238 C. Ligeikis and R. Christenson Fig. 27.1 Two DOF mass-spring system Fig. 27.2 Mean (magnitude and phase) of the FRFs evaluated at 500 sample points by the true system model and the metamodel this methodology works, a numerical example is provided. The system in question is the two DOF quarter car model shown in Fig. 27.1. Vibrations are suppressed using a magneto-rheological (MR) fluid damper. MR dampers are devices with variable damping characteristics that are controlled by adjusting a supplied current. In this system, the sprung mass m1 and the current supplied to the damper i are treated as uniformly distributed random variables ranging from 400 to 600 kg and 0–0.25 amps, respectively. The un-sprung mass m2 and the spring stiffnesses k1 and k2 are considered to be deterministic with values of 50 kg, 25,000 N/m, and 150,000 N/m, respectively. The MR damper is simulated using the hysteretic model proposed by Kwok et al. [4]. In a real RTHS implementation, the MR damper would be the physical substructure and the two DOF system would be the numerical substructure. The system is excited by a 20 Hz band-limited white noise base displacement xb input. Using Latin hypercube sampling on the random input parameters m1 and i, 50 simulations are performed using Simulink. Next, the frequency response functions (FRFs) relating the input base displacement xb to the output sprung mass displacement x1 are estimated for each of the 50 simulations using thetfestimatefunction in MATLAB. These FRFs represent the vector-valued response output of the model. The goal is to predict the FRF for a given m1 and i pair without having to conduct an additional RTHS test. This is accomplished via a statistical metamodel. First, following the methodology proposed by Yaghoubi et al., the size of the response output vector is reduced by Principal Component Analysis (PCA) [5]. PCA is used to transform the very long (>1000 points) FRF vectors of correlated system outputs into much shorter (<10 points) vectors of uncorrelated variables that represent the core statistical features of the response output. The MATLAB based uncertainty quantification software framework UQLab is then used to build independent Kriging metamodels for each of these uncorrelated PCA variables [6]. Using these metamodels, a vector of PCA variables can be predicted for a desired set of random inputs. These variables are then transformed back into the full length FRF output vector for the unknown m1 and i pair. To evaluate the effectiveness of this methodology, a mean FRF is computed for 500 MC samples using both the full system model and the metamodel as shown in Fig. 27.2. These mean FRFs are almost identical. Further, Fig. 27.3 shows the full FRFs found using the true model and metamodel for these 500 samples along with the means shown in black. These results demonstrate that this approach is effective at accurately predicting this system’s probabilistic behavior.
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