结构动力学逐步积分算法稳定性讨论
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摘要
美国著名计算结构力学专家、地震工程专家Ray W Clough和Anil K Chopra分别写有在全世界范围内有很大影响的专著《DYNAMICS OF STRUCTURES》.两部著作在关于逐步积分算法的论述中都曾指出加速度由当前位移和速度代入运动方程求解将能得到更好的结果.但这一点存在可商榷之处,为避免读者误用,从算法的稳定性出发,经分析指出:专著观点应用于Newmark法增量格式时确实使得算法的谱半径有所减小;然而,应用于Newmark法全量格式时,算法特性并未发生改变,结果得不到改善;此外,书中给出了增量格式Wilsonθ法的推导步骤,按此推导的算法相对于全量格式稳定界限发生了改变——只有当θ≥1.5时算法才是无条件稳定的.如果误将加速度由当前位移和速度代入运动方程求解推广到Wilsonθ法全量格式,算法的稳定性将变得非常差.
Ray W. Clough and Anil K. Chopra, the famous experts of computer structural mechanics and earthquake engineering, have written two monographs with the same name: DYNAMICS OF STRUCTURES, which have a large influence around the world. A viewpoint about the step-by-step integration method, whose result comes out better when acceleration is calculated by the equations of motion according to displacement and velocity, is emphasized in both books. However, this point of view needs to be discussed carefully. After some analyses to the stability of the methods mentioned in the books, we point out that the application of this method to the increment format of Newmark method reduces the spectral radius as expected; while its application to the whole quantity format of Newmark method indicates that characteristics of the algorithm are not changed, and there’s no improvement to the result. This paper also provides the steps to deduce the increment format of Wilson θ method. The algorithm, which is whole quantity format, deduced by these steps changes the stability limit comparing with the algorithm discussed in Article 1. And the algorithm is unconditional stable only when θ is greater than or equals to 1.5. The stability of the algorithm becomes very bad when acceleration is replaced by current displacement and velocity to be used in motion equation and further proceeded to the whole quantity format of Wilson θ method.
Ray W. Clough and Anil K. Chopra, the famous experts of computer structural mechanics and earthquake engineering, have written two monographs with the same name: DYNAMICS OF STRUCTURES, which have a large influence around the world. A viewpoint about the step-by-step integration method, whose result comes out better when acceleration is calculated by the equations of motion according to displacement and velocity, is emphasized in both books. However, this point of view needs to be discussed carefully. After some analyses to the stability of the methods mentioned in the books, we point out that the application of this method to the increment format of Newmark method reduces the spectral radius as expected; while its application to the whole quantity format of Newmark method indicates that characteristics of the algorithm are not changed, and there’s no improvement to the result. This paper also provides the steps to deduce the increment format of Wilson θ method. The algorithm, which is whole quantity format, deduced by these steps changes the stability limit comparing with the algorithm discussed in Article 1. And the algorithm is unconditional stable only when θ is greater than or equals to 1.5. The stability of the algorithm becomes very bad when acceleration is replaced by current displacement and velocity to be used in motion equation and further proceeded to the whole quantity format of Wilson θ method.
引文
[1]晏启祥,刘浩吾,何川.一种改进的Wilson-θ法及算法稳定性[J].土木工程学报,2003,36(10):76-79.
[2]李小军,刘爱文.动力方程求解的显示积分格式及其稳定性与适用性[J].世界地震工程,2000,16(2):8-12.
[3]陈学良,袁一凡.求解振动方程的一种显式积分格式及其精度与稳定性[J].地震工程与工程振动,2002,22(3):9-14.
[4]周正华,周扣华.有阻尼振动方程常用显示积分格式稳定性分析[J].地震工程与工程振动,2001,21(3):22-28.
[5]吴斌,保海娥.实时子结构实验Chang算法的稳定性和精度[J].地震工程与工程振动,2006,26(2):41-48.
[6]R W克拉夫,J彭津.结构动力学[M].北京:科学出版社,1981.
[7]CLOUGH R W,PENZIEN J.Dynamics of Structures[M].2ndEdition Revised Edition.Berkeley:Computers and Structures,Inc.1995.
[8]CHOPRA A K.Dynamics of Structures[M].2ndEdi-tion,影印版.北京:清华大学出版社,2005.
[9]BATHE K J WILSONE L.Stability and accuracy analy-sis of direct integration methods[J].Earthquake Engi-neering and Structural Dynamics,1973,1:283-291.
[10]HILBER HM.,HUGHES TJ R,TAYLOR R L.Colloca-tion,dissipation and‘overshoot’for time integration schemes in structural dynamics[J].Earthquake Engi-neering and Structural Dynamics.1978,6:99-117.
[11]WANG H D,ZHANG Y S,WANG W.A high order single step-βmethod for nonlinear structural dynamic analysis[J].Journal of Harbin Institute of Technology,2003,10(2):113-119.
[2]李小军,刘爱文.动力方程求解的显示积分格式及其稳定性与适用性[J].世界地震工程,2000,16(2):8-12.
[3]陈学良,袁一凡.求解振动方程的一种显式积分格式及其精度与稳定性[J].地震工程与工程振动,2002,22(3):9-14.
[4]周正华,周扣华.有阻尼振动方程常用显示积分格式稳定性分析[J].地震工程与工程振动,2001,21(3):22-28.
[5]吴斌,保海娥.实时子结构实验Chang算法的稳定性和精度[J].地震工程与工程振动,2006,26(2):41-48.
[6]R W克拉夫,J彭津.结构动力学[M].北京:科学出版社,1981.
[7]CLOUGH R W,PENZIEN J.Dynamics of Structures[M].2ndEdition Revised Edition.Berkeley:Computers and Structures,Inc.1995.
[8]CHOPRA A K.Dynamics of Structures[M].2ndEdi-tion,影印版.北京:清华大学出版社,2005.
[9]BATHE K J WILSONE L.Stability and accuracy analy-sis of direct integration methods[J].Earthquake Engi-neering and Structural Dynamics,1973,1:283-291.
[10]HILBER HM.,HUGHES TJ R,TAYLOR R L.Colloca-tion,dissipation and‘overshoot’for time integration schemes in structural dynamics[J].Earthquake Engi-neering and Structural Dynamics.1978,6:99-117.
[11]WANG H D,ZHANG Y S,WANG W.A high order single step-βmethod for nonlinear structural dynamic analysis[J].Journal of Harbin Institute of Technology,2003,10(2):113-119.
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