摘要
基于忽略了梁截面剪切变形和转动惯量效应的Euler-Bernoulli梁理论,研究了轴向力作用下轴向功能梯度变截面梁的横向自由振动问题,将轴向功能梯度Euler-Bernoulli梁自由振动固有频率和临界荷载的计算转化为变系数常微分方程特征值问题。运用插值矩阵法可一次性计算出轴向功能梯度变截面梁各阶振动固有频率和临界荷载,分析了轴向荷载对轴向功能梯度Euler-Bernoulli梁自由振动固有频率的影响,即轴向压力使梁的第1阶固有频率降低,轴向拉力使梁的第1阶固有频率增大。在简支-简支梁(H-H)边界条件下、不同截面宽锥度系数c_b和截面高锥度系数c_h,且区间划分点数n为40时,本文计算结果与已有文献计算结果之间的最大相对误差不超过0.00768%;在简支-简支梁(H-H)、固端-自由梁(C-F)、固端-固端梁(C-C)这三种不同边界条件下,不同c_b和c_h,且n为40时,最大相对误差不超过0.101%,说明了本文方法的有效性和良好的计算精度。
Based on the Euler-Bernoulli beam theory in which the effects of shear deformation and rotary inertia are ignored, the paper deals with the free transverse vibration analysis of axially functionally graded beams subjected to a constant compressive loading. Then the governing equations of free transverse vibration and critical buckling load analysis of axially functionally graded Euler-Bernoulli beams are transformed into a set of nonlinear characteristic ordinary differential equations. By applying the interpolating matrix method(IMM) to the established equations, all the natural frequencies of free transverse vibration and critical buckling load of axially functionally graded beam with variable cross-section are calculated at one time.The effect of axial load on the natural frequency of free vibration of axially functionally graded Euler-Bernoulli beams is analyzed, that is to say, the axial pressure causes the first natural frequencies of the beam to decrease, and the axial tension causes the first natural frequencies of the beam to increase. Under hinged-hinged(H-H), and considering different wide taper coefficient c_b and high taper coefficient c_h, the maximum relative error between the calculated results and the existing ones is not more than 0.00768% when the interval division numbers is 40. Under three different boundary conditions, such as hinged-hinged(H-H), Clamped-Free(C-F) and Clamped-Clamped(C-C),and considering different wide taper coefficient c_b and high taper coefficient c_h, the maximum relative error between the calculated results and the existing ones is not more than 0.101% when the interval division numbers is 40. Therefore the validity and accuracy of the proposed method are illustrated.
出处
《应用力学学报》
CSCD
北大核心
2017年第5期875-880,共6页
Chinese Journal of Applied Mechanics
基金
国家自然科学基金(11272111)
安徽省高校自然科学研究重点项目(KJ2016A055)
关键词
变截面梁
横向振动
固有频率
插值矩阵法
功能梯度材料
variable cross-section beam
transverse vibration
natural frequency
the interpolating matrix method
functionally graded material(FGM)
