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

TL;DR

This work addresses the challenge of computing resonance and anti-resonance backbone curves in large nonlinear FE models with non-polynomial nonlinearities. It combines a constrained optimization on the Harmonic Balance residual with a novel analytical Hessian derived via an extended Alternating Frequency/Time (AFT) method, plus a coordinate-transformation framework for localized nonlinearities. The approach enables robust, fast tracing of backbone curves and simultaneous extraction of nonlinear mode shapes across energy levels, validated on a 2-DOF cubic system, a beam with cubic/tanh nonlinearities, and a compressor blisk with a tanh friction damper. The results demonstrate accurate backbone prediction, efficient Jacobian assembly, and scalability to industrial-scale FE models, providing a practical tool for frequency-domain analysis of general nonlinear structures.

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.

Figures (16)

• Figure 1: Schematic representation of the 3-DOF nonlinear element and the definition of local and global coordinate systems.
• Figure 2: A schematic diagram of a forced, 2-DOF oscillator with cubic nonlinearity.
• Figure 3: Nonlinear frequency response analysis of the two-degree-of-freedom oscillator. The computed backbone curves (thick lines) are superimposed on the reference forced response curves (thin orange lines) obtained via NLvib. The thick red line denotes the first near-linear branch used for initialization. Markers on the red line indicate branch points for the 1st resonance ($\bullet$), 2nd resonance ($\blacklozenge$), and anti-resonance ($\blacksquare$). The resulting thick black and blue lines represent the resonance and anti-resonance backbone curves, respectively. Purple triangles ($\blacktriangledown$) denote unstable solutions.
• Figure 4: Forced response surface with superimposed backbone curves. The color-mapped surface illustrates the steady-state vibration amplitude versus excitation frequency and force magnitude. The computed resonance backbone curves (black lines) and anti-resonance backbone curve (blue line) are overlaid on the surface. Notably, the backbone curves precisely correspond to the topological ridges (local maxima) and the trench (local minima) of the response surface, demonstrating the global characterization of the nonlinear dynamics.
• Figure 5: Comparison of computational efficiency between the proposed analytical Hessian and numerical differentiation. The purple bars denote the proposed method using the analytical Hessian, while the orange bars represent the conventional method based on numerical differentiation.