A deep neural network integrated harmonic balance identification method for high-dimensional bistable structures

Abstract The bistable structures are widely utilized in energy harvesting, vibration isolation, and morphing applications due to the beneficial effects of negative stiffness. However, the jump phenomena and complex nonlinearity of bistable restoring forces make parameter identification challenging, particularly in high-dimensional systems with localized bistability. To overcome this, paper introduces a novel integrated framework that synergistically combines harmonic balance identification with deep neural networks, capable of identifying bistable nonlinear restoring forces without prior assumption of their analytical form. The proposed approach requires only a limited set of harmonic excitation responses to achieve high-fidelity identification. The linear mass, damping, and stiffness matrices are directly estimated via harmonic coefficient balancing in the frequency domain. For high-dimensional bistable structures, fully connected deep neural networks are constructed to represent each local nonlinear restoring force, enabling scalable and flexible identification beyond conventional parametric methods. Numerical simulations conducted on three typical bistable cases demonstrate that the method accurately identifies system parameters and nonlinear forces under noise levels up to 40 dB. Experimental validations are performed on a bistable nonlinear energy sink with different potential well depths, confirming that the identified model closely matches the measured frequency response, restoring force, and reconstructed random response under band-limited noise. Compared to the pure deep neural networks-based approach, the identification efficiency is improved by a factor of nine. The results verify that the proposed integrated approach offers a generalizable and experimentally feasible solution for identifying complex bistable nonlinear systems.