Abstract
During flight test programmes structural response data may only be available in terms of accelerations, but relating these accelerations to stresses can be difficult without extensive analysis. Presented here is a method to quickly and accurately generate allowable acceleration levels to prevent fatigue failure. The method applies to lightly damped structures primarily excited at their fundamental mode such as antennae, radomes, and panels, but may be able to be conservatively extended if multiple modes exist.
The method relies on the structure primarily responding at its fundamental mode, and therefore that frequency can be used for the upwards crossing rate and also used to govern the relationship between acceleration and displacement. Furthermore, at the fundamental mode the relationship between displacement and detail stress can often be estimated based upon quasi-static considerations. Accurate understanding of the damping in the system is not required. Together these allows simplifications to be made to relate accelerations with stress and then fatigue damage.
Accuracy of the results is evaluated and quantified, and the situations where acceptable accuracy achieved is defined. Finally, a nomogram is provided to aid the rapid calculation of allowable acceleration levels.
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Notes
- 1.
Noting that Miles’ equation accuracy reduces at higher damping ratios and where the input PSD is not flat and care needs to be taken as any error may be compounded in this case, because we are relying on the ratio of Miles’ acceleration and Miles’ displacement (see details in the Verification Section).
- 2.
In many cases, perhaps a majority, we’ll be concerned with an applied force rather than a base input acceleration. In this case there is a Miles’ solution for displacement response, but the acceleration response is unbounded (see (Irvine, Derivation of Miles Equation for an Applied Force, Rev C., 2008a) for details). Therefore, the accuracy of the acceleration to displacement result will be affected by the bandwidth. This will be explored in the Verification Section.
- 3.
Where the nth spectral moment is calculated by \( m_{n} = \mathop \smallint \limits_{0}^{\infty } f^{n} S\left( f \right)df \).
- 4.
Note that this is the difference in UZC rate between broadband and narrowband signals. The UZC response for Base Input Acceleration and Applied Force PSDs will differ more, as is seen in Table 1.
- 5.
XRMS from (Irvine, Derivation of Miles Equation for an Applied Force, Rev C., 2008a) eq. 95 converted to a FRMS by multiplying through by stiffness k.
- 6.
Note: Although normal modes FEM runs cannot be used to assess stresses, then can be used to establish σ/δ relationships. The mode shapes are normalised, but here we are interested in the relationship between displacement and stress, so recovering both for the mode shape of concern allows an accurate σ/δ relationship to be developed.
- 7.
In the time domain sampling at least 10 times the highest frequency of interest is very important, but in the frequency domain this is less important. A level above the Nyquist frequency is, of course, a minimum requirement. The DOF is calculated as per (Irvine 2000).
- 8.
It is recommended that measured PSD data be examined in both Log-Log and linear domains in order to identify potentially damaging accelerations at lower frequencies that may seem less significant than larger magnitude results at higher frequencies.
- 9.
With a 100-hour desired safe life and 102 Hz effective frequency this means we’ll be looking at cycles above 1E7, so this is reasonable.
- 10.
For a more exact solution we could have used the model to generate an FRF between APSD and σPSD and then evaluated the σPSD using the Steinberg method or a more accurate solution like Dirlik (Halfpenny 1999). These results will be invariant with bandwidth assumptions but will require an accurate structural model.
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Acknowledgements
We thank Leif Roschier, developer of PyNomo software, for his assistance during the development of the GRMS nomogram.
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Dosman, S., Gorman, J. (2020). Rapid Calculation of Safe Acceleration Values for Aircraft Structures Under Flight Test. In: Niepokolczycki, A., Komorowski, J. (eds) ICAF 2019 – Structural Integrity in the Age of Additive Manufacturing. ICAF 2019. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-21503-3_18
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