454 S. Ilbeigi and D. Chelidze 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Forcing Amplitude, q0 Displacement, x30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Forcing Amplitude, q0 Displacement, x30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Forcing Amplitude, q0 Displacement, x30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Forcing Amplitude, q0 Displacement, x30 Fig. 41.5 Bifurcation diagram for ROMs: of the nonlinear system; for harmonic forcing with ! D 7:97Hz: six-dimensional POD (top left); six-dimensional separated POD (top right); four-dimensional SOD (bottom left); four-dimensional separated SOD (bottom right) 41.5 Conclusions The separated multivariate analysis using POD and SOD was proposed to deal with the rank-deficiency of nonlinear dynamical systems. Following the mathematical formulation of the separated POD- and SOD-based MOR, a nonlinear massspring-damper was studied. The full-scale state-space model of the system was obtained and the data matrices were extracted. The performance of four methods for MOR including POD, SOD, separated POD and separated SOD was investigated in terms of stability and accuracy. The results confirmed that SOD was able to capture the dynamics of the system in a lower dimensional space. Separated multivariate analysis was shown to improve the accuracy and stability of the POD-based MOR. However, it had a worsening effect on the SOD-based MOR. Acknowledgements The authors would like to thank National Science Foundation under Grant No. 1100031. References 1. Balajewicz, M., Amsallem, D., and Farhat, C.: Projection-based model reduction for contact problems. Int. J. Numer Meth Eng (2015). 2. Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems, vol. 45. Springer, Berlin (2005) 3. Ilbeigi, S., Chelidze, D.: Model order reduction of nonlinear euler-bernoulli beam. In: Nonlinear Dynamics, vol. 1, pp. 377–385. Springer, Berlin (2016) 4. Antoulas, A.C., et al.: Model order reduction: Methods, concepts and properties. In: Coupled multiscale simulation and optimization in nanoelectronics, pp. 159–265. Springer, Berlin (2015)
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