﻿ 环形桁架结构径向振动的等效圆环模型
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 力学学报  2016, Vol. 48 Issue (5): 1184-1191  DOI: 10.6052/0459-1879-16-076 0

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Liu Fushou , Jin Dongping . EQUIVALENT CIRCULAR RING MODEL FOR THE RADIAL VIBRATION ANALYSIS OF HOOP TRUSS STRUCTURES[J]. Chinese Journal of Ship Research, 2016, 48(5): 1184-1191. DOI: 10.6052/0459-1879-16-076.
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文章历史

2016-03-22 收稿
2016-06-01 录用
2016-06-08网络版发表

1 环形桁架结构等效圆环模型

 图 1 环形桁架结构及其坐标系 Figure 1 Hoop truss structure and its coordinate system
 图 2 等效圆环模型及其坐标系 Figure 2 Equivalent ring model and its coordinate system

 $\frac{EA}{{{R}^{2}}}\left( \frac{{{\partial }^{2}}{{u}_{x}}}{\partial {{\theta }^{2}}}-\frac{\partial {{u}_{y}}}{\partial \theta } \right)+\frac{E{{I}_{z}}}{{{R}^{4}}}\left( \frac{{{\partial }^{2}}{{u}_{x}}}{\partial {{\theta }^{2}}}+\frac{{{\partial }^{3}}{{u}_{y}}}{\partial {{\theta }^{3}}} \right)+{{q}_{x}}=\rho A\frac{{{\partial }^{2}}{{u}_{x}}}{\partial {{t}^{2}}}$ (1a)
 $\frac{EA}{{{R}^{2}}}\left( \frac{\partial {{u}_{x}}}{\partial \theta }-{{u}_{y}} \right)-\frac{E{{I}_{z}}}{{{R}^{4}}}\left( \frac{{{\partial }^{3}}{{u}_{x}}}{\partial {{\theta }^{3}}}+\frac{{{\partial }^{4}}{{u}_{y}}}{\partial {{\theta }^{4}}} \right)+{{q}_{y}}=\rho A\frac{{{\partial }^{2}}{{u}_{y}}}{\partial {{t}^{2}}}$ (1b)

 ${{q}_{x}}=\sum\limits_{i=1}^{M}{\frac{{{q}_{xi}}(t)}{R}\delta (\theta -\theta _{i}^{*})}$ (2a)
 ${{q}_{y}}=\sum\limits_{j=1}^{N}{\frac{{{q}_{yj}}(t)}{R}\delta (\theta -\theta _{j}^{*})}$ (2b)

 ${{\varphi }_{z}}=\frac{{{u}_{x}}}{R}+\frac{1}{R}\frac{\partial {{u}_{y}}}{\partial \theta }$ (3)

 ${{\varepsilon }_{x}}=\frac{\partial {{u}_{x}}}{R\partial \theta }-\frac{{{u}_{y}}}{R},{{\kappa }_{z}}=\frac{\partial {{\phi }_{z}}}{\partial \theta }$ (4)

 $u_x (0,t) = u_x (2\pi,t) = 0$ (5a)
 $u_y (0,t) = u_y (2\pi,t) = 0$ (5b)
 $\varphi _z (0,t) = \varphi _z (2\pi,t) = 0$ (5c)
2 圆环运动方程求解

 $\frac{{{\partial }^{6}}{{u}_{x}}}{\partial {{\theta }^{6}}}+2\frac{{{\partial }^{4}}{{u}_{x}}}{\partial {{\theta }^{4}}}+\frac{{{\partial }^{2}}{{u}_{x}}}{\partial {{\theta }^{2}}}-\frac{{{R}^{4}}}{E{{I}_{z}}}\left( \frac{\partial {{q}_{y}}}{\partial \theta }-{{q}_{x}} \right)+\frac{\rho A{{R}^{4}}}{E{{I}_{z}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left( \frac{{{\partial }^{2}}{{u}_{x}}}{\partial {{\theta }^{2}}}-{{u}_{x}} \right)=0$ (6)

 $\left. \begin{array}{*{35}{l}} {{v}_{1}}=\frac{\partial {{u}_{x}}}{\partial t}, & {{v}_{2}}=\frac{\partial {{u}_{y}}}{\partial t} \\ {{\eta }_{1}}=\frac{\partial {{u}_{x}}}{R\partial \theta }-\frac{{{u}_{y}}}{R}, & {{\eta }_{2}}=\frac{{{u}_{x}}}{R}+\frac{\partial {{u}_{y}}}{R\partial \theta } \\ \end{array} \right\}$ (7)

 $\dfrac{\partial v_1 }{\partial t} = k_1 \dfrac{\partial \eta _1 }{\partial \theta } + k_2 \dfrac{\partial \eta _3 }{\partial \theta } + \dfrac{q_x }{\rho A}$ (8a)
 $\dfrac{\partial v_2 }{\partial t} = k_1 \eta _1 - k_2 \dfrac{\partial \eta _4 }{\partial \theta } + \dfrac{q_y }{\rho A}$ (8b)
 $\dfrac{\partial \eta _1 }{\partial t} = \dfrac{\partial v_1 }{R\partial \theta } - \dfrac{v_2 }{R}$ (8c)
 $\dfrac{\partial \eta _2 }{\partial t} = \dfrac{v_1 }{R} + \dfrac{\partial v_2 }{R\partial \theta }$ (8d)
 $\eta _3 = \dfrac{\partial \eta _2 }{\partial \theta }$ (8e)
 $\eta _4 = \dfrac{\partial \eta _3 }{\partial \theta }$ (8f)

 $v_1 (0,t) = v_1 (2 \pi,t) = 0$ (9a)
 $v_2 (0,t) = v_2 (2\pi,t) = 0$ (9b)
 $\eta _2 (0,t) = \eta _2 (2\pi,t) = 0$ (9c)

 $\frac{\partial \widehat{x}(s,\theta )}{\partial \theta }=A_{1}^{-1}{{A}_{0}}\hat{x}(s,\theta )+A_{1}^{-1}\hat{b}(s,\theta )$ (10)

 ${ A}_1 = \left[\!\!\begin{array}{cccccc} 0 & 0 & {k_1 } & 0 & {k_2 } & 0 \\ 0 & 0 & 0 & 0 & 0 & { - k_2 } \\ {1/R} & 0 & 0 & 0 & 0 & 0 \\ 0 & {1/R} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\!\! \right]$ (11a)
 ${ A}_0 = \left[\!\!\begin{array}{cccccc} s & 0 & 0 & 0 & 0 & 0 \\ 0 & s & { - k_1 } & 0 & 0 & 0 \\ 0 & {1/R} & s & 0 & 0 & 0 \\ { - 1/R} & 0 & 0 & s & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \!\! \right]$ (11b)

 $\hat{ b}(s,\theta ) = - \left\{ \!\! \begin{array}{cccccc} {v_1^0 + \dfrac{\hat {q}_x }{\rho A}} & {v_2^0 + \dfrac{\hat {q}_y }{\rho A}} & {\eta _1^0 } & {\eta _2^0 } & 0 & 0 \end{array}\!\! \right\}^{\rm T}$ (12)

 ${{\Sigma }_{0}}\hat{x}(s,0)+{{\Sigma }_{1}}\hat{x}(s,2\pi )=0$ (13)

 ${ \varSigma}_0 = \left[\!\!\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \!\!\right]$ (14a)
 ${ \varSigma}_1 = \left[\!\!\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \!\! \right]$ (14b)

 $\hat{x}(s,\theta )=\int_{0}^{2\pi }{{{G}_{r}}}(s,\theta ,\xi )A_{1}^{-1}\hat{b}(s,\xi )\text{d}\xi$ (15)

 ${ G}_r (s,\theta ,\xi ) = \\ \left\{ \!\!\begin{array}{ll} - { N}(s,\theta ){ \varSigma}_1 { \varPhi} (s,2\pi - \xi ) , & 0 ≤ \theta ≤ \xi \\ { N}(s,\theta ){ \varSigma} _0 { \varPhi} (s,- \xi ) , & \xi ≤ \theta ≤ 2 \pi \end{array} \right.$ (16)

 ${\varPhi} (s,\theta ) = \exp ({ A}_1 ^{ - 1}{ A}_0 \theta )$ (17)
 ${ N}(s,\theta ) = {\varPhi} (s,\theta )\left[{{ \varSigma}_0 + { \varSigma}_1 {\varPhi } (s,2 \pi )} \right]^{ - 1}$ (18)

 $\hat {u}_x = \dfrac{\hat {v}_1 + u_x^0 }{s} ,\ \hat {u}_y = \dfrac{\hat {v}_2 + u_y^0 }{s}$ (19)

 $\left( {{ \varSigma}_0 + { \varSigma}_1 {\varPhi} (s,2 \pi )} \right){\psi} = 0$ (20)

 $\det \left( {{ \varSigma }_0 + { \varSigma }_1 {\varPhi} (s,2 \pi )} \right) = 0$ (21)

 ${ X}_n (\theta ) = {\varPhi} (s_n ,\theta ){\psi}_n$ (22)

 $\hat{ b}(s,\theta ) = \left\{ { - \dfrac{\delta (\theta - \xi )}{\rho AR},0,0,0,0,0} \right\}^{\rm T}$ (23a)

 $\hat{ b}(s,\theta ) = \left\{ {0,- \dfrac{\delta (\theta - \xi )}{\rho AR},0,0,0,0} \right\}^{\rm T}$ (23b)

 $\hat{ x}(s,\theta ) = { G}_r (s,\theta ,\xi ){ A}_1 ^{ - 1}\bar { b}$ (24)

 $\bar{ b} = \left\{ { - \dfrac{1}{\rho AR},0,0,0,0,0} \right\}^{\rm T}$ (25a)

 $\bar{ b} = \left\{ {0,- \dfrac{1}{\rho AR},0,0,0,0} \right\}^{\rm T}$ (25b)

 ${ G}(s,\theta ,\xi ) = { G}_r (s,\theta ,\xi ){ A}_1 ^{ - 1}\bar { b}$ (26)
3 数值算例

 图 3 环形桁架结构径向振动的固有振型(俯视图，"$\bullet$"：固支点)(续) Figure 3 Mode shapes for the radial vibration of hoop truss (top view,"$\bullet$": fixed point)(continued)

 ${\rm MAC}\left( { { \varphi}_i ,{ \psi}_i } \right) = \dfrac{\left| {{ \varphi}_i ^{\rm T}{ \psi}_i } \right|^2}{\left( {{ \varphi}_i ^{\rm T}{ \varphi}_i } \right)\left( {{ \psi}_i ^{\rm T}{ \psi}_i } \right)}$ (27)

 图 4 $A$点径向力与$B$点径向位移之间的传递函数 Figure 4 Transfer function between the radial force at point $A$ and the radial displacement at point $B$
 图 5 $A$点径向力与$C$点切向位移之间的传递函数 Figure 5 Transfer function between the radial force at point $A$ and the tangential displacement at point $C$

 图 6 环形桁架高度变化时等效圆环模型的精度 Figure 6 Precision of the equivalent ring model with the variation of the height of hoop truss

4 结论

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EQUIVALENT CIRCULAR RING MODEL FOR THE RADIAL VIBRATION ANALYSIS OF HOOP TRUSS STRUCTURES
Liu Fushou, Jin Dongping
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: In large deployable mesh antenna, the dynamic property of hoop truss is vital to the working performance of the whole antenna. For large space truss structures, the model simplification of these structures with simple elastic continuum model is always the focus of dynamics research. By regarding the hoop truss as a hoop periodic structure composed of repetitive planar truss elements, and based on the equivalent beam model of the repeated truss element, an equivalent circular ring model for the radial vibration analysis of the hoop truss structure is presented. By variable substitution, the fourth-order partial differential equations (PDEs) for the radial vibration of theffcircular ring are reduced to first-order PDEs, then the reduced PDEs are transform to ordinary differential equations via Laplace transform, Green's function method is utilized to solve the dynamic response of theffcircular ring in complex frequency domain. Furthermore the characteristic equations for natural vibration and the expression of transfer function of the equivalent ring model were derived. At last, a numerical example is used to compare the natural frequencies, mode shapes and transfer functions of the finite element model and the equivalent ring model of the hoop truss. The results verifies the feasibility of using equivalent circular ring model for radial vibration analysis of hoop truss structure.
Key words: hoop truss structure    equivalent circular ring model    Laplace transform    transfer function    natural vibration