Nonlinear vibration analysis of forced response for rubbing problems using the automatic differential frame

The multi-harmonic balance method is widely applied to obtain the forced responses of nonlinear systems undergoing rubbing problems. Despite large-scale time savings compared with the time marching method, it suffers from the complicated derivations of the Jacobian matrix. To solve this problem, this paper focuses on applying the automatic differentiation frame to the multi-harmonic balance method to implement the nonlinear vibration analysis of systems subjected to the rub phenomena. By establishing computational graph and utilizing the automatic differentiation process, tedious works such as the derivations of the complicated analytical expressions of the Jacobian matrix are avoided, which guarantees the efficiency and applicability of the presented method. A single-degree-of-freedom system with nonlinear force in the form of cubic is used to verify the accuracy of the method, and numerical analysis results reveal that the method is accurate enough compared with the time marching method. Furthermore, for the purpose of application, the responses of two common friction models, which are of great concern in some practical engineering fields, including a two-degree-of-freedom system containing a friction damper and a rotor disk system with circumferential rubbing, are obtained utilizing the presented approach. The influences of several model parameters on their responses are investigated as well. Numerical investigations demonstrate that the automatic differential solution framework developed in this research for solving nonlinear vibration equations has high accuracy and eliminates the need for a complicated partial derivative analytical formula derivation.