分数阶粘弹性地基上裂纹梁的自由振动  

Free Vibration of Cracked Beams on a Fractional⁃Order Viscoelastic Foundation

作　　者：张骏 魏培君 周小利[1,3] 王刚[1] ZHANG Jun;WEI Peijun;ZHOU Xiaoli;WANG Gang(College of Civil Engineering,Hebei University of Engineering,Handan 056038,Hebei,China;Department of Applied Mechanics,University of Science and Technology Beijing,Beijing 100083,China;The Center for Computational Mechanics and Engineering Applications,Hebei University of Engineering,Handan 056038,Hebei,China)

机构地区：[1]河北工程大学土木工程学院,河北邯郸056038 [2]北京科技大学应用力学系,北京100083 [3]河北工程大学计算力学与工程应用研究中心,河北邯郸056038

出　　处：《力学季刊》2024年第3期877-886,共10页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(12072022);河北省教育厅项目(QN2018061)。

摘　　要：本文研究了分数阶粘弹性Pasternak地基上含多裂纹Euler梁的自由振动问题.首先,引入分数阶导数概念,建立粘弹性Pasternak地基模型,推导出地基应力-应变本构的复模量以及地基反力.其次,将裂纹等效成无质量扭转弹簧,借助于裂纹能量释放率和应力强度因子的关系,推导出梁的局部柔度.然后,基于传递矩阵方法,建立含多裂纹梁的分段传递矩阵进而得到总体传递矩阵.最后,基于梁的边界条件,建立求解裂纹梁的复固有频率及振型的线性代数方程组,数值求解得到含多裂纹梁的固有频率及振型.以含双裂纹的两端简支Euler梁为例,数值计算了复固有频率和振型.并基于曲率模态分析裂纹位置对复固有频率和振型的影响.数值结果揭示了粘弹性地基的分数阶系数、粘性系数以及裂纹位置和裂纹深度对复固频率和振型的影响规律.In this paper,the free vibration problem of a Euler beam with multiple cracks mounted on the fractional-order viscoelastic Pasternak foundation is studied.Firstly,the fractional order derivative is introduced to establish the viscoelastic foundation model.The complex modulus of the stress-strain constitutive equation and the subgrade reaction force of the fractionalorder viscoelastic Pasternak foundation are derived.Next,the cracks are modelled as massless torsional springs.The local flexibility of the crack is derived based on the relation between the energy release rate and the stress intensity factor.Further,by means of the transfer matrix method,the partial transfer matrix of each segment and the total transfer matrix of beam with multiple cracks are obtained.Finally,the linear algerbric equations of cracked Euler beam are derived based on the boundary conditions to determine the complex natural frequency and the vibration mode.As a numerical example,the Euler beam with two cracks is investigated and the complex natural frequency and the vibration mode are calculated.The curvature modes are used to analyze the effect of cracks.Besides,the effects of fractional order and viscosity coefficient of foundation as well as the location and depth of cracks on the natural frequency and vibration mode are also discussed based on the numerical results.

关 键 词：裂纹梁 分数阶导数 粘弹性地基 复固有频率 局部柔度法 传递矩阵法 

分 类 号：O326[理学—一般力学与力学基础]