Data-driven reduced-order modeling of nonlinear multivalued FRFs: Applications to beam and SDOF gear systems

We present a novel approach to developing non-intrusive Reduced Order Models (ROMs) for predicting nonlinear, multivalued Frequency Response Functions (FRFs). Such multivalued behavior often arises in nonlinear dynamic systems due to phenomena like softening and hardening effects, where the FRF exhibits multiple amplitude values for the same input frequency. To handle this issue, we introduce a parametric spline interpolation technique that maps both amplitude and frequency onto an auxiliary axis. This parametrization process converts the multivalued FRF into two separate single-valued functions. Using these single-valued functions, we construct consistent snapshot matrices. The spline interpolation serves as a post-processing step for Full-Order Model (FOM) solutions obtained via the Harmonic Balance Method (HBM). An autoencoder then reduces the system's dimensionality to a latent space, while a Polynomial Chaos Kriging (PCK) surrogate models the dynamics in this space. The surrogate maps the input parameters to single-valued frequency and amplitude functions. We validate the proposed approach on a Bernoulli beam with cubic spring nonlinearity and a two-degrees-of-freedom gear model, comparing it against FOM solutions. Results demonstrate that the developed approach efficiently reduces complex nonlinear FRFs without requiring prior system knowledge. This method can accelerate the robust design optimization tasks of large-scale dynamic systems, such as gear transmissions, and facilitates uncertainty propagation through the PCK surrogate.