﻿ 径向基点插值法在旋转柔性梁动力学中的应用
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 力学学报  2015, Vol. 47 Issue (2): 279-288  DOI: 10.6052/0459-1879-14-334 0

### 引用本文 [复制中英文]

[复制中文]
Du Chaofan, Zhang Dingguo, Hong Jiazhen. A MESHFREE METHOD BASED ON RADIAL POINT INTERPOLATION METHOD FOR THE DYNAMIC ANALYSIS OF ROTATING FLEXIBLE BEAMS[J]. Chinese Journal of Ship Research, 2015, 47(2): 279-288. DOI: 10.6052/0459-1879-14-334.
[复制英文]

### 文章历史

2014-10-24收稿
2014-11-25录用
2014–12–12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

1 旋转柔性梁动力学模型 1.1 系统的动能与势能

 图 1 旋转柔性梁变形示意图 Fig.1 The deformation of rotating flexible beam

 $r = \Theta ({r_A} + {\rho _0} + u)$ (1)

 ${ r}_{ A} = (a,0)^{\rm T}$ (2)
 ${\rho}_0 = (x,0)^{\rm T}$ (3)
 ${ u} = (u_x ,u_y )^{\rm T}$ (4)
 $\Theta = \left( \begin{array}{l} \cos \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \sin \theta \\ \sin \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cos \theta \end{array} \right)$ (5)

 $u = \left[\begin{array}{l} {u_x}\\ {u_y} \end{array} \right] = \left[\begin{array}{l} {w_1} + {w_{\rm{c}}}\\ {w_2} \end{array} \right]$ (6)

 ${w_{\rm{c}}} = - \frac{1}{2}\int {_0^x} {\left( {\frac{{\partial {w_2}}}{{\partial \zeta }}} \right)^2}{\rm{ d}}\zeta {\rm{ }}$ (7)

 $\dot r = \dot \Theta ({r_A} + {\rho _0} + u) + \Theta \dot u$ (8)

 $T = \frac{1}{2}{J_{{\rm{oh}}}}{\dot \theta ^2} + \frac{1}{2}\int {_{_V}} \rho {\dot r^{\rm{T}}}{\dot r^{\rm{T}}}\dot rdV$ (9)

 $U = \frac{1}{2}\int {_0^L} ES{\left( {\frac{{\partial {w_1}}}{{\partial x}}} \right)^2}{\rm{ d}}x + \frac{1}{2}\int {_0^L} EI{\left( {\frac{{{\partial ^2}{w_2}}}{{\partial {x^2}}}} \right)^2}{\rm{ d}}x$ (10)
1.2 无网格径向基点插值法离散

 $\begin{array}{l} u(x) = \sum\limits_{i = 1}^n {{R_i}(x){a_i}} + \sum\limits_{i = 1}^m {{p_i}(x){b_i}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R^{\rm{T}}}(x)a + {p^{\rm{T}}}(x)b \end{array}$ (11)

 $r = \sqrt {(x - x_i )^2}$ (12)
 $r = {\rm{ }}\sqrt {{{(x - {x_i})}^2} + {{(y - {y_i})}^2}}$ (13)

 $R_i ({ x}) = [r_i^2 + (\alpha _c d_c )^2]^q$ (14)

 ${ p}^{\rm T}({ x}) = [1 \ \ x],\qquad m = 2$ (15)
 ${ p}^{\rm T}({ x}) = [1 \ \ x \ \ y],\qquad m = 3$ (16)

 ${ p}^{\rm T}({ x}) = [ 1 \ \ x \ \ {x^2}] ,\qquad m = 3$ (17)
 ${ p}^{\rm T}({ x}) = [ 1 \ \ x \ \ y \ \ {x^2} \ \ {xy} \ \ {y^2}],\qquad m = 6$ (18)

 ${ p}^{\rm T}({ x}) = [1 \ \ x \ \ {x^2} \ \ \cdots \ \ {x^{p - 1}} \ \ {x^p}]$ (19)
 ${ p}^{\rm T}({ x}) = [1 \ \ x \ \ y \ \ {x^2} \ \ {xy} \ \ {y^2} \ \ \cdots \ \ {x^p} \ \ {y^p}]$ (20)

 $d_s = \alpha _s d_c$ (21)
 图 2 无网格法中计算点的支持域 Fig.2 The support domain of a point of interest

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} {u_1} = \sum\limits_{i = 1}^n {{R_i}({x_1}){a_i}} + \sum\limits_{i = 1}^m {{p_i}({x_1}){b_i}} \\ {u_2} = \sum\limits_{i = 1}^n {{R_i}({x_1}){a_i}} + \sum\limits_{i = 1}^m {{p_i}({x_1}){b_i}} \\ \vdots \\ {u_n} = \sum\limits_{i = 1}^n {{R_i}({x_n}){a_i}} + \sum\limits_{i = 1}^m {{p_i}({x_1}){b_i}} \end{array} \end{array}} \right\}$ (22)

 ${{\rm{U}}_{\rm{s}}} = {R_0}a + {P_m}b$ (23)

 ${u_1 } \ \ {u_2 } \ \ {u_3 } \ \ \cdots \ \ {u_n } \}^{\rm T}$ (24)

 ${R_0}{\rm{ = }}{\left[\begin{array}{l} {R_1}({r_1}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_2}({r_1}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_n}({r_1})\\ {R_1}({r_2})\;{R_2}({r_2})\;\; \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_n}({r_2})\\ \vdots \;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\; \ddots \;\;\;\;\; \vdots \;\\ {R_1}({r_n})\;{R_2}({r_n})\;\; \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_n}({r_n}) \end{array} \right]_{n \times n}}$ (25)

 ${ a} = \left\{ {a_1 } \ \ {a_2 } \ \ {a_3 } \ \ \cdots \ \ {a_n } \right\}^{\rm T}$ (26)

 ${p_m}{\rm{ = }}{\left[\begin{array}{l} {p_1}({x_1}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_2}({x_1}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_m}({x_1})\\ {p_1}({x_2})\;{p_2}({x_2})\;\; \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_m}({x_2})\\ \vdots \;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\; \ddots \;\;\;\;\; \vdots \;\\ {p_1}({x_n})\;{p_2}({x_n})\;\; \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_m}({x_n}) \end{array} \right]_{n \times m}}$ (27)

 ${ b} = \left\{ {b_1 } \ \ {b_2 } \ \ {b_3 } \ \ \cdots \ \ {b_m } \right\}^{\rm T}$ (28)

 $r_k = \sqrt {(x_k - x_i )^2}$ (29)
 ${r_k} = {\rm{ }}\sqrt {{{({x_k} - {x_i})}^2} + {{({y_k} - {y_i})}^2}}$ (30)

 $\sum\limits_{i = 1}^m {{p_j}({x_i}){a_i}} = P_m^{\rm{T}}a = 0,\;\;\;j = 1,2,\cdots ,m$ (31)

 ${\tilde U_s} = \left[{\begin{array}{*{20}{c}} \begin{array}{l} {U_s}\\ {\bf{0}} \end{array} \end{array}} \right] = \left[\begin{array}{l} {R_{0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {P_m}\\ P_m^T{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right]\left[{\begin{array}{*{20}{c}} \begin{array}{l} a\\ b \end{array} \end{array}} \right] = G{a_0}$ (32)

 ${a_0} = {G^{ - 1}}{\tilde U_s}$ (33)

 $u(x) = \left\{ {{R^{\rm{T}}}(x)\;\;{p^{\rm{T}}}(x)} \right\}{G^{ - 1}}{\tilde U_s} = {\rm{ }}{\tilde \Phi ^{\rm{T}}}(x){\tilde U_s}$ (34)

 ${\tilde \Phi ^{\rm{T}}}(x) = \left\{ {{\phi _1}(x)\;\;{\phi _2}(x)\;\; \cdots \;{\phi _{n + m}}(x)} \right\}{\rm{ }}$ (35)

 ${\Phi ^{\rm{T}}}(x) = \left\{ {{\phi _1}(x)\;\;{\phi _2}(x)\;\; \cdots \;{\phi _n}(x)} \right\}$ (36)

 ${\phi _i}(x = {x_j}) = \left\{ \begin{array}{l} 1,i = j,i,j = 1,2 \cdots ,n\\ 0,i \ne j,i,j = 1,2 \cdots ,n \end{array} \right.$ (37)

 $0 = x_1 ＜ x_2 ＜ \cdots ＜ x_N = L$ (38)
 图 3 无网格法中柔性梁的离散 Fig.3 The discrete of the flexible beam

 ${w_1} = \sum\limits_{i = 1}^n {{\Phi _{xi}}} (x){d_i}(t) = {\Phi _x}d\\ {w_2} = \sum\limits_{i = 1}^{2n} {{\Phi _{yi}}(x){d_i}(t) = {\Phi _y}d}$ (39)

 ${\Phi _{xi}}(x) = \left( {{\phi _{xi}}\;\;0\;\;0} \right){\rm{ }}\\ {\Phi _{yi}}(x) = \left( {0\;\;{\phi _{yi}}\;\;{\phi _{\theta i}}} \right){\rm{ }}\\ {d_i} = {\left( {{u_{xi}}\;\;{u_{yi}}} \right)^{\rm{T}}}$ (40)

 $u = \left[ {\begin{array}{*{20}{l}} \begin{array}{l} {u_x}\\ {u_y} \end{array} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} \begin{array}{l} {\Phi _x}d - \frac{1}{2}{d^{\rm{T}}}Hd\\ {\Phi _y}d \end{array} \end{array}} \right]$ (41)

 $H(x) = {\rm{ }}\int {_0^x} {\Phi ^\prime }_y^{\rm{T}}(\zeta ){\Phi ^\prime }_y(\zeta )\Phi \zeta {\rm{ }}$ (42)

 $\dot { u} = \left[\!\!\begin{array}{l} \dot {u}_x \\ \dot {u}_y \\ \end{array} \!\!\right] = \left[\!\!\begin{array}{l} {\Phi}_x \dot{ d} - { d}^{\rm T}{ H}\dot{ d} {\Phi}_y \dot{ d} \end{array} \!\!\right]$ (43)
1.3 系统的动力学方程

 $\frac{{\rm{d}}}{{dt}}\left( {\frac{{\partial T}}{{\partial \dot q}}} \right) - \frac{{\partial T}}{{\partial q}} = - \frac{{\partial U}}{{\partial q}} + {F_q}$ (44)

 $\left( \begin{array}{l} {M_{11}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {M_{12}}{\kern 1pt} \\ {M_{21}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {M_{22}} \end{array} \right)\left( {\begin{array}{*{20}{c}} \begin{array}{l} {\ddot \theta }\\ {\ddot d} \end{array} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} \begin{array}{l} {Q_\theta }\\ {Q_d} \end{array} \end{array}} \right)$ (45)

 $\begin{array}{l} {M_{11}} = {J_{{\rm{oh}}}} + {J_{{\rm{ob}}}} + 2{S_x}d + {d^{\rm{T}}}({M_1} + {M_2})d - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \underline {{d^{\rm{T}}}(aC + D){d^{\rm{T}}}} \end{array}$ (46)
 ${ M}_{21} = { M}_{12} ^{\rm T} = ({ M}_3 ^{\rm T} - { M}_3 ){ d}-{ S}_y ^{\rm T}$ (47)
 $\begin{array}{l} {M_{22}} = {M_1} + {M_2} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int {_0^L} \rho S\Phi _x^{\rm{T}}{\Phi _x}dx + \int {_0^L} \rho S\Phi _y^{\rm{T}}{\Phi _y}dx \end{array}$ (48)
 $\begin{array}{l} {Q_\theta } = \tau - 2\dot \theta [{S_x}\dot d + {d^{\rm{T}}}({M_1} + {M_2})\dot d] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \underline {2\dot \theta {d^{\rm{T}}}(aC + D)\dot d}] \end{array}$ (49)
 $\begin{array}{l} {Q_d} = {{\dot \theta }^2}S_x^{\rm{T}} + 2\dot \theta ({M_3} - M_3^{\rm{T}})\dot d + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [{{\dot \theta }^2}({M_{22}}\underline { - aC - D} ) - K]d \end{array}$ (50)

 $\int {_\Omega } G{\rm{d}}\Omega = {\rm{ }}\sum\limits_k^{{n_c}} {\int {_{\Omega k}} } GD\Omega$ (51)

 $\int {_\Omega } G{\rm{d}}\Omega = {\rm{ }}\sum\limits_k^{{n_c}} {\int {_{\Omega k}} } GD\Omega = \sum\limits_k^{{n_c}} {\sum\limits_{i = 1}^{{n_g}} {{{\mathop w\limits^\frown }_i}G({x_{Qi}})\left| {{J_{ik}}} \right|{\rm{ }}} }$ (52)

 $\begin{array}{l} {K_{IJ}} = \sum\limits_k^{{n_c}} {\sum\limits_{i = 1}^{{n_g}} {\underbrace {ES{{\mathop w\limits^\frown }_i}\phi _{xI}^{{\rm{,T}}}({x_{Qi}})\phi _{xJ}^,({x_{Qi}})|{J_{ik}}|}_{K_{IJ}^{ik}} + } } \qquad \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_k^{{n_c}} {\sum\limits_{i = 1}^{{n_g}} {\underbrace {EI{{\mathop w\limits^\frown }_i}\phi _{yI}^{{\rm{,,T}}}({x_{Qi}})\phi _{yJ}^{,,}({x_{Qi}})|{J_{il}}|}_{K_{IJ}^{il}}} } \end{array}$ (53)

 $K = \left[\begin{array}{l} {K_{11}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{12}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{1N}}\\ {K_{21}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{22}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{2N}}\\ {K_{N1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{N2}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_{NN}} \end{array} \right]$ (54)

 $\begin{array}{l} {S_y} = \sum\limits_k^{{n_c}} {\sum\limits_{i = 1}^{{n_g}} {\underbrace {\rho S{{\mathop w\limits^\frown }_i}\phi _{xI}^{'{\rm{T}}}(a + {x_{Qi}})\phi _{yI}^{\rm{T}}({x_{Qi}})|{J_{ik}}|}_{s_I^{ik}}} } = \qquad \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_k^{{n_c}} {\sum\limits_{i = 1}^{{n_g}} {S_l^{ik}} } ,\;\;\;I = 1,2, \cdots ,N \end{array}$ (55)

2 大范围运动已知的动力学仿真

 ${M_{22}}\ddot d = {Q_d} - {M_{21}}\ddot \theta$ (56)

 $\dot \theta = \left\{ \begin{array}{l} \frac{{{\Omega _0}}}{T}t - \frac{{{\Omega _0}}}{{2\pi }}\sin \left( {\frac{{2\pi }}{T}t} \right),0 \le t \le T\\ {\Omega _0}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t{\rm{ > }}T \end{array} \right.$ (57)

 图 4 柔性梁末端纵向变形($\Omega _0 = 4$rad/s) Fig.4 Tip deformation of the beam along $x$ direction ($\Omega _0 = 4$rad/s)
 图 5 柔性梁末端横向变形($\Omega _0 = 4$rad/s) Fig.5 Tip deformation of the beam along $y$ direction ($\Omega _0 = 4$rad/s)
 图 6 柔性梁末端横向变形速度($\Omega _0 = 4$rad/s) Fig.6 Tip deformation rate of the beam along $y$ direction
 图 7 柔性梁末端纵向变形($\Omega _0 = 20$rad/s) Fig.7 Tip deformation of the beam along $x$ direction
 图 8 柔性梁末端横向变形($\Omega _0 = 20$rad/s) Fig.8 Tip deformation of the beam along $y$ direction
 图 9 柔性梁末端横向变形速度($\Omega _0 = 20$rad/s) Fig.9 Tip deformation rate of the beam along $y$ direction
 图 10 柔性梁末端横向变形($\Omega _0 = 4$rad/s, $E=5$GPa) Fig.10 Tip deformation of the beam along $y$ direction

3 3种时间积分方法的对比

4 径向基点插值法形状参数的影响

 图 11 径向基点插值法形状参数对仿真结果的影响 Fig.11 The influence of shape parameter of RPIM on results
5 结 论

(1)模态截断数取3阶的假设模态法计算效率最高，模态截断数越多，计算效率越低；径向基点插值法比有限元法计算效率稍高. 随着 变形的增大，假设模态法精度降低，即使增加模态阶数，精度也不会明显提高，但计算效率会降低明显，而径向基点插值法与有限元法 仿真结果基本一致.

(2)在处理大变形问题的研究中，假设模态法结果发散，不能用于求解大变形问题. 径向基点插值法与有限元法结果收敛且基本一致，说 明这两种方法适用于求解大变形问题.

(3)求解动力学方程时，隐式方法纽马克方法效率最高，4阶龙格库塔法次之，而亚当姆斯预报校正法最低. 因此，对于柔性悬臂梁动力 学方程的求解优先选用隐式方法.

(4)径向基点插值法形状参数$\alpha _c =1$时，误差很大，当$\alpha _c$取值超过7时，将会导致径向基点插值法力矩矩阵奇异而计算失败. $\alpha _c$取$2 \sim 7$可得到令人满意的结果.

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A MESHFREE METHOD BASED ON RADIAL POINT INTERPOLATION METHOD FOR THE DYNAMIC ANALYSIS OF ROTATING FLEXIBLE BEAMS
Du Chaofan, Zhang Dingguo, Hong Jiazhen
1. School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China;
2. Dept. of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China
Fund: The project was supported by the National Natural Science Foundation of China (11272155，11132007), the 333 Project of Jiangsu Province, China (BRA2011172) , and the Fundamental Research Funds for the Central Universities of China (30920130112009).
Abstract: A meshfree method based on radial point interpolation method (RPIM) is proposed for dynamic analysis of a rotating flexible beam. The RPIM is used to describe the deformation of the flexible beam. The longitudinal and transverse deformations of the beam are both considered, and the coupling term of the deformation which is caused by the transverse deformation is included in the longitudinal deformation of the beam. The rigid-flexible coupling dynamic equations of the system are derived via employing Lagrange's equations of the second kind. Simulation results of the RPIM are compared with those obtained by using finite element method (FEM) and assumed modes method and show the limitations of assumed modes method. It is demonstrated that the meshfree method as a discrete method of the flexible body can be extended in the field of multibody system dynamics. Meanwhile, the influence of the radial basis shape parameters is discussed. What's more, three integration methods (Newmark method, fourth-order Runge—Kutta method and Adams prediction correction method) are used to solve the dynamic equations, and it shows that the Newmark method is the fastest with the computational effciency.
Key words: flexible beam    radial point interpolation method    meshfree method    dynamics    integration methods