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 （南京工业大学，土木工程学院，南京 211816）
[收稿日期]： 2017-11-25

1 微分求积法基本原理

 $f(x) \approx \sum\limits_{j = 0}^m {{q_j}(x)f({x_j})}$ (1)

 ${f^{(k)}}({x_i}) \approx \sum\limits_{j = 0}^m {q_j^{(k)}({x_i})f({x_j}){\rm{ }}}, \;\;i = 0, \;1, \cdots \cdots, m$ (2)

$a_{ij}^{(k)} = q_j^{(k)}({x_i}), {f^{(k)}}({x_i}) = f_i^{(k)}, f({x_j}) = {f_j}$，则：

 $f_i^{(k)} \approx \sum\limits_{j = 1}^N {{a^{(k)}}_{ij}{f_j}{\rm{ }}}, \;\;i = 0, \;1, \cdots \cdots, m$ (3)

 $f(x) = {x^r}, r = 0, 1, \cdots \cdots, m$ (4)
 $f(x) = \prod\limits_{i = 0\atop i \ne j}^m {\frac{{x - {x_i}}}{{{x_j} - {x_i}}}} \;\;{\rm{, }}\;\;j = 0, 1, \cdots \cdots, m$ (5)

 ${{\bf{A}}^{k}}={({\bf{A}})^k}$ (6)

2 结构动力反应微分求积分析方法

 $\frac{{{{\rm{d}}^2}u}}{{{\rm{d}}{t^2}}} + 2\xi \omega \frac{{{\rm{d}}u}}{{{\rm{d}}t}} + {\omega ^2}u = p(t)$ (8)

 ${\ddot u_i} + 2\xi \omega \Delta t{\dot u_i} + {(\Delta t)^2}{\omega ^2}u{}_i = {(\Delta t)^2}p{}_i, \;\;i = 0, \;1, \cdots \cdots, m$ (10)

 ${\dot u_i} = \sum\limits_{j = 0}^N {{a_{ij}}{u_j}}, \;\;i = 0, \;\;1, \cdots \cdots, \;m$ (11)
 ${\ddot u_i} = \sum\limits_{k = 0}^m {{a_{ik}}{{\dot u}_k} = } \sum\limits_{k = 0}^m {{a_{ik}}\sum\limits_{j = 0}^m {{a_{kj}}{u_j}} }, \;\;i = 0, \;\;1, \cdots \cdots, \;m$ (12)

 ${\bf{\dot u}}={\bf{Au}}$ (13)
 ${\bf{\ddot u}} = {{\bf{A}}^2}{\bf{u}}$ (14)

 ${{\bf{\dot u}}_{\bf{s}}} = {{\bf{G}}_{\bf{0}}}u{}_0 + {\bf{G}}{{\bf{u}}_{\bf{s}}}$ (15)
 ${{\bf{\ddot u}}_{\bf{s}}} = {{\bf{G}}_{\bf{0}}}v{}_0\Delta t + {{\bf{G}}_{\bf{0}}}{\bf{G}}{u_0} + {{\bf{G}}^{\bf{2}}}{{\bf{u}}_{\bf{s}}}$ (16)

 ${\bf{\hat k}}{{\bf{u}}_{\bf{s}}}={\bf{\hat p}}$ (17)

 ${{\bf{v}}_{\bf{s}}} = \frac{1}{{\Delta t}}({{\bf{G}}_{\bf{0}}}{u_0} + {\bf{G}}{{\bf{u}}_{\bf{s}}})$ (18)

 $\frac{{{{\rm{d}}^2}u}}{{{\rm{d}}{t^2}}} = p(t) - 2\xi \omega \frac{{{\rm{d}}u}}{{{\rm{d}}t}} - {\omega ^2}u$ (19)
3 算例

 图 1 简谐荷载激励下体系1的位移反应 Fig. 1 Displacement response of system 1 under simple harmonic load
 图 2 简谐荷载激励下体系2的位移反应 Fig. 2 Displacement response of system 2 under simple harmonic load
 图 3 简谐荷载激励下体系3的位移反应 Fig. 3 Displacement response of system 3 under simple harmonic load

4 结论

（1）采用微分求积法求解结构在动力荷载激励下的反应合理可行。在较大的节点距离下依然能够得到精确的结果，计算精度高，且具有普适性，对不同自振周期的结构、不同频率的动力荷载都适用。

（2）用DQM进行动力分析时，能一次性求得多个时刻的反应。相比传统的每次只能求1个时刻的逐步积分法，计算效率得到提高，计算成本也得到了降低。

（3）在时步长度Δt一定时，DQM求解动力反应的计算精度和数值稳定性与时步分段数m有关。排除一些失稳飘移的情况，一般m越大，计算精度越高，但计算量也越大。综合考虑，对于均匀网格离散方案，实际计算时取m=10相对较优。

（4）使用DQM进行实际结构动力反应分析时，时间步长Δt可选为动力荷载的等效周期，然后将各时步等分成10段来计算，这样可获得较满意的计算结果。

The Application of Differential Quadrature Method in Structural Dynamic Analysis
Ren Zhengzheng*, Mei Yuchen, Li Hongjing*
 (College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China)
Abstract

The differential quadrature method (DQM) is a numerical technique of solving initial/boundary value problems of differential equations and capable of obtaining a higher numerical accuracy with a smaller calculation workload. This method is often used to solve the problems of structural static analysis of beams and slabs or eigenvalue analysis when it is applied to the engineering fields, which is to solve the differential equation of the boundary value problems. Dynamic analysis of structures is an initial value problem as well as particular loads and structural response. As a result, applying the DQ method of solving the boundary value problem directly cannot obtain solution of problem. The principle of differential quadrature is applied to establish the specific method of solving structural dynamic response in this paper. By analogizing the idea of unit method, the duration of the load is divided into many time steps and dynamic load and structure are discretized in each time step, then the response of the system can be solved in the whole time domain by employing the DQ method step by step. The feasibility of this method and the characteristics of high precision and high efficiency are expounded by calculating the response of three linear elastic single-freedom-degree system of different natural vibration periods excited by simple harmonic loads of different frequencies. By means of numerical experiment, the optimal meshing scheme is determined and the suggestion for the time step is given.

《震灾防御技术》, DOI：10.11899/zzfy20180410