252 R. Locke et al. Subscript t indicates the present vector values for acceleration, velocity, and displacement. The change in displacement from the present time stept to the future time step(t + t) is represented by{ z}. The parameters e0−e4 are numerical integration constants; these values, and others, are determined using the equations below: e0 = 1 β t2 e1 = 1 β t e2 = 1 2β − 1 e3 =(1−γ) t e4 =γ t e5 = γ βt e6 = γ β − 1 e7 = t 2 γ β − 2 . (28.16) The parameters βandγ represent the variation in acceleration during the incremental time step t, and numerical or artificial damping introduced by discretization in the time domain, respectively. For this study, the average acceleration Newmark method was utilized, meaning β = 1 4 and γ = 1 2. As previously mentioned, the average acceleration method was ideal for this study because it is conditionally stable for any size time step, and provides accurate results for a “small enough” time step [5]. In this study the numerical time integration step was t =0.0005s. Once the equations for the future acceleration, velocity, and displacement vectors were known, they were entered into Eq. (28.11) and matrix algebra was performed to solve for the unknown change in displacement { z}. Equation (28.17) indicates the new equation of motion derived to solve for { z}. { z}=[ ]−1{ q}t (28.17) [ ] =e0[M]+e5[C]+[K] (28.18) {q}t ={F}+[M](e1{ ˙z}t +e2{ ¨z}t) +[C](e6{ ˙z}t +e7{ ¨z}t) −[K]{z}t (28.19) When the change in displacement { z} was calculated in Eq. (28.17), it was entered back into Eqs. (28.13)–(28.15) to solve for the future acceleration, velocity, and displacement vector values. After these values were known, the integration process started over with the future time step (t + t) now becoming the present time step t. This process was repeated until the system entered a steady state at a total simulation time of t =2 s. During this process, the total suspension force Fsd and total foot force Fef were calculated for each time step using the equations in Eqs. (28.2)–(28.3). 28.6 Uncertainty Quantification 28.6.1 Sensitivity Analysis As mentioned, the primary focus of this research project was to investigate uncertainty in the six 2DOF cases that are mathematical representations of the MAFDS. The purpose of having multiple cases is to identify the minimum level of model complexity required to capture the governing physics and mechanics of the physical system in the dynamic outputs. Before solving this problem, however, it was necessary to first perform a sensitivity analysis on each model to identify the parameters that had the least impact on output response variations. In this study, sensitivity analysis methodologies were employed to identify parameters that contributed an insignificant variance to the output response features and eliminate them from the calibration space (i.e. hold at a constant mean value). Low sensitivity variables were eliminated from the calibration space to prevent numerical ill conditioning and to reduce computational costs associated with statistical inversion algorithms [3]. Analysis of Variation (ANOVA) and the Coefficient of Determination R 2 were utilized as statistical screening methodologies in this paper to evaluate the sensitivity of each coefficient in Sects. 28.3 and 28.4. The later R2 value provided a measure for how varying an independent parameter between its Lbounds affects the variability in the output response, which is expressed in Eq. (28.20) as R 2 p,n =1− L i=1 LN−1 j=1 (yp,ij − ˆyp,i) 2 LN k=1(yp,k − ¯yp) 2 , (28.20) where p is a vector from one to eight representing the response features of interest in Fig. 28.2, and n is a vector from {1, 2, . . . ,N} representing the parameters from a subject model. The overall mean value for a given responsepis represented
RkJQdWJsaXNoZXIy MTMzNzEzMQ==