T型裂纹梁的自振特性分析

Analysis of natural vibration characteristics of cracked T-beams

从二维弹性力学出发,采用Chebyshev-Ritz法研究含单裂纹T型梁的自振特性。根据弹性理论中应变相同且总内力不变原则,利用转换截面法将T型截面梁等效为由两层不同材料特性组成的矩形截面梁。将等效后的梁沿裂纹和层界面划分成四个子域。使用Rayleigh-Ritz法得到各子域的振动特征方程,结合子域间界面处的位移连续条件导出整个T型裂纹梁的振动特征方程。采用Chebyshev多项式构建各子域的位移试函数,由Chebyshev多项式的正交完备性可获得快速收敛解。与实际T型截面梁的有限元分析结果对比验证了该方法的正确性。以两端固支T型裂纹梁为例,分析了裂纹位置和裂纹深度对结构振动特性的影响。

Based on the 2-D elasticity theory,natural vibration characteristics of T-beams with a crack were studied using Chebyshev-Ritz method.Firstly,according to the principle of the same strain and the unchanged total internal force,the beam was equivalent to a rectangular cross-section beam composed of two material layers with different characteristics using the transfer section method.Then,the equivalent beam was divided into 4 sub-domains along the crack and layer interface.The natural vibration characteristic equation of each sub-domain was deduced using Rayleigh-Ritz method,and the natural vibration characteristic equation of the whole cracked T-beam was derived with displacement continuity conditions on interfaces of sub-domains.Chebyshev polynomials were applied to construct displacement trial functions of various sub-domains,and the fast convergence solutions were obtained using the orthogonality and completeness of Chebyshev polynomials.The correctness of the proposed method was verified by comparing its solving results with those using the finite element analysis for actual T-beams.Finally,a cracked T-beam fixed at both ends was taken as an example,effects of crack position and its depth on this beam’s natural vibration characteristics were analyzed.