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

Abstract

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.