非齐次边界条件下轴向运动梁的非线性振动
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摘要
轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.
The transverse nonlinear vibration of the axial moving system has become one of the hot subjects at home and abroad. At present, most of the related studies are on the homogeneous boundary conditions. However, the nonhomogeneous boundary conditions are more common in the engineering practice. There are relatively few publications on the axial moving system with homogeneous boundary conditions. In order to study the effect of nonhomogeneous boundary conditions on the transverse nonlinear vibration of an axial moving beam, the nonlinear parametric vibrations of an axially accelerating viscoelastic Euler beam under nonhomogeneous boundary conditions are studied in this paper.The variable tension caused by the axial acceleration is introduced. A nonlinear integro-partial-differential equation and corresponding nonhomogeneous boundary conditions of an axially accelerating viscoelastic beam are presented. The effects of nonhomogeneous boundary conditions are highlighted. The method of multiple scales is used to establish the solvability conditions. The steady-state response of the beam was obtained from the solvability condition. According to the Routh-Hurvitz criterion, the stability of the response was determined. Some numerical examples are introduced to demonstrate the effect of the viscoelastic coefficient, mean speed, axial speed fluctuation amplitude, nonlinear coefficient,and the nonhomogeneous boundary conditions on the steady-state response. The larger viscoelastic coefficient leads to the smaller instability interval of the trivial solutions and the smaller stable steady-state response amplitude especially for the second mode. The instability interval and the stable steady-state response amplitude with the nonhomogeneous boundary conditions are larger than that with the homogeneous boundary conditions. The differential quadrature scheme is introduced to confirm the approximate analytical results. The numerical results show reasonable agreement with the approximate analytical results.
The transverse nonlinear vibration of the axial moving system has become one of the hot subjects at home and abroad. At present, most of the related studies are on the homogeneous boundary conditions. However, the nonhomogeneous boundary conditions are more common in the engineering practice. There are relatively few publications on the axial moving system with homogeneous boundary conditions. In order to study the effect of nonhomogeneous boundary conditions on the transverse nonlinear vibration of an axial moving beam, the nonlinear parametric vibrations of an axially accelerating viscoelastic Euler beam under nonhomogeneous boundary conditions are studied in this paper.The variable tension caused by the axial acceleration is introduced. A nonlinear integro-partial-differential equation and corresponding nonhomogeneous boundary conditions of an axially accelerating viscoelastic beam are presented. The effects of nonhomogeneous boundary conditions are highlighted. The method of multiple scales is used to establish the solvability conditions. The steady-state response of the beam was obtained from the solvability condition. According to the Routh-Hurvitz criterion, the stability of the response was determined. Some numerical examples are introduced to demonstrate the effect of the viscoelastic coefficient, mean speed, axial speed fluctuation amplitude, nonlinear coefficient,and the nonhomogeneous boundary conditions on the steady-state response. The larger viscoelastic coefficient leads to the smaller instability interval of the trivial solutions and the smaller stable steady-state response amplitude especially for the second mode. The instability interval and the stable steady-state response amplitude with the nonhomogeneous boundary conditions are larger than that with the homogeneous boundary conditions. The differential quadrature scheme is introduced to confirm the approximate analytical results. The numerical results show reasonable agreement with the approximate analytical results.
引文
1刘延柱.大变形轴向运动梁的精确动力学模型.力学学报,2012,44(5):832-838(Liu Yanzhu.Exact dynamic model of axially moving beam with large deformation.Chinese Journal of Theoretical and Applied Mechanics,2012,44(5):832-838(in Chinese))
2宫苏梅,张伟.平带系统非线性振动实验研究.动力学与控制学报,2014,12(4):368-372(Gong Sumei,Zhang Wei.Experimental study on nonlinear vibration of flat-belt system.Journal of Dynamics and Control,2014,12(4):368-372(in Chinese))
3陈红永,陈海波,张培强.轴向受压运动梁横向振动特性的数值分析.振动与冲击,2014,33(24):101-105(Chen Hongyong,Chen Haibo,Zhang Peiqiang.Numerical analysis of free vibration of an axially moving beam under compressive load.Journal of Vibration and Shock,2014,33(24):101-105(in Chinese))
4王延庆,郭星辉,梁宏琨等.凸肩叶片的非线性振动特性与运动分岔.力学学报,2011,43(4):755-764(Wang Yanqing,Guo Xinghui,Liang Honghui,et al.Nonlinear vibratory characteristics and bifurcations of shrouded blades.Chinese Journal of Theoretical and Applied Mechanics,2011,43(4):755-764(in Chinese))
5 Chen LH,Zhang W,Yang FH.Nonlinear dynamics of higher dimensional system for an axially accelerating viscoelastic beam.Journal of Sound and Vibration,2010,329:5321-5345
6 Yang XD,Tang YQ,Chen LQ,et al.Dynamic stability of axially accelerating timoshenko beams:Averaging method.European Journal of Mechanics A/Solids,2010,29:81-90
7 Sahoo B,Panda LN,Pohit G.Parametric and internal resonances of an axially moving beam with time-dependent velocity.Modelling and Simulation in Engineering,2013,2013(5):1-18
8胡璐,闫寒,张文明等.黏性流体环境下Ⅴ型悬臂梁结构流固耦合振动特性研究.力学学报,2018,50(3):643-653(Hu Lu,Yan Han,Zhang Wenming,et al.Analysis of flexural vibration of V-shaped beams immersed in viscous fluids.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):643-653(in Chinese))
9 Ghayesh MH,Ghayesh HA,Reid T.Sub-and super-critical nonlinear dynamics of a harmonically excited axially moving beam.International Journal of Solids and Structures,2012,49:227-243
10 Ghayesh MH,Amabili M.Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support.Mechanism and Machine Theory,2013,67(67):1-16
11吕海炜,李映辉,李亮等.轴向运动软夹层梁横向振动分析.振动与冲击,2014,33(2):41-46(L¨u Haiwei,Li Yinghui,Li Liang,et al.Analysis of transverse vibration of axially moving soft sandwich beam.Journal of Vibration and Shock,2014,33(2):41-46(in Chinese))
12华洪良,廖振强,张相炎.轴向移动悬臂梁高效动力学建模及频率响应分析.力学学报,2017,49(6):1390-1398(Hua Hongliang,Liao Zhenqiang,Zhang Xiangyan.An efficient dynamic modeling of an axially moving cantilever beam and frequency response analysis.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1390-1398(in Chinese))
13 Chen SH,Huang JL,Sze KY.Multidimensional lindstedt-poincar′e method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2007,306:1-11
14 Li YH,Dong YH,Qin Y,et al.Nonlinear forced vibration and stability of an axially moving viscoelastic sandwich beam.International Journal of Mechanical Sciences,2018,138:131-145
15 Wang YB,Ding H,Chen LQ.Nonlinear vibration of axially accelerating hyperelastic beams.International Journal of Non-Linear Mechanics,2018,99:302-310
16陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报,2005,37(1):57-63(Chen Shuhui,Huang Jianliang.On internal resonance of nonlinear vibration of axially moving beams.Acta Mechanica Sinica,2005,37(1):57-63(in Chinese))
17 Lü HW,Li L,Li YH.Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity.Applied Mathematical Modelling,2018,53:83-105
18 Hua HL,Liao ZQ,Zhang XY.The self-excited vibrations of an axially retracting cantilever beam using the Galerkin method with fitted polynomial basis functions.Journal of Mechanical Science and Technology,2018,32(1):29-36
19 Hua HL,Qiu M,Liao ZQ.Dynamic analysis of an axially moving beam subject to inner pressure using finite element method.Journal of Mechanical Science and Technology,2017,31(6):2663-2670
20 Kelleche A,Tatar NE,Khemmoudj A.Stability of an axially moving viscoelastic beam.Journal of Dynamical and Control Systems,2017,23:283-299
21胡宇达,张立保.轴向运动导电导磁梁的磁弹性振动方程.应用数学和力学,2015,36(1):70-77(Hu Yuda,Zhang Libao.Magnetoelastic vibration equations for axially moving conductive and magnetic beams.Applied Mathematics and Mechanics,2015,36(1):70-77(in Chinese))
22唐有绮.轴向变速黏弹性Timoshenko梁的非线性振动.力学学报,2013,45(6):965-973(Tang Youqi.Nonlinear vibrations of axially acceleratiing viscoelastic Timoshenko beams.Chinese Journal of Theoretical and Applied Mechanics,2013,45(6):965-973(in Chinese))
23 Tang YQ,Zhang DB,Gao JM.Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions.Nonlinear Dynamics,2016,83(1-2):401-418
24 Zhang DB,Tang YQ,Chen LQ.Irregular instability boundaries of axially accelerating viscoelastic beams with 1:3 internal resonance.International Journal of Mechanical Sciences,2017,133:535-543
25 Tang YQ,Zhang DB,Rui M,et al.Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions.Applied Mathematics and Mechanics(English Edition),2016,37(12):1-22
26 Chen LQ,Tang YQ,Lim CW.Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams.Journal of Sound and Vibration,2010,329:547-565
27唐有绮.轴向运动梁和面内平动板横向振动的建模与分析.[博士论文].上海:上海大学,2011(Transverse vibrations of axially moving beams and in-plane translating plates:Modeling and analysis.[PhD Thesis].Shanghai:Shanghai University,2011(in Chinese))
28 Tang YQ,Chen LQ.Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed.European Journal of Mechanics A/Solids,2013,37:106-121
29 Chen LQ,Zu JW.Solvability condition in multi-scale analysis of gyroscopic continua.Journal of Sound and Vibration,2008,309:338-342
30王冬梅,张伟,李慕荣等.用微分求积法分析轴向移动黏弹性梁的非平面非线性振动.动力学与控制学报,2015(1):23-27(Wang Dongmei,Zhang Wei,Li Murong,et al.Using DQM to analyze the nonplanar nonlinear vibrationa of an axially moving viscoelastic beam.Journal of Dynamics and Control,2015(1):23-27(in Chinese))
31丁虎.数值仿真轴向运动黏弹性梁非线性参激振动.计算力学学报,2012,29(4):545-550(Ding Hu.Numerical investigation into nonlinear parametric resonance of axially moving accelerating viscoelastic beams.Chinese Journal of Computational Mechanics,2012,29(4):545-550(in Chinese))
32 Bert CW,Malik M.The differential quadrature method in computational mechanics:A review.Applied Mechanics Reviews,1996,49:1-28
33 Malik M,Bert CW.Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s:Application to free vibration of plates.International Journal for Numerical Methods Engineering,1996,39:1237-1258
2宫苏梅,张伟.平带系统非线性振动实验研究.动力学与控制学报,2014,12(4):368-372(Gong Sumei,Zhang Wei.Experimental study on nonlinear vibration of flat-belt system.Journal of Dynamics and Control,2014,12(4):368-372(in Chinese))
3陈红永,陈海波,张培强.轴向受压运动梁横向振动特性的数值分析.振动与冲击,2014,33(24):101-105(Chen Hongyong,Chen Haibo,Zhang Peiqiang.Numerical analysis of free vibration of an axially moving beam under compressive load.Journal of Vibration and Shock,2014,33(24):101-105(in Chinese))
4王延庆,郭星辉,梁宏琨等.凸肩叶片的非线性振动特性与运动分岔.力学学报,2011,43(4):755-764(Wang Yanqing,Guo Xinghui,Liang Honghui,et al.Nonlinear vibratory characteristics and bifurcations of shrouded blades.Chinese Journal of Theoretical and Applied Mechanics,2011,43(4):755-764(in Chinese))
5 Chen LH,Zhang W,Yang FH.Nonlinear dynamics of higher dimensional system for an axially accelerating viscoelastic beam.Journal of Sound and Vibration,2010,329:5321-5345
6 Yang XD,Tang YQ,Chen LQ,et al.Dynamic stability of axially accelerating timoshenko beams:Averaging method.European Journal of Mechanics A/Solids,2010,29:81-90
7 Sahoo B,Panda LN,Pohit G.Parametric and internal resonances of an axially moving beam with time-dependent velocity.Modelling and Simulation in Engineering,2013,2013(5):1-18
8胡璐,闫寒,张文明等.黏性流体环境下Ⅴ型悬臂梁结构流固耦合振动特性研究.力学学报,2018,50(3):643-653(Hu Lu,Yan Han,Zhang Wenming,et al.Analysis of flexural vibration of V-shaped beams immersed in viscous fluids.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):643-653(in Chinese))
9 Ghayesh MH,Ghayesh HA,Reid T.Sub-and super-critical nonlinear dynamics of a harmonically excited axially moving beam.International Journal of Solids and Structures,2012,49:227-243
10 Ghayesh MH,Amabili M.Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support.Mechanism and Machine Theory,2013,67(67):1-16
11吕海炜,李映辉,李亮等.轴向运动软夹层梁横向振动分析.振动与冲击,2014,33(2):41-46(L¨u Haiwei,Li Yinghui,Li Liang,et al.Analysis of transverse vibration of axially moving soft sandwich beam.Journal of Vibration and Shock,2014,33(2):41-46(in Chinese))
12华洪良,廖振强,张相炎.轴向移动悬臂梁高效动力学建模及频率响应分析.力学学报,2017,49(6):1390-1398(Hua Hongliang,Liao Zhenqiang,Zhang Xiangyan.An efficient dynamic modeling of an axially moving cantilever beam and frequency response analysis.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1390-1398(in Chinese))
13 Chen SH,Huang JL,Sze KY.Multidimensional lindstedt-poincar′e method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2007,306:1-11
14 Li YH,Dong YH,Qin Y,et al.Nonlinear forced vibration and stability of an axially moving viscoelastic sandwich beam.International Journal of Mechanical Sciences,2018,138:131-145
15 Wang YB,Ding H,Chen LQ.Nonlinear vibration of axially accelerating hyperelastic beams.International Journal of Non-Linear Mechanics,2018,99:302-310
16陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报,2005,37(1):57-63(Chen Shuhui,Huang Jianliang.On internal resonance of nonlinear vibration of axially moving beams.Acta Mechanica Sinica,2005,37(1):57-63(in Chinese))
17 Lü HW,Li L,Li YH.Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity.Applied Mathematical Modelling,2018,53:83-105
18 Hua HL,Liao ZQ,Zhang XY.The self-excited vibrations of an axially retracting cantilever beam using the Galerkin method with fitted polynomial basis functions.Journal of Mechanical Science and Technology,2018,32(1):29-36
19 Hua HL,Qiu M,Liao ZQ.Dynamic analysis of an axially moving beam subject to inner pressure using finite element method.Journal of Mechanical Science and Technology,2017,31(6):2663-2670
20 Kelleche A,Tatar NE,Khemmoudj A.Stability of an axially moving viscoelastic beam.Journal of Dynamical and Control Systems,2017,23:283-299
21胡宇达,张立保.轴向运动导电导磁梁的磁弹性振动方程.应用数学和力学,2015,36(1):70-77(Hu Yuda,Zhang Libao.Magnetoelastic vibration equations for axially moving conductive and magnetic beams.Applied Mathematics and Mechanics,2015,36(1):70-77(in Chinese))
22唐有绮.轴向变速黏弹性Timoshenko梁的非线性振动.力学学报,2013,45(6):965-973(Tang Youqi.Nonlinear vibrations of axially acceleratiing viscoelastic Timoshenko beams.Chinese Journal of Theoretical and Applied Mechanics,2013,45(6):965-973(in Chinese))
23 Tang YQ,Zhang DB,Gao JM.Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions.Nonlinear Dynamics,2016,83(1-2):401-418
24 Zhang DB,Tang YQ,Chen LQ.Irregular instability boundaries of axially accelerating viscoelastic beams with 1:3 internal resonance.International Journal of Mechanical Sciences,2017,133:535-543
25 Tang YQ,Zhang DB,Rui M,et al.Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions.Applied Mathematics and Mechanics(English Edition),2016,37(12):1-22
26 Chen LQ,Tang YQ,Lim CW.Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams.Journal of Sound and Vibration,2010,329:547-565
27唐有绮.轴向运动梁和面内平动板横向振动的建模与分析.[博士论文].上海:上海大学,2011(Transverse vibrations of axially moving beams and in-plane translating plates:Modeling and analysis.[PhD Thesis].Shanghai:Shanghai University,2011(in Chinese))
28 Tang YQ,Chen LQ.Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed.European Journal of Mechanics A/Solids,2013,37:106-121
29 Chen LQ,Zu JW.Solvability condition in multi-scale analysis of gyroscopic continua.Journal of Sound and Vibration,2008,309:338-342
30王冬梅,张伟,李慕荣等.用微分求积法分析轴向移动黏弹性梁的非平面非线性振动.动力学与控制学报,2015(1):23-27(Wang Dongmei,Zhang Wei,Li Murong,et al.Using DQM to analyze the nonplanar nonlinear vibrationa of an axially moving viscoelastic beam.Journal of Dynamics and Control,2015(1):23-27(in Chinese))
31丁虎.数值仿真轴向运动黏弹性梁非线性参激振动.计算力学学报,2012,29(4):545-550(Ding Hu.Numerical investigation into nonlinear parametric resonance of axially moving accelerating viscoelastic beams.Chinese Journal of Computational Mechanics,2012,29(4):545-550(in Chinese))
32 Bert CW,Malik M.The differential quadrature method in computational mechanics:A review.Applied Mechanics Reviews,1996,49:1-28
33 Malik M,Bert CW.Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s:Application to free vibration of plates.International Journal for Numerical Methods Engineering,1996,39:1237-1258