﻿ 旋转薄板的一种高次动力学模型与频率转向
«上一篇
 文章快速检索 高级检索

 力学学报  2016, Vol. 48 Issue (1): 173-180  DOI: 10.6052/0459-1879-15-194 0

### 引用本文 [复制中英文]

[复制中文]
Fang Jianshi, Zhang Dingguo. A HIGH-ORDER RIGID-FLEXIBLE COUPLING MODEL AND FREQUENCY VEERING OF A ROTATING CANTILEVER THIN PLATE[J]. Chinese Journal of Ship Research, 2016, 48(1): 173-180. DOI: 10.6052/0459-1879-15-194.
[复制英文]

### 文章历史

2015-05-29 收稿
2015-08-21 录用
2015-08-28 网络版发表

1. 南京工程学院材料工程学院, 南京 211167;
2. 南京理工大学理学院, 南京 210094

1 旋转柔性薄板的变形描述

 ${\pmb r} = ( R + x + u ) {\pmb i} + ( y + v ) {\pmb j} + w{\pmb k}$ (1)

 图 1 做旋转运动中心刚体--柔性薄板模型 Fig. 1 Model of a rotating hub--flexible thin plate

 $u = u_1 + u_{\rm c} ,\ \ \ v = v_1 + v_{\rm c}$ (2)

 $\left. \begin{array}{l} {u_{\rm{c}}} = - \frac{1}{2}\int {_0^x} {\left( {\frac{{\partial w(\xi ,y,t)}}{{\partial \xi }}} \right)^2}{\rm{d}}\xi \\ {v_{\rm{c}}} = - \frac{1}{2}\int {_0^y} {\left( {\frac{{\partial w(x,\eta ,t)}}{{\partial \eta }}} \right)^2}{\rm{d}}\eta \end{array} \right\}$ (3)

 $w( x,y,t ) = \sum\limits_{j = 1}^J {\phi _j ( x,y ) q_j ( t ) }$ (4)

 $\phi _j ( x,y ) = F( x,y ) P_r ( x ) P_s ( y )$ (5)

 $\left. P_r ( x ) = \cos \Big[(r - 1) {\rm arcos} \Big (2\dfrac{x}{a} - 1 \Big) \Big] \\ P_s ( y ) = \cos \Big[(s - 1) {\rm arcos} \Big(2\dfrac{y}{b} -1 \Big)\Big] \right\}$ (6)

2 系统动力学方程

 $\dot{\pmb r} = ( \dot {u}_{\rm c} + \varOmega w ) {\pmb i} + \dot {v}_{\rm c} {\pmb j} + [\dot {w} -\varOmega ( R + x + u_{\rm c} )] {\pmb k}$ (7)

 $T = \dfrac{1}{2}J_{\rm oh} \varOmega ^2 + \dfrac{1}{2}\int {} _V \rho\dot {\pmb r} \cdot \dot{\pmb r} \rm d V$ (8)

 $U = \dfrac{1}{2}\int {} _0^b \int {} _0^a D\left[{\left( {\dfrac{\partial^2w}{\partial x^2}} \right)^2 + 2\mu \left( {\dfrac{\partial ^2w}{\partial x^2}} \right)\left({\dfrac{\partial ^2w}{\partial y^2}} \right)} \right. + \\ \qquad \left. { \left( {\dfrac{\partial^2w}{\partial y^2}} \right)^2 + 2 (1 - \mu ) \left( {\dfrac{\partial ^{2}w}{\partial x\partial y}}\right)^2} \right] \rm d x \rm d y$ (9)

 ${\pmb M}\ddot {\pmb q} = {\pmb Q}$ (10)

 ${ M} = \left[\!\!\begin{array}{cc} {M^{11}} & {{ M }^{12}} \\ {{ M }^{21}} & {{ M }^{22}}\end{array}\!\! \right] ,\ \ Q = \left[\!\!\begin{array}{c} {Q_\theta } \\ {{ Q }^2}\end{array} \!\!\right]$ (11)

 $M^{11} = J_{\rm oh} + J_1 + \sum\limits_{i = 1}^J \sum\limits_{j = 1}^JM_{ij} q_i q_j \underline{ - \sum\limits_{i = 1}^J \sum\limits_{j = 1}^J C_{ij} q_i q_j } + \\ \qquad \underline{\underline { \dfrac{1}{4}\sum\limits_{i =1}^J \sum\limits_{j = 1}^J \sum\limits_{k = 1}^J \sum\limits_{l = 1}^JG_{ijkl}^X q_i q_j q_k q_l } }$ (12)

 $M_i^{12} = M_i^{21} = - S_i \underline{\underline { - \sum\limits_j^J{\sum\limits_k^J {E_{ijk} q_j q_k } } }} + \\ \qquad \underline{\underline { \dfrac{1}{2}\sum\limits_j^J {\sum\limits_k^J {E_{jki} q_j q_k } } }}$ (13)

 $M_{ij}^{22} = M_{ij} \underline{\underline { + \sum\limits_{k = 1}^J{\sum\limits_{l = 1}^J {G_{ijkl}^X q_k q_l } } + \sum\limits_{k = 1}^J{\sum\limits_{l = 1}^J {G_{ijkl}^Y q_k q_l } } }}$ (14)

 $Q_\theta = \tau - 2\dot {\theta }\left( {\sum\limits_{i = 1}^J{\sum\limits_{j = 1}^J {M_{ij} q_i \dot {q}_j } } \underline{ -\sum\limits_{i = 1}^J {\sum\limits_{j = 1}^J {C_{ij} q_i \dot {q}_j } } }}\right) - \\ \qquad {\underline{\underline { \dot {\theta }\sum\limits_{i = 1}^J {\sum\limits_{j = 1}^J {\sum\limits_{k =1}^J {\sum\limits_{l = 1}^J {G_{ijkl}^X q_i q_j q_k \dot {q}_l } } } } + \sum\limits_{i = 1}^J {\sum\limits_{j = 1}^J {\sum\limits_{k= 1}^J {E_{ijk} \dot {q}_i \dot {q}_j q_k } } } }} }$ (15)

 $Q_i^2 = \dot {\theta }^2\left( {\sum\limits_{j = 1}^J {M_{ij} q_j }\underline{ - \sum\limits_{j = 1}^J {C_{ij} q_j } }} \right) -\sum\limits_{j = 1}^J {K_{ij}^B q_j } + \\ \qquad {\underline{\underline { 2\dot {\theta }\left( {\sum\limits_{j = 1}^J {\sum\limits_{k = 1}^J{E_{ijk} q_j \dot {q}_k } } - \sum\limits_{j = 1}^J {\sum\limits_{k = 1}^J {E_{jki} q_j \dot {q}_k } } } \right)}} } + \\ \qquad {\underline{\underline { \dfrac{1}{2}\dot {\theta}^2\sum\limits_{j = 1}^J {\sum\limits_{k = 1}^J {\sum\limits_{l = 1}^J{G_{ijkl}^X q_j q_k q_l } } } - \sum\limits_{j = 1}^J {\sum\limits_{k = 1}^J{\sum\limits_{l = 1}^J {G_{ijkl}^X q_j \dot {q}_k \dot {q}_l } } } }} }- \\ \qquad {\underline{\underline { \sum\limits_{j = 1}^J {\sum\limits_{k = 1}^J {\sum\limits_{l = 1}^J{G_{ijkl}^Y q_j \dot {q}_k \dot {q}_l } } } }} }$ (16)

3 系统刚柔耦合动力学分析

 $\varOmega = \left\{ \begin{array}{ll} \varOmega _s \Big( \dfrac{t}{T} - \dfrac{1}{ 2\pi }\sin \dfrac{2\pi t}{T} \Big) ,\ & 0 \leqslant t \leqslant T \\ \varOmega _s ,& t > T \end{array} \right.$ (17)

 图 2 柔性薄板自由端面中点的变形位移($\varOmega _s = 20$ rad/s) Fig. 2 The lateral deflection of middle point of the free end of the thinplate ($\varOmega _s = 20$ rad/s)

 $\tau ( t ) = \left\{ \begin{array}{ll} \tau _0 \sin \Big(\dfrac{2\pi t}{T} \Big ) ,& 0 \leqslant t \leqslant T \\ 0 ,& t > T \end{array} \right.$ (18)

 图 3 柔性薄板自由端面中点的变形位移 Fig. 3 The lateral deflection of middle point of the free end of the thin plate

 $T + U = {\rm const}$ (19)

 图 4 柔性薄板自由转动的能量时变曲线 Fig. 4 The energy loci of the thin plate

 图 5 薄板和梁自由端面中点变形位移的比较 Fig. 5 The lateral deflection of a thin plate and a beam
4 频率转向与振型转换

 $\sum\limits_{j = 1}^J \Big[M_{ij} \ddot {q}_j - \dot {\theta }^2( M_{ij} \underline{ - C_{ij} } ) q_j + K_{ij}^{B} q_j \Big] = 0$ (20)

 $\varsigma = t/ {T^ * } ,\ \ \xi = x/a ,\ \\zeta = y / b ,\ \ \kappa _i = {q_i } / a \\ \delta = R / a ,\ \ \sigma = a / b ,\ \\gamma = \varOmega T^ *$

 $\sum\limits_{j = 1}^J \Big[\bar {M}_{ij} \ddot {\kappa }_j - \gamma ^2( \bar {M}_{ij} \underline{ - \bar {C}_{ij} }) \kappa _j + \bar {K}_{ij}^{ B} \kappa _j \Big] = 0$ (21)

 $\begin{array}{l} {{\bar M}_{ij}} = \int {_0^1} \int {_0^1} {\psi _i}{\psi _j}{\rm{d}}\xi {\rm{d}}\zeta \\ {{\bar C}_{ij}} = \int {_0^1} \int {_0^1} (\delta + \xi )\bar H_{ij}^X{\rm{d}}\xi {\rm{d}}\zeta \\ \bar K_{ij}^B = \int {_0^1} \int {_0^1} \left[ {{\psi _{i,\xi \xi }}{\psi _{j,\xi \xi }} + {\psi _{i,\zeta \zeta }}{\psi _{j,\zeta \zeta }} + \mu {\psi _{i,\xi \xi }}{\psi _{j,\zeta \zeta }}} \right. + \\ \qquad \left. {\mu {\psi _{i,\zeta \zeta }}{\psi _{j,\xi \xi }} + 2\left( {1 - \mu } \right){\psi _{i,\xi \zeta }}{\psi _{j,\xi \zeta }}} \right]{\rm{d}}\xi {\rm{d}}\zeta \end{array}$ (22)

 $( \hat {\pmb K} - \omega ^2 \hat {\pmb M} ) {\pmb \varTheta } = 0$ (23)

 $\hat {M}_{ij} = \bar {M}_{ij} ,\ \ \hat {K}_{ij} = \gamma ^2( \bar {C}_{ij} - \bar {M}_{ij} ) + \bar{K}_{ij}^{ B}$ (24)

 图 6 旋转悬臂柔性方板的前8 阶无量纲固有频率随角速度的变化曲线 Fig. 6 The first eight dimensionless natural frequencies versus angular speed

 图 7 第4 阶模态振型节线随无量纲角速度的变化规律 Fig. 7 Variation of the fourth mode shape nodal lines versus dimensionless rotational speed

 图 8 第5 阶模态振型节线随无量纲角速度的变化规律 Fig. 8 Variation of the fifth mode shape nodal lines versus dimensionless rotational speed

 图 9 第6 阶模态振型节线随无量纲角速度的变化规律 Fig. 9 Variation of the sixth mode shape nodal lines versus dimensionless rotational speed

 图 10 旋转悬臂柔性方板的模态振型节线随无量纲角速度的变化规律 Fig. 10 Variation of the fifth and sixth mode shape nodal lines versus dimensionless rotational speed
5 结 论

 1 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): 139-151 2 Cai GP, Hong JZ, Yang SX. Dynamic analysis of a flexible hubbeam system with tip mass. Mechanics Research Communications,2005, 32(2): 173-190 3 王新栋,邓子辰,王艳等.基于时间有限元方法的旋转叶片动力学响应分析.应用数学与力学, 2015, 35(4): 353-363 (Wang Xindong, Deng Zichen, Wang Yan, et al. Dynamic behavior analysis of rotational flexible blades based on time-domain finite element method. Applied Mathematics and Mechanics, 2015, 35(4): 353-363 (in Chinese)) 4 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): 279-288 (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)) 5 Yoo HH, Chung J. Dynamics of rectangular plates undergoing prescribed overall motion. Journal of Sound and Vibration, 2001,239(1): 123-137 6 刘锦阳,洪嘉振.做大范围运动矩形薄板的建模理论和有限元离散方法.振动工程学报, 2003, 16(2): 175-179 (Liu Jinyang, Hong Jiazhen. Dynamic modeling theory and finite element method for a rectangular plate undergoing large overall motion. Journal of Vibration Engineering, 2003, 16(2): 175-179 (in Chinese)) 7 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)) 8 邹凡,刘锦阳.大变形薄板多体系统的动力学建模.应用力学学报, 2010, 27(4): 740-745 (Zou Fan, Liu Jinyang. Dynamic formulation for thin plate system with large deformation. Chinese Journal of Applied Mechanics, 2010, 27(4): 740-745 (in Chinese)) 9 Southwell R, Gough F. The free transverse vibration of airscrew blades. British A. R. C. Report and Memoranda, 1921, 766 10 Li L, Zhang DG, Zhu WD. Free vibration analysis of a rotating hubfunctionally graded material beam system with the dynamic sti ening e ect. Journal of Sound and Vibration, 2014, 333: 1526-1541 11 Saurabh K, Anirban M. Large amplitude free vibration analysis of axially functionally graded tapered rotating beam by energy method. Mechanisms and Machine Science, 2015, 23: 473-483 12 Yoo HH, Pierre C. Modal characteristic of a rotating rectangular cantilever plate. Journal of Sound and Vibration, 2003, 259(1): 81-96 13 Zhao J, Tian Q, Hu HY. Modal analysis of a rotating thin plate via absolute nodal coordinate formulation. Journal of Computational and Nonlinear Dynamics, 2011, 6(041013): 1-8
A HIGH-ORDER RIGID-FLEXIBLE COUPLING MODEL AND FREQUENCY VEERING OF A ROTATING CANTILEVER THIN PLATE
Fang Jianshi, Zhang Dingguo
1. School of Materials Engineering, Nanjing Institute of Technology, Nanjing 211167, China;
2. School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China
Abstract: The rigid-flexible coupling dynamics and frequency veering of a thin flexible plate on a rotating rigid body are further studied. The high-order coupling (HOC) dynamic model is derived by Lagrange's equations. The in-plane longitudinal shortening terms caused by lateral deformation, generally called non-linear coupling deformation terms, are considered here. Furthermore, all derived items associated with the non-linear coupling terms are retained completely in the HOC model. The HOC model can not only be applied in the small deformation case, but also in the large deformation case, and makes up the deficiency of the first-order approximation coupling (FOAC) model in the large deformation case. In addition, the frequency veering phenomena along with the corresponding mode shape variations are exhibited and discussed in detail. When two frequency loci veer, the nodal line patterns of the mode shapes switch their shapes each other, and the changes of the nodal line patterns are continuous in the mild veering region, while those in the abrupt veering region are discontinuous. The transitive frequency veerings among the multiple modes accompanied by mode shape transfer are also exhibited.
Key words: rotating thin plate    high-order coupling model    dynamics    frequency veering    mode shape interaction