端部带质量和弹簧约束悬臂梁振动响应的解析解

根据端部带质量和弹簧约束悬臂梁的特征值条件，提出了一种特征变换方法，获得了带约束悬臂梁广义质量和振动响应的解析解。通过分析根部弯矩、端部位移、速度和加速度放大系数的变化特征可知，端部弹簧的刚度对静态和一阶载荷响应有明显的影响，减载设计时可以放宽对端部质量的限制，载荷响应分析阶次介于速度和加速度的分析阶次之间。提出的特征变换方法可应用于求解其他载荷分布、边界条件和端部约束悬臂梁的振动响应解析解。 

Vibration response analytical solutions of cantilever beam with tip mass and spring constraints

According to the eigenvalue condition of cantilever beam with tip mass and spring, this paper proposes a characteristic transformation method, and obtains the analytical solutions of generalized mass and vibration response of cantilever beam with constraints. By analyzing the variational regularities of amplification factors of root bending moment, tip displacement, tip velocity, and tip acceleration for this cantilever beam, the results indicate that the stiffness of tip spring has notable effect on static and first-order load responses, the restriction of tip mass can be relaxed in the load reduction design, and the analysis order of load response is between the analysis orders of velocity and acceleration. The proposed characteristic transformation method can be used to obtain the vibration response analytical solutions of cantilever beam with other loading distribution, boundary conditions and tip constraints.