利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。

IDENTIFICATION OF CRACK PARAMETERS IN A STEPPED CANTILEVER BEAM USING TRANSFER MATRIX METHOD

The problem of crack parameter identification in a stepped cantilever beam studyed.The local effect of "softening" at the crack location can be simulated by an equivalent spring connecting the two segments of the beam.Combined with the Bernoulli-Euler theory for beams,the characteristic equation of a stepped cantilever beam model is derived by using transfer matrix method.The resulted frequency equation of a cracked beam is generally depends on the boundary conditions including the closer(elastic) end supports and can be formulated analytically by using symbolic codes.The procedure proposed simply operates on some fourth order determinants so that the computational efforts for calculating natural frequencies are significantly reduced.Based on this characteristic equation,two identification methods,the contour curve method and the minimizing object function one,are developed to identify the location and the depth of the crack,which only requires the first three tested natural frequencies.The proposed approaches are verified through an example with a single-stepped cantilever beam.It is shown that the location and the depth of the crack can be worked out as long as the first three natural frequencies are known.It is also shown that making a "zero setting" may offer a tremendous improvement in identification of crack parameters.Moreover,the minimizing object function method can easily be adapted for complex boundary conditions and for multi-cracks located in any segments of the beam.