﻿ 考虑曲率纵向变形效应的大变形柔性梁刚柔耦合动力学建模与仿真
«上一篇
 文章快速检索 高级检索

 力学学报  2016, Vol. 48 Issue (3): 692-701  DOI: 10.6052/0459-1879-15-385 0

### 引用本文 [复制中英文]

[复制中文]
Zhang Xiaoshun, Zhang Dingguo, Hong Jiazheny. RIGID-FLEXIBLE COUPLING DYNAMIC MODELING AND SIMULATION WITH THE LONGITUDINAL DEFORMATION INDUCED CURVATURE EFFECT FOR A ROTATING FLEXIBLE BEAM UNDER LARGE DEFORMATION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 692-701. DOI: 10.6052/0459-1879-15-385.
[复制英文]

### 文章历史

2015-10-21 收稿
2016-03-07 录用
2016-03-09 网络版发表.

1. 南京理工大学理学院, 南京 210094;
2. 上海交通大学工程力学系, 上海 200240

0 引言

1 刚柔耦合动力学方程 1.1 物理模型

 图1 刚体-柔性梁变形示意图 Fig.1 The deformation of the hub-beam
1.2 系统动力学方程

 $\text{r=(a+x +}{{\text{u}}_{\text{1}}}\text{)}{{\text{e}}_{\text{x}}}\text{+}{{\text{w}}_{\text{2}}}{{\text{e}}_{\text{y}}}$ (1)

 ${{w}_{\text{c}}}(x,t)=-\frac{1}{2}\int_{0}^{x}{(}\frac{\partial {{w}_{2}}}{\partial \xi }{{)}^{2}}d\xi$ (2)

 \begin{align} & \dot{r}=({{{\dot{w}}}_{1}}+{{{\dot{w}}}_{c}}-{{w}_{2}}\dot{\theta }){{e}_{x}}+ \\ & [(a+x+{{w}_{1}}+{{w}_{c}})\dot{\theta }+{{{\dot{w}}}_{2}}]{{e}_{y}} \\ \end{align} (3)

 $T=\frac{1}{2}{{J}_{oh}}{{\dot{\theta }}^{2}}+\frac{1}{2}\int_{0}^{L}{\rho S{{{\dot{r}}}^{T}}\dot{r}}dx$ (4)

 $V={{V}_{\text{G}}}+{{V}_{\text{E}}}$ (5)

 \left. \begin{align} & {{V}_{\text{G}}}=0 \\ & V={{V}_{\text{E}}}={{V}_{l}}+{{V}_{\kappa }} \\ \end{align} \right\} (6)
 ${{V}_{l}}=\frac{1}{2}\int_{0}^{L}{ES{{\left( \frac{\partial {{w}_{1}}(x,t)}{\partial x} \right)}^{2}}}dx$ (7)
 ${{V}_{\kappa }}=\ \frac{1}{2}\int_{0}^{L}{EI{{\left( \frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}} \right)}^{2}}}dx$ (8)

 $\left. \begin{array}{*{35}{l}} \phi (x,t)=x+{{w}_{1}}(x,t)+{{w}_{c}}(x,t) \\ \psi (x,t)={{w}_{2}}(x,t) \\ \end{array} \right\}$ (9)

 $\kappa (x)=\frac{\left| \frac{\partial \varphi (x,t)}{\partial x}\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}-\frac{{{\partial }^{2}}\varphi (x,t)}{\partial {{x}^{2}}}\frac{\partial \psi (x,t)}{\partial x} \right|}{{{\left[ {{(\frac{\partial \varphi (x,t)}{\partial x})}^{2}}+{{(\frac{\partial \psi (x,t)}{\partial x})}^{2}} \right]}^{3/2}}}$ (10)

 \begin{align} & {{V}_{\kappa }}=\frac{1}{2}\int_{0}^{L}{EI{{\kappa }^{2}}(x)}dx=\frac{1}{2}\int_{0}^{L}{EI}\cdot \\ & \frac{{{\left| \frac{\partial \varphi (x,t)}{\partial x}\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}-\frac{{{\partial }^{2}}\varphi (x,t)}{\partial {{x}^{2}}}\frac{\partial \psi (x,t)}{\partial x} \right|}^{2}}}{{{\left[ {{(\frac{\partial \varphi (x,t)}{\partial x})}^{2}}+{{(\frac{\partial \psi (x,t)}{\partial x})}^{2}} \right]}^{3}}}dx \\ \end{align} (11)
 \begin{align} & \frac{\partial \varphi (x,t)}{\partial x}=1+\frac{\partial {{w}_{1}}(x,t)}{\partial x}+\frac{\partial {{w}_{c}}(x,t)}{\partial x}= \\ & 1+\frac{\partial {{w}_{1}}(x,t)}{\partial x}-\frac{1}{2}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}} \\ \end{align} (12)
 $\frac{\partial \psi (x,t)}{\partial x}=\frac{\partial {{w}_{2}}(x,t)}{\partial x}$ (13)

 $\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}=\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}}$ (14)
 $\frac{{{\partial }^{2}}\phi (x,t)}{\partial {{x}^{2}}}=-\frac{\partial {{w}_{2}}(x,t)}{\partial x}\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}}$ (15)
 \begin{align} & {{(\frac{\partial \phi (x,t)}{\partial x})}^{2}}+{{(\frac{\partial \psi (x,t)}{\partial x})}^{2}}= \\ & 1+2\frac{\partial {{w}_{1}}(x,t)}{\partial x}-\frac{\partial {{w}_{1}}(x,t)}{\partial x}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}}+ \\ & {{(\frac{\partial {{w}_{1}}(x,t)}{\partial x})}^{2}}+\frac{1}{4}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{4}} \\ \end{align} (16)

 ${{(\frac{\partial \phi (x,t)}{\partial x})}^{2}}+{{(\frac{\partial \psi (x,t)}{\partial x})}^{2}}=1+2\frac{\partial {{w}_{1}}(x,t)}{\partial x}$ (17)
 \begin{align} & \frac{\partial \phi (x,t)}{\partial x}\frac{{{\partial }^{2}}\psi (x,t)}{\partial {{x}^{2}}}-\frac{{{\partial }^{2}}\phi (x,t)}{\partial {{x}^{2}}}\frac{\partial \psi (x,t)}{\partial x}= \\ & \frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}}+\frac{1}{2}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}}\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}} \\ \end{align} (18)
 \begin{align} & {{V}_{\kappa }}=\frac{1}{2}\int_{0}^{L}{EI{{\kappa }^{2}}\left( x \right)}dx= \\ & \frac{1}{2}\int_{0}^{L}{EI\frac{{{\left| \frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}}+\frac{1}{2}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}}\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}} \right|}^{2}}}{{{\left[ 1+2\frac{\partial {{w}_{1}}(x,t)}{\partial x} \right]}^{3}}}}dx \\ \end{align} (19)

 \begin{align} & {{\left( 1+2\frac{\partial {{w}_{1}}(x,t)}{\partial x} \right)}^{-3}}=1-6\frac{\partial {{w}_{1}}(x,t)}{\partial x}+ \\ & 12{{(\frac{\partial {{w}_{1}}(x,t)}{\partial x})}^{2}}+\cdots \\ \end{align} (20)

 \begin{align} & {{V}_{\kappa }}=\frac{1}{2}\int_{0}^{L}{E}I\{[1+\frac{\partial {{w}_{1}}(x,t)}{\partial x}+ \\ & \frac{1}{2}{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}}]\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}}{{\}}^{2}}\cdot \\ & (1-6\frac{\partial {{w}_{1}}(x,t)}{\partial x})\text{d}x= \\ & \frac{1}{2}\int_{0}^{L}{E}I{{(\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}})}^{2}}dx+ \\ & \underline{\frac{1}{2}\int_{0}^{L}{E}I{{(\frac{\partial {{w}_{2}}(x,t)}{\partial x})}^{2}}{{(\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}})}^{2}}dx}- \\ & \underline{\underline{2\int_{0}^{L}{EI\frac{\partial {{w}_{1}}(x,t)}{\partial x}{{(\frac{{{\partial }^{2}}{{w}_{2}}(x,t)}{\partial {{x}^{2}}})}^{2}}}dx}} \\ \end{align} (21)

 \left. \begin{align} & {{w}_{1}}(x,t)={{\Phi }_{x}}(x)A(t) \\ & {{w}_{2}}(x,t)={{\Phi }_{y}}(x)B(t) \\ \end{align} \right\} (22)

 ${{w}_{\text{c}}}=-\frac{1}{2}{{B}^{\text{T}}}H(x)B$ (23)

 $H(x)=\int_{0}^{x}{{{\Phi }^{\prime }}_{y}^{\text{T}}}(\xi ){{{\Phi }'}_{y}}(\xi )d\xi$ (24)
$\Phi '_y (x)$表示${\Phi } _y (x)$对$x$求一次导，取广义坐标$q = \left( {\theta ,{ A}^{\rm T},{ B}^{\rm T}}\right)^ {\rm T}$，将式(4) $\sim$式(7)，式(21)代入第二类拉格朗日方程，即可得到系统的动力学方程
 $M\ddot{q}=Q$ (25)

 \left. \begin{align} & M=\left[ \begin{matrix} {{M}_{11}} & {{M}_{12}} & {{M}_{13}} \\ {{M}_{21}} & {{M}_{22}} & {{M}_{23}} \\ {{M}_{31}} & {{M}_{32}} & {{M}_{33}} \\ \end{matrix} \right] \\ & Q=\left[ \begin{matrix} {{Q}_{\theta }} \\ {{Q}_{A}} \\ {{Q}_{B}} \\ \end{matrix} \right]+\left[ \begin{matrix} \tau \\ \mathbf{0} \\ \mathbf{0} \\ \end{matrix} \right] \\ \end{align} \right\} (26)

 \begin{align} & {{M}_{11}}={{J}_{oh}}+{{J}_{ob}}+2{{S}_{x}}A+{{A}^{\text{T}}}{{M}_{x}}A+{{B}^{\text{T}}}{{M}_{y}}B- \\ & \underline{{{B}^{\text{T}}}CB}+\underline{\underline{\frac{1}{4}\int_{0}^{L}{\rho S({{B}^{\text{T}}}HB{{B}^{\text{T}}}HB)}dx}}- \\ & \underline{\underline{\int_{0}^{L}{\rho S({{A}^{\text{T}}}\Phi _{x}^{\text{T}}{{B}^{\text{T}}}HB)}dx}} \\ \end{align} (27)
 ${{\text{M}}_{\text{22}}}\text{ = }{{\text{M}}_{\text{x}}}$ (28)
 ${{M}_{33}}={{M}_{y}}+\underline{\underline{\int_{o}^{L}{\rho S(HB{{B}^{\text{T}}}H)}dx}}$ (29)
 ${{M}_{21}}=M_{12}^{\text{T}}=-{{M}_{xy}}B$ (30)
 \begin{align} & {{M}_{31}}=M_{13}^{\text{T}}=S_{y}^{\text{T}}+M_{xy}^{\text{T}}A+ \\ & \underline{\underline{\int_{0}^{L}{\rho }S(HB{{\Phi }_{y}}B-\frac{1}{2}\Phi _{y}^{\text{T}}{{B}^{\text{T}}}HB)dx}} \\ \end{align} (31)
 ${{M}_{32}}=M_{23}^{\text{T}}=\underline{\underline{-\int_{0}^{L}{\rho S(HB{{\Phi }_{x}}})dx}}$ (32)
 \begin{align} & {{Q}_{\theta }}=-2\dot{\theta }({{S}_{x}}\dot{A}+{{A}^{\text{T}}}{{M}_{x}}\dot{A}+{{B}^{\text{T}}}{{M}_{y}}\dot{B}-\underline{{{B}^{\text{T}}}C\dot{B}})- \\ & \underline{\underline{\int_{0}^{L}{\rho S({{B}^{\text{T}}}\Phi _{y}^{\text{T}}{{{\dot{B}}}^{\text{T}}}H\dot{B})}dx}}+ \\ & \underline{\underline{\dot{\theta }\int_{0}^{L}{\rho }S({{{\dot{A}}}^{\text{T}}}\Phi _{x}^{\text{T}}{{B}^{\text{T}}}HB+2{{A}^{\text{T}}}\Phi _{x}^{\text{T}}{{B}^{\text{T}}}H\dot{B}}}- \\ & \underline{\underline{{{B}^{\text{T}}}HB{{B}^{\text{T}}}H\dot{B})dx}} \\ \end{align} (33)
 \begin{align} & {{Q}_{A}}={{{\dot{\theta }}}^{2}}(S_{x}^{\text{T}}+{{M}_{x}}A)+2\dot{\theta }{{M}_{xy}}\dot{B}-{{K}_{1}}A- \\ & \underline{\underline{\frac{1}{2}{{{\dot{\theta }}}^{2}}\int_{0}^{L}{\rho S(\Phi _{x}^{\text{T}}{{B}^{\text{T}}}HB})dx}}+ \\ & \underline{\underline{\int_{0}^{L}{\rho }S(\Phi _{x}^{\text{T}}{{{\dot{B}}}^{\text{T}}}H\dot{B})dx}}+ \\ & \underline{\underline{\underline{\underline{2\int_{0}^{L}{E}I\Phi {{'}_{x}}^{T}{{B}^{T}}{{\Phi }_{y}}''(x){{\Phi }_{y}}''Bdx}}}} \\ \end{align} (34)
 \begin{align} & {{Q}_{B}}={{{\dot{\theta }}}^{2}}({{M}_{y}}\underline{-C})B-2\dot{\theta }M_{xy}^{\text{T}}\dot{A}-{{K}_{2}}B+ \\ & \underline{\underline{{{{\dot{\theta }}}^{2}}\int_{0}^{L}{\rho }S(\frac{1}{2}HB{{B}^{\text{T}}}HB-HB{{\Phi }_{x}}A)dx}}- \\ & \underline{\underline{2\dot{\theta }\int_{0}^{L}{\rho S}(HB{{\Phi }_{y}}\dot{B}-\Phi _{y}^{\text{T}}{{B}^{\text{T}}}H\dot{B})dx}}- \\ & \underline{\underline{\int_{0}^{L}{\rho S}(HB{{{\dot{B}}}^{\text{T}}}H\dot{B})dx}}- \\ & \underline{\underline{\underline{\mathop{\int }_{0}^{L}EI\Phi '{{'}_{y}}^{T}{{\Phi }_{y}}''B{{B}^{T}}\Phi {{'}_{y}}^{T}{{\Phi }_{y}}'Bdx-}}} \\ & \underline{\underline{\underline{\int_{0}^{L}{E}I{{B}^{T}}\Phi '{{'}_{y}}^{T}{{\Phi }_{y}}''B\Phi {{'}_{y}}^{T}{{\Phi }_{y}}'Bdx+}}} \\ & \underline{\underline{\underline{\underline{4\int_{0}^{L}{E}I{{\Phi }_{x}}'A\Phi '{{'}_{y}}^{T}{{\Phi }_{y}}''Bdx}}}} \\ \end{align} (35)

 ${{J}_{ob}}=\int_{0}^{L}{\rho S{{(a+x)}^{2}}}dx$ (36)
 ${{S}_{x}}=\int_{0}^{L}{\rho }S(a+x){{\Phi }_{x}}(x)dx$ (37)
 ${{S}_{y}}=\int_{0}^{L}{\rho }S(a+x){{\Phi }_{y}}(x)dx$ (38)
 ${{M}_{x}}=\int_{0}^{L}{\rho S}\Phi _{x}^{\text{T}}(x){{\Phi }_{x}}(x)dx$ (39)
 ${{M}_{y}}=\int_{0}^{L}{\rho S}\Phi _{y}^{\text{T}}(x){{\Phi }_{y}}(x)dx$ (40)
 ${{M}_{xy}}=\int_{0}^{L}{\rho }S\Phi _{x}^{\text{T}}(x){{\Phi }_{y}}(x)dx$ (41)
 $C=\int_{0}^{L}{\rho }S(a+x)H(x)dx$ (42)
 ${{K}_{1}}=\int_{0}^{L}{E}S\Phi {{'}_{x}}^{T}(x)\Phi {{'}_{x}}(x)dx$ (43)
 ${{K}_{2}}=\int_{0}^{L}{E}I\Phi \text{ }\!\!'\!\!\text{ }\!\!'\!\!\text{ }_{y}^{\text{T}}(x)\Phi \text{ }\!\!'\!\!\text{ }{{\text{ }\!\!'\!\!\text{ }}_{y}}(x)dx$ (44)

2 算例 2.1 做大范围旋转运动刚体-悬臂梁系统

 $\tau (t)=\left\{ \begin{array}{*{35}{l}} {{\tau }_{0}}\sin (\frac{2\pi }{T}t), & 0\le t\le T \\ 0, & t>T \\ \end{array} \right.$ (45)

 图2 柔性梁末端变形 Fig.2 The tip deformation of the beam
2.2 自由下落柔性单摆

 图3 柔性单摆 Fig.3 The flexible pendulum

 ${{V}_{\text{G}}}=-{{g}^{\text{T}}}W\int_{0}^{L}{\rho }Srdx=-{{g}^{\text{T}}}WR$ (45)

 $g={{[{{g}_{X}},{{g}_{Y}}]}^{\text{T}}}$ (46)
 $W=\left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right]$ (47)
 $R=\int_{0}^{L}{\rho }Srdx$ (48)

 图4 柔性单摆的末端变形，$E =68.952$ GPa Fig.4 The tip deformation of the pendulum at $E =68.952$ GPa
 图5 柔性单摆的末端变形，$E =6.895 2$ GPa Fig.5 The tip deformation of the pendulum at $E =6.895 2$ GPa

 图6 柔性单摆的末端变形，$E =3.447 6$ GPa Fig.6 The tip deformation of the pendulum at $E =3.447 6$ GPa

 图7 柔性单摆的末端变形，$E =1.723 8$ GPa Fig.7 The tip deformation of the pendulum at $E =1.723 8$ GPa
 图8 大柔度单摆末端变形，$E =1.149 2$ GPa Fig.8 The tip deformation of the large flexibility pendulum at $E =1.149 2$ GPa

 图9 单摆的能量曲线，$E =3.447 6$ GPa Fig.9 Energy curves of the pendulum at $E =3.447 6$ GPa
3 结论

 [1] Likins PW. Finite element appendage equations for hybrid coordinate dynamic analysis. Journal of Solids and Structures, 1972, 8:790-831 [2] Kane TR, Ryan RR, Banerjee AK. Dynamics of a cantilever beam attached to a moving base. Journal of Guidance, Control and Dynamics,1987, 10(2): 139-150 [3] Wallrapp O, Schwertassek R. Representation of geometric stiffening in multi-body system simulation. International Journal for Numerical Methods in Engineering, 1991, 32: 1833-1850 [4] Bakr EM, Shabana AA. Geometrically nonlinear analysis of multibody system. Computer and Structures, 1986, 23(6): 739-751 [5] Wu SC, Haug E. Geometric non-linear substructuring for dynamics of flexible mechanical systems. International Journal for Numerical Methods in Engineering, 1988, 26: 2211-2226 [6] Wallrapp O, Schwertassek R. Representation of geometric stiffening in multi-body system simulation. International Journal for Numerical Methods in Engineering, 1991, 32: 1833-1850 [7] 洪嘉振,尤超蓝. 刚柔耦合系统动力学研究进展. 动力学与控制学报,2004,2(2): 1-6 (Hong Jiazhen, You Chaolan. Advances in dynamics of rigid-flexible coupling system. Journal of Dynamics and Control, 2004, 2(2): 1-6 (in Chinese)) [8] Liu JY, Hong JZ. Geometric stiffening effect on rigid-flexible coupling dynamic of an elastic beam. Journal of Sound and Vibration,2004, 278: 1147-1162 [9] 蔡国平,洪嘉振. 旋转运动柔性梁的假设模态方法研究. 力学学报,2005,37(1): 48-56 (Cai Guoping, Hong Jiazhen. Assumed mode method of a rotating flexible beam. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(1): 48-56 (in Chinese)) [10] 章定国,余纪邦. 作大范围运动的柔性梁的动力学分析. 振动工程学报,2006,19(4): 475-479 (Zhang Dingguo, Yu Jibang. Dynamical analysis of a flexible cantilever beam with large overall motions. Journal of Vibration Engineering, 2006,19(4): 475-479 (in Chinese)) [11] 刘锦阳,李彬,洪嘉振. 做大范围空间运动柔性梁的刚-柔耦合动力学. 力学学报,2006,38(2): 276-282 (Liu Jinyang, Li Bin, Hong Jiazhen. Rigid-flexible coupling dynamics of a flexible beam with three-dimensional large overall motion. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(2): 276-282 (in Chinese)) [12] 邓峰岩,和兴锁,李亮等. 耦合变形对大范围运动柔性梁动力学建模的影响. 计算力学学报,2006,23(5): 599-606 (Deng Fengyan, He Xingsuo, Li Liang, et al. Dynamics modeling of the elastic beam undergoing large overall motion considering coupling effect in deformation. Chinese Journal of Computational Mechanics, 2006,23(5): 599-606 (in Chinese)) [13] 和兴锁,邓峰岩,王睿. 具有大范围运动和非线性变形的空间柔性梁的精确动力学建模. 物理学报,2010,59(3): 1428-1436 (He Xingsuo, Deng Fengyan, Wang Rui. Exact dynamic modeling of a spatial flexible beam with large overall motion and nonlinear deformation. Acta Physica Sinica, 2010, 59(3): 1428-1436 (in Chinese)) [14] 吴胜宝,章定国. 大范围运动刚体-柔性梁刚柔耦合动力学分析. 振动工程学报,2011,24(1): 1-7 (Wu Shengbao, Zhang Dingguo. Rigid-flexible coupling dynamic analysis of hub-flexible beam with large overall motion. Journal of Vibration Engineering, 2011, 24(1):1-7 (in Chinese)) [15] 陈思佳,章定国. 中心刚体-变截面梁系统的动力学特性研究. 力学学报,2011,43(4): 790-794 (Chen Sijia, Zhang Dingguo. Dynamics of hub-variable section beam systems. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(4): 790-794 (in Chinese)) [16] 方建士,章定国. 旋转悬臂梁的刚柔耦合动力学建模与频率分析. 计算力学学报,2012,29(3): 333-339 (Fang Jianshi, Zhang Dingguo. Rigid-flexible coupling dynamic modeling and frequency analysis of a rotating cantilever beam. Chinese Journal of Computational Mechanics, 2012, 29(3): 333-339 (in Chinese)) [17] Shabana AA. An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies. Technical Report. no. MBS96-1-UIC, 1996 [18] Berzeri M, Shabana AA. Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. Journal of Sound and Vibration, 2000, 235(4): 539-565 [19] Berzeri M, Shabana AA. Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation. Multibody System Dynamics, 2002, 7: 357-387 [20] Omar MA, Shabana AA. A two-dimensional shear deformable beam for large rotation and deformation problems. Journal of Sound and Vibration, 2001, 243(3): 565-576 [21] Shabana AA, Mikkola AM. Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity. Transactions of the ASME, Journal of Mechanical Design, 2003, 125: 342-350 [22] Yoo HH, Kim JM, Chung J. Equilibrium and modal analyses of rotating multibeam structures employing multiple reference frames. Journal of Sound and Vibration, 2007, 302: 789-805 [23] Gerstmayr J, Irschik H. On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. Journal of Sound and Vibration, 2008,318(3): 461-487 [24] Liu C, Tian Q, Hu HY. New spatial curved beam and cylindrical shell elements of gradient-deficient absolute nodal coordinate formulation. Nonlinear Dynamics, 2012, 70(3): 1903-1918 [25] Liu JY, Hong JZ. Geometric stiffening effect on rigid-flexible coupling dynamic of an elastic beam. Journal of Sound and Vibration,2004, 278: 1147-1162 [26] Liu ZY, Hong JH, Liu JY. Complete geometric nonlinear formulation for rigid-flexible coupling dynamics. Journal of Central South University of Technology, 2009, 16: 119-124 [27] 陈思佳,章定国,洪嘉振. 大变形旋转柔性梁的一种高次刚柔耦合动力学模型. 力学学报,2013,45(2): 251-256 (Chen Sijia, Zhang Dingguo, Hong Jiazhen. A high-order rigid-flexible coupling model of a rotating flexible beam under large deformation. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 251-256 (in Chinese)) [28] 范纪华,章定国. 旋转悬臂梁动力学的B 样条差值方法. 机械工程学报,2012,48(23): 59-64 (Fan Jihua, Zhang Dingguo. Bspline interpolation method for the dynamics of rotating cantilever beam. Journal of Mechanical Engineering , 2012, 48(23): 59-64 (in Chinese)) [29] 范纪华,章定国. 旋转柔性悬臂梁动力学的Bezier 插值离散方法研究. 物理学报,2014,63(15): 154501 (Fan Jihua, Zhang Dingguo. Bezier interpolation method for the dynamics of rotating flexible cantilever beam. Acta Physica Sinica, 2014, 63(15): 154501 (in Chinese)) [30] 杜超凡,章定国,洪嘉振. 径向基点插值法在旋转柔性梁动力学中的应用. 力学学报,2015,47(2): 279-88 (Du Chaofan, Zhang Dingguo, Hong Jiazhen. A meshfree method based on radial point interpolation method for the dynamic analysis of rotating flexible beams. Chinese Journal of Theoretical and Applied Mechanics,2015, 47(2): 279-288 (in Chinese)) [31] 李彬,刘锦阳. 大变形柔性梁系统的绝对坐标方法. 上海交通大学学报,2005,39(5): 827-831 (Li Bin, Liu Jingyang. Application of absolute nodal coordination formulation in flexible beams with large deformation. Journal of Shanghai Jiaotong University, 2005,39(5): 827-831 (in Chinese))
RIGID-FLEXIBLE COUPLING DYNAMIC MODELING AND SIMULATION WITH THE LONGITUDINAL DEFORMATION INDUCED CURVATURE EFFECT FOR A ROTATING FLEXIBLE BEAM UNDER LARGE DEFORMATION
Zhang Xiaoshun, Zhang Dingguo, Hong Jiazheny
1. School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China;
2. Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: The dynamics of a flexible beam which is rotating in a plane is further studied in this paper. The dynamic model of the rigid-flexible coupling system under large deformation is established and here by the dynamic simulation is also carried out. In this dynamic model not only the transversal bending deformation and the longitudinal deformation (including the axial stretching deformation and the longitudinal shortening term caused by the transversal bending deformation) of the flexible beam are considered, but also the curvature e ect induced by the longitudinal deformation is included. In the previous studies the bending deformation energy of the beam is usually expressed in terms of the bending deformation directly without considering the longitudinal deformation e ect. To take into account the influence due to the longitudinal deformation on the bending deformation energy of the beam the precise curvature formula in the form of parametric equation expressed in the floating frame of reference is used to calculate the bending deformation energy. And consequently the rigid-flexible coupling dynamic model of the system with the said the longitudinal deformation induced curvature e ect (LDICE) model is obtained. To validate the algorithm presented in this paper, several dynamic simulation examples are given. The results show that the dynamic model presented in this paper can not only be used in the analysis of the small deformation dynamics, but also in the large deformation dynamics, and indicate that the dynamic model with the curvature e ect obtained in this paper is more suitable to solve for the large deformation dynamic problems than the high-order coupled (HOC) model presented in the existing literature. The results obtained using the proposed model are compared with the results obtained using the absolute nodal coordinate formulation (ANCF) which is suitable for large deformation problems, and consequently validate the dynamic model proposed in this paper.
Key words: flexible beam    rigid-flexible coupling    dynamics    large deformation    longitudinal deformation induced curvature effect