Hessian-Enhanced Alternating Frequency/Time method for Computing Resonance Backbone Curves

TL;DR

This paper introduces a Hessian-enhanced optimization method for efficiently computing resonance backbone curves in complex nonlinear mechanical systems, overcoming limitations of polynomial-based approaches and enabling large-scale model analysis.

Contribution

It develops an analytical Hessian tensor for generic nonlinear elements and integrates it into a direct optimization framework using an extended AFT method, improving robustness and computational efficiency.

Findings

Accurately computes resonance and anti-resonance curves for complex nonlinear systems.

Reduces computational runtime significantly compared to numerical differentiation.

Validated on benchmarks including an industrial-scale finite element model.

Abstract

Computing resonance and anti-resonance backbone curves in complex nonlinear mechanical systems is of high engineering relevance but remains computationally challenging, especially for large finite-element (FE) models. Existing manifold-based approaches often rely on polynomial parameterizations, limiting their effectiveness for general smooth, non-polynomial nonlinearities. To overcome these limitations, we develop a direct optimization framework that employs a Lagrange multiplier formulation to determine the resonance backbone curve on the response surface constrained by the harmonic balance governing equations. Crucially, solving this formulation efficiently requires second-order sensitivity information. Therefore, the primary innovation of this work is the derivation of a analytical Hessian Tensor for generic $C^2$-continuous nonlinear elements. This is achieved by combining an extended Alternating Frequency/Time (AFT) method for computing second-order derivatives with local-coordinate tensor transformations. By integrating this analytical Hessian into the solver, the proposed framework ensures robust convergence and significantly reduces runtime, making it practical for large-scale models where numerical differentiation is computationally prohibitive. The method is validated on three benchmarks of increasing complexity: a two-degree-of-freedom (2-DOF) system with cubic nonlinearity, a beam with cubic stiffness or hyperbolic tangent (tanh) friction nonlinearities, and an industrial-scale finite element model of a compressor bladed disk (blisk) with a friction ring damper. Results demonstrate that the proposed framework accurately and efficiently computes both resonance and anti-resonance backbone curves, providing a robust frequency-domain tool for structures with non-polynomial nonlinearities.