﻿ 一种无条件稳定的结构动力学显式算法
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 力学学报  2015, Vol. 47 Issue (2): 310-319  DOI: 10.6052/0459-1879-14-209 0

### 引用本文 [复制中英文]

[复制中文]
Du Xiaoqiong, Yang Dixiong, Zhao Yongliang. AN UNCONDITIONALLY STABLE EXPLICIT ALGORITHMFOR STRUCTURAL DYNAMICS[J]. Chinese Journal of Ship Research, 2015, 47(2): 310-319. DOI: 10.6052/0459-1879-14-209.
[复制英文]

### 文章历史

2014-07-16 收稿
2014-08-18 录用
2014–12–12 网络版发表

1 显式算法的设计方法

 $\label{eq1} m\ddot{x}_i+c\dot{x}_i+kx_i=f_i$ (1)

 $\label{eq2} \dot{x}_{i + 1} = \dot{x}_i + \alpha _1 h\ddot{x}_i,\ \ x_{i + 1} = x_i + h\dot{x}_i + \alpha _2 h^2 \ddot{x}_i$ (2)

 $\label{eq3} \alpha _1 = \alpha _2 = 4 / (\varOmega ^2 + 4\xi \varOmega + 4 )$ (3)

 $\left.\begin{array}{l} \dot{x}_{i + 1} = \dot{x}_i + 0.5h ( \ddot{x}_i + \ddot{x}_{i + 1} )\\ x_{i + 1} = x_i + \beta _1 h \dot{x}_i + \beta _2 h^2 \ddot{x}_i \end{array}\right\}$ (4)

 $\left.\begin{array}{l} \beta _1 = 4(1 + \xi \varOmega ) / (\varOmega ^2 + 4\xi \varOmega + 4)\\ \beta _2 = 2 / (\varOmega ^2 + 4\xi \varOmega + 4) \end{array}\right\}$ (5)

 $\left.\begin{array}{l} \beta _1 = 2(1 + \xi \varOmega ) / (\varOmega ^2 + 2\xi \varOmega + 2) \\ \beta _2 = (1 - \xi \varOmega ) / (\varOmega ^2 + 2\xi \varOmega + 2) \end{array}\right\}$ (6)

CR算法递推式中的系数是通过离散控制论推导出的,而常氏算法中的系数是依据经验试凑得到的.

 $\label{eq7} \lambda^3-2A_1\lambda^2+A_2\lambda-A_3=0$ (7)

 $\label{eq8} \left.\begin{array}{l} Z(\ddot{x}_i)=\ddot{X}(z),\ \ Z(\dot{x}_i)=\dot{X}(z)\\ Z({x}_i)={X}(z),\ \ Z(f_i)=F(z) \\ Z(\ddot{x}_{i+1})=z\ddot{X}(z),\ \ Z(\dot{x}_{i+1})=z\dot{X}(z) \\ Z({x}_{i+1})=z{X}(z),\ \ Z({f}_{i+1})=zF(z) \end{array}\right\}$ (8)

 $\label{eq9} G(z) = \frac{X(z)}{F(z)} = \frac{n_2 z^2 + n_1 z + n_0 }{d_2 z^2 + d_1 z + d_0 }$ (9)

 $p_{1,2} = \frac{4 - \varOmega ^2\pm 4\varOmega \sqrt {\xi ^2 - 1} }{\varOmega^2 + 4\xi \varOmega + 4}$

 $p_{1,2} = \frac{2\pm \varOmega \sqrt {4\xi ^2 - 4 - \varOmega ^2} }{\varOmega ^2 + 2\xi \varOmega + 2}$

 $\label{eq10} d_1^{{\rm CR}}= d_{\rm 1}^{{\rm Chang}} d_2^{{\rm CR}}/d_{\rm 2}^{{\rm Chang}}, \ \ d_0^{{\rm CR}}=d_0^{{\rm Chang}} d_2^{{\rm CR}}/d_{\rm 2}^{{\rm Chang}}$ (10)

 $\label{eq11} \alpha _1 = \alpha _2 = 2 / (\varOmega ^2 + 2\xi \varOmega + 2 )$ (11)

 $\label{eq12} \alpha _1 = \alpha _2 = s / (\varOmega ^2 + s\xi \varOmega + s)$ (12)

2 新显式算法性能分析 2.1 线性系统的稳定性分析

 图 1 开环系统的结构图 Fig. 1 Block diagram of open-loop system

 $\begin{array}{l} {p_{1,2}} = \frac{{2{\Omega ^2}/s - {\Omega ^2} + 2 \pm \Omega \sqrt {{\Omega ^2} - 4 - 4{\Omega ^2}/s} }}{{2 + 2{\Omega ^2}/s}} = \\ \qquad \;P \pm Qi \end{array}$ (13)

 $\label{eq14} P = \frac{2\varOmega ^2 / s - \varOmega ^2 + 2}{2(1 + \varOmega ^2 / s)},\ \ Q = \frac{\varOmega \sqrt {4\varOmega ^2 / s - \varOmega ^2 + 4} }{2(1 + \varOmega ^2 / s)}$ (14)

2.2 非线性稳定性分析

 图 2 闭环系统的结构图 Fig. 2 Block diagram of closed-loop system

 $\label{eq15} G(z)=\frac{\Delta X(z)}{\Delta F(z)}=\frac{G'(z)}{1+k_tG'(z)}$ (15)

 $\label{eq16} {G}'(z) = \frac{X(Z)}{L(Z)} = \frac{\Delta X(z)}{\Delta L(z)} = \frac{{n}'_2 z^2 + n_1 ^\prime z + {n}'_0 }{d_2 ^\prime z^2 + d_1 ^\prime z + {d}'_0 }$ (16)

$G'(z)$为闭环系统的开环传递函数,其极点为$p'_{1,2}$,零点为$z'_{1,2}$. 式(15)中$G(z)$分母为零对应的特征方程为

 $\label{eq17} 1 + k_t {G}'(z) = 1 + k_t \frac{{n}'_2 z^2 + n_1 ^\prime z + {n}'_0 }{d_2 ^\prime z^2 + d_1 ^\prime z + {d}'_0 } = 0$ (17)

 $\label{eq18} G(z) = \frac{{G}'(z)}{1 + k{G}'(z)}$ (18)

 图 3 新算法特征方程的根轨迹 Fig. 3 Root-locus of characteristic equation of new algorithm

 $k_t \leqslant 4k / s + 4m / h^2\ \ {\rm or}\ \ k_t / k \leqslant 4 / s + 4 / \varOmega ^2$ (19)

2.5 精度分析

 $\label{eq19} \tau = h^{{\rm - }2}[x(i + 1) - 2A_1 x(i) + A_2 x(i - 1)]$ (20)

 $\label{eq20} \tau = \left( - \frac{\omega ^4}{s}x_i - \frac{2\xi \omega ^3}{s}\dot{x_i} + \frac{s\xi \omega }{3}{x}"'_i + \frac{1}{12}{x}""_i \right)h^2 \nonumber\\ +\; o(h^3)$ (20)

 $\label{eq21} p_{1,2} = P\pm Q{\rm i} = \exp \left[\bar {\varOmega }\left( - \bar {\xi }\pm {\rm i}\sqrt {1 - \bar {\xi }^2} \right)\right]$ (22)

 $\label{eq22} AD = 1 - \exp ( - 2\pi \bar {\xi }),\ \ PE = \varOmega / \bar {\varOmega } - 1$ (23)

 图 4 不同的s值对应的PE值 Fig. 4 PE values corresponding to different s

 图 5 $s$取$10\sim 12$时$PE$值 Fig. 5 $PE$ values corresponding to $s$ with $10\sim 12$
2.4 超调特性分析

 $\label{eq23} x_1 = c_{11} x_0 + c_{12} v_0 h,\ \ v_1 = c_{21} x_0 / h + c_{22} v_0$ (24)

 $c_{11} = 1 - \dfrac{\varOmega ^2}{\varOmega ^2 / s + \varOmega + 1} ,\ \ c_{12} = 1 - \frac{2\varOmega \xi}{\varOmega ^2 / s + \varOmega + 1}\\ c_{21} = - \frac{\varOmega^2}{\varOmega ^2 / s + \varOmega + 1},\ \ c_{22} = 1 - \frac{2\varOmega \xi }{\varOmega ^2 / s + \varOmega + 1}$

3 算例分析

 图 6 步长$h=0.01$ s时位移响应时程曲线 Fig. 6 Time histories of displacement response with $h=0.01$ s
 图 7 $h=0.05$ s时位移响应时程曲线 Fig. 7 Time histories of displacement response with $h=0.05$ s
 图 8 $h=0.1$ s时位移响应时程曲线 Fig. 8 Time histories of displacement response with $h=0.1$ s

 图 9 Fig. 9

 图 10 双层剪切框架结构 Fig. 10 Two-story shear-type frame structure

 图 11 算例3底层位移时程曲线 ($h=0.02$ s) Fig. 11 Displacement time histories of bottom story for Example 3 ($h=0.02$ s)
 图 12 算例3顶层位移时程曲线 ($h=0.02$ s Fig. 12 Displacement time histories of top story for Example 3 ($h=0.02$ s)
 图 13 顶层位移时程局部放大图 ($h=0.02$ s) Fig. 13 Local zooming view for displacement time histories of top story ($h=0.02$ s)
 图 14 顶层位移时程曲线 ($h=0.03$ s) Fig. 14 Displacement time histories of top story ($h=0.03$ s)
 图 15 顶层位移时程曲线 ($h=0.04$ s) Fig. 15 Displacement time histories of top story ($h=0.04$ s)

(1) 取$h=0.02$ s,0.03 s,0.04 s,$s=2$,4,6,8,10分别得到结构的位移响应与"精确解"对比(图11 $\sim$图15).

(2) 取$h=0.01$ s,$s=2$,4,10分别得到第2种工况下结构的位移地震响应与"精确解"对比.

 图 16 顶层位移时程曲线 Fig. 16 Displacement time histories of top story
 图 17 底层位移时程曲线 Fig. 17 Displacement time histories of bottom story

 图 18 算例4顶层位移时程曲线($h=0.01$ s) Fig. 18 Displacement time histories of top story for Example 4 ($h=0.01$ s)
 图 19 顶层位移局部放大图($h=0.01$ s) Fig. 19 Local zooming view for displacement time histories of top story ($h=0.01$ s)
 图 20 顶层位移时程曲线($h=0.02$ s) Fig. 20 Displacement time histories of top story ($h=0.02$ s)
4 结 论

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AN UNCONDITIONALLY STABLE EXPLICIT ALGORITHMFOR STRUCTURAL DYNAMICS
Du Xiaoqiong, Yang Dixiong, Zhao Yongliang
Department of Engineering Mechanics, State Key Laboratory for Structural Analysis of Industrial Equipment Dalian University of Technology, Dalian 116024, China
Fund: The project was supported by the National Natural Science Foundation of China (51478086, 11332004) and Key Laboratory Foundation of Science and Technology Innovation in Shaanxi Province (2013SZS02-K02).
Abstract: This paper proposes an unconditionally stable explicit algorithm for time integration of structural dynamics by utilizing the discrete control theory. New algorithm adopts the recursive formula of velocity and displacement of CR algorithm, and obtains the respective transfer function based on Z transformation. Further, the specific expressions of coeffcients of recursive formula are derived according to the pole condition. Then, a variable s in the coeffcients to control the period elongation is introduced, which is applied to adjust the accuracy of new algorithm. Theoretical analysis indicate that the new proposed unconditionally stable explicit algorithm possesses the properties of second accuracy, zero amplitude decay, non-overshoot and self-starting, and its period elongation can be controlled by the variable s. Moreover, the CR algorithm is a special case of the proposed algorithm. Finally, the stability limit of nonlinear stiffening system is determined, and variable interval corresponding to the higher accuracy of new algorithm is presented. Numerical examples demonstrate that in this interval of variable s, the accuracy of new algorithm is superior to that of Newmark constant average acceleration and CR algorithm.
Key words: explicit algorithm for structural dynamics    discrete control theory    algorithm design    controllable accuracy    unconditionally stable