Precise Time-Integration Method with Dimensional Expanding for Structural Dynamic Equations
Precise Time-Integration Method with Dimensional Expanding for Structural Dynamic Equations
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DOI：
10.2514/2.1248
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发表时间：
2001
期刊：
影响因子：
2.5
通讯作者：
Yuanxian Gu;Biaosong Chen;Hongwu W. Zhang;Z. Guan
中科院分区：
文献类型：
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作者：
Yuanxian Gu;Biaosong Chen;Hongwu W. Zhang;Z. Guan
针对线性定常动力系统提出的精细时程积分方法，可以在积分点处给出逼近精确解的精确数值结果。然而，当该算法用于非齐次动态系统时，由于需要求逆矩阵计算而出现困难。本文提出了一种用维数展开法将非线性动力方程转化为非线性动力方程的新算法。通过这种转换，在精细时程积分法中不需要逆矩阵计算。新算法通过编程实现和数值稳定性的改善，增强了精细时程积分法的计算能力，提高了计算效率。数值算例验证了算法的有效性。公式Ai =傅里叶级数解的系数ai =系数向量Bi =傅里叶级数解的系数bi =系数向量C =结构节点上载荷的大小D =由非齐次向量满足的常微分方程系数矩阵艾德d =一个级数的项数f =载荷向量G =阻尼矩阵H =结构动力系统系数矩阵H？=展开结构动力系统系数矩阵I =单位矩阵K =时间步长数K =刚度矩阵k =时间步长阶数l =动力模态数M =质量矩阵m = 2个数N N =精细时程积分算法参数n =结构动力系统维数p =变换后的系统状态变量q =位移向量r =结构动力系统的非齐次向量Ta =矩阵的一小部分指数Tc =根据矩阵C的矩阵的一部分指数Td =根据矩阵D的矩阵的一部分指数t =时间v =结构动力系统的状态变量v <$=扩展结构动力系统的状态变量向量
The precise time-integration method proposed for a linear time-invariant dynamic system can give precise numerical results approaching the exact solution at the integration points. However, dife culties arise when the algorithm is used for nonhomogeneous dynamic systems due to the inverse matrix calculation required. A new algorithm isproposedto convertnonhomogeneousdynamicequationsinto homogeneousequationsbymeansofthe dimensional expanding method. With this conversion, the inverse matrix calculation is not required in the precise time-integration method. The new algorithm has enhanced the precise time-integration method by benee ting the programming implementation and the numerical stability; it has improved the computational efe ciency as well. Numerical examples are given to demonstrate the validity and efe ciency of the algorithm. Nomenclature Ai = coefe cient of the solution of Fourier series ai = coefe cient vector Bi = coefe cient of the solution of Fourier series bi = coefe cient vector C = magnitude of the load on the nodes of the structure D = coefe cient matrix of ordinary differential equations satise ed by the nonhomogeneous vector d = number of terms of one series f = load vector G = damping matrix H = coefe cient matrix of structural dynamic system H ¤ = coefe cient matrix of expanding structural dynamic system I = identity matrix K = number of time steps K = stiffness matrix k = order of time step l = number of dynamic modes M = mass matrix m = number of 2 N N = algorithm parameter of precise time integration n = dimension of the structural dynamic system p = transformed system state variable q = displacement vector r = nonhomogeneous vector of structural dynamic system Ta = small part of the matrix exponential Tc = part of matrix exponential according to matrix C Td = part of matrix exponential according to matrix D t = time v = state variable of structure dynamic system v ¤ = state variable vector of expanding structural