6. A Covariance-Based Approach for Recovering Phase-Consistent Loads from Random Vibration Analysis

In the field of aerospace analysis, it is common practice to apply the root-mean-square (RMS) interface loads from a system-level random vibration analysis as static inputs to high-fidelity stress models. This process serves as an efficient means for evaluating several designs or design iterations; however, because the interface loads typically exhibit different magnitude and phase relationships at different times and frequencies, it is unrealistic to assume that all loads should be applied simultaneously at their maximum, in-phase magnitudes (often assumed to be the three-sigma or three times RMS loads). A more realistic approach for enveloping the problem space is to define a load vector (load case) for each interface degree of freedom (DOF), with a selected DOF applied at its maximum magnitude and all other DOFs applied at their phase-consistent magnitudes. These phase-consistent loads would seem easy to calculate through transient analysis by identifying the peak or valley with the desired magnitude for each DOF (e.g., three-sigma) and extracting the phase-consistent loads for the other DOFs; however, each peak or valley will produce a different load vector, since each peak or valley is merely a single data point in a statistical distribution. Thus, the preferred approach is to obtain phase-consistent loads in the frequency domain, such that the load vectors represent the expected magnitudes based on the statistical distribution of loads. This paper presents a frequency-domain, covariance-based approach for obtaining phase-consistent static loads from random vibration analysis. The load vectors obtained using this approach reasonably and realistically envelope the problem space and provide an efficient means for conducting high-fidelity random vibration stress analysis.