非保守功能梯度材料简支梁在过屈曲附近的振动响应
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摘要
研究了受沿轴线分布切向随动载荷作用下功能梯度材料简支梁在过屈曲前后的自由振动问题.基于可伸长梁的大变形理论,建立了功能梯度材料梁在随动非保守载荷下的大挠度动力学控制方程.其中,假设功能梯度材料性质沿梁厚度方向按成分含量百分比的幂指数形式连续变化.采用打靶法求解振动问题的控制方程,得到了前三阶固有频率的数值解,给出了不同材料梯度指数下前三阶固有频率随载荷变化的关系曲线,分析和讨论了梁的材料梯度指数对梁振动响应的影响.
The free vibration of a simply supported beam made from functionally gradedmaterials(FGM)before and after the buckling subjected to uniformly and tangentiallycompressing follower force were studied.Based on the large deformation theory forextensible beams,governing equations for free vibration of FGM beams with the largeamplitude under a non-conservative load were formulated.It is assumed that the materialproperties of functionally graded material vary continuously in the thickness directionand obey a simple power law of volume fraction of the constituents.By using shootingmethod to solve the governing equations for vibration,numerical solutions of the firstthree natural frequencies depending on load parameter were obtained and characteristicscurves of the lower non-dimensional frequencies versus load parameter of FGM beamsunder the different graded index were plotted.The effects of material gradient propertyon the vibration of the beam were discussed in detail.
The free vibration of a simply supported beam made from functionally gradedmaterials(FGM)before and after the buckling subjected to uniformly and tangentiallycompressing follower force were studied.Based on the large deformation theory forextensible beams,governing equations for free vibration of FGM beams with the largeamplitude under a non-conservative load were formulated.It is assumed that the materialproperties of functionally graded material vary continuously in the thickness directionand obey a simple power law of volume fraction of the constituents.By using shootingmethod to solve the governing equations for vibration,numerical solutions of the firstthree natural frequencies depending on load parameter were obtained and characteristicscurves of the lower non-dimensional frequencies versus load parameter of FGM beamsunder the different graded index were plotted.The effects of material gradient propertyon the vibration of the beam were discussed in detail.
引文
[1]仲政,吴林志,陈伟球.功能梯度材料与结构的若干力学问题研究进展[J].力学进展,2010,40(5):528-541Zhong Zheng,Wu Linzhi,Chen Weiqiu.Progress in the study on mechanics problems of functionally graded materials and structures[J].Advances in Mechanics,2010,40(5):528-541
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[4]Farzad Ebrahimi,Mohadese Mokhtari.Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method[J].Journal of the Brazilian Society of Mechanical Sciences and Engineering,2015,37(4):1435-1444
[5]Chen W R,Chang H.Close-Form solutions for free vibration frequencies of functionally graded Euler-Bernoulli beam[J].Mechanics of Composite Materials,2017,53(1):79-98
[6]Aleksandar Nikolic'.Free vibration analysis of a non-uniform axially functionally graded cantilever beam with a tip body[J].Archive of Applied Mechanics,2017,87(7):1227-1241
[7]李清禄,李世荣.功能梯度材料梁在后屈曲构形附近的自由振动[J].振动与冲击,2011,30(9):76-78Li Qinglu,Li Shirong.Free vibration of FGM Euler beam with post-buckling configuration subjected to axial force[J].Journal of Vibration and Shock,2011,30(9):76-78
[8]焦常科,李爱群,伍小平.竖向动压力对FPS隔震连续梁桥地震响应影响研究[J].应用基础与工程科学学报,2017,25(4):733-741Jiao Changke,Li Aiqun,Wu Xiaoping.Research on the influence of dynamic vertical pressure on the seismic response of continuous girder bridge isolated with FPS[J].Journal of Basic Science and Engineering,2017,25(4):733-741
[9]李清禄,杨静宁.非保守力作用下简支梁在屈曲附近的自由振动[J].力学与实践,2014,36(3):333-336Li Qinglu,Yang Jingning.Free vibration of a simply supported beam under a non-conservation distributed load[J].Mechanics in Engineering,2014,36(3):333-336
[10]李清禄.功能梯度材料梁和圆板在随动载荷作用下的静动态响应[D].兰州:兰州理工大学博士论文,2012Li Qinglu.Static and dynamic response of functionally graded material beams and circular plates subjected to follower force[D].Lanzhou:Lanzhou University of Technology,2012
[11]Li Q L,Zhang J H.Vibration and post-buckling of a functionally graded beam subjected to non-conservation forces.Journal of vibroengineering,2016,18(8):4901-4913
[12]Li Q L,Luan W D,Zhu Z Q.Shooting method for free vibration of FGM Reissner-Mindlin circular plates resting on elastic foundation in thermal environment[J].Journal of Vibroengineering,2017,19(6):4423-4439
[13]Yang J,Shen H S.Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments[J].Journal of Sound and Vibration,2002,255:579-602
[2]Mao Y Q,Ai S,Chen C,et al.Nnlinear dynamic response and damage analysis for functionally graded metal shallow spherical shell under low-velocity impact[J].Archive of Applied Mechanics,2015,85(11):1627-1647
[3]Nuttawit W,Jarruwat C.Vibration characteristics of stepped beams made of FGM using differential transformation method[J].Meccanica,2015,50(4):1089-1101
[4]Farzad Ebrahimi,Mohadese Mokhtari.Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method[J].Journal of the Brazilian Society of Mechanical Sciences and Engineering,2015,37(4):1435-1444
[5]Chen W R,Chang H.Close-Form solutions for free vibration frequencies of functionally graded Euler-Bernoulli beam[J].Mechanics of Composite Materials,2017,53(1):79-98
[6]Aleksandar Nikolic'.Free vibration analysis of a non-uniform axially functionally graded cantilever beam with a tip body[J].Archive of Applied Mechanics,2017,87(7):1227-1241
[7]李清禄,李世荣.功能梯度材料梁在后屈曲构形附近的自由振动[J].振动与冲击,2011,30(9):76-78Li Qinglu,Li Shirong.Free vibration of FGM Euler beam with post-buckling configuration subjected to axial force[J].Journal of Vibration and Shock,2011,30(9):76-78
[8]焦常科,李爱群,伍小平.竖向动压力对FPS隔震连续梁桥地震响应影响研究[J].应用基础与工程科学学报,2017,25(4):733-741Jiao Changke,Li Aiqun,Wu Xiaoping.Research on the influence of dynamic vertical pressure on the seismic response of continuous girder bridge isolated with FPS[J].Journal of Basic Science and Engineering,2017,25(4):733-741
[9]李清禄,杨静宁.非保守力作用下简支梁在屈曲附近的自由振动[J].力学与实践,2014,36(3):333-336Li Qinglu,Yang Jingning.Free vibration of a simply supported beam under a non-conservation distributed load[J].Mechanics in Engineering,2014,36(3):333-336
[10]李清禄.功能梯度材料梁和圆板在随动载荷作用下的静动态响应[D].兰州:兰州理工大学博士论文,2012Li Qinglu.Static and dynamic response of functionally graded material beams and circular plates subjected to follower force[D].Lanzhou:Lanzhou University of Technology,2012
[11]Li Q L,Zhang J H.Vibration and post-buckling of a functionally graded beam subjected to non-conservation forces.Journal of vibroengineering,2016,18(8):4901-4913
[12]Li Q L,Luan W D,Zhu Z Q.Shooting method for free vibration of FGM Reissner-Mindlin circular plates resting on elastic foundation in thermal environment[J].Journal of Vibroengineering,2017,19(6):4423-4439
[13]Yang J,Shen H S.Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments[J].Journal of Sound and Vibration,2002,255:579-602