Rapid Frequency Domain Vibration Analysis Part 3 – Method and Verification

In parts 1 and 2 we looked at how to identify suitable fatigue SN data and how to relate stress to displacement – now we’ll get into the meat of the method – how to assess fatigue life from a measured acceleration. We’ll also briefly cover verification of the method, but you really need to see our paper [1] to get the detailed verification results and accuracy charts.

Links --- [Part 1] --- [Part 2] --- [Part 3] --- [Part 4]

Rapid Frequency Domain Vibration Analysis Method and Verification

Acceleration does not generate stresses in structures, and it is the stresses that we are considering here. High acceleration at high frequency leads necessarily to very small displacements, and it is the displacement that relates directly to stress; generally, either by extending a structure in an axial fashion, or by generating curvature or shear, say a beam in bending/shear. Therefore, if we can express the relationship between displacement at the accelerometer position for the fundamental mode and the local most critical stress and if we have a way of converting acceleration to displacement then we have a means of relating accelerations to stresses.   This mode shape relationship between displacement and stress can be found by standard texts, by FEM {1}, or through approximations.

With this information available, and knowledge of Miles’ Equation {2} (ref [2]) and the assumption that the structure is primarily acting as a SDOF oscillator, then the following relationships are appropriate:

acceleration {3} [2], ASD is the acceleration spectral density in g2/Hz, and Q is the transmissibility at fn, the natural frequency

So together

inches/s^2 basis. This is the exact solution for a sinusoidal excitation, but only approximate for an ASD input (see verification in [1] for further discussion). 

When the damage law follows the Basquin curve, and when it is presumed that the upward zero crossings frequency is the same as the dominant mode. If other damage laws are appropriate, then similar relations can be generated and used.

The relationship between stress at the critical detail and displacement at the accelerometer location for the mode shape in question is

Leading to

Alternatively, if one is using RMS based SN data and has a defined s_RMS allowable then:

Verification

Detailed verification of the method and definitive accuracy figures is outside the scope of this article – for that see my paper, ref [1]; however some example results are provided here to give a feel for how the assumptions in Miles’ equation effect the accuracy of the results.

Example – Applied Force vs Applied Acceleration:

Here we have modelled a SDOF system, mass = 0.1 [kg], stiffness K = 1E5 [N/m], 0.05 Critical Damping, and applied 0.01 [g2/Hz] PSD base acceleration or 0.001 [lb2/Hz] (0.0198 [N2/Hz]) applied force over one octave each side of the natural frequency (fn = (K/m)0.5 = 1000 [rad/s] = 159.2 [Hz]). This gives us the results in Figure 1 and Figure 2, with the RMS summaries provided in Table 1. It can be seen that the mass and stiffness lines (i.e. the slopes above and below the resonance) are very different between the two, and this affects the accuracy of the Miles’ Equation based ratio used here.

Figure 1 - Example PSD for SDOF under random APSD loading, Flat 1 octave each side of fn

Figure 2 - Example PSD for SDOF under random FPSD loading, Flat 1 octave each side of fn

(Note the differences in the the Mass and Stiffness Lines between the two figures)

Table 1 - Summary of RMS results for example SDOF system

In this example the Base Input Acceleration situation we can see that for a 1-octave wide band each side of the resonance the Mile’s equation overestimates the displacement to acceleration ratio very slightly, thus slightly overestimating stress for a given acceleration, which will be proportional to displacement (For the exact version of Miles’ equation the spectral results and Miles’ are equal). We can’t do the same comparison for an applied force, as Miles’ equation acceleration RMS for white noise is infinite (see [3] for discussion);

Summary

The primary result here is equation (12) which relates G_RMS to a safe life based upon the natural frequency, the detail fatigue parameters, and the relationship between stress and displacement for the dominant mode shape at the natural frequency.

In the final instalment we’ll bring everything together to provide a general method of assessment.

Links --- [Part 1] --- [Part 2] --- [Part 3] --- [Part 4]

? Stephen Dosman, 2019. Unauthorized use and/or duplication of this material without express and written permission from the author is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Stephen Dosman and this site with appropriate and specific direction to the original content.

Footnotes

{1} A NASTRAN SOL103 modal analysis can be used to generate the stress to displacement functions for specific eigenvectors without the need to generate a full frequency response analysis for example.

{2} 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).

{3} 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 [3] 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.

Credit: xkcd, Resonance

References

[1] S. Dosman, & J. Gorman “Rapid Calculation of Safe Acceleration Values for Aircraft Structures under Flight Test,” in 30th ICAF Symposium, Krakow, 2019.

[2] R. Simmons, “Miles' Equation,” 2001. [Online]. Available: https://femci.gsfc.nasa.gov/random/MilesEqn.html.

[3] T. Irvine, “Derivation of Miles Equation for an Applied Force, Rev C.,” Vibrationdata, Madison, Alabama, 2008.

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