Let the mean-zero, Gaussian stress process $\left\{X(t)\right\}$ be the response of an SDF oscillator excited by white noise. For a specimen with $m = 4$ , compare the fatigue lives predicted by the Rayleigh approximation, the peak-approximation method, the single-moment method, the formula of Ortiz and Chen, and the formula of Wirsching and Light.

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From the results in Section 6.8, we can write the autospectral density for [latex] \left\{X(t)\right\} [/latex] as

[latex] S_{XX}(\omega )=S_{0}m^{-2}\left([\omega ^{2}_{0}-\omega ^{2}]^{2}+[2\zeta \omega _{0}\omega ]^{2}\right) ^{-1} [/latex]

in which [latex] \omega _{0} [/latex] and [latex] \zeta [/latex] are the resonant frequency and the damping of the oscillator and [latex] S_{0} [/latex] is the autospectral density of the white noise excitation. From the results in either Chapter 5 or 6, we know the values of [latex] \lambda _{0} [/latex] and [latex] \lambda _{2} [/latex] to be

[latex] \lambda _{0}\equiv \sigma ^{2}_{X}=\frac{\pi S_{0}}{2m^{2}\zeta \omega ^{3}_{0}}, \lambda _{0}\equiv \sigma ^{2}_{\dot{X}}=\frac{\pi S_{0}}{2m^{2}\zeta \omega _{0}}, [/latex]

Using these two spectral moments in Eqs. 11.50 [latex]E(\Delta D)=K^{-1}(2)^{3m/2}\lambda _{0}^{m/2}\Gamma \left(1+\frac{m}{2} \right) [/latex] and 11.51 [latex] E(T)= \frac{2\pi \sigma _{X}}{E(\Delta D)\sigma _{X}} =\frac{2\pi }{E(\Delta D)}\left(\frac{\lambda _{0}}{\lambda _{2}} \right) ^{1/2}[/latex] with [latex] m = 4 [/latex] gives the Rayleigh approximation result as

[latex] E(T_{Ray})=\frac{2\pi K}{2^{6}\sigma ^{4}_{X}\Gamma (3)\omega _{0}} =\frac{Km^{4}\zeta ^{2}\omega ^{5}_{0}}{16\pi S^{2}_{0}} [/latex]

It is not possible to apply some of the other spectral methods to this particular problem. In particular, this [latex] S_{XX}(\omega ) [/latex] gives [latex] \lambda _{4}=\infty [/latex] , as was noted in Example 7.9. Thus, we have [latex] IF=0 [/latex] , for which Eq. 11.54 [latex] E(T)=\frac{2\pi K}{2^{m}E[P^{m}(t)\mid P(t)\geq 0]} \frac{2(IF)}{(1+IF)} \frac{\sigma _{X}}{\sigma _{\dot{X} }} = \frac{4\pi K(IF)}{2^{m}(1+IF)E[P^{m}(t) \mid P(t)\geq 0]} \left(\frac{\lambda _{0}}{\lambda _{2}} \right) ^{1/2}[/latex] gives [latex] E(T_{peak})=0 [/latex] , and Eq. 11.58 [latex]CF=a+(1-a)\left(1-[1-(IF)^{2}]^{1/2}\right) ^{b}[/latex] gives [latex] RR=(CF)^{-1}=a^{-1}\approx 1.26 [/latex] for the approximation of Wirsching and Light. Similarly, Eq. 11.56 56 [latex] E(T)=\frac{2\pi K (\lambda _{2+2/m})^{m/2}}{(2)^{3m/2}\Gamma \left(1+\frac{m}{2} \right)(\lambda _{2/m})^{m/2} \lambda _{2}^{(m-1)}\lambda _{4}^{1/2}} [/latex] gives [latex] E(T_{oc})=0 [/latex] for the Ortiz and Chen approximation, because [latex] \lambda _{4}=\infty [/latex] and the other pertinent spectral moments are finite.

To apply the single-moment method to a situation with [latex] m = 4 [/latex] , we must calculate the [latex] \lambda _{2/m}=\lambda _{0.5} [/latex] spectral moment. For our autospectral density, this can be written as

[latex] \lambda _{0.5}=2\int_{0}^{\infty }{\omega ^{1/2}} S_{XX}(\omega )d\omega =\frac{2S_{0}}{m^{2}\omega ^{5/2}_{0}} \int_{0}^{\infty }{\frac{\Omega ^{1/2}d\Omega }{(1-\Omega ^{2})^{2}+(2\zeta \Omega )^{2}} } [/latex]

in which the dimensionless variable [latex] \Omega [/latex] is [latex] \omega /\omega _{0} [/latex] . This integral can be evaluated numerically to obtain values that can be used in Eq. 11.55 [latex] E(T)=\frac{2\pi K}{(2)^{3m/2}\Gamma \left(1+\frac{m}{2} \right)(\lambda _{2/m})^{m/2} } [/latex] to give single-moment results. One convenient way to present these numerical results is as the [latex] RR [/latex] Rayleigh ratio of Eq. 11.57 [latex] RR=\frac{E(T)by \cdot alternative \cdot approximation}{E(T) by \cdot Rayleigh \cdot approximation} [/latex] . These [latex] RR [/latex] values have been computed for [latex] \zeta [/latex] values varying from zero to unity, and it is found that they agree with [latex] RR=1+\zeta [/latex] , which implies that

[latex] \lambda _{0.5}=\frac{\pi S_{0}}{2m^{2}\omega ^{5/2}_{0}\zeta (1+\zeta )^{1/2}} [/latex]

For [latex] IF=0 [/latex] , the Wirsching and Light formulas give a value of [latex] RR [/latex] that depends only on [latex] m [/latex] , not on the properties of the autospectral density function. This empirical approximation, thus, gives the rather unusual prediction that the rainflow fatigue life will be 26% greater than the Rayleigh approximation, regardless of the amount of damping in the oscillator. This value of [latex] RR [/latex] agrees with the

single-moment result only for [latex] \zeta \approx 0.26 [/latex] . The various results are sketched versus damping value.

This is an example in which the single-moment method is the only one of the alternative spectral methods that gives what one might consider to be a bandwidth correction for the Rayleigh approximation. That is, the bandwidth of

the oscillator response is generally considered to depend on [latex] \zeta [/latex] , and only the single-moment method gives an [latex] RR [/latex] value that depends on [latex] \zeta [/latex] .

Question 11.10: Let the mean-zero, Gaussian stress process { X(t) } be the r...

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