起重臂动力稳定分析中的非线性三结点梁单元
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摘要
针对起重机箱形伸缩臂在轴向周期载荷作用下的动力稳定性和传统两结点梁单元在结构分析中精度不够的问题,依据有限元插值理论采用五次Hermite插值,构造考虑二阶效应的非线性三结点Euler-Bernoulli梁单元横向位移场,推导出新型三结点梁单元的单元矩阵.运用Lagrange方程建立周期载荷作用下伸缩臂在小变形情况下的参数振动方程,给出了起重机伸缩臂的动力不稳定区域边界,并讨论阻尼对伸缩臂动力稳定性的影响.分析结果表明,非线性三结点梁单元与传统两结点梁单元划分3～4个单元具有相同的计算精度,应用有限单元法求解动力稳定问题是有效的;随着阻尼系数的增大,伸缩臂动力不稳定区域逐渐减小,阻尼的存在有利于伸缩臂的动力稳定性.
In order to study the dynamic stability of crane's telescopic booms under periodic load,and examine the poor accuracy of the traditional two-node beam element in structural analysis,this paper proposes to implement the quintic Hermite interpolation to establish the transverse displacement field of non-linear 3-node Euler-Bernoulli beam element,that considers the second-order effect,thus developing matrixes of 3-node beam element.The parametric vibration equation of stepped columns with small deformation under periodic load was established by the Lagrange equation;the dynamic instability region of telescopic booms was obtained,and then the effect of damping on the dynamic stability of telescopic booms was discussed.The results of the analysis indicate that the non-linear 3-node beam element has the same accuracy as traditional 2-node beam element dividing into 3 to 4 elements.By utilizing the finite element method to analyze the dynamic stability,it is shown effective;even more,the dynamic instability region reduces as the damping coefficient increases,which means damping is advantageous to the structural dynamic stability.
In order to study the dynamic stability of crane's telescopic booms under periodic load,and examine the poor accuracy of the traditional two-node beam element in structural analysis,this paper proposes to implement the quintic Hermite interpolation to establish the transverse displacement field of non-linear 3-node Euler-Bernoulli beam element,that considers the second-order effect,thus developing matrixes of 3-node beam element.The parametric vibration equation of stepped columns with small deformation under periodic load was established by the Lagrange equation;the dynamic instability region of telescopic booms was obtained,and then the effect of damping on the dynamic stability of telescopic booms was discussed.The results of the analysis indicate that the non-linear 3-node beam element has the same accuracy as traditional 2-node beam element dividing into 3 to 4 elements.By utilizing the finite element method to analyze the dynamic stability,it is shown effective;even more,the dynamic instability region reduces as the damping coefficient increases,which means damping is advantageous to the structural dynamic stability.
引文
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[9]殷学纲.梁的动力稳定性分析的有限元方法[J].应用力学学报,1995,12(4):81-87.YIN Xuegang.Finite element method for dynamic stabilityanalysis of beams[J].Chinese Journal of Applied Mechan-ics,1995,12(4):81-87.
[10]GURKAN S,MUSTAFA S.Dynamic stability of a rotatingasymmetric cross-section blade subjected to an axial peri-odic force[J].International Journal of Mechanical Sci-ence,2003,45(9):1467-1482.
[11]MARTIN S,SEBASTION P M.Free vibration and dynam-ic stability of rotating thin-walled composite beams[J].European Journal of Mechanics-A/Solids,2011,30(3):432-441.
[12]曲淑英.大型空间框架结构的动力稳定性分析[J].世界地震工程,1998,14(3):44-48.QU Shuying.Analysis for dynamic stability of large spatialframe structure[J].World Information on Earthquake En-gineering,1998,14(3):44-48.
[2]孙强,杨大军.弹性介质中杆的动力稳定性研究[J].工程力学,1997,14(1):87-91.SUN Qiang,YANG Dajun.Research on the dynamic stabil-ity of bars in elastic medium[J].Engineering Mechanics,1997,14(1):87-91.
[3]孙强.竖向谐振载荷下摩擦桩的横向振动性能研究[J].振动与冲击,2006,25(4):85-87.SUN Qiang.Study on transverse vibration behavior of a fric-tion pile under vertical harmonic loads[J].Journal of Vi-bration and Shock,2006,25(4):85-87.
[4]RATKO P,PREDRGA K,PREDRGA R,et al.Dynamicstability of a thin-walled beam subjected to axial loads andend moments[J].Journal of Sound and Vibration,2007,301(3-5):690-700.
[5]YANG Xiaodong,TANG Youqi,CHEN Lijun,et al.Dy-namic stability of axially accelerating Timoshenko beam av-eraging method[J].European Journal of Mechanics-A/Sol-ids,2010,29(1):81-90.
[6]李晓东,王秀丽.屈曲约束支撑的动力稳定性分析[J].华中科技大学学报:城市科学版,2008,25(4):223-226.LI Xiaodong,WANG Xiuli.Dynamic stability analysis ofbuckling restrained brace[J].Journal of Huazhong Univer-sity of Science and Technology:Urban Science Edition,2008,25(4):223-226.
[7]ROSA M A,AUCIELLO N M.Dynamic stability analysisand DQM for beams with variable cross-section[J].Me-chanics Research Communications,2008,35(3):187-192.
[8]夏鹭平.截锥形直杆的动力稳定性研究[J].合肥工业大学学报:自然科学版,2001,24(2):194-197.XIA Luping.Research on the dynamic of cut-off-cone bar[J].Journal of Hefei University of Technology:Natural Sci-ence,2001,24(2):194-197.
[9]殷学纲.梁的动力稳定性分析的有限元方法[J].应用力学学报,1995,12(4):81-87.YIN Xuegang.Finite element method for dynamic stabilityanalysis of beams[J].Chinese Journal of Applied Mechan-ics,1995,12(4):81-87.
[10]GURKAN S,MUSTAFA S.Dynamic stability of a rotatingasymmetric cross-section blade subjected to an axial peri-odic force[J].International Journal of Mechanical Sci-ence,2003,45(9):1467-1482.
[11]MARTIN S,SEBASTION P M.Free vibration and dynam-ic stability of rotating thin-walled composite beams[J].European Journal of Mechanics-A/Solids,2011,30(3):432-441.
[12]曲淑英.大型空间框架结构的动力稳定性分析[J].世界地震工程,1998,14(3):44-48.QU Shuying.Analysis for dynamic stability of large spatialframe structure[J].World Information on Earthquake En-gineering,1998,14(3):44-48.
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