基于实模态的非比例阻尼体系复模态叠加法

Complex mode superposition method of non-proportional damping system based on real modes

为提高非比例阻尼体系强迫振动时复模态叠加法的计算效率,基于模态摄动法基本原理,提出一种利用无阻尼体系实模态的复模态叠加法。该方法由模态摄动法将非比例阻尼体系的复模态表示为实模态线性组合,在此基础上,建立了强迫振动复模态叠加法的实模态的线性组合解。以一个带附加阻尼的强非比例阻尼三层框架结构为例进行地震反应分析,计算结果表明当附加实模态数不小于8时,模态摄动法所得的特征值误差小于2%;在复模态叠加法模态截断方面,基于累积振型参与质量和累积振型贡献系数所得的模态数偏少,建议采用首层平动的累积振型加速度贡献系数作为模态截断的依据,此时,复模态叠加法所得水平位移的峰值误差小于10%,累积误差小于15%,显示了良好的计算精度;且该方法的计算时间小于基于状态空间法的复模态叠加法。以一个钢结构框架办公楼地震反应分析为例,验证了该方法的适用性。综合来看,对于强非比例阻尼体系,强制解耦的实模态叠加法计算误差较大,而该方法可兼顾计算效率与精度。

A complex mode superposition method based on real modes of the undamped system is proposed to improve the calculation efficiency of forced vibration of the non-proportional damping system.The complex eigenvalues and modes are represented by the linear combination of real modes through the modal perturbation method.So the forced vibration solution can be expressed by the real modes.Then,the seismic response is analyzed by a three-story frame structure with additional damping which is a strong non-proportional damping system.The result shows that eigenvalue errors are less than 2%when the number of additional real modes exceeds 8.The numbers of complex modes calculated from the cumulative mode participating mass and modal contribution factor error of displacement are not enough,and the modal contribution factor error of acceleration is suggested to be a better modal truncation index.Compared with the state-space method,the proposed method is more efficient,the peak errors of the displacement response are less than 10%,and the cumulative errors are less than 15%,by a few low modes determined by modal contribution factor error of acceleration.Finally,the applicability of the proposed method is verified by seismic response analysis of an office building with a steel frame.Overall,the proposed method can balance calculation efficiency and accuracy for highly non-proportional damping systems,while the real mode superposition method with forced decoupling shows large errors.