基于弹性力学的端部有裂缝悬臂梁的自由振动分析  

Free vibration of a cantilever beam with a crack at clamped end based on elasticity theory

作　　者：蒋杰 周叮[1] 胡朝斌[1] JIANG Jie;ZHOU Ding;HU Chaobin(College of Civil Engineering,Nanjing Tech University,Nanjing 211816,China)

机构地区：[1]南京工业大学土木工程学院

出　　处：《振动与冲击》2019年第15期196-201,共6页Journal of Vibration and Shock

基　　金：国家自然科学基金(11372127);江苏省交通厅重大专项课题(2014Y01)

摘　　要：基于小变形的二维线弹性力学理论,采用Chebyshev-Ritz法分析端部有裂纹悬臂梁的自由振动特性。在裂缝尖端处将裂纹梁沿长度方向划分为不同边界条件的两层子梁,利用第一类切比雪夫多项式与边界特征函数的乘积作为每层子梁的位移试函数,通过利兹法分别求得每层子梁的振动方程,再根据上下层在交界面处的位移连续性条件得到整个裂缝梁的振动方程。数值结果与有限元解以及文献数据比较显示了很好的一致性,最后详细分析了高跨比、裂缝深度对无量纲自振频率和振型的影响。Here, based on the 2-D linear elasticity theory, free vibration characteristics of a cantilever beam with a crack at clamped end were analyzed using Chebyshev-Ritz method. Firstly, at crack tip along the beam length direction, the cracked beam was divided into two sub-beam layers with different boundary conditions. The 1 st kind Chebyshev polynomials multiplied by boundary characteristic functions satisfying geometric boundary conditions were chosen as each sub-beam layer’s displacement trial functions. Ritz method was used to acquire vibration equations of two sub-beam layers, respectively. Then, utilizing the displacement continuity condition at the interface between two sub-beam layers, the whole beam’s vibration equation was derived. The numerical results were compared with those using the finite element method and available ones in literature, and it was shown that all of them agree well with each other. Finally, effects of height to span ratio and crack depth on non-dimensional natural frequencies and modal shapes were analyzed in detail.

关 键 词：二维线弹性理论 裂缝悬臂梁 自由振动 Chebyshev-Ritz法 

分 类 号：TU311.3[建筑科学—结构工程]