过滤高阶失稳振型满足显式算法的稳定性
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摘要
以中心差分显式算法为例,证明了算法传递矩阵的特征向量与结构自振向量的一致性,以及显式算法步长过大时的失稳,是传递矩阵的特征值和特征向量两个因素共同作用的结果.在证明的基础上得到了过滤高阶失稳振型来满足算法稳定性的方法,也就是在每步按通常的显式算法得到计算结果后,增加了从结果位移向量中过滤振型参与系数很小的高阶失稳振型的计算步骤,使得计算步长即使取在稳定域外也能使算法不会失稳.这种方法,既大大提高了通常采用的显式方法的计算效率,同时对求解的精度也影响不大.算例证明了这种方法的有效性.为求解刚性常微分方程组提供一个思路.
The consistency between transferring matrix's characteristic vector and structure's natural vibration shape was proved by taking central differential method as an example.The explicit method's destabilization was caused by a combination of the transferring matrix's characteristic value and the corresponding characteristic vector.A new method was proposed by filtrating the high destabilization vibration mode from result displacement vectors at different time,as a result,the wide time step size beyond the stability field will not lead to a calculation destabilization.With the proposed method,the calculation efficiency is improved with the same precision.Numerical model validates the method,which develops a new way to solve the stiff differential equations.
The consistency between transferring matrix's characteristic vector and structure's natural vibration shape was proved by taking central differential method as an example.The explicit method's destabilization was caused by a combination of the transferring matrix's characteristic value and the corresponding characteristic vector.A new method was proposed by filtrating the high destabilization vibration mode from result displacement vectors at different time,as a result,the wide time step size beyond the stability field will not lead to a calculation destabilization.With the proposed method,the calculation efficiency is improved with the same precision.Numerical model validates the method,which develops a new way to solve the stiff differential equations.
引文
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[3]杜修力,王进廷.阻尼弹性结构动力计算的显式差分法[J].工程力学,2000,17(5):37.DU Xiuli,WANG Jinting.An explicit difference formulation ofdynamic response calculation of elastic structure with damping[J].Engineering Mechanics,2000,17(5):37.
[4]周正华,周扣华.有阻尼振动方程常用显式积分格式稳定性分析[J].地震工程与工程振动,2001,21(3):22.ZHOU Zhenghua,ZHOU Kouhua.Stability analysis of explicitintegral methods for damped vibration equation[J].Earthquake Engineering and Engineering Vibration,2001,21(3):22.
[5]王进廷,杜修力.有阻尼体系动力分析的一种显式差分法[J].工程力学,2002,19(3):109.WANG Jinting,DU Xiuli.An explicit difference method fordynamic analysis of a structure system with damping[J].Engineering Mechanics,2002(3):109.
[6]张晓志,程岩,谢礼立.结构动力反应分析的三阶显式方法[J].地震工程与工程振动,2002,22(3):1.ZHANG Xiaozhi,CHENG Yan,XIE Lili.A new explicitsolution of dynamic response analysis[J].EarthquakeEngineering and Engineering Vibration,2002(3):1.
[7]王进廷,张楚汉,金峰.有阻尼动力方程显式积分方法的精度研究[J].工程力学,2006,23(3):1.WANG Jinting,ZHANG Chuhan,JIN Feng.On the accuracyof several explicit integration schemes for dynamic equationwith damping[J].Engineering Mechanics,2006,23(3):1.
[8]李小军,唐晖.结构体系动力方程求解的显式积分格式的能耗特性[J].工程力学,2007,24(2):28.LI Xiaojun,TANG Hui.Numerical dissipation property of anexplicit integration scheme for dynamic equation of structuralsystem[J].Engineering Mechanics,2007,24(2):28.
[9]林家浩,曲乃泗,孙焕纯.计算结构动力学[M].北京:高等教育出版社,1989.LIN Jiahao,QU Naisi,SUN Huanchun.Computationalstructural dynamics[M].Beijing:Higher EducationPress,1989.
[10]克拉夫,彭津.结构动力学[M].北京:高等教育出版社,2006.Ray Clough,Joseph Penzien.Structural dynamics[M].Beijing:Higher Education Press,2006.
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