28 Applying Uncertainty Quantification to Structural Systems: Parameter Reduction for Evaluating Model Complexity 245 differing levels of complexity. These cases serve the purpose of assessing what degree of model complexity is required to effectively represent the MAFDS in the 2DOF mathematical model. The system is assumed to be subjected to a free and full homogeneous field of gravitational acceleration g. When dropped from a height hf, the initial time step t measurement does not occur until right before the system hits the ground, meaning the initial displacements, zu(t),zl(t), and velocities, ˙zu(t), ˙zl(t), at time t =0 are: zu(0) =zl(0) =0, ˙zu(0) = ˙zl(0) =,2ghf. (28.1) The dynamic outputs of interest, as indicated in Fig. 28.4 are calculated using the equations below. Relative Displacementszr Suspension ForcesFsd Elastic Foot ForcesFef zr,max =max|zu −zl| Fsd,max =max|Fks +Fbs| Fef,max =max|Fkef +Fbef| zr,min =(zu,end −zl,end), t =tend Fsd,min =min|Fks +Fbs|, tmax < t ≤tend Fef,min =min|Fkef +Fbef|, tmax < t ≤tend Fsd,rest =Fks +Fbs, t =tend Fef,rest =Fkef +Fbef, t =tend (28.2) The stiffness Fks and damping Fbs forces are calculated by Mallapur and Platz [9]: Suspension Elastic Foot Fks =ks(zu −zl) =kszr Fkef =kefzl Fbs =bs( ˙zu − ˙zl) =bs ˙zr Fbef =bef ˙zl (28.3) 28.3 Stiffness Regression Models Before creating a variety of different mathematical models to represent the behavior of the entire MAFDS, data from the static system first had to be obtained to model the stiffness behavior of the suspension system. In this study, a series of static tests were performed by increasing the added payload from 0 to 200 kg in increments of 10 kg. During each incremental increase, measurements for the upper force Fsd and relative displacements zr were recorded using the sensors illustrated in Fig. 28.3. For each test, the recorded force was divided by the average relative displacement to calculate the suspension system’s stiffness. Figure 28.5a displays the relation between the measured force Fsd and relative displacements zr. The resulting Fks(zr) curve indicates that the linear relationship between force and relative displacement transitions when zr is approximately 0.068 m. This bi-linear relationship, also observed in previous studies [4, 9], indicates there are two potential slope values Fig. 28.5 Experimental stiffness (a) force and (b) coefficient curves of the spring-damper system
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