不同初始应力及变形对梁自振频率的影响
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摘要
应用Hamilton原理,建立了同时考虑初始荷载产生的弯曲和轴向变形影响时梁的控制微分方程,利用Galerkin方法推导出了考虑两种初始变形影响时简支梁自振频率的解析表达式,讨论了初始荷载大小、梁自身刚度及约束情况等不同参数对这两种初始变形对梁动力反应影响的敏感度,采用产生相同变形能的方式,定量地比较了各种边界条件下两种初始变形对梁自振频率的影响。
Based on the Hamilton principle, the governing differential equations were established for beams, considering the effects of both axial and flexural deformations induced by initial load and the analytical expression for the natural frequencies of a simply-supported beam under these effects was derived by the Galerkin method. A discussion on the analytical expression demonstrates the susceptivity of the two types of effects to the magnitude of initial load, beam stiffness and restraint conditions. Under the same amount of deformation energy produced, the effects of the two types of initial deformations on the natural frequencies of beams with various boundary conditions were compared quantitatively.
Based on the Hamilton principle, the governing differential equations were established for beams, considering the effects of both axial and flexural deformations induced by initial load and the analytical expression for the natural frequencies of a simply-supported beam under these effects was derived by the Galerkin method. A discussion on the analytical expression demonstrates the susceptivity of the two types of effects to the magnitude of initial load, beam stiffness and restraint conditions. Under the same amount of deformation energy produced, the effects of the two types of initial deformations on the natural frequencies of beams with various boundary conditions were compared quantitatively.
引文
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[15] 张家玮,刘合敏,杨治国.考虑初始荷载影响下梁动力特性的有限元分析[J].振动与冲击,2012,31(12):120-124.ZHANG Jiawei,LIU Hemin,YANG Zhiguo,el at.Finite element analysis for effect of initial load on dynamic characteristic of a beam [J].Journal of Vibration and Shock,2012,31(12):120-124.
[16] 周世军.板恒载效应非线性分析的刚度法[J].振动与冲击,2007,26(2):33-36.ZHOU Shijun.Stiffness method for nonlinear analysis of effect of dead loads on plates[J].Journal of Vibration and Shock,2007,26(2):33-36.
[17] 周世军,刘德贵.任意形状板考虑初始荷载效应的动力特性有限元分析[J].振动与冲击,2014,33(12):15-23.ZHOU Shijun,LIU Degui.Finite element analysis for dynamic characteristics of arbitrary shape plates with initial loads[J].Journal of Vibration and Shock,2014,33(12):15 -23.
[18] WASHIZU K.Variational methods in elasticity and plasti-city[M].New York:Pergaman Press,1982.
[2] BIOT M A.Increase of torsional stiffness of a prismatical bar due to axial tension[J].Journal of Applied Physics,1939,10(12):860-864.
[3] BRUNELLE E J.The statics and dynamics of a transversely isotropic Timoshenko beam[J].Journal of Composite Materials,1970,4(3):404-416.
[4] 石庆华,向锦武.大挠度梁静态变形对动态特性影响分析[J].振动与冲击,2007,26(1):73-76.SHI Qinghua,XIANG Jinwu.Effect of static large deformation on dynamic characteristic of a beam [J].Journal of Vibration and Shock,2007,26(1):73-76.
[5] 梁枢平,陈文.考虑轴力的简支梁非线性静力问题的DQ解[J].华中理工大学学报,1998,26(2):65-67.LIANG Shuping,CHEN Wen.The DQ solution of nonlinear static problem of simply-supported beam in consideration of axial force [J].Journal of Huazhong University of Science and Technology,1998,26(2):65-67.
[6] 梁枢平,刘洋,杨永波,等.梁的轴力非线性静力问题的进一步研究[J].华中科技大学学报(自然科学版),2005,33(2):116-118.LIANG Shuping,LIU Yang,YANG Yongbo,et al.Improvement of the geometrically nonlinear analysis of immovably simply supported beams by the DQM[J].Journal of Huazhong University of Science and Technology (Nature Science) 2005,33(2):116-118.
[7] TAKABATAKE H.Effects of dead loads in static beams[J].Journal of Structural Engineering ASCE,1990,116(4):1102-1120.
[8] TAKABATAKE H.Effects of dead loads on natural frequencies of beams[J].Journal of Structural Engineering ASCE,1991,117(4):1039-1052.
[9] ZHOU S J,ZHU X.Analysis of effect of dead loads on natural frequencies of beams using finite element techniques[J].Journal of Structural Engineering ASCE,1996,122 (5) :512-516.
[10] 周世军,朱唏.恒载对梁自振频率影响的分析[J].铁道学报,1995,17(4):98-103.ZHOU Shijun,ZHU Xi.Analysis of effect of dead loads on natural frequencies of beams [J].Journal of the China Railway Society,1995,17(4):98-103.
[11] 朱唏,周世军.分析恒载效应的有限元方法[J].工程力学,1996,13(3):54-60.ZHU Xi,ZHOU Shijun.A finite element method for analyzing the effect of dead loads[J].Engineering Mechanics,1996,13(3):54-60.
[12] 张家玮,周世军.恒载对拱形梁自振频率的影响分析[J].振动与冲击,2009,28(8):163-167.ZHANG Jiawei,ZHOU Shijun.Analysis of effect of dead loads on natural frequencies of arch beams[J].Journal of Vibration and Shock,2009,28(8):163-167.
[13] 周世军,张家玮.恒载效应对拱形梁的影响分析[J].工程力学,2010,27(7):120-125.ZHOU Shijun,ZHANG Jiawei.Analysis of the effect of dead loads on static arch beams[J].Engineering Mechanics,2010,27(7):120-125.
[14] 张家玮,周世军.考虑恒载效应的拱形梁静力近似解[J].计算力学学报,2010,27(4):665-660.ZHANG Jiawei,ZHOU Shijun.Approximate solutions of static arch beams considering dead loads effect[J].Chinese Journal of Computational Mechanics,2010,27(4):665-660.
[15] 张家玮,刘合敏,杨治国.考虑初始荷载影响下梁动力特性的有限元分析[J].振动与冲击,2012,31(12):120-124.ZHANG Jiawei,LIU Hemin,YANG Zhiguo,el at.Finite element analysis for effect of initial load on dynamic characteristic of a beam [J].Journal of Vibration and Shock,2012,31(12):120-124.
[16] 周世军.板恒载效应非线性分析的刚度法[J].振动与冲击,2007,26(2):33-36.ZHOU Shijun.Stiffness method for nonlinear analysis of effect of dead loads on plates[J].Journal of Vibration and Shock,2007,26(2):33-36.
[17] 周世军,刘德贵.任意形状板考虑初始荷载效应的动力特性有限元分析[J].振动与冲击,2014,33(12):15-23.ZHOU Shijun,LIU Degui.Finite element analysis for dynamic characteristics of arbitrary shape plates with initial loads[J].Journal of Vibration and Shock,2014,33(12):15 -23.
[18] WASHIZU K.Variational methods in elasticity and plasti-city[M].New York:Pergaman Press,1982.