Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.

中文翻译：

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非线性弹簧质量系统支撑的轴向加载梁的动态行为分析

具有非线性支撑的 Euler-Bernoulli 梁的动态分析近年来受到了越来越多的研究兴趣。目前的研究通常将边界和内部非线性支撑分开考虑，通常忽略系统旋转约束。然而，对于这种非线性问题，考虑轴向载荷、集中质量和内部支撑同时存在的研究很少。受此限制的启发，本文求解并研究了由非线性弹簧质量系统支撑的轴向加载梁的动态行为。在非线性控制微分方程的 Galerkin 离散化中，采用带有线弹性支撑的轴向加载 Euler-Bernoulli 梁的模态函数作为轨迹函数。采用伽辽金截断法求解非线性弹簧质量系统支撑的轴向加载梁的稳态响应，并通过有限差分法进行验证。结果表明，非线性弹簧质量系统参数和边界条件对系统动力学行为有显着影响。此外，适当的非线性参数可以有效地在单周期状态和准周期状态之间切换系统行为。