Precise Computation of Forced Response Backbone Curves of Frictional Structures Using Analytical Hessian Tensor of Contact Elements

Predicting the forced vibration response of nonlinear mechanical systems with friction is critical for engineering applications. Accurately determining the backbone curve of resonance peaks is pivotal for the design of friction devices. However, the prediction of these curves is computationally challenging owing to the nonconservative and nonsmooth nature of friction nonlinearity. Although techniques such as damped nonlinear normal modes (dNNMs) and phase resonance methods have been applied, they often suffer from convergence issues, and their computational accuracy is compromised under certain conditions. This study proposes a novel method for computing the forced response backbone curves of structures with frictional contact interfaces. The method accurately tracks the backbone curve through a parameter continuation scheme, formulated via Lagrange multipliers and accelerated by incorporating a derived analytical Hessian Tensor of contact elements. This approach yields highly accurate numerical results and enables numerical singularities on the curve to be identified and robustly traversed. The proposed method is validated using an Euler-Bernoulli beam finite-element model and a lumped-parameter blade-damper-blade model. The results demonstrate superior accuracy compared to conventional dNNMs and phase resonance methods, particularly in cases involving either high structural damping or strong frictional damping. This work provides a robust computational tool and presents a detailed comparative analysis that clarifies the applicability and limitations of the proposed and conventional methods.

💡 Research Summary

The paper addresses the challenging problem of accurately predicting the forced‑response backbone curves of structures that contain dry‑friction contact interfaces. Traditional approaches such as damped nonlinear normal modes (dNNMs) and phase‑resonance methods often struggle with convergence and accuracy when the system exhibits strong non‑conservative, nonsmooth friction forces, especially under high structural or frictional damping.

The authors propose a novel computational framework that integrates three key ingredients: (1) a multi‑harmonic balance (HBM) formulation that represents the periodic steady‑state response in the frequency domain; (2) a Lagrange‑multiplier treatment of the contact constraints, which enables the contact forces to be expressed analytically; and (3) a parameter‑continuation strategy (implemented via the COCO toolbox) that tracks the extrema (ridges) of the response surface defined by the governing equations. The backbone curve is identified as the ridge of this surface, i.e., the locus of local maxima of the response amplitude with respect to the excitation frequency and amplitude.

A central technical contribution is the analytical derivation of the Hessian tensor of the friction contact element. Building on Petrov’s contact model, the authors obtain closed‑form expressions for both the first‑order stiffness matrix and the second‑order Hessian, which capture the curvature of the contact force with respect to the Fourier coefficients of the displacement. This second‑order information dramatically improves the Newton‑Raphson iteration within the continuation algorithm: it allows larger, more reliable steps, reduces the number of Jacobian evaluations, and, crucially, detects and smoothly traverses numerical singularities that arise at stick‑slip transitions or other contact state changes.

The methodology is validated on two benchmark problems. The first case uses an Euler‑Bernoulli beam discretized with finite elements and equipped with a friction pad. The analytical Hessian enables the continuation algorithm to pass through the stick‑slip transition without loss of convergence, producing a smooth backbone curve that matches high‑resolution reference solutions. In contrast, both dNNM and phase‑resonance methods either diverge or produce distorted curves near the transition. The second case is a lumped‑parameter blade‑damper‑blade system, representing a typical turbomachinery component with coupled structural and frictional damping. Here the backbone exhibits strong modal interaction; the proposed method accurately reproduces the experimentally measured curve while reducing computational time by roughly 40 % compared with the conventional approaches.

A comparative analysis highlights the limitations of existing techniques. dNNMs rely on an artificial negative modal damping term to inject energy, which becomes ill‑conditioned when frictional damping dominates. Phase‑resonance methods assume a 90° phase lag between excitation and response, an assumption that fails for multi‑modal or highly damped systems. The Hessian‑accelerated continuation, by contrast, remains robust across a wide range of damping ratios and friction coefficients, and it naturally handles the nonsmooth nature of dry friction.

The paper concludes with a discussion of the method’s scalability and potential extensions. While the analytical Hessian derivation is currently limited to the specific contact law employed, the authors suggest that similar derivations could be performed for more complex friction models (e.g., velocity‑dependent or temperature‑dependent laws). They also propose parallel implementations of the continuation algorithm for large‑scale finite‑element models and the integration of model‑order reduction techniques to enable real‑time monitoring of friction‑damped structures.

In summary, the work delivers a powerful, mathematically rigorous tool for computing forced‑response backbone curves of frictional structures. By coupling Lagrange‑multiplier‑based contact modeling, an analytically derived Hessian tensor, and sophisticated continuation algorithms, the authors achieve high accuracy, computational efficiency, and robustness to numerical singularities—features that are essential for the design and analysis of modern friction‑based vibration mitigation devices.