A Methodology for Feature Selection and Electrical Capability Prediction of a Coupled Shaker-DUT Model 71 Coupling the example matrix 8 with the stiffness matrix from equation 1 at DOF 3 gives the example coupled matrix of k11 k12 k13 0 0 0 k21 k22 k23 0 0 0 k31 k32 k33 +kglue −kglue 0 0 0 0 −kglue k1 +kglue −k1 0 0 0 0 −k1 k1 +k2 Kf 0 0 0 0 0 R . (9) Because the overarching goal of this project was to model a shaker’s ability to run an environmental test with a given DUT, a flat ASD of 0.0125(m/s2)2/Hz was controlled at a specified point on the BARC, on both the physical and modeled BARC. The specified point was the retained node (see Section 3.1) on the removable component of the BARC. The FRFs from this data were calculated using methods described above (see Section 3.3), and were algebraically manipulated to solve for the electromotive force, resulting in VSD: power spectral density (V2/Hz) as a function of frequency (Hz). VSD is the form of data that was used to compare classes within the Bayesian classifier. Additionally, the coupled system model was validated against the experimental ground truth by comparing their VSDs and showing they qualitatively matched (see section 4 and Figure 5). Bayesian classification methodology Bayesian classification theory A Na¨ıve Bayesian classifier works by classifying an object Xas class C when the posterior probability is highest, that is, when P(Ci|X) for i =1, ...,mis maximum. Bayes’ Theorem shows that P(Ci|X)= P(X|Ci)P(Ci) P(X) . (10) P(X), the probability that Xis the observation chosen, is constant in equation 10. All classes are assumed to be equally likely, that is, P(C1)=P(C2)=... =P(Cm) - a valid assumption since a constant number of observations was used from each class. Therefore, in equation 10, the only non-constant value to be maximized on the right-hand side is P(X|Ci) [12]. In Na¨ıve Bayesian Classification, it is assumed that all mpoints in each observation are independent of each other. The densities of each class’s multivariate normal distribution were calculated under this assumption. These densities were normalized by dividing each class by the total density of all classes, resulting in the probability that the observation came from class C, or P(X|Ci). The class with the maximum probability/density for the observation was selected as the best class. Application of bayesian classification As mentioned above, this project used Bayesian classification to select features of a shaker-DUT model that most closely matched the physical system, with the goal of determining if a shaker is adequate for an experimental test. As voltage was one of the limiting factors of a shaker test, VSDs were chosen as the medium to compare models. That is, the Bayesian classifier would select the class of model whose VSDs most closely matched the ground truth VSD. As an initial verification, coupled systems using three different ROMs of the BARC were compared to a sampled VSD coming from the VSD distribution of the shaker-DUT model using Craig-Chang for the ROM. Once the classifier was verified to correctly choose the Craig-Chang representation of the BARC, three joint model stiffnesses were compared to the experimental VSD. As mentioned previously, the classifier needs to know the distribution of VSDs from each class of shaker-DUT model being compared in order to determine the probability the physical data comes from each model class’s VSD distribution. There was uncertainty in the shaker parameters as well as the BARC model, so these parameters were each assigned a normal distribution according to the uncertainty of the parameter. In Table 1, there is a list of these varied parameters, as well as the mean and variance of the normal distribution they were selected from. Mean was determined by the measured/optimized values, and standard deviation was 10% of the mean value. For each shaker-DUT model class, Latin hypercube sampling was used to sample from all parameter distributions and calculate the VSD resulting from each sampled set of parameters. Note that the parameters were varied from the same distributions shown in Table 1 for every model class. The resulting coupled system VSDs for each class maintained a multivariate normal distribution. The VSD distributions for each ROM are shown in Figure 2. It was also found that classifying with only the frequencies corresponding to the first eight minima of the VSD, which usually correspond to the natural frequencies, was sufficient for distinction in classification.
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