﻿ 基于二阶摄动法求解区间参数结构动力响应
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 力学学报  2015, Vol. 47 Issue (1): 147-153  DOI: 10.6052/0459-1879-14-088 0

### 引用本文 [复制中英文]

[复制中文]
Li Qi, Qiu Zhiping, Zhang Xudong. SECOND-ORDER PARAMETER PERTURBATION METHOD FOR DYNAMIC STRUCTURES WITH INTERVAL PARAMETERS[J]. Chinese Journal of Ship Research, 2015, 47(1): 147-153. DOI: 10.6052/0459-1879-14-088.
[复制英文]

### 文章历史

2014-04-01收稿
2014-09-19录用
2014-10-29网络版发表

1 问题描述

n自由度结构系统的振动控制方程可以表示为
 ${ K x}(t)+{ C}\dot{ x}(t)+{ M}\ddot{ x}(t)={ F}(t)$ (1)

 $b_k=b^{\rm c}_k+\delta_k$ (2)

 $\left. \begin{gathered} K(b) = {K_0} + \Delta K = {K_0} + \sum\limits_{k = 1}^p {{K_k}{\delta _k}} \hfill \\ C(b) = {C_0} + \Delta C = {C_0} + \sum\limits_{k = 1}^p {{C_k}{\delta _k}} \hfill \\ M(b) = {M_0} + \Delta M = {M_0} + \sum\limits_{k = 1}^p {{M_k}{\delta _k}} \hfill \\ F(t,b) = {F_0}(t) + \Delta (t) = {F_0}(t) + \sum\limits_{k = 1}^p {{F_k}(t){\delta _k}} \hfill \\ \end{gathered} \right\}$ (3)

 $\begin{gathered} ({K_0} + \Delta K)x(t) + ({C_0} + \Delta C)\dot x(t) + \hfill \\ ({M_0} + \Delta M)\ddot x(t) = {F_0}(t) + \Delta F(t) \hfill \\ \end{gathered}$ (4)

 $\left. \begin{gathered} \bar x(t) = {G_{\max }}\left\{ {x(t):Kx(t) + C\dot x(t) + M\ddot x(t) = F(t)} \right\} \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} (t) = {G_{\min }}\left\{ {x(t):Kx(t) + C\dot x(t) + M\ddot x(t) = F(t)} \right\} \hfill \\ \end{gathered} \right\}$ (5)

2 基于二阶摄动展开法求解区间参数结构 2.1 采用摄动法求解区间参数结构响应

 $x(t) = {x_0}(t) + {x_1}(t) + {x_2}(t) + {x_3}(t) + \cdots$ (6)

 ${K_0}{x_0}(t) + {C_0}{\dot x_0}(t) + {M_0}{\ddot x_0}(t) = {F_0}(t)$ (7)
${x_1}(t)$表示当动力系统参数存在不确定量 $\delta_k$,$k = 1,2,\cdots,p$时的动力响应的一阶小量； ${x}_2 (t)$表示动力响应的二阶小量；式(6)中其他项以此类推. 设$\Delta { x}_0 (t) = { x}(t) - { x}_0(t)$. 将${ x}(t) = { x}_0 (t) + \Delta { x}_0 (t)$ 代人方程(4)中，可以得到
 $\begin{gathered} ({K_0} + \Delta K)\Delta {x_0}(t) + ({C_0} + \Delta C)\Delta {{\dot x}_0}(t) + \hfill \\ ({M_0} + \Delta M)\Delta {{\ddot x}_0}(t) = {F_1}(t) \hfill \\ \end{gathered}$ (8)

 $\begin{gathered} {F_1}(t) = \Delta F(t) - (\Delta K{x_0}(t) + \Delta C{{\dot x}_0}(t) + \Delta M{{\ddot x}_0}(t)) = \hfill \\ \sum\limits_{k = 0}^p {{\delta _k}({F_k}(t) - {K_k}{x_0}(t) - {C_k}{{\dot x}_0}(t) - {M_k}{{\ddot x}_0}(t))} \hfill \\ \end{gathered}$ (9)

 ${K_0}{x_1}(t) + {C_0}{\dot x_1}(t) + {M_0}{\ddot x_1}(t) = {F_1}(t)$ (10)

 ${x_1}(t) = {\text{ }}\sum\limits_{k = 1}^p {{\delta _k}{x_{1k}}(t)}$ (11)

 $\begin{gathered} {K_0}{x_{1k}}(t) + {C_0}{{\dot x}_{1k}}(t) + {M_0}{{\ddot x}_{1k}}(t) = \hfill \\ {F_k}(t) - ({K_k}{x_0}(t) + {C_k}{{\dot x}_0}(t) + \hfill \\ {M_k}{{\ddot x}_0}(t)),k = 1,2,\cdots ,p \hfill \\ \end{gathered}$ (12)

 $x(t) = {x_0}(t) + {x_1}(t) + \Delta {x_1}(t)$ (13)

 ${K_0}{x_2}(t) + {C_0}{\dot x_2}(t) + {M_0}{\ddot x_2}(t) = {F_2}(t)$ (14)

 $\begin{gathered} {F_2}(t) = - (\Delta K{x_1}(t) + \Delta C{{\dot x}_1}(t) + \Delta M{{\ddot x}_1}(t)) = \hfill \\ - \sum\limits_{k = 1}^p {{\delta _k}({K_k}{x_1}(t) + {C_k}{{\dot x}_1}(t) + {M_k}{{\ddot x}_1}(t))} = \hfill \\ - \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}({K_k}{x_{1l}}(t) + {C_k}{{\dot x}_{1l}}(t) + {M_k}{{\ddot x}_{1l}}(t))} } \hfill \\ \end{gathered}$ (15)

 ${x_2}(t) = {\text{ }}\sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{2kl}}(t){\text{ }}} }$ (16)
${x_{2kl}}(t)$满足
 $\begin{gathered} {K_0}{x_{2kl}}(t) + {C_0}{{\dot x}_{2kl}}(t) + {M_0}{{\ddot x}_{2kl}}(t) = \hfill \\ - ({K_k}{x_{1l}}(t) + {C_k}{{\dot x}_{1l}}(t) + {M_k}{{\ddot x}_{1l}}(t)),\hfill \\ k,l = 1,2,\cdots ,p \hfill \\ \end{gathered}$ (17)

 $x(t) = {x_0}(t) + {\text{ }}\sum\limits_{k = 1}^p {{\delta _k}{x_{1k}}(t)} + \sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{2kl}}(t)}$ (18)

2.2 基于优化分析得到动力响应的界限

 $\left. \begin{gathered} \bar x(t) = {x_0}(t) + {G_{\max }}(\sum\limits_{k = 1}^p {{\delta _k}{x_{1k}}(t)} + \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{2kl}}(t)} } ) \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} (t) = {x_0}(t) + {G_{\min }}(\sum\limits_{k = 1}^p {{\delta _k}{x_{1k}}(t)} + \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{2kl}}(t)} } ) \hfill \\ \end{gathered} \right\}$ (19)

 $\begin{gathered} {{\bar x}_i}(t) = {x_{i,0}}(t) + {G_{\max }}(\sum\limits_{k = 1}^p {{\delta _k}{x_{i,1k}}(t)} + \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{i,2kl}}(t)} } \hfill \\ {\text{s}}.{\text{t}}.\; - \Delta {b_j} \leqslant {\delta _j} \leqslant \Delta {b_j},j = 1,2,\cdots ,p \hfill \\ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_i}(t) = {x_{i,0}}(t) + {G_{\min }}((\sum\limits_{k = 1}^p {{\delta _k}{x_{i,1k}}(t)} + \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {{\delta _k}{\delta _l}{x_{i,2kl}}(t)} } ) \hfill \\ {\text{s}}.{\text{t}}.\; - \Delta {b_j} \leqslant {\delta _j} \leqslant \Delta {b_j},j = 1,2,\cdots ,p \hfill \\ \end{gathered}$ (20)

 $\begin{gathered} {G_{\min }}({G_{\max }}){x_{i,0}}(t) + f(Y) = {x_{i,0}}(t) + \frac{1}{2}{Y^{\text{T}}}AY + {d^{\text{T}}}Y \hfill \\ {\text{s}}.{\text{t}}. - \Delta {b_j} \leqslant {y_j} \leqslant \Delta {b_j} \hfill \\ \end{gathered}$ (21)

DC算法是Pham Dinh Tao[15]于1988年提出的基于局部最优性条件和DC对偶理论的算法.由于该方法只涉及到内积计算，计算速度相对很快，得到了广泛的关注与应用.采用DC算法可以有效在多项式时间内求解在箱型约束下二次规划问题的全局最优解[10].基于DC算法求解箱型约束下的二次规划问题的方法，主要有3个组成部分：(1)基于DC算法求解包含箱型可行域的外接超球约束下二次规划问题的全局最优值作为原问题全局最优值的下界[16](以求解全局最小值为例)；(2)基于DC算法求解箱型约束下的二次规划问题的全局最优值的上界；(3)配合恰当的分支定界方法不断分割可行域得到全局最优解.那么结构动力响应的上界与下界分别为
 $\left. \begin{gathered} {{\bar x}_i}(t) = {x_{i,0}}(t) + {G_{\max }}(f(Y)),\;\;i = 1,2,\cdots ,n \hfill \\ 3mm{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_i}(t) = {x_{i,0}}(t) + {G_{\min }}(f(Y)),\;\;i = 1,2,\cdots ,n \hfill \\ \end{gathered} \right\}$ (22)

3 本文方法与其他区间分析方法的对比

 ${x^{\text{I}}}(t) = {x_0}(t) + _{k = 1}^p\delta _k^{\text{I}}{x_{1k}}(t)$ (23)

 ${x^{\text{I}}}(t) = {x_0}(t) + \sum\limits_{k = 1}^p {\delta _k^{\text{I}}{x_{1k}}(t) + \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {\delta _k^{\text{I}}\delta _l^{\text{I}}{x_{2kl}}(t)} } }$ (24)

 $\left. \begin{gathered} {a^I} + {b^I} = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\bar a + \bar b] \hfill \\ {a^I} - {b^I} = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} - \bar b,\bar a - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}] \hfill \\ {a^I} \times {b^I} = [\min \{ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \bar b,\bar a\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\bar a\bar b\} ,\max \{ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \bar b,\bar a\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\bar a\bar b\}] \hfill \\ {a^I}/{b^I} = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\bar a] \times [1/\bar b,1/\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}],0 \notin {b^I} \hfill \\ \end{gathered} \right\}$ (25)

 $\begin{gathered} {{\bar x}_i}(t) = {x_{i,0}}(t) + \sum\limits_{k = 1}^p {\Delta {b_k}\left| {{x_{i,1k}}(t)} \right|} ,\;\;I = 1,2,\cdots ,n \hfill \\ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_i}(t) = {x_{i,0}}(t) - \sum\limits_{k = 1}^p {\Delta {b_k}\left| {{x_{i,1k}}(t)} \right|} ,\;\;i = 1,2,\cdots ,n \hfill \\ \end{gathered}$ (26)

 $\begin{gathered} {{\bar x}_i}(t) = {x_{i,0}}(t) + _{k = 1}^p\Delta {b_k}\left| {{x_{i,1k}}(t)} \right| + \hfill \\ \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {\Delta {b_k}\Delta {b_l}\left| {{x_{i,2kl}}(t)} \right|} } ,\;\;i = 1,2,\cdots ,n \hfill \\ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_i}(t) = {x_{i,0}}(t) - _{k = 1}^p\Delta {b_k}\left| {{x_{i,1k}}(t)} \right| - \hfill \\ \sum\limits_{k = 1}^p {\sum\limits_{l = 1}^p {\Delta {b_k}\Delta {b_l}\left| {{x_{i,2kl}}(t)} \right|} } ,\;\;i = 1,2,\cdots ,n \hfill \\ \end{gathered}$ (27)

4 算例

 图 1 20自由度弹簧质量块系统 Fig.1 Twenty degree of freedom spring mass system

 图 2 采用一阶区间分析方法和蒙特卡洛方法得到的质量块6 动力响应上界的之间的比较 Fig.2 Comparison of the upper bounds of dynamic response of Mass 6 by the first-order interval analysis method and Monte Carlo simulation method

 图 3 采用基于区间运算的二阶区间分析方法、本文方法、蒙特卡洛方法得到的质量块6动力响应上界之间的比较 Fig.3 Comparison of the upper bounds of dynamic response of Mass 6 by the second-order method based on interval operations,the proposed method and Monte Carlo simulation method

 图 4 采用本文方法和蒙特卡洛方法得到的质量块6动力响应界限之间的比较 Fig.4 Comparison of the bounds of the dynamic response of Mass 6 by the proposed method and Monte Carlo simulation method

 图 5 60 杆空间桁架 Fig.5 Sixty-bar space truss

 图 6 采用一阶区间分析方法和蒙特卡洛方法得到的结点24动力响应上界之间的比较 Fig.6 Comparison of the upper bounds of dynamic response of Node 24 by the first-order interval analysis method and Monte Carlo simulation method

 图 7 采用基于区间运算的二阶区间分析方法、本文方法、蒙特卡洛方法得到的结点24动力响应上界之间的比较 Fig.7 Comparison of the upper bounds of dynamic response of Node 24 by the second-order method based on interval operations,the proposed method and Monte Carlo simulation method

 图 8 采用本文方法和蒙特卡洛方法得到的结点24动力响应界限之间的比较 Fig.8 Comparison of the bounds of the dynamic response of Node 24 by the proposed method and Monte Carlo simulation method
5 结论

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SECOND-ORDER PARAMETER PERTURBATION METHOD FOR DYNAMIC STRUCTURES WITH INTERVAL PARAMETERS
Li Qi, Qiu Zhiping, Zhang Xudong
School of Aeronautic Science and Technology, Beijing University of Aeronautics and Astronaut ics, Beijing 100191, China
Fund: The project was supported by the 111 Project (B07009),the National Natural Science Foundation of China (11372025,11002013),the Defense Industrial Technology Development Program (A0820132001,JCKY2013601B) and Aeronautical Science Foundation of China (2012ZA51010).
Abstract: When considering the problem of the dynamic responses of structures with interval parameters, previous interval analysis methods are mostly restricted to its first-order. But if the uncertainties of the parameters are fairly large, the response region obtained using the first-order interval analysis method would fail to contain the real region of the dynamic response of uncertain structures. Therefore, the second-order analysis method should be considered. However, the second-order analysis method relating to operations of interval may result in an exorbitantly overestimated dynamic response region, which makes the result useless for practical engineering problems. To circumvent this drawback, firstly the general function of the dynamic response of structures in terms of structural parameters is obtained based on the second-order parameter perturbation method. Then via solving the maximum and minimum of the function, the problem of determining the bounds of the dynamic response of uncertain structures is changed into a series of low dimensional box constrained quadratic problems, and these box constrained quadratic programming problems can be solved using the DC algorithm (difference of convex functions algorithm) effectively. The proposed method can avoid the exorbitant overestimate of the dynamic response region of uncertain structures, while does not introduce much more computational expense. A numerical example is used to illustrate the accuracy and the efficiency of the proposed method when comparing with other methods.
Key words: interval parameters    second-order parameter perturbation method    dynamical response of uncertain structures    quadratic problems    DC algorithm