Model Uncertainty Factors (MUFs)
Model Uncertainty Factors (MUFs) are multiplicative scaling factors applied to analysis results to account for Finite Element Model (FEM) errors. The MUF accounts explicitly for modal gain errors in the predicted system response. For flight vib analysis, it typically takes the form of a frequency dependent scaling factor that is multiplied into the predicted response. For tonal disturbances, this is done in the frequency domain for each harmonic prior to collapsing the responses from each harmonic into the response versus wheelspeed. For time simulations, the MUF is applied to the observation or force influence matrices as a mode-by-mode scale factor.
Typically the MUF is determined by looking at prediction versus measurement accuracy from previous programs, by comparing predicted to measured Frequency Response Functions (FRFs). Generally, there are at least two separate frequency regimes, a low-frequency regime where there is separation between modes, and a high frequency regime where the modal spacing is dense. In the former, the error is in the modeshape itself, where it can be more disturbable or more observable than predicted. In the higher regime, the modeshape uncertainty is augmented by the coupling between adjacent modes (they can add to one another or subtract), so that shifts in modal frequency alone can cause an increase in the amplification from the adjacent modes. As a general rule observed across many missions, CLA minimum-frequency requirements result in system modes that are fairly uncoupled below about 20 Hz, and by about 40 Hz there is sufficient modal density to require protection against adjacent mode frequency separation shift. For the transition region 20-40 Hz, a linear ramp can be used. These basic parameters can be adjusted, for example with a larger transition range, and through the use of a smoothed off transition.
The specific values for the low- and high-frequency regimes will often be adjusted over the mission, starting high when models are coarser and reducing over time as component models are gradually test-verified. As an example, the low-frequency value may start at 3 in the architecture definition phase and be reduced to 1.5 by the pre-ship analysis, and the high-frequency value may start at 6 and end at 3.
Individual missions can tailor these basic parameters based on their needs. For example, a high-value mission that is currently under development performed a physical parameter Monte Carlo analysis to determine the variability in optical response. Based on this they use a 3-regime MUF curve, with low-, mid-, and high-frequency values.
Other error sources should be accounted for in the analysis, separately from the MUF:
Modal frequency shifts: adjust the input frequency range where appropriate to account for the possibility of specific modal frequencies being wrong in the FEM. For example, if the RWA drive range is up to 40 rev/sec (2400 RPM), analyze up to 44 rev/sec to account for critical modes just outside the drive frequency range in the model, that may in reality fall inside the drive range.
Damping uncertainties: typically a conservative value for damping will be used in the analysis. Typical values are 0.1% for room temperature and 0.02% for cryogenic structures. Note that this is much lower than typical values used for launch loads analysis, so that there must be a robust model management process that ensures that all lower level models are delivered with damping values appropriate to both regimes, and that system-level FE solutions are run with the correct damping values. Conversely, damping for jitter may be higher than damping values assumed for Attitude Control System (ACS) stability analysis, where the possibility of controller instability warrants a bounding assumption (lowest expected with a factor of safety).
Disturbance input levels: there is uncertainty in the forcing amplitudes. For example, the tonal disturbance signature of RWAs will vary from wheel to wheel, and can change after the RWA is subjected to high loading conditions, such as during vibe testing and launch. Typically a single bounding wheel model is developed from multiple RWA Exported Force/Torque (EFT) measurements, encompassing all the wheels, pre- and post-vibe. The bounding model may be a max-over-all-datasets, or can be a 1- or 2-sigma model assessed from mean and variance, or could be a specified percentile by eliminating a certain number of peak values over the datasets.
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