254 R. Locke et al. Table 28.8 Average R 2 value Param Case 1 Param Case 2 Param Case 3 Param Case 4 Param Case 5 Param Case 6 a0,1 12.62 a0,1 5.73 b0 2.02 b0 0.63 c0,1 8.13 c0,1 2.77 a1,1 27.03 a1,1 21.88 b1 36.99 b1 23.68 c1,1 25.04 c1,1 10.58 a0,2 1.26 a0,2 1.05 b2 30.41 b2 30.72 c2,1 6.53 c2,1 2.15 a2,1 28.35 a1,2 19.33 b3 16.07 b3 14.95 c0,2 0.23 c0,2 0.29 m (a) 1,1 6.61 v (a) 1,1 1.19 m (b) 1,1 0.41 v (b) 1,1 0.62 c1,2 22.00 c1,2 23.40 m (a) 1,2 5.12 v (a) 2,1 18.14 m (b) 1,2 1.76 v (b) 2,1 12.42 c2,2 10.16 c2,2 9.90 bef 19.00 v (a) 3,1 20.60 bef 12.35 v (b) 3,1 14.17 m (c) 1,1 5.26 v (c) 1,1 1.05 bef 12.07 bef 2.83 m (c) 1,2 4.58 v (c) 2,1 17.24 bef 18.05 v (c) 3,1 20.74 bef 11.88 Table 28.9 Reduced set of input parameters for cases 1–6 of the suspension stiffness and damping coefficients Number of full set of input parameters Number of reduced set of input parameters Case 1, F (d) bs,a 7 6 Case 2, F (e) bs,a 8 6 Case 3, F (d) bs,b 7 4 Case 4, F (e) bs,b 8 6 Case 5, F (d) bs,c 9 8 Case 6, F (e) bs,c 10 8 could be removed from the calibration space and held constant. Table 28.9 indicates the size of the reduced parameter space for each case. 28.6.2 Uncertainty Quantification Frameworks Moving forward, inverse modeling will be applied to the 2DOF mathematical model for the six 2DOF cases. Inverse modeling or inversion is the process of using the known dynamic outputs to solve for the set of input variables in Table 28.8 when their true values are unknown. As part of inverse modeling, two Bayesian frameworks will be presented and deployed in a later study to evaluate model-form uncertainty for each of the six 2DOF cases using the output responses in Eq. (28.2) and the reduced set of input variables based on the results from Fig. 28.10 and Table 28.8. The foundations of their methodologies are derived from Bayes’ theorem, P(m| d) = P({doutput}|{xinput})P({xinput}) P({doutput}) ∝P({doutput}|{xinput})P({xinput}), (28.21) where {xinput} is vector of input variables to the 2DOF mathematical model; {doutput} is the vector of dynamic outputs; P({xinput}|{doutput}) is the posterior distribution; P({doutput}|{xinput}) is the likelihood function, P({xinput}) is the prior distribution; and, P({doutput}) is the data distribution, which is held constant for the fixed data set. The posterior distribution P({xinput}|{doutput}) is the probability that the given set of input variables produced the data set {doutput}, and the likelihood function L({xinput}|{doutput}) or P({doutput}|{xinput}) is the probability {doutput} produced the given set of input variables {xinput}. Finally, P({xinput}) is the prior knowledge about the distribution of the input variables, such as its shape, and lower and upper limits. The prior distribution of the input variables from Table 28.8 influences how the parameters are expected to vary in the model space, like a uniform or multivariate Gaussian distribution, and if they are treated as dependent (correlated) or independent (uncorrelated) variables.The first case (1) will be a delayed-acceptance single component adaptive Metropolis (DA-SCAM) algorithm with ordinary kriging of the likelihood, and the second case (2) will be the Gaussian Process Model
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