Compare the ηX(U) values from Eqs. 11.17, 11.23, 11.25, and 11.26 for a stationary, mean-zero, Gaussian, narrowband process with α1= 0.995. (One particular process with this value of α1 is the response to white noise of an SDF oscillator with approximately 0.8% of critical damping.)

Question

Compare the [latex] \eta _{X}(u) [/latex] values from Eqs. 11.17 [latex] \eta _{X}(u,t)\approx v_{X}^{+}(u,t)[/latex], 11.23 [latex] \eta _{X}(u,t)\approx \frac{v_{X}^{+}(u,t)}{F_{X(t)}(u)} [/latex] , 11.25 [latex] \eta _{A}(u,t)\approx v_{A}^{+}(u,t) [/latex] , and 11.26 [latex] \eta _{A}(u,t)\approx \frac{v_{A}^{+}(u,t)}{F_{A(t)}(u)} [/latex] for a stationary, mean-zero, Gaussian, narrowband process with [latex] \alpha _{1}=0.995 [/latex] . (One particular process with this value of [latex] \alpha _{1} [/latex] is the response to white noise of an SDF oscillator with approximately 0.8% of critical damping.)

Accepted Answer

The basic upcrossing rate [latex] u ^{+}_{X}(u) [/latex] for this process was found in Example 7.1 to be

[latex] u ^{+}_{X}(u)=\frac{\omega _{c2}}{2\pi}\exp \left(\frac{-u^{2}}{2\sigma ^{2}_{X}} ight) [/latex]

in which [latex] \omega _{c2}=\sigma _{\dot{X}}/\sigma _{X} [/latex] . This, then, is the approximation of [latex] \eta _{X}(u) [/latex] for Eq. 11.17

[latex] \eta _{X}(u,t)\approx v_{X}^{+}(u,t)[/latex]

Because this is the most commonly used approximation, we will present our comparisons for each of the other three approximations as the ratio [latex] \eta _{X}(u)/ u ^{+}_{X}(u) [/latex] . (This normalization of the results has been commonly used since Crandall et al. in 1966.) The [latex] \eta _{X}(u)/ u ^{+}_{X}(u) [/latex] ratio for Eq. 11.23

[latex] \eta _{X}(u,t)\approx \frac{v_{X}^{+}(u,t)}{F_{X(t)}(u)} [/latex]

of course, is simply [latex] [F_{X}(u)]^{-1} [/latex] , which is [latex] [\Phi _{X}(u/\sigma _{X})]^{-1} [/latex] for a mean-zero Gaussian process.

The amplitude upcrossing rates for this process were found in Examples 7.12 and 7.13, respectively, for both the Cramer and Leadbetter and the energy-based definitions. The result for the energy-based amplitude, though, depends on [latex] \alpha _{2} [/latex] , which is not given in the statement of this problem and cannot be determined from the given value of [latex] \alpha _{1} [/latex] . (Recall, for example, that [latex] \alpha _{2}=0 [/latex] for the response of any SDF oscillator excited by white noise.) The Cramer and Leadbetter result for Eq. 11.25

[latex] \eta _{A}(u,t)\approx v_{A}^{+}(u,t)[/latex]

is

[latex] \frac{\eta _{X}(u)}{ u ^{+}_{X}}=\frac{ u ^{+}_{A_{1}}(u)}{ u ^{+}_{X}(u)}=\frac{u}{\sigma _{X}}[2\pi (1-\alpha ^{2}_{1})]^{1/2}\approx 0.250\frac{u}{\sigma _{X}} [/latex]

Because the amplitude has the Rayleigh distribution, the additional term needed to obtain the approximation of Eq. 11.26

[latex] \eta _{A}(u,t)\approx \frac{v_{A}^{+}(u,t)}{F_{A(t)}(u)} [/latex]

is

[latex] F_{A}(u)=1-\exp \left(\frac{-u^{2}}{2\sigma ^{2}_{X}} ight) [/latex]

The accompanying sketch shows the values of [latex] \eta _{X}(u)/ u ^{+}_{X}(u) [/latex] versus [latex] u/\sigma _{X} [/latex] for Eqs. 11.17 and 11.23, and for Eqs. 11.25 and 11.26 using the Cramer and Leadbetter amplitude.

From the plot we see that the approximations give quite different results for small values of [latex] u [/latex] , which may

not be of too much practical significance because failure is generally related to large values of [latex] u [/latex] . We also see, though, that the [latex] X [/latex] crossing and the amplitude-crossing results behave very differently for the crucial region with large values of [latex] u [/latex] . In particular, the [latex] u ^{+}_{A}(u)/ u ^{+}_{X}(u) [/latex] ratio grows linearly with [latex] u [/latex] , so the amplitudecrossing result eventually becomes more conservative (i.e., predicts larger probabilities of upcrossings) than the original Poisson approximation.

Question 11.4: Compare the ηX(U) values from Eqs. 11.17, 11.23, 11.25, and ...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions