Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis
-
摘要: 基于修正的偶应力理论并考虑Lagrange应变张量所给出的几何非线性,运用Hamilton原理建立了微尺度悬臂管平面振动的积分-微分方程通过Galerkin方法将原积分-微分方程离散成常微分方程组,研究了临界流速-质量比曲线的不同阶Galerkin近似解与精确解的符合程度以及它们对材料长度尺寸参数的依赖性对不同的模态截断数,运用基于中心流形-范式理论的投影法计算了临界流速处系统的第一Lyapunov(李雅谱诺夫)系数和临界特征值关于流速的变化率,以此为基础分析了系统的分岔模式,探讨了模态截断数对系统动力学性质的影响临界流速-质量比曲线的滞后部分及交点处的动力学性质表明,系统存在不同的分岔方向,用6个模态的Galerkin离散化方程作分岔图对此进行了验证,并通过理论分析及数值方法分别计算了颤振的固有频率.Abstract: Based on the modified couple stress theory, the integro-differential equations of motion for micro-scale cantilever pipes were derived by means of Hamilton’s principle. The geometric nonlinearity, arising from the Lagrangian strain tensor, was taken into account. The integro-differential equations were transformed into ordinary differential equations with the Galerkin method. With different numbers of modes in the Galerkin discretization, the diagrams of critical flow velocity vs. mass ratio were given. The difference between the Galerkin approximation results and the exact solutions to the 2-point boundary problem was investigated and the effect of the internal material length scale parameter on the graphs of critical flow velocity vs. mass ratio was studied. For different numbers of modes, the first Lyapunov’s coefficient was calculated and the critical eigenvalue with respect to the flow velocity was derived with the projection method based on the center manifold theory and the normal form method, therefrom, the bifurcation model was analyzed and the effect of the number of modes on the dynamical behaviors was examined. The dynamics of hysteresis and intersection points of the curves of critical flow velocity vs. mass ratio was also investigated and then bifurcation diagrams in different directions were found. Finally, the 6-mode ordinary differential equations of the Galerkin discretization were employed to construct the bifurcation diagrams and verify the relevant results obtained, and the natural frequencies of flutter were calculated through the theoretical analysis and with the numerical method, respectively.
-
[1] BENJAMIN T B. Dynamics of a system of articulated pipes conveying fluid I: theory[J]. Proceedings of the Royal Society of London(Series A): Mathematical and Physical Sciences,1961,261(1307): 457-486. [2] GREGORY R W, PADOUSSIS M P. Unstable oscillation of tubular cantilevers conveying fluid I: theory[J]. Proceedings of the Royal Society of London(Series A): Mathematical and Physical Sciences,1966,293(1435): 512-527. [3] BAJAJ A K, SETHNA P R, LUNDGREN T S. Hopf bifurcation phenomena in tubes carrying a fluid[J]. SIAM Journal on Applied Mathematics,1980,39(2): 213-230. [4] SEMLER C, LI G X, PADOUSSIS M P. The non-linear equations of motion of pipes conveying fluid[J]. Journal of Sound and Vibration,1994,169(5): 577-599. [5] 徐鉴, 杨前彪. 流体诱发水平悬臂输液管的内共振和模态转换(I)(II)[J]. 应用数学和力学, 2006,27(7): 819-832.(XU Jian, YANG Qianbiao. Flow-induced internal resonances and mode exchange in horizontal cantilevered pipe conveying fluid (I)(II)[J]. Applied Mathematics and Mechanics,2006,27(7): 819-832.(in Chiense)) [6] CHEN Liqun, ZHANG Yanlei, ZHANG Guoce, et al. Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed[J]. International Journal of Non-Linear Mechanics,2014,58: 11-21. [7] GAN Chunbiao, JING Shuai, YANG Shixi, et al. Effects of supported angle on stability and dynamical bifurcations of cantilevered pipe conveying fluid[J]. Applied Mathematics and Mechanics (English Edition),2015,36(6): 729-746. [8] 毛晓晔, 丁虎, 陈立群. 3∶1内共振下超临界输液管受迫振动响应[J]. 应用数学和力学, 2016,37(4): 345-351.(MAO Xiaoye, DING Hu, CHEN Liqun. Forced vibration responses of supercritical fluid-conveying pipes in 3∶1 internal resonance[J]. Applied Mathematics and Mechanics,2016,37(4): 345-351.(in Chinese)) [9] 徐鉴, 王琳. 输液管动力学分析和控制[M]. 北京: 科学出版社, 2015.(XU Jian, WANG Lin. Dynamics and Control of Fluid-Conveying Pipe Systems [M]. Beijing: Science Press, 2015.(in Chinese)) [10] RINALDI S, PRABHAKAR S, VENGALLATORE S, et al. Dynamics of microscale pipes containing internal fluid flow: damping, frequency shift, and stability[J]. Journal of Sound and Vibration,2010,329(8): 1081-1088. [11] NAJMZADEH M, HAASL S, ENOKSSON P. A silicon straight tube fluid density sensor[J]. Journal of Micromechanics and Microengineering,2007,17(8): 1657-1663. [12] BHIRDE A A, PATEL V, GAVARD J, et al. Targeted killing of cancer cells in vivo and in vitro with EGT-directed carbon nanotube-based drug delivery[J]. ACS Nano,2009,3(2): 307-316. [13] DELADI S, BERENSCHOT J W, TAS N R, et al. Fabrication of micromachined fountain pen with in situ characterization possibility of nanoscale surface modification[J]. Journal of Micromechanics and Microengineering,2005,15(3): 528-534. [14] KIM K H, MOLDOVAN N, ESPINOSA H D. A nano fountain probe with sub-100 nm molecular writing resolution[J]. Small,2005,1(6): 632-635. [15] FLECK N A, MULLER G M, ASHBY M F, et al. Strain gradient plasticity: theory and experiment[J]. Acta Metallurgica et Materialia,1994,42(2): 475-487. [16] LAM D C C, YANG F, CHONG A C M, et al. Experiments and theory in strain gradient elasticity[J]. Journal of the Mechanics and Physics of Solids,2003,51(8): 1477-1508. [17] MCFARLAND A W, COLTON J S. Role of material microstructure in plate stiffness with relevance to microcantilever sensors[J]. Journal of Micromechanics and Microengineering,2005,15(5): 1060-1067. [18] YANG F, CHONG A C M, LAM D C C, et al. Couple stress based strain gradient theory for elasticity[J]. International Journal of Solids and Structures,2002,39(10): 2731-2743. [19] WANG Y G, LIN W H, LIU N. Nonlinear free vibration of a microscale beam based on modified couple stress theory[J]. Physica E,2013,47: 80-85. [20] ASGHARI M, KAHROBAIYAN M, AHMADIAN M. A nonlinear Timoshenko beam formulation based on the modified couple stress theory[J]. International Journal of Engineering Science,2010,48(12): 1749-1761. [21] GHAYESH M H, AMABILI M, FAROKHI H. Three-dimensional nonlinear size-dependent behavior of Timoshenko microbeams[J]. International Journal of Engineering Science,2013,71: 1-14. [22] DAI H L, WANG Y K, WANG L. Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory[J]. International Journal of Engineering Science,2015,94: 103-112. [23] MOHAMMAD-ABADI M, DANESHMEHR A R. Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions[J]. International Journal of Engineering Science,2014,74: 1-14. [24] GHAYESH M H, FAROKHI H, ALICI G. Subcritical parametric dynamics of microbeams[J]. International Journal of Engineering Science,2015,95: 36-48. [25] WANG L. Size-dependent vibration characteristics of fluid-conveying microtubes[J]. Journal of Fluids and Structures,2010,26(4): 675-684. [26] XIA W, WANG L. Microfluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory[J]. Microfluidics and Nanofluidics,2010,9(4/5): 955-962. [27] WANG L, LIU H T, NI Q, et al. Flexural vibrations of microscale pipes conveying fluid by considering the size effects of micro-flow and micro-structure[J]. International Journal of Engineering Science,2013,71: 92-101. [28] YANG T Z, JI S D, YANG X D, et al. Microfluid-induced nonlinear free vibration of microtubes[J]. International Journal of Engineering Science,2014,76: 47-55. [29] HOSSEINI M, BAHAADINI R. Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory[J]. International Journal of Engineering Science,2016,101: 1-13. [30] BAHAADINI R, HOSSEINI M. Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid[J]. Computational Materials Science,2016,114: 151-159. [31] TANG Min, NI Qiao, WANG Lin, et al. Nonlinear modeling and size-dependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory[J]. International Journal of Engineering Science,2014,84: 1-10. [32] LI G X, PADOUSSIS M P. Stability, double degeneracy and chaos in cantilevered pipes conveying fluid[J]. International Journal of Non-Linear Mechanics,1994,29(1): 83-107. [33] JIN J D, SONG Z Y. Parametric resonances of supported pipes conveying pulsating fluid[J]. Journal of Fluids and Structures,2005,20(6): 763-783. [34] JIN J D. Stability and chaotic motions of a restrained pipe conveying fluid[J]. Journal of Sound and Vibration,1997,208(3): 427-439. [35] JIN J D, ZOU G S. Bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid[J]. Journal of Sound and Vibration,2003,260(5): 783-805. [36] PADOUSSIS M P, GHAYESH M H. Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array[J]. International Journal of Non-Linear Mechanics,2010,45(5): 507-524. [37] MODARRES-SADEGHI Y, SEMLER C, WADHAM-GAGNON M, et al. Dynamics of cantilevered pipes conveying fluid—part 3: three-dimensional dynamics in the presence of an end-mass[J]. Journal of Fluids and Structures,2007,23(4): 589-603. [38] MODARRES-SADEGHI Y, PADOUSSIS M P. Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[J]. Computers and Structures,2013,122: 192-201. [39] GHAYESH M H, PADOUSSIS M P, MODARRES-SADEGHI Y. Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass[J]. Journal of Sound and Vibration,2011,330(12): 2869-2899. [40] CHANG C H, MODARRES-SADEGHI Y. Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation[J]. Journal of Sound and Vibration,2014,333(18): 4265-4280. [41] 尤里·阿·库兹涅佐夫. 应用分支理论基础[M]. 金成桴, 译. 北京: 科学出版社, 2010.(KUZNETSOV YURI A. Elements of Applied Bifurcation Theory [M]. JIN Chengfu, transl. Beijing: Science Press, 2010.(Chinese version)) [42] IOOSS G, JOSEPH D D. Elementary Stability and Bifurcation Theory [M]. 2nd ed. Springer-Verlag, 1990.
点击查看大图
计量
- 文章访问数: 928
- HTML全文浏览量: 86
- PDF下载量: 733
- 被引次数: 0