A complex-domain harmonic balance method for parameter identification of nonlinear fractional rheological models of viscoelastic materials

Excessive vibrations induced by wind or seismic excitation pose serious risks to the safety and serviceability of civil engineering structures. Viscoelastic dampers (VEDs) are widely implemented as passive energy dissipation devices due to their robustness and efficiency, yet their predictive performance depends critically on accurate constitutive modeling of viscoelastic materials. Classical rheological models, based on linear springs and dashpots, are often insufficient to capture broad-spectrum damping and nonlinear elastic effects observed in practice. In this study, we adopt nonlinear fractional Kelvin and Zener rheologies with a cubic spring as a parsimonious yet expressive law for viscoelastic media. A parameter identification procedure is developed using the harmonic balance method (HBM), formulated in the complex domain to improve numerical robustness. The methodology is validated through synthetic and noisy datasets as well as laboratory experiments. Fit quality is quantified using two complementary figures of merit: the absolute least-squares objective J (complex residuals) and the root-mean-square relative error (RMSRE, %) computed from force magnitudes; for the Kelvin model these criteria yield different optimal fractional orders, yet both calibrations produce practically indistinguishable reconstructions across the tested amplitudes and frequencies. The framework is computationally efficient and directly applicable to system-level analyses in the frequency domain, or via internal-variable surrogates in the time domain.