基于摄动法的多条裂纹欧拉梁特征模态分析

基于摄动理论推导了带多条开口裂纹的欧拉梁的特征模态参数的理论计算公式。采用最直接的方式将梁开口裂纹模拟成梁微段内的横截面折减并用δ函数表达了带开口裂纹的梁沿轴线的截面惯矩和线质量等物理参数。基于此,建立了裂纹梁动力微分方程,并采用一阶摄动理论推导得到了梁的模态频率和振型计算公式。简支梁及悬臂梁算例研究表明,该方法具有很好的精度,与有限元模拟结果及实验结果都能很好地吻合。并采用此方法分析了裂纹深度和位置对带多条开口裂纹梁的特征模态参数的影响。结果表明,裂纹对各阶模态频率虽然影响有限,但其引起的各阶频率变化有着明显的模式,可用于结构损伤定位;裂纹对模态振型影响不明显,但对模态曲率影响比较大,可用于结构损伤位置和程度的诊断。

Modal Analysis for Euler-Bernoulli Beam with Multiply Open Cracks Based on Perturbation Method

This paper presented a method for modal analysis of Euler-Bernoulli beam with open cracks based on perturbation method. Beam cracks are firstly modeled as the cross section reduction of the cracked segment. The moment of inertia and mass per unit length of the beam are then expressed using function. The equation of motion for the cracked Euler-Bernoulli beam is formulated. Based on perturbation theory, beam natural frequencies and mode shapes are expressed using the first order perturbation terms. Case studies on a simply supported beam and a cantilever beam tell that the presented method is of good precision to match the results of finite element simulation and experimental tests. Based on these formulations, the effects of crack size and location on the variation of beam modal parameters are analyzed. The results tell that although micro-crake will induce small variation on structural natural frequencies, the variations of multiple modes are of a pattern to indicate crack location. The mode shape is not sensitive to the size and location variation of crack. However, the curvature mode shape is quite sensitive to indicate the location and relative severity of cracks.