﻿ 应变能中耦合项对旋转悬臂梁振动频率的影响
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 力学学报  2015, Vol. 47 Issue (2): 362-366  DOI: 10.6052/0459-1879-14-207 0

### 引用本文 [复制中英文]

[复制中文]
Zhao Guowei, Wu Zhigang. EFFECTS OF COUPLING TERMS IN STRAIN ENERGY ON FREQUENCY OF A ROTATING CANTILEVER BEAM[J]. Chinese Journal of Ship Research, 2015, 47(2): 362-366. DOI: 10.6052/0459-1879-14-207.
[复制英文]

### 文章历史

2014-09-18 收稿
2014-10-31 录用
2014–11–02 网络版发表

1. 大连理工大学, 工业装备结构分析国家重点实验室, 大连116024;
2. 大连理工大学, 航空航天学院, 大连116024

1 运动学描述 1.1 浮动坐标系

 图 1 平面梁的位移描述 Fig. 1 Displacement description of planar beam
 $r = B\left( {{\rho _0} + u} \right){\mkern 1mu} ,\;\;\;B = \left[\begin{array}{l} \cos \theta \;\;\;\;\; - \sin \theta \\ \sin \theta \;\;\;\;\;\;\;\cos \theta \; \end{array} \right]$ (1)

 $v = \dot r = \dot B\left( {{\rho _0} + u} \right) + B\mathop u\limits^ \circ$ (2)

1.2 变形描述

 $\begin{array}{l} u = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {u_1}\\ {u_2} \end{array} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {w_1} + {w_{{\rm{1r}}}} + {w_{\rm{c}}}\\ {w_2} + {w_{{\rm{2r}}}} \end{array} \end{array}} \right\} = \\ \;\;\;\;\;\left\{ {\begin{array}{*{20}{c}} {{w_1} - y\frac{{\partial {w_2}}}{{\partial x}} - \frac{1}{2}{{\int_0^x {\left( {\frac{{\partial {w_2}}}{{\partial \xi }}} \right)} }^2}d\xi {w_2} - \frac{1}{2}y{{\left( {\frac{{\partial {w_2}}}{{\partial x}}} \right)}^2}} \end{array}} \right\} \end{array}$ (3)

2 刚柔耦合动力学方程

 $T = \frac{1}{2}\int_V \gamma {\dot r^{\rm{T}}}\dot rdV$ (4)

 ${\varepsilon _{xx}} = u_{\rm{1}}^, + \frac{1}{2}\left[ {u_1^{,2} + u_2^{,2}} \right]$ (5)

 $\begin{array}{*{20}{l}} {U = \frac{1}{2}\int_V E \varepsilon _{xx}^2dV = }\\ {\;\;\;\;\;\frac{1}{2}\int_L {} \left[ {EA\left( {\underline {w_1^{,2}} + w_1^{,3} + \frac{1}{4}w_1^{,4} + \frac{1}{{64}}w_2^{,8} + \frac{1}{4}w_1^{,2}w_2^{,4}} \right.} \right.}\\ {\;\;\;\; + \left. {\frac{1}{4}w_1^,w_2^{,4} - w_1^{,2}w_2^{,2} + \frac{1}{8}w_1^{,2}w_2^{,4} - \frac{1}{2}w_1^{,3}w_2^{,2} - \frac{1}{8}w_1^,w_2^{,6}} \right) + }\\ {\;\;\;\;EI\left( {\underline {w_2^{,,2}} + 3w_1^,w_2^{,,2} + \frac{3}{2}w_1^{,2}w_2^{,,2} - w_2^{.2}w_2^{,,2} + \frac{3}{8}w_2^{,4}w_2^{,,2}} \right. - }\\ {\;\;\;\left. {\left. {\frac{3}{2}w_1^,w_2^{,2}w_2^{,,2}} \right)} \right]dx + \frac{1}{2}\int_V E \frac{1}{4}{y^4}w_2^{,,4}dV} \end{array}$ (6)

 $\begin{array}{*{20}{l}} {\gamma A\left[ {{{\ddot w}_1} - {{\dot \theta }^2}\left( {x + {w_1}} \right) - 2\dot \theta {{\dot w}_2} - \ddot \theta {w_2}} \right] - }\\ {\;\;\;\;EA\left( {w_1^{,,} - w_2^{,2}w_1^{,,} + \frac{3}{8}w_2^{,4}w{{_1^{,,}}_1}} \right) - EI(\frac{3}{2}w_2^{,,2}w_1^{,,}) = 0} \end{array}$ (7)
 $\begin{array}{*{20}{l}} {\gamma A({{\ddot w}_2} - {{\dot \theta }^2}{w_2} + {{\dot \theta }^2}{w_1}{w_2} + {{\dot \theta }^2}xw_2^, - }\\ {{{\dot \theta }^2}\frac{{{L^2} - {x^2}}}{2}w_2^{,,}) + \gamma I\left( {{{\dot \theta }^2}w_2^{,,} - \ddot w_2^{,,}} \right) + }\\ {6w_1^{,,}w_2^{,,,} + 6w_1^,w_1^{,,}w_2^{,,,} + 3w_1^{,,2}w_2^{,,} + 3w_1^{,,,}w_2^{,,} + }\\ {3w_1^,w_1^{,,,}w{{_2^{,,}}_2}) + EA(w_1^{,2}w_2^{,,} + \frac{1}{2}w_1^{,3}w_2^{,,} + }\\ {2w_1^,w_1^{,.}w_2^, + \frac{3}{2}w_1^{,2}w_1^{,,}w_2^,) = 0} \end{array}$ (8)

 $\gamma {\dot \theta ^2}x + E{w''_1} = 0$ (9)

 ${w_1}\left( x \right) = \frac{{\gamma {{\dot \theta }^2}}}{{6E}}\left( {3{L^2}x - {x^3}} \right)$ (10)

 $\left( {K - {\omega ^2}M} \right)h = 0$ (11)

 $\begin{array}{*{20}{l}} {K = \int_L {} [\gamma A{{\dot \theta }^2}({w_1}{\beta ^{\rm{T}}}{\beta ^,} - {\beta ^{\rm{T}}}\beta + x{\beta ^{\rm{T}}}{\beta ^,} - }\\ {\;\;\;\;\;\qquad \frac{{{L^2} - {x^2}}}{2}{\beta ^{\rm{T}}}{\beta ^{,,}}) + \gamma I{{\dot \theta }^2}{\beta ^{\rm{T}}}{\beta ^{,,}} + }\\ {\;\;\;\;\;\;\;EI\left( {{\beta ^{\rm{T}}}{\beta ^{,,,,}} + } \right.\underline {3w_1^{,,}{\beta ^{\rm{T}}}{\beta ^{,,,,}}} + \underline {\frac{3}{2}w_1^{,,2}{\beta ^{\rm{T}}}{\beta ^{,,,,}}} + }\\ {\;\;\qquad \underline {6w_1^{,,}{\beta ^{\rm{T}}}{\beta ^{,,,}} + 6w_1^,w_1^{,,}{\beta ^{\rm{T}}}{\beta ^{,,,}} + 3w_1^{,,2}{\beta ^{\rm{T}}}{\beta ^{,,}}} + }\\ {\qquad \underline {3w_1^{,,,}{\beta ^{\rm{T}}}{\beta ^{,,}} + 3w_1^,w_1^{,,,}{\beta ^{\rm{T}}}{\beta ^{,,}}} ) + EA\left( {\underline {w_1^{,2}{\beta ^{\rm{T}}}{\beta ^{,,}}} + } \right.}\\ {\;\;\;\;\;\;\left. {\left. {\underline {\frac{1}{2}w_1^{,3}{\beta ^{\rm{T}}}{\beta ^{,,}}} + \underline {2w_1^,w_1^{,,}{\beta ^{\rm{T}}}{\beta ^,}} + \underline {\frac{3}{2}w_1^{,2}w_1^{,,}{\beta ^{\rm{T}}}{\beta ^,}} } \right)} \right]dx}\\ {\;\;M = \int_L {\left( {\gamma A{\beta ^{\rm{T}}}\beta - \gamma I{\beta ^{\rm{T}}}{\beta ^{,,}}} \right)dx} } \end{array}$

3 动力学特性分析

 图 2 一阶弯曲固有频率随转速变化 Fig. 2 Angular velocity and bending natural frequencies
4 结 论

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EFFECTS OF COUPLING TERMS IN STRAIN ENERGY ON FREQUENCY OF A ROTATING CANTILEVER BEAM
Zhao Guowei, Wu Zhigang
1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China;
2. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China
Fund: The project was supported by the National Natural Science Foundation of China (11072044), the Research Fund for the Doctoral Program of Higher Education (20110041130001), and the Program for New Century Excellent Talents in University (NCET-11-0054).
Abstract: Dynamic modeling of a rotating cantilever beam is very important for dynamic characteristics analysis and controller design. In literature, the coupling terms between axial and transverse deformation in strain energy are ignored in first order approximate model. However, as shown by this paper, these terms have a significant impact on the dynamic characteristics. By discussing how to select the strain energy, this paper takes into account the coupling terms. Based on Hamilton's principle, the coupling vibration equations of the rotating cantilever beam are obtained. Using Rayleigh- Ritz method, the bending mode shapes of the beam without rotational motion are selected as basic functions to derive characteristic equation. A numerical example is presented to show that the evident difference of bending frequencies between the models with and without the coupling terms in strain energy.
Key words: cantilever beam    rigid-flexible coupling    floating reference frame    Rayleigh—Ritz method