246 R. Locke et al. that can describe the stiffness behavior of the system. On the contrary, Fig. 28.5b indicates that the upper suspension stiffness varies non-linearly with the relative displacement, and, as a result, there are more than two potential values for modeling the system. It was determined that this nonlinear behavior was not observed in the previous studies [4, 9] as a result of biased data produced from improper sensor calibration, and because of this, new suspension stiffness models needed to be developed. Three new independent models of varying complexity were fitted to the stiffness coefficient curve to address the linearity-nonlinearity discrepancy: (1) the most simple model was a piecewise function composed of two linear polynomials; (2) the intermediate model was a cubic polynomial; and (3) the most complex model was a piecewise function composed of two power functions. The following subsections discuss how each method in (1)–(3) was developed. 28.3.1 Piecewise Linear Polynomials The first and most simple stiffness case was fit to the experimental data in Fig. 28.5b with a piecewise function composed of two linear polynomials. The function is founded on the assumption that the stiffness curve varies bi-linearly with relative displacement at the transition point zr,tp when zr ∈ [0m, 0.068m] and zr ∈ [0.068m,0.082m]. Each polynomial consisted of two unknown coefficients, which resulted in a total of four unknown stiffness coefficients. The transition point zr,tp is the location where the rate of change in the stiffness coefficient curve increases and is used to derive the fitted suspension stiffness ks,a(zr) as a function of relative displacement zr in Eq. (28.4). ks,a(zr) =7a0,1 +a1,1zr, zr ≤0.068m a0,2 +a1,2zr, zr >0.068m (28.4) To determine the values for the coefficients in Eq. (28.4), the data in Fig. 28.5b was divided at the transition point zr,tp and a linear regression analysis was performed on each individual set of data. Table 28.1 indicates the calculated mean and standard deviation for each of the subject parameters. 28.3.2 Cubic Polynomial For the second stiffness case, the complexity of the regression increased as the model form changed from linear polynomials to a third order polynomial. Model complexity did not increase from the number of unknown variables that had to be solved, but rather from the assumption that the stiffness coefficient curve is continuously smooth and nonlinear. Similar to the piecewise linear polynomials case, the cubic polynomial case had a total of four unknown coefficients from fitting a third order polynomial to the stiffness coefficient data. The fitted model ks,b(zr) is developed in Eq. (28.5). ks,b(zr) =b0 +b1zr +b2z 2 r +b3z 3 r (28.5) Similar to the piecewise linear polynomials case, the mean and standard deviation values for the cubic polynomials were calculated by performing a polynomial regression analysis on the experimental data in Fig. 28.5b. Table 28.2 indicates the mean and standard deviation for each of the cubic polynomial parameters in Eq. (28.5). Table 28.1 Piecewise first-order polynomial stiffness coefficients, ks,a(zr) Polynomial coefficients Mean Standard deviation a0,1, in kN m 28 0.50 a1,1, in kN m2 73 11.56 a0,2, in kN m −1.58 0.95 a1,2, in kN m2 516 12.50
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