﻿ 光滑节点插值法:计算固有频率下界值的新方法
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 力学学报  2015, Vol. 47 Issue (5): 839-847  DOI: 10.6052/0459-1879-15-146 0

### 引用本文 [复制中英文]

[复制中文]
Du Chaofan, Zhang Dingguo. NODE-BASED SMOOTHED POINT INTERPOLATION METHOD: A NEW METHOD FOR COMPUTING LOWER BOUND OF NATURAL FREQUENCY[J]. Chinese Journal of Ship Research, 2015, 47(5): 839-847. DOI: 10.6052/0459-1879-15-146.
[复制英文]

### 文章历史

2015-05-12 收稿
2015-07-09 录用
2015–07–09 网络版发表

1 静力学分析 1.1 欧拉-伯努利梁的弱式方程

 $EId\frac{{{d^4}v}}{{d{x^4}}} = {b_y}{\rm{ }}$ (1)

 $v({x_0}) = {v_\Gamma }在边界{\Gamma _v}上$ (2)
 $- \frac{{dv({x_0})}}{{dx}} = {\theta _\Gamma }{\mkern 1mu} ,\;\;在边界{\Gamma _\theta }上$ (3)
 $M({x_0}) = - EI\frac{{{d^2}v}}{{{\rm{d}}{x^2}}} = {M_\Gamma }_M{\mkern 1mu} ,\;\;在边界{\Gamma _M}上$ (4)
 $Q({x_0}) = - EI\frac{{{d^3}v}}{{{\rm{d}}{x^3}}} = {Q_\Gamma }{\mkern 1mu} ,\;\;在边界{\Gamma _Q}上$ (5)

 $\begin{array}{l} \int_\Omega {EI} \frac{{{d^2}(\delta v)}}{{d{x^2}}}\frac{{{d^2}v}}{{d{x^2}}}dx = \\ \int_\Omega {\delta v{b_y}} dx + {Q_\Gamma }\delta v{|_{{\Gamma _Q}}} - {M_\Gamma }\frac{{(\delta v)}}{{dx}}{|_{{\Gamma _M}}} \end{array}$ (6)

1.2 迦辽金方程的弱-弱形式

 图 1 柔性梁的离散及光滑域的形成 Fig. 1 The discrete of the flexible beam and the form of smoothing domain

 $\frac{{\overline {{d^2}v} }}{{d{x^2}}} = \frac{1}{{l_i^s}}\int_{\Gamma _i^S} {} \left( {n(x)\frac{{dv}}{{dx}}} \right)dx$ (7)

 $\begin{array}{l} \sum\limits_{i = 1}^{{N_n}} {\frac{1}{{l_i^s}}} \left[{\int_{\Gamma _i^s} {\left( {n(x)\frac{{d\delta v}}{{dx}}} \right)} dx} \right]\left[{EI\int_{\Gamma _i^s} {\left( {n(x)\frac{{d\delta v}}{{dx}}} \right)} } \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\int_\Omega \delta v{b_y}dx + {\left. {{Q_\Gamma }\delta v} \right|_{{\Gamma _Q}}} - {M_\Gamma }{\left. {\frac{{d\delta v}}{{dx}}} \right|_{{\Gamma _M}}} \end{array}$ (8)
1.3 光滑节点插值法公式

 $v(x) = \left\{ {{\phi _n}(x)\;\;{\phi _{n + 1}}(x)} \right\}\left\{ {\begin{array}{*{20}{c}} {{v_n}}\\ {{v_{n + 1}}} \end{array}} \right\}{\mkern 1mu} ,\;\;x \in \Omega _n^c$ (9)

 $\left. {\begin{array}{*{20}{c}} {{\phi _n}(x) = 1 - (x - {x_n})/l_n^c}\\ {{\phi _{n + 1}}(x) = (x - {x_n})/l_n^cn} \end{array},x \in \Omega _n^c} \right\}$ (10)

 $\begin{array}{l} \frac{1}{{l_i^s}}\int_{\Gamma _i^s} {\left( {n(x)\frac{{dv}}{{dx}}} \right)dx} = \frac{1}{{l_i^s}}\underbrace {{{\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} - {\phi _{(n - 1),x}}({x_{n - 1/2}})\\ \left( {{\phi _{n,x}}({x_{n + 1/2}}) - {\phi _{n,x}}({x_{n - 1/2}})} \right)\\ {\phi _{(n + 1),x}}({x_{n + 1/2}}) \end{array} \end{array}} \right\}}^{\rm{T}}}}_{{B_n}}\underbrace {\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {v_{n - 1}}\\ {v_n}\\ {v_{n + 1}} \end{array} \end{array}} \right\}}_{{d_n}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;{B_n}{d_n}{\mkern 1mu} ,\;\;\;\;\;n = 2,3 \cdots ,{N_n} - 1 \end{array}$ (11)

 $\frac{1}{{l_1^s}}\int_{\Gamma _l^s} {\left( {n(x)\frac{{dv}}{{dx}}} \right)dx} = \frac{1}{{l_1^s}}\underbrace {{{\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \left( {{\phi _{1,x}}({x_{1 + 1/2}}) - {\phi _{1,x}}({x_1})} \right)\\ \left( {{\phi _{2,x}}({x_{1 + 1/2}}) - {\phi _{2,x}}({x_1})} \right) \end{array} \end{array}} \right\}}^{\rm{T}}}}_{{B_1}}\underbrace {\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {v_1}\\ {v_2} \end{array} \end{array}} \right\}}_{{d_1}} = {B_1}{d_1}$ (12)

 $\begin{array}{l} \frac{1}{{l_{{N_n}}^s}}\int_{\Gamma _{Nn}^s} {\left( {n(x)\frac{{dv}}{{dx}}} \right)dx} = \\ \frac{1}{{l_{{N_n}}^s}}\underbrace {{{\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\phi _{({N_n} - 1),x}}({x_{{N_n}}}) - {\phi _{({N_n} - 1),x}}({x_{{N_n} - 1/2}})\\ {\phi _{{N_n},x}}({x_{{N_n}}}) - {\phi _{{N_n},x}}({x_{{N_n} - 1/2}}) \end{array} \end{array}} \right\}}^{\rm{T}}}}_{{B_{{N_n}}}}\underbrace {\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {v_{{N_n} - 1}}\\ {v_{{N_n}}} \end{array} \end{array}} \right\}}_{{d_{{N_n}}}} = \\ \;\;\;\;\;\;\;\;\;\;\;{B_{{N_n}}}{d_{{N_n}}} \end{array}$ (13)

 $KU{\rm{ = }}F$ (14)

 $\begin{array}{l} F = {\int_\Omega \Phi ^{\rm{T}}}(x){b_y}dx{ + ^{\rm{T}}}(x){Q_\Gamma }d\Gamma - \\ \;\;\;\;\;\;\int_{{\Gamma _M}} \Phi _{,x}^{\rm{T}}(x){M_\Gamma }d\Gamma \end{array}$ (15)

 $K = \sum\limits_{n = 1}^{{N_n}} {{K_n}}$ (16)

 ${K_n} = EI{({B_n})^{\rm{T}}}{B_n}{l_n}$ (17)
1.4 数值算例

 图 2 受分布力与集中力作用的悬臂梁 Fig. 2 Cantilever beam with distributed and concentrated load
 图 3 悬臂梁中各点挠度 Fig. 3 The deflection of cantilever beam
 图 4 梁末端挠度随模态阶数的变化 Fig. 4 Tip deflection with different modes
 图 5 应变能的变化 Fig. 5 The change of strain energy
2 频率分析 2.1 悬臂梁的自由振动

 $EI\frac{{{d^4}v}}{{d{x^4}}} + \rho S\frac{{{d^2}v}}{{d{x^2}}} = 0$ (18)

 $\int_\Omega {EI} \frac{{{d^2}(\delta v)}}{{d{x^2}}}\frac{{{d^2}v}}{{d{x^2}}}dx + {\rm{ }}\int_\Omega \rho S\delta v\frac{{{d^2}v}}{{d{t^2}}}dx = 0$ (19)

 $\begin{array}{l} \sum\limits_{i = 1}^{{N_n}} {\frac{1}{{l_i^s}}} \left[{\int_{\Gamma _i^s} {\left( {n(x)\frac{{d\delta v}}{{dx}}} \right)dx} } \right]\left[{EI\int_{\Gamma _i^s} {\left( {n(x)\frac{{dv}}{{dx}}} \right)dx} } \right] + \\ \;\;\;\;\;\;\;\int_\Omega \rho S\delta v\frac{{{d^2}v}}{{d{t^2}}}dx = 0 \end{array}$ (20)

 $KU + M\ddot U = {\bf{0}}$ (21)

 $M = {\rm{diag}}\left( {{m_1}\;\;{m_2}\;\; \cdots \;\;{m_{{N_n}}}} \right)$ (22)

 ${m_n} = \rho Sl_n^s$ (23)

2.2 柔性旋转悬臂梁的横向弯曲固有频率分析

 图 6 旋转柔性梁物理模型 Fig. 6 The model of rotating flexible beam

 $r = \Theta ({\rho _0} + u)$ (24)

 $\left. {\begin{array}{*{20}{c}} {{\rho _0} = {{\left\{ {x,0} \right\}}^{\rm{T}}}}\\ {u = {{\left\{ {{u_x},{u_y}} \right\}}^{\rm{T}}}}\\ {\Theta = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - \sin \theta }\\ {\sin \theta } \end{array}}&\begin{array}{l} \cos \theta \\ \cos \theta \end{array}&\begin{array}{l} \\ \end{array} \end{array}} \right\}} \end{array}} \right\}$ (25)

 $u = \left\{ {\begin{array}{*{20}{l}} {{u_x}}\\ {{u_y}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{l}} {{w_1} + {w_{\rm{c}}}}\\ {{w_2}} \end{array}} \right\}$ (26)

 ${w_{\rm{c}}} = - \frac{1}{2}\int_0^x {{{\left( {\frac{{\partial {w_2}}}{{\partial \zeta }}} \right)}^2}} d\zeta$ (27)

 $T = \frac{1}{2}\int_V \rho {\dot r^{\rm{T}}}\dot rdV$ (28)

 $\begin{array}{l} U = \frac{1}{2}\int_0^L E S{\left( {\frac{{\partial {w_1}}}{{\partial x}}} \right)^2}dx + \\ \;\;\;\;\frac{1}{2}\int_0^L E I{\left( {\frac{{{\partial ^2}{w_2}}}{{\partial {x^2}}}} \right)^2}dx \end{array}$ (29)

 $\left. {\begin{array}{*{20}{c}} {{w_1} = \left\{ {{\phi _i}(x)\;\;{\phi _{i + 1}}(x)} \right\}\left\{ {\begin{array}{*{20}{c}} {{u_{xi}}}\\ {{u_{x(i + 1)}}} \end{array}} \right\} = }\\ {\Theta (x){A^i}(t)}\\ {{w_2} = \left\{ {{\phi _i}(x)\;\;{\phi _{i + 1}}(x)} \right\}\left\{ {\begin{array}{*{20}{c}} {{u_{yi}}}\\ {{u_{y(i + 1)}}} \end{array}} \right\} = }\\ {\begin{array}{*{20}{l}} {{\Theta ^i}(x){B^i}(t)} \end{array}{\mkern 1mu} } \end{array},\;x \in \Omega _i^c} \right\}$ (30)

 $\left. {\begin{array}{*{20}{c}} {A = {{\left\{ {{u_{x1}}\;\;{u_{x2}}\;\; \cdots \;\;{u_{x{N_n}}}} \right\}}^{\rm{T}}}}\\ {B = {{\left\{ {{u_{y1}}\;\;{u_{y2}}\;\; \cdots \;\;{u_{y{N_n}}}} \right\}}^{\rm{T}}}} \end{array}} \right\}$ (31)

 ${A^i} = {R^i}A{\mkern 1mu} ,\;\;\;\;{B^i} = {R^i}B$ (32)

 ${R^i}\left. { = \left\{ \begin{array}{l} 0\;\;\;\;\;0\;\;\;\; \cdots \;\;\;1\;\;\;\;0\;\;\;\; \cdots \;\;\;{\rm{0}}\\ {\rm{0}}\;\;\;\;\;{\rm{0}}\;\;\;\; \cdots \;\;\;{\rm{1}}\;\;\;\;{\rm{0}}\;\;\;\; \cdots \;\;\;0 \end{array} \right.} \right\}$ (33)

 $\left. {\begin{array}{*{20}{c}} {{w_1} = \Phi (x)A(t)}\\ {{w_2} = \Phi (x)B(t)} \end{array},\;\;\;x \in \Omega _i^c} \right\}$ (34)

 $u = \left\{ {\begin{array}{*{20}{l}} {{u_x}}\\ {{u_y}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \Phi (x)A(t) - \frac{1}{2}{B^{\rm{T}}}(t)H(x)B(t)\\ \Phi (x)B(t) \end{array} \end{array}} \right\}$ (35)

 $H(x) = {R^{i{\rm{T}}}}{\int_0^x \Phi ^{i'{\rm{T}}}}(\zeta ){\Phi ^{i'}}(\zeta )d\zeta {R^i}$ (36)

 $\frac{d}{{dt}}\left( {\frac{{\partial T}}{{\partial \mathop q\limits^. }}} \right) - \frac{{\partial T}}{{\partial q}} = - \frac{{\partial U}}{{\partial q}} + {F_q}$ (37)

 $\left. {\left\{ \begin{array}{l} {M_{11}}\;\;\;\;{M_{12}}\;\;\;\;\;{M_{13}}\;\;\\ {M_{21}}\;\;\;\;{M_{22}}\;\;\;\;\;0\\ M31\;\;\;\;0\;\;\;\;\;\;\;\;{M_{33}}\; \end{array} \right.} \right\}\left\{ \begin{array}{l} {\ddot \theta }\\ \mathop A\limits^{..} \\ \mathop B\limits^{..} \end{array} \right\} = \left\{ \begin{array}{l} {Q_\theta }\\ {Q_A}\\ {Q_B} \end{array} \right\}$ (38)

 $\begin{array}{l} {M_{11}} = \frac{1}{3}\rho S{L^3} + 2SA + {A^{\rm{T}}}MA + \\ \;\;\;\;\;\;\;\;\;{B^{\rm{T}}}MB\underline { - {B^{\rm{T}}}DB} \end{array}$ (39)
 ${M_{21}} = M_{12}^{\rm{T}} = - MB$ (40)
 ${M_{31}} = M_{13}^{\rm{T}} = - {S^{\rm{T}}} + {M^{\rm{T}}}A$ (41)
 ${M_{22}} = {M_{33}} = M$ (42)
 ${Q_\theta } = \tau - 2\dot \theta [S\mathop A\limits^. + {A^{\rm{T}}}M\mathop A\limits^. + {B^{\rm{T}}}M\mathop B\limits^. \underline { - {B^{\rm{T}}}D\mathop B\limits^. }]{\rm{ }}$ (43)
 ${Q_A} = {\dot \theta ^2}{S^{\rm{T}}} + 2\dot \theta M\mathop B\limits^. + ({\dot \theta ^2}M - {K_1})A$ (44)
 ${Q_B} = {\dot \theta ^2}(M\underline { - D} )B - 2\dot \theta M\mathop A\limits^. - {K_2}B$ (45)
 ${K_1} = ES{\int_0^L \Phi ^\prime }^{\rm{T}}{\Phi ^\prime }dx$ (46)
 ${K_2} = ES{\int_0^L \Phi ^{\prime \prime }}^{\rm{T}}{\Phi ^{\prime \prime }}dx$ (47)

 ${M_{33}}\ddot B + [{\dot \theta ^2}(D - {M_{33}}) + {K_2}]B = {\bf{0}}$ (48)

 $\varsigma = t/T{\mkern 1mu} ,\;\xi = x{\rm{ / }}L{\mkern 1mu} ,{\rm{ }}{\kappa _2} = B/L{\mkern 1mu} ,\;\gamma = T\dot \theta$ (49)

 图 7 不同转速下前三阶固有频率比较 Fig. 7 The natural frequencies with different angular velocity ratio
3 结 论

(1)光滑节点插值法通过光滑梯度处理，可降低形函数对连续性的要求. 对于4阶微分方程，使用简单的线性插值即可. 相对有限元，该方法结构更简单，且在同等离散条件下自由度数减少一半.

(2)光滑节点插值法虽使用线性插值，但仍有很高的精度，且能提供应变能的上界值. 假设模态法提供应变能的下界值，进一步研 究发现，通常认为的模态阶数取的越多结果越精确的说法并不成立，当模态阶数大于9时结果将发散. 对于欧拉-伯努利梁取3$\sim$6阶模态即可.

(3)在频率分析中，通过与解析解的对比，发现光滑节点插值法的重要特性：能提供固有频率的下界值，而有限元法和假设模态 法只能提供固有频率的上界值. 在柔性旋转悬臂梁系统中，随着转速的增大，假设模态法有发散的趋势，已不适用. 对于如柔性旋转梁等不能求解析解的问题，有限元法结合光滑节点插值法可以从固有频率的上下界最大程度地逼近真实解，从而达 到提高精度的目的，在柔性多体系统动力学中具有重要的应用价值.

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NODE-BASED SMOOTHED POINT INTERPOLATION METHOD: A NEW METHOD FOR COMPUTING LOWER BOUND OF NATURAL FREQUENCY
Du Chaofan, Zhang Dingguo
School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, 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 called node-based smoothed point interpolation method (NS-PIM) is proposed for static analysis of cantilever beam and dynamic analysis of rotating flexible beam for the first time. Gradient smoothing technique is utilized to perform the numerical integration required in the weakened weak (W2) form formulation. The shape functions are approximated using linear interpolation functions, which can be used to solve the 4th order differential equation. In static problems, the cantilever beams with two loading conditions are analyzed, and the results are compared with the analytic solution, which shows a high accuracy of this method even if using linear shape functions. A further study shows that if more than 9 modes were used in the assumed mode method, the result will be divergent. In dynamic problem, the natural frequencies of a rotating flexible beam are analyzed. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed modes method (AMM). It is found that NS-PIM can provide unique lower bounds of natural frequencies, while FEM and AMM can provide upper bounds of natural frequencies. That means we can get more accurate results for the problems by using FEM and NS-PIM in case that exact solution can't be obtained. The NS-PIM has easier shape functions and less independent variable than FEM, and can provide lower bounds of natural frequencies, with a great value of application and dissemination.
Key words: flexible beam    node-based smoothed point interpolation method    meshfree method    dynamics    lower bound of natural frequency