ The WPI of Eq. (16) possesses a probability distribution on the path space as its integrand, which is denoted by W½wðtÞ and is called probability density functional. Note that the probability density functional for the white noise process wðtÞ is given by (e.g., [13,28]) Z tm 1 w2 ðtÞ dt ð17Þ W½wðtÞ ¼ C exp − tm−1 2 2πS0 where C is a normalization coefficient. Following next, the approach proposed in [15], Eq. (10) is substituted into Eq. (17) and the probability density functional W½wðtÞ for wðtÞ is interpreted as the probability density functional W½xðtÞ for xðtÞ. This yields Z tm 1 ð¨ x þ β eq;m x˙ þ ω2eq;m xÞ2 dt ð18Þ W½xðtÞ ¼ C exp − 2πS0 tm−1 2 In Eq. (18) and in the ensuing analysis, it is assumed that the time interval ½tm−1 ; tm is relatively small, i.e., tm − tm−1 → 0; thus, β eq ðtÞ ¼ β eq ðtm Þ ¼ β eq;m and ω2eq ðtÞ ¼ ω2eq ðtm Þ ¼ ω2eq;m for t ∈ ½tm−1 ; tm . Further, note that even if the probability density functional is constructed, the analytical solution of the WPI of Eq. (16) is a rather challenging task. To address this challenge, a variational formulation is invoked in the following for determining the transition PDF pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ in an approximate manner; see [13–15] for a more detailed presentation. In this regard, for the oscillator of Eq. (11) and for tm − tm−1 → 0, a Lagrangian function is defined as Lðx; x˙ ; x¨ Þ ¼

1 ð¨x þ β eq;m x˙ þ ω2eq;m xÞ2 2 2πS0

ð19Þ

Next, focusing on Eq. (19), the largest contribution to the WPI comes from the trajectory for which the integral in the exponential becomes as small as possible. Variational calculus rules (e.g., [29]) dictate that this trajectory with fixed end points is subject to the condition Z t m Lðxc ; x˙ c ; x¨ c Þdt ¼ 0 ð20Þ δ tm−1

where xc is the most probable path, namely, the most probable trajectory connecting points ðxm−1 ; x˙ m−1 ; tm−1 Þ and ðxm ; x˙ m ; tm Þ. Equation (20) yields a corresponding Euler–Lagrange equation of the form ∂L ∂ ∂L ∂ 2 ∂L − þ ¼0 ∂xc ∂t ∂ x˙ c ∂t2 ∂ x¨ c

ð21Þ

where D is a normalization coefficient. Clearly, the primary approximation of the technique relates to the fact that only the most probable path xc is considered in the evaluation of the functional integral of Eq. (16) instead of all the possible paths fxm−1 ; x˙ m−1 ; tm−1 ; xm ; x˙ m ; tm g. It can be argued that the concept of the most probable path can be viewed as something equivalent to the fact that the most probable value of a random variable is the one corresponding to the maximum value of the PDF. Substituting next Eq. (19) into Eq. (21) yields d4 x c d2 x þ 2ð1 − 2ζ 2eq;m Þω2eq;m 4c þ ω4eq;m xc ¼ 0 4 dt dt

ð24Þ

where β eq;m ¼ 2ζ eq;m ωeq;m . Equation (24) is a fourth-order linear ODE that can be readily solved analytically to obtain xc ðtÞ ¼ C1 expðλ1 tÞ þ C2 expðλ2 tÞ þ C3 expðλ3 tÞ þ C4 expðλ4 tÞ ð25Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ1 ¼ ζ eq;m þ i 1 − ζ 2eq;m ωeq;m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 ¼ ζ eq;m − i 1 − ζ 2eq;m ωeq;m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ3 ¼ −ζ eq;m þ i 1 − ζ 2eq;m ωeq;m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ4 ¼ −ζ eq;m − i 1 − ζ 2eq;m ωeq;m

ð26Þ

and C1 ; C2 ; C3 ; C4 are constants to be determined by utilizing the boundary conditions of Eq. (22). In this regard, analytical expressions for C1 ; C2 ; C3 ; C4 have been obtained by utilizing the symbolic toolbox of MATLAB®; these are provided in the Appendix. Further, Eqs. (25) and (26) are substituted into Eq. (23). Next, relying on the assumption that tm − tm−1 → 0, a Taylor series expansion is employed for the most probable path (Eq. (25)) around point t ¼ tm−1 , yielding a closed-form expression for the transition PDF of the form pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ n n ¼ 4;m 7;m expð−½ðn1;m xm−1 þ n2;m x˙ m−1 þ n3;m xm þ n4;m x˙ m Þ2 π þ ðn5;m xm−1 þ n6;m x˙ m−1 þ n7;m xm Þ2 Þ ð27Þ The analytical expressions of the constants n1;m ; n2;m ; n3;m ; n4;m ; n5;m ; n6;m ; n7;m in Eq. (27) are provided in the Appendix. Equivalently, using a vectorial notation, Eq. (27) can be cast into the Gaussian PDF form pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ 1 ¼ ð2πÞ−1 jΣt j−1=2 exp − ðXt − μt ÞT Σ−1 ðX − μ Þ t t t 2

ð28Þ

where

in conjunction with the boundary conditions

Xt ¼ ðxm ; x˙ m ÞT ; σ2x;t Σt ¼ ρt σx;t σx˙ ;t

xc ðtm−1 Þ ¼ xm−1 ; x˙ c ðtm−1 Þ ¼ x˙ m−1 ; xc ðtm Þ ¼ xm ; x˙ c ðtm Þ ¼ x˙ m ð22Þ Solving the boundary value problem of Eq. (21) together with Eq. (22) yields a closed-form expression for the transition PDF pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ, i.e., Z tm Lðxc ; x˙ c ; x¨ c Þdt pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ ¼ D exp −

μx;t

μt ¼ ðμx;t ; μx˙ ;t ÞT ; ρt σx;t σx˙ ;t σ2x˙ ;t

n n ¼ − 5;m xm−1 þ 6;m x˙ m−1 n7;m n7;m

ð29Þ

ð30Þ

n3;m n5;m n1;m n n n xm−1 þ 3;m 6;m − 2;m x˙ m−1 ð31Þ − n4;m n7;m n4;m n4;m n7;m n4;m

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tm−1

ð23Þ

μx˙ ;t ¼


1 σx;t ¼ pffiffiffi 2n7;m

σx˙ ;t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n23;m þ n27;m ¼ pffiffiffi 2n7;m n4;m

ð32Þ

¼ ð33Þ

−∞

ð35Þ

holds true. Note that the Gaussian form for the short-time transition PDF is in agreement with the concept of time-local Gaussian processes introduced by Dekker [31]. In this regard, it was shown that even for a nonlinear system, subject to the condition tm − tm−1 → 0, a Gaussian form for the transition PDF together with the C–K equation (Eq. (35)) can lead, in an exact manner, to the corresponding F–P equation. In fact, the discretized version of the C–K equation in conjunction with a Gaussian form of the transition PDF has been the core of several numerical path integral solution schemes that have been developed recently (e.g., [19–23,32]). These schemes have proven to be highly accurate. Nevertheless, they appear to be computationally demanding, mainly because high-dimensional numerical integration needs to be performed for every time step. To address this challenge, for the quite general system considered herein, i.e., the nonlinear oscillator of Eq. (1), the aforementioned numerical integration is circumvented by analytically evaluating the involved integrals; thus, the joint transition and nonstationary response PDF of the oscillator can be obtained at minimum computational cost. Specifically, starting from an initial state ðx0 ; x˙ 0 ; t0 Þ with short-time transition PDFs pðx1 ; x˙ 1 ; t1 jx0 ; x˙ 0 ; t0 Þ and pðx2 ; x˙ 2 ; t2 jx1 ; x˙ 1 ; t1 Þ of the form of Eq. (27) (or, alternatively, Eq. (28)) and utilizing the C–K Eq. (35), analytical evaluation of the involved convolution integral yields the transition PDF pðx2 ; x˙ 2 ; t2 jx0 ; x˙ 0 ; t0 Þ of the form pðx2 ; x˙ 2 ; t2 jx0 ; x˙ 0 ; t0 Þ ¼

k4,2 k7,2 expð−½ðk1,2 x0 þ k2,2 x˙ 0 þ k3,2 x2 þ k4,2 x˙ 2 Þ2 π þ ðk5,2 x0 þ k6,2 x˙ 0 þ k7,2 x2 Þ2 Þ

pðxm ; x˙ m ; tm jx0 ; x˙ 0 ; t0 Þ Z þ∞ Z þ∞ ¼ pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ −∞

−∞

× pðxm−1 ; x˙ m−1 ; tm−1 jx0 ; x˙ 0 ; t0 Þdxm−1 d˙xm−1

ð38Þ

can be used to determine the transition PDF at time t ¼ tm ; i.e., analytical evaluation of the integral of Eq. (38) yields pðxm ; x˙ m ; tm jx0 ; x˙ 0 ; t0 Þ ¼

k4;m k7;m expð−½ðk1;m x0 þ k2;m x˙ 0 þ k3;m xm þ k4;m x˙ m Þ2 π þ ðk5;m x0 þ k6;m x˙ 0 þ k7;m xm Þ2 Þ

ð39Þ

Equivalently, using a vectorial notation, Eq. (39) can be cast into the Gaussian PDF form

pðxmþ1 ; x˙ mþ1 ; tmþ1 jxm−1 ; x˙ m−1 ; tm−1 Þ Z þ∞ Z þ∞ ¼ pðxmþ1 ; x˙ mþ1 ; tmþ1 jxm ; x˙ m ; tm Þ × pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þdxm d˙xm

ð37Þ

ð34Þ

Obviously, the joint transition PDF of Eq. (27) is Gaussian as anticipated given that the system of Eq. (10) is linear. Further, note that in comparison with alternative approximate expressions of the transition PDF based on a stochastic averaging treatment [30], the herein determined transition PDF of Eq. (28) based on the WPI technique takes into account the correlation of the processes x and x˙ via the correlation coefficient ρt . This is important for the accuracy of the response analysis, especially during the transient phase where the oscillator response displacement and velocity are correlated (e.g., [27]). Further, invoking the Markov property of the response process x, the C–K equation

−∞

k4;m−1 k7;m−1 expð−½ðk1;m−1 x0 þ k2;m−1 x˙ 0 þ k3;m−1 xm−1 π þ k4;m−1 x˙ m−1 Þ2 þ ðk5;m−1 x0 þ k6;m−1 x˙ 0 þ k7;m−1 xm−1 Þ2 Þ

Utilizing the short-time transition PDF form of Eq. (27), the C–K equation

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n23;m ρt ¼ − n23;m þ n27;m

pðxm−1 ; x˙ m−1 ; tm−1 jx0 ; x˙ 0 ; t0 Þ

ð36Þ

The analytical expressions of the constants k1;m ; k2;m ; k3;m ; k4; m; k5;m ; k6;m ; k7;m in Eq. (36) can be found in the Appendix. Obviously, in this manner, the nonstationary joint response PDF of the original nonlinear oscillator can be advanced in short-time steps at essentially zero computational cost. Specifically, for a given time instant t ¼ tm−1 , the transition PDF pðxm−1 ; x˙ m−1 ; tm−1 jx0 ; x˙ 0 ; t0 Þ has an expression similar to Eq. (36); i.e.,

pðxm ; x˙ m ; tm jx0 ; x˙ 0 ; t0 Þ ¼ ð2πÞ−1 jΣm j−1=2 1 × exp − ðX − μÞT Σ−1 ð40Þ m ðX − μÞ 2 where μ ¼ ðμx;m ; μx˙ ;m ÞT ; X ¼ ðxm ; x˙ m ÞT ; σ2x;m ρm σx;m σx˙ ;m Σm ¼ ρm σx;m σx˙ ;m σ2x˙ ;m μx;m ¼ − μx˙ ;m ¼

k5;m k x0 þ 6;m x˙ 0 k7;m k7;m

k3;m k5;m k1;m k k k x0 þ 3;m 6;m − 2;m x˙ 0 − k4;m k7;m k4;m k4;m k7;m k4;m 1 σx;m ¼ pffiffiffi 2k7;m

σx˙ ;m and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k23;m þ k27;m ¼ pffiffiffi 2k7;m k4;m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ρm ¼ − 2 3;m 2 k3;m þ k7;m

ð41Þ

ð42Þ

ð43Þ

ð44Þ

ð45Þ

ð46Þ

Note that for the case x0 ¼ x˙ 0 ¼ 0 and integrating with respect to x˙ m yields the oscillator nonstationary response displacement PDF pðxm ; tm Þ of the form 1 x2 pffiffiffiffiffiffi exp − m ð47Þ pðxm ; tm Þ ¼ 2σ2x;m σx;m 2π

2.3 Nonlinear Oscillator Survival Probability Determination. The WPI technique developed in Sec. 2.2, besides determining the nonlinear oscillator transition and nonstationary response PDFs

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efficiently, can be used for determining the oscillator reliability/ first-passage statistics as well without additional significant computational effort. In this regard, an approximate analytical technique is developed in this section for determining the survival probability PB ðtÞ of the nonlinear oscillator of Eq. (1). This is defined as the probability that the system response displacement x stays within the bounded interval ½−B; B over the time interval ½t0 ; T; i.e.,

where pðxm−1 ; x˙ m−1 ; tm−1 Þ is given by Eq. (39) and pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ is given by Eq. (27). 2.4 Mechanization of the WPI-Based Technique. The mechanization of the developed technique involves the following steps: 1. Numerical solution (e.g., standard Runge–Kutta integration scheme) of the first-order ODE (Eq. (15)) to determine the system response variance cðtÞ. 2. Determination of the equivalent linear time-dependent damping β eq ðtÞ and stiffness ωeq ðtÞ elements via Eqs. (8) and (9), respectively. 3. Determination of the oscillator short-time joint transition PDF in the form of Eq. (27) by utilizing the analytical expressions of the constants n1;m ; n2;m ; n3;m ; n4;m ; n5;m ; n6;m ; n7;m (see Appendix). 4. Determination of the oscillator nonstationary joint response PDF in the form of Eq. (39) in short-time steps by utilizing the analytical expressions of the constants k1;m ; k2;m ; k3;m ; k4;m ; k5;m ; k6;m ; k7;m (see Appendix). Note that steps (1)–(4) constitute an efficient scheme for determining approximately the nonlinear oscillator joint transition PDF and the oscillator nonstationary joint response PDF. Further, these steps can be used as a basis for determining the nonlinear oscillator reliability statistics; i.e., 5. Determination of parameters H m−1 and Qm−1;m via Eqs. (53) and (56), respectively. 6. Determination of the survival probability PB ðTÞ via Eq. (51) and of the corresponding first-passage PDF pB ðTÞ via Eq. (50).

PB ðTÞ ¼ Probf−B < xðtÞ < B; t0 < t ≤ Tjxðt0 Þ ¼ x0 ; x˙ ðt0 Þ ¼ x˙ 0 g ð48Þ In general, it is rather challenging to calculate the survival probability exactly as it has been defined in Eq. (48) with its state in continuous time; see also [8,25]. Thus, in the following, the survival probability is calculated numerically by adopting the discretization in time introduced in Sec. 2.2. In this regard, Eq. (48) becomes PB ðT ¼ tM Þ ¼ Probf−B < xðtm Þ < B; m ¼ 1; : : : ; Mjxðt0 Þ ¼ x0 ; x˙ ðt0 Þ ¼ x˙ 0 g

ð49Þ

where tm ¼ t0 þ mΔt; m ¼ 1; : : : ; M and Δt ¼ ðT − t0 Þ=M. Note that Eq. (49) can approximate the survival probability as closely as desired by appropriately choosing Δt. Further, it can be readily shown that the corresponding first-passage PDF pB ðTÞ can be determined as pB ðTÞ ¼ −

dPB ðTÞ dT

ð50Þ

Taking into account the discretization of Eq. (49), the survival probability PB ðTÞ is obviously given by the equation PB ðT ¼ tM Þ ¼

M Y

Fm

ð51Þ

m¼1

where Fm denotes the probability that xðtÞ stays within the range ½−B; B in the time interval ½tm−1 ; tm , given that no crossings have occurred prior to time tm−1 . Next, invoking the Markov property for the process xðtÞ and utilizing the standard definition of conditional probability yields Fm ¼

Prob½jxðtm Þj < B ∩ jxðtm−1 Þj < B Qm−1;m ¼ Prob½jxðtm−1 Þj < B Hm−1

where

Z Hm−1 ¼

and

Z Qm−1;m ¼

B

Z

−B

B −B

B

−B

pðxm−1 ; tm−1 Þdxm−1

pðxm−1 ; tm−1 ; xm ; tm Þdxm−1 dxm

ð52Þ

ð53Þ

ð54Þ

Note that the probabilities Hm−1 ; Qm−1;m can be readily determined via the technique developed in Sec. 2.2. Specifically, H m−1 can be evaluated by utilizing Eqs. (53) and (47). Further, taking into account the Markov property for the response process, the joint response PDF pðxm−1 ; x˙ m−1 ; tm−1 ; xm ; x˙ m ; tm Þ is expressed as pðxm−1 ; x˙ m−1 ; tm−1 ; xm ; x˙ m ; tm Þ ¼ pðxm−1 ; x˙ m−1 ; tm−1 Þpðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ

ð55Þ

Considering Eq. (55), Qm−1;m becomes Z þ∞ Z þB Z þ∞ Z þB pðxm−1 ; x˙ m−1 ; tm−1 Þ Qm−1;m ¼ −∞

−B

−∞

−B

× pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þdxm−1 d˙xm−1 dxm d˙xm ð56Þ

In comparison with alternative, albeit more versatile, numerical path integral schemes for determining first-passage PDFs (e.g., [32]), the herein developed technique appears significantly more efficient computationally. This is because the computationally demanding task of numerically integrating, for every time step, the high-dimensional convolution integrals involved in the C–K equation has been circumvented. In this regard, the computational cost is kept at a minimum level as it is restricted, in essence, to the numerical integration of Eq. (15) via standard schemes (e.g., Runge–Kutta) and to the numerical integration involved in Eqs. (53) and (56).

3

Numerical Examples

The Duffing hardening and the bilinear hysteretic oscillators are considered in this section to demonstrate the reliability of the technique. For this purpose, the nonstationary response PDFs, the survival probabilities, and the first-passage PDFs obtained via the developed approximate analytical WPI technique are compared with response PDF, survival probability, and first-passage PDF estimates obtained via pertinent MCSs (10,000 realizations). A standard fourth-order Runge–Kutta numerical integration scheme is employed for solving the nonlinear oscillator differential equation of motion (Eq. (1)), whereas the barrier level B is expressed as a fraction λ of the corresponding linear oscillator stationary response standard deviation, i.e., B ¼ λσ where σ2 ¼ πS0 =2ζ 0 ω30 (e.g., [27]). Further, the value Δt ¼ tm − tm−1 ¼ 0.1 is chosen for the time discretization of the WPI technique, whereas the initial distributions chosen for the response displacement and velocity PDFs are the Dirac delta functions, i.e., pðxðt0 Þ; t0 ¼ 0Þ ¼ δðx0 Þ and pð˙xðt0 Þ; t0 ¼ 0Þ ¼ δð˙x0 Þ, assuming the system is initially at rest. In the ensuing analysis, a seventh-order order Taylor series expansion is chosen for determining the coefficients n1;m ; n2;m ; n3;m ; n4;m ; n5;m ; n6;m ; n7;m in Eq. (27). 3.1 Duffing Nonlinear (Hardening) Oscillator. Consider a Duffing nonlinear oscillator whose equation of motion is given by


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x¨ ðtÞ þ β 0 x˙ ðtÞ þ ω20 xðtÞ þ εω20 x3 ðtÞ ¼ wðtÞ

ð57Þ

where the parameter ε > 0 represents the magnitude of the nonlinearity. Further, the nonlinear restoring function zðt; x; x˙ Þ of Eq. (1) becomes zðt; x; x˙ Þ ¼ ω20 xðtÞ þ εω20 x3 ðtÞ

ð58Þ

Substituting Eq. (58) into Eqs. (6) and (7) yields βðAÞ ¼ β 0

ð59Þ

3 ω2 ðAÞ ¼ ω20 1 þ εA2 4

ð60Þ

and

Next, substituting Eqs. (59) and (60) into Eqs. (8) and (9), and considering Eq. (14), yields β eq ðcðtÞÞ ¼ β 0 and

ð61Þ

3 ω2eq ðcðtÞÞ ¼ ω20 1 þ εcðtÞ 2

ð62Þ

In Fig. 1, the nonstationary response variance cðtÞ determined by solving Eq. (15) is plotted for a Duffing oscillator with parameter values S0 ¼ 0.0637; ω20 ¼ 1; β 0 ¼ 0.2; ε ¼ 0.2 (Case 1), and S0 ¼ 0.0637; ω20 ¼ 1; β 0 ¼ 0.2; ε ¼ 1 (Case 2). It can be readily seen that the degree of nonlinearity is significant, especially for Case 2 where the stationary response variance ðlimt→þ∞ cðtÞÞ is approximately half of that of a corresponding linear oscillator (i.e., limt→þ∞ cðtÞ ¼ σ2 ¼ 1). In Fig. 2, the time-varying equivalent linear natural frequency ωeq ðtÞ determined via Eq. (9) is plotted for Case 1 and Case 2. Further, in Figs. 3 and 4, the nonstationary response displacement PDF is plotted for various time instants for Case 1 and Case 2, respectively. It is seen that the approximate WPI technique exhibits satisfactory accuracy when compared with MCS-based estimates, even for the high nonlinearity case (Case 2). Finally, in Figs. 5 and 6, the survival probability and the corresponding first-passage PDF for Case 1 for various barrier levels are plotted, respectively. Similarly, in Figs. 7 and 8, the survival probability and the corresponding first-passage PDF for Case 2 for various

Fig. 1 Nonstationary response variance ct of a Duffing oscillator under white noise excitation with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 0.2 (Case 1) and S 0 0.0637, ω20 1, β0 0.2, ε 1 (Case 2); comparison with pertinent Monte Carlo simulations (10,000 realizations).

Fig. 2 Time-varying equivalent linear natural frequency ωeq t for a Duffing oscillator under white noise excitation with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 0.2 (Case 1) and S 0 0.0637, ω20 1, β0 0.2, ε 1 (Case 2).

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barrier levels are plotted, respectively. Comparisons with pertinent MCS (10,000 realizations) are included as well, demonstrating a quite satisfactory accuracy. It is noted that the irregular/nonsmooth shape of the WPI-based first-passage PDFs is due to the differentiation of the survival probability (Eq. (50)). In this regard, the survival probability (Eq. (51)) is assumed to have constant values over the time intervals ½tm−1 ; tm resulting in a nonsmooth representation. Obviously, the level of nonsmoothness increases when differentiation takes place. 3.2 Bilinear Hysteretic Oscillator. A bilinear hysteretic oscillator is considered next, which has been widely studied in conjunction with earthquake engineering applications (e.g., [33–36]). In this regard, its equation of motion takes the form x¨ ðtÞ þ β 0 x˙ ðtÞ þ

aω20 xðtÞ

þ ð1 −

aÞω20 xy uðtÞ

¼ wðtÞ

ð63Þ

where a denotes the post-elastic-to-elastic stiffness ratio; xy is the yield displacement of the system; and uðtÞ is an additional variable controlling the evolution of the plastic behavior in the structure via the differential equation

˙ ¼ x˙ ðtÞ½1 − Hð˙xðtÞÞHðuðtÞ − 1Þ − Hð−˙xðtÞÞHð−uðtÞ − 1Þ xy uðtÞ ð64Þ In Eq. (64), Hð·Þ represents the Heaviside function defined as HðxÞ ¼

1 for x ≥ 0 0

for x < 0

ð65Þ

Further, the nonlinear restoring function zðx; x˙ ; tÞ of Eq. (1) becomes zðx; x˙ ; tÞ ¼ aω20 xðtÞ þ ð1 − aÞω20 xy uðtÞ

ð66Þ

Taking into account Eqs. (6) and (7) as well as Eq. (66), the equivalent linear damping and stiffness elements take the form βðAÞ ¼ β 0 þ

ð1 − aÞω20 S ðAÞ AωðAÞ h

ð67Þ

and

Fig. 3 Response displacement PDF for a Duffing oscillator under white noise excitation with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 0.2 (Case 1) for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Fig. 4 Response displacement PDF for a Duffing oscillator under white noise

excitation with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 1 (Case 2) for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations).


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Fig. 5 Survival probability for a Duffing oscillator under white noise excita-

tion with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 0.2 (Case 1) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Fig. 6 First-passage PDF for a Duffing oscillator under white noise excitation

with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 0.2 (Case 1) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

Fig. 7 Survival probability for a Duffing oscillator under white noise excitation

with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 1 (Case 2) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

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Fig. 8 First-passage PDF for a Duffing oscillator under white noise excitation

with parameter values S 0 0.0637, ω20 1, β0 0.2, ε 1 (Case 2) for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

C ðAÞ ω2 ðAÞ ¼ ω20 a þ ð1 − aÞ h A

ð68Þ

where Ch ðAÞ; Sh ðAÞ are given via the expressions [27,36] 8 < A ½A − 0.5 sinð2ΛÞ A > x y Ch ðAÞ ¼ π : A A ≤ xy 8 < 4xy 1 − xy A > xy π A Sh ðAÞ ¼ : 0 A ≤ xy

4xy ð1 − aÞω20 πcðtÞ

Z

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0 a þ ð1 − aÞ Λ−0.5πsinð2ΛÞ xy A2 × 1− exp − dA ð72Þ A 2cðtÞ

β eq ðcðtÞÞ ¼ β 0 þ

þ∞

xy

ð69Þ

and

ð70Þ

x2y ω2eq ðcðtÞÞ ¼ ω20 a þ ð1 − aÞ 1 − exp − 2cðtÞ

Z þ∞ 1 A2 ðΛ − 0.5 sinð2ΛÞÞA exp − dA þ πcðtÞ xy 2cðtÞ ð73Þ

where cosðΛÞ ¼ 1 −

2xy A

ð71Þ

Further, substituting Eqs. (67)–(71) into Eqs. (8) and (9), and considering Eq. (14), yields

In Fig. 9, the nonstationary response variance cðtÞ determined by solving Eq. (15) is plotted for a bilinear hysteretic oscillator with parameter values S0 ¼ 0.0637; a ¼ 0.6; β 0 ¼ 0.1; ω0 ¼ 1; xy ¼ 1. In Figs. 10 and 11, the time-varying equivalent linear natural frequency ωeq ðtÞ of Eq. (9) and damping β eq ðtÞ of Eq. (8) are plotted,

Fig. 9 Nonstationary response variance ct of a bilinear hysteretic oscillator under white noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1; comparison with pertinent Monte Carlo simulations (10,000 realizations).


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Fig. 10 Time-varying equivalent linear natural frequency ωeq t for a bilinear hysteretic oscillator under white noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1.

Fig. 11 Time-varying equivalent linear damping βeq t for a bilinear hysteretic

oscillator under white noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1.

Response displacement PDF for a bilinear oscillator under white noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1 for various time instants; comparison with pertinent Monte Carlo simulations (10,000 realizations). Fig. 12

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Fig. 13 Survival probability for a bilinear hysteretic oscillator under white

noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1 for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations).

First-passage PDF for a bilinear hysteretic oscillator under white noise excitation with parameter values S 0 0.0637, a 0.6, β0 0.1, ω0 1, x y 1 for various barrier levels; comparison with pertinent Monte Carlo simulations (10,000 realizations). Fig. 14

respectively. Further, in Fig. 12, the nonstationary response displacement PDF is plotted for various time instants. Comparisons with pertinent MCS data (10,000 realizations) demonstrate a satisfactory level of accuracy. Finally, in Figs. 13 and 14, the survival probability and corresponding first-passage PDF for various barrier levels are plotted, respectively. Comparisons with the pertinent MCS (10,000 realizations) are included as well demonstrating a quite satisfactory agreement.

4

Concluding Remarks

A WPI-based technique for determining the nonstationary response PDF, the survival probability, and the first-passage PDF of nonlinear/hysteretic oscillators subject to stochastic excitation has been developed. Specifically, based on a stochastic averaging/ linearization treatment of the problem, the nonlinear oscillator has been cast into an equivalent linear time-variant oscillator. In this regard, equivalent linear time-dependent stiffness and damping elements have also been determined as part of the solution procedure. Further, relying on a variational formulation and on the

concept of the most probable path, a closed-form analytical expression has been derived for the oscillator short-time transition PDF. Next, utilizing the short-time transition PDF and the C–K equation, a closed-form expression for the oscillator nonstationary joint response PDF has been derived as well. Thus, the solution can propagate in short-time steps, yielding not only the nonstationary response PDF but also the survival probability and the first-passage PDF of the nonlinear oscillator. In comparison with existing, albeit more versatile, numerical path integral schemes, a significant advantage of the proposed WPI technique is that the computationally demanding task of numerically integrating, for every time step, the high-dimensional convolution integrals involved in the C–K equation has been circumvented. This is because closed-form analytical expressions have been derived for the involved multidimensional convolution integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators have been considered in the Sec. 3. Comparisons with pertinent MCSs have demonstrated the reliability of the technique for various nonlinearity magnitudes and barrier levels.


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Appendix In this Appendix, analytical expressions are provided for several coefficients used in expressions derived in the main text. In this regard, with the aid of the symbolic toolbox of MATLAB®, the analytical expressions for the coefficients C1 , C2 , C3 , C4 of Eq. (25) are given as follows: C1 ¼ ð˙xm−1 expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ sin tm−1 ωeq;m − x˙ m expð3ζ eq;m tm ωeq;m Þ sin ωeq;m tm − x˙ m−1 expð3ζ eq;m tm−1 ωeq;m Þ sin ωeq;m tm−1 þ x˙ m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ sin tm ωeq;m − x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cos tm−1 ωeq;m − x˙ m ζ eq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cos tm ωeq;m − xm−1 ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cos tm−1 ωeq;m − xm ω0 expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cos tm ωeq;m þ x˙ m ζ eq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cosðtm − 2tm−1 Þωeq;m þ x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cosð2tm − tm−1 Þωeq;m þ xm−1 ωeq;m expð3ζ eq;m ωeq;m tm−1 ÞÞ cos tm−1 ωeq;m þ xm ωeq;m expð3ζ eq;m ωeq;m tm Þ cos tm ωeq;m þ xm ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cosðtm − 2tm−1 Þωeq;m − xm−1 ζ eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ sin ωeq;m ð2tm − tm−1 Þ þ xm−1 ωeq;m ζ eq;m expð3ζ eq;m tm−1 ωeq;m Þ sin ωeq;m tm−1 þ xm ωeq;m ζ eq;m expð3ζ eq;m tm ωeq;m Þ sin ωeq;m tm þ xm−1 ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cosð2tm − tm−1 Þωeq;m þ xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ sinðtm − 2tm−1 Þωeq;m − xm−1 ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cos ωeq;m tm−1 − xm ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cos ωeq;m tm Þ=ðωeq;m ðexpð4ζ eq;m tm ωeq;m Þ þ expð4ζ eq;m tm−1 ωeq;m Þ − 2 expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ − 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ þ 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ cos 2ωeq;m ðtm − tm−1 ÞÞ ðA1Þ

C2 ¼ −ð˙xm−1 expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ cos tm−1 ωeq;m − x˙ m expð3ζ eq;m tm ωeq;m Þ cos ωeq;m tm − x˙ m−1 expð3ζ eq;m tm−1 ωeq;m Þ cos ωeq;m tm−1 þ x˙ m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cos tm ωeq;m − xm−1 ωeq;m expð3ζ eq;m ωeq;m tm−1 Þ sin tm−1 ωeq;m − xm ωeq;m expð3ζ eq;m ωeq;m tm Þ sin tm ωeq;m þ x˙ m−1 ζ expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ sin tm−1 ωeq;m þ x˙ m ζ eq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ sin tm ωeq;m þ xm−1 ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ sin tm−1 ωeq;m þ xm ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ sin tm ωeq;m þ x˙ m ζ eq;m expððtm þ 2tm−1 Þζ eq;m ωeq;m ÞÞ sinðtm − 2tm−1 Þωeq;m − x˙ m−1 ζ eq;m expðð2tm þ tm−1 Þζ eq;m ωeq;m ÞÞ sinð2tm − tm−1 Þωeq;m þ xm−1 ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ tm−1 ÞÞ sin tm−1 ωeq;m þ xm ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ sin ω0 tm − xm−1 ωeq;m ζ eq;m expðð2tm þ tm−1 Þζ eq;m ωeq;m ÞÞ cosð2tm − tm−1 Þωeq;m þ xm ωeq;m ζ 2eq;m expððtm þ 2tm−1 Þζ eq;m ωeq;m ÞÞ sinðtm − 2tm−1 Þωeq;m þ xm−1 ζ eq;m ωeq;m expð3ζ eq;m tm−1 ωeq;m Þ cos ωeq;m tm−1 þ xm ζ eq;m ωeq;m expð3ζ eq;m ωeq;m tm Þ cos tm ωeq;m − xm−1 ζ 2eq;m ωeq;m expðζ 2eq;m ωeq;m ð2tm þ tm−1 ÞÞ sinð2tm − tm−1 Þωeq;m − xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 2tm−1 ÞÞ cosðtm − 2tm−1 Þωeq;m Þ=ðωeq;m ðexpð4ζ eq;m tm ωeq;m Þ þ expð4ζ eq;m tm−1 ωeq;m Þ − 2 expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ − 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ þ 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ cos 2ωeq;m ðtm − tm−1 ÞÞ ðA2Þ

C3 ¼ −ð˙xm−1 expðζ eq;m ωeq;m ð4tm þ tm−1 ÞÞ sin tm−1 ωeq;m − x˙ m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin ωeq;m tm − x˙ m−1 expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sin ωeq;m tm−1 þ x˙ m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ sin tm ωeq;m þ x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos ωeq;m ð2tm − tm−1 Þ − xm−1 ωeq;m expðζ eq;m ωeq;m ð4tm þ tm−1 ÞÞ cos tm−1 ωeq;m − xm ωeq;m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ cos tm ωeq;m − x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos tm−1 ωeq;m − x˙ m ζ eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos tm ωeq;m þ xm−1 ωeq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos tm−1 ωeq;m þ xm ωeq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos tm ωeq;m þ x˙ m ζ eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos ωeq;m ðtm − 2tm−1 Þ þ xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sinðtm − 2tm−1 Þωeq;m þ xm−1 ζ 2eq;m ωeq;m expðζωeq;m ð2tm þ 3tm−1 ÞÞ cos ωeq;m tm−1 þ xm ωeq;m ζ 2eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos ωeq;m tm − xm ωeq;m ζ 2eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos ωeq;m ðtm − 2tm−1 Þ − xm−1 ζ eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sinð2tm − tm−1 Þωeq;m þ xm−1 ζ eq;m ωeq;m expðζ eq;m ωeq;m ð4tm þ tm−1 ÞÞ sin tm−1 ωeq;m þ xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ sin ωeq;m tm − xm−1 ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos ωeq;m ð2tm − tm−1 ÞÞ =ðωeq;m ðexpð4ζ eq;m tm ωeq;m Þ þ expð4ζ eq;m tm−1 ωeq;m Þ − 2 expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ − 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þω0 Þ þ 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ cos 2ωeq;m ðtm − tm−1 ÞÞ 021006-12 / Vol. 1, JUNE 2015


ðA3Þ Transactions of the ASME

C4 ¼ −ð˙xm−1 expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos tm−1 ωeq;m þ x˙ m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cos ωeq;m tm − x˙ m−1 expðζ eq;m ωeq;m ð4tm þ tm−1 ÞÞ cos ωeq;m tm−1 − x˙ m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ cos tm ωeq;m þ x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sin ωeq;m ð2tm − tm−1 Þ − xm−1 ωeq;m expðζ eq;m ω0 ð4tm þ tm−1 ÞÞ sin tm−1 ωeq;m − xm ωeq;m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ sin tm ωeq;m − x˙ m−1 ζ eq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sin ti ωeq;m − x˙ m ζ eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin tm ωeq;m þ xm−1 ωeq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sin tm−1 ωeq;m þ xm ωeq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin tm ωeq;m − x˙ m ζ eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin ωeq;m ðtm − 2tm−1 Þ þ xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ cosðtm − 2tm−1 Þωeq;m þ xm−1 ζ 2eq;m ωeq;m expðζ 2eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ sin ωeq;m tm−1 þ xm ω0 ζ 2eq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin ωeq;m tm þ xm−1 ωeq;m ζ eq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ cos ωeq;m ð2tm − tm−1 Þ − xm−1 ζ eq;m ωeq;m expðζ eq;m ωeq;m ð4tm þ tm−1 ÞÞ cos tm−1 ωeq;m − xm ζ eq;m ωeq;m expðζ eq;m ωeq;m ðtm þ 4tm−1 ÞÞ cos tmi ωeq;m þ xm ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð3tm þ 2tm−1 ÞÞ sin ωeq;m ðtm − 2tm−1 Þ − xm−1 ζ 2eq;m ωeq;m expðζ eq;m ωeq;m ð2tm þ 3tm−1 ÞÞ × sin ωeq;m ð2tm − tm−1 ÞÞ=ðωeq;m ðexpð4ζ eq;m tm ωeq;m Þ þ expð4ζ eq;m tm−1 ωeq;m Þ − 2 expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ − 2ζ 2eq;m expð2ζ 2eq;m ðtm þ tm−1 Þωeq;m Þ þ 2ζ 2eq;m expð2ζ eq;m ðtm þ tm−1 Þωeq;m Þ cos 2ωeq;m ðtm − tm−1 ÞÞ

ðA4Þ

Further, to determine the analytical expression of the transition PDF of Eq. (27), a Taylor series expansion has been employed to expand the expression of Eq. (25) for the most probable path around the point t ¼ tm−1 . For instance, for a fifth-order order Taylor expansion of the form fðtÞ ≈ fðtm−1 Þ þ f 0 ðtm−1 Þðt − tm−1 Þ þ

f 0 0 ðtm−1 Þ f ð3Þ ðtm−1 Þ f ð3Þ ðtm−1 Þ fð4Þ ðtm−1 Þ ðt − tm−1 Þ2 þ ðt − tm−1 Þ3 þ · · · ðt − tm−1 Þ3 þ ðt − tm−1 Þ4 2! 3! 3! 4! ðA5Þ

pðxm ; x˙ m ; tm jxm−1 ; x˙ m−1 ; tm−1 Þ takes the form of Eq. (27), whereas the coefficients n1;m , n2;m , n3;m , n4;m , n5;m , n6;m , n7;m are given by n4;m ¼ ½ð506ζ 6eq;m Δt6 ω6eq;m þ 1581ζ 5eq;m Δt5 ω5eq;m þ 146ζ 4eq;m Δt6 ω6eq;m þ 2804ζ 4eq;m Δt4 ω4eq;m þ 190ζ 3eq;m Δt5 ω5eq;m þ 2380ζ 3eq;m Δt3 ω3eq;m þ 14ζ 2eq;m Δt6 ω6eq;m þ 36ζ 2eq;m Δt4 ω4eq;m þ 1500ζ 2eq;m Δt2 ω2eq;m þ 15ζ eq;m Δt5 ω5eq;m − 8ζ eq;m Δt3 ω3eq;m þ 540ζ eq;m Δtωeq;m − 2Δt6 ω6eq;m þ 22Δt4 ω4eq;m − 52Δt2 ω2eq;m þ 480Þ=ð120πSΔtÞ0.5

ðA6Þ

n1;m ¼ ð1328ζ 7eq;m Δt7 ω70 þ 5041ζ 6eq;m Δt6 ω60 þ 432ζ 5eq;m Δt7 ω7eq;m þ 5852ζ 5eq;m Δt5 ω5eq;m − 657ζ 4eq;m Δt6 ω6eq;m þ 4938ζ 4eq;m Δt4 ω4eq;m − 384ζ 3eq;m Δt7 ω7eq;m − 1104ζ 3eq;m Δt5 ω5eq;m þ 7200ζ 3eq;m Δt3 ω3eq;m − 492ζ 2eq;m Δt6 ω6eq;m þ 476ζ 2eq;m Δt4 ω4eq;m þ 6000ζ 2eq;m Δt2 ω2eq;m − 64ζ eq;m Δt7 ω7eq;m þ 80ζ eq;m Δt5 ω5eq;m − 144ζ eq;m Δt3 ω3eq;m þ 2880ζ eq;m Δtωeq;m − 24Δt6 ω6eq;m þ 64Δt4 ω4eq;m − 216Δt2 ω2eq;m þ 720Þ=ð480πSΔt2 n4;m Þ

ðA7Þ

n3;m ¼ −ð367ζ 7eq;m Δt7 ω7eq;m þ 1314ζ 6eq;m Δt6 ω6eq;m þ 170ζ 5eq;m Δt7 ω7eq;m þ 1498ζ 6eq;m Δt5 ω5eq;m − 80ζ 4eq;m Δt6 ω6eq;m þ 1247ζ 4eq;m Δt4 ω4eq;m − 51ζ 3eq;m Δt7 ω7eq;m − 322ζ 3eq;m Δt5 ω5eq;m þ 1800ζ 3eq;m Δt3 ω3eq;m − 63ζ 2eq;m Δt6 ω6eq;m − 46ζ 2eq;m Δt4 ω4eq;m þ 1500ζ 2eq;m Δt2 ω2eq;m − 14ζ eq;m Δt7 ω7eq;m þ 56ζ eq;m Δt5 ω5eq;m − 96ζ eq;m Δt3 ω3eq;m þ 720ζ eq;m Δtωeq;m − 7Δt6 ω6eq;m þ 41Δt4 ω4eq;m − 84Δt2 ω2eq;m þ 180Þ=ð120πSΔt2 n4;m Þ

ðA8Þ

n2;m ¼ ð518ζ 6eq;m Δt6 ω6eq;m þ 1742ζ 5eq;m Δt5 ω5eq;m þ 342ζ 4eq;m Δt6 ω6eq;m þ 1993ζ 4eq;m Δt4 ω40 þ 404ζ 3eq;m Δt5 ω5eq;m þ 1280ζ 3eq;m Δt3 ω30 þ 4ζ 2eq;m Δt6 ω6eq;m þ 192ζ 2eq;m Δt4 ω4eq;m þ 1020ζ 2eq;m Δt2 ω2eq;m þ 16ζ eq;m Δt5 ω5eq;m þ 144ζ eq;m Δt3 ω3eq;m þ 480ζ eq;m Δtωeq;m − 8Δt6 ω6eq;m þ 64Δt4 ω4eq;m − 64Δt2 ω2eq;m þ 120Þ=ð2400πSΔtn4;m Þ

ðA9Þ

n7;m ¼ ½ð743ζ 8eq;m Δt8 ω8eq;m þ 2024ζ 7eq;m Δt7 ω7eq;m þ 166ζ 6eq;m Δt8 ω8eq;m − 260ζ 6eq;m Δt6 ω6eq;m − 1898ζ 5eq;m Δt7 ω7eq;m − 1032ζ 5eq;m Δt5 ω5eq;m − 709ζ 4eq;m Δt8 ω8eq;m − 2088ζ 4eq;m Δt6 ω6eq;m þ 5432ζ 4eq;m Δt4 ω4eq;m − 1004ζ 3eq;m Δt7 ω7eq;m þ 1192ζ 3eq;m Δt5 ω5eq;m þ 8160ζ 3eq;m Δt3 ω3eq;m − 92ζ 2eq;m Δt8 ω8eq;m − 62ζ 2eq;m Δt6 ω6eq;m þ 776ζ 2eq;m Δt4 ω4eq;m þ 6240ζ 2eq;m Δt2 ω2eq;m − 82ζ eq;m Δt7 ω7eq;m þ 272ζ eq;m Δt5 ω5eq;m − 264ζ eq;m Δt3 ω3eq;m þ 2880ζ eq;m Δtωeq;m þ 4Δt8 ω8eq;m − 52Δt6 ω6eq;m þ 188Δt4 ω4eq;m − 336Δt2 ω2eq;m þ 720Þ=ð240πSΔt3 Þ − n23;m 0.5 ðA10Þ n6;m ¼ ½−ð1104ζ 7eq;m Δt7 ω7eq;m þ 2597ζ 6eq;m Δt6 ω6eq;m þ 464ζ 5eq;m Δt7 ω7eq;m þ 284ζ 5eq;m Δt5 ω5eq;m − 1117ζ 4eq;m Δt6 ω6eq;m − 86ζ 4eq;m Δt4 ω40 − 576ζ 3eq;m Δt7 ω7eq;m − 976ζ 3eq;m Δt5 ω5eq;m þ 5280ζ 3eq;m Δt3 ω3eq;m − 724ζ 2eq;m Δt6 ω6eq;m þ 1540ζ 2eq;m Δt4 ω4eq;m þ 6000ζ 2eq;m Δt2 ω2eq;m − 96ζ eq;m Δt7 ω7eq;m þ 144ζ eq;m Δt5 ω5eq;m þ 816ζ eq;m Δt3 ω3eq;m þ 2880ζ eq;m Δtωeq;m − 56Δt6 ω6eq;m þ 256Δt4 ω4eq;m − 216Δt2 ω2eq;m þ 720Þ=ð480πSΔt2 Þ − n3;m n2;m =n7;m ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering Downloaded From: http://risk.asmedigitalcollection.asme.org/ on 04/23/2015 Terms of Use: http://asme.org/terms

ðA11Þ

JUNE 2015, Vol. 1 / 021006-13

n5;m ¼ ½−ð349ζ 8eq;m Δt8 ω8eq;m þ 1002ζ 7eq;m Δt7 ω7eq;m þ 62ζ 6eq;m Δt8 ω8eq;m − 125ζ 6eq;m Δt6 ω6eq;m − 936ζ 5eq;m Δt7 ω70 − 546ζ 5eq;m Δt5 ω5eq;m − 361ζ 4eq;m Δt8 ω8eq;m − 895ζ 4eq;m Δt6 ω6eq;m þ 2071ζ 4eq;m Δt4 ω4eq;m − 462ζ 3eq;m Δt7 ω7eq;m þ 708ζ 3eq;m Δt5 ω5eq;m þ 4080ζ 3eq;m Δt3 ω3eq;m − 30ζ 2eq;m Δt8 ω8eq;m − 30ζ 2eq;m Δt6 ω6eq;m þ 238ζ 2eq;m Δt4 ω4eq;m þ 3120ζ 2eq;m Δt2 ω2eq;m − 16ζ eq;m Δt7 ω7eq;m þ 24ζ eq;m Δt5 ω5eq;m − 192ζ eq;m Δt3 ω3eq;m þ 1440ζ eq;m Δtωeq;m þ 4Δt8 ω8eq;m − 28Δt6 ω6eq;m þ 64Δt4 ω4eq;m − 168Δt2 ω2eq;m þ 360Þ=ð120πSΔt3 Þ − n1;m n3;m =n7;m

ðA12Þ

Note that the transition PDF does not depend on the initial tm−1 and final tm time points, but only on the time interval Δt ¼ tm − tm−1 . Furthermore, with the aid of the symbolic toolbox of MATLAB®, the integration in Eq. (31) can be performed analytically to yield the nonstationary response PDF of Eq. (32), with the coefficients k1;m , k2;m , k3;m , k4;m , k5;m , k6;m , k7;m given by k4;m ¼ ððn24;m ðk23;m−1 n26;m þ k24;m−1 k27;m−1 − 2k4;m−1 k3;m−1 n6;m n5;m þ k24;m−1 n25;m þ k27;m−1 n26;m ÞÞ=ðk23;m−1 n22;m þ k23;m−1 n26;m þ k24;m−1 k27;m−1 − 2k4;m−1 k3;m−1 n2;m n1;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k24;m−1 n21;m þ k24;m−1 n25;m þ k27;m−1 n22;m þ k27;m−1 n26;m þ n21;m n26;m − 2n2;m n1;m n6;m n5;m þ n22;m n25;m ÞÞ0.5

ðA13Þ

k1;m ¼ −ðn4;m ðk5;m−1 n1;m k24;m−1 k7;m−1 þ k1;m−1 n2;m k4;m−1 k27;m−1 − k3;m−1 k5;m−1 n2;m k4;m−1 k7;m−1 − k1;m−1 n1;m k4;m−1 n6;m n5;m þ k1;m−1 n2;m k4;m−1 n25;m þ k5;m−1 n1;m k7;m−1 n26;m − k5;m−1 n2;m k7;m−1 n6;m n5;m þ k1;m−1 k3;m−1 n1;m n26;m − k1;m−1 k3;m−1 n2;m n6;m n5;m ÞÞ=ðððn24;m ðk24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m ÞÞðk24;m−1 k27;m−1 þ k24;m−1 n21;m þ k24;m−1 k25;m−1 − 2k4;m−1 k3;m−1 n2;m n1;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n22;m þ k23;m−1 n26;m þ k27;m−1 n22;m þ k27;m−1 n26;m þ n21;m n26;m − 2n2;m n1;m n6;m n5;m þ n22;m n25;m ÞÞ0.5 Þ

ðA14Þ

k3;m ¼ ðn4;m ðn3;m k24;m−1 k27;m−1 þ n3;m k24;m−1 n25;m − n1;m n7;m k24;m−1 n5;m − 2n3;m k4;m−1 k3;m−1 n6;m n5;m þ n1;m n7;m k4;m−1 k3;m−1 n6;m þ n2;m n7;m k4;m−1 k3;m−1 n5;m þ n3;m k23;m−1 n26;m − n2;m n7;m k23;m−1 n6;m þ n3;m k27;m−1 n26;m − n2;m n7;m k27;m n6;m ÞÞ=ðððn24;m ðk24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m ÞÞðk24;m−1 k27;m−1 þ k24;m−1 n21;m þ k24;m−1 k25;m−1 − 2k4;m−1 k3;m−1 n2;m n1;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n22;m þ k23;m−1 n26;m þ k27;m−1 n22;m þ k27;m−1 n26;m þ n21;m n26;m − 2n2;m n1;m n6;m n5;m þ n22;m n25;m ÞÞ0.5 Þ

ðA15Þ

k2;m ¼ −ðn4;m ðk6;m−1 n1;m k24;m−1 k7;m−1 þ k2;m−1 n2;m k4;m−1 k27;m−1 − k3;m−1 k6;m−1 n2;m k4;m−1 k7;m−1 − k2;m−1 n1;m k4;m−1 n6;m n5;m þ k2;m−1 n2;m k4;m−1 n25;m þ k6;m−1 n1;m k7;m−1 n26;m − k6;m−1 n2;m k7;m−1 n6;m n5;m þ k3;m−1 k2;m−1 n1;m n26;m − k3;m−1 k2;m−1 n2;m n6;m n5;m ÞÞ=ðððn24;m ðk24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m ÞÞðk24;m−1 k27;m−1 þ k24;m−1 n21;m þ k24;m−1 k25;m−1 − 2k4;m−1 k3;m−1 n2;m n1;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n22;m þ k23;m−1 n26;m þ k27;m−1 n22;m þ k27;m−1 n26;m þ n21;m n26;m − 2n2;m n1;m n6;m n5;m þ n22;m n25;m ÞÞ0.5 Þ k7;m ¼

ðA16Þ

k24;m−1 k27;m−1 n27;m k24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m

0.5

ðA17Þ

k6;m ¼ −ðk4;m−1 k7;m−1 n7;m ðk2;m−1 k7;m−1 n6;m − k3;m−1 k6;m−1 n6;m þ k4;m−1 k6;m−1 n5;m ÞÞ=ððk24;m−1 k27;m−1 n27;m ðk24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m ÞÞ0.5 Þ

ðA18Þ

k5;m ¼ −ðk4;m−1 k7;m−1 n7;m ðk1;m−1 k7;m−1 n6;m − k3;m−1 k6;m−1 n6;m þ k4;m−1 k5;m−1 n5;m ÞÞ=ðððk24;m−1 k27;m−1 n27;m ðk24;m−1 k27;m−1 þ k24;m−1 n25;m − 2k4;m−1 k3;m−1 n6;m n5;m þ k23;m−1 n26;m þ k27;m−1 n26;m ÞÞ0.5 Þ

ki;1 ¼ ni;1 ;

i ¼ 1,2; : : : ; 6,7

021006-14 / Vol. 1, JUNE 2015


ðA19Þ

ðA20Þ Transactions of the ASME

Acknowledgment The first author gratefully acknowledges the financial support from China Scholarship Council.

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