基于Chebyshev-Ritz法分析多裂纹梁自振特性

基于弹性力学平面应力理论，利用Chebyshev-Ritz法分析多裂纹梁的自振特性. 根据裂纹情况将裂纹梁分成若干个梁段，用边界函数与第一类Chebyshev多项式的乘积构造各梁段的位移函数，具有很好的收敛性，能够适用于不同的几何边界条件. 用Ritz法得到各梁段的振动方程，根据各梁段之间的位移连续条件整合方程，建立整个裂纹梁的振动特征方程. 计算结果与有限元分析和相关文献数据吻合很好. 分析裂纹深度和位置对自振特性的影响. 随着裂纹深度的增大，裂纹梁的频率减小，振型的幅值变大，且影响的程度会受裂纹的位置影响.

Free vibration characteristics of multi-cracked beam based on Chebyshev-Ritz method

The free vibration characteristics of multi-cracked beam were analyzed based on the plane stress theory of elasticity by using Chebyshev-Ritz method. The cracked beams were divided into several sections according to their cracks. The products of boundary functions and Chebyshev polynomials were taken as the functions of the displacement, which had good convergence, making the method applicable for different geometric boundary conditions. The vibration equation of each sub-beam could be obtained by using Ritz method. The vibration characteristic equation of the whole cracked beam was established by the continuity conditions of displacements between adjacent sub-beams. The calculation results accorded well with those available from the literature and the finite element analysis. The effects of the structural parameters such as crack depth and location on the natural vibration characteristics of the beam were analyzed. As the crack depth increases, the natural frequency of the cracked beam decreases, the amplitude of the mode shape increases, and the degree of influence is affected by the location of the crack.