基于裂纹等效扭转弹簧模型的裂纹梁振动分析

doi:10.12066/j.issn.1007-2861.1978

摘要/Abstract

摘要：

基于开裂纹的等效扭转弹簧模型, 研究了裂纹梁动力特性和动力响应的计算方法. 在给出裂纹梁等效抗弯刚度的基础上, 建立了一种新的裂纹梁动力控制方程通解的求解方法, 给出了具有任意条裂纹 Euler-Bernoulli 梁振动模态的统一显示表达式. 数值分析了简支、悬臂和两端固支裂纹梁的自振频率和振动模态, 并研究了简支裂纹梁在集中简谐载荷作用下的动力响应, 考察了裂纹条数和深度等对裂纹梁动力特性和动力响应的影响. 结果表明: 随着裂纹深度和条数的增加, 裂纹梁的自振频率减小, 且当裂纹较深时, 裂纹深度对自振频率的影响更为显著; 裂纹梁的模态曲线在裂纹处呈现尖点, 其尖点处斜率的改变随裂纹深度的增加而增加, 且当裂纹处的弯矩为 0 时, 裂纹对梁的模态和频率没有影响; 由于裂纹梁的模态仍满足正交性, 因此可采用模态叠加法分析裂纹梁的动力响应.

关键词: 裂纹 Euler-Bernoulli 梁, 等效扭转弹簧模型, 广义函数, 动力特性, 动力响应

Abstract:

Based on an equivalent rotational spring model of crack, computation methods of dynamic characteristics and dynamic responses of cracked beams were investigated. On the basis of the equivalent flexural rigidity of the cracked beam, a method for obtainingageneral solution to the dynamic governing equation of cracked beam is established. A unified explicit expression of the vibration mode of the Euler-Bernoulli beam with an arbitrary number of cracks is presented. Natural frequencies of simply-supported, cantilever and clamped-clamped cracked beams are analyzed numerically. The dynamic response of the simply-supported cracked beam subject to a concentrated harmonic load is studied. Influences of depth and number of cracks on the dynamic characteristics and dynamic response are examined, revealing that the natural frequencies decrease with increased depth and number of cracks, and influence of the crack depth on the natural frequencies is more remarkable when the crack depth is large. There is a cusp on the mode curve of the cracked beam at the crack location, and slope change of the mode curve at the crack location increases with the increase of the crack depth.The crack has no influence on the natural frequencies and the modes of the cracked beam when the bending moment of the beam at the crack location is zero. Furthermore, the mode superposition method can be usedto analyze dynamic responses of acracked beam due to orthogonality of the modes of the cracked beam.

Key words: cracked Euler-Bernoulli beam, equivalent rotational spring model, generalized function, dynamic characteristic, dynamic response

中图分类号:

戴缘, 王天宇, 杨骁. 基于裂纹等效扭转弹簧模型的裂纹梁振动分析[J]. 上海大学学报(自然科学版), 2019, 25(6): 965-977.

Yuan DAI, Tianyu WANG, Xiao YANG. Vibration analysis of cracked beam based on crack's equivalent rotational spring model[J]. Journal of Shanghai University（Natural Science Edition）, 2019, 25(6): 965-977.

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