Stability and bifurcation analysis of a bevel gear system supported by finite-length squeeze film dampers

To analyze the dynamic characteristics of gear systems supported by squeeze film dampers (SFD), the nonlinear oil-film force of SFD is usually obtained by the short bearing approximation (SBA), long bearing approximation (LBA) or finite difference method (FDM). However, the SBA and LBA methods only hold for the cases of infinitely short and infinitely long SFD, which may be not true in practice. Additionally, the FDM method is generally applied to the case of the regular film boundary. Hence, the present work proposes a finite element method to achieve the film pressure of finite-length SFDs (FLSFD) based on the variational principle. The proposed method is not plagued with the boundary conditions and is verified by the comparison with the classic methods. Then, a seven-degree-of-freedom dynamic model of a bevel gear system with FLSFD is developed incorporating the nonlinear film force. Based on Gram–Schmidt QR-decomposition, a strategy to calculate the Lyapunov spectrum of the high-dimensional gear system is presented, and the characteristic multipliers of the system are obtained by solving the eigenvalues of the monodromy matrix. The Lyapunov exponents, characteristic multipliers, and bifurcation diagrams, as well as phase portraits and Poincaré sections, are utilized to qualify the nonlinear behaviors of the bevel gear system with and without FLSFD. The results show that the application of FLSFD can effectively reduce the occurrences of saddle-node bifurcation, Hopf bifurcation, and period-doubling, and suppress nonlinear characteristics like the bistable response and jump phenomenon.