Harmonic balance-automatic differentiation method: A practical nonlinear dynamics solver

The Harmonic Balance-Alternating Frequency-Time domain method is widely used for analyzing nonlinear system dynamics, but its application to high-dimensional complex systems is often limited by the computational burden of deriving Jacobian matrices in Newton-Raphson iterations. To overcome this, we propose the Harmonic Balance-Automatic Differentiation method. This approach integrates Automatic Differentiation into the Harmonic Balance formulation to compute exact Jacobians, eliminating the need for error-prone manual derivation and enhancing generality for complex nonlinearities. By leveraging deep learning infrastructures, our implementation harnesses native parallel computing and CUDA acceleration. Combined with arc-length continuation, the Harmonic Balance-Automatic Differentiation method allows users to supply only the system's dynamic equations to autonomously trace the complete panorama of periodic responses, including stable and unstable branches. Computational experiments on a rotor system with a squeeze-film damper validate the method's capability in handling complex nonlinear expressions. For a high-dimensional aero-engine rotor-bearing-casing system, Harmonic Balance-Automatic Differentiation method achieved a 17-fold speedup over the conventional Harmonic Balance-Alternating Frequency-Time domain method and was 144 times faster than the Newmark method. Furthermore, results obtained via the HB-AD method show good agreement with experimental data from a nonlinear vibration isolator, corroborating the method's validity and practical applicability.