基于变长度单元ANCF的轴向伸展悬臂梁振动分析
|
摘要
伸展悬臂梁系统中含有刚体运动和大变形运动,属于时变刚柔耦合非线性动力学问题。采用Shabana提出的绝对节点坐标法(ANCF),建立了一种变长度的Euler-Bernoulli梁单元模型。从大变形条件下准确的曲率和Green-Lagrangian正应变出发,基于考虑惯性力的虚功原理,得到了轴向伸展悬臂梁的单元非线性动力学方程组,以及组装后的伸展悬臂梁系统的非线性动力学方程组。最后,通过算例分析了材料特性参数(弹性模量、密度)和伸展规律(匀速伸展、匀加速伸展)对伸展悬臂梁系统的末端非线性挠度响应的影响。
A deploying cantilever beam system has rigid body motion and large deformation motion, so its vibration problem is time-varying and rigid-flexible coupled nonlinear dynamic one. Here, the absolute node coordinates formulation(ANCF) proposed by Shabana was adopted to establish a length-varying Euler-Bernoulli beam element model. Starting from the accurate curvature under the condition of large deformation and Green-Lagrangian normal strain, based on the virtual work principle considering inertia force, nonlinear dynamic equations of a beam element, and those of a deploying cantilever beam system after element-assembling were derived. Finally, the effects of material characteristic parameters including elastic modulus and mass density and deploying laws including constant speed deploying and constant acceleration one on nonlinear deflection response at free end of the deploying cantilever beam system were analyzed through numerical examples.
A deploying cantilever beam system has rigid body motion and large deformation motion, so its vibration problem is time-varying and rigid-flexible coupled nonlinear dynamic one. Here, the absolute node coordinates formulation(ANCF) proposed by Shabana was adopted to establish a length-varying Euler-Bernoulli beam element model. Starting from the accurate curvature under the condition of large deformation and Green-Lagrangian normal strain, based on the virtual work principle considering inertia force, nonlinear dynamic equations of a beam element, and those of a deploying cantilever beam system after element-assembling were derived. Finally, the effects of material characteristic parameters including elastic modulus and mass density and deploying laws including constant speed deploying and constant acceleration one on nonlinear deflection response at free end of the deploying cantilever beam system were analyzed through numerical examples.
引文
[1] SIVAKUMAR S K.Tadikonda and haim baruh.dynamics and control of a translating flexible beam with a prismatic joint[J].Journal of Dynamic Systems,Measurement,and Control,1992,114(3):422-427.
[2] STYLIANOU M,TABARROK B.Finite element analysis of an axially moving beam,Part I:Time integration[J].Journal of Sound and Vibration,1994,178(4):433-453.
[3] STYLIANOU M,TABARROK B.Finite element analysis of an axially moving beam,Part II:Stability analysis[J].Journal of Sound and Vibration,1994,178(4):455-481.
[4] AL-BEDOOR B O,KHULIEF Y A.Finite element dynamic modeling of a translating and rotating flexible link[J].Computer Methods in Applied Mechanics & Engineering,1996,131(1/2):173-189.
[5] PIOVAN M T,SAMPAIO R.Vibrations of axially moving flexible beams made of functionally graded materials[J].Thin-Walled Structures,2008,46(2):112-121.
[6] ROGERS K S,FERGUSON N S,PERRYMAN A A,et al.Modelling axially moving beams of varying length using the finite element method[C].7th Euromech Solid Mechanics Conference,2009:7-11.
[7] WANG L H,HU Z D,ZHONG Z,et al.Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length[J].Acta Mechanica,2010,214(3):225-244.
[8] CHANG J R,LIN W J,HUANG C J,et al.Vibration and stability of an axially moving Rayleigh beam[J].Applied Mathematical Modelling,2010,34(6):1482-1497.
[9] 罗炳华,高跃飞,刘荣华,等.轴向运动梁受移动载荷作用的横向动力响应[J].振动与冲击,2011,30(12):59- 63.LUO Binghua,GAO Yuefei,LIU Ronghua,et al.Lateral dynamic response of an axially moving beam under a moving load[J].Journal of Vibration and Shock,2011,30(12):59- 63.
[10] DOWNER J D,PARK K C.Formulation and solution of inverse spaghetti problem:application to beam deployment dynamics[J].Aiaa Journal,2012,31(2):339- 347.
[11] PARK S,HONG H Y,CHUNG J.Vibrations of an axially moving beam with deployment or retraction[J].AIAA Journal,2013,51(51):686-696.
[12] YANG X D,LIU M,ZHANG W,et al.Invariant and energy analysis of an axially retracting beam[J].Chinese Journal of Aeronautics,2016,29(4):952-961.
[13] 赵亮,胡振东.轴向运动功能梯度悬臂梁动力学分析[J].振动与冲击,2016,35 (2):124-128.ZHAO Liang,HU Zhendong.Dynamic analysis of an axially translating functionally graded cantilever beam[J].Journal of Vibration and Shock,2016,35 (2):124-128.
[14] 杨鑫,陈海波.热冲击作用下轴向运动梁的振动特性研究[J].振动与冲击,2017,36 (1):8-15.YANG Xin,CHEN Haibo.Vibration characteristics of an axially moving beam under thermal shocks[J].Journal of Vibration and Shock,2017,36 (1):8-15.
[15] 谭霞,丁虎,陈立群.超临界轴向运动Timoshenko梁横向受迫振动[J].振动与冲击,2017,36 (22):1-5.TAN Xia,DING Hu,CHEN Liqun.Transverse forced vibration of an axially moving Timoshenko beam at a supercritical speed[J].Journal of Vibration and Shock,2017,36 (22):1-5.
[16] SHABANA A A.Definition of the slopes and the finite element absolute nodal coordinate formulation[J].Multibody System Dynamics,1997,1(3):339-348.
[2] STYLIANOU M,TABARROK B.Finite element analysis of an axially moving beam,Part I:Time integration[J].Journal of Sound and Vibration,1994,178(4):433-453.
[3] STYLIANOU M,TABARROK B.Finite element analysis of an axially moving beam,Part II:Stability analysis[J].Journal of Sound and Vibration,1994,178(4):455-481.
[4] AL-BEDOOR B O,KHULIEF Y A.Finite element dynamic modeling of a translating and rotating flexible link[J].Computer Methods in Applied Mechanics & Engineering,1996,131(1/2):173-189.
[5] PIOVAN M T,SAMPAIO R.Vibrations of axially moving flexible beams made of functionally graded materials[J].Thin-Walled Structures,2008,46(2):112-121.
[6] ROGERS K S,FERGUSON N S,PERRYMAN A A,et al.Modelling axially moving beams of varying length using the finite element method[C].7th Euromech Solid Mechanics Conference,2009:7-11.
[7] WANG L H,HU Z D,ZHONG Z,et al.Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length[J].Acta Mechanica,2010,214(3):225-244.
[8] CHANG J R,LIN W J,HUANG C J,et al.Vibration and stability of an axially moving Rayleigh beam[J].Applied Mathematical Modelling,2010,34(6):1482-1497.
[9] 罗炳华,高跃飞,刘荣华,等.轴向运动梁受移动载荷作用的横向动力响应[J].振动与冲击,2011,30(12):59- 63.LUO Binghua,GAO Yuefei,LIU Ronghua,et al.Lateral dynamic response of an axially moving beam under a moving load[J].Journal of Vibration and Shock,2011,30(12):59- 63.
[10] DOWNER J D,PARK K C.Formulation and solution of inverse spaghetti problem:application to beam deployment dynamics[J].Aiaa Journal,2012,31(2):339- 347.
[11] PARK S,HONG H Y,CHUNG J.Vibrations of an axially moving beam with deployment or retraction[J].AIAA Journal,2013,51(51):686-696.
[12] YANG X D,LIU M,ZHANG W,et al.Invariant and energy analysis of an axially retracting beam[J].Chinese Journal of Aeronautics,2016,29(4):952-961.
[13] 赵亮,胡振东.轴向运动功能梯度悬臂梁动力学分析[J].振动与冲击,2016,35 (2):124-128.ZHAO Liang,HU Zhendong.Dynamic analysis of an axially translating functionally graded cantilever beam[J].Journal of Vibration and Shock,2016,35 (2):124-128.
[14] 杨鑫,陈海波.热冲击作用下轴向运动梁的振动特性研究[J].振动与冲击,2017,36 (1):8-15.YANG Xin,CHEN Haibo.Vibration characteristics of an axially moving beam under thermal shocks[J].Journal of Vibration and Shock,2017,36 (1):8-15.
[15] 谭霞,丁虎,陈立群.超临界轴向运动Timoshenko梁横向受迫振动[J].振动与冲击,2017,36 (22):1-5.TAN Xia,DING Hu,CHEN Liqun.Transverse forced vibration of an axially moving Timoshenko beam at a supercritical speed[J].Journal of Vibration and Shock,2017,36 (22):1-5.
[16] SHABANA A A.Definition of the slopes and the finite element absolute nodal coordinate formulation[J].Multibody System Dynamics,1997,1(3):339-348.