*This guest contribution on the Altair Blog is written by **the CAEfatigue Team, **a member of the Altair Partner Alliance.*

The following blog post will explain the need for good frequency definition in both the Solver FRF analysis and the Input PSD so that proper resonances are found in the Transfer Function an frequency content is matched up between the FRF analysis and the Input PSD. Understanding these concepts will ensure that the intended conditions are defined to calculate accurate fatigue damage in the Frequency Domain using CAEfatigue VIBRATION (CFV).

Let’s use the example below:

In order to calculate fatigue damage and fatigue life in the frequency domain, we first need to generate the **RESPONSE PSD** that is the result of multiplying the **TRANSFER FUNCTION** by the **INPUT PSD**. The Transfer Function is calculated within CFV using the solver FRF stress data, and the Input PSD is defined directly within CFV. Below is an example of the two PSD’s and connecting transfer function as presented by the PSD Plotter in the CFV GUI (CFG).

As can be seen in the plot above, the shape and frequency content of the Response PSD is heavily dependent on the shape and frequency content of the Transfer Function. If care is not taken in the Solver FRF calculation, it would be very possible to miss, for example, the peak amplitude at 14 Hz in the Transfer Function and thus, it would completely change the shape and content of the Response PSD. We will look at this more closely in a different example below.

**Frequency Resolution and Resonance Detection**

Below are two Transfer Functions that were generated within CFV from two different Solver FRF analyses. In one case, the peak amplitudes near 40-45 Hz are recognized and accounted for by the user (blue, solid line) whereas, in the second case, the user failed to recognize the peak contribution (resonance) and bypassed the 40-45 Hz frequency in the FRF analysis settings (orange, dashed line).

If we now use these Transfer Functions with a simple Input PSD, we can see the influence this mistake has on the shape and frequency content of the Response PSDs.

It is clear to see that the Response PSDs have different shapes and peak values. Using the correct Transfer Function (blue, solid line), the frequency resolution in the FRF analysis has picked up the response peak at 40-45 Hz and the correct amplitude values are passed through to the Response PSD.

However, in the second case (orange, dashed line), the Response PSD falsely shows that the max peak response is at 25 Hz and completely misses the significant resonances at 40-45 Hz.

This mistake by the user, will likely cause a significant difference in the fatigue damage and life calculations because the cumulative areas under both curves are very different, which significantly changes the M_{o} value used by the Dirlik method (most common damage calculation method used in frequency domain fatigue calculations). Most solvers have parameters (like ** FREQ1** and

**in Nastran), which guide the user in choosing correct points around such resonances. It is very important to pay close attention to the FRF analysis to accurately reflect the response of the model.**

*FREQ4***Frequency Matching**

A keen observer may also have notice something else that is incorrect in our example above. Although the blue, solid line Transfer Function does accurately reflect the response of the model, the user failed to recognize that the points in the Transfer Function do not match up well with the points in the Input PSD, especially at 5 Hz, i.e. there is no Transfer Function frequency point at 5 Hz to match up with the 5 Hz point in the Input PSD. Nearest points in the Transfer function are at 0.5 Hz and 25 Hz.

Within CFV, the Response PSD is generated by multiplying the value of the Transfer Function PSD points by the corresponding value of the Input PSD points. When a Transfer Function point does not align with an Input PSD point, the nearest points are ** interpolated to generate a new value** at the frequency value from the Transfer Function. This creates the green dot on Input PSD.

However, the reverse does not happen; i.e. if a point on the Input PSD does not line up with a point on the Transfer Function, that Input PSD point is ** ignored** in the Response PSD calculation (red arrow, empty red circle on Response PSD).

CFV does not change the Solver FRF stress results to add in or take away points to account for the Input PSD frequencies. This could, in theory, be attempted for single input (real) PSD’s with corresponding real transfer functions but there may be difficult theoretical issues to resolve. However, it cannot be done for multi-inputs because of the difficulty to interpolate across complex cross PSD’s and the corresponding complex transfer functions. Hence, CFV requires the user to ensure the resolution is sufficient in the FRF to match the Input PSD as well as match the true response of the model.

In this example, the Transfer Function does not have a defined point at 5 Hz so the contribution of the Input PSD at 5 Hz is ignored. Hence, this frequency mismatch produces an incorrect Response PSD at 5 Hz. Effectively, the Input PSD is treated differently based on the Transfer Function points and behaves like the modified Input PSD below where the red dots line up with the Transfer Function points and the blue dots are ignored.

Therefore, to ensure the fatigue damage and life calculations are accurate, the user must consider frequency resolution ** and** frequency matching between the Solver FRF analysis and the Input PSD definition. Below are the correct Input PSD, Transfer Function and Response PSD for this example, which reflect appropriate frequency content and matching.

Good frequency content and good frequency matching can be accomplished by a careful understanding of the expected resonances in the model and the desired PSD input loading that will be applied as part of the fatigue analysis. Once understood, an appropriate choice of solver parameters (such as “FREQ1” and “FREQ4” entries for Nastran) can be made to ensure the desired frequency resolution is defined to pick up resonances and where the loading changes significantly in the Input PSD, i.e. for typical road load data, the Input PSD will likely require good solver FRF frequency resolution in the 1.0 Hz to 10.0 Hz range.

This post was intended to provide an understanding of how frequency domain fatigue analysis is conducted in CFV. Remember, providing accurate data in, will ensure accurate results out.

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