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.

为了分析由挤压膜阻尼器（SFD）支撑的齿轮系统的动力特性，通常通过短轴承近似（SBA），长轴承近似（LBA）或有限差分法（FDM）获得SFD的非线性油膜力。 。但是，SBA和LBA方法仅适用于无限短和无限长SFD的情况，在实践中可能并非如此。另外，FDM方法通常应用于规则膜边界的情况。因此，目前的工作提出了一种基于变分原理的有限元方法来实现有限长度SFDs（FLSFD）的薄膜压力。该方法不受边界条件的困扰，并与经典方法进行了比较验证。然后，利用非线性薄膜力，开发了带有FLSFD的锥齿轮系统的七自由度动力学模型。基于Gram–Schmidt QR分解，提出了一种计算高维齿轮系统Lyapunov谱的策略，并通过求解单峰矩阵的特征值来获得系统的特征乘数。利用Lyapunov指数，特征乘数和分叉图以及相图和庞加莱截面，可以对带或不带FLSFD的锥齿轮系统的非线性行为进行鉴定。结果表明，FLSFD的应用可以有效减少鞍结分叉，霍普夫分叉和周期倍增的发生，并抑制双稳态响应和跳跃现象等非线性特性。