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Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique


An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.

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H. Vanvinckenroye was supported by the National Fund for Scientific Research of Belgium. This research has been conducted during a visit of H. Vanvinckenroye at Columbia University

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Comparison between Galerkin scheme and analytical expression for the average and mean square first-passage times

Comparison between Galerkin scheme and analytical expression for the average and mean square first-passage times

Considering a linear undamped oscillator under white noise excitation (\(S_{w}(\omega ,t)=S_{0}\)), since \({\dot{T}}_{i}(t)=-\pi \lambda _{i}S_{0}T_{i}(t)\) and \({\dot{c}}_{i}(t)=0\), Eq. (44) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {E}}\left\{ \tau _{H_{c}}\right\} = &{} \sum _{i}A_{i}\varPhi _{i}(H)\ \text {with}\ A_{i}=\frac{1}{\pi \lambda _{i}S_{0}}\frac{F_{1,i}}{F_{2,i}}\\ {\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} = &{} \sum _{i}B_{i}\varPhi _{i}(H)\ \text {with}\ B_{i}=\frac{2}{\pi ^{2}\lambda _{i}^{2}S_{0}^{2}}\frac{F_{1,i}}{F_{2,i}} \end{array}\right. }\nonumber \\ \end{aligned}$$

Analytical results are derived for this particular case in [7, 8] for the mean and mean square first-passage times \({\mathcal {E}}\left\{ \tau _{H_{c}}\right\} \) and \({\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} \) based on a stochastic averaging treatment and are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {E}}\left\{ \tau _{H_{c}}\right\} = &{} \frac{1}{\pi S_{0}}\varDelta H\\ {\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} = &{} \frac{1}{\pi ^{2}S_{0}^{2}}\varDelta H\left( H+\frac{3}{2}\varDelta H\right) \end{array}\right. } \end{aligned}$$

where \(\varDelta H=H_{c}-H\) is the energy increase. The equalities to be proven are

$$\begin{aligned} \frac{1}{\pi S_{0}}\sum _{i}\frac{1}{\lambda _{i}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)=\frac{1}{\pi S_{0}}\varDelta H \end{aligned}$$

for the average first-passage time, and

$$\begin{aligned}&\frac{2}{\pi ^{2}S_{0}^{2}}\sum _{i}\frac{1}{\lambda _{i}^{2}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)\nonumber \\&\quad =\frac{1}{\pi ^{2}S_{0}^{2}}\varDelta H\left( H+\frac{3}{2}\varDelta H\right) \end{aligned}$$

for the mean square first-passage time. The lemmas (57) and (58) are demonstrated by projection of the two sides of the equations on the basis of eigenfunctions \(\varPhi _{j}(H)\). Equality is proven by showing that these projections are identical for all \(\varPhi _{j}(H)\).

First, Eq. (57) is proven. Then, the second demonstration for Eq. (58) results in a similar way from the first one.

Invoking the orthogonality property (20) with \(f(H)=1\) , the projection of Eq. (57) on the eigenfunctions \(\varPhi _{j}(H)\) yields

$$\begin{aligned}&\frac{1}{\pi S_{0}}\int _{0}^{H_{c}}\left( \sum _{i}\frac{1}{\lambda _{i}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)\right) \varPhi _{j}(H)\text {d}H \nonumber \\&\quad =\frac{F_{1,j}}{\pi \lambda _{j}S_{0}}=\frac{1}{\pi S_{0}}\int _{0}^{H_{c}}\left( H_{c}-H\right) \varPhi _{j}(H)\text {d}H \end{aligned}$$

that can be rewritten, according to (23), as

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H=\int _{0}^{H_{c}}\lambda _{j}\varPhi _{j}(H)\left( H_{c}-H\right) \text {d}H. \end{aligned}$$

Considering the definition of the Bessel function (38), one finds

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\int _{0}^{H_{c}}\left( H\varPhi _{j}''(H)+\varPhi _{j}'(H)\right) \left( H_{c}-H\right) \text {d}H\nonumber \\&\quad =-\int _{0}^{H_{c}}\left( H\varPhi _{j}'(H)\right) '\left( H_{c}-H\right) \text {d}H. \end{aligned}$$

Integration by parts of (61) leads to

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} \left[ \left( H\varPhi _{j}'(H)\right) \left( H_{c}-H\right) \right] _{0}^{H_{c}}\nonumber \\&-\int _{0}^{H_{c}}H\varPhi '(H)\text {d}H. \end{aligned}$$

The first term in the right-hand side is trivially equal to zero while a second integration by parts of the second term leads to

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} -\left[ H\varPhi _{j}(H)\right] _{0}^{H_{c}}\nonumber \\&+\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H. \end{aligned}$$

Accounting for the boundary condition \(\text {BesselJ}(0,\sqrt{4\lambda _{i}H_{c}})=0\) (21), equality is proven.

The demonstration is similarly done for the mean square first-passage time. After projection on the eigenfunction \(\varPhi _{j}(H)\), Eq. (58) yields

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =\frac{1}{4}\int _{0}^{H_{c}}\left( H_{c}-H\right) \left( 3H_{c}-H\right) \lambda _{j}^{2}\varPhi _{j}(H)\text {d}H.\nonumber \\ \end{aligned}$$

Considering the definition of the Bessel function (38), one finds

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\frac{1}{4}\int _{0}^{H_{c}}\left( H_{c}-H\right) \left( 3H_{c}-H\right) \lambda _{j}\nonumber \\&\qquad \times \left( H\varPhi _{j}''(H)+\varPhi _{j}'(H)\right) \text {d}H. \end{aligned}$$

A first integration by parts leads to

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\frac{1}{2}\lambda _{j}\left[ H\varPhi '(H)\left( H_{c}-H\right) \left( 3H_{c}-H\right) \right] _{0}^{H_{c}}\nonumber \\&\qquad -\frac{\lambda _{j}}{2}\int _{0}^{H_{c}}\left( H-2H_{c}\right) H\varPhi _{j}'(H)\text {d}H. \end{aligned}$$

The first term in the right-hand side is equal to zero, while the second one can be developed using (38) and a second integration by parts and yields

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} \left[ H\varPhi '(H)(H-H_{c})\right] _{0}^{H_{c}}\nonumber \\&-\int _{0}^{H_{c}}H\varPhi _{j}'(H)\text {d}H, \end{aligned}$$

which proves equality.

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Vanvinckenroye, H., Kougioumtzoglou, I.A. & Denoël, V. Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique . Nonlinear Dyn 95, 293–308 (2019).

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  • Galerkin scheme
  • Reliability assessment
  • First-passage time
  • Nonlinear oscillator
  • Evolutionary excitation
  • Backward Kolmogorov equation