| 17 | 0 | 36 |
| 下载次数 | 被引频次 | 阅读次数 |
本研究采用插值矩阵法（IMM）分析了沿着厚度方向功能梯度材料（FGM）Timoshenko梁的横向自由振动特性。首先，基于Hamilton原理，推导出功能梯度材料Timoshenko梁的运动微分方程组；接着，基于插值矩阵法原理，将Timoshenko梁的运动微分方程组的特征值问题转化为标准型广义代数方程组特征值问题；最后，采用QR法计算广义代数方程组，获得Timoshenko梁横向自由振动的固有频率，并一次性地给出了前10阶固有频率及其相应的振型函数。数值算例表明，梁长上离散单元数越多计算精度越高，随着离散单元数成倍增加，本研究计算值与精确解加速收敛，当取离散单元数n为40时，IMM近似解与精确解完全收敛，证明了插值矩阵法的可行性和精确性，也是可以满足工程实际需要的；同时，计算结果还表明，Timoshenko梁的高跨比δ=h/L对功能梯度材料Timoshenko梁的低阶固有频率影响较小，但对高阶固有频率影响还是比较明显。
Abstract：The lateral free vibration characteristics of Timoshenko beams made of functionally graded materials(FGM) along the thickness direction are analyzed by the interpolation matrix method(IMM). Firstly, based on Hamilton's principle, the motion differential equation system for functionally graded material Timoshenko beams is derived. Then, relying on the principle of the interpolation matrix method, the eigenvalue problem of the motion differential equation system for Timoshenko beams is transformed into a standard generalized algebraic equation eigenvalue problem. Finally, the QR method is adopted to solve the generalized algebraic equation system,thereby obtaing the natural frequency of the transverse free vibration of Timoshenko beams. In this paper, the first10 natural frequencies and their corresponding mode functions are presented simultaneously. Numerical examples show that the greater the number of discrete elements along the beam length, the higher the calculation accuracy.When the number of discrete elements is doubled, the convergence between the calculated values and exact solutions in this paper is accelerated. Specifically, when the number of discrete elements n is set to 40, the IMM approximate solutions and exact solutions in this paper achieve full convergence, which verifies the feasibility and accuracy of the IMM and confirms its ability to meet the practical engineering requirements. Meanwhile, the calculation results also indicate that the height-to span ratio of Timoshenko beams(δ=h/L) exerts a relatively minor impact on the low-order natural frequencies of functionally graded material Timoshenko beams, but has a more significant impact on the high-order natural frequencies.
[1]KADOLI R, AKHTAR K, GANESAN N. Static analysis of functionally graded beams using higher order shear deformation theory[J].Applied Mathematical Modelling,2007, 32(12):2509-2525.
[2]SIMSEK M. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories[J].Composite Structures, 2009, 92(4):904-917.
[3]HSU Y S. Enriched finite element methods for Timoshenko beam free vibration analysis[J]. Applied Mathematical Modelling, 2016, 40(15-16):7012-7033.
[4]MOALLEMI-OREH A, KARKON M. Finite element formulation for stability and free vibration analysis of timoshenko beam[J].Advances in Acoustics and Vibration,2013,2013(2):143-152.
[5]TONG X, TABARROK B, YEH KY. Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section[J].Journal of Sound&Vibration, 1995,186(5):821-835.
[6]SORRENTINO S, FASANA A, MARCHESIELLO S.Analysis of non-homogeneous Timoshenko beams with generalized damping distributions[J].Journal of Sound&Vibration, 2007, 304(3-5):779-792.
[7]RAJASEKARAN S, NOROUZZADEHTOCHAEI E.Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order[J]. Meccanica,2014, 49(4):995-1009.
[8]SHAHBA A, ATTARNEJAD R, TAVANAIE MARVI M,et al. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions[J].Composites Part B:Engineering, 2011, 42(4):801-808.
[9]SANKAR B.V. An elasticity solution for functionally graded beams[J].Composites Science&Technology,2001, 61(5):689-696.
[10]CHAKRABORTY A, GOPALAKRISHNAN S, REDDY J N. A new beam finite element for the analysis of functionally graded materials[J].International Journal of Mechanical Sciences, 2003; 45(3):519-539.
[11]CHING H K, YEN S C. Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads[J].Composites Part B Engineering, 2004, 36(3):223-240.
[12]CHING H K, YEN S C. Transient thermos-elastic deformations of 2D functionally graded beams under nonuniformly convective heat supply[J].Composite Structures, 2006, 73(4):381-393.
[13]QIAN L F, CHING H K. Static and dynamic analysis of 2-D functionally graded elasticity by using meshless local Petrov-Galerkin method[J].Journal of the Chinese Institute of Engineers, 2004, 27(4):491-503.
[14]XIANG H J, YANG J. Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction[J].Composites Part B Engineering,2007, 39(2):292-303.
[15]ZHU H, SANKAR B V. Analysis of sandwich TPS panel with functionally graded foam core by Galerkin method[J].Composite Structures, 2005, 77(3):280-287.
[16]SANKAR B V, TZENG J T. Thermal stresses in functionally graded beams[J]. Aiaa Journal, 2002,40(6):1228-1232.
[17]ZHU H, SANKAR B V. A combined Fourier series-galerkin method for the analysis of functionally graded beams[J].Journal of Applied Mechanics, 2004,71(3):421-425.
[18]陈琦,马连生,郭章新.功能梯度材料梁自由振动的线性与非线性振动[J].力学与实践, 2023, 45(3):520-525.
[19]ZHONG Z, YU T. Analytical solution of a cantilever functionally graded beam[J].Composites Science and Technology, 2007, 67(3-4):481-488.
[20]NIRMALA K, UPADHYAY P C, PRUCZ J, et al.Thermo-elastic stresses in composite beams with functionally graded layer[J].Journal of Reinforced Plastics and Composites, 2006, 25(12):1965-1977.
[21]王珍,朱少平.一类非线性微分方程三阶三点边值问题正解的存在性[J].井冈山大学学报(自然科学版),2023,44(4):7-10.
[22]葛仁余,张金轮,韩有民,等.轴向功能梯度变截面Timoshenko梁自由振动的研究[J].振动与冲击, 2017,36(22):158-165.
[23]SIMSEK M. Bi-directional functionally graded materials(BDFGMs)for free and forced vibrationof Timoshenko beams with various boundary conditions[J].Composite Structures, 2015,133(12):968-978.
[24]SINA S A, NAVAZI H M, HADDADPOUR H. An analytical method for free vibration analysis of functionally graded beams[J]. Materials and Design, 2009,30(3):741-747.
[25]SIMSEK M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories[J]. Nuclear Engineering and Design, 2009,240(4):697-705.
基本信息:
中图分类号:O327
引用信息:
[1]刘羽天,葛仁余.插值矩阵法分析FGM矩形截面梁的自由振动[J].井冈山大学学报(自然科学版),2026,47(03):17-25.
基金信息:
安徽省自然科学基金项目（1808085ME147）; 安徽商贸职业技术学院重点项目（2024KZR11）
2026-05-10
2026-05-10