Vibration analysis of nonlinear damping systems by the discrete incremental harmonic balance method

An improved incremental harmonic balance method (IHBM) is proposed by Wang (J Sound Vib 441:111-125, 2019) to solve the periodic responses of the continuous nonlinear stiffness systems. However, the nonlinear damping systems remain unsolved. This paper aims to investigate the nonlinear damping parts by the proposed IHBM method, which is based on the principle that any continuous curve can be approximated by a piecewise-linear curve with discrete nodes. The piecewise-linear function can be considered a unified benchmark function that can convert the complex IHBM Galerkin process of arbitrary nonlinear damping systems to that of unified piecewise-linear damping systems. The general process of the proposed method for this piecewise-linear system is derived considering the stability of the solutions. Then, a polynomial nonlinear damping system is investigated to validate the accuracy of the method. Furthermore, five typical cases of single-degree-of-freedom (SDOF) nonlinear damping systems are carried out, and this method is also extended to multi-degree-of-freedom (MDOF) systems where each nonlinear force in the systems is expressed by the function of only one independent DOF. The results illustrate that the proposed method shows convenience and accuracy in obtaining the dynamics of nonlinear systems.