复振型叠加法截断误差及改进

摘要

根据复振型叠加理论,对具有非比例阻尼特性的线性体系,分析了复振型截断误差的来源,给出了误差公式.与比例阻尼体系的高阶实振型响应类似,通过反应比表达式可以看出随着输入频率与自振频率比的减小,高阶复振型的响应也趋于荷载的静态响应.基于此推导出了与传统修正方法形式一致的复振型静力修正方法和振型加速度方法,同时根据复振型退化为实振型的条件(实部与虚部成比例关系)建立了实、复振型修正方法的统一.用数值算例对修正方法进行了验证,结果表明:复振型修正方法能够很好地提高振型叠加的计算精度,特别是当结构主要固有频率大于输入卓越频率时修正效果更为显著;还发现附加阻尼较大时,导致部分高阶振型自振频率减小,在一定程度上会降低修正效果.%Based on the complex modal superposition theory,the original error of complex mode truncation,which is used to analyze the non-classically damped linear system,is studied and the error formula is obtained.Similar to higher modal responses of classical damping systems,the form of response ratio shows that with the decrease of the ratio of input frequency and natural frequency the displacements of higher complex modal equations can also represented by static displacement of dynamic load.Therefore,because of this fact,static correction method and modal acceleration approach about non-classical damping systems are derived,whose forms are the same as that of the conventional methods.In addition,the correction methods of both real and complex modal superposition approaches are unified according to the condition of complex modes degenerating to real modes (real part is proportional to imaginary part).At last,the effectiveness of correction methods is verified through a numerical example,particularly for the case that the major natural frequencies are beyond the input significant frequency range;in addition,it is observed that rising the added damping will reduce,to some degree,the efficiency of the modified method since the added damping decreases some high-order natural frequencies.