9 Modeling Nonlinear Structures Using Physics-Guided, Machine-Learnt Models 73 frequency and S-shaped response curves [4] for the studied system. The training, testing, and validation data set was built using families of periodic solutions, where each training point contained the state variables and the value of the excitation force. Figures 9.1 and 9.2 show an excellent agreement between the model of the system and the trained LNN. The LNN can also be used to extract the stiffness and damping characteristics of the system, informing more detailed physics-based modeling of the system of interest. Fig . 9. 1 (Left) Lagrangian and Hamiltonian of the trained LNN and the analytical model. (Right) Comparison between the response obtained from the model (solid green) and the response obtained from the LNN (dashed black). For both models, shooting and pseudo-arc length continuation were used to obtain the response curves. Crosses highlight the location of the fold points Fig . 9. 2 Comparison between the stiffness and damping forces obtained from the model (dashed-red) and derived from the LNN (solid black). The shaded area represents displacement/velocity amplitudes outside the training data set
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