248 R. Locke et al. Fig. 28.7 Elastic foot force curve from manufacturer’s data Table 28.4 Elastic foot stiffness coefficients, kef(zl) Polynomial coefficients Mean d0, in kN m 148 d1, in kN m2 −1717 d2, in kN m3 7.45e+5 28.3.4 Elastic Foot Stiffness To reduce the number of unknown calibration parameters, and because the manufacturer provided the stiffness force versus displacement data, the stiffness coefficients for the elastic foot were treated as known parameters [10]. The coefficient values were calculated by taking the derivative of the force versus displacement curve seen in Fig. 28.7, Eq. (28.7) displays the resulting equation for kef(zl). Table 28.4 indicates the stiffness coefficient values for kef(zl). kef (zl) =d0 +d1zl +d2z 2 l (28.7) 28.4 Damping Regression Models To determine the damping coefficient bs of the suspension system, a series of 35 dynamic tests were performed by varying the drop height between 0 and 0.1 m and the added payload between 0 and 100 kg. Similar to the static tests, total force Fsd and relative displacement zr measurements were recorded using the upper suspension force sensor and the displacement sensors illustrated in Fig. 28.3. For each test, the damping force Fbs was computed by taking the difference between the total measured force Fsd, and the stiffness force Fks fitted to each stiffness case (see Eq. (28.8)). Figure 28.8 provides a purely figurative example to illustrate how a damping force curve is developed from the combination of 35 experimental drop tests. As can be seen in Fig. 28.8, a characteristic hysteresis curve is formed, which is assumed to be attributable to the compressibility of oil and/or cavitation within the suspension [11, 12]. In this study, these effects are ignored, and single curve models are fitted to the experimental data via regression analysis to “average” out the effect of hysteresis. bs(˙zr) = Fbs ˙zr = Fsd − Fks ˙zr (28.8) Two different cases of varying model complexity were used to model the relationship between the damping force Fbs versus relative velocity ˙zr , which are a (1) piecewise function with two linear polynomials, Fbs,d, and (2) cubic polynomial, Fbs,e. All together, there are a total of six 2DOF cases that vary in complexity for modeling suspension stiffness and damping (refer to Table 28.5). The following three subsections further discuss the development of each damping case with respect to its subject stiffness case.
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