28 Applying Uncertainty Quantification to Structural Systems: Parameter Reduction for Evaluating Model Complexity 249 Fig. 28.8 An illustrative example showing the development of the experimental damping force curve of the suspension strut system for (a) one and (b) all thirty-five experimental drop tests Table 28.5 Cases for suspension stiffness and damping force curves 2-Piecewise damping, Fbs,d Third order damping, Fbs,e 2-Piecewise stiffness, Fks,a Case 1 Case 2 Third order stiffness, Fks,b Case 3 Case 4 Power 2-piecewise stiffness, Fks,c Case 5 Case 6 Table 28.6 Piecewise first-order polynomial damping coefficients, bs,d(˙zr) Polynomial Standard Polynomial Standard Polynomial Standard coefficients Mean deviation coefficients Mean deviation coefficients Mean deviation Case 1 Case 3 Case 5 m (a) 1,1 , in kNs m 4.81 4.50e−02 m (b) 1,1 , in kNs m 4.92 4.82e−02 m (c) 1,1 , in kNs m 4.86 4.39e−02 m (a) 1,2 , in kNs m 1.09 1.88e−02 m (b) 1,2 , in kNs m 1.08 1.93e−02 m (c) 1,2 , in kNs m 1.08 1.83e−02 28.4.1 Piecewise Linear Polynomials The first case for modeling damping involved fitting two linear polynomials in Eq. (28.9a) to the damping force data Fbs at the transition point ˙zr =0 m s . This is the simplest case because it assumes damping does not exhibit hysteretic behavior, and changes linearly at different rates when ˙zr < 0 m s and ˙zr ≥ 0 m s . There are two unknown polynomial coefficients, which are the slopes of the first-order polynomials in Eq. (28.9a). Because the damping force should be zero at ˙zr =0 m s , it was assumed that the intercepts m(j) 0,1 ,m (j) 0,2 =0, j =a,b,c for all stiffness cases. In other words, by taking the first derivative of Fbs,d(˙zr), the damping coefficients bs,d(˙zr) of the system in Eq. (28.9b) can be calculated from Eq. (28.9a). F (j) bs,d (˙zr) =7m (j) 0,1 +m (j) 1,1˙ zr, ˙zr <0m m (j) 0,2 +m (j) 1,2˙ zr, ˙zr ≥0m j =a,b,c (28.9a) b (j) s,d (˙zr) =7m (j) 1,1 , ˙zr <0 m s m (j) 1,2 , ˙zr ≥0 m s j =a,b,c (28.9b) Table 28.6 displays the mean values for m(j) 1,1 ,m (j) 1,2 , j =a,b,c and their standard deviations that were used to construct bs,d( ˙zr).
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