﻿ 弹性连接旋转柔性梁动力学分析
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 力学学报  2016, Vol. 48 Issue (4): 963-971  DOI: 10.6052/0459-1879-16-083 0

### 引用本文 [复制中英文]

[复制中文]
Huang Yixin , Tian Hao , Zhao Yang . DYNAMIC ANALYSIS OF A ROTATING FLEXIBLE BEAM WITH ELASTIC BOUNDARY CONDITIONS[J]. Chinese Journal of Ship Research, 2016, 48(4): 963-971. DOI: 10.6052/0459-1879-16-083.
[复制英文]

### 文章历史

2016-03-30 收稿
2016-05-13 录用
2016-05-15网络版发表

0 引言

1 旋转柔性梁动力学模型 1.1 系统动力学方程

 图 1 弹性连接旋转柔性梁 Fig. 1 Rotating flexible beam with elastically restrain

 $\left. {\matrix{ {\rho A\left( {\ddot u - 2\Omega \dot v - {\Omega ^2}u - \dot \Omega v} \right) - EAu'' = } \cr {\rho A{\Omega ^2}\left( {R + x} \right)} \cr {\rho A\left( {\ddot v + 2\Omega \dot u + \dot \Omega u - {\Omega ^2}v} \right) - EA{{\left( {u'v'} \right)}^\prime } + EIv'''' = } \cr { - \rho A\dot \Omega \left( {R + x} \right)} \cr } } \right\}$ (1)

1.2 Chebyshev谱方法

Chebyshev谱方法以Chebyshev多项式插值为基础的方法.本文采用第一类Chebyshev多项式，它是一类递归正交多项式，其第$k$项可表示为[38]

 $T_k (x) = \cos (k\arccos x) ,\quad k = 1,2,3,\cdots$ (7)

 $T_k (x) = 2xT_{k - 1} (x) - T_{k - 2} (x)$ (8)

 $y_N (x) = \sum_{k = 0}^{N - 1} a_k T_k (x)$ (9)

 ${ y }^{(n)} = { \Gamma }_{\rm B} { D }^n{ \Gamma }_{\rm B}^{ - 1} { y } = { Q }_n { y }$ (14)

${ Q }_n $$n阶Chebyshev微分矩阵. 函数y ( x )的积分可表示为  \int_{l_1 }^{l_2 } {y_N (x)} d x = \sum_{k = 0}^N {a_k } \int_{l_1 }^{l_2 } {T_k (x)} d x = { v }^{\rm T}{ a }_N (15) 式中{ v }$$N$维列向量，称作Chebyshev积分向量. 在区间$\left[{l_1 ,l_2 } \right]$平方可积的函数$f ( x )$$g ( x )的内积可表示为  \int_{l_1 }^{l_2 } {f(x)g(x)d x = { f}^{\rm T}} { V g} (16) { V}$$N\times N$ Chebyshev内积矩阵，${ f}$${g}分别为函数f\left( x \right)$$g\left( x \right)$的Gauss-Lobattoa节点值列向量. 同理，函数$f\left( x \right)$$g\left( x \right)关于函数h\left( x \right)的加权内积可表示为  \int_{l_1 }^{l_2 } {f\left( x \right)} g\left( x \right)h\left( x \right)d x = { f }^{\rm T}{ V }_h { g } (17) 式中{ V}_h 是关于函数h\left( x \right)$$N\times N$ Chebyshev加权内积矩阵.

1.3 动力学方程离散

 ${ q} = { P}\tilde { q} + { R b}$ (29)

 $\tilde{ M}\ddot {\tilde { q}} + \tilde{ G}\dot {\tilde { q} } + \tilde { K}\tilde{ q} = {\bf 0}$ (32)
 $\tilde{ M} = { P }^{\rm T}{ M P } ,\quad \tilde{ G} = { P }^{\rm T}{ G P } ,\quad \tilde{ K } = { P }^{\rm T}{ K P }$ (33)

 ${ z } = \left[ \tilde{ q }^{\rm T} \dot {\tilde { q} }^{\rm T} \right]^{\rm T}$ (34)

 ${ C}\dot { z} + { D z} = {\bf 0}$ (35)

 ${ C} = \left[\!\! \begin{array}{cc} \tilde{ G} & \tilde{ M} \\ \tilde{ M} & {\bf 0} \end{array}\!\! \right] , \quad { D} = \left[\!\! \begin{array}{cc} \tilde{ K} & {\bf 0} \\ {\bf 0} & -\tilde{ M} \\ \end{array} \!\! \right]$ (36)

 $\left( {s{ C} + { D} } \right){ Z} = {\bf 0}$ (37)

2 动力学特性分析 2.1 弹性连接刚度影响

 \eqalign{ & \cr & \left. {\matrix{ {\delta = {R \over L},\quad \alpha = \sqrt {{{A{L^2}} \over I}} ,\quad \gamma = \Omega {L^2}\sqrt {{{\rho A} \over {EI}}} } \cr {f = \omega {L^2}\sqrt {{{\rho A} \over {EI}}} ,\quad \kappa = {{{k_{\rm{c}}}L} \over {EI}}} \cr } } \right\} \cr} (38)

 图 2 无量纲固有频率计算结果收敛性 Fig. 2 The convergence of dimensionless natural frequencies

 图 4 角速度比率对模态频率与振型的影响 Fig. 4 Variation of dimensionless frequencies and mode shapes with dimensionless rotating speed

 图 5 系统径长比对无量纲固有频率的影响 Fig. 5 Variation of dimensionless frequencies with ratio of hub's radius and length of beam

 图 6 梁的长细比对无量纲固有频率影响 Fig. 6 Variation of dimensionless frequencies with slenderness ratio of beam
3 结论

(1) Chebyshev谱方法应用于柔性旋转梁动力学分析结果与有限元及加权残余法结果基本一致，采用较少的多项式阶数即可获得与有限元方法相当的精度，采用投影矩阵方法可以方便地用同样形式处理固定及弹性边界条件.

(2) 出现频率转向与振型转换的原因是系统弯曲模态、拉伸模态随系统参数变化规律不一致，变化快的低价模态频率增大并超过原高阶模态频率.

(3) 系统弯曲模态频率随连接刚度呈阶梯状变化，拉伸模态频率不随连接刚度变化而变化，二者频率变化曲线交叉，出现频率转向与振型转换现象，为保证系统动力学特性稳定，连接刚度应大于关键刚度$\kappa _{\rm c}$.

(4) 系统弯曲模态频率随系统径长比增大而增大，拉伸模态频率不随径长比变化而变化，二者频率变化曲线交叉，出现频率转向与振型转换现象.

(5) 拉伸模态频率随梁的长细比的增大而增大，弯曲模态频率保持不变，二者频率变化曲线交叉，出现频率转向与振型转换现象.

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DYNAMIC ANALYSIS OF A ROTATING FLEXIBLE BEAM WITH ELASTIC BOUNDARY CONDITIONS
Huang Yixin, Tian Hao, Zhao Yang
School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
Abstract: The dynamic characteristics of a flexible beam rotating in a plane with elastic boundary condition are investigated by the Chebyshev spectral method. The discrete equations of motion are obtained based on Gauss-Lobatto sampling and Chebyshev polynomials. Employing projection matrices, fixed and elastic boundary conditions are incorporated in the same form. Numerical solutions of natural frequencies and mode shapes are gained by Chebyshev spectral method and compared with the results of finite element and weighted residual methods to verify its correctness. The effects of various parameters, such as connection sti ness, angular velocity, hub radius ratio and slenderness ratio of the beam, on the vibration of the beam are analyzed. The results show that there is a veering phenomenon of natural frequencies loci accompanied by exchanges of the corresponding mode shape, due to the difference in sensitivity to system parameters between bending mode and strength mode. With the increasing of connection sti ness, angular velocity and hub radius ratio, a lower bending mode frequency will surpass its adjacent higher strength mode frequency. Similarly, strength mode frequencies will also surpass their adjacent higher bending mode frequencies with the increasing of slenderness ratio of the beam.
Key words: rotating flexible beam    Chebyshev spectral method    rigid-flexible coupling    frequency veering    dynamic characteristics