Abstract
Generalized Frequency Response Functions (GFRF) are a higher-order analog to the well-known frequency response functions and play a central role in understanding the behavior of nonlinear systems, such as the occurrence of intermodulation. Existing probing algorithms for estimating GFRFs from input–output time series data rely on lengthy manual derivations or the use of symbolic computation for setting up harmonic balances. However, these approaches become intractable for higher-order or multi-scale systems commonly found among engineering applications as the number of intermodulation terms grows substantially. Starting from a polynomial NARX model (Nonlinear AutoRegressive model with eXogenous input) trained on input/output data, this paper presents a probing algorithm for estimating the GFRFs, in which harmonic balances are computed in a numerical manner suitable for parallel computing. The new approach constructs the harmonic balances in the frequency domain by exploiting the linearity of the Volterra system response with respect to the GFRFs. Therefore, solving for the unknown GFRF reduces to solving a linear equation involving the spectral residual constructed with respect to the unknown GFRF. This new method is verified in two examples. The first one is the well-studied Duffing oscillator. The second one, of great interest for marine hydrodynamics, is the estimation of the quadratic transfer function linking incoming waves to the low-frequency wave loading on a floating structure.