含复阻尼振动系统的逐步积分法稳定性研究  

Stability of step-by-step methods in solving complex damping vibration systems

作　　者：潘玉华[1] 王元丰[1] 

机构地区：[1]北京交通大学,北京100044

出　　处：《土木工程学报》2010年第S1期206-210,共5页China Civil Engineering Journal

基　　金：国家自然科学基金(50978015)

摘　　要：逐步积分法对于黏滞阻尼振动系统是条件稳定的,但用于求解复阻尼振动方程时,无一例外地出现不稳定现象。首先,针对复阻尼理论应遵循对偶复化的原则,给出Newmark-β法求解复阻尼振动方程的逐步列式。然后,根据理论推导,探讨Newmark-β法求解黏滞阻尼系统的稳定性条件。最后,通过理论分析和数值试验表明,对于复阻尼振动系统,求解不稳定的原因并非由于数值方法的不稳定,而是源于复阻尼本构方程本身的病态。稳定性条件与结构自振周期、荷载持续时间以及复阻尼常数的大小有关,当荷载持续时间与结构自振周期的比值小于某临界稳定值时,可以得到稳定的数值解。Step-by-step method is conditionally stable for viscous damping systems.However,it is instable without exception when used in complex damping vibration equations.First,based on the dual principle of complex damping theory,the step-by-step equations of Newmark-β method were presented to solve the stability problems of vibration systems with complex damping.Then,the stability conditions of Newmark-β method in solving viscous damping systems were studied.Finally,by means of theoretical and numerical analysis,it was found that for complex damping vibration systems the instability of numerical solutions was not resulted from the algorithm itself but the divergent complex constitutive equation.The stability conditions of complex damping systems have relations with the period of structure vibration,the duration of exciting force and the value of the complex damping coefficient.A stable numerical solution can be obtained when the practical ratio of the duration of exciting force to the period of structure is smaller than the critical stable ratio.

关 键 词：复阻尼 逐步积分法 NEWMARK-Β法 稳定性 

分 类 号：TU311.3[建筑科学—结构工程]