一般边界条件下功能梯度梁三维振动特性研究

    陈玉坤 靳国永 叶天贵

    

    

    

    摘要: 基于卡莱拉统一公式(CUF)建立了一般边界条件下功能梯度(FGM)梁的高阶统一动力学模型和分析方法。利用二维泰勒公式对FGM梁截面位移函数进行高阶拟合,经典梁理论可以视为一阶泰勒公式的特殊形式。采用Voigt线性混合模型分别考虑了两种功能梯度材料分布形式:材料属性仅沿宽度或厚度单一方向发生变化;材料属性同时沿宽度和厚度方向发生变化。通过罚函数法将FGM梁的边界条件量化为边界能量的形式,实现了对边界条件的参数化分析,并消除了位移容许函数对边界条件的依赖性。利用瑞利-利兹法和勒让德多项式函数对FGM梁的振动问题进行求解。通过与文献中结果对比,验证了此方法的有效性和正确性。最后,研究了几何尺寸、材料属性和边界条件对FGM梁振动特性的影响规律。

    关键词: 结构振动; 功能梯度梁; 一般边界条件; 卡莱拉统一公式(CUF); 罚函数方法

    中图分类号: O327; O326 ?文献标志码: A ?文章编号: 1004-4523(2020)04-0756-08

    DOI:10.16385/j.cnki.issn.1004-4523.2020.04.014

    引 言

    功能梯度材料(Functionally Graded Materials, FGMs)是将两种或两种以上的材料按照一定的比例混合起来,其材料属性沿一定方向上呈现微观的连续性变化,可有效降低和避免一般层合材料由于层间材料参数差距过大而产生的应力集中现象。近年来,随着材料技术和加工工艺的进步,各种FGMs也渐渐出现[1]。为此,对FGM梁结构的动力学特性研究具有十分重要的意义。

    首先,本文以固支边界(C-C)条件下长宽比L/b=10的FGM梁为例,验证本文方法的收敛性。表2给出了两种不同材料属性分布的FGM梁无量纲化频率随截断级数的变化情况,表中每一行代表FGM梁不同类型的振动模态,每一列代表了不同的勒让德多项式项数。从表中可以较为清楚地看到随着勒让德多项式项数的增加,本文方法的计算结果快速收敛至稳定值。故而在接下来的计算分析中,勒让德多项式项数统一取为M=14。图3给出了FGM梁(Type 2)的几类特殊模态的振型图,红色网格代表弯曲模态的弯曲中性面。从图3中可以看出,前两阶弯曲模态的弯曲中性面不再为xy或yz平面,而是绕着y轴发生一定的旋转,旋转角度与功能梯度材料属性有关,由于本案例所选用的功能梯度材料属性关于梁截面對角线呈对称分布,所以其弯曲中性面与截面对角线重合。另外,而且可以看出本文方法不仅可以用于预测FGM梁的弯曲模态,而且可以较为准确地预测FGM梁的扭转模态和纵振模态。

    为了进一步验证本文方法的正确性,表3给出了简支边界(S-S)条件下三种不同长宽比的FGM梁的无量纲频率参数。同时表3也列出了文献[11]中的计算结果,在该文献中,作者利用有限元方法进行求解。通过与文献[11]对比,可以发现,对于不同长宽比的FGM梁结构,本文方法均具有较高的计算精度。对于经典梁理论模型,当结构长宽比较小时,其计算结果往往会产生较大的误差。而本文方法由于不存在对应力应变分布规律的任何假设,故而可以适用于任意长宽比的FGM梁结构。

    3 结 论

    本文基于CUF理论建立了一般边界条件下FGM梁的高阶统一动力学模型;同时考虑了两种功能梯度材料分布形式:沿宽度单一方向变化和沿截面任意方向变化。通过罚函数法将FGM梁的边界条件量化为边界能量的形式,实现了对边界条件的参数化分析,并消除了位移容许函数对边界条件的依赖性。利用瑞利-利兹法对FGM梁的三维振动问题进行求解,并验证了该方法的快速收敛性和正确性。最后,本文重点研究了几何尺寸、材料属性和边界条件对功能梯度梁振动特性的影响。针对本文所研究的FGM梁,其无量纲频率参数随着长宽比的增大和材料梯度指数的增加而迅速减小,直至趋于稳定。在利用罚函数方法对FGM梁边界条件的研究中可以发现,罚因子参数存在一弹性区间,当罚因子参数的取值在此区间外时,FGM梁各阶无量纲频率参数保持不变;当罚因子参数取值在此区间内时,无量纲频率随特定的罚因子参数迅速增大。

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    Abstract: In this study, a uniform dynamic model is established using Carrera unified formulation for the vibration analysis of FGM beams with general boundary conditions. Higher-order Taylor expansions are used to simulate the cross-sectional displacement functions, and the classical beam theories can be seen as particular cases of the first-order Taylor expansion. Two types of material distribution over the beams cross-section are considered by utilizing the Voigt model and power-law function. The penalty method is applied and three groups of penalty factor are used for the boundary conditions of the FGM beam. In this way, the energy stored in the boundary conditions can be considered as a part of the total energy function, releasing the limitation of geometrical boundary restraints on the selection of admissible functions. The Rayleigh-Ritz method and Legendre polynomial functions are utilized to address the vibration problem of the FGM beams. Several numerical examples are carried out to demonstrate the effectiveness and correctness of the present method. Finally, the effects of the geometrical dimensions, material parameters and boundary conditions on the vibration characteristics of the FGM beams are studied.

    Key words: structural vibration; FGM beams; general boundary conditions; Carrera unified formulation; penalty method