250 R. Locke et al. Table 28.7 Cubic polynomial damping coefficients, bs,e(˙zr) Polynomial Standard Polynomial Standard Polynomial Standard coefficients Mean deviation coefficients Mean deviation coefficients Mean deviation Case 2 Case 4 Case 6 v (a) 1,1 , in kNs m 3.05 2.08e−02 v (b) 1,1 , in kNs m 3.16 2.25e−02 v (c) 1,1 , in kNs m 3.06 2.07e−02 v (a) 2,1 , in kNs2 m2 −2.62 4.57e−02 v (b) 2,1 , in kNs2 m2 −2.91 4.95e−02 v (c) 2,1 , in kNs2 m2 −2.66 4.54e−02 v (a) 3,1 , in kNs3 m3 0.89 2.60e−02 v (b) 3,1 , in kNs3 m3 1.03 2.81e−02 v (c) 3,1 , in kNs3 m3 0.91 2.58e−02 28.4.2 Cubic Polynomial For the second damping case, the complexity of the system increased as the model form changed from linear polynomials to a third order polynomial. A third order polynomial Fbs, e( ˙zr) was selected for the nonlinear damping coefficient curve because it was better at capturing the transient behavior with a smoothly varying curve from˙zr <0 m s to ˙zr ≥0 m s . A total of four unknown polynomial coefficients must be optimized to fit a curve Fbs, e( ˙zr) to Fbs( ˙zr) in Eq. (28.10a). The intercept parameter v (j) 0,1 , j =a,b,c, however, was assumed to be zero based on the assumption that the damping force should be zero at ˙zr =0 m s . Table 28.7 displays the mean values for v (j) 1,1 ,v (j) 2,1 ,v (j) 3,1 , j =a,b,c and their standard deviations that were used to construct bs,e( ˙zr). F (j) bs,e (˙zr) =v (j) 0,1 +v (j) 1,1 ˙ zr +v (j) 2,1 z 2 r +v (j) 3,1 ˙ zr 3 , j =a,b,c (28.10a) b (j) s,e(˙zr) =v (j) 1,1 + 2v (j) 2,1˙ zr +3v (j) 3,1 ˙ zr 2 , j =a,b,c (28.10b) The six damping models for each test case are the damping force plotted against the relative velocity in Fig. 28.9a–c. These results display Fbs(˙zr) from the two damping cases, F (j) bs,d and F (j) bs,e , j =a,b,c, which were derived from the stiffness cases Fks,a, Fks,b and Fks,c. 28.4.3 Elastic Foot Damping Because damping data was not provided by the elastic foot manufacturer, the damping coefficient bef was assumed to be a constant viscous damping coefficient. The prior value for the damping coefficient bef was approximated to follow a uniform probability distribution with values ranging from 0 Ns m to1000 Ns m. 28.5 Solving the Equation of Motion Once the stiffness and damping properties were approximated for both the suspension and elastic foot for each of the regression cases in Table 28.5, they were indexed into the equation of motion for the 2DOF system, [M] ⎧ ⎨ ⎩ ¨zu ¨zl ⎫ ⎬ ⎭t+ t +[B] ⎧ ⎨ ⎩ ˙zu ˙zl ⎫ ⎬ ⎭t+ t +[K] ⎧ ⎨ ⎩ zu zl ⎫ ⎬ ⎭t+ t ={F}, (28.11)
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