28 Applying Uncertainty Quantification to Structural Systems: Parameter Reduction for Evaluating Model Complexity 247 Table 28.2 Cubic polynomial stiffness coefficients, ks,b(zr) Polynomial coefficients Mean Standard deviation b0, in kN m 23 0.52 b1, in kN m2 601 48 b2, in kN m3 −1.49e+04 1.25e+03 b3, in kN m4 1.24e+05 9.45e+03 Table 28.3 Piecewise power function stiffness coefficients, ks,c(zr) Polynomial coefficients Mean Standard deviation c0,1, in kN m 6.55 0.33 c1,1, in kN m2 17 0.42 c2,1, unitless 0.10 2.50e−03 c0,2, in kN m 4.43 0.13 c1,2, in kN m2 0.19 4.83e−03 c2,2, unitless 1.19 3.70e−03 Fig. 28.6 Three fitted regression model to stiffness coefficient data when zr ≤zr,tp and zr >zr,tp 28.3.3 Piecewise Power Functions The third stiffness model is the final and most complex case, consisting of a piecewise function composed of two power functions and a total of six unknown stiffness coefficients. The fitted model ks,c(zr) is developed in Eq. (28.6), and again, the mean and standard deviations of the parameters were calculated through a regression analysis of the data in Fig. 28.5b. Table 28.3 indicates the values for each coefficient in Eq. (28.6). During the regression analysis, the power functions had the lowest residual error of all three stiffness model cases, which strongly implies that the suspension stiffness data behaved most like a power function. ks,c(zr) =7c0,1 +c1,1z c2,1 r , zr ≤0.068m c0,2 +c1,2z c2,2 r , zr >0.068m (28.6) The results in Fig. 28.6 shows how well the three regression models fit the experimental stiffness coefficient data before and after the transition point zr,tp = 0.068 m. Initially, the first piecewise power function and the third-order polynomial underestimate the stiffness coefficient data whenzr is approximately less than 0.018 m. The first piecewise linear polynomial, however, overestimates the stiffness coefficient data when zr < 0.018m. Once zr ≥ 0.018 m, the third-order polynomial overestimates the data while the first piecewise linear polynomial underestimates when zr ∈ [0.018 m,0.048 m]. From zr ∈ [0.048m, 0.082m], all models closely approximate the stiffness data before and after the transition point zr, tp.
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