Use the modified Vanmarcke approximation to estimate the probability of first-passage failure of the linear oscillator considered in Example 11.3. It has resonant frequency ω0 and damping ζ = 0.01, and it is excited by stationary, Gaussian, white noise with mean zero and autospectral density S0.

Question

Use the modified Vanmarcke approximation to estimate the probability of first-passage failure of the linear oscillator considered in Example 11.3. It has resonant frequency [latex] \omega _{0} [/latex] and damping [latex] \zeta =0.01 [/latex] , and it is excited by stationary, Gaussian, white noise with mean zero and autospectral density [latex] S_{0} [/latex] . Failure occurs if [latex] X(t) [/latex] exceeds the level [latex] 4\sigma _{stat}=4(\pi S_{0})^{1/2}/(2m^{2}\zeta \omega ^{3}_{0})^{1/2} [/latex] within the time interval [latex] 0\leq \omega _{0} t\leq 250 [/latex] .

Accepted Answer

Considering [latex] \left\{X(t) ight\} [/latex] to have zero initial conditions requires that we use the nonstationary [latex] \eta _{X}(u,t) [/latex] from Eq. 11.30

[latex] \eta _{X}(u,t)\approx -v_{A}^{+}(u,t) \left\lgroup 1-\exp \left[ \frac{-v_{A_{1}}^{+}(u,t)}{v_{X}^{+}(u,t)} ight] ight group \left\lgroup1-\frac{v_{X}^{+}(u,t)}{v_{X}^{+}(0,t)} ight group ^{-1} [/latex]

In addition to the nonstationary crossing rate for [latex] \left\{X(t) ight\} [/latex] , we must now use the corresponding crossing rate for [latex] \left\{A_{1}(t) ight\} [/latex] .

This was given in Example 7.14, but we now modify it for the current purpose by adding a power of 1.2 to the [latex] (1-\alpha ^{2}_{1}) [/latex] term, as discussed in Section 11.4, giving

[latex] u ^{+}_{A_{1}}(u,t)=\frac{u}{\sigma _{X}(t)}\exp \left(-\frac{u^{2}}{2\sigma ^{2}_{X}(t)} ight) U(u)\frac{\sigma _{\dot{X}}(t)}{\sigma _{X}(t)} imes [/latex]

[latex] \left(\frac{ ho _{X\dot{X}}(t)u}{\sigma _{X}(t)}\Phi \left[\frac{ ho _{X\dot{X}}(t)u}{\sigma _{X}(t)\left([1-\alpha ^{2}_{1}(t)]^{1.2}- ho ^{2}_{X\dot{X}}(t) ight)^{1/2} } ight] + \left(\frac{[1-\alpha ^{2}_{1}(t)]^{1.2}- ho ^{2}_{X\dot{X}}(t)}{2\pi} ight)^{1/2}\exp \left[-\frac{1}{2}\left(\frac{ ho ^{2}_{X\dot{X}}(t)u^{2}}{\sigma ^{2}_{X}(t)\left([1-\alpha ^{2}_{1}(t)]^{1.2}- ho ^{2}_{X\dot{X}}(t) ight) } ight) ight] ight) [/latex]

Most of the time-varying terms in this expression were given in Example 11.3 and are the same here. The new information that is needed is the nonstationary bandwidth parameter [latex] \alpha _{1}(t) [/latex] , which was investigated in Example 7.10. Evaluation of [latex] \alpha _{1}(t)=\lambda _{1}(t)l[\lambda _{0}(t)\lambda _{2(t)}]^{1/2} [/latex] requires a formula for [latex] \lambda _{1}(t) [/latex] , and the expression for this term was given in Example 7.10 as

[latex] \lambda _{1}(t)=2S_{0}\left([1+e^{-2\zeta \omega _{0}t}]\int_{0}^{\infty }{\left|H_{x}(\omega ) ight|^{2} }\omega d\omega +\omega ^{-1}_{d}e^{-\omega _{0}t}\sin (\omega _{d}t) imes \left[\omega ^{2}_{0}\int_{0}^{\infty }{\left|H_{x}(\omega ) ight|^{2}\sin (\omega t)d\omega +\int_{0}^{\infty }{\left|H_{x}(\omega ) ight|^{2}\omega ^{2}\sin (\omega t)d\omega } } ight]-2e^{-\zeta \omega _{0}t}\cos (\omega _{d}t) \int_{0}^{\infty }{\left|H_{x}(\omega ) ight|^{2}\omega \cos (\omega t)d\omega } ight) [/latex]

Using this formula along with the approximations in Example 11.3 allows evaluation of the conditional rate of upcrossings in Eq. 11.30:

[latex] \eta _{X}(u,t)\approx u ^{+}_{X}(u,t)\left(1-\exp \left[\frac{- u ^{+}_{A_{1}}(u,t)}{ u ^{+}_{X}(u,t)} ight] ight) \left(1-\frac{ u ^{+}_{X}(u,t)}{ u ^{+}_{X}(0,t)} ight) ^{-1} [/latex]

Numerical integration of Eq. 11.13

[latex]L _{X}(u,t)= L_{X}(u,0)\exp \left(-\int_{0}^{t}{\eta _{X}(u,s)ds} ight)[/latex]

then gives [latex] L(u,250/\omega _{0})=0.99663 [/latex] , and the probability of survival is [latex] P_{f}\approx 0.0034 [/latex] . This value is approximately 55% of that obtained in Example 11.3 by the Poisson approximation.

Comparison of [latex] \eta _{X}(u,t) [/latex] and [latex] u ^{+}_{X}(u,t) [/latex] reveals a possible simplified approximation

of this problem. In particular, the accompanying sketch shows that [latex] \eta _{X}(u,t) [/latex] is almost proportional to [latex] u ^{+}_{X}(u,t) [/latex] . Thus, one might use an approximation of [latex] \eta _{X}(u,t)\approx u ^{+}_{X}(u,t)[\eta _{X}(u,\infty )/ u ^{+}_{X}(u,\infty )] [/latex] . For the current problem the ratio of stationary crossing rates is 0.5166, so [latex] \eta _{X}(u,t)\approx 0.5166 u ^{+}_{X}(u,t) [/latex] and Eq. 11.13 yields

[latex] L_{Vanm}\approx (L_{poisson})^{0.5166}=0.9968 [/latex] , [latex] P_{f,Vanm}\approx 0.0032 [/latex]

in which we have used the Poisson result from Example 11.3.The result is seen to be quite accurate, and it can be obtained with the much simpler integration of the unconditional nonstationary crossing rate, in place of the conditional rate.

Question 11.6: Use the modified Vanmarcke approximation to estimate the pro...

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